Universit¨at Paderborn, Germany
Reducing Energy Consumption of
Radio Access Networks
Matthias Herlich
September 2013
Dissertation
submitted to the
Faculty of Electrical Engineering,
Computer Science, and Mathematics
in partial fulfillment of the requirements for the degree of
Doctor rerum naturalium (Dr. rer. nat.)
Referees:
Prof. Dr. Holger Karl
Prof. Dr. Friedhelm Meyer auf der Heide
Additional committee members:
Prof. Dr. Marco Platzner
Prof. Dr. Heike Wehrheim
Prof. Dr. Hans Kleine B¨uning
Submitted October 6, 2013
Examination February 18, 2014
Published February 25, 2014
Paderborn, Germany
Abstract
Radio access networks (RANs) have become one of the largest energy consumers of
communication technology [LLH+13] and their energy consumption is predicted to in-
crease [FFMB11]. To reduce the energy consumption of RANs different techniques have
been proposed. One of the most promising techniques is the use of a low-power sleep
mode. However, a sleep mode can also reduce the performance. In this dissertation,
I quantify how much energy can be conserved with a sleep mode and which negative
effects it has on the performance of RANs. Additionally, I analyze how a sleep mode
can be enabled more often and how the performance can be kept high.
First, I quantify the effect of power-cycle durations on energy consumption and latency
in an abstract queuing system. This results in a trade-off between energy consumption
and latency for a single base station (BS). Second, I show that considering a network as
a whole (instead of each BS individually) allows the energy consumption to be reduced
even further. After these analyses, which are not specific for RANs, I study RANs for
the rest of the dissertation.
RANs need to both detect and execute the requests of users. Because detection
and execution of requests have different requirements, I analyze them independently.
I quantify how the number of active BSs can be reduced if the detection ranges of
BSs are increased by cooperative transmissions. Next, I analyze how more BSs can be
deactivated if the remaining active BSs cooperate to transmit data to the users.
However, in addition to increasing the range, cooperative transmissions also radiate
more power. This results in higher interference for other users which slows their trans-
missions down and, thus, increases energy consumption. Therefore, I describe how the
radiated power of cooperative transmissions can be reduced if instantaneous channel
knowledge is available. Because the implementation in real hardware is impractical for
demonstration purposes, I show the results of a simulation that incorporates all effects
I studied analytically earlier.
In conclusion, I show that a sleep mode can reduce the energy consumption of RANs if
applied correctly. To apply a sleep mode correctly, it is necessary to consider power-cycle
durations, power profiles, and the interaction of BSs. When this knowledge is combined
the energy consumption of RANs can be reduced with only a slight loss of performance.
Because this results in a trade-off between energy consumption and performance, each
RAN operator has to decide which trade-off is preferred.
Zusammenfassung
Mobilfunknetze sind zu einem der gr¨
oßten Energieverbraucher der Informations- und
Telekommunikationsbranche geworden [LLH+13] und es wird erwartet, dass ihr Ver-
brauch weiter steigt [FFMB11]. Um den Energieverbrauch von Mobilfunknetzen zu ver-
ringern, wurden verschiedene Techniken vorgeschlagen. Eine der vielversprechendsten
Techniken ist der Einsatz eines stromsparenden Schlafmodus. Allerdings kann dieser
Schlafmodus die Leistungsf¨
ahigkeit reduzieren. Meine Dissertation beschreibt, wie viel
Energie durch einen Schlafmodus eingespart werden kann und welche Auswirkungen
dieser auf die Leistungsf¨
ahigkeit hat. Zus¨
atzlich untersuche ich, wie der Schlafmodus
m¨
oglichst h¨
aufig genutzt werden kann und trotzdem die Leistungsf¨
ahigkeit erhalten
bleibt.
Als erstes quantifiziere ich den Effekt der Ein- und Ausschaltdauer auf die Latenz
und den Energieverbrauch in einem abstrakten Warteschlangensystem. Dies resultiert
in einer Austauschbeziehung zwischen Energieverbrauch und Latenz f¨
ur eine einzelne
Basisstation. Als zweites demonstriere ich, dass der Energieverbrauch weiter reduziert
werden kann, wenn man das gesamte Netz betrachtet (im Gegensatz zu jeder Basissta-
tion einzeln). Nach diesen beiden Analysen, welche nicht speziell auf Mobilfunknetze
zugeschnitten sind, betrachte ich im Rest der Dissertation nur noch Mobilfunknetze.
Mobilfunknetze m¨
ussen sowohl die Anforderungen der Nutzer detektieren als auch
durchf¨
uhren. Weil Detektion und Durchf¨
uhrung unterschiedliche Anforderungen haben,
analysiere ich diese getrennt. Ich quantifiziere, wie die Anzahl der aktiven Basisstatio-
nen gesenkt werden kann, wenn ihre Detektierungsreichweite durch Kooperation erh¨
oht
wird. Als n¨
achstes stelle ich dar, wie mehr Basisstationen abgeschaltet werden k¨
onnen,
wenn die restlichen aktiven Basisstationen kooperieren, um Daten zu den Nutzern zu
¨
ubertragen.
Allerdings erh¨
ohen kooperative ¨
Ubertragungen nicht nur die Reichweite, sondern strah-
len auch mehr Leistung ab. Dies resultiert in h¨
oherer Interferenz bei anderen Nut-
zern, welche die ¨
Ubertragungen verlangsamen und somit den Energieverbrauch wieder
erh¨
ohen. Deshalb beschreibe ich, wie mit augenblicklichem Kanalwissen die abgestrahlte
Leistung verringert werden kann. Weil die Implementierung in echter Hardware f¨
ur De-
monstrationszwecke zu aufw¨
andig ist, zeige ich zuletzt die Ergebnisse einer Simulation,
welche alle vorher analytisch betrachteten Effekte zusammenfasst.
Mein Fazit ist, dass ein Schlafmodus den Energieverbrauch eines Mobilfunknetzes re-
duzieren kann, falls er korrekt verwendet wird. Um einen Schlafmodus korrekt zu verwen-
den, ist es notwendig Ein- und Ausschaltdauer, Leistungsprofile und die Interaktion von
Basisstationen zu ber¨
ucksichtigen. Wenn dieses Wissen kombiniert wird, kann der Ener-
gieverbrauch von Mobilfunknetzen reduziert werden, ohne dass die Leistungsf¨
ahigkeit
nennenswert beeintr¨
achtigt wird. Weil dies eine Abw¨
agung zwischen Energieverbrauch
und Leistungsf¨
ahigkeit darstellt, muss jeder Betreiber eines Mobilfunknetzes selbst ent-
scheiden, was er vorzieht.
Table of contents
List of acronyms xi
List of symbols xiii
List of figures xix
List of tables xxii
1. Introduction 1
1.1. Radioaccessnetworks ............................. 2
1.2. Load, energy consumption, and latency . . . . . . . . . . . . . . . . . . . 5
1.2.1. Powerprofiles.............................. 6
1.2.2. Energy-latency trade-off . . . . . . . . . . . . . . . . . . . . . . . . 10
1.2.3. Consumption vs. efficiency . . . . . . . . . . . . . . . . . . . . . . 11
1.3. Splitting signaling and data traffic . . . . . . . . . . . . . . . . . . . . . . 11
1.4. Cooperative transmissions . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5. Generalrelatedwork.............................. 14
1.5.1. Energy consumption on the BS level . . . . . . . . . . . . . . . . . 14
1.5.2. Other wireless techniques . . . . . . . . . . . . . . . . . . . . . . . 15
1.5.3. Energy consumption on the network level . . . . . . . . . . . . . . 15
1.5.4. Prediction................................ 15
1.5.5. Moving and reducing load . . . . . . . . . . . . . . . . . . . . . . . 16
1.5.6. Effects on the power grid . . . . . . . . . . . . . . . . . . . . . . . 16
1.6. Contributions and chapter overview . . . . . . . . . . . . . . . . . . . . . 16
2. Conserving energy at each BS individually 20
2.1. Introduction................................... 20
2.2. Relatedwork .................................. 21
2.3. Model ...................................... 22
2.3.1. Queuingsystem............................. 22
2.3.2. Server states and timing . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.3. Policies and metrics . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.4. Poissonarrivals................................. 25
2.4.1. Greedypolicy.............................. 25
2.4.2. Accumulate & fire policy . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.3. Energy-minimizing policy . . . . . . . . . . . . . . . . . . . . . . . 26
2.4.4. Latency-minimizing policy . . . . . . . . . . . . . . . . . . . . . . . 27
vi
2.4.5. Comparison............................... 29
2.5. Adversary-controlled arrivals . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5.1. Latency ratio: greedy policy . . . . . . . . . . . . . . . . . . . . . . 31
2.5.2. Latency ratio: accumulate & fire policy . . . . . . . . . . . . . . . 33
2.5.3. Energy ratio: greedy policy . . . . . . . . . . . . . . . . . . . . . . 34
2.5.4. Energy ratio: accumulate & fire policy . . . . . . . . . . . . . . . . 34
2.5.5. Impossible trade-offs between energy and latency . . . . . . . . . . 35
2.5.6. Arbitrarily distributed random variables . . . . . . . . . . . . . . . 36
2.6. Results...................................... 37
2.7. Conclusion ................................... 44
3. Conserving energy by coordinating sleep modes network-wide 45
3.1. Introduction................................... 45
3.2. Relatedwork .................................. 46
3.3. Model ...................................... 47
3.3.1. Networkgraph ............................. 47
3.3.2. Power consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3.3. Latency ................................. 48
3.4. Analysis of metrics for latency aggregation . . . . . . . . . . . . . . . . . . 50
3.4.1. Relationships between the latency metrics . . . . . . . . . . . . . . 50
3.4.2. Bandwidth-delay product and latency . . . . . . . . . . . . . . . . 51
3.5. Optimizationmodel .............................. 53
3.6. Results...................................... 55
3.7. Conclusion ................................... 57
4. Conserving energy in signaling transmissions 60
4.1. Introduction................................... 60
4.2. Relatedwork .................................. 61
4.3. Model ...................................... 63
4.3.1. Non-cooperative range . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3.2. Cooperativerange ........................... 63
4.4. Signaling with optimal BS spacing . . . . . . . . . . . . . . . . . . . . . . 64
4.4.1. Nocooperation............................. 64
4.4.2. Limited 2-cooperation . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.4.3. Unlimited 2-cooperation . . . . . . . . . . . . . . . . . . . . . . . . 66
4.4.4. Infinite cooperation . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.5. Results...................................... 70
4.6. Deactivating opportunities with fixed BS spacing . . . . . . . . . . . . . . 72
4.7. Comparing ergodic and outage capacity . . . . . . . . . . . . . . . . . . . 75
4.8. Conclusion ................................... 77
5. Conserving energy in data transmissions 78
5.1. Introduction................................... 78
5.2. Relatedwork .................................. 79
vii
5.3. Model ...................................... 80
5.3.1. Cooperation............................... 80
5.3.2. Metrics ................................. 81
5.4. Analysis..................................... 81
5.4.1. Nocooperation............................. 81
5.4.2. Cooperation with densely placed BSs . . . . . . . . . . . . . . . . . 86
5.4.3. Cooperation with sparsely placed BSs . . . . . . . . . . . . . . . . 88
5.5. Numericalresults................................ 90
5.6. Conclusion ................................... 94
6. Radiated power 95
6.1. Introduction................................... 95
6.2. Relatedwork .................................. 96
6.3. Model ...................................... 96
6.3.1. Effective radiated power . . . . . . . . . . . . . . . . . . . . . . . . 96
6.3.2. Static vs. dynamic association . . . . . . . . . . . . . . . . . . . . . 97
6.4. Outageprobability............................... 97
6.4.1. Static association . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.4.2. Dynamic association . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.4.3. Strictly superior cooperation schemes . . . . . . . . . . . . . . . . 100
6.5. Effective radiated power . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.5.1. Static association . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
6.5.2. Dynamic association . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.6. Results......................................106
6.6.1. Scenarios with high gain . . . . . . . . . . . . . . . . . . . . . . . . 106
6.6.2. Outage probability and channel gain . . . . . . . . . . . . . . . . . 107
6.6.3. Effective radiated power . . . . . . . . . . . . . . . . . . . . . . . . 110
6.6.4. Relating ERP and outage probability . . . . . . . . . . . . . . . . 110
6.7. Conclusion ...................................112
7. Network simulation 114
7.1. Introduction...................................114
7.2. Relatedwork ..................................115
7.3. Model ......................................116
7.3.1. BSdeployment.............................116
7.3.2. UE deployment and load generation . . . . . . . . . . . . . . . . . 117
7.3.3. Radiomodel ..............................117
7.3.4. Powermodel ..............................118
7.3.5. Cooperative diversity . . . . . . . . . . . . . . . . . . . . . . . . . 118
7.4. Algorithms ...................................120
7.4.1. Always on and always off . . . . . . . . . . . . . . . . . . . . . . . 120
7.4.2. Greedy and accumulate & fire . . . . . . . . . . . . . . . . . . . . . 121
7.4.3. Setcover ................................121
7.5. Results......................................122
viii
7.6. Conclusion ...................................128
8. Final thoughts 129
8.1. Summary ....................................129
8.2. Futurework...................................130
8.3. Conclusion ...................................131
A. Proofs for the stretch metrics 133
A.1. Equality of stretch metrics for the geometric mean . . . . . . . . . . . . . 133
A.2. Possible orders of metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
A.3.Impossibleorders................................134
A.4. Bounds between metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
A.4.1. Maximum of stretches . . . . . . . . . . . . . . . . . . . . . . . . . 135
A.4.2. Stretch of maximum . . . . . . . . . . . . . . . . . . . . . . . . . . 136
A.4.3. Stretch of average . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
A.4.4. Average of stretches . . . . . . . . . . . . . . . . . . . . . . . . . . 136
A.4.5. Geometric mean of stretches . . . . . . . . . . . . . . . . . . . . . 137
A.5.Latencyinrings ................................137
A.5.1. Limit in odd-length rings . . . . . . . . . . . . . . . . . . . . . . . 138
A.5.2. Limit in even-length rings . . . . . . . . . . . . . . . . . . . . . . . 139
B. Bibliography 140
C. Glossary 157
ix
List of acronyms
BDP bandwidth-delay product. 46, 50–52
BS base station. iii, xii–xvi, 1–6, 8–18, 20, 21, 23, 24, 44,
48, 59–70, 72–75, 77–132, 157–159
CC coherent combining. 13, 14, 99
DU dense urban. 116, 117, 123–127
ERP effective radiated power. xv, 95–97, 101–105, 108,
110–113, 119, 120, 130
LTE-A Long Term Evolution Advanced. 13–15, 118, 131
MILP mixed integer linear programming. 45, 53, 73
MIMO multiple-input and multiple-output. 12, 15, 96, 97,
117, 120
MRC maximal-ratio combining. 13, 14, 62, 76, 77, 99
MWC minimum wireless coverage. 79
NP nondeterministic polynomial time. 45, 46, 79
PTAS polynomial-time approximation scheme. 79
QoS quality of service. 132
RAN radio access network. iii, 1–6, 9–12, 14–20, 44–46,
59–62, 73, 79, 81, 90, 94–96, 107, 114–116, 124, 128–
132, 157–159
SDU sparse dense urban. 117, 124, 125
SINR signal-to-interference-(plus-)noise ratio. 101, 117–
121, 125
SNR signal-to-noise ratio. xvi, 4, 5, 63, 65, 68, 70, 76, 97,
110, 158
xi
List of symbols
! Factorial. 138, 139
∞Infinity. 25, 36, 49, 50, 69–71, 105, 106, 137–139
A Area of coverage. 64–67, 70, 71, 80, 83
A1Area a BS can cover. 82, 83, 89, 90
A1e Area that can be covered by exactly one BS. 82–87
A1ec Area that can be covered by exactly one BS using
cooperative transmissions. 87, 88
A2Area that each of two BSs can cover. 82, 83, 85
A2e Area that can be covered by exactly two BSs. 82–86,
88
A3Area that each of three BSs can cover. 82–86, 88
AdArea that can be covered by exactly two cooperating
BSs or one other single BS. 85, 87, 88
AhArea a BS has to cover if the user equipments (UEs)
are assigned to the closest BS. 82–84, 89, 90
AI(d, r, R) Size of intersection area of two circles with distance
dand radii rand R. 83, 87
αRate of activation of a BS. 24–27, 29, 31, 37
ARandom variable that describes the activation dura-
tion of a BS. xii, 11, 23–25, 28–37, 41–43, 118, 122,
126, 127
AExpected value of the activation duration E[A]. 24,
31, 37, 40, 42–44
AP(k) Accumulate and fire policy with an activation thresh-
old if k. xiii, 24, 26, 33, 35, 38, 41, 121, 123, 124
AsArea that one BS has to be active to cover using
sparse deployment. 89, 90
Ase Area that can be covered by exactly two cooperative
BSs in a sparse deployment. 89, 90
avg Arithmetic mean operator (average). 49–53
C Configuration. 47–53, 133, 135, 138, 157
cMaximum number of BSs that can cooperate on a
single transmission. xiv, 74–76, 98, 99, 101, 103, 105–
107, 112, 113
cap Capacity of a BS or edge. 47, 48, 51–53
CLLatency-minimal configuration. 47–51, 133, 135, 138
xiii
CP Critical point, a point with the lowest received signal
strength in the plane. 64, 67–70
CRLCompetitive ratio for latency. 25, 31–33, 35–37
CRPCompetitive ratio for power consumption. 25, 34–37
D Set of demands. 47–53, 133
d(a, b) Euclidean distance between points aand b. 5, 63,
68–70, 73
DDData rate users demand for data transmissions. xv,
xvi, 4, 5, 118
δPath-loss exponent describing the loss of signal
strength of an electromagnetic wave over distance.
5, 63–66, 68–71, 73, 75, 76, 80, 108
dr(·) Function that maps from SNR to data rate. 5, 117–
120
DSData rate necessary for transmission of signaling traf-
fic. xv, xvi, 4, 5, 63, 72
e Base of the natural logarithm, also known as Euler’s
number (≈2.71828). 29, 50, 80, 137–139
EA(A) Function mapping the size of an area Ato the prob-
ability of activity in this area. 80, 84, 86, 88, 90
EGEdges of the graph G. xiii, 47, 48, 52, 53
EP Energy-minimizing policy. 24, 31, 36
E[·] Expected value operator. xii–xiv, xvii, 24, 25, 37
FFactor between average channel gains of BSs (sorted
from high to low). 107–110, 113
G A graph with nodes VGand edges EG. 47
gGain of covered area compared to not using cooper-
ation. 65, 66, 68, 70, 71
Γ Expected channel gain (loss) E[γ]. 96, 102, 104, 107
γInstantaneous channel gain (loss). xiii, xiv, xvi, 96–
100, 102–107, 109, 111, 113
geo Geometric mean operator. 50, 133, 138
GP Greedy policy. 24, 25, 31, 33, 34, 36–39, 41–43, 121,
123–127
IC Infinite cooperation scheme. 68, 70, 71
J Joule, the unit of energy of the International System
of Units (SI). 11, 110
kActivation threshold of the accumulate and fire policy
AP(k). xii, 24, 26, 27, 33–35, 38, 121, 123
L Latency. 25, 26, 28, 31, 36, 38–43, 47–53, 123–127,
133, 138, 158
λRate or density of arrivals of a (space-)time Poisson
process representing user requests. 4, 6, 24–29, 31,
37–40, 44, 56–58, 80, 93
xiv
LC Limited cooperation scheme. 65, 66, 71
LRandom variable that describes the interarrival du-
ration of requests. xiv, 24, 25, 28, 29
LExpected value of the interarrival duration E[L]. 24,
29
LP Latency-minimizing policy with an oracle. 24, 28, 29,
31, 36, 39, 41–43
m Meter, the unit of length of the International System
of Units (SI). 4, 117
MiB Mebibyte, which is equal to 220 byte. 117
MRandom variable that describes the processing dura-
tion of requests. xiv, 24, 25, 30–37
MExpected value of the processing duration E[M]. 24,
27, 37
µProcessing rate of requests. 24–29, 31, 37
NMean noise power. xvi, 5, 97, 119, 158
NC No cooperation scheme. 64, 65, 70, 71
OOutage probability. 75, 97, 100, 103, 109, 111–113
ODOutage probability under dynamic association. 97,
99, 100, 102, 103
ODe Probability that exactly cout of nBSs need to co-
operate to serve a UE. 103, 106
OffP Always-off policy. 120, 123, 124, 126
ωRate of deactivation of a BS. 24–27, 29, 31, 37
OnP Always-on policy. 24, 28, 31, 36, 48, 120, 122–126
OSOutage probability under static association. 97, 98,
100
OSp Outage probability under static association with
power control. 101
p(·) Probability density function of the instantaneous
channel gain γ. 96, 98, 99, 105, 106
φ(·) Function from source and destination of demand to
it size. 47–53, 133–137
πRatio between circumference and diagonal in every
circle (≈3.14159). 65, 83
P Power consumption. 6, 25, 26, 29, 31, 38, 39, 41, 42,
47, 48, 53, 56–58, 74–76, 81, 84–86, 88, 90, 92, 123,
126, 127
P[X] Probability of event X. 28, 29, 75, 97–99, 103, 106
Pst[S] Probability to be in state S. 25, 28, 29
PRLPoisson ratio for latency. 25, 31
PRPPoisson ratio for power consumption. 25, 31
PIdle power consumption as fraction of maximal power
consumption. 48, 53, 56–58, 118
xv
Q Expected number of active BSs per area. 81, 84, 91–
93, 124, 125
REffective radiated power (ERP) of all BSs combined.
97, 110–113
rEffective radiated power (ERP) of a single BS, which
is equivalent to the mean transmit power. 96, 97,
101, 103–105
RDRadiated power under the dynamic association
scheme. 103
rDRange of a BS in which it can provide data traffic
with a data rate DD. 159
RDp Radiated power under the dynamic association
scheme with power control. 105, 106
RSet of real numbers. 47
RSRadiated power under the static association scheme.
101
rSRange of a BS in which it can provide signaling traffic
with a data rate DS. 63–67, 69–73, 159
RSp Radiated power under static association scheme with
power control. 101
s Second, the unit of time of the International System
of Units (SI). 4, 118, 123–127
SABS state of being active. 23, 25–27, 157
SAS Aggregated stretch determined by arithmetic mean
of stretches. 49, 51, 53, 55–57, 134–137
SDBS state of deactivating. 23, 25–27, 157
SGS Aggregated stretch determined by geometric mean of
stretches, equal to SSG. xvi, 50, 51, 133, 136–138
σ(e) Function mapping an edge eto the amount of data
flow over it. 47, 48, 52, 53
skew Skew operator, returns ratio of highest to lowest ele-
ment. 51, 135–137
SMS Aggregated stretch determined by maximum of
stretches. 48–50, 53, 135–137
SnSymmetric group, that is, the set of all permutations
of the natural numbers (1, . . . , n). xvi, 99, 105, 106
SNR Signal-to-noise ratio. xiii, 4, 5, 14, 73, 75, 97
SP(A) Set cover policy using average activity in areas. 122–
127
SP(C) Set cover policy using currently active UEs. 122, 123
SSBS state of sleeping. 23, 25–27, 159
SSA Aggregated stretch determined by stretch of arith-
metic mean. 49, 51, 53, 55–57, 134–137
xvi
SSG Aggregated stretch determined by stretch of geomet-
ric mean, equal to SGS. xv, 50, 133
SSM Aggregated stretch determined by stretch of maxi-
mum. 48, 49, 51, 53, 134–137
SUBS state of activating (starting up). 23, 25–27, 157
TThreshold of signal-to-noise ratio (SNR) necessary
for a UE to reliably decode transmissions. 4, 5, 13,
14, 63, 75
τA permutation out of the symmetric group Sn. 99,
105, 106
TDThreshold of SNR necessary for a UE to reliably de-
code transmissions at the demanded data rate DD.
xvi, 5, 97
TγThreshold of the channel gain (loss) for a successful
transmission, which is equal to TDN. 97–100, 102–
107, 109, 111, 113
TSThreshold of SNR necessary for a UE to reliably de-
code transmissions at the rate necessary for signaling
DS. 5, 65, 68, 70, 73
U Utilization of a BS or edge. 48, 51–53
UC Unlimited cooperation scheme. 66–68, 71
VGVertices (nodes) of the graph G. xiii, 47, 53
W Watt, the unit of power of the International System
of Units (SI). 1, 118, 123–127
ξSpacing of BSs, also known as inter-site distance
(ISD). 64–70, 82–85, 87, 89, 92, 159
ξmax Maximum spacing for the non-cooperative transmis-
sions. 82, 83, 86, 91
ξmaxD Maximum considered spacing for the dense coopera-
tive deployment. 85, 86
ξmaxSL Maximum spacing for the sparse deployment with
limited cooperation. 88, 89
ξmaxSU Maximum spacing for the sparse deployment with un-
limited cooperation. 88, 89
ξmin Minimum considered spacing for the non-cooperative
transmissions. 82, 83
ξminD Minimum considered spacing for the dense coopera-
tive deployment. 85, 86
ξminS Minumum considered spacing for the sparse deploy-
ment with cooperation. 88, 89
ZRandom variable that describes the deactivation du-
ration of BSs. xvii, 11, 23–25, 28, 29, 31–37, 41–43,
118, 122, 126, 127
xvii
List of figures
1.1. Base station and user equipment overview . . . . . . . . . . . . . . . . . . 2
1.2. Categories of power profiles . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3. Power profile composed by balancing over space . . . . . . . . . . . . . . . 7
1.4. Power profile composed by aggregating over space . . . . . . . . . . . . . 7
1.5. Power profiles composed over time . . . . . . . . . . . . . . . . . . . . . . 8
1.6. Daily traffic pattern in mobile networks . . . . . . . . . . . . . . . . . . . 9
1.7. Trade-offs between energy consumption and latency . . . . . . . . . . . . 10
1.8. Cooperative transmission increases range of BSs . . . . . . . . . . . . . . 13
1.9. Chapter interdependency guide . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1. Queuing model for single server processing . . . . . . . . . . . . . . . . . . 22
2.2. Power states of a server . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3. Greedy policy Markov model . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4. Accumulate and fire policy Markov model . . . . . . . . . . . . . . . . . . 27
2.5. Conserving energy using the latency-minimizing policy . . . . . . . . . . . 28
2.6. Greedy policy can consume less energy than the latency-minimizing policy 30
2.7. Latency-minimizing policy can consume less energy than the greedy policy 30
2.8. Competitive ratio for latency of the greedy policy . . . . . . . . . . . . . . 32
2.9. Case distinction for job arrivals . . . . . . . . . . . . . . . . . . . . . . . . 33
2.10. Competitive ratio for power for the greedy policy . . . . . . . . . . . . . . 34
2.11. Competitive ratios and possible trade-offs . . . . . . . . . . . . . . . . . . 36
2.12. Analytic and simulation results of latency . . . . . . . . . . . . . . . . . . 38
2.13. Analytic and simulation results of power consumption . . . . . . . . . . . 38
2.14. Power consumption depending on load . . . . . . . . . . . . . . . . . . . . 39
2.15. Latency depending on load . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.16. Differentiating the sources of latency . . . . . . . . . . . . . . . . . . . . . 40
2.17. Trade-offs of the accumulate and fire policy . . . . . . . . . . . . . . . . . 41
2.18. Power consumption depending on state change duration . . . . . . . . . . 41
2.19. Latency depending on power-cycle duration . . . . . . . . . . . . . . . . . 42
2.20. Power consumption depending on distribution of power-cycle duration . . 42
2.21. Latency depending on distribution of power-cycle duration . . . . . . . . . 43
3.1. An 8-Ring with a single inactive edge . . . . . . . . . . . . . . . . . . . . 49
3.2. Relationship between BDP and latency . . . . . . . . . . . . . . . . . . . 52
3.3. The (4 ×4)-grid ................................ 54
3.4. The 4-dimensional hypercube . . . . . . . . . . . . . . . . . . . . . . . . . 54
xix
3.5. The nobel-germany network ......................... 55
3.6. Power consumption in the nobel-germany network............. 56
3.7. Power consumption depending on upper bound for latency . . . . . . . . . 56
3.8. Power consumption with a limit on stretch depending on load . . . . . . . 57
3.9. Power models and deactivation strategies in the hypercube . . . . . . . . 58
3.10. Power models and deactivation strategies in the grid . . . . . . . . . . . . 58
4.1. Coveringanarea ................................ 60
4.2. Cooperation increases covered area . . . . . . . . . . . . . . . . . . . . . . 61
4.3. Non-cooperative coverage . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.4. Covering the plane using limited cooperation . . . . . . . . . . . . . . . . 65
4.5. Covering the plane using unlimited cooperation . . . . . . . . . . . . . . . 67
4.6. Reason unlimited cooperation covers plane . . . . . . . . . . . . . . . . . . 67
4.7. Comparing limited and unlimited cooperation . . . . . . . . . . . . . . . . 68
4.8. Hexagonalnumbering ............................. 69
4.9. Gain depending on path-loss . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.10. Fixed deployment scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.11. Power consumption depending on excess range . . . . . . . . . . . . . . . 74
4.12. Power consumption depending on path-loss . . . . . . . . . . . . . . . . . 75
4.13. Power consumption depending on degree of cooperation . . . . . . . . . . 76
4.14. Coverage of ergodic and outage capacity . . . . . . . . . . . . . . . . . . . 76
4.15. Coverage of selection and combining . . . . . . . . . . . . . . . . . . . . . 77
5.1. Overview of areas of non-cooperative transmissions . . . . . . . . . . . . . 82
5.2. Areas of non-cooperative coverage . . . . . . . . . . . . . . . . . . . . . . 82
5.3. Sizes of multiply covered areas . . . . . . . . . . . . . . . . . . . . . . . . 84
5.4. Overview of areas of dense cooperative coverage . . . . . . . . . . . . . . . 85
5.5. Areas of dense cooperative coverage . . . . . . . . . . . . . . . . . . . . . 87
5.6. Calculating the area of cooperative coverage . . . . . . . . . . . . . . . . . 87
5.7. Overview of sparsely cooperative areas . . . . . . . . . . . . . . . . . . . . 89
5.8. Areas of sparse cooperative coverage . . . . . . . . . . . . . . . . . . . . . 89
5.9. Re-tiling with one third of BSs active . . . . . . . . . . . . . . . . . . . . 90
5.10. Re-tiling with one fourth of BSs active . . . . . . . . . . . . . . . . . . . . 91
5.11. Activity probability depending on spacing . . . . . . . . . . . . . . . . . . 92
5.12. Expected number of active BSs per area depending on spacing . . . . . . 92
5.13. Expected active BSs per area depending on user density . . . . . . . . . . 93
5.14. Relative active BSs per area depending on user density . . . . . . . . . . . 93
6.1. Better association scheme can depend on threshold . . . . . . . . . . . . . 100
6.2. Optimal power allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.3. Outage probability for different power distributions . . . . . . . . . . . . . 103
6.4. On-offallocation ................................104
6.5. Reallocatingpower...............................105
6.6. Probability for static selection to select best BSs . . . . . . . . . . . . . . 107
xx
6.7. Factor between channel gains . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.8. Outage probability with different average channel gains . . . . . . . . . . 109
6.9. Outage probability with different thresholds . . . . . . . . . . . . . . . . . 109
6.10. ERP with power control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
6.11. Outage probability with different distributions of ERP . . . . . . . . . . . 111
6.12. ERP with power control for different thresholds . . . . . . . . . . . . . . . 111
6.13. Outage probability and ERP . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.14. Overview of static and dynamic association strategies . . . . . . . . . . . 113
7.1. Simulation scenario overview . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.2. Combining different degrees of cooperation . . . . . . . . . . . . . . . . . 119
7.3. Strategycomparison ..............................123
7.4. Accumulate and fire thresholds . . . . . . . . . . . . . . . . . . . . . . . . 123
7.5. Strategies in sparse BS deployment . . . . . . . . . . . . . . . . . . . . . . 124
7.6. Cooperation in sparse BS deployment . . . . . . . . . . . . . . . . . . . . 125
7.7. Deactivating macro BSs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.8. Effect of power-cycle durations for set cover . . . . . . . . . . . . . . . . . 126
7.9. Effect of power-cycle durations for set cover and greedy . . . . . . . . . . 127
7.10.NumberofpicoBSs ..............................127
xxi
List of tables
1.1. Initiator and direction of transmissions . . . . . . . . . . . . . . . . . . . . 3
1.2. Effects considered by chapter . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1. Modeling decisions for the wired example . . . . . . . . . . . . . . . . . . 50
4.1. Coverable area summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2. Gainfactorsummary.............................. 71
xxii
1. Introduction
Information and communication technology is responsible for 2% of the world carbon-
dioxide equivalent emissions. About 75% of these come from the electrical energy con-
sumed during the use of the equipment [WO08]. The other 25% are consumed during
production of the equipment.
It is sometimes stated that the carbon dioxide equivalent emissions of the information
and communications technology have surpassed that of aviation. This comparison is not
fair because it includes the cost of hardware manufacturing for communication technol-
ogy, but not for aviation [MML+10, MBL13]. But as the communication sector grows
faster than the aviation sector [Ols01, Cis13], the communication sector will overtake the
aviation sector in dioxide equivalent emissions, if current trends (in growth and dioxide
equivalent emissions) keep the same.
Additionally, reducing the energy consumption is not only an environmental concern,
but also a cost factor. The electricity cost of the information and communication sector
is already at an all time high (both absolute and relative to total electrical energy
consumption) [LLH+13]. The price of electricity is predicted to further increase [U.S13,
Aus12]. This will result in further increasing costs for operating electrical equipment.
One of the largest consumers of electrical energy in the information and commu-
nication sector are network operators [LLH+13]. The operators of radio access net-
works (RANs) face high power consumption [KAK+11] and high growth of user de-
mands [Cis13]. The global energy consumption of all RANs is predicted to be 99 TWh
in the year 2020 [FFMB11] resulting from a strong increase in end user demand in the
next years [Rui11]. This is equivalent to 11.3 GW, which is close to the output of 12
nuclear power generating units (at a mean output of 975 MW [U.S12]). Future ubiqui-
tous RANs that are able to support the growing number of users require to reduce their
energy consumption.
The energy consumption of RANs is to a large degree determined by their base stations
(BSs) [FFMB11]. The power consumption of BSs ranges from 10 W for small BSs to over
1 kW for large BSs [AGD+11]. In the year 2007 there were approximately 3.3 million
BSs operating worldwide [FFMB11].
Methods to reduce their power consumption can be applied at different levels and
using different techniques, such as more efficient power amplifiers and transmission tech-
nologies. In this dissertation, I compare different ways to reduce the energy consumption
of entire RANs and not just individual BSs. The techniques I focus on are the use of a
sleep mode and deploying BSs with different distances between each other. I focus on
analytical approaches instead of simulations. While simulations can be more detailed
(and thus realistic) the analytic approach provides more insights into the fundamental
dependencies and relationships. I use these techniques to reduce the energy consumption
1
of RANs while considering how long users have to wait for their request to be processed.
I create models which focus on different aspects and analytically evaluate them individ-
ually. Finally, I combine all aspects in a simulation to determine their interaction.
In this chapter, I first describe what RANs are and define what they do (section 1.1).
To compare different RANs, I describe two performance metrics (energy consumption
and latency) in section 1.2. Next, section 1.3 describes the signaling and data traffic
split which allows me to analyze the need to provide data traffic to users and the need
to reach every possible location with signaling traffic in two different models. As I use
cooperative transmissions for both signaling and data traffic, I provide an overview of
cooperative transmissions next (section 1.4). In section 1.5, I describe which alternative
approaches to conserving energy in RANs exist and what distinguishes them from my
approach. This section only summarizes the general approaches and is supplemented by
a related work section in each of the following chapters. Finlly, I provide an overview of
the contribution of the individual chapters and their approaches in section 1.6.
1.1. Radio access networks
In this section, I will provide an overview of what RANs are, what features they provide
and how they usually fulfill them. Moreover, I describe how I model a RAN and which
assumptions I make. These assumptions hold for the rest of the dissertation unless
declared otherwise in the individual chapters. The model explained here forms the
common basis for all chapters and is expanded in the individual chapters where necessary.
Base station (BS)
User equipment (UE)
Backhaul network
Figure 1.1.: UEs connect to close BSs, which provide connectivity to the internet and
between UEs using a backhaul network.
I define a radio access network (RAN) as a telecommunication system that allows users
2
to wirelessly connect to the internet. Users can do this using, for example, mobile phones,
notebooks, and tablets, which are collectively called user equipment (UE). I denote the
plural as UEs as is standard in the wireless communication community, even though
the work “equipment” is uncountable. A RAN consists of stationary radio transceivers
called base station (BS). The BSs are connected with each other and to the internet by
a backhaul network. The backhaul network is a mix of wired (e.g., copper or fiber optic
connections) and wireless (e.g., microwave) links. Figure 1.1 illustrates the interactions
of UEs, BSs, and the backhaul network.
UEs can, for example, be used to make phone calls, read eMails, and surf the world-
wide web. For my work there is no need to distinguish between these different uses as,
from the point of view of the RAN, they all are data transfers (with different sizes).
When a UE uses a RAN it executes the following steps. It creates a request, which
I define as the need of a UE to exchange data. To grant this request the RAN decides
to which BS the UE has to connect. I understand a connection as being the logical
association of a UE with a BS, which allows them to exchange data. I define transmission
as the act of exchanging data between UE and BS.
The most important aspect to determine the quality of radio transmissions is the
path loss of wirelessly transmitted signals over distance. Because of path loss, wireless
communication between UEs and BSs is only possible if the communicating partners are
close together. Therefore, it is necessary to spread the BSs of a RAN over the area to
which they have to provide their service (called covering an area).
A typical UE neither sends nor receives data (called idle) most of the time [FMK+10,
Pau11]. Hence, there is no need for the BSs to maintain connections to the UEs at
all times. Typically, a UE sends or receives data (called active) for a short time and
becomes idle again thereafter. This activity period can be triggered both from the UE
(e.g., an outgoing call) or from the network (e.g., an incoming call). Hence, both UE
and BS must be able to initiate the connection between UE and BS. This is usually
implemented by both partners listening for transmissions from the other partner.
Table 1.1.: All combinations of initiator and prevalent direction of traffic are possible,
but video streaming is prominent.
Prevalent data direction Initiated by UE Initiated by the BS
More uplink data Sending eMail Remote access
Symmetric Outgoing (video) call Incoming (video) call
More downlink data Watching video stream Receiving eMail
While a transmission can be initiated both by a UE or by the BS, this does not
necessarily mean that the initiator is also the source of the resulting transmission. Ta-
ble 1.1 provides an overview with examples of the different possibilities to initiate and
transfer data. Because video streaming is becoming the predominant consumer of data
rate [Cis13], I focus on this. Using video streaming, traffic is initiated by the UE but is
3
mainly transmitted from the BS to the UE.
Initiating connections and transmitting data directly corresponds to the two distinct
duties for each BS: (1) receive UE transmission requests (called signaling traffic) and
(2) execute the transmissions (called data traffic). This distinction is important be-
cause I consider providing them from different BSs (section 1.3) and analyze both duties
separately.
When the network wants to initiate a connection to a UE the network has to know
which BS can connect to the UE. This can for example be achieved by a broadcast from
all BSs the UE has been connected to recently [Sau10]. The BS which gets a response
then establishes a connection to the UE. I simplify this and assume that a RAN always
knows which BS is in range of a UE.
When the UE wants to initiate a connection, it sends a request, which is received by
a nearby BS and the connection is established. When the connection is established the
data is transmitted regardless of the traffic direction.
I separate both duties in the following way: the RAN has to (1) provide a small
signaling data rate DSto every possible UE location and (2) fulfill transmission requests
of UEs at a higher data rate DD. While the signaling traffic has to be provided to every
possible UE location all the time, data traffic has only to be provided when demanded. I
model these demands as being generated by a space-time Poisson process [Kin93]. That
is, each request for a transmission occurs at a point in space and time. All request
arrivals are independent of each other. One possibility to create such a space-time
Poisson process is to create a Poisson process over time with exponentially distributed
inter-arrival times and assign each arrival to a location. The location is determined by
randomly selecting a position in the considered area, where every position is equally
likely. In case of a space-time Poisson process the arrival rate λis measured in requests
per area and time (e.g., 1 request/km2/s).
I assume both UEs and BSs reside in a 2-dimensional plane. I consider both as
points without size. In reality idle UEs move and continuously or periodically have
to be provided with signaling traffic. From their current positions they then generate
requests, randomly distributed in time. However, I do not model idle UEs and, thus,
define requests as appearing randomly.
In chapters where I do not model time, I simplify this model to a spatial Poisson
process. A spatial Poisson process places a Poisson-distributed number of requests in
a given area. All requests are placed independently and with equal likelihood for each
location in the area. In case of a spatial Poisson process the arrival rate λis measured
in requests per area (e.g., 1 request/km2).
Poisson processes have been shown to adequately model the timing of arriving trans-
mission requests in RANs [WMBW09]. Another empirical study [Pau11] shows that the
spatial correlation between load of BSs is low, which indicates that a Poisson modeling
of user locations is acceptable.
Providing a given data rate to a UE (both for signaling and data traffic) requires the
signal-to-noise ratio (SNR) to be higher or equal than a certain threshold T. Hence, a
transmission is successful if
SNR ≥T. (1.1)
4
I assume the mean noise power is constant. This assumption results in a lower bound on
the received signal strength. The required data rates for signaling DSand data traffic
DDare different and, thus, also the required SNR to reliably transmit them. Therefore,
I define the threshold of SNR for a transmission to be reliable as TSfor signaling traffic
and TDfor data traffic. This mean I define the data rate by ergodic capacity instead
of outage capacity (for a definition and comparison see section 4.7). This also means
I consider the long term mean data rate and not the instantaneous data rate, which is
influenced by fading. The conversion of the demanded data rate DDto the threshold
TDdepends for example on the modulation of the the transmission scheme, but is not
in the scope of this dissertation. I assume a function dr to map the mean SNR to the
resulting data rate. However, in all chapters except chapter 7 I do not use the function.
In the other chapters I assume the required thresholds TDand TSto be given directly.
The signal strength mostly depends on the distance to the BS. While I only consider
path loss analytically, I consider further effects such as fading and shadowing in the
simulation in chapter 7. I model the loss of signal strength by the log-distance path-loss
model [Sey05]. This results in the mean SNR to be calculated by
SNR = d(BS,UE)δ/N, (1.2)
where d(a, b) is the Euclidean distance of aand b,Nis the mean noise power, and δis
the path-loss exponent. As the log-distance path-loss model needs a reference distance,
but it only scales the result I assume it is 1. I define the highest distance at which the
threshold Tis still reached as range of a BS.
Some BSs have several antenna sectors, which can transmit independently. This in-
creases the maximum capacity by a factor of 3 or 6, respectively. The increase in capacity
is only important under high load. Because I consider low load scenarios and signaling
traffic, I consider all BSs as omnidirectional. That is they have only a single antenna
transmitting in all directions. The only exception to this is chapter 7, where I also con-
sider high loads and, thus, also sectorized antennas. Because interference is also more
important in high load scenarios I also include it in chapter 7. For all other chapters I
assume the channels are noise-limited and ignore interference.
1.2. Load, energy consumption, and latency
The most important performance metrics I analyze are the energy consumption and the
latency of demands. The energy consumption is the sum of the energy consumed by
all BSs. The latency is the time that passes between a UE’s request for a transmission
and the end of the corresponding transmission. Both energy consumption and latency
depend on the ratio of work it currently executes to the work it can maximally execute.
As BSs in industrial countries are all connected to the power grid (as opposed to
some solar or Diesel-powered BSs in developing countries) [MLOH10], I consider only
the total consumed energy and do not differentiate where the energy is consumed or
how it is generated. Therefore, I define the energy consumption of a RAN as the total
electrical energy consumed by all BSs . Also, I do not differentiate when energy is
5
consumed (e.g., to compensate for changing electricity prices). I do not consider the
energy consumption of the UEs because they are not under control of the RAN operator.
Moreover, most of the energy consumption UEs are responsible for is consumed during
their production and not during their operation [HHA+11]. In contrast to this, most of
the energy consumption BSs are responsible for is during their operation and not during
their production [HHA+11]. Although the backhaul is under control of the operator
I do not consider its energy consumption because it is small compared to the energy
consumption of the BSs [FFMB11].
1.2.1. Power profiles
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Mean power consumption P
Load λ
Binary
Super-linear
Linear
Sub-linear
Sleep
Figure 1.2.: These different categories of power profiles are typical for electrical devices.
The energy consumption of electrical devices is usually not constant but changes with
the tasks it currently handles. An important parameter to determine energy consump-
tion is the load. The load of a device is the ratio of work it currently executes to the
work it can maximally execute. Different types of devices have different characteristic
functions mapping load to power consumption. I call this function the power profile of
a device. Figure 1.2 illustrates four different categories of power profiles: binary, super-
linear, linear, and sub-linear. In a linear power profile the power consumption is always
proportional to the load. In a sub-linear power profile the power consumption is lower
than using the linear profile. In a super-linear power profile the power consumption is
higher than using the linear profile. The binary power profile is a special case of the
super-linear power profile which has constant power consumption if not in sleep mode.
On the smallest scales of time and space (i.e., the smallest component) digital equip-
ment is usually either fully loaded or idle. On larger scales (of time and space) the power
profile is composed from smaller scales. Methods to reduce the energy consumption on
one scale result in a different power profile on a larger scale. Depending on how the
load of a partially loaded component is distributed to its sub-components the composed
6
Sub-component 1
Load
Power
Sub-component 2
Load
Power
Sub-component 3
Load
Power
Requests Load
Balancer
Composed
power profile
Load
Power
Figure 1.3.: The composed power profile of a component depends on the load distribution
to its sub-components and their power profile. Load balancing recreates the
power profile of the sub-components.
Sub-component 1
Load
Power
Sub-component 2
Load
Power
Sub-component 3
Load
Power
Requests Load
Aggregator
Composed
power profile
Load
Power
Figure 1.4.: The composed power profile of a component depends of the load distribution
to its sub-components and their power profile. Load aggregation approaches
a linear power profile if the number of sub-components approaches infinity.
7
power profile will look different.
If the load of a large component is equally distributed over all its sub-components the
power profile of the large component will be the same as of its sub-components. For a
component with a sub-linear power profile the power consumption is low for most loads
and, hence, reproducing it is a good idea. But deadlines, unknown future demands, and
unsplittable demands make it hard or impossible to perfectly balance load [KCLMT12].
If the power profile is super-linear, load balancing also recreates the power profiles
of the sub-components. A component with a super-linear power profile will have high
power consumption for most loads. However, in this case there is an alternative with
less power consumption: load aggregation. Load can be aggregated in space or in time
and ideally uses each sub-component at full load or not at all. In case it is not used
at all it can be placed in a sleep mode which consumes less energy than running idle.
However, in sleep mode it cannot serve any traffic (signaling or data) to UEs.
Aggregating the load results in a power profile that is linear between the power con-
sumed in sleep mode and full power under full load. This is good for super-linear power
profiles, but bad for sub-linear power profiles. Figures 1.3 and 1.4 illustrate an example
for load balancing and aggregation over three sub-components (i.e., in space). Figure 1.5
illustrates load distribution over three time slots. This example assumes load can be
freely moved in time and space, which is not true in most systems. Real systems will
have deadlines which prevent load from being moved in time and the sub-components
will have different functionalities which prevent load from being moved in space. There-
fore, it is important to consider which negative effects the movement of load has. Moving
load in time can, for example, be quantified by latency.
Time slot 1
Load
Power
Time slot 2
Load
Power
Time slot 3
Load
Power
Load balancing
Load
Power
Load aggregation
Load
Power
Figure 1.5.: The aggregated power profile of a large time slot depends on the load distri-
bution to smaller time slots. Load balancing recreates the power profile of
the smaller time slots. Load aggregation approaches a linear power profile.
To apply methods to conserve energy, it is necessary to first determine the power
profile of the considered components. The power profile of a typical BS has been close
to binary in the past [AGD+11, ARFB10, CH08, LGP12]. While the power profile of
8
future BSs will get closer to the linear profile, they will be far from achieving it [Fis07].
But a power profile closer to linear and higher-level energy conservation methods can be
combined for even less energy consumption than each of them individually (see chap-
ter 3). Additionally, the absolute energy savings by the techniques described in this
dissertation will increase over time because the number of deployed BSs is expected to
increase [FFMB11].
Figure 1.6.: The mobile traffic pattern repeats daily with a slight variation during week-
ends. The plot is based on measured data from China Mobile of Tianjin;
both data and image are from Shu et al. [SYLY03].
To be most energy-efficient a system with a super-linear power profile should always
run under full load. But because the traffic demands of the users of a RAN change over
time, it is only possible to run a RAN under full load if demands are moved in time or
dropped. Both are only acceptable in very limited amounts. The load changes over time
in different forms:
•short-term (minutes and shorter) random fluctuations,
•medium-term (hours to days) periodic changes due to daily cycle (see figure 1.6),
and
•long-term (months and longer) changes due to changes in usage behavior and
population density.
The power profile of a complete RAN can be determined from the power profiles of
the BSs and the load distribution scheme. By summarizing the power profiles of all
BSs, it is possible to create a power profile of an entire RAN. If UEs are assigned to
BSs without any concern for energy consumption, the load will be equally distributed
to all BSs. Then all BSs will be under the same load as the RAN. Therefore, the power
profile of the RAN will look the same as the power profile of each BS. But it is possible
to change the power profile of the RAN to be closer to the linear power profile. The
greatest difference between the super-linear power profile of a BS and the linear power
profile is usually at low load [AGD+11]. Therefore, the greatest potential for energy
9
saving is achievable at low load. Because the fraction of time spent under low load is
not negligible [SYLY03] I focus on low load. However, it is important to keep in mind
that the RAN must still be able to handle high load when necessary.
1.2.2. Energy-latency trade-off
Energy consumption
Latency
Energy-minimizing policy
Possible trade-off
Latency-minimizing policy
Pareto front
Figure 1.7.: Depending on the requirements each policy at the Pareto front is a viable
candidate for the best policy.
Under high load either requests have to be dropped or the latency will increase. An
alternative is to reduce reduce the load before this happens. However, I do not consider
reducing the total traffic because those efforts are mostly application-specific. If the
total amount of traffic is fixed, load can be moved in two ways: (1) move it in time
and (1) move it in space. Most traffic cannot be freely moved in time but the loss of
performance can be measured in latency. For a single type of technology this usually
results in a trade-off between energy consumption and latency. Figure 1.7 shows how
the trade-off looks like conceptually.
One possibility to move traffic in space is to move users. However, it is impractical
to move users. But, due to the wireless nature of the transmission, most users can
communicate with several nearby BSs. Hence, the load of one BS can be shifted to other
BSs. On the one hand, this usually results in a worse channel, which results in slower
transmissions and higher latency. On the other hand, it allows the deactivation of now
idle BSs which in turn results in a lower energy consumption.
To reduce the energy consumption of a BS, it needs to be deactivated [ABH11]. Ac-
tivation and deactivation consume time. During this time the BS also consumes energy.
This delay and additional energy consumption are two reasons which make it non-trivial
to decide when and which BSs to deactivate.
10
I assume the activation time Ato be either constant or exponentially distributed.
I make the same assumption for the deactivation time Z. I call the time needed for
activation and deactivation power-cycle duration and assume the BS consumes full power
during that time. Model and analysis work the same way if activation and deactivation
consumptions are different, but this makes results more complex.
Except for chapter 7, I only consider one type of BS in each chapter. Therefore, I
normalize the maximum power consumption of a BS to 1. Additionally, I assume a
binary power profile and the power consumption in sleep mode to be 0.
1.2.3. Consumption vs. efficiency
If the total traffic can be changed, reducing the energy consumption is not a useful
goal because deactivating the system and ignoring all traffic would reduce the energy
consumption to zero. For this reason, in cases where the traffic can be changed usually
the energy efficiency of a system is considered. The energy efficiency can be defined as
consumed (electrical) energy per unit of work done (e.g., J/bit). An alternative definition
is to use the inverse: the amount of work done per unit of consumed energy (e.g.,
bit/J). Both definitions are technically equivalent, but differ in their ease of use. Which
definition is easier to use depends on the scenario, but in general it is more convenient to
have the quantity which is fixed in the denominator. This makes comparisons between
the energy efficiency of systems easier because the gains are proportional to the increase
in efficiency (in contrast to inversely proportional) [Kah11].
Because I consider the amount of traffic handled by the RAN to be fixed (with a small
exception in chapter 7), changing the energy consumption of a RAN is equivalent to
changing the energy efficiency. This is also equivalent to a reduction of the mean con-
sumed power of the RAN. Because all energy and power consumption metrics discussed
in this section are equivalent for my model I choose the one that is easiest to calculate:
energy consumption.
1.3. Splitting signaling and data traffic
A BS can be deactivated only if another BS is in range of each UE and every location
can still be reach with signaling traffic. When every BS is needed to provide signaling
traffic to the UEs, no BSs can be put into sleep mode.
A new approach [CSF12] places two different types of BSs: macro BSs to provide
signaling traffic and pico BSs to provide data traffic. Macro BSs have high ranges and
only need to provide low data rates to the UEs. Pico BSs have small ranges but can
provide high data rates to the users.
By splitting the BSs in two groups which fulfill two different duties, it is possible to
deactivate the pico BSs if they do not transmit data and activate them again via the
backhaul network when necessary.
When signaling and data traffic is served by different types of BSs it becomes easier
to serve different amounts of data requests in an area by simply deploying more or fewer
pico BSs. This is especially important when providing high data rates to hotspots.
11
Hotspots are areas in which the rate of request generation is considerably higher than
in the surrounding area. Because the traffic in hotspots changes it is important to
deactivate the pico BSs, which have been placed to provide high data rates, when the
demands are low (e.g., at night).
To be able to conserve energy with a pico BS sleep mode a RAN has to be able to
(1) determine the channel quality between a UE and nearby BSs, (2) connect a UE to
another BS, possible in mid-transmission (called handover), and (3) deactivate idle pico
BSs. The hardest part is to estimate the channel quality between a UE and a deactivated
pico BS. This could for example be done by periodic transmission of a beacon from
deactivated BSs.
These requirements can be fulfilled by modifying traditional handover schemes from
one BS to the next. But redesigning the structure of RANs allows the macro BSs to
consume low energy, to have long range, and to be designed to provide only the data
rates needed for signaling. I assume the RAN operates using split signaling and data
traffic. While BSs with other ranges exist [CHS08] I consider only macro and pico BSs
for simplicity.
The split of signaling and data traffic between macro and pico BSs allows the energy
consumption of signaling and data traffic to be analyzed independently from each other
(chapters 4 and 5).
1.4. Cooperative transmissions
When each UE can only be served by a single BS there is no freedom to decide which
BSs to activate and which to deactivate. Increasing the number of available pico BSs a
UE can be assigned to can be achieved by deploying more BSs. This not only increases
the setup costs of the RAN, but also its effect on the energy consumption is unclear
because the maximum consumed power also increases (I analyze the effect of increasing
the number of pico BSs on the energy consumption in chapters 5 and 7).
Another possibility of increasing the freedom to assign UEs to BSs is to increase the
transmit power. This might not only be counterproductive for energy consumption, but
is also largely prevented by law and inter-cell interference.
Another possibility to increase the range of BSs is to use cooperative transmissions
from several BSs. The idea of cooperative transmissions is to transmit the same signal
from more than one BS and combine the signal at the receiver. The combined signal is
stronger than any of the individual signals and can thus reach a UE that is not in range
of any of the individual BSs. Figure 1.8 illustrates how such a range extension works.
Cooperation is sometimes also referred to as network multiple-input and multiple-output
(MIMO) or distributed MIMO. In contrast to classical MIMO, where the antennas are
close together, the antennas used for cooperation are further apart.
When the ranges of BSs are increased, fewer BSs are needed to cover an area. For
every power profile which does not consume zero power when idle, reducing the number
of BSs also reduces the idle power consumption of the RAN. Larger ranges increase
the flexibility to serve traffic from more distant BSs. Therefore, it becomes possible to
12
0 T
0.5 T
1 T
1.5 T
2 T
Mean signal strength
Left BS
Both
Right BS
Threshold
Figure 1.8.: Using cooperation the signal of two BSs is combined. Thus, they can provide
service in an area where none of them individually could.
aggregate load on fewer BSs more often. If the load is aggregated to fewer BSs more
BSs can be deactivated and thus less energy is consumed.
For downlink transmissions different implementations of cooperative transmissions
are possible. One way to achieve this is to synchronously transmit from the cooperating
BSs so that the signals constructively interfere at the receiver. This is called coherent
combining (CC) [Gol05]. Another possibility is to transmit both signals at different
times, record them in the receiver, and combine them inside the receiver. This is called
maximal-ratio combining (MRC) [Pro01].
For uplink transmissions the receiving BSs can communicate the received signal to a
single location (using the backhaul network) where the transmission is then decoded.
One possibility to implement this in Long Term Evolution Advanced (LTE-A) is joint
detection [MF11].
If two BSs cooperate, they both send the transmission to the UE, which combines the
two transmissions into one. This allows the UE to decode the transmission, even if it
could not decode a single transmission on its own. A similar method can be used for the
uplink.
The requirements for cooperative transmissions differ for the different implementa-
tions and can be quite high (especially on the data rates and the latency of the backhaul
network [BSC+12]). I ignore these requirements and the difficulties in their implemen-
tation and use cooperative transmissions as a tool and determine what can be gained by
their use. I determine the energy gain of cooperation for both signaling (chapter 4) and
data transmissions (chapter 5).
In chapter 4 I analyze how cooperation reduces the number of macro BSs needed to
cover an area and how many macro BSs need to cooperate to achieve this. Additionally,
I determine how cooperation can be used to decrease the fraction of time a pico BS is
13
needed to transmit data to UEs.
I focus on scenarios in which the network load is low, for example at night and in rural
areas. In such scenarios the demanded data rates of UEs can easily be fulfilled, but the
need to cover an area prevents macro BSs from being deactivated.
The ideas presented here are applicable to state of the art RANs, for example Universal
Mobile Telecommunications System (UMTS) with macro diversity [SFH04] and LTE-A
with coordinated multipoint transmission/reception (CoMP) [PDF+08].
I model cooperation as the possibility of a UE to decode a transmission if the sum of
the received SNR(i) of BS iis at least T:
X
i∈S
SNR(i)≥T, (1.3)
where Sis the set of cooperating BSs. This is a valid model for coherent combining
(CC) [Gol05] and maximal-ratio combining (MRC) [Pro01].
As the shapes of the areas covered by these definitions are hard to determine geo-
metrically (see section 4.7) I define a more restrictive, but geometrically easier to use
type of cooperation. I call the type of cooperation defined by equation 1.3 unlimited
cooperation. In contrast to this I define limited cooperation as
∀i∈S: SNR(i)≥T
|S|,(1.4)
where |S|is the number of cooperating BSs.
1.5. General related work
In this section, I give an overview of other work which reduces the energy consumption of
RANs. In this section, I summarize work which is related to all chapters. More specific
related work sections are in every chapter.
1.5.1. Energy consumption on the BS level
Mancuso and Alouf [MA11] describe how the energy efficiency of each BS can be in-
creased individually by using more energy-efficient antennas and power amplifiers. Song
et al. [SDA+11] describe how deactivating carriers in a single BS can reduce their power
consumption. Son and Krishnamachari [SK12] describe how individual components of
BSs can be slowed down to conserve energy. While these papers describe how to reduce
the energy consumption of an individual BS, I analyze how the energy consumption can
be reduced on the network level.
An alternate approach by Fehske and Marsch [FMF10, MFF10] determines the energy
efficiency of cooperative and dense deployments in terms of cost per bit. In contrast to
my work they do not consider sleep modes and only consider fully loaded RANs. Other
alternatives to considering the total energy consumption is to increase the energy effi-
ciency of transmissions [CZXL11] and increasing the achieved data rates [PF11] without
increasing energy consumption.
14
Son and Krishnamachari [SK12] describe how the speed of components of BSs can be
scaled to apply dynamic speed scaling algorithms [Alb09]. In contrast to their work,
I do not need to modify the internals of BSs and just use the sleep mode to conserve
energy. The results of Son and Krishnamacharis work are an example of lowering power
profiles, which I consider in chapter 3 .
1.5.2. Other wireless techniques
While cooperative transmissions are sent from antennas of different BSs, MIMO trans-
missions use similar principles but from different antennas of the same BS. While both
are available in LTE-A [She10], I only consider cooperation between antennas of dif-
ferent BSs. An alternative I do not consider is cooperative transmission from at least
two UEs to a single BS [NH04]. This is not only a problem because UEs are under
user control, but most importantly it is impractical during low load because only few
cooperation partners will be available. Another alternative is relaying of transmission,
which in contrast to cooperation does not combine both signals at the receiver, from BS
or UE [FSC11, HCM12, LWS+10, NYZR06].
Other applications of cooperation [NH04] usually consider the additional problem
of getting the transmission to the cooperating partner, but because BSs are usually
connected by high-bandwidth, low-latency wired networks I assume they are able to do
so reliably.
1.5.3. Energy consumption on the network level
My goal is to reduce the energy consumption of a single RAN of one technology. Alterna-
tives include reducing the energy consumption of a network of mixed technologies [Ism11]
and reducing the total energy consumption of several RANs by cooperation between op-
erators [Ser12]. Marsan and Meo [MM11] describe how roaming UE between different
operators can conserve energy. I do not consider fairness [KLZ09] to be of high priority
because I only consider low load scenarios where the demands of most UEs can be fully
met most of the time.
I do not explicitly consider the backhaul network [TZJ11]. While approaches exist that
consider the limitations of the backhaul for cooperation [BSC+12], adding backhaul to
my models would increase the complexity. Because the BSs consume more energy than
the backhaul network in RANs [VH11], I ignore the energy consumption of the backhaul.
1.5.4. Prediction
Paul et al. [Pau11] show empirically that the load of RAN is periodic and thus easy to
predict, but the load of individual BSs is not. I only use predictions of aggregated future
traffic to determine which type of traffic will be most dominant (video) [Cis13] to create
a model that can represent this type of traffic.
It is possible to determine the intensity of a Poisson process for an area and track it
over time [FN04, Lee91, Rat13, RB03]. While such techniques are necessary to determine
15
locations of hotspots and the change of network load over time I assume to know the
location of hotspots.
1.5.5. Moving and reducing load
I consider the traffic the UEs generate to be only slightly movable in time and not to have
(known) deadlines. I determine the mean latency of requests from the UEs. Alternative
formulations of the problem consider the traffic to be fully movable in time [Sir02] and
can thus be moved into low load periods [GBW95]. Other alternatives include deadlines
for requests [CLMW07, CCLL07, Li09].
Another possibility to reduce the energy consumption of RANs is to reduce the de-
mands it has to serve. These range from better content dissemination [BBEE08] in
Content-Delivery networks [BPV08] to information-centric networking [KAK10]. Lee et
al. [LRH10, LRKH11] consider the effects of information-centric networking on energy
consumption, while I assume the demands to be fixed.
1.5.6. Effects on the power grid
My goal is to reduce the energy consumed by RANs. Other methods which are not
necessarily specific to RANs or even computer networking try to consume power in areas
where it is cheaper [QWB+09]. While this does not reduce the energy consumption it
reduces the cost and environmental impact of the energy consumption. Hashimoto et
al. [HYT05] describe how to build BSs which are independent of external energy sources.
Felter et al. [FRKR05] describe how to reduce peak power consumption because reducing
the variance in power consumption leads to a more efficient power grid [FMXY12].
While I consider how to run a communication network with less power, it is also
possible to increase the efficiency of the power grid with the help of a computer net-
work [Far10]. Last but not least computer networks can help to reduce the energy con-
sumption and greenhouse gas emissions of other industrial or residential sectors [WO08].
The gained efficiency in other sectors is projected to be even larger than the gain in
RANs [WO08], but for these gains to be useful RANs need to run energy efficiently
themselves.
1.6. Contributions and chapter overview
In this section, I provide an overview of the contributions of the following chapters.
Additionally I reference the publications they are based on.
Chapter 2 I begin by considering an individual BS as the server in a queuing system.
Because this model does not depend on any specifics of a RAN, the results can be
applied to a large class of other systems. Using this queuing model I quantify the
statement that considering an individual BS with a sleep mode only reduces the
energy consumption if power cycles are short.
16
•M. Herlich and H. Karl. Average and Competitive Analysis of Latency and
Power consumption of a Queuing System with a Sleep Mode. In Proceedings
of the International Conference on Future Energy Systems: Where Energy,
Computing and Communication Meet, pages 14:1—-14:10, New York, NY,
USA, 2012. ACM
Chapter 3 As considering each BS individually does not conserve much energy for long
power cycles, I describe how all devices of a network can coordinate to reduce the
energy consumption even if power cycles are long. I use the example of a wired
network to ignore the complexities of wireless transmissions. I quantify the trade-
off between the energy consumption and latency considering a network as a whole
(instead of each device individually).
•M. Herlich and H. Karl. The Trade-Off between Power Consumption and
Latency in Computer Networks. In Vicente Casares-Giner, Pietro Manzoni,
and Ana Pont, editors, Proceedings of the Networking Workshops, volume
6827 of Lecture Notes in Computer Science, pages 273–280. Springer Berlin /
Heidelberg, 2011
For the rest of the dissertation I apply the same idea of network-wide coordination to
RANs. I use the split of signaling and data traffic (see section 1.3) to analyze the energy
consumption of signaling independent from data traffic. The independent analysis makes
it possible to analyze these models analytically. An analysis would be far more complex
in a combined model.
Chapter 4 I describe how much energy can be conserved in RANs if BSs cooperatively
transmit to extend their signaling range. Firstly, I determine the reduction in
energy consumption when BSs are placed specifically for energy efficient signaling.
Secondly, I determine the reduction when BSs are placed but can be deactivated.
•M. Herlich and H. Karl. Reducing Power Consumption of Mobile Access Net-
works with Cooperation. In Proceedings of the International Conference on
Energy-Efficient Computing and Networking, pages 77–86, New York, USA,
2011. ACM
Chapter 5 I analyze the reduction of energy consumption to provide data traffic using
cooperation. I quantify the activity probability for a BS depending on the activity
of UEs.
•M. Herlich and H. Karl. Energy-Efficient Assignment of User Equipment to
Cooperative Base Stations. In Proceedings of the International Symposium
on Wireless Communication Systems, 2013
Both the reduction of consumed energy in signaling and data traffic depend cooperative
transmissions. If cooperative transmissions are not coordinated correctly the radiated
power becomes interference at other receivers. This reduced channel quality will in-
crease the time necessary for their transmissions and thus increase energy consumption.
17
Therefore, it is necessary to keep the radiated power low to prevent disrupting other
transmissions.
Chapter 6 I describe how interference from cooperating BSs can be reduced by selecting
which BSs actually transmit based on instantaneous channel knowledge instead of
average channel knowledge.
•M. Herlich and H. Karl. Analytic Quantification of Outage Probability and
Radiated Power of Cooperative Base Stations. In preparation
Chapter 7 As each of the analytical chapters ignore some aspects of a real RAN, I
present the results of simulating a RAN which includes all these aspects. This al-
lows me to determine how all aspects interact. I determine the energy consumption
of the different approaches with realistic dense urban parameters.
•M. Herlich, T. Hohenberger, and H. Karl. Activation Strategies for Low-Power
Radio Access Networks. In preparation
Figure 1.9 shows which chapters introduce the concepts used in another chapter. It
also provides an overview which chapters are specifically for RANs and which are not.
Moreover, it shows which chapters provide analytical results and which are based on
simulations. Table 1.2 shows in detail which effects I consider in which chapter.
18
Table 1.2.: I consider different effects in each chapter. Xmeans explicitly considered,
(X) means implicitly considered. - means not applicable
Effect\Chapter 2 3 4 5 6 7
Path loss - - X X (X)X
Fading - - X(X)
Antenna sectors - - X
Cooperative transmissions - - X X X X
Coordinated deactivation - X X X X
Progressing time X(X) (X) - X
Interference - - X X
Overload X X X
Shifting load in time X(X)
Shifting load in space - X X X X
2 3
4 5
6
7
RAN
independent
RAN
dependent
Analytic
(some optimization)
Simulative
Figure 1.9.: The interdependency guide shows which chapter provides the basis for other
chapters.
19
2. Conserving energy at each BS
individually
All methods to reduce the energy consumption at each base station (BS) will directly
translate into a reduction of energy consumption of the complete radio access network
(RAN). To conserve energy in a RAN it is, thus, natural to try to conserve energy at
each BS. In this chapter, I determine the limits of sleep modes for reducing the energy
consumption if the sleep modes of the BSs are not coordinated.
I quantify the effect of times on latency and power consumption for a BS with a sleep
mode. The calculations in this chapter do not depend on any special properties of BSs
and, hence, can be applied to any single server queuing system with a sleep mode.
When a BS is in sleep mode it consumes only negligible power. But changing into
sleep mode takes time and during that time power is also consumed. This power-cycle
duration can also introduce additional latency into the system because jobs have to wait
for the BS to become active again. Deactivating the BS more will often decrease the
energy consumption but also increase latency. I first analyze the trade-off between energy
consumption and latency for Poisson arrivals. Because the traffic a BS has to serve does
not actually form a Poisson process, I also do a competitive analysis for worst-case
arrivals. This allows me to describe whether the results are specific for Poisson arrivals
or can be generally applied.
To calculate the latency and power consumption for Poisson arrivals I use Markov
chains. For the competitive analysis I present arrival patterns which result in high
latency and energy consumption and prove that these patterns are the worst case.
2.1. Introduction
To determine the increased latency I use a classical queuing system in which the server
has the additional ability change into sleep mode. Being in this sleep mode consumes
less power than being active, but the power cycle consumes time and energy.
Different policies can be applied to change between active and sleep mode. For ex-
ample, a greedy policy deactivates the server as soon as no jobs are in the system and
activates the server again when a job arrives.
I want to determine expressions for the worst-case and average latency and power
consumption, depending on the power-cycle duration of a BS. Also, I want to quantify
the following (intuitively plausible) statements for the different assumptions about the
job arrivals: “It is easy to conserve energy, but hard (or impossible) to keep the latency
low” and “With higher power-cycle durations, using a sleep mode becomes less and less
practical.”
20
As the assumptions of this chapter are valid for a wide range of queuing systems I
formulate them independently of user equipments (UEs) and BSs. Hence, I use the
terminology which is usually used in queuing systems (jobs and servers).
2.2. Related work
Chen et al. [CX07, CXSY09] describe a Markov chain to calculate the power consumption
of the greedy policy and the accumulate and fire policy. My model is the same as theirs,
but I expand upon their work and additionally calculate the mean latency of a job and
compare this to the latency obtained when using an oracle. While they focus on the
average case for greedy and accumulate and fire policy, I compare this to the optimal
values and calculate the competitive ratio.
Bredenbals analyzed the behavior of system with delayed activation and deactivation
in his master’s thesis [Bre13]. It is based on the same publication [HK12] as this chapter.
An interesting result of his work is that it is possible to select activation and deactivation
delays, such that the resulting system uses less energy than the greedy policy and at the
same time has a lower mean latency.
Irani et al. [ISG02] analyze the competitive ratio of systems with multiple power saving
states. In their model the only penalty for a state change is the consumed energy; the
model does not include the time necessary for the change. Thus, they make no statements
about the effect of conserving energy on the latency. In contrast to this, I describe both
latency and power consumption to characterize the trade-off between them.
Ren et al. [RKM05] focus on hierarchical scenarios with non-stationary service re-
quests, but also provide a review of the dynamic power management problem. This
includes the power consumption and performance degradation of a system with con-
stant state-change times.
Andrew et al. [AWT09] investigate how to minimize a linear combination of energy
and response time. They show that Shortest Remaining Processing Time scheduling is
2-competitive. Stidham [Sti70] also converts waiting time and energy cost into a single
metric which then can be minimized. Lam et al. [LLT+09] minimize the sum of energy
and latency. Zhang and Chanson [ZC05] limit the average delay and try to minimize
the consumed power. In contrast to my model, none of these models considers the time
necessary to change states.
Authors providing overview papers of energy-efficient queuing include: Pruhs [Pru07]
and Albers [Alb09], who provide overviews of competitive analyses of scheduling prob-
lems, Irani and Pruhs [IP05], who provide a description of both open and solved algorith-
mic problems with respect to power management, and Lu and De Micheli [De 01], who
describe an oracle and its application to different traces. Benini et al. [BBPD99] describe
how dynamic power management schemes can be modeled using Markov models.
Others [TzJwHj04, Ped99] propose alternative policies to conserve energy. The new
policies are usually compared to other polices, while I compare the greedy policy to the
optimal values.
Li [Li09] analyses the competitive ratio of a finite-capacity queue for jobs with hard
21
deadlines. While this model is more detailed than mine it does not consider energy
consumption. Baptiste et al. [BCD07] provide a polynomial time algorithm to calculate
the minimum-energy schedule for a given set of jobs. While they provide an offline
algorithm, I analyze online algorithms.
Instead of limiting the latency by deadlines, Chan et al. [CLMW07] limit the energy
consumption and calculate the competitive ratio for throughput. I do not limit any
resource but analyze the interaction between latency and energy consumption described
by competitive ratios and averages.
Heyman [Hey69] models similar problems as an M/G/1 with (de-)activation costs and
converts waiting time into the same costs. He determines that the optimal policy is
the accumulate and fire policy used by Chen [CX07] and in this dissertation. Heyman
considers the arrivals to be a Poisson process, while I also consider competitive ratios.
Methodically different approaches have been considered to develop and analyze power
management policies as well: machine learning [BGN+10], Petri nets [QW00], and model
checking [NP02]. Each of these uses different techniques to approach problems related
to mine.
Another type of work focuses on realistic values for power consumption and job arrival
patterns. Prominent examples are web servers [BEK+02] and hard drives [SBdM00].
They use traces to compare different energy conservation methods, while I focus on the
worst case for arbitrary systems.
2.3. Model
2.3.1. Queuing system
Queue Server
Power Manager
Jobs Jobs
Status Status Commands
Figure 2.1.: In my model jobs arrive at the queue to be processed by the server. The
power manager observes both the queue and the server and initiates state
changes of the server.
Figure 2.1 shows the model that I use for power cycles. It consists of jobs, a queue, a
server, and a power manager. The jobs arrive at the system and have to be processed
by the server. Because the server cannot handle more than a single job at a time, the
queue buffers arriving jobs until the server can process them in a first-in-first-out (FIFO)
manner. The size of the queue is unlimited. The power manager observes the server and
the queue and initiates activation and deactivation of the server.
22
While the model that only a single job can be processed at a time is widely used in
computing it does not hold for BSs which process transmission requests concurrently.
They will usually use a scheduling policy to fairly distribute the radio resource to all
currently running transmissions. But under the assumption that the durations of jobs
are exponentially distributed, processing one job after the other and processing all jobs
in parallel is equivalent. The reason for this is that the exponential distribution is mem-
oryless and, thus, the finishing rate of jobs is the same independently of the processing
strategy. Therefore, the distribution of number of jobs in the system is the same and,
thus, also the mean latency (see Little’s law [Nel95]). Note that this is only valid as long
as the server does not know the processing time in advance and can base its scheduling
on it.
Because the M/M/1 system I describe models that a job is finished faster when it is
alone in the system compared to a crowded system, it is a good representation for data
transfers. As the duration of voice (and video) calls does not depend on the load of the
BS, a M/M/1 is not a good model for it. For voice traffic a M/M/kmodel (without
queue) is more suited, where kis the maximum number of concurrent calls a BS can
handle. However, as I consider data transfers a M/M/1 is the more suitable model.
2.3.2. Server states and timing
Active (SA)Deactivation (SD)
Sleep (SS) Activation (SU)
Triggered by power manager
Z
Triggered by power manager
A
Figure 2.2.: State changes of the server are triggered by the power manager or take a
random time to complete – depending on the type of state change.
Figure 2.2 shows all states of the power cycle of the server: Sleeping (SS), activating
(starting up) (SU), active (SA), and deactivating (SD). The changes from deactivating
to sleeping and from activating to active are determined by the (random) variables A
and Z, while the other two are triggered by the power manager. I assume the power
manager can only issue the state changes stated above. It cannot change the state from
deactivating to activating or change the order of jobs in the queue. Other models for
example allow the server to abort the deactivation and directly enter the active state
without the penalty of going through the activation state again. However, I chose my
model as it is the most restrictive and the results can also be applied to the other models,
but not the other way around.
23
The following four values describe the behavior of the system. (1) The inter-arrival
time of jobs L(at rate λand mean E[L] = L). (2) The time Mthe server needs to
process a job (at rate µand mean E[M] = M). (3) The time Zthe server needs to
deactivate (at rate ωand mean E[Z] = Z). (4) The time Athe server needs to activate
(at rate αand mean E[A] = A).
I analyze scenarios in which the four variables are determined in different ways:
(1) random variables that follow exponential distributions, (2) constant values, and
(3) values selected by an adversary for construction of the worst case. For all different
cases I only consider scenarios which have infinite length to ignore transient effects. This
is usually done in case (1) and easy to construct by repeating a scenario infinitely often
in cases (2) and (3).
Using Markov chains the exponential distribution is easiest to analyze. For the com-
petitive analysis constant times are easiest to analyze. But in addition to being easier or
more complicated to analyze, different systems will have different distributions of power-
cycle durations. For very simple system (e.g., a hand-held calculator) the deactivation
is zero because the power can simply be turned off. More complex systems (e.g., a per-
sonal computer) will have to perform tasks before it can be go into sleep mode. This can
include informing connected devices that it is going into sleep mode and preparing the
sleep mode itself. The time this preparation takes might depend on the system state,
for example, the time it takes for a computer to suspend-to-disk depends on the amount
of RAM that has to be written to the disk.
The activation time can also vary depending on what the system has to prepare to
fulfill its functions. A computer might run periodic integrity and virus scans or upgrade
the system. Networked systems (such as BSs) need time to activate their networking
components and synchronize their state with their neighbors or the rest of the network.
Because many of these effects are hard to predict and are usually outside the control
of the power manager, using a randomly distributed power-cycle duration is reasonable.
The reason to pick exponentially distributed power-cycle durations is that they can be
modeled easily in Markov chains.
2.3.3. Policies and metrics
I compare four different policies of the power manager: (1) a greedy policy GP, which
deactivates the server as soon as no jobs are in the system and activates it as soon as
a new job arrives. (2) The “accumulate and fire” policy AP(k) [CX07], which behaves
like greedy but waits for kjobs to activate the server again. (3) An energy-minimizing
policy EP with an oracle, which I use as a reference for energy consumption. (4) A
latency-minimizing policy LP with an oracle, which I define to use the minimal amount
of energy to achieve the same latency as an always-on policy OnP. I allow the energy-
and latency-minimizing policies to use an oracle, which has knowledge of all future values
of all random variables, that is inter-arrival time, processing time, activation time, and
deactivation time. I show in section 2.4.3 that the energy-minimizing policy does not
need the oracle. This makes the latency-minimizing policy the only policy that uses the
oracle.
24
I consider different metrics to compare a policy Xto others: (1) power consumption
P(X) of the server, averaged over time, (2) latency L(X) as the mean time a job is in
the system (waiting time plus processing time), (3) the ratio of latency compared to
the always-on policy and, (4) the ratio of power consumption compared to the energy-
minimizing policy. For both ratios I calculate the worst case compared to an offline
algorithm (competitive ratio [BEY05] CR(X)) and the mean under Poisson arrivals
(Poisson ratio PR(X)). I denote the ratios for latency with an index L (CRL(X) and
PRL(X)) and the ratios for power consumption with an index P (CRP(X) and PRP(X)).
To calculate the power consumption of the system, I assume the server is the main
consumer of power and ignore all other components. To further simplify the analysis I
assume the server consumes 1 unit of power while active, activating, and deactivating,
and zero power while sleeping. If these assumptions do not apply, the results can be
scaled accordingly.
2.4. Poisson arrivals
In this section, I analyze the policies under the assumption that all variables (namely,
A,L,M, and Z) are exponentially distributed. This allows the construction of a con-
tinuous-time Markov chain [Bol98] to calculate the mean power consumption and mean
latency. Chen et al. [CX07] provide a similar analysis for the power consumption. I
summarize their results and explain how I derive the mean latency. I used Maxima
5.25.1 to do the calculations I present in this chapter.
2.4.1. Greedy policy
Chen et al. [CX07] model the greedy policy as a Markov chain as shown in figure 2.3.
The states in the chain are labeled as (State of server, Number of jobs in the system). I
denote the probability to be in state Xin the stationary distribution as Pst[X].
Using the same steps as Chen et al. [CX07] I express the probability that the server
is consuming power P using the greedy policy GP as
P(GP) = 1 −Pst[SS,0] = λ(µ(ω+α)λ+ω(µω +αω +αµ))
µ((ω+α)λ(λ+ω) + αω2).(2.1)
As I assume that the server consumes no power when in sleep mode and 1 unit of power
else, this is also the mean power consumption of the greedy policy over time.
I calculate the mean number of users in the system by
E[Users in System] = ∞
X
i=1
i(Pst[SA, i] + Pst[SD, i] + Pst[SU, i]) .(2.2)
With this result and Little’s law [Nel95] I calculate the average latency of a job to be
L(G) = U1λ3
ω+ω
α−µ
ωU1λ2−(µU1+α)λ−(µ+α)ω
(ω+α)λ3
ω+ω−αµ
ω−µ+αλ2+ U2λ−αµω ,(2.3)
where U1=ω
α+α
ω+ 1 and U2=αω −µω −αµ.
25
SD,1
SU,1
SA,1
SD,2
SU,2
SA,2
SD,3
SU,3
SA,3
SD,0
SS,0
λ
λ
λ
µ
λ
λ
λ
µ
λ
λ
λ
µ
ω
α
ω
α
ω
α
µ
ω
λ
λ
Deactivating
Sleeping
Activating
Active
Users in system
Figure 2.3.: A Markov model for the behavior of the greedy policy [CX07] allows both
latency and power consumption to be determined.
2.4.2. Accumulate & fire policy
A generalization of the greedy policy is the accumulate and fire policy. It deactivates
the server as soon as no jobs are left in the system and activates it again when the queue
contains kjobs [CX07]. Figure 2.4 shows the Markov chain of the accumulate and fire
policy. The probability to consume power and thus also the mean power consumption
using accumulate and fire using my model is [CX07]
P(AP(k)) = αλk+2 −αλ(λ+ω)kλ−µ
α+kω−µ
µω (λ+αk) (λ+ω)k+αµλk+1 .(2.4)
Using the same method to calculate the mean number of jobs in the system as used
for the greedy policy and Little’s law [Nel95] I calculate the mean latency to be
L(AP(k)) = (λ+ω)kω2V1+ 2λkαλV2
2αωλ (λ−µ)(λ+αk)ω(λ+ω)k+αλk+1,(2.5)
where V1= 2λ3+ 2 (αk −µ−α)λ2+αk (αk −2µ−3α)λ+α2(1 −k)kµ and V2=
(ω+α)λ2+ ((−µ+αk −α)ω−αµ)λ−αkµω.
2.4.3. Energy-minimizing policy
To minimize the energy consumption the server needs to spend as little time doing state
changes and running idle as possible. This can be achieved by letting the parameter
26
SD,0 SD,1 . . . SD,k−1 SD,kSD,k+1 . . .
SS,0 SS,1 . . . SS,k−1
SU,kSU,k+1 . . .
SA,1 . . . SA,k−1 SA,kSA,k+1 . . .
λλλλ λ λ
λλλ
λ
λλ
λλλ λ λ
µ
µµ µ µ
ω ω ω
ωω
α α
µ
Figure 2.4.: A Markov model for the behavior of accumulate and fire policy [CX07] allows
both latency and power consumption to be determined.
kof the accumulate and fire policy go towards infinity. Intuitively the power manager
waits for a lot of jobs to be scheduled before activating the server up to process them,
amortizing the power-cycle cost over the jobs that are processed at once. Note that this
policy does not need an oracle because all the information to implement it is already
available without an oracle.
As every policy needs to spend at least λMenergy per time unit on average just
to process the jobs, no policy can consume less than λM=λ/µ power on average
and still process all jobs. Because the mean power consumption using the accumulate
and fire policy when ktends towards infinity becomes exactly λ/µ, this limiting case is
the energy-minimizing policy. With increasing k, however, the mean latency also tends
towards infinity.
As the latency is unbounded, the factor by which the latency is higher than the
minimum possible latency is also unbounded. In contrast, to this the power consumption
of any policy can be at most 1 (the maximum power consumed by the server). Because
the most energy-efficient policy consumes λ/µ power, the ratio for power consumption
for any policy cannot be higher than µ/λ.
2.4.4. Latency-minimizing policy
I analyze the latency-minimizing policy under the assumption that it can access an oracle
that knows all future values of all random variables and not only their distributions.
First, I assume all random variables are exponentially distributed and their realizations
27
are known to the oracle. Then, I apply the same calculation to constant power-cycle
durations. In the second case, the oracle is still necessary to predict the inter-arrivals
times.
The minimum latency is achieved by always keeping the server active. Hence, the
latency of the latency-minimizing policy LP is the same as of the always-on policy OnP.
It is [Bol98]
L(LP) = L(OnP) = 1/µ
1−(λ/µ)=1
µ−λ.(2.6)
Time
Arrivals
Latency-
minimizing
policy
Next arrival
I II
Job Active
Sleeping
Deactivation Activation
No work
Figure 2.5.: The latency-minimizing policy can only deactivate the server during idle pe-
riods that are long enough. It can only do this reliably because it has access
to an oracle for the knowledge of inter-arrival and power-cycle durations.
To reach the same latency, the latency-minimizing policy must finish each job at the
same time and, thus, also start processing each job at the same time as the always-
on policy would. The only time when the latency-minimizing policy can save energy is
during idle periods that are long enough: when the server is idle and the time for a power
cycle is smaller than the time until the next arrival, it can be deactivated. Figure 2.5
illustrates this.
I calculate the probability that the latency-minimizing policy can deactivate the server
between two arrivals from the probability that the server is idle after processing the job
and the probability that there is enough time for a power cycle. The probability that
a job leaves the system empty is the same as the probability that a job finds an empty
system because the M/M/1 is reversible [Nel95]. And because “Poisson arrivals see time
averages” (PASTA) [Nel95], this is the same as the time average Pst[0] = 1−λ/µ. Hence,
the probability that a deactivation is possible after the server has processed a job is
P[Deactivation possible] =
Job leaves system empty
z}|{
Pst[0]P[A+Z<L]
=Pst[0]P[Z<L]
| {z }
Enough time to deactivate
P[A+Z<L|Z <L]
| {z }
Enough time to activate again
.(2.7)
I derived the third part of the equation with law of the total probability and the
knowledge that P[A+Z<L|Z >L] = P[A<0] = 0. When Lis exponentially
28
distributed and thus memoryless, I can simplify this to
P[Deactivation possible] = Pst[0]P[Z<L]P[A<L].(2.8)
With probability P[Deactivation possible] the server can be deactivated for a time of
(L−A−Z|A+Z<L) = L. Because the average time the server spends in sleep
mode between two arrivals is P[Deactivation possible]Land the mean time between two
arrivals is L,P[Deactivation possible] is also the fraction of time spent in sleep mode
in the steady state distribution with the latency-minimizing policy. Given the binary
energy consumption model, this results in a mean power consumption of
P(LP) = λ+ω+α+αω
µ
λ+ω+α+αω
λ
,(2.9)
when all random variables are exponentially distributed.
To calculate P[Deactivation possible] I only needed to assume that Aor Zare expo-
nentially distributed for the last step. Using, as example, constant values for Aand Z,
the mean power consumption simplifies to
PC(LP) = 1 −
Time not processing jobs
z }| {
1−λ
µe−(Z+A)λ
| {z }
Fraction of that time spent in sleep mode
(2.10)
A similar analysis can be made for constant service times: The probability that a
job leaves the system behind empty is given by the queue-length distribution of an
M/D/1 [Nel95].
2.4.5. Comparison
In this section, I first show that both the greedy policy and the latency-minimizing
policy can consume less energy than the other depending on the arrival pattern. Then I
show that the latency-minimizing policy consumes less energy than the greedy policy, in
the long run, when all random variables (inter-arrival time, processing time, power-cycle
durations) are exponentially distributed.
Figure 2.6 shows an arrival pattern that causes the latency-minimizing policy to con-
sume more energy than the greedy policy as it does not deactivate the server between
the arrivals. The greedy policy will process the first job later and the second right after
the first one. In contrast to this, Figure 2.7 shows an arrival pattern in which the future
knowledge of the latency-minimizing policy prevents it from deactivating the server be-
cause this would delay the second job. The idle time of length is necessary to give the
greedy policy time to initiate the deactivation. The latency-minimizing policy not only
reduces latency, but also conserves energy in this case because it deactivates the server
for a longer period later.
Given these two examples, it is clear that both policies can be more energy-efficient
than the other for given examples. Next I determine which of these effects outweighs
the other when arrivals are given by a Poisson process.
29
Arrivals
Latency-
minimizing
policy
Greedy
policy
A+M
I II
I II
Figure 2.6.: This is an arrival pattern in which the greedy policy consumes less energy
than the latency-minimizing policy.
Arrivals
Latency-
minimizing
policy
Greedy
policy
A+M+
I II
I II
Figure 2.7.: This is an arrival pattern in which the latency-minimizing policy consumes
less energy than the greedy policy.
30
I already calculated the power consumption of the greedy policy P(GP) and the power
consumption of the latency-minimizing policy P(LP). Subtracting the power consump-
tion of the latency-minimizing policy from the greedy policy and simplifying yields a
strictly positive result (with the reasonable assumptions α, ω > 0 and µ > λ > 0).
Hence, the greedy policy consumes more power on average than the latency-minimizing
policy when the arrivals are a Poisson process.
Given the mean values for the latency of the greedy policy and the always-on policy,
I calculate the factor PRL(GP) = L(GP)/L(OnP) by which greedy policy’s latency is
higher than that of the always-on policy
PRL(GP) = 1 + (A+Z)(µ−λ) + R, (2.11)
where
R=AZ2λ3−µAZ2λ2+AZλ2−µAZλ+Zλ−µZ
Z2λ2+AZλ2+Zλ+Aλ+ 1 <0 (2.12)
is always negative, and thus PRL(GP) grows at most linear in Aand Z. Analogously
the factor for mean power consumption of the greedy policy PRP(GP) = P(GP)/P(EP)
is
PRP(GP) = 1 + µ−λ
λW, (2.13)
where
W= 1 −1
A+ZλZλ+ 1+ 1 (2.14)
takes values between 0 and 1 and describes how much of the power that the energy-
minimizing policy saves is conserved by the greedy policy.
2.5. Adversary-controlled arrivals
In this section, I compare the mean latency and the mean power consumption of the
greedy policy to the respective optima. I do this for an adversary who has control over
the inter-arrival times of new jobs. Letting an adversary try to maximize latency or
power consumption by selecting values for the variables is the same as calculating the
worst case for a random variable, because the adversary can select the worst case.
I begin by considering both the job processing time and the (de-)activation durations
to be constant values, because this is easiest to analyze. Later, I generalize the result to
arbitrary distributions.
2.5.1. Latency ratio: greedy policy
I show that the competitive ratio of the greedy policy for latency is CRL(GP) = 1+ A+Z
M.
To do this I provide an example of job inter-arrival times in which the mean latency of
the greedy policy is a factor of 1 + A+Z
Mhigher than the latency of the always-on policy.
Later I show that this is the maximum for all inter-arrival times.
31
Arrivals
Latency-
minimizing
policy
Greedy
policy
I
A+M+
II
M
III
M
IV
M
V
M
I II III
Figure 2.8.: This is the arrival pattern with the worst competitive ratio for latency CRL
for the greedy policy.
The basic idea for the example is to create the highest possible latency for the greedy
policy and never let it catch up with the always-on policy again. Figure 2.8 illustrates
the idea. It is achieved by creating an idle time gap of size between the end of the
first job and the arrival of the second job. While the server of the always-on policy
will stay active, the greedy policy will undergo a full power cycle and finish processing
the first job A+Z − later than the always-on policy. For the rest of the (infinitely
long) example, the arrival times of new jobs equal the finish times of the last job of the
always-on policy. Hence, every following job will be finished A+Z −later using the
greedy policy than the always-on policy.
Using this allocation of job inter-arrival times, the always-on policy will finish pro-
cessing each job Mtime units after it arrived, while the greedy policy will finish all
(infinitely many) but the first job after A+Z −+Mtime. Or, equivalently, a factor
of 1 + A+Z
Mlater than the always-on policy for going to zero.
To show that this is the worst the greedy policy can behave, I show that the greedy
policy can never be more than A+Zlate when finishing a job. To do this I show that
the invariant “the arriving job is finished at most A+Zlater than using the always-
on policy” holds for all job arrivals during any sequence of inter-arrival times using
mathematical induction.
First I show that the invariant holds at the first arrival time t0: The always-on policy
will immediately start processing the job if it arrives at t0, while the greedy policy will
initiate a state change at t0and finish at t0+A. Hence, the job is finished at time
t0+Musing the always-on policy and t0+A+Musing the greedy policy.
Now I show that the invariant also holds for all subsequent arrivals. I make a case
distinction over how many jobs are in the system of the always-on policy when the new
job arrives, called n. For n > 0, both policies will start processing the new job right
after the last job finishes. Hence, the new job will have the same delay as the previous,
which by the invariant is at most A+Z.
When a new job arrives and no other jobs are in the always-on system, I define the
32
Always-on
policy
Greedy
policy
Case 1 Case 2a 2b 2c
I
I
dZ
Figure 2.9.: The different possibilities for arriving jobs need to be considered individually
to analyze the latency of the greedy policy GP.
time between the previous job finished in the always-on system and in the greedy system
as d(note that time dmay or may not have passed when the new job arrives). Figure 2.9
illustrates the different possible time windows in which a new job can arrive. If dhas
not passed, the new job will start processing immediately using the always-on policy and
after dhas passed using the greedy policy. Hence, this new job will be finished at most
dlater than using the always-on policy, because d < A+Zholds by the invariant.
If the new job arrives after dhas passed, the greedy policy will power cycle the server
and the delay will not be larger than A+Z; using the always-on policy processing starts
as soon as the new job arrives.
This shows that no job will be finished more that A+Zlater using the greedy policy
than the same job using the always-on policy. Because each job has a running time of
M, each job is in the system Mtime units compared to A+Z+M, which is equivalent
to a maximum increase in latency by a factor of 1 + A+Z
M. Hence, the competitive ratio
for latency of the greedy policy is CRL(GP) = 1 + A+Z
M.
2.5.2. Latency ratio: accumulate & fire policy
As the greedy policy is equal to the accumulate and fire policy with k= 1, the com-
petitive ratio for latency CRL(AP(k)) is 1 + A+Z
Mfor k= 1. I now show that for all
larger k, the competitive ratio for latency of the accumulate and fire policy is infinite by
providing an arrival pattern that can create arbitrarily large latency ratios. The idea is
to let the always-on policy handle the first job without delay while the accumulate and
fire policy waits for the second job to activate the server. The time nbetween arrivals
can be selected arbitrarily large.
The i-th job arrives at time i·n. The accumulate and fire policy will only activate
the server when kjobs have arrived, to process them in a batch. Hence, when ngoes
towards infinity the mean latency of a job goes towards infinity.
33
2.5.3. Energy ratio: greedy policy
Again, first I give a series of job inter-arrival times that result in the worst case energy
consumption for the greedy policy; then I prove that this is the worst case.
The basic idea for the scenario is to let the greedy policy power cycle for each job while
the energy-minimizing policy waits until a large batch of jobs has arrived and processes
them in one cycle.
Arrivals
Energy-
minimizing
policy
Greedy
policy
A+M+Z A+M+Z A+M+Z
I II
I II . . .
Figure 2.10.: This is the arrival with the worst competitive ratio for power CRPfor the
greedy policy.
The job arrivals are at i(A+M+Z), with i∈ {0, . . . , n}. The greedy policy will
power cycle between each arrival and thus consume n(A+M+Z) units of energy. This
is illustrated in figure 2.10. The energy-minimizing policy will activate once when all
jobs have arrived, process all jobs, and deactivated after that. Hence, it will consume
A+nM+Zunits of energy. When napproaches infinity, the ratio between both energy
consumptions approaches 1 + A+Z
M.
In the worst case, the greedy policy will do a full power cycle for each job. Hence,
consuming A+M+Zunits of energy for each job. To process a job, the server must be
active for at least Munits of time, and thus consumes at least Munits of energy. The
ratio between the two energy consumptions shows that the competitive ratio for power
CRP(GP) consumption of the greedy policy is 1 + A+Z
M. Note that the competitive ratio
for power is the same for the latency-minimizing policy because it also activates the
server for every single arriving job.
2.5.4. Energy ratio: accumulate & fire policy
First, I describe an arrival pattern that results in this ratio and, second, I show that no
pattern can be worse. Both arguments closely follow the same idea as for the greedy
policy, which figure 2.8 shows; the only difference is that the accumulate and fire policy
processes kjobs in each cycle.
The job arrivals are at i(A+kM+Z), with i∈ {0, . . . , kn}. The accumulate and fire
policy will power cycle between every karrivals and thus consume n(A+kM+Z) units
34
of energy. The energy-minimizing policy will activate once when all jobs have arrived,
process all jobs, and deactivated after. Hence, it will consume A+nkM+Zunits
of energy. When napproaches infinity, the ratio between both energy consumptions
approaches 1 + A+Z
kM.
As the accumulate and fire policy only initiates state changes when jobs arrive or
finish, it will do a full power cycle every kjobs. Hence, it consumes A+kM+Zunits
of energy for kjobs. To process kjobs, the server must be active for at least kMunits
of time, and thus consumes at least kMunits of energy. The ratio between the two
energy consumptions shows that the competitive ratio for power consumption of the
Accumulate and Fire policy CRP(AP(k)) is 1 + A+Z
kM.
Note that the competitive ratio for power CRL(AP(k)) for the accumulate and fire pol-
icy approaches 1 when kapproaches infinity. The reason for this is that the accumulate
and fire policy becomes the energy-minimizing policy when kapproaches infinity.
2.5.5. Impossible trade-offs between energy and latency
In this section, I prove two theorems that give lower bounds on the competitive ratios
for algorithms without an oracle.
Theorem 2.1. No single policy (without an oracle) can have a competitive ratio for
latency CRLstrictly lower than 1 + A+Z
Mand a finite competitive ratio for power CRP.
Proof. To reach a finite competitive ratio for power, every policy has to deactivate the
server at some point in time. The same example from the calculation of the competitive
ratio CRLof the greedy policy (figure 2.8) can now be used again: after the deactivation
has been initiated the adversary can then make jobs arrive at the system. When the
server is active again it has a delay of A+Z−compared to the always-on policy. From
then on the adversary can keep the server utilized all the time, thus never letting any
policy catch up with the always-on policy. Hence, all jobs will finish after A+Z−+M,
while the always-on policy finishes them after Mtime units. For →0, this results in
a competitive ratio for latency of CRL= 1 + A+Z
M.
Theorem 2.2. No single policy (without an oracle) can have a finite competitive ratio
for latency CRLand a competitive ratio for power CRPstrictly lower than 1 + A+Z
M.
The idea of the proof is that in order to have a competitive ratio CRPfor energy
lower than 1 + A+Z
M, the policy must process more than one job during one power cycle
on average. However, because jobs can be arbitrarily far apart, this will result in an
arbitrarily large mean latency.
Proof. An arrival pattern for which no policy can perform under both boundaries is the
example already provided to show that any accumulate and fire policy with k > 1 has
infinite competitive ratio for latency: The job iarrives at time in, where napproaches
infinity. A policy that executes each job in a single power cycle will consume A+M+Z
energy for each job and thus have a CRPof 1+ A+Z
M. Thus, it violates the first boundary.
Define the mean number of jobs executed in a power cycle as c > 1. The competitive
35
ratio for power is 1+ A+Z
cM, which fulfills the first boundary. To process cjobs in a single
power cycle, c−1 jobs must be delayed by at least nunits of time. Hence, the mean
latency of all jobs must be at least
L≥
Non-delayed jobs
z }| {
(0 + M)1
c+
Delayed jobs
z }| {
(n+M)c−1
c.(2.15)
For every fixed c > 1, this sum tends towards infinity, when ntends toward infinity.
Competitive ratio for power CRP
Competitive ratio for latency CRL
11 + A+Z
M∞
1
1 + A+Z
M
∞
Impossible
by definition
Impossible
for online algorithms
Energy-minimizing policy EP
Greedy policy GP
Latency-minimizing policy
with oracle LP
Always-on policy OnP
Figure 2.11.: This is an illustration of the competitive ratios of the different policies.
Note that this is not to any scale.
Figure 2.11 gives an overview of the different competitive ratios and which combina-
tions of ratios are not possible for online algorithms. The figure is only meant as a visual
reminder of the possible trade-offs of the different policies; it is not to any scale. The
results only hold for deterministic algorithms. An analysis for randomized algorithms
that is based on my work can be found in Nico Bredenbals’ masters’s thesis [Bre13].
2.5.6. Arbitrarily distributed random variables
If the adversary has control over the inter-arrival times of new jobs and all other values
(A,M,Z) are general random variables or also under adversarial control, the competitive
36
ratios become
CRP(GP) = E[A+Z)]
E[M](2.16)
and
CRL(GP) = max(A+Z)
E[M].(2.17)
This holds for all distributions of A,Mand Z. I assume the adversary knows all
realizations of all random variables not under its control. If the adversary has control over
A,M,or Z, both the expected value Eand the maximum value max can be calculated
from the choices I allow the adversary. If the numerator is infinite or the denominator
approaches zero, the corresponding competitive ratio is infinite.
As the proofs of these statements are similar to the proofs done in the earlier sections,
I only outline the differences here: The maximum power cycle time determines the
competitive ratio for latency and the mean power cycle time determines the competitive
ratio for power. The reason for this is that a single long power cycle can result in high
latency for all following jobs, but only consumes additional energy once.
To create the worst ratio for power consumption, the adversary provides the jobs so
sparsely that the greedy policy will power cycle for each job, thus consuming A+M+Z
energy while the energy-minimizing policy consumes Menergy asymptotically.
The worst ratio for latency can be achieved by letting the greedy policy power cycle
repeatedly. Once a power cycle of maximum (or any arbitrarily large) length starts, the
rest of the arrivals can be scheduled just when the always-on policy finished the previous
job. This results in a mean latency of the always-on policy of M, while the mean latency
for the greedy policy is max(A+Z) + M. It is possible to show this using the same
distinction as in section 2.5.1.
The statements made in this section hold for all arrival patterns of infinite length.
They are not true for arrival patterns of finite length, but the longer the arrival pattern,
the more it approaches the values calculated in this section. The reason for this is that
the adversary needs time and in some cases repeated tries to generate the worst case.
When the time is limited the amount of tries to for example find a long power cycle is
limited and, hence, he cannot create the worst case reliably.
2.6. Results
Unless noted otherwise, I arbitrarily selected the arrival rate λ= 1/4, the processing
rate µ= 1, the activation rate α= 2, and the deactivation rate ω= 2 for the plots
shown in this section.
To gain confidence in the correctness of the analytic derivation of the energy consump-
tion and the latency, I implemented a simple event-based simulation from the model.
Figures 2.12 and 2.13 show that the results derived in the previous section closely match
the results from the simulation. Hence, I conclude that the analytical results are correct.
Additonally, both figures show that a higher activation threshold increaes latency, but
the effect on power consumption is negligible.
37
0
2
4
6
8
10
0 0.2 0.4 0.6 0.8 1
Mean latency L
Load λ
GP = AP(1)
AP(2)
AP(3)
Figure 2.12.: The analytic results for the latency match my simulation. Note that the
latency increases when the load approaches 0 for the accumulate and fire
policy for k > 1 because often single jobs wait for others to arrive and
activate the server.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Mean power consumption P
Load λ
GP = AP(1)
AP(2)
AP(3)
Figure 2.13.: The analytic results for the power consumption match my simulation.
38
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Mean power consumption P
Load λ
Greedy policy GP
Latency-minimizing
policy LP
Figure 2.14.: The greedy policy conserves nearly as much power as the latency-
minimizing policy.
0
2
4
6
8
10
0 0.2 0.4 0.6 0.8 1
Mean latency L
Load λ
Greedy policy GP
Latency-minimizing policy LP
Figure 2.15.: The greedy policy has a higher latency than the latency-minimizing policy.
39
Figure 2.14 and 2.15 compare the latency and the power consumption of the greedy
policy and the latency-minimizing policy. They show that the greedy policy conserves
nearly as much power as the latency-minimizing policy, but pays for this with increased
latency.
0
5
10
15
20
25
30
35
40
45
0 0.2 0.4 0.6 0.8 1
Mean latency L
Load λ
Other jobs
On/off
Full
On/off + other jobs
Figure 2.16.: The S-Curve of the full model is the sum of two different effects: Waiting for
the server to become active, which dominates under low load, and waiting
for the server to finish other jobs, which dominates at high load. I use
A=Z= 20 to show both effects in this plot.
Figure 2.16 shows an S-curve that appears when the activation and deactivation times
are higher (A=Z= 20 in this plot). The S-curve is the result of the overlap of two
different reasons for latency: Waiting for the server to reach the active state and waiting
for other jobs to finish processing. For this comparison I calculated the time a job waits
on other jobs to finish from the always-on policy; and the time waiting for the server to
become active from a Markov chain described in section 2.4 that only has one customer.
Figure 2.16 shows that the sum of these two effects closely describes the effect seen in
the full model.
Figure 2.17 illustrates the possible trade-offs that the greedy policy and the accumulate
and fire policies allow. It shows how latency and power can be traded against each other
and where the theoretical boundaries are.
Figures 2.18 and 2.19 show how higher activation and deactivation durations influence
the power consumption and the latency of the greedy policy and the latency-minimizing
policy. The power consumption of the greedy policy is influenced more by power cy-
clepower-cycle durations than the latency-minimizing policy. While the latency of the
latency-minimizing policy is independent of activation and deactivation durations, the
latency of the greedy policy increases approximately linearly (see section 2.4).
Figure 2.20 shows what happens if the time required for a power cycle is not equally
distributed between activation and deactivation phases. The left side shows small ac-
40
0
2
4
6
8
10
0 0.2 0.4 0.6 0.8 1
Mean latency L
Mean power consumption P
AP(1) = GP
AP(2)
AP(3)
AP(4)
AP(5)
Boundaries
Figure 2.17.: The greedy policy and the accumulate and fire policies allow different trade-
offs between latency and power consumption. The bounds are the theoret-
ical lower bounds.
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5
Mean power consumption P
Mean time for a state change A=Z
Greedy policy GP
Latency-minimizing policy LP
Figure 2.18.: The power consumption increases with increasing power state change times.
The effect using the greedy policy is stronger than using the latency-
minimizing policy.
41
0
2
4
6
8
10
0 1 2 3 4 5
Mean latency L
Mean time for a state change A=Z
Greedy policy GP
Latency-minimizing
policy LP
Figure 2.19.: The latency increases with increasing power state change times using the
greedy policy, but not using the latency-minimizing policy.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Mean power consumption P
Mean activation duration A
Mean power-cycle duration A+Z
Greedy policy GP
Latency-minimizing policy LP
Figure 2.20.: While the power consumption with the latency-minimizing policy with an
oracle is symmetric, it is asymmetric with the greedy policy. I fixed A+Z=
10 here to highlight the asymmetry.
42
tivation durations and high deactivation durations and the opposite is on the right. I
normalized the state change durations so that the expected value of their sum is the
same.
The power consumption of the latency-minimizing policy is symmetric with respect to
the activation and deactivation durations: The latency-minimizing policy will produce
the same result if the expected values for Aand Zare exchanged. More generally,
the power consumption depends only on the distribution of A+Z, but on neither one
individually. For exponentially distributed Aand Zthis only holds if their expected
values are switched; if the sum of their expected values is the same, the distribution will
generally not be the same (formally: Xi∼exp(λi)⇒(X1+X2)∼(e
X1+e
X2) if and
only if λ1=e
λ1∧λ2=e
λ2or λ1=e
λ2∧λ2=e
λ1).
In contrast to the latency-minimizing policy the power consumption of the greedy
policy is not symmetric: If the time of the power cycle is mostly spent in activation
phases the power consumption is lowest. The reason for this is that long activation
durations will create batches of jobs that will be processed at once. At the extreme of
zero deactivation duration, the power consumption of the greedy policy is the same as of
the latency-minimizing policy. However, creating batches of jobs introduces additional
waiting time. This effect is shown in figure 2.21. If the activation is the dominant
part of the power cycle, the example presented in figure 2.7 results in a higher power
consumption of the greedy policy.
0
2
4
6
8
10
12
14
0 0.2 0.4 0.6 0.8 1
Mean latency L
Mean activation duration A
Mean power-cycle duration A+Z
Greedy policy GP
Latency-minimizing policy LP
Figure 2.21.: While the mean latency of the latency-minimizing policy with an oracle
power is independent of the power-cycle durations, the mean latency of the
greedy policy changes. I fixed A+Z= 10 here to highlight the asymmetry.
Figure 2.21 shows how the mean latency using the greedy policy increases if the power-
cycle durations are dominated by activation durations. It additionally shows that the
mean latency does not have its minimum when the activation time is lowest: When the
mean activation time is low, the mean deactivation time is high and the probability for
a job to arrive during the deactivation of the previous job increases more than what the
43
lower activation time compensates for. The minimum mean latency is achieved when
the mean activation time is
A=3S2λ2+ 2Sλ + 3 −(Sλ + 1) pS2λ2+ 2Sλ + 9
4S2λ2S, (2.18)
where S=A+Zis the fixed expected time for a power cycle. Note that the mean
latency of the latency-minimizing policy does not depend on the distribution between
activation and deactivation times (not even their actual values) because it guarantees
the same mean latency as the always-on policy.
2.7. Conclusion
In this chapter, I analyzed how the power-cycle durations of a single BS influence the
mean power consumption and mean latency. I calculated the mean latency and power
consumption both under Poisson and worst case arrivals. In the worst case, both the
latency and the power consumption are linear in power-cycle durations. For Poisson
arrivals, the power-cycle duration determines both to a large extent. However, the
result for Poisson arrivals is asymmetric for other distributions of power-cycle duration
to activation and deactivation durations. In summary, I quantified how a high ratio of
power cycle to job processing time makes the sleep mode impractical.
Disabling a BS without coordinating with its neighbors is only useful, if (1) the jobs
can tolerate a high amount of delay or (2) the power cycle of the BS is fast compared to
processing a job. But neither of these are guaranteed to hold in future RANs. Hence, I
need to look beyond a single BS.
The results of this chapter apply only when considering a single BS, but not when
considering a complete RAN in its entirety. To show how a RAN can conserve more
energy, I first show how a system composed of multiple smaller components can conserve
energy without depending on fast power-cycle durations in the next chapter.
44
3. Conserving energy by coordinating sleep
modes network-wide
The propagation of radio waves is complex due to effects such as path loss, fading, shad-
owing, and interference. Hence, it is hard to analyze the effects of energy-conservation
methods in a radio access network (RAN). To show these effects in a simpler example, I
consider a wired network in this chapter. I selected the example to show that the effects
of energy-conservation methods on latency can be measured with different latency met-
rics. Also, this chapter shows how a system can adapt to traffic with sleep modes even if
the power cycles are long. Long power cycles in this context means that the power cycles
are so long that adapting to single demands of the users (as in the previous chapter) is
impractical. In this chapter, I assume that power-cycle durations are short compared to
changes of average load.
With this example I also show how different methods to conserve energy interact. I
analyze the influence of deactivating network components on power consumption and
latency of data transfers. Then, I compare the power consumption necessary to bound la-
tency without depending on the power-cycle durations of the devices. This results in de-
scription of the trade-off between power consumption and latency. It also illustrate how
a complex system can approach a linear power profile even though all sub-components
have a binary power profile.
The previous chapter showed an example how demands can be moved in time. This
chapter is an example how demands can be moved in space by rerouting. In both cases I
measure the effect for the users as increased latency. In the previous chapter the increase
in latency was due to queuing effects and in this chapter they are due to the increased
distance the signals have to travel.
3.1. Introduction
In this chapter, I describe how energy can be conserved in a wired network not operating
under full load. To reduce the energy consumption I allow traffic to be rerouted away
from some connections and then deactivate the line cards of these connections. Rerouting
traffic increases the latency of the rerouted traffic and, thus, reduces the users experience.
I formalize the problem of deactivating lines to conserve power while fulfilling all
demands. Then I define network configurations that minimize power consumption and
configurations that minimize latency as well as trade-offs in between. Unfortunately, the
corresponding decision problem (capacitated fixed-charge multi-commodity minimum-
cost flow network) is nondeterministic polynomial time (NP)-complete [JLR78]. Because
it is not feasible to find the optimal solution for NP-complete problems, I use mixed
45
integer linear programming (MILP) to approximate the solutions for a given realistic
network. The combinatorial explosion of possible states that makes this NP-complete is
typical for energy conservation methods in networks and will be important in the later
analysis of the energy consumption of RANs.
To analyze the trade-off between power consumption and latency, I begin by formal-
izing the problem as a graph. Then, I prove a bound between different possible metrics
of latency and explain the relationship between latency and bandwidth-delay product
(BDP). After that I create an optimization model from the formalization of the problem
and show the results I obtain with it. My results show that it is possible to conserve
considerable power by deactivating line cards in wired networks while increasing the
latency only slightly.
3.2. Related work
Many papers present work on minimizing the cost in fixed-charge networks [HS89, KF91],
but not many of them consider the effects of deactivation on latency. One example which
does is the work of Lin et al. [LS12]. While they focus on developing efficient heuristics
I use an optimization model and compare the different ways to measure the increased
latency of the demands.
Chiaraviglio et al. [CMN09a, CMN09b] describe several centralized heuristics to ap-
proximate the most power-efficient topology for the binary power-consumption model.
They assume that line cards as well as routers can be deactivated and demands of end-
to-end flows are known. The heuristics provide results that are close to minimal power
consumption, but they do not analyze latency.
Vasi´c and Kosti´c [VK10] describe a distributed algorithm that uses adaptive link rates
to reduce power consumption by modifying both topology and multi-path routing. Their
idea is to distribute the load so that lines can be set to low-speed low-power operating
modes. Their work provides a possible approach to actually implement power-saving
measures, but does not focus on its effects on latency.
Revirigo et al. [RMHL10] analyze how adaptive link rates and burst transmissions
can reduce the idle power consumption of single links. This method considers each link
individually while I consider the whole network.
Other heuristics for finding solutions to the fixed-charge network flow problem ex-
ist [HNS09] which provably find high-quality solutions quickly, but these do not address
the latency as I consider it.
Magnanti and Wong [MW84] provide an overview under which assumptions the net-
work design problem is equivalent to other known problems and their complexity. This
overview allows a quick estimation of how complex it would be to find energy-efficient
configurations of models which consider different effects. An alternative formulation by
Lin et al. [LS12], for example considers to deactivate unnecessary cables in bundled links.
The mathematical concept of k-spanners [CC95, PS89] formalizes the maximal stretch
that the shortest path between two nodes may have in a subgraph. Analysis of k-
Spanners for weighted graphs exist [ADD+93] as well as k-Spanners with fault toler-
46
ance [CLP09], but because k-Spanners have no notion of capacity they cannot fulfill all
constraints that are needed for a feasible configuration. The results for k-Spanners can
thus only be used if no edge reaches is capacity limit. I focus on the general case in
which not all capacities can be assumed to be sufficient for all demands. In addition,
k-Spanners only consider the maximum stretch, while I also consider different metrics,
such as the average stretch.
Mean distances in graphs have been analyzed in general [DG77] and together with
the minimum degree [KW97]. The difference between these two publications and the
analysis of this chapter is that they focus on proving bounds for the mean distance and
ignore capacities. Dankelmann and Entringer [DE00] show that spanning trees with
certain mean distances exist, but do not take the capacity restrictions I consider into
account either. Plesnik [Ple84] gives general results for the sum of all distances.
Another research area that uses topology control is ad hoc and sensor networks [FS07].
Topology control in this chapter as well as in ad hoc and sensor networks deactivate edges
in a network graph to conserve power while maintaining desired properties such as low
latency. So both have similar goals but the assumptions for the underlying network are
different: While quality of a link to a neighbor can change rapidly in a wireless network,
this will only rarely happen in wired networks. Additionally, reducing the node degree
is not important in a wired network because I assume a network to be built only with
node degrees it is able to handle.
3.3. Model
In this section, I describe the modeling assumptions I make for this chapter. The mod-
eling in this chapter is different from the following chapters because it considers a wired
network instead of wireless network.
3.3.1. Network graph
I formalize the network as a graph G with the vertexes VGrepresenting routers and the
edges EGrepresenting line cards and wirings that connect routers. For each edge ethe
capacity cap(e) is the maximal data rate, the latency L(e) as the propagation delay, and
P(e) is the power consumption under full load. The network has to serve unicast, static
demands D ⊆VG×VG; the function φ: D →R+maps the demands to the required
data rates.
As I analyze propagation delay caused by edges, I consider only the power consumption
of edges. Thus, to conserve power I deactivate edges and call the resulting states active
and sleep. I represent the status of edges using configurations: a configuration C is a set
of active edges together with routes for the demands using only active edges. A special
configurations is a (not necessarily unique) configurations CLwhich activates every edge
and routes every demand on its shortest path to minimize latency. Additionally, I define
the amount of data of a flow of demand don an edge eas σd
C(e) and the utilization of
47
edge eas
UC(e) = Pd∈Dσd
C(e)
cap(e).(3.1)
This is equivalent to the always-on policy OnP and routing on shortest paths.
3.3.2. Power consumption
The power models for base stations (BSs) (see section 1.2) also apply well to wired
network hardware. The binary power model models today’s non-load-adaptive hardware
well [CSB+08]. The linear model is motivated by techniques like adaptive link rates and
burst transmissions that may reduce idle power consumption [RMHL10].
An algorithm which controls the topology to conserve power must consider idle power
consumption, lest it might consume more power [PVC+09]. The reason for this is:
Deactivating edges conserves power, but rerouting increases power consumption. If the
later is greater than the former the total power consumption increases.
I apply the same power model I describe for BSs for lines in this chapter. The notation
is as follows. I assume an active edge econsumes PP(e) power when idle, where P ∈
[0,1] is the fraction of maximum power consumed when idle. Power consumption scales
linearly in utilization UC(e) from PP(e) when idle to P(e) under full load.
3.3.3. Latency
As I restrict my model of latency to propagation delays, the latency of a demand dis the
sum of the latencies of its constituting edges; I call it LC(d). To compare the latencies
in two different configurations, I define the stretch of a configurations as the increase
factor in all latencies caused by rerouting.
There are at least five different, intuitively reasonable ways to define the stretch of
a configurations C compared to the latency-minimizing configurations CL. I introduce
each metric and give its value for a simple example: a ring-shaped network with 8 nodes
where I deactivate one edge to conserve energy. Figure 3.1 shows this example network.
Every edge e∈EGhas latency L(e) = 1 and there is a demand of φ(d) = 1 from every
node to every other node. I assume the capacities are large enough that an edge can be
deactivated. In the following paragraphs, I give reasons why each metric is useful and
give their values for the ring example with n= 8 (and the value it approaches when n
approaches infinity).
The first two metrics express that no demand should suffer from a high latency due
to power conservation. I define SMS as the maximum stretch a single demand suffers in
configurations C:
SMSC(D) = max
d∈DLC(d)
LCL(d).(3.2)
In contrast, SSM describes the stretch the maximum of all demands suffers in configu-
48
N1
N2
N3
N4N5
N6
N7
N8
active edge
inactive edge
Figure 3.1.: An 8-Ring with a single inactive edge is an 8-Path. I use this example to
demonstrate the different ways to compare latencies.
rations C:
SSMC(D) =
max
d∈DLC(d)
max
d∈DLCL(d).(3.3)
While SMS compares each latency to its own latency in the latency-minimizing configura-
tion, SSM compares the maxima of all latencies. In the 8-Ring, SMS is 7 (limn→∞ SMS =
∞), because the highest increase in latency is from 1 (1) to 7 (n−1) and SSM is 7/4=1.75
(limn→∞ SSM = 2) the highest latency increases from 4 (bnc) to 7 (n−1).
To describe the tendency of the latencies, the next two metrics use the weighted arith-
metic mean (weighted by the amount of transferred data φ(d)) instead of the maximum.
This is reasonable if it is acceptable for some demands to suffer from a high latency
as long as the average stays low. Analogous to SMS and SSM, I define (a) SAS as the
weighted arithmetic mean of the stretch and (b) SSA as the stretch of the weighted
arithmetic mean latency:
SASC(D) = avg
d∈D
LC(d)
LCL(d)=P
d∈D
LC(d)
LCL(d)φ(d)
P
d∈D
φ(d)(3.4)
and
SSAC(D) =
avg
d∈D
LC(d)
avg
d∈D
LCL(d)=P
d∈D
LC(d)φ(d)
P
d∈D
LCL(d)φ(d).(3.5)
For the 8-ring example SAS is 10/7≈1.43 (limn→∞ SAS = 1.5) and SSA is 21/16 =
1.3125 (limn→∞ SSA = 4/3≈1.33).
49
One of the problems with the first four metrics is that they give different values when
the order of combining (avg and max) and stretch calculation is reversed. The idea to
use the weighted geometric mean SGS is that this order is irrelevant and thus SGS =SSG
(see section A.1). They are defined by
SGSC(D) = geo
d∈D
LC(d)
LCL(d)= Y
d∈DLC(d)
LCL(d)φ(d)!1
P
d∈D
φ(d)
(3.6)
and
SSGC(D) =
geo
d∈D
LC(d)
geo
d∈D
LCL(d)=Q
d∈D
LC(d)φ(d)1
P
d∈D
φ(d)
Q
d∈D
LCL(d)φ(d)1
P
d∈D
φ(d)
(3.7)
and are approximately 1.22 (limn→∞ SGS = 2/√e≈1.21, see section A.5) in the 8-Ring.
The values for the different metrics show that most metrics do not differ by much in
this example. In the next section I analyze the relationships between them in general
and give a bound for their ratio.
Table 3.1.: My modeling decisions for the wired example are highlighted in bold).
Option Possible choices
Topology control deactivated edges deactivated routers
Change of demands static dynamic
Power consumption model binary linear
Considered delay propagation queuing
Topology control and routing distinct combined
Location of calculations distributed centralized
Knowledge of demands edge utilization end-to-end demands
Table 3.1 summarizes the modeling choices I have mode for this chapter.
3.4. Analysis of metrics for latency aggregation
In this section, I first analyze the relationships between the different latency metrics in
the same situation. Then, I describe the relation of BDP and latency.
3.4.1. Relationships between the latency metrics
First note that SMS is always larger than or equal to any of the other four metrics.
Between any two of the other metrics, only one comparative statement holds in general:
50
SGS ≤SAS. This is a direct implication of generalizing the inequality of arithmetic and
geometric means to weighted means [Ste04]. While all possible orders of the metrics
SSM,SAS, and SSA exist (see section A.2) not all are possible with two unweighted
demands (see section A.3).
I define the skew of configurations C skew(C) as the ratio between maximal and
minimal latency:
skew(C) =
max
d∈DLC(d)
min
d∈DLC(d).(3.8)
The bound
A≤skew(C) ·skew(CL)·B(3.9)
holds for all combinations of Aand Bfrom the five metrics (see section A.4). It implies
that all five metrics yield similar values when the ratio between maximum and minimum
latency in both the latency-minimizing configurations CLand in configurations C is
small. For some pairs of metrics tighter bounds are possible, but because examples exist
that are arbitrarily close to this bound it is the tightest possible general bound. Note
that this bound depends on neither number nor size of demands and thus the number
of demands can be arbitrarily large and all metrics still fulfill the inequality above.
I conclude that the metrics are similar for low skew and are bounded independently of
the number and size of demands. I analyze how much the metrics differ for a practical
scenario in section 3.6. Next, I show a connection between BDP and latency.
3.4.2. Bandwidth-delay product and latency
The bandwidth-delay product (BDP) is the product of the data rate of an edge with its
latency:
BDP(e) = cap(e)L(e).(3.10)
I define the used BDP of an edge eas the latency times the actually used data rate.
In contrast, the unused BDP is the BDP which is not used to transfer data and sleeping
BDP is the BDP of edges in sleep mode. Theorem 3.1 shows that if the used BDP is
low, the average latency will be low and so will SSA.
Theorem 3.1. The mean used bandwidth-delay product (BDP) of all edges is propor-
tional to the mean latency of all demands.
Proof. The idea of the proof is that the order in which the total sum latency in the system
of determined (sum over flows and edges or over edges and flows) does not matter.
Recall the definition of the weighted arithmetic mean latency
avg
d∈D
LC(d) = X
d∈D
LC(d)φ(d)/X
d∈D
φ(d).(3.11)
I consider the used BDP averaged over all edges, captured per edge as the product of
its utilization UCin configuration C it capacity cap and its latency L. It is proportional
51
to mean latency, because
avg
e∈EG
used BDP on edge e
z }| {
UC(e)cap(e)L(e) = 1
|EG|X
e∈EG
σ(e)L(e)
=1
|EG|X
e∈EGX
d∈D
σd
C(e)L(e)
=1
|EG|X
d∈DX
e∈EG
σd(e)L(e)
=1
|EG|X
d∈D
LC(d)φ(d)
=Pd∈Dφ(d)
|EG|avg
d∈D
LC(d)
∼avg
d∈D
LC(d)
|{z}
latency of demand d
(3.12)
The following related theorem follows directly from the definition the binary power
profile.
Theorem 3.2. When the power consumption of each edge is proportional to its total
BDP and the binary model for power consumption is assumed, the BDP of sleeping edges
is proportional to the conserved power.
Total BDP
Deactivating edges
Sleeping BDP (∼conserved power)
Unused BDP
Used BDP ∼avg latency
Figure 3.2.: Deactivating edges and rerouting demands conserves power, but increases
latency. Both of which can be quantified in terms of BDP.
Figure 3.2 illustrates how BDP, power consumption, and latency are related as de-
scribed by the theorems 3.1 and 3.2. Each of the two vertical bars represents the total
BDP that is available in a network and shows how it is used. Rerouting traffic and
deactivating edges will increase the used BDP, but also allows edges to be deactivated.
52
3.5. Optimization model
To get a general understanding of the power consumption of practical networks I for-
mulate the problem as an MILP. It is based on the following assumptions: A single
algorithm controls routing and topology, the algorithm has global knowledge of end-to-
end flows, the demands are static, and power consumption follows the linear model (with
the binary power model a special case).
∀e∈EG:X
d∈D
max(0, σd
C(e)) ≤cap(e)UC(e) (3.13)
∀e∈EG: UC(e)≤xC(e) (3.14)
∀d∈D; u, v ∈VG:σd
C((u, v)) = −σd
C((v, u)) (3.15)
∀d∈D, u ∈VG:X
v∈N(u)
σd
C((v, u)) −X
v∈N(u)
σd
C((u, v))=
−φ(d),if u= src(d)
φ(d),if u= dest(d)
0,else
(3.16)
X
e∈EG
UC(e)cap(e)L(e)/avg
d∈D
minLat(d)≤BSA ·X
d∈D
φ(d) (3.17)
X
d∈DX
e∈EG
L(e) max(0, σd
C(e))/minLat(d)≤BAS ·X
d∈D
φ(d)(3.18)
min X
e∈EG
P(e)((1 −P)UC(e) + PxC(e)) (3.19)
I use a network flow model to calculate allocation of demands to edges. To express
whether an edge is active or sleeping I use the binary variable xC(e). Using the nota-
tion introduced in section 3.3, I define the capacity constraint in equation 3.13. I use
max(0, f) to model full duplex links. Equation 3.14 guarantees that only active edges
can transfer data. Equation 3.15 is the skew symmetry. I define the flow conservation in
equation 3.16 so that the allocation of flows meets all demands. N(v) denotes the set of
vertices incident to the vertex v. Note that these definitions allow multi-path routing.
To specify an upper bound BSA (BAS) on the SSA (SAS) metric I use equation 3.17
(3.18). I only use one of these constraints at a time. Here minLat(d) is the minimal
latency necessary to route demand d. I minimize the power consumption specified in
Term 3.19.
Using the MILP I am able to specify upper bounds for the average stretch metrics SAS
and SSA. Because the geometric upper bound cannot be written as a linear constraint,
I am not able to use it in the optimization model. Because my network model is based
on the idea of flows, which allows multi-path routing, I cannot calculate the maximum
latency in the linear program either. Hence, I cannot bound the maximum stretch
metrics SSM and SMS.
53
Figure 3.3.: The (4 ×4)-grid I use as an example network. The capacity, latency and
energy consumption of all edges is 1. The demands are created from every
node to every other node with equal sizes.
Figure 3.4.: The 4-dimensional hypercube is topologically equivalent to (and easiest to
draw in 2 dimensions as) a (4 ×4)-grid with wrap-around edges. The ca-
pacity, latency and energy consumption of all edges is 1. The demands are
created from every node to every other node with equal sizes.
54
Figure 3.5.: The nodel-germany network is from SNDlib [OPTW07]. The demands,
capacity and energy consumption of all edges is provided in the model. I
determined the latency from the geometric distances.
3.6. Results
I use my model to calculate the power consumption in different networks: a two-
dimensional Grid (see figure 3.3), a hypercube (see figure 3.4), and as a practical network
I use the nobel-germany (see figure 3.5) network from the Survivable fixed telecom-
munication Network Design library (SNDlib) [OPTW07]. The SNDlib includes power
consumption (cost), capacity and a demand pattern, while I estimate the latencies from
the physical distances of the routers (locations are present in the model). For both
theoretical networks I set all capacities, latencies, and power consumption values to 1
and assume equal demands from every nodes to every other node. I scale the network
load with a single scalar factor and solve the optimization problems for minimal power
consumption with the GNU Linear Programming Kit (v4.44) and the Gurobi Optimizer
4.0. The error bars show the upper and lower bounds provided by the solver.
Figure 3.6 compares both the most power-efficient configurations and the latency-
minimizing configurations as well as the power models. Because the difference between
the two power models is small, I use the simpler binary model for further analysis. Addi-
tionally, Figure 3.6 shows that it is possible to conserve up to 45% of power. The amount
of conserved power depends on the type of network considered, because a minimum-
power spanning tree is needed for connectivity between all nodes. Because the load on
the edges is inhomogeneous and I use a single scale factor, it is possible to deactivate
edges even at the highest load the network can transfer.
Figure 3.7 illustrates the power necessary to keep the stretch below a given bound.
Allowing the latency to increase by 20% saves up to 39% of power. Although both
average metrics, SAS and SSA, are generally different, they produce similar results in
this scenario. Figure 3.8 shows the difference when bounding the metrics by 1.2 for
55
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Mean power consumption P
Load λ
No deactivating (linear)
Optimal deactivating (binary)
Optimal deactivating (linear)
Figure 3.6.: Power consumption with different models using the nobel-germany network;
The idle power consumption Pis 0.98 in the linear model.
0
0.2
0.4
0.6
0.8
1
1 1.04 1.08 1.12 1.16 1.2
Mean power consumption P
Allowed stretch B
SAS
SSA
Figure 3.7.: Power consumption for different upper bounds for the stretch metrics SAS
and SSA in the nobel-germany network
56
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Mean power consumption P
Load λ
SAS ≤1.2
SSA ≤1.2
Unlimited stretch
Figure 3.8.: Power consumption increases only slightly for SAS and SSA bounds of 20%
increased latency in the nobel-germany network.
different load levels. Again both metrics do not differ much and allowing a stretch of
1.2 consumes less than 5% more power than not limiting the stretch at all.
Figure 3.9 (3.10) shows the power consumption of different methods to conserve power
in a 4-dimensional hypercube ((4×4)-grid). I compare the optimal deactivating of edges
to a method of reducing the idle power consumption Pof all edges to 50% and the
combination of both. The results show that both approaches reduce the consumed
power in both networks and the power consumption is further reduced by combining
both approaches. This exemplifies how methods to reduces the energy consumption on
different levels interact and are necessary to conserve most energy.
3.7. Conclusion
I formalized the problem of conserving power in wired networks, applied my model to
different networks, and analyzed power consumption and latency. Assuming the binary
power-consumption model, this allows the consumed power to be reduced by 39% while
increasing the latency by only 20% in the nobel-germany network. I conjecture that for
most networks and demand patterns configurations exist that conserve large amounts of
power and increase latency only slightly.
There are many ways to describe the cost in latency that the users have to accept for
reducing the energy consumption. I analytically showed that they cannot be arbitrarily
different and their difference depends on the skewness of the latency of the demands. I
also showed that for a practical example the difference between the different metrics is
negligible.
In future work more complex models can be analyzed, for example arbitrary power
profiles. Another approach is to consider not only propagation delays, but to include
57
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Mean power consumption P
Load λ
Reducing idle power
(P= 0.5)
Deactivating edges
Both
Figure 3.9.: Different ways to conserve power are illustrated for the example of the Hy-
percube of dimension 4.
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Mean power consumption P
Load λ
Reducing idle power
(P= 0.5)
Deactivating edges
Both
Figure 3.10.: Different ways to conserve power are illustrated for the example of the
(4 ×4)-grid.
58
queuing delays into the model and test how they interact with power conservation meth-
ods.
In this chapter, I showed how a system that is composed of smaller sub-systems
can have a different power profile than the sub-systems. In this example the power
consumption of the network which consists the power consumption of the edges is closer
to the linear power profile than the power profile of the edges, which have a (nearly)
binary power profile. To do this the power-cycle durations do not have to be in the same
order as request processing times (as needed for chapter 2), but only on the order of
changes of the average traffic.
In the rest of the dissertation I apply the same idea to deactivating selected BSs in
RAN. I consider the energy consumption of signaling and data transmission indepen-
dently first (chapters 4 and 5) and later combine all effects in a simulation (chapter 7).
59
4. Conserving energy in signaling
transmissions
In this chapter, I answer the question of how much power can be conserved in radio access
networks (RANs) when some base stations (BSs) use cooperative transmissions to signal
at extended range. In addition, I analyze the effect of different path-loss exponents and
varying number of cooperating BSs.
First, I analytically show how it is possible to cover an area with fewer BSs when
their range is extended by cooperation. To do this, I analytically determine the optimal
spacing of BSs under the assumption that the BSs can be freely placed. Second, I create
and solve an optimization model that describes the power necessary to cover a given
area using linear programming. In the optimization model I assume the BSs are already
placed and it is only possible to select which of them to activate.
4.1. Introduction
BSs must provide signaling traffic to every location at all times. Hence, they cannot
simply be deactivated under low load. A BS can only be deactivated when the neigh-
boring BSs cover its area. Because BSs can only cover the area they are physically close
to, methods to extend their ranges are necessary. While different such methods exist, I
consider the use of cooperative transmissions (see section 1.4) to extend the ranges of
BSs.
A B C
BS Covered area
Figure 4.1.: This illustrates three BSs and the area they cover without cooperation.
Figure 4.1 shows the BSs A, B, and C and the area to which they can provide signaling
traffic without cooperation. Figure 4.2 shows how the BSs A and C can use cooperation
60
A B C
Deactivated BS
Cooperatively covered area
Area eligible for cooperation
Figure 4.2.: BSs A and C can cooperate to cover the area of BS B (not to scale).
to cover the area previously covered by BS B. This allows BS B to be deactivated and
thereby conserve power.
In this chapter, I calculate a theoretically possible increase in covered area under
the assumption that the BSs are placed to maximize the covered area when taking
cooperation into account. Because BSs are impractical to move in reality, I determine
how to deactivate BSs so they can conserve energy when the load is low. I create an
optimization model to determine how many BSs, which are placed for non-cooperative
coverage, can be deactivated when cooperation is introduced. Up to this point I only
consider ergodic capacity. Finally, I briefly compare the area BSs cover when I consider
outage capacity instead of ergodic capacity to determine the ranges for signaling.
4.2. Related work
The BS deployment problem is well known and analyzed, for example by Lev-Tov and
Peleg [LTP02]. They analyze in which given locations to build BSs, but I determine
the best locations when the BSs can be freely placed. Also Lev-Tov and Peleg do not
consider cooperative transmissions.
An abstraction that covers both the design time and run-time decisions is the fixed-
charge facility location problem [Noz01]. It is also well studied but does not consider
cooperation of BSs. While there is other work that analyzes cooperation in RANs [DD08,
SSBNS06, SSPS09], it does not consider its use for energy conservation.
61
Niu et al. [NWGY10] explain how changing the cell sizes (called cell zooming) can
be used to reduce the power consumption of RANs. They focus on the effects that
changeable cell sizes have on the energy consumption but only briefly state examples
how the cell size can actually be changed. The BS cooperation I analyze in this chapter
is one of their examples, but they do not provide a detailed analysis.
Richter et al. [RFF09] consider how to minimize the power needed to cover an area
with micro and macro cells. They focus on calculating the optimal number of micro BSs.
In contrast to this I try to deactivate BSs to conserve power. They assume the power
consumption of macro BSs to be static and the power consumption of micro BSs to be
dynamic, which is also supported by Arnold et al. [ARFB10]. Using these assumptions
they show that in low load scenarios micro BSs are more energy-efficient. This effect
is orthogonal to my investigation because both aspects can be combined into a single
model.
Sadek et al. [SHL06] propose a protocol to increase the range of BSs by cooperation
of user equipments (UEs). They use UEs as cooperation partner and have to consider
an additional channel to get the transmission to the partner. I use another BS as
cooperation partner and consider only two wireless channels because I assume the BSs
are connected by a wired backhaul network. My assumption is realistic (for cooperating
BSs) and gives a lower error probability.
Viterbi and Gilhousen [VGZ94] as well as Staehle et al. [SLH+01] make an analysis
that is similar to mine for soft handovers. Soft handovers use selection combining, while
I consider maximal-ratio combining (MRC).
In the final report of “Understanding the Environmental Impact of Communication
Systems” [FDM+09] the authors hint at a method to “Power-down during quiet periods”
for GSM that saves around 25% of the consumed energy, but no further references or
details are provided.
Lichte et al. [LFK10] consider how cooperation can be used in wireless sensor networks,
but focus on different aspects, like building broadcast trees and determining a useful
partner for cooperation, while I focus on covering an area with the least amount of
power.
Woldegebreal et al. [WK08] analyze how cooperation can increase the range of trans-
missions in wireless sensor networks. They consider a three-node scenario (sender, relay,
receiver) and require the cooperation partner (relay) to receive the transmission using a
wireless channel. Woldegebreal et al. focus on reducing the outage probability, while I
focus on reducing the power consumption, which is different from radiated power.
Blundon [Blu10] analyzes a variant of the circle-covering problem, which requires every
point to be covered by a given number of circles. The key difference is that I only need
a location to be covered by several BSs if no BS is close enough to provide connectivity
alone.
62
4.3. Model
In this chapter, I analytically determine the highest distance at which cooperating BSs
can be placed so that the BSs can still provide the signaling data rate DSto every
possible point. The radio model I use here considers only the path loss over distance. I
do not consider time, load, UEs, and interference in this chapter.
The signaling traffic UEs generate is created by a number of (possibly moving) users.
Because the signaling traffic contains only small amounts of data and the need to transmit
signaling traffic can arise nearly everywhere I model this as a need to transmit with the
signaling data rate DSto every possible UE location. Because this ignores the number
of UEs which want to transmit signaling data, the modeling is only valid as long as the
signaling data rate is small enough to not overload BSs.
4.3.1. Non-cooperative range
Using non-cooperative transmission the requirements for signaling explained in chapter 1
define a location uas covered when it closer to a BS Bthan the signaling range of the
BS:
d(u, B)≤rS,(4.1)
where d(u, B) is the Euclidean distance between uand B.
4.3.2. Cooperative range
A simple way to guarantee a UE reaches this threshold Tis to make sure it receives a
mean signal-to-noise ratio (SNR) of at least T/2 from two different BSs. I call this type
of cooperation 2-cooperation because it allows two BSs to cooperate and limited because
I set a lower bound on the received SNR per BS such that receiving such power from two
BSs guarantees that the sum of their SNRs is always at least T. Limited cooperation
is more restrictive than necessary, but it provides a simple abstraction. The range at
which a base station can provide an SNR of at least T/2 is 21/δrwhere δis the path-loss
exponent and ris the range of a non-cooperating BSs. Hence, for the special case of
δ= 2, the cooperative range is √2r. For a limited cooperation of nBSs, every BS needs
to provide at least T/n of the mean SNRs.
For unlimited cooperation, I remove the lower limit of T /n and require only the sum
of the mean SNRs of the cooperating BSs to be higher than the threshold T. Unlike
limited cooperation the resulting problem cannot be stated as purely geometric problem.
For each possible location of a UE uthe definition from the introduction (section 1.4)
is equivalent to
X
i∈SrS
d(u, i)δ
≥1.(4.2)
The equivalent form makes it easier to determine the increase in covered area by using
cooperation.
63
4.4. Signaling with optimal BS spacing
In this section, I define different types of cooperation and analytically show how the
covered area is increased when cooperation is taken into account for the deployment of
the BSs. The size of this area is a proxy for power consumption, because a greater area
can be covered with fewer BSs if their cells are larger. In this section, I consider the
constraint for connection to be based on ergodic capacity.
In the next subsection, I calculate the cell area without cooperation as a reference for
later comparison. Second, I give a simplified cooperative model which describes coverage
as a purely geometrical problem. And third, I describe a more powerful abstraction of
cooperation, in which optimal deployment of BSs is not a purely geometric problem
anymore.
4.4.1. No cooperation
Spacing ξNC(2) ≈1.732rSSignaling range rS
Critical point CP
Cell boundary
Cell area ANC(2) ≈2.598rS2
Figure 4.3.: The best non-cooperative coverage is achieved with a hexagonal pattern.
Given my assumptions a single BS can cover the area of a disk. The most efficient
pattern to cover an infinite plane with disks is the hexagonal pattern [Ker39, Tot72],
which figure 4.3 shows. I define the distance of two neighboring BSs in the hexagonal
pattern as spacing. The hexagonal pattern allows neighboring BSs to be placed with a
spacing of ξ=√3rSand still cover the infinite plane. Where rSis the signaling range
of the BSs. The cell area is uniquely defined for a given spacing ξas I only consider
hexagonal deployments. In general I use ξX(δ) to denote the highest spacing which
can provide every location in the plane with signaling traffic, where Xis the type of
cooperation used and δis the path-loss exponent of the scenario. A critical point CP is
a point which receives just enough power to decode the signaling transmission and thus
will lose its connectivity when the spacing is increased. This is the point that has the
highest distance to any BSs.
64
Basic geometry limits the signaling cell area of a BS to πrS2. But when a larger area
has to be covered, cells overlap and the area served by a particular BS of combination of
BSs (called cell area) ANC(2) is only 33/2rS2/2. In the following subsections, I use this
area as a reference value and define the gain factor gX(δ) of a cooperation type Xas
the factor by which the cell area A increases compared to not using cooperation.
4.4.2. Limited 2-cooperation
Spacing ξLC(2) ≈2.189rS
50% Power range ∼1.732rS
Critical point
Cell area ALC(2) ≈4.149rS2
Figure 4.4.: Using limited cooperation of 2 BSs the area each BS is responsible for can
be increased by 59.7% when the path-loss exponent δis 2.
As defined in section 1.4 a location is covered by limited 2-cooperation, when two BSs
each provide at least half of the required signal strength. To cover the infinite plane with
limited 2-cooperation each point must be closer than rSto a BS or closer than √2rS
to two different BSs. A possible deployment of BSs is shown in figure 4.4. Using only
basic geometry I derive the spacing of ξLC(2) = p√21 + 5rS/√2. Hence, the cell area is
ALC(2) = 3√7rS2/4+5√3rS2/4, which is equivalent to a gLC(2) = √21 + 5/6 increase
in cell area over not using cooperation. The BS increases its range by cooperating with
the corresponding neighbor at the edges of the cell. Further increasing the spacing will
not cover the plane anymore because the critical point would be further away from all
BSs than rSand would not have two BSs in cooperative range.
For path-loss exponents δ > 2 the range at which a UE receives an SNR of at least
TS/2 decreases in comparison to the range at which it receives SNR TS. Hence, the
65
gain using limited 2-cooperation also decreases for higher path-loss exponents. The
cooperative range of 21/δrSand the same geometry used for the special case δ= 2 allow
me to calculate the maximum spacing with an arbitrary path-loss exponent δ:
ξLC(δ) = q√3p22/δ+2 −1+22/δ+1 + 1
√2rS.(4.3)
This directly results in a cell area of
ALC(δ) = 3p22/δ+2 −1 + √322/δ+1 +√3
4rS2(4.4)
and thus a gain factor over not using cooperation of
gLC(δ) = √3p22/δ+2 −1+22/δ+1 + 1
6.(4.5)
This deployment provides a substantial gain over non-cooperative deployment. Be-
cause a proof of its optimality is not evident, I do not know if this deployment is the best
deployment possible. While no hexagonal deployment with higher spacing is possible,
still other non-hexagonal deployments might need fewer BSs. But because the cell area
without cooperation is proven to be optimal [Ker39, Tot72] this is a lower bound for the
gain that can be achieved using limited 2-cooperation.
4.4.3. Unlimited 2-cooperation
Figure 4.5 shows a deployment of BSs with a spacing of ξUC(2) = √6rSwhich allows
each of the BSs to cover a cell of size AUC(2) = 33/2rS2. This is a 100% increase in
covered area over not using cooperation.
To understand why this deployment covers the complete plane, assume a UE uis
connected by unlimited 2-cooperation to the two BSs A and B. Without loss of generality
I assume that A is closer to uthan B. The following two lemmas can be proven easily:
Lemma 4.1. Every UE u0that is closer to both BS A and B than uis connected by
unlimited 2-cooperation.
Lemma 4.2. Every UE u0that is on a line between uand BS A is connected by unlimited
2-cooperation.
In lemma 4.1 the signal strength of each BSs is higher in u0than in uand, thus, the
total signal strength must also be higher.
For lemma 4.2 it is necessary to show that moving closer to the closest BSs increases
total signal strength. To prove this it is important that the increase in distance to BS
B is at most the decrease in distance to BS A. Therefore, for a path loss exponent of at
least 1 (which is true for all practical purposes) the total signal increases from uto u0.
Every point in the plane can be reached from a critical point, which is connected, by
using each of the two lemmas once. Figure 4.6 illustrates how starting at a critical point
66
and moving along the lines that have coverage granted by the two lemmas reaches any
point. Thus, every location in figure 4.5 is covered.
For general path-loss exponents δit is possible to increase the spacing to ξUC(δ) =
√3 21/δ and still cover the plane. This results in a cell area of AUC(δ)=33/222/δ−1and
a gain factor of gUC(δ) = 22/δ, which can be shown using geometry and lemmata 4.1
and 4.2.
No coop
Limited coop
Unlimited coop
Figure 4.7.: The area that limited cooperation can cover fully contained in the area that
unlimited cooperation can cover.
To illustrate the differences between the limited and unlimited cooperation, Figure 4.7
shows the area that can be covered without cooperation, by limited 2-cooperation, and
by unlimited 2-cooperation.
4.4.4. Infinite cooperation
The next step to generalize cooperation is to allow more than two BSs to cooperate. I
call the cooperation of nBSs n-cooperation. When each BS has to provide at least TS/n
SNR, I call it limited n-cooperation and if only the sum of the SNRs has to be greater
than TSI call it unlimited n-cooperation.
I do not calculate the gain of n-cooperation for any given nhere, but analyze how
coverage changes when napproaches infinity. Note that the difference between limited
and unlimited cooperation vanishes when napproaches infinity, because the individual
threshold a BS has to achieve tends towards zero. I call this infinite cooperation and
denote it as IC.
As the received power is lowest at the critical points, I calculate the received power
for a critical point depending on the spacing. Figure 4.8 shows how I number the BSs
as seen from a critical point. For every i∈Nand j∈ {1, ..., 3i−2}a BS (i, j) exists.
Note that this is exactly one third of all BSs. BS (i, j) is located at
Location of BS(i, j) = (3i−2−3(j−1)/2)ξ.√3, ξ(j−1)/2.(4.6)
The distance between the critical point and BS (i, j) is
d(CP,BS(i, j)) = p3j2−9i j + 9 i2−3i+ 1 ·ξ.√3 (4.7)
68
(1,1) (2,1)
(2,2)
(2,3)
(2,4)
(3,1)
(3,2)
(3,3)
(3,4)
(3,5)
(3,6)
(3,7)
Critical point CP
Figure 4.8.: The way I use to number a third of all BSs is rotational symmetric to the
other two thirds.
and the expression
(3i/2−1)ξ/√3≤d(CP,BS(i, j)) ≤d(CP,BS(i, 1)) = (3i−2)ξ/√3 (4.8)
bounds the distance of the critical point CP from BS (i, j) independent of j, where ξis
the spacing of the hexagonal pattern.
For a UE at the critical point to be connected, the following equivalent to unlimited
2-cooperation must hold:
3∞
X
i=1
3i−2
X
j=1 rS
d(CP,BS(i, j))δ
≥1.(4.9)
For a path-loss exponent δ= 2 a lower bound for equation 4.9 is
3∞
X
i=1
3i−2
X
j=1 rS
d(CP,BS(i, j))δ
≥3∞
X
i=1
3i−2
X
j=1 r√3
ξ(3i−2)!2
= 9rS2ξ−2∞
X
i=1
3i−2
X
j=1 1
3i−22
= 9rS2ξ−2∞
X
i=1
1
3i−2(4.10)
As the previous sum is unbounded (see harmonic series), this value will be larger than
1 for every value of ξ. Thus, for a path-loss exponent δ= 2 the covered area and the
gain factor are infinite.
69
For a path-loss exponent δ > 2, however, an upper bound is
3∞
X
i=1
3i−2
X
j=1 rS
d(CP,BS(i, j))δ
≤3∞
X
i=1
3i−2
X
j=1 √3rS
ξ(3i/2−1)!δ
= 2−δ3δ/2+1rSδξ−δ∞
X
i=1
3i−2
X
j=1 1
3i−2δ
= 2−δ3δ/2+1rSδξ−δ∞
X
i=1 1
3i−2δ−1
= 2−δ32−δ/2rSδξ−δ∞
X
i=1 1
i−2/3δ−1
(4.11)
The sum is finite for any δ > 2 according to the convergence of the Hurwitz zeta func-
tion [Ivi03] (also known as generalization of the Riemann zeta function) and, thus, the
total received SNR will be less than TSfor a spacing that is large enough.
I determine the gain factor for the cell area of infinite cooperation by finding the
largest spacing sthat satisfies equation 4.9:
3∞
X
i=1
3i−2
X
j=1 √3·rS
ξp3j2−9i j + 9 i2−3i+ 1!δ
≥1 (4.12)
which is
ξIC(δ) = rS31+δ/2Z1/δ (4.13)
with
Z=∞
X
i=1
3i−2
X
j=1 3j2−9i j + 9 i2−3i+ 1−δ/2(4.14)
and results in a covered area A of
AIC(δ) = 32/δ+3/2rS2Z2/δ/2.(4.15)
AIC(δ) is the following multiple of the non-cooperative cell area ANC(δ)=33/2rS2/2:
gIC(δ) = 32/δZ2/δ (4.16)
As no closed form for Zis evident, no closed form is evident for equations 4.15 and 4.16
either.
4.5. Results
The tables 4.1 and 4.2 summarize the results of the previous section. Figure 4.9 shows
the gain that can be achieved with different types of cooperation for different values of
70
Table 4.1.: This shows the coverable area A with path-loss exponent δ= 2 and range
rS= 1.
Type of cooperation Coverable area A(2) Approx.
No cooperation NC 33/2/2 2.598
Limited 2-cooperation LC 3√7/4+5√3/4 4.149
Unlimited 2-cooperation UC 33/25.196
Infinite cooperation IC ∞ ∞
Table 4.2.: The gain factor gdescribes how much more area can be covered by use of
cooperation for a path-loss exponent δ= 2 and in general.
Coop. type g(2) g(δ)
Limited 2-cooperation LC √21+5
6√3p22/δ −1+22/δ + 1/6
Unlimited 2-cooperation UC 2 22/δ
Infinite cooperation IC ∞see equation 4.16
1
1.5
2
2.5
3
2 2.5 3 3.5 4 4.5 5
Gain factor g
Path-loss exponent δ
∞-coop IC
Unlimited
2-coop UC
Limited 2-coop LC
Figure 4.9.: The area that is possible to cover increases when using cooperation.
71
path-loss exponents: the lower the path-loss exponent, the greater the gain from the use
of cooperation. The greater the covered area, the fewer BSs are needed to cover an area
and, thus, also less power is consumed.
An alternative deployment that would work under my modeling assumptions is to
place two BSs at each location and increase the spacing compared to the non-cooperative
deployment. Next, I explain why this is not efficient. The two co-located BSs can provide
signaling to every point inside their cooperative range together. Hence, for a path-loss
exponent of 2 the spacing can be √2 times the spacing of non-cooperative deployment.
This results in the same number of BSs as the non-cooperative deployment because
two BSs are needed at every location. For higher path-loss exponents the co-located
deployment is even worse. The reason for this is that the area which is already covered
by a single BSs is covered two times with co-located BSs. This shows that it is more
efficient to spread signaling BSs instead of co-locating them.
4.6. Deactivating opportunities with fixed BS spacing
In the previous section, I showed that it is possible to cover a larger area with cooperation
when BSs are placed specifically for cooperation. In this section, I assume BSs are already
placed to cover an area without cooperation. I analyze how cooperation allows some BSs
to be deactivated while still covering the area. Again I consider ergodic capacity as the
constraint for connectivity.
I create an optimization model with the goal to cover a given area with BSs that have
already been placed in a hexagonal pattern. I assume the power consumption of BSs to
be binary. Note that this implies that the power consumption does not depend on the
network load. A more detailed model [ARFB10] would also have been an alternative,
but because I only consider signaling traffic in this chapter this would yield the same
results.
As I only consider signaling traffic in this chapter and the power consumption of the
BSs is independent of their load, I do not model the capacity of the BSs. In my model
the only constraint is to provide the signaling data rate DSto every location of the area.
To make sure every location in the scenario is connected I place a fine rectangular grid of
UEs in the area to cover. I assume that when each of the UEs is connected, the complete
area is covered.
To model cooperation I use the limited n-cooperation as introduced in section 4.4
because it is the more restrictive than unlimited cooperation and, thus, is able to achieve
at least the same performance as limited cooperation.
I use the following constraints to calculate the necessary power to cover the given area.
Equation 4.17 guarantees that each UE receives a signal high enough to be connected.
Because I place the UEs in a fine grid on the area I want to cover, this approximates
covering the complete area. I use equation 4.18 to allow only active BSs to transmit
signal to the UEs and equation 4.19 to calculate the received signal at the UE: (1) a BS
can provide the necessary signal for connectivity to a UE when its distance is smaller
than rS, (2) it provides a part of the signaling when further away, but still in cooperative
72
range, and (3) provides no signal at all, when further away. The optimization target is
to minimize the number of active BSs and, thus, power consumption; it is expressed in
term 4.20.
∀m∈M:X
b∈B
SNR[m, b]≥TS(4.17)
∀b∈B:X
m∈M
SNR[m, b]≤x[b]·|M|·TS(4.18)
∀m∈M, b ∈B:
SNR[m, b]≤
TS,if d(m, b)≤rS
TSrSδ
d(m,b)δ,if rS<d(m, b)≤rSn1/δ
0,if rSn1/δ <d(m, b)
(4.19)
min X
b∈B
x[b] (4.20)
The optimization problem belongs to the mixed integer linear programming (MILP)
class. Next I show how much power can be conserved in a RAN by using the GNU
Linear Programming Kit (GLPK LP/MIP Solver, v4.44) and the Gurobi Optimizer 4.0
to solve different scenarios. The ranges in the following plots are the upper and lower
bounds of the optimal solution the solver provides.
For all my analyses I consider a fixed rectangular area filled with 2500 UEs in a regular
grid and 56 BS located in a hexagonal structure. Each of the 2500 UEs represents only a
possible UE location and thus must be able to receive the signaling rate rS. Figure 4.10
shows the BSs as well as the rectangular area I want to cover.
In real scenarios the ranges of the BSs are larger than strictly necessary to cover an
area for different reasons: (1) to compensate for non-homogeneous signal propagation,
(2) to allow soft hand-overs and (3) to compensate for misplacement of BSs, because
they cannot always be placed in the ideal location. I call the factor by which the real
ranges are larger than strictly necessary to cover the area excess range. I determine the
number of BSs necessary to provide signaling traffic to the area for different values of
excess range next.
Figure 4.11 shows that the higher the excess range in a scenario the more power can
be conserved using cooperation. Also it shows that allowing up to two or three BSs to
cooperate already reduces the power consumption. As I consider only a finite scenario
the curves are not smooth and the larger bounds are higher for higher excess ranges
as the complexity of MILP is higher. I conclude that cooperation allows BSs to be
deactivated even for a low degree of cooperation. Also the gain increases with increasing
excess range.
The range extension of BSs using cooperation is influenced by the path-loss exponent.
Therefore, I analyze the effect of different path-loss exponents next. I assume the area
the BSs can cover is twice as large as necessary to cover the area. That is, the excess
73
0
0.2
0.4
0.6
0.8
1
2 2.5 3 3.5 4 4.5 5
Mean power consumption P
Path-loss exponent δ
No coop
c= 2
c= 3
Figure 4.12.: The power necessary to cover the scenario depends on path-loss exponent
δand the number of cooperating BSs c.
range is √2 including the necessary adjustment for the different path-loss exponents.
Figure 4.12 shows how the different path-loss exponents of the different scenarios influ-
ence the potential power savings by cooperation. The area around a BS that is too far
away to provide enough power to make a direct reception possible, but still close enough
to be usable for cooperation, is small when the path-loss exponent is high. Hence, I con-
clude that scenarios with a low path-loss exponent are better suited to conserve power
by cooperation.
Figure 4.11 shows that cooperation of three BSs allows more power to be conserved
than the cooperation of only two BSs. Additionally using three BSs to cooperate is
more robust against higher path-loss exponents. How much power is conserved when
even more BSs can cooperate is shown in figure 4.13. The power consumption decreases
when the degree of cooperation is increased, but the gain becomes insignificant at about
four in my scenario. I conclude that it is sufficient to let a few BSs cooperate to conserve
power.
4.7. Comparing ergodic and outage capacity
As the channel changes over time, there are two different possibilities to assert its ca-
pacity to transfer data: ergodic capacity and outage capacity [MA05]. Ergodic capacity
is the long-term average data rate that can be transmitted over the channel, assuming
the channel state is always known. Outage capacity is the highest data rate that can be
transmitted over the channel such that the probability for a transmission failure is lower
than the outage probability O– formally, P[SNR > T ]>1−O.
In all earlier sections I considered a lower limit on ergodic capacity. In this section, I
take a brief look at what happens when I change the requirement for a lower bound on
75
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5 6 7 8 9 10
Mean power consumption P
Allowed cooperation c
δ= 4
δ= 3
δ= 2
Figure 4.13.: The power necessary to cover the scenario depends on the degree of limited
cooperation and path-loss exponent δ.
ergodic capacity to a lower bound on outage capacity. To compare both requirements I
normalize them in such a way that they cover the same area without cooperation.
No coop
Ergodic
Outage
Figure 4.14.: The covered areas differ when defining connectivity by ergodic and by out-
age capacity thresholds.
Figure 4.14 illustrates the difference between covered area when considering ergodic
capacity and outage capacity and unlimited cooperation is used: It shows only a small
difference. This provides confidence that the results of my analysis for ergodic capacity
will provide similar results when outage capacity is considered.
Figure 4.15 shows the area covered without cooperation, with selection combining, and
with MRC. This figure illustrates that the gain ofMRC instead of selection combining is
largest when two signals are nearly of equal SNR. Note that the areas also depend on
the whether the needed data rate is defined by outage or ergodic capacity.
This is a hint that the results I calculated in sections 4.4 and 4.6 will also be valid
when considering outage capacity instead of ergodic capacity.
76
No coop
Selecting
Combining
Figure 4.15.: The covered areas depend on the combination method of the UE (selection
combining and maximal-ratio combining (MRC)).
4.8. Conclusion
Firstly, I analytically showed how cooperation increases the covered area of a BS. Sec-
ondly, I created a model to calculate the power necessary to cover an area with given
BSs and analyzed how cooperation reduces the power consumption. My analysis shows
that using a few BSs to cooperate on the transmission to each UE is enough to conserve
significant amounts of power. Both the analytical and the optimization model show
that scenarios with a low path-loss exponent are better suited to conserve power by
cooperation.
In the future this has to be verified for more complex channel models and a more
detailed form of cooperation to be sure it can be applied in reality. In addition, the
necessary protocols to activate and deactivate BSs as well as their cooperation need to
be developed.
In this chapter, I quantified that fewer BSs are necessary when cooperation is used to
transmit signaling traffic to a given area and how many BSs can be deactivated when
already placed BSs can cooperate. Owing to my definition of signaling traffic this is
independent of actual user activity. In contrast to this, the BSs activity I determine in
the next chapter depends on the user activity to serve data traffic.
77
5. Conserving energy in data transmissions
In contrast to the previous chapter, the requirement in this chapter is not to provide
signaling traffic to every possible location, but to provide data traffic to every active
user equipment (UE). Because the requests for data traffic arise randomly in time and
space it is not necessary that all base stations (BSs) are always active.
The idea of this chapter is to activate BSs only when necessary to serve data traffic
to UEs. In chapter 2 I already analyzed how a single BS can do this when considering
power-cycle durations. In this chapter, I do not consider power-cycle durations, but
focus on the interaction of neighboring BSs via the areas that both can serve.
The configuration of BSs I determined by optimization at the end of the last chapter
describes a configuration that can be used until the load of the network changes. In
contrast to this, the configurations I determine in this chapter are determined to provide
data traffic to the UEs that are currently active. Therefore, the configurations deter-
mined in this chapter change faster, and, thus, I am only interested in their average
power consumption.
Just as with signaling, using cooperative transmissions from BSs allows UEs to be
reached which are not in range of any single BS. In this chapter, I quantify the reduction
of energy consumption when BSs can cooperate compared to not cooperating to provide
data traffic to all active UEs.
5.1. Introduction
One way to reduce the energy consumption of the BSs is to decrease the fraction of time
they are active. Because all active UEs still have to be covered by active BSs, extending
the data range of the BSs reduces the necessary activity of neighboring BSs. In this
chapter, I consider the same cooperative transmission used in the previous chapter but
to transfer data traffic instead of signaling traffic.
I use the notion of activity probability as a proxy for power consumption in this
chapter. The activity probability is the fraction of time a BS is serving data traffic to
UEs. If the power-cycle duration is low and the BSs have binary power profile, the
activity probability is equal to the mean power consumption. One possibility to include
the power-cycle durations is to determine the energy consumption from the activity
probability analogous to calculations in chapter 2.
To determine the activity probability of data BSs most related work uses simulations.
In contrast to this, I determine it analytically. To be able to do this, I assume the
BSs are placed in a hexagonal deployment and have a circular radius in which they can
provide the requested data rate to the UE. While this is not a realistic model, I only
78
need this model to determine the sizes of overlapping areas and do not consider their
actual shape. This allows me to contribute an analytic description of the possible gain
in energy efficiency when enabling cooperation in a radio access network (RAN).
I determine an approximation of the activity probability. The same method I use can
be used to determine upper and lower bounds on the activity probability, but for these
bounds to be practically relevant the number of considered BSs has to be impractically
high. Also note that the activity probability I determine is only an approximation of the
actual value as I ignore some dependencies between probabilities to be able to determine
the results analytically. I do not determine the quality of my approximation as the
complexity to do this would be comparable to evaluating the upper and lower bounds.
5.2. Related work
Ashraf et al. [ABH11] provide an overview of possible strategies to put small BSs into
sleep mode and activate them again when necessary. Ismail and Zhuang [Ism11] describe
how energy can be saved on different levels of the network. While they compare the
different approaches and provide overviews, I analytically determine the gain which is
not specific to any implementation. Other work [BK97, Blu10] describes how to cover
an area redundantly.
Wan et al. [WXW11] provide a polynomial-time approximation scheme (PTAS) for
the nondeterministic polynomial time (NP)-hard problem of minimum wireless coverage
(MWC). MWC seeks to find the minimum number of disks to cover a set of UEs. While
this optimally solves a single instance of my problem, I determine the expected number
of active BSs, when the UEs are distributed according to a spatial Poisson process. The
proof of its NP-hardness is by a transformation from planar 3SAT [Joh82]. While a PTAS
exists no fully polynomial-time approximation scheme (FPTAS) is known [WXW11].
Hohenberger determines the effects cooperation and queuing on shared area of BSs in
his bachelor’s thesis [Hoh12]. Based on it we published a paper [HHK13] which compares
strategies to assign the UEs to BSs which take queuing effects into account. I do not
consider queuing effects in this chapter because I consider a low-load scenario in which
these effects are unimportant.
Another group of related work uses stochastic geometry [BB09] to determine outage
probabilities, data rates and power consumption under the assumption that both UEs
and BSs are placed by Poisson processes. Suryaprakash et al. [SFdSF12], for example,
prove that putting BSs into sleep mode is more energy-efficient than varying the available
bandwidth. I assume BSs to be placed in a hexagonal grid instead of by a Poisson process
and determine activity probabilities of BSs.
Son et al. [SKYK11] and Zhou et al. [ZGY+09] describe heuristics to activate BSs and
associate the UEs to BSs. Vereecken et al. [VDC+12] consider a network of macro and
femto BSs. In contrast to my analytical results, they provide results by simulations.
Richter et al. [RFF09] determine the optimal number of pico BSs per macro BS by
comparing different deployments. Alternatively, the problem can be formulated as an
optimization problem. Gonzalez-Brevis et al. [GBGF11] provide an example for such an
79
optimization.
Conte et al. [CFC11] describe how changing cell sizes and deactivating BSs can con-
serve energy. Auer et al. [AGD+11, AGG11] focus on detailed power and deployment
models and determine their results using simulations. In contrast to this, I use a simpler
power and signal model but determine my results analytically.
Marsan et al. [MC09] as well as Oh and Krishnamachari [OK10] describe how the
changes in daily load can be approximated and determine the reduction in energy con-
sumption that is possible by adapting to the changes. In contrast to the work I describe
in this chapter they do not consider the effects of overlapping areas and the random
occurrence of user requests.
5.3. Model
My model consists of an infinite plane with BSs placed in a hexagonal grid, because this
is optimal to cover the plane [Ker39]. I assume that a higher layer of signaling BSs exists,
which detects the presence and activity of UEs. In the previous chapter I described how
these signaling BSs can be placed.
In this chapter, I model only the pico BSs, which provide data to the UEs and do not
cover the area with signaling traffic. I assume the macro BSs do not provide data traffic
at all and thus the pico BSs need to be able to reach every location. Because pico BSs
are usually omnidirectional, I do not consider any directional properties of the antennas.
I model only a single moment in time and not the progression of time in this chapter.
The plane is populated with UEs by a 2-dimensional spatial Poisson process with a
density λ. Because I assume the UEs to be distributed by a Poisson process, the expected
activity EA(A) = 1 −e−Aλis the probability that at least 1 UE is in an area (I call this
“activity in the area”) of size A.
Each UE must be associated to a single active BS or to two cooperating active BSs.
5.3.1. Cooperation
In this chapter, I consider only limited 2-cooperation because the areas of overlap it
creates are simple geometric shapes. Additionally, using unlimited cooperation or a
higher degree of cooperation is superior to using limited 2-cooperation. Therefore, the
activity probability determined by limited 2-cooperation is an upper bound for the ac-
tivity probability (and, thus, power consumption) using unlimited cooperation. In this
chapter, I assume the path-loss exponent δto be equal to 2 for simplicity. This results
in a cooperative range which is √2 times the non-cooperative range. The calculations I
do in this chapter are also possible for other path-loss exponents.
For the purpose of this chapter, I consider only the power consumption of the BSs
during periods of low load. As a result of this, I assume that every BS is always able
to handle the requested load. This allows me to ignore any limits on the number of
UEs that are associated to a BS. Depending on the type of traffic this might result in
longer transmission (e.g., file transfers) or not (e.g., video stream). When a transmission
80
takes longer it will stay active longer, and will thus increase the energy consumption. I
consider it in chapter 7.
5.3.2. Metrics
I consider two different metrics of power consumption: (1) the activity probability of a
BS P and (2) the expected number Q of active BSs per covered area. To fairly compare
scenarios with different spacing, I consider both activity probability of a single BS and
the expected number of active BSs per area. Note that the mean power consumption of
a BS is equal to the activity probability under the binary power model and I, thus, use
the same symbol.
I denote all functions used with an index “n” for non-cooperative transmission. Be-
cause the use of cooperative transmissions allows greater spacing (see chapter 4) I distin-
guish two different ranges of spacing under cooperation. To directly compare cooperative
assignments with non-cooperative assignment I use the same spacing and denote a coop-
erative assignment as “d” to denote densely placed BSs. As cooperative transmissions
allow for a higher spacing I use “s” for sparsely placed BSs with cooperation.
5.4. Analysis
In this section, I determine the probability that a BS has to be active when all UEs
have to be covered. The parameters I analyze are the spacing between the BSs and
the different transmission technologies. This allows me to determine the gain of using
cooperation in a network which is built for non-cooperative transmission and a network
that is built for cooperative use.
To determine the activity probability, I divide the plane into areas that summarize
all UEs which can be connected to the same set of BSs. From the size of these areas I
determine the probability that active UEs are in the area and from that the probability
that a BS has to be active.
I first determine the activity probability for spacings which are able to provide service
to every location without cooperation. This reflects a RAN which was built without
cooperative transmissions in mind. Later I also consider higher spacings which become
possible when the network is built for the use of cooperation (see chapter 4). Therefore,
I do not need to determine the activity probability for all possible spacings. Determining
the results for other spacings would be merely routine work.
5.4.1. No cooperation
Using non-cooperative transmission, each UE has to be within 1 unit distance of an
active BS. Because I am interested in the activity probability of a BS I look at the UE
from the perspective of a BS.
81
Spacing ξ= 1.4
Covered by 1 BS: A1
Covered by 2 BSs: A2
Covered by 3 BSs: A3
Figure 5.1.: The areas of non-cooperative transmissions with spacing ξmin ≤s≤ξmax
overlap.
A1A2A3
A2e A1e Ah
Figure 5.2.: The areas and the names I assign them for non-cooperative transmissions
are shown here for a spacing of ξ= 1.4.
82
Areas
In this section, I describe the size of the areas which arise from using non-cooperative
transmission. The largest spacing of non-cooperative BSs that still covers the complete
plane is ξmax =√3. In general small spacings are not practical because a high number
of BSs would be needed to cover a given area. Therefore, arbitrarily small spacings are
not practical. I consider ξmin = 2/√3 as the smallest spacing. The reason I use exactly
this spacing is that further reducing the spacing would create areas of 4 overlapping BSs.
I denote sizes of areas by A and add an index for the number of BSs nthat can
cover it. I additionally add the label e when the area can be covered by exactly nBSs.
As a reference, the area a BS covers when each UE is assigned to the closest BS is
Ah. Figure 5.1 gives an overview which types of overlapping areas occur and figure 5.2
highlights the areas together with their name.
The size of the area that can be covered by 1 BS is:
A1=π, (5.1)
The size of an area covered by two BSs is:
A2= AI(ξ, 1,1) = 2 arccos ξ
2−ξ√2−ξ√ξ+ 2
2,(5.2)
where the size of the intersection of two circles with distance dand radii rand R[Weia]
is
AI(d, r, R) = r2arccos d2+r2−R2
2dr +R2arccos d2+R2−r2
2dR
−1
2p(−d+r+R)(d+r−R)(d−r+R)(d+r+R).(5.3)
The overlap of 3 BSs is an equilateral circular triangle [Few06] with area:
A3=√3
4c2+ 3 arcsin c
2−c
4p4−c2,(5.4)
where cis the distance between two corners of the equilateral circular triangle, which
obeys:
c2= 3 −ξ2
2−ξq3−3ξ2/4.(5.5)
This results in the size of the areas that can be covered by exactly 2 BSs to be:
A2e = A2−2A3,(5.6)
and the area that can be covered by exactly one BS:
A1e = A1−6A2e −6A3.(5.7)
83
0
0.2
0.4
0.6
0.8
1
1.2 1.3 1.4 1.5 1.6 1.7
Fraction of area
Spacing
1 BS
2 BSs
3 BSs
Figure 5.3.: The fraction of the area of the plane that is covered by a given number of
BSs depends on the spacing ξ.
To calculate a reference value and to determine the number of BSs per area I addi-
tionally need the hexagonal Voronoi area that is closest to a BS:
Ah=√3ξ2/2.(5.8)
Hence, there are 1/AhBSs per unit area.
Figure 5.3 illustrates which fraction of the plane is covered by exactly 1, 2, and 3 BSs.
Note that this is different from the fractions of area inside the range of a single BS. The
reason for this is that areas A2e are seen by two BSs and areas A3are seen by 3 BSs
and are, thus, counted two and three times, respectively, when considering areas each
BS “sees”.
Activity probability
In this section, I determine (1) the activity probability of a BS and (2) the expected
number of active BSs per area for non-cooperative BSs.
As a reference value against which to compare the energy-efficient assignment of UEs
I first determine the activity probability when each UE is assigned to its closest BS.
When each UE is assigned to its closest BS, a BS has to be active if and only if there is
activity in Ah. Hence, the activity probability of a BS under closest assignment is:
Ph= EA (Ah).(5.9)
For any activity probability P of a BS the expected number of active BSs per area is:
Q = P/Ah.(5.10)
84
Spacing ξ= 1.7
Covered by
a single BS or
2 cooperating BSs: Ad
Figure 5.4.: In a dense cooperative deployment, areas can be covered by a single BS or
two cooperating BSs. This is valid for spacings ξminD ≤ξ≤ξmaxD.
Determining the exact activity probability of a BS for an energy-efficient assignment
is not directly possible because it depends on the activity probability of its neighbors,
which in turn depends on the activity probability of their neighbors and so on. Because
my scenario is symmetric with respect to all BSs, the activity probability P of all BSs
is the same as long as the scheme for assignment of UE considers all BSs equally. Next,
I determine an equation for the activity probability of a BS depending on the activ-
ity probability of its neighbors. Because all these probabilities are equal assuming a
homogenous Poisson process, I solve the equation for it.
Consider a BS B: it has to cover at least its central area A1e. Additionally, consider
a neighbor N: If Nis active (activity probability for non-cooperative assignment Pn),
Bcan ignore the area which is shared between both Band N(size A2e). If Nis not
active (probability 1−Pn), the area has to be covered by Bif there is activity. The same
argument works for the area A3: If neither of the two neighbors that also cover the area
are active, Bhas to cover it in case of activity. These arguments allow me to calculate
the activity probability Pnof BS B, based on the activity probability of its neighboring
BSs. I do not need to recursively continue because from the symmetry of the scenario
all BSs have the same activity probability.
Note that using this approach, I ignore the dependencies between activity of neigh-
boring BSs and dependencies between probability of activity in an area and one of its
neighboring BSs. For example, if I know that neighbor Nis active, the activity proba-
bility of Bwill be slightly lower than Pnbecause BS Nalready covers the users in the
area shared by Band N. Hence, I only estimate the activity probability and not the
exact value.
When the overlapping areas (A2and A3) are small the error done ignoring the depen-
dencies is small. When the size of the overlapping areas is greater the potential for error
is greater, but my approximation does not necessarily have to be worse. Because deploy-
ing BSs at their maximum range is generally more efficient (as fewer BSs are needed), I
consider only spacing close to the highest possible spacing here. For these high spacings
the error of my approximation is limited as the size of the overlapping areas is limited.
85
To determine the activity probability, I encode the activity configurations of the neigh-
bors in the binary digits of variable i: 0 means deactivated and 1 means active. I denote
the jth binary digit of ias i[j] = bi/2jcmod 2. I iterate over all possible configurations
of activity of the neighbors and determine the activity probability of BS Bfor each
configurations by computing the size of the area that it has to cover. This results in the
following equation:
Pn=
26−1
X
i=0
5
Y
j=0
Probability that neighbor jis in state i[j]
z }| {
i[j]Pn+1−i[j]1−Pn(5.11)
·EA
A1e + A2e
5
X
j=01−i[j]
| {z }
Number of A2e areas to cover
+ A3
5
X
j=01−i[j]1−i[j+ 1 mod 6]
| {z }
Number of A3areas to cover
.
Note that simply using the binomial formula to determine the probability for a given
number of active neighbors is enough to determine the number of areas A2e that have
to be covered. But this approach does not work for the areas A3because not only the
number of active neighbors determines the number of areas A3, but also their relative
position has to be taken into account.
Equation 5.11 can be simplified to a polynomial of degree 6 in Pn. While this is too
complex for a closed form solution, Newton’s method provides me with the means to
determine its solution numerically. This solution is the activity probability Pnof a single
BS for the scenario without cooperative transmissions.
While the method just explained determines the same activity probability for each
BS, this does not necessarily result in the minimal activity probability, when averaged
over all BSs because I did not prove that an asymmetric activity pattern does not result
in a lower average activity probability. I propose that every asymmetric scheme can
be converted to a symmetric scheme by randomly selecting variants of the asymmetric
scheme which have been translated or mirrored. But as this is not in the scope of my
dissertation I did not pursue this idea. A good starting point to follow up on this idea
could be the mathematics of tilings and colorings [Soi08].
5.4.2. Cooperation with densely placed BSs
To compare the activity probability of two association schemes they must have the same
spacing. Because cooperative transmissions allow a higher spacing, I call the deployment
with the same spacing as the non-cooperative deployment dense.
Areas
Figure 5.4 provides an overview of the involved areas when the spacing is between ξminD =
p8/3 and ξmaxD =ξmax =√3. Figure 5.5 highlights the relevant areas and provides
86
AdA1ec
Figure 5.5.: These are the shapes and names of regions for cooperative BSs with dense
deployment of spacing ξ= 1.7.
Ad
2=
AI ξ, √2,√2
−2
AI ξ, 1,√2
+
AI(ξ, 1,1)
Figure 5.6.: The area Adcan be calculated from overlapping areas of different sizes.
their names. Note that I only consider limited 2-cooperation for dense cooperative
deployments due to the more complex shape of areas with unlimited cooperation.
I call the area that can be covered by exactly two cooperating BSs A and B or one
other fixed BS C (A 6= B 6= C) Ad. It can be calculated from of intersection of circles.
Figure 5.6 shows that Adcan be calculated as:
Ad=1
2AI ξ, √2,√2−2AI ξ, 1,√2+AI ξ, 1,1.(5.12)
The area that is only covered by a single BS has size:
A1ec = A1e −6Ad.(5.13)
87
Activity probability
The equation for the activity probability with limited cooperation can be determined in
the same way the one as without cooperation by additionally considering the area Ad::
Pd=
26−1
X
i=0
5
Y
j=0
i[j]Pn+1−i[j]1−Pn
EA
A1ec
+ A2e
5
X
j=01−i[j]
+ A3
5
X
j=01−i[j]1−i[j+ 1 mod 6]
+ Ad
5
X
j=0
1−i[j]i[j+ 1 mod 6]
| {z }
Number of cooperative areas Adto cover
.(5.14)
5.4.3. Cooperation with sparsely placed BSs
In addition to the dense deployment, cooperating BSs can also be placed sparsely. The
reason why this deployment allows all UEs in the plane to be served is the same as
already explained in section 4.4.
Areas
The geometric illustration in figure 5.7 shows the areas I consider for cooperative trans-
missions with a minimum spacing of ξminS = 2. This lower limit prevents overlap of
non-cooperative regions and thus simplifies the calculations. The maximum spacing for
limited cooperation is
ξmaxSL =q√21 + 5√2.(5.15)
The same scheme explained here also works for unlimited cooperation with a spacing up
to ξmaxSU =√6 (see chapter 4).
The way I determine the sizes of the areas squander some potential to reduce the
activity probability because the areas that can actually be reached are larger (compare
the size of covered areas by limited and unlimited cooperation in figure 4.7). The benefit
of my method is that I can use the same expressions of sparse area sizes for limited and
unlimited cooperation and it holds even as I increase the spacing up to ξmaxSU. Note that
I squander even more potential for unlimited cooperation than for limited cooperation.
88
Spacing ξ= 2.2
Covered by
2 cooperative BSs
Figure 5.7.: These are the areas in a sparse cooperative deployment (with spacing
ξminS ≤ξ≤ξmaxSU for unlimited cooperation and ξ≤ξmaxSL for limited
cooperation).
A1Ah
Ase As
Figure 5.8.: These are the shapes and names of regions for sparse cooperative deployment
of BSs with spacing of ξ= 2.2.
89
A lower bound for the size of the area that is covered by exactly two BSs, which is
highlighted in figure 5.8, is:
Ase =Ah−A1
3.(5.16)
Hence, total area for which a BS has to be active is:
As= A1+ 6Ase.(5.17)
Activity probability
The activity of a BS can be directly determined from the activity in the areas, because
no association choices have to be made. The activity probability is:
Ps= EA(As).(5.18)
5.5. Numerical results
In this section, I illustrate the analytical results from the previous section with the help
of the computer algebra system Maxima 5.28.0.
Active BS
Sleeping BS
Original cell border Re-tiled cell border
Figure 5.9.: The original hexagonal tiling can be re-tiled to new hexagonal tiling with a
√3 larger spacing of BSs by deactivating 2 of 3 BSs.
While it is impractical to move BSs to adapt the parameters of a RAN to the current
load, it is possible to transform a hexagonal deployment of BSs to another hexagonal
deployment with a different spacing simply by removing BSs. For the purpose of energy
adaption this just means deactivating and ignoring the removed BSs. Examples for
90
Figure 5.10.: An alternative re-tiling with the double spacing can be achieved by deac-
tivating 3 of 4 BSs.
possible re-tilings of the hexagonal deployments are: (1) removing 2 of 3 BSs results
in a hexagonal tiling with a spacing that is larger by a factor of √3 (figure 5.9) and
(2) removing 3 of 4 BSs results in a hexagonal tiling with a spacing that is larger by a
factor of 2 (figure 5.10). While other re-tilings are possible [BSW97], these have higher
spacings. These examples show that a hexagonal deployment can be transformed into
another hexagonal deployment with higher spacing by deactivating BSs. The factors
between spacings for these examples are √3 and 2. Note that because each BS has a
capacity limit, re-tiling is only possible when the total traffic in the network is low.
Figure 5.11 shows how cooperative schemes reduce the activity probability for a given
spacing. Because mean power consumption not only depends on the activity probability,
but also on the number of placed BSs, this is not a fair comparison. The expected number
of active BSs per area Q considers this. Figure 5.12 illustrates how cooperative schemes
reduce the expected active BSs per area Q. Non-cooperative schemes have a maximum
spacing of ξmax =√3. Cooperative schemes can cover the plane for spacings between √3
and 2,but for my analysis it was not necessary to determine the activity probability in
this range. I conclude that it is more energy-efficient to operate BSs at a higher spacing,
as long as they are still able to serve all UEs with the necessary data rates.
Figure 5.13 shows that the optimal selection of a BS activation scheme depends on
the user density. When the user density increases, it converges to the number of BSs
per area, because all BSs have to be active in this case.
Figure 5.14 shows the expected number of active BSs per area compared to the non-
cooperative assignment. The dense cooperating scheme is always lower than 1 and,
hence, better than the non-cooperative assignment. The sparse cooperative assignment
91
0
0.2
0.4
0.6
0.8
1
1.2 1.4 1.6 1.8 2 2.2 2.4
Activity probability
Spacing
closest
non-coop
dense coop
sparse coop
Figure 5.11.: The activity probability of a BS increases, when the spacing is increased.
0
0.1
0.2
0.3
0.4
0.5
0.6
1.2 1.4 1.6 1.8 2 2.2 2.4
Expected enabled BSs per unit area
Spacing
closest
non-coop
dense coop sparse coop
Figure 5.12.: The expected number of active BSs per area decreases when the spacing is
increased. The arrow shows a change that can be implemented by deacti-
vating BSs in the network (re-tiling) instead of physically moving BSs.
92
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 0.2 0.4 0.6 0.8 1
Expected enabled BSs per unit area
User density
closest
non-coop
dense coop
sparse coop
Figure 5.13.: The expected number of active BSs asymptotically approaches a limit for
each of the schemes when the user density is increased. In this case all
schemes use their optimal spacing.
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2
Relative number of enabled BSs per area
User density
dense coop
sparse coop
Figure 5.14.: The expected number of active BSs per area for the cooperative schemes at
their maximum spacings compared to the non-cooperative scheme. Note
that the dense cooperative scheme uses the same spacing and thus the same
BSs.
93
on the contrary has more expected active BSs because a single UE can potentially require
activating 2 BSs to cooperatively provide it with data. However, when the user density
increases, the sparse deployment becomes more efficient because it uses fewer BSs. Note
that this analysis only holds as long as the sparse cooperative BSs can provide enough
data rate to all users.
5.6. Conclusion
I first determined the sizes of intersections of ranges for a hexagonal deployment of BSs
in a RAN. I then calculated the probability that at least one UE is in such an area under
the assumption that the UEs are distributed according to a Poisson process. Based on
this I calculated the activity probability of the BSs for cooperative and non-cooperative
association schemes. Using this result, I determined the expected number of active BSs
per area.
My results show that the reduction in power consumption by allowing cooperative
transmissions without changing the spacing of the BSs is between 0% and 11% depending
on the load. When additionally changing the spacing the power consumption per area
can be reduced by additional 39%. I concluded that fewest active BSs are needed if the
BSs are placed as far apart as possible, but still cover the area and are able to provide
the necessary data rates. Cooperative transmission from the BSs allows the BSs to be
placed further apart than non-cooperative transmission would allow.
Future work will include the limit of data rates BSs can provide and, hence, determine
when the assignment schemes will be data rate-limited instead of coverage-limited.
94
6. Radiated power
In the previous two chapters, I determined how base stations (BSs) can cooperate based
on the distance to user equipments (UEs). The distance (together with other factors)
determines only the average channel quality. In this chapter, I demonstrate how knowl-
edge about the instantaneous channel gain can reduce the outage probability and the
radiated power. Because the radiated power becomes interference at other UEs it is
important to keep it as low as possible, especially in dense urban environments.
To reduce outage probability and radiated power, I compare strategies which select
which BSs cooperate based on average and instantaneous channel gains. During the
analysis it also becomes clear why BSs selection and transmit power control produce
similar results.
6.1. Introduction
An important decision in cooperative radio access networks (RANs) is the selection of
the cooperating BSs from all possible BSs. This selection can be based on average or
instantaneous channel quality. Basing this decision on instantaneous knowledge cannot
be worse and it is reasonable to assume it can be better. Related work has quantified the
overhead and the gain of instantaneous knowledge. But most of this work is based on
simulations, while I analytically compare BS selection with instantaneous and average
channel knowledge. In contrast to other work, I do not consider distances, but directly
base my decisions on channel quality, which is also more realistic to know in a real
system.
I differentiate between two uses of instantaneous channel knowledge: (1) instanta-
neously changing the effective radiated power (ERP) of BSs and (2) selecting the BSs
that cooperate to transmit the signal to the UE. I ignore the technical requirements
(see section 1.4) needed to cooperate and compare the outage probability and the mean
ERP. I use ERP because it is proportional to the interference generated at other UEs.
The only assumption I make about timing is that BSs which use instantaneous channel
knowledge need to be able to track it fast enough.
The contribution of this chapter is the analytic description of outage probability and
ERP for cooperating BSs in a RAN with and without instantaneous channel knowledge.
Additionally, I show that the highest gain from instantaneous channel knowledge can be
achieved when the possibly cooperating BSs have similar average channel gains.
I show that instantaneous selection of cooperating BSs can reduce the ERP while
keeping the outage probability the same.
95
6.2. Related work
Tse and Viswanath [TV05] describe waterfilling, which maximizes the ergodic capacity
of a cooperative transmission with instantaneous channel knowledge and limited ERP.
In contrast to this, I measure the outage probability and compare it under instantaneous
and average channel knowledge.
Hoydis et al. [Hoy11] analyze the optimal fraction of coherence time of a channel to
be used to determine the instantaneous channel quality. I assume the instantaneous
channel quality is known and analyze how ERP and outage probability can be traded
off when using different cooperative schemes.
Park et al. [PSS09] describe how an adaptive use of cooperative and non-cooperative
schemes can maximize the network’s capacity. Zakhour and Hanly [ZH10] maximize the
minimum data rate. Instead of data rate I use the outage probability as a quality metric
for the resulting transmission.
Other work considers scenarios with relaying UEs [AK04, LT04, NH04] and multi-
hop communication [LFK10]. Biermann et al. [BSC+12] compare how well different
back-haul topologies are suited for cooperative transmissions. I assume a list of all
possibly cooperating BSs to be available from which a selection can be made. Maaref and
A¨ıssa [MA05] describe the outage and ergodic capacity of multiple-input and multiple-
output (MIMO) Systems. I determine the outage probability and the ERP of cooperating
BSs.
While I assume the channel state to be determined without overhead, Ramprashad
and Caire [RC09] consider the overhead of collecting this information in a MIMO Sys-
tem. Similarly, Goldenbaum et al. [GAV11] consider the effect of delayed channel state
information. Another group of work [BK12, LSC12] focuses on the practical aspects of
cooperative transmissions while I provide analytical results.
6.3. Model
I consider only the connection of a single UE to the RAN and only downlink transmissions
from the BSs to the UE. I denote the instantaneous channel gain from BS ito the UE as
γiand its probability density function as piwith the mean Γi. Note that I do not make
any assumption about the distribution of the instantaneous channel gain and, hence, the
results are valid for all block-fading fading environments. For simplicity, I assume the
BSs to be ordered by decreasing average channel gain Γ.
6.3.1. Effective radiated power
I define the ERP riof a single BS to be the power that is transmitted from the sender’s
antenna including all gains and losses of the sender and its antenna as well as the gain
of the receiving antenna. For simplicity, I normalize the ERP to be between 0 and 1.
Power control is the ability of a BS to change the ERP in the interval [0,1] for each
fading block, while a BS without power control is limited to the values 0 and 1. Note
that mean ERP, in addition to being a metric (if the BS adapts the ERP based on the
96
channel quality), can also be seen as a parameter (if setting ERP to a constant value).
When not further specified I assume all BSs transmit at full power.
I do not explicitly consider the number of transmit and receive antennas at each BS,
but as long as the signals of different BSs can be additively combined my results also
hold for MIMO systems. Because BS clustering and the back-haul network constrain
the free selection of BSs, I assume a list of possibly cooperating BSs is available from
which the actually cooperating BSs are selected.
6.3.2. Static vs. dynamic association
I compare two different types of cooperative schemes: static and dynamic. Using static
association, the UE is associated to the cBSs with the highest average channel gain.
Using dynamic association, the UE is associated to the cBSs with the highest instanta-
neous channel gain. I use nto denote the total number of available BSs while cdenotes
the number of BSs that can actively cooperate.
I assume all users require a fixed minimum data rate. I model this as a threshold TD
of signal-to-noise ratio (SNR) above which the transmission succeeds and below which
it fails. The probability that this threshold is not met is called outage probability O:
P[Outage] = Priγi
N
|{z}
SNR
< TD=P(riγi< Tγ),(6.1)
where Nis the mean power of the noise. I denote the outage probability using static
association by OSand using dynamic association by ODand power control by a “p” in
the index. For simplicity, I define the power threshold Tγ=TDNas the threshold of
ERP rand channel gain γfor a successful transmission. This threshold Tγis system-
dependent but a constant for the purpose of my analysis. I denote the sum of the ERP
of all BSs averaged over time by R.
6.4. Outage probability
In this section, I describe the outage probability for static and dynamic association when
all cooperating BSs transmit at maximum power. While this minimizes the outage prob-
ability it will also radiate more power than necessary. These calculations provide a lower
bound for the possible outage probability when the ERP is reduced. The calculation of
outage probability using static association is not new [Gol05], but I included it in this
chapter, using the notation of the rest of the dissertation, as a reference for dynamic
association.
6.4.1. Static association
While I focus on cooperative transmissions from the BSs, non-cooperative transmission
is the special case of cooperative transmission when only one BS transmits.
97
Static association without cooperation
Without cooperation and with static association, the lowest outage probability is achieved
by associating the UE to the base station with the highest average channel gain. Be-
cause I number the BSs in decreasing order of average channel gain, BS 1 has the highest
average channel gain.
The outage probability for static selection of a single BS can be calculated as:
OS(1) = P[γ1< Tγ] = ZTγ
0
p1(x)dx. (6.2)
Static association with two cooperating BSs
If I allow two BSs to cooperate and select these BSs based only on the average channel
gain, it is best to select the two BSs with the highest average channel gain to maximize
the their sum. Because I number the BS in decreasing order of average channel gain,
these are BS 1 and 2. The resulting outage probability with static selection of two BSs
is:
OS(2) = P[γ1+γ2< Tγ] = ZTγ
0
p1(x1)ZTγ−x1
0
p2(x2)dx2dx1.(6.3)
Static association with ccooperating BSs
If a UE is statically associated to cBSs, it is best to select the cBSs with the highest
average channel gain to maximize their sum (note that this only minimizes the outage
probability when all quantiles of the distribution with the higher mean are also higher
than the corresponding quantiles of the distribution with the lower mean, which is true
for the exponential distribution). Because I number the BSs in decreasing order of
average channel gain, these are BSs 1 to c. The resulting outage probability with static
selection of cBSs is:
OS(c) = P c
X
i=1
γi< Tγ!=sc(Tγ),(6.4)
where si(t) is the probability that the sum of the channel gains of BSs 1 to iis lower
than t(i.e., the convolution all cinvolved γi):
si(t) = (Rt
0pi(xi)si−1(t−xi)dxiif i > 0,
1 else. (6.5)
Under the Rayleigh fading assumption, the outage probability of static association
OS(c) can be calculated from the Erlang distribution of shape cwhen all average channel
gains are equal. The hypoexponential or generalized Erlang distribution [Ros09] is a
generalization which allows the average channels gains to be different. Its cumulative
distribution function and probability density function can also be calculated as a special
case of the phase-type distribution [Neu81].
98
6.4.2. Dynamic association
With dynamic association, I select the best cBSs out of npossible BSs based on their
instantaneous channel gain. This results in an outage probability of:
OD(n, c) = P max
τ∈Sn
c
X
i=1
γτ(i)< T!=X
τ∈Sn
hτ(n, c)
=X
τ∈Sn
P c
X
i=1
γτ(i)< T ∧∀i∈ {2, ..., n}:γτ(i−1) > γτ(i)!,(6.6)
where Snis the symmetric group, that is, the set of all possible permutations τof the
natural numbers (1, . . . , n) and τ(i) is the ith element of τ. I use τto describe the order
of instantaneous channel gains. I derive a formula for OD(n, c) by applying the law of
total probability over all possible permutations τof channel gains using the function h.
The function hcalculates the probability that the threshold is not met when the best
cBSs cooperate and the order of the instantaneous channel gains is τ. I calculate it from
the probability that the sum of the clargest instantaneous channel gains is smaller than
Tγand they are in the order τ(with the function v) and that the n−cother channel
gains are in the order τ(with the function q):
hτ(n, c) = ZTγ/c
0
fτ(c)(xc)vτ(c−1, Tγ−xc, xc)qτ(n, c + 1, xc)dxc.(6.7)
The function qcalculates the probability that the channel gains from BSs ito n(which
do not contribute to the combined signal) are in the order τ:
qτ(n, i, x) = (Rx
0pτ(i)(xi)qτ(n, i + 1, xi)dxiif i≤n,
1 else. (6.8)
The function vcalculates the probability that the channel gains of BSs 1 to iare in
order τand their sum is lower than Tγ. The parameter xis the channel gain of the BS
i+1 and is a lower bound for the channel gain of BS i. The parameter t=Tγ−Pc
k=i+1 xk
is the channel gain that is left for the BSs 1 to ito not go over threshold Tγ:
vτ(i, t, x) = (Rt/i
xpτ(i)(xi)vτ(i−1, t −xi, xi)dxiif i > 0,
1 else. (6.9)
Note that it is not necessary to split the calculation over all different orders of the
n−cnot selected BSs, but doing so gives a uniform way to state the probability. My
formulation has the property that it does not need any case distinction inside the inte-
grals, for example, for values smaller than 0. This allows easy analytical evaluation of
the integrals using a computer algebra system.
In addition to being an abstraction for maximal-ratio combining (MRC) and coherent
combining (CC), my formulation can also be used to calculate the outage probability for
selection combining [Gol05] when using dynamic association with c= 1.
99
6.4.3. Strictly superior cooperation schemes
In this section, I show which cooperation schemes provide lower outage probability than
others for all possible channel situations.
Theorem 6.1. For natural numbers a≤bthe following holds:
OS(b) = OD(b, b)≤OD(b, a)≤OD(a, a) = OS(a).(6.10)
Proof. OS(n) = OD(n, n): Dynamic selection of nout of nBSs is by definition the same
as static selection of all nBSs.
OD(b, b)≤OD(b, a): Because both schemes select the best BSs based on instantaneous
channel gain, OD(b, b) will always select the aBSs that OD(b, a) selects. As all channel
gains are positive and the quality of the cooperative transmission depends on the sum
of the individual channel gains, OD(b, b) must be lower than OD(b, a).
OS(b, a)≤OD(a, a): analogously.
1
1 10
Outage probability O
Threshold Tγ
Static 3 Dyn. 1 of 10
Figure 6.1.: Depending on the threshold each of two cooperative schemes can achieve a
lower outage probability at the UE.
This result gives a lower and an upper bound on the outage probability of dynamic
association, which are the two bounding static associations. Note that not for all pairs
of cooperative schemes one is strictly superior to the other: Whether static association
to 3 BSs or dynamic association to 1 of 10 BSs results in a lower outage probability
depends on the threshold Tγ. In contrast to the strict superiority of one scheme above
the other, Figure 6.1 shows an example in which the threshold determines which scheme
provides the lower outage probability.
100
6.5. Effective radiated power
In this section, I calculate the total ERP (i.e., from all BSs) averaged over time for the
different cooperation schemes with and without power control.
Today’s BSs have a nearly binary power profile. Hence, a lower transmission power
will not directly reduce the power consumption of a BS. However, the power consumption
of future BSs which are developed with energy efficiency in mind will depend more on
the radiated power than today (see linear power profile in section 1.2). When such BSs
are used, the reduction in radiated power will also reduce the consumed power.
However, the main focus of this chapter is that the signal-to-interference-(plus-)noise
ratio (SINR) of other nearby transmissions will be higher, when less power is radiated.
When the SINR is higher the transmissions are and, thus, provide more opportunities
to deactivate BSs.
Without instantaneous channel knowledge, reducing the ERP will increase the out-
age probability. Hence, there is a trade-off between ERP and outage probability. With
instantaneous channel knowledge this is not necessarily the case. That is, with instanta-
neous channel knowledge, reducing the ERP at the correct times does not increase the
outage probability.
Reducing the ERP will result in less interference for other transmissions and thus bet-
ter channel quality in a multi-user environment. I do not directly quantify the effect on
other transmissions in this chapter, but only use ERP to compare different transmission
schemes.
6.5.1. Static association
Without instantaneous channel knowledge, it is not possible to adapt the ERP to the
actually necessary value. Hence, reducing the ERP can only be done by reducing the
ERP in all channel situations.
Calculating the outage probability with reduced ERP can be done with the same
equations as for full transmit power: The channel gain with reduced ERP is equal to
the channel gain multiplied by the ERP rat full transmit power. This holds both for
average and instantaneous values. The total ERP of all BSs is RSp(c) = Pc
i=1 riwhen
BS itransmits with ERP ri. In the special case of full power transmission this becomes:
RS(c) = c. I use these as reference values to compare with dynamic association and
instantaneous power control.
The calculation of RSp(c) and OSp(n, c) enables a calculation of the outage probability
for a given allocation of ERPs to the BSs. However, this does not provide a closed formula
for the optimal distribution of total ERP to minimize the outage probability.
When the instantaneous channel gains are known it is best to assign as much power to
the channel with the highest gain as possible (theorem 6.2). However, this does not hold
if only the averages of the channel gains are known. That is, allocating all transmit power
the the channel with the highest average channel gain does not necessarily minimize the
outage probability. It is possible to use Newton’s method to find a local minimum in
the distribution of total ERP. I present an example for this next.
101
0
0.2
0.4
0.6
0.8
1
1Tγ10Tγ
Fraction of Power allocated to BS 1
Mean channel gain of BS 2 Γ2
Figure 6.2.: The optimal fraction of power allocated to BS 1 depends on the channel
gain of BS 2. The channel gain of BS 1 is 10Tγ.
Figure 6.2 shows how the optimal power distribution between two BSs depends on
the channel gain of the BSs. I determined the optimal value using Newton’s method.
It shows that even when the average channel gain of BS 1 is higher than the average
channel gain of BS 2 it is necessary to allocate ERP to BS 2 to minimize the outage
probability.
One drawback of this method is that the found local minimum is not always the global
minimum: For example using 2 BSs with Γ1= 15Tγand Γ2= 0.995Tγ, a local minimum
is at about 0.14 of the power allocated to BS 1, while the global minimum is at 0. While
the actual difference in outage probability is small (at least in this example), the problem
has to be considered when applying Newton’s method to find the optimal allocation of
ERP to the BSs. For the remainder of this chapter, I assume all BSs which transmit
using static association to transmit at full power. Hence, the power does not need to be
distributed, but all BSs transmit at full power.
Figure 6.3 shows that the actual difference is small, but a local minimum is not
necessarily a global minimum for a distribution of ERP to the BSs.
6.5.2. Dynamic association
At first I calculate the ERP with dynamic association without power control. That is,
the BSs do not transmit at all when the sum of the channel gains will be lower than the
necessary threshold. When the threshold can be reached only those BSs transmit which
are needed to reach the threshold. With OD(n, 0) = 1, the expected ERP without power
102
0.06449
0.06450
0.06451
0.06452
0.06453
0.06454
0.06455
0.06456
0 0.05 0.1 0.15 0.2
Outage probability O
Fraction of power allocated to BS 2
Figure 6.3.: The outage probability depends on the distribution of the ERP. This is
an example that the local minimum t 0.14 does not have to be the global
minimum at 0. The BSs have a channel gain of 15Tγand 0.995Tγ.
control can be written as:
RD(n, c) =
c
X
i=1
iP[iBSs are necessary and sufficient]
=
c
X
i=1
i(
iBSs are sufficient
z }| {
(1 −OD(n, i)) −
i−1 BSs are not
z }| {
(1 −OD(n, i −1)))
=
c
X
i=1
i(OD(n, i −1) −OD(n, i))
=
c
X
i=1
iODe(n, i),(6.11)
where ODe(n, c) = OD(n, c −1) −OD(n, c) is the probability that exactly cBSs have to
cooperate.
With power control, it is possible to transmit only with the ERP necessary to reach
the threshold Tγ. The following definition and its use describe how much total ERP is
necessary to reach the threshold Tγand how to distribute it.
I define an allocation of power to be an on-off allocation if the following condition is
met:
∃j∈ {1, . . . , n}:∀i<j:ri= 1 ∧∀i>j:ri= 0.(6.12)
An intuition of the on-off allocation is that it tries to transmit as much power as
possible over the best channels. This not only seems reasonable, but is also the best
method to allocate the power. Note that this is only holds when the instantaneous
channel gains are known, but not for the average channel gains.
103
Lower instantaneous channel gain
BS 1:
r= 1 . . . BS j−1:
r= 1
BS j:
r= ?
BS j+ 1:
r= 0 ... BS n:
r= 0
Full power No power
Any power
Figure 6.4.: The on-off allocation maximizes the ERP on the channels with the highest
instantaneous channel gain.
Figure 6.4 illustrates the on-off power allocation: a BS jexists that transmits at some
ERP, all BSs with a higher channel gain transmit at full ERP, and all BSs with a lower
channel gain do not transmit. Note that the on-off allocation is uniquely defined for a
given total ERP, when all average channel gains Γiare different. Else they are unique
except for permutation.
Theorem 6.2. If the instantaneous channel gains γiare known, the threshold Tγis
reachable with minimum sum of radiated power if the ERP is distributed according to
the on-off allocation sorted by instantaneous channel gains.
Proof. Assume an allocation Awhich is not an on-off allocation radiates the least total
power and reaches the threshold Tγat the UE. Select jto be the index of the BS with
the lowest channel gain with ERP rj>0. Because all BSs iwith smaller channel gain
have pi= 0 the statement ∀i>j:ri= 0 is fulfilled. Hence, there must be a BS kwith
k < j and rk<1 (else it would be an on-off allocation).
Now construct a power allocation that reaches the threshold and radiates less power
than allocation Aand, thus, prove it cannot have been the one that radiates least power.
Because I only change the ERP of BS iand kI ignore all others. In allocation Athe
BS iand kgenerate a total received channel gain of riγi+pkγkat the receiver. Note
that γk> γiby the way kand jwere selected. Select the new values r∗
i=ri−γi/γk
and r∗
k=rk+, with ε < min(ri,1−rk). This results in the same received channel
gain r∗
iγi+r∗
kγkas the allocation A:riγi+rkγk. But because γi< γkthe total ERP
is lower.
Figure 6.5 illustrates how shifting ERP to a BS with higher instantaneous channel
gain reduces total ERP, but still reaches the same threshold of channel quality. Note
that this is different from waterfilling power allocation [TV05], which maximizes ergodic
capacity instead of minimizing ERP for a given outage probability.
An implication from the optimality of the on-off allocation is the following: In a set of
cooperating BSs with instantaneous power control, all BSs that transmit will transmit
at full power, with the single exception of the BS with the lowest instantaneous channel
gain. Hence, adapting the ERP will only make a difference at one BS, namely the BS
that is actively transmitting and has the lowest instantaneous channel gain.
104
Lower instantaneous channel gain
BS 1:
γ1= 5
r1= 1
BS 2:
γ2= 4
r2= 0.8
+0.1
BS 3:
γ3= 3
r3= 0.5
BS 4:
γ4= 2
r4= 0.2
−0.2
BS 5:
γ5= 1
r5= 0
Shift and reduce power by a factor of γ4/γ2
Figure 6.5.: Allocating ERP from BS 4 to BS 2 reduces the total ERP as the increase
at BS 2 is lower than the decrease at BS 4, while the signal strength at the
receiver does not change. This is an illustration of theorem 6.2
The mean ERP for a single BS which can instantaneously adapt its power can be
calculated as:
RDp(1,1) = Z∞
T
p(x)
ERP
z}|{
Tγ/x dx. (6.13)
Calculating the optimal allocation of power in a given situation with instantaneous
channel knowledge can be derived from the on-off allocation. The mean ERP if selecting
1 out of nBSs is:
RDp(n, 1) = X
τ∈SnZ∞
Tγ
pτ(1)(x1)Tγ
x1
qτ(n, 2, x1)dx1,(6.14)
where the function qτis from section 6.4. For c > 1 BSs the following formula describes
the mean total ERP:
RDp(n, c) = RDp(n, c −1) + X
τ∈Sn
uτ(n, c, 1, Tγ, Tγ),(6.15)
where the function udescribes the ERP if cof nBSs are necessary and sufficient to not
be in outage and instantaneous channel gains are in order τ:
uτ(n, c, i, t, x) = (Rmin(t,x)
t/(c−i+1) pτ(i)(xi)u(n, c, i + 1, t −xi, xi)dxiif i < c,
Rx
tfτ(i)(xi)(c−1 + t/xi)
| {z }
ERP
qτ(n, c + 1, xi)dxielse, (6.16)
where tis the amount of channel gain that the BSs ito chave to provide. The channel
gain of BS imust be smaller than the channel gain of BS i−1, which is represented as
parameter x. The formula includes the calculation of a minimum in the boundaries of
the integral, which must be replaced with a case distinction to evaluate it. This makes
evaluation more complex, but I was unable to find a solution without such a distinction.
105
Bounds for RDp(n, c), which are easier to evaluate than the exact formula, can be
expressed with the help of the on-off allocation:
(c−1)ODe(n, c)≤X
τ∈Sn
uτ(n, c, Tγ,1, Tγ)≤cODe(n, c).(6.17)
The lower bound ignores the power radiated from the BS that is not transmitting at
full power. The upper bound considers it to transmit at full power. Note that it is
only necessary to replace the exact terms of uτin equation 6.15 by the boundaries when
three or more BSs cooperate because the exact terms for one and two BSs are easy to
evaluate.
6.6. Results
In this section, I describe implications from the results of the previous section. Now, I
assume the instantaneous conditions of the channel to be described by Rayleigh fading,
making the instantaneous channel gains exponentially distributed [TV05]. The results
shown in this section are analytical results derived with the help of the computer algebra
system Maxima 5.27.0 (and not results of simulations).
6.6.1. Scenarios with high gain
First, I explain why the benefit of dynamic association becomes greater the more similar
the average channel gains are. Dynamic association will always select the BSs with the
highest instantaneous channel gain to cooperate and, hence, will always use the best
BSs. In contrast to this, static association selects the BSs based on average channel
knowledge and, hence, can make non-optimal selections of BSs.
If the order based on the average and the instantaneous channel gains of all BSs is the
same, both methods will select the same BSs. More precisely, if and only if the cBSs
with highest average channel gains also are the cBSs with the highest instantaneous
channel gains, both schemes will pick the same BSs.
The probability that one exponentially distributed random variable is smaller than
another is P[X < Y ] = λX
λX+λY[Ros09]. Hence, the probability that the order of two
channel gains is different for the average and instantaneous values is smaller when their
expected values are further apart. Given this, the probability that static association
selects the best BSs is:
P[Static selects best BSs] = P
^
1≤i≤c
c<j≤n
γi≥γj
=P
^
c<j≤nmin
1≤i≤cγi≥γj
=Z∞
0
pΣ(y)
n
Y
j=c+1 Zy
0
pj(x)dxdy, (6.18)
106
where fΣ(y) is the probability density function of min1≤i≤cγi, which is exponentially
distributed with a mean of ΓΣ= 1/Pc
i=1 1
Γi.
While dynamic association will always select the best BSs it will not guarantee a
successful transmission, but if static and dynamic association select the same BSs their
outage state will be the same. When they select different BSs there are two possibilities:
(1) if the dynamic association is in outage, the static association must also be in outage,
because it cannot select better channels and (2) if the dynamic association is not in
outage, the static association may or may not be in outage, depending on the selection
of BSs. Hence, if the static association selects non-optimal BSs it does not mean that it
performs worse than the dynamic association, but it is a necessary condition to do so. I
will analyze the full effect next.
I conclude that the gain from using instantaneous knowledge to associate UEs to BSs is
greatest if the average channel gains are similar. Or put differently: using instantaneous
knowledge for association is unnecessary if the average channel gains are very different.
6.6.2. Outage probability and channel gain
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5
Probability to select correct BSs
Factor of channel gain between BSs F
1 of 2
1 of 3 2 of 3
3 of 6
Figure 6.6.: The probability that static association selects the correct BSs depends on
factor Fby which the average channel gains of the BSs are apart.
Figure 6.6 shows the probability that static association selects the best cof nBSs if
the average channel gains of the BSs are a geometric progression with the common ratio
1/F. The average channel gain of BS iis 10Tγ/Fi. For example, for a path-loss exponent
of 2 and successive BSs being at the double distance Fequals 4. This means that on
the left side of the plot the average channel gains are close to each other while they are
different on the right side. The figure shows that the probability to select non-optimal
BSs is lower the more different the average channel gains are.
In RANs the users with the worst data rates are on the edge cells. That is, they are
107
close to the boarder between two cells and thus approximately equally far from two BSs.
Therefore, the UEs which have the worst data rate benefit the most from cooperation
because they usually have several BSs with similar average channel gain in range.
0
2
4
6
8
10
0 0.1 0.2 0.3 0.4 0.5
Mean factor between channel gains F
Cell edge fraction of UEs
δ= 5
δ= 2
Figure 6.7.: The average factor between channel gains for UEs at the cell edge is close to
1. When the fraction of UEs considered as belonging to the edge increases
the factor between channel gains Fincreases.
To get an understanding of the relevant sizes of Fconsider UEs on a line between two
BSs (as in figure 1.8). Figure 6.7 shows the factor between the average channel gains
Fdepending on the fraction of UEs considered to belong to the cell edge. It shows
that even for relatively large fractions of considered locations of UEs the average factor
between channel gains is low. It also shows that for higher path-loss exponents δthe
factor Fis higher for the same fraction.
Figure 6.8 illustrates the effect that instantaneous BSs selection is better than static
selection especially if the channel gains are similar. Additionally, it shows that dynami-
cally selecting 2 out of 3 BSs is nearly as good as always using 3 BSs in terms of outage
probability.
Figure 6.9 shows the effect of Theorem 6.1: Some schemes provide lower outage prob-
ability than others for all possible thresholds. Note that the dynamic association to 2
out of 3 BSs is closer to the static selection of 3 BSs than to the static selection of 2 BSs.
This shows that selecting 2 out of 3 BSs generates nearly the same outage probability
as always using all 3 BSs, but is better than statically selecting 2 BSs.
Figure 6.10 shows the difference in ERP between static and dynamic association be-
comes smaller the more different the channel gains of the cooperating BSs are. These
results match the results which determine the probability to select the correct BSs (fig-
ure 6.6). Note that figure 6.10 does not show anything about the efficiency of the
compared schemes because they have different outage probabilities. I relate the ERP to
108
0.0001
0.001
0.01
0.1
1
1 10
Outage probability O
Factor between channel gains F
Static 2
Dyn. 2
of 3
Static 3
Figure 6.8.: The outage probability of different cooperation schemes depends on the
factor between individual average channel gains F.
1e-05
0.0001
0.001
0.01
0.1
1
0.001 0.01 0.1 1 10
Outage probability O
Threshold Tγ
Static 3
Dyn. 2
of 3
Static 2
Figure 6.9.: Outage probability of static and dynamic association of 3 BSs depends on
the thresholds Tγat the UE.
109
0
0.1
0.2
0.3
0.4
0.5
1 10
Mean radiated power R
Channel gain factor between BSs F
Static 2
Dyn. 2 of 3
Figure 6.10.: The mean ERP of cooperative schemes with power control depends on
factors between the channel gain of individual BSs.
the outage probability in the next section.
6.6.3. Effective radiated power
The following plots show how the distribution of ERP to two different BSs influence the
outage probability if the total ERP is fixed.
Figure 6.11 shows that distributing the ERP between BSs can reduce the outage
probability. Note that this is in contrast to theorem 6.2 that distributes power optimally
if the instantaneous channel conditions are known. I have not been able to find a
closed form solution for the optimal ERP distribution without instantaneous channel
knowledge. However it is possible to determine the optimum numerically.
More power is radiated if a higher threshold has to be reached at the receiver. Because
a higher SNR at the receiver has to be reached. Figure 6.12 however, shows that when
the threshold reaches a certain level, further increasing the threshold reduces the mean
ERP, because the channel is in outage more and more often and no power is radiated in
this case. This illustrates that considering only the ERP without the outage probability
is not meaningful.
6.6.4. Relating ERP and outage probability
For each individual transmission scheme a trade-off between ERP and outage probability
exists. Finding the most energy-efficient scheme (in J/bit) is not helpful, because as the
most energy-efficient scheme will only transmit when the channel is extremely good. This
would result in unusably low data rates. Hence, I do not compare the energy efficiency
of the different cooperative schemes.
110
0.0001
0.001
0.01
0.1
0 0.2 0.4 0.6 0.8 1
Outage probability O
Fraction of power allocated to BS 2
Equal channel
Factor 10
Factor 100
Figure 6.11.: The outage probability of different cooperative schemes depends on the
distributions of ERP between the two available BSs. Here the average
channel gain of BS 1 is fixed and the average channel gain of BS 2 is lower
by a factor of 1 (10 and 100, resp.).
0
0.1
0.2
0.3
0.4
0.5
0.6
0.01 0.1 1 10
Mean radiated power R
Threshold Tγ
Static 1
Static 2
Dyn. 1
of 2
Figure 6.12.: The ERP of different cooperation schemes with power control depends on
the receive thresholds at the UE. Note that for high thresholds the channel
is in outage more often and the power control prevents all transmissions in
this case.
111
1e-05
0.0001
0.001
0.01
0.1
1
1 2 3 4 5 6 7 8 0
0.2
0.4
0.6
0.8
1
Outage probability O
Mean radiated power R
Number of cooperating BSs c
Power lower bound
Power upper bound
Outage
Figure 6.13.: The outage probability and mean ERP depend on the number of cooperat-
ing BSs.In this scenario dynamic association is used to determine the BSs
best suited for cooperation from a pool of 8 BSs with equal average channel
gain.
Figure 6.13 shows how outage probability and ERP are related. The outage proba-
bility can be decreased greatly by letting more BSs cooperate. Because dynamic asso-
ciation only uses the additional BS when necessary, the ERP is kept low. Additionally,
figure 6.13 illustrates the quality of the bounds for ERP.
Figure 6.14 summarizes both ERP and outage probability for a scenario with 10
BSs in the range of the UE. The static association with power adaption achieves the
lowest ERP because it only transmits if the channel is not in outage; but as its outage
probability is about 63% in case c= 1 it does not transmit most of the time. The static
association with power adaption and the dynamic association become more similar the
higher the allowed number of cooperating BSs gets and becomes the same at c= 10
because both select all 10 BSs to cooperate and determine the necessary ERP using the
on-off allocation.
I conclude that the use of instantaneous channel knowledge to select the cooperating
BSs provides a large reduction in ERP, while additionally using power control reduces
the ERP further but not by as much. Figure 6.14 also shows that a higher number of
cooperating BSs does not radiate much more power than non-cooperative transmission
if instantaneous channel knowledge can be used to control the ERP.
6.7. Conclusion
In this chapter, I provided the means to calculate the outage probability and ERP for
cooperative schemes with and without instantaneous channel knowledge. The results
allow me to quantify the gain of providing cooperative BSs with instantaneous channel
112
0.1
1
10
1e-05 0.0001 0.001 0.01 0.1 1
Mean radiated power R
Outage probability O
Static
Dynamic
Sta.+PC
Dyn.+PC
Higher c c = 1
Figure 6.14.: The different schemes for static and dynamic association have different
outage probabilities and ERP. In this scenario 10 BSs are available, which
all have an average channel gain of Tγ. The intervals mark upper and lower
bounds. The lines connect the different cooperative degrees of the same
scheme: static and dynamic association with and without power control
(PC). The cooperative degree starts at c= 1 on the right and increases to
the left by 1 for each drawn point.
knowledge. The formulas I provides can be used to quantify the different trade-offs
between ERP, outage probability, and number of cooperating BSs.
The on-off allocation and the evaluations show that using instantaneous channel knowl-
edge to select the cooperating BSs is similar to using it for power control because it only
makes a differences at one BS. I also showed that instantaneous channel knowledge pro-
vides the greatest gain over average channel knowledge if the average channel gains of
possibly cooperating BSs are the same.
Possible future work includes extending the two special cases (average channel knowl-
edge and instantaneous channel knowledge) to a continuum of cases, that is, quantifying
the effect of non-perfect and delayed channel knowledge. Moreover, it would be inter-
esting to determine the average factor channel gains of BSs Fexperimentally.
113
7. Network simulation
In the previous chapters, I determined the the latency and power consumption of radio
access networks (RANs) analytically. I analyzed effects auch as power-cycle durations
and cooperative transmission individually. The next step is to combine all effects in a
single model and determine how they interact. However, determining a solution to a
single model that contains all aspects is impractical. Therefore, I show the results of an
event-based simulation in this chapter.
The goal of this chapter is to determine if the methods to conserve power, which
I described in the previous chapters can be applied to a more realistic scenario. As
a realistic scenario I choose the dense urban scenario of the mobile working group of
the GreenTouch consortium. This scenario is based on the 3GPP E-UTRA simulation
model [3GP10]. Because I try conserve energy with a sleep mode, I consider only the
scenario with low load. In this scenario the potential gain from deactivating is greatest.
The work of this chapter was partially funded by the GreenTouch consortium. For the
work I present in this chapter Till Hohenberger supported me by implementing large
parts of the simulation. Additionally I thank Daniel Roeske for finding and removing
bugs in the simulator.
7.1. Introduction
In this chapter, I analyze how to decide which base stations (BSs) should be active. I
compare both local and central algorithms to make this selection. In contrast to related
work, I do not develop new heuristics, but take algorithms for which a analytic analyses
exists and apply them to a realistic simulation. These algorithms are in part those which
I explained in the earlier chapters.
I consider the local “accumulate and fire” algorithm and its variants (see section 2.3.3).
They consider power cycle durations, but have no notion of interaction between BSs. I
also analyze a set cover approach (see chapter 5), which considers interaction between
BSs but has no notion of power cycle durations. In the theoretical analysis of these
algorithms some effects were ignored (interaction of BSs and power cycles, resp.). This
means that assumptions of the theoretical analyses do apply here. However, the theo-
retical analyses provide hints at their performance.
In contrast to the theoretical analyses, I consider all these effects for all algorithms in
this chapter. The contribution of this chapter is to determine which of the algorithms
performs best in a realistic scenario. Additionally, I compare both the achieved data
rates and the energy consumption of scenarios with less macro BSs (see chapter 4).
114
7.2. Related work
Fehske et al. [FFMB11] describe the global energy consumption of RANs. Other au-
thors [CYZK11, HHA+11, HA13, MMSL11, OKLN11] provide overviews and explain
different ideas how to reduce the energy consumption of RANs. Others [ABH11, BZB10,
CZB10, HB11] categorize approaches differently (e.g., by network layer or needed BSs ca-
pabilities). All overview chapters describe the problems and possible approaches, while I
pick one possible approach (sleep mode) to reduce the energy consumption and compare
different methods to use it.
Cai et al. [CXY+03], Hossain et al. [HMJ11], Falconetti et al. [FHG13] and Niu et
al. [NWGY10] implement a scenario similar to ours in a simulator and compare different
versions of their own heuristics. In contrast to their work I compare theoretically well
understood algorithms to each other and to the optimum. This gives me confidence that
my algorithms also behave well in scenarios I have not simulated.
Other simulations consider only macro BSs [BEK+10] or more different types of
BSs [AGD+11, CFC11]. They determine the achieved data rates for different network
configurations. Their work does not consider how to determine which BSs to activate
and which to put to sleep.
McLaughlin et al. [MG11] determine which transmission modes for multi antenna
systems are more efficient and use only short sleep cycles (on time scales of milliseconds).
They determine the effect for only a single BS while I focus on the interaction of BSs.
Also, I use only a single mode and determine how to use this sleep mode with longer
sleep cycles (on time scales of seconds).
Others [SFdSF12, TGAA13] analyze similar models analytically. While their re-
sults are analytical instead of based on simulations, the models they use are simpler
than ours and for example do not include activation and deactivation times. Oth-
ers [LCB11, VDC+12] use such simplified models to formulate the problem as an integer
linear problem and solve it optimally. Because my model is too complex to find optimal
solutions I simulate heuristics instead.
Ericson [Eri11] determines the energy consumption of RANs analytically by estimating
the fraction of time a BS will spend in sleep mode for different protocols. Both Marsan et
al. [MC09, MCCM12, MM11, RRMF13] and Oh and Krishnamachari [OK10] determine
the fraction of time a BS can be switched off from the daily traffic patterns. In contrast
to this, I compare different algorithms which adapt the activity of BSs depending on the
channel quality to nearby user equipments (UEs).
Others [FMF10, MFF10, RFGB10, SER11] compare different strategies to place BSs
and different numbers of macro and pico BSs. They try to reduce the energy consumption
under maximum load (where a sleep mode is not needed), while I try to reduce the energy
consumption under low load in RANs where BSs need to be put into sleep mode. Son
and Krishnamachari [SK12] determine how distributing the load of BSs differently and
adapting the processing speed of BSs can reduce energy consumption. While they base
their energy-conservation method on speed scaling I use a sleep mode.
To reduce the total energy consumption of a RAN it is possible it reduce the en-
ergy consumption of each BS on its own. This can be done by changing the timing of
115
transmissions [HAH11, ZHS04], the number of active radio units [HA11, SDA+11], and
the interaction of sub-components of a BS [MA11]. Torrea-Duran et al. [TDD12] con-
sider adapting the transmission parameters to the channel conditions to conserve energy.
While all these methods also reduce the energy consumption of RANs, I consider the
possibilities on a higher layer of abstraction, where I coordinate the sleep modes of BSs
instead of reducing the energy consumption of each BS individually.
Further approaches to the problem of reducing energy consumption of RANs in-
clude: Auctions to distribute radio resources [CMP09] and mechanism design for pricing
schemes [CA11]. These two approaches assume that the users of a RANs are (greedy)
adversaries which do not agree on the best usage of the resources of a RAN. While they
seek ways to achieve an agreement, I assume this is dictated by the RANs. While I
only consider a single RAN, Ismail and Zhuang [Ism11] describe how different RANs can
cooperate to reduce the total energy consumption.
7.3. Model
My model is based on the 3GPP E-UTRA Model [3GP10] and is used in all system-
wide simulations of the mobile working group of the GreenTouch consortium. Refer
to Blume et al. [BAWB13] for a more detailed model description. This scenario is the
default scenario for dense urban (DU) areas with low load for the year 2020 of the mobile
working group of the GreenTouch consortium.
Macro BS
Pico BS
Figure 7.1.: The scenario consists of 7 macro BSs and two pico BSs per macro BS sector.
7.3.1. BS deployment
The model consists of a scenario of seven macro BSs with three sectors each; Figure 7.1
illustrates this deployment. Cells look slightly different due to antenna lobe considera-
tions in the simulation and due not have sharp edges due to shadowing. If nothing else
116
is said, I model 2 pico BSs per sector. They radiate and consume less power and have
only a single omnidirectional antenna.
The macro BSs are placed in a hexagonal deployment (figure 7.1). Each sector covers
one third of the BS’ coverage. Both scenarios, DU and sparse dense urban (SDU), consist
of 7 macro BSs; the distance between the macro BSs is 500 m in DU and 1000 m in SDU.
The pico BSs’ deployment depends on the positions of hotspots. At first, each hotspot
will be covered by one pico BS. The remaining pico BSs will be distributed uniformly at
random, equally into each sector of a macro BS.
7.3.2. UE deployment and load generation
I model only active UEs and only downlink data transfers, because they will generate
most traffic in a real network (caused by internet video [Cis13]). Hence, I do not model
signaling traffic in this chapter. I consider all UEs to be stationary during data transfers.
The model consists of a fixed number of hotspots. The positions of hotspots are inde-
pendently and randomly selected with equal probability for each location. Two thirds
of the total UE placed at hotspots with a 2-dimensional Gaussian distribution (indepen-
dent multivariate normal distribution with standard deviation σx=σy= 100 m). The
remaining traffic is generated by a homogenous Poisson process. The DU scenario has a
size of 2.16 km2(see figure 7.1) with two hotspots, and 8.66 km2with eight hotspots for
the SDU scenario.
The density of the Poisson process that creates the UEs demands is constant over
time. To reflect the changing load over the course of a day, I let the simulation run
with different densities, but do not change it during a single run. UEs demand a 2 MiB
large transmission. If the transmission is not finished within 4 seconds they abort the
transmission. These values are used, for example, by the GreenTouch consortium which
based its decision on the 3GPP simulation parameters [BAWB13].
7.3.3. Radio model
To determine the average channel quality between a BS and UEs I consider the path-
loss based on distance, the antenna gain (based on both horizontal and vertical angle),
log-normal shadowing with a 0.5 correlation from a UE to all BSs, a constant wall-
penetration loss (to compensate for walls which I do not model explicitly), receiver
noise, and interference from all other transmitting BSs of the same type.
The decision whether a UE can be served from a BS depends not only on the channel
between them but also on the interference from other BSs and the number of other UEs
which are also connected to the BS. For this purpose I assume every BS has constant
radiated power, if it transmits to at least one UE and zero radiated power otherwise.
To determine the achieved data rate from the mean signal-to-interference-(plus-)noise
ratio (SINR) I use a look-up table. It was generated based on detailed channel models
by members of the GreenTouch consortium, but is not publicly available. It includes
multiple-input and multiple-output (MIMO) gain and fading effects. For modeling ab-
stractions I denote it as a function dr mapping SINR to data rate. If nUEs are connected
117
to a BS each gets 1/n-th of the radio resources and, thus, 1/n-th of the data rate it would
get if it were the only UE at the BSs.
To prevent border effects I implemented a wrap-around. That is, assuming the hexag-
onal deployment of BSs is infinite, each UE considers the 7 closest BSs. I map the 7
closest BSs to the 7 BSs in my model.
7.3.4. Power model
Each BS consumes a constant idle power (macro: 132 W, pico: 4 W) when it is active
and linearly scales its power consumption in the load to maximum load (macro: 638 W,
pico: 11 W). Mean is the fraction of power consumed when idle is P ≈ 0.21 for macro
BSs and P ≈ 0.36 for pico BSs. The power profiles of the BSs are not assumed to be
binary in this simulation, because BSs in the year 2020 are assumed to be closer to the
linear power profile.
Note that pico BSs always transmit at full power or not at all, which makes their
power profile binary. Because the same holds for each of the three sectors of a macro
BSs their load changes in steps of one third of the maximum load. I assume the time
it takes to change a BS from activated to sleep A(and the other way Z) is constant
(1 s) if not specified otherwise. I assume BSs do not transmit during activation and
deactivation and, thus, also do not create interference.
7.3.5. Cooperative diversity
In my model I allow BSs to cooperatively transmit to UEs to increase the data rate. I
extended the GreenTouch model by cooperative diversity to determine its effects. When
two or more BSs cooperate they transmit the same data to the UE. I assume the BSs are
synchronized so that the signal at the UE constructively interferes. Hence, the received
signal of a cooperative transmission is equivalent to a single transmission with a signal
that has the sum of all signal strengths. This is an abstraction of joint processing
used in Coordinated Multipoint transmission of Long Term Evolution Advanced (LTE-
A) [SK10], which is an implementation of coherent combining [Gol05].
I assume that cooperation can only be done with BSs of the same type, because pico-
and macro BSs operate on different frequencies. As the cooperating BSs can assign differ-
ent fractions of their radio resources to a UE I have to determine the final data rate. Let
C={BS1,...,BSn}denote the set of cooperating BSs and F={f(BS1), . . . , f(BSn)}
the set of fractions of the radio resources these BSs assign to a given UE. Let min
i(K)
describe the i-th order statistic (i.e, the i-th smallest element of a set K) and define
min
0(K) = 0 for all K. The function dr maps SINR values to data rate (using the table
described above). The data rate DDwhen using cooperation is computed as:
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d=
n
X
i=1
Fraction of time exactly iBSs cooperate
z }| {
min
i(F)−min
i−1(F)·dr
Sum SINR of i cooperating BSs
z }| {
P
f(i)≥min
i(F)
Signal(i)
N+P
f(i)<min
i(F)
Signal(i)
| {z }
Interference
.(7.1)
For two signals with the fraction of radio resources the BSs provide F1> F2this
simplifies to:
d=
Only BS 1 transmits
z }| {
(F1−F2)dr Signal(1)
N+ Signal(2) + Other signals+
Cooperation of BS 1 and 2
z }| {
F2·dr Signal(1) + Signal(2)
N+ Other signals .
(7.2)
Fraction of allocated resources f
BS 1 BS 2 BS 3
Signal a
Signal b
Signal c
0
1No cooperation:
Signal a
Interference b+c
2 cooperating BSs:
Signal a+b
Interference c
3 cooperating BSs:
Signal a+b+c
Figure 7.2.: Combining of cooperative transmissions results in a higher data rate than
using only a single transmission.
This means that each BS cooperates as much as it can (depending on the fraction of
time it is working on this transmission) with the others. Figure 7.2 illustrates how the
fractions of the cooperating BSs determine the final data rate. While this model is more
complex than needed for the current simulations (with maximum of two cooperating BSs)
it allows future simulations to take advantage from more complex forms of cooperation
(with more than two cooperating BSs).
Reducing the effective radiated power (ERP) as described in chapter 6 is only partially
considered in this chapter. The function dr that maps the SINR to the data rate includes
119
effects such as MIMO transmissions and multi-user diversity. This includes transmitting
data over the best channel and not transmitting over bad channels. However I do not
consider the effect of reduced ERP as reduced interference. To do this it would be
necessary to simulate the instantaneous channel conditions, which would increase the
time need to run the simulations.
7.4. Algorithms
This section introduces the algorithms that choose which BSs will be activated and which
will be deactivated or stays unchanged. In all simulations, I do not change the state of
macro BSs dynamically to guarantee that signaling traffic can be received.
The theoretical analyses of the algorithms have no notion of overlap of cells or power-
cycle durations which need to be considered in reality. For each algorithm I describe
how I implemented the algorithm to incorporate these effects. While these changes
would make the theoretical analysis far more complex their implementation changes
only slightly and I can determine the results from my simulation.
The strategies are split into two main parts. (1) Making the (de-)activation decision
(called policy); (2) assigning the UEs to BSs (called scheme) when a UEs creates a new
demand or a BS changes its state (i.e., creating handovers).
For the association scheme I only simulate a greedy approach that assigns each UE to
the active BS with the highest SINR. Handovers are possible to change the assignment
if a BSs with a higher SINR becomes active.
To decide which BSs cooperate I determine the SINR between the best BS and the
UE. If the second best SINR is less than a fixed factor (default value 2) worse (I call this
coop factor), these two BSs cooperate. Formally: cfmin
n−1(SINR) ≥min
n(SINR), where cf
is the coop factor. Furthermore, the second best BS will only join the cooperation if it
would be idle otherwise. This means that I do not activate BSs specifically to cooperate
but only cooperate when a cooperation partner happens to be active.
Next I describe the (de-)activation policies I compare.
7.4.1. Always on and always off
The simplest policies always-on and always-off keep all pico BSs switched on or off,
respectively. The UEs will be assigned to the BS (macro or pico) with the highest SINR.
Because dr is monotonic this is equivalent to deciding by data rate. I denote Always-
on by OnP and Always-off by OffP. I also used the always-on policies as a reference
in previous chapters. In earlier chapters the always-off policy was not feasible because
it would not have served any requests. However, because I only apply the activation
policies to pico BS, macro BS can still sever requests.
Osagami [Oso05] describes the analytical background for the queuing effects of strate-
gies in which idle BSs (servers) help others to process their jobs without the use of sleep
modes. While Osagami’s analysis is helpful to understand the fundamental interactions
120
of load sharing, the methods are too complex to be used in scenarios of larger scale.
Hence, I compare it to others in my simulation.
7.4.2. Greedy and accumulate & fire
The greedy policy GP activates a pico BS Bas soon as it is the BS with the highest
SINR for a UE that demands a transmission. I assume I can estimate SINRs even for
BSs in sleep mode. Once no UEs are assigned to a pico BS, it is deactivated immediately.
As BSs need time to activate, during activation a UE is assigned to the already active
BS with the highest SINR and is reassigned once the activation has finished. Once Bis
activated, all UEs best served by Bwill be assigned to it. Note that the all assignment
orders result in the same assignment. In case that Bfinishes its activation and no UE
will be optimally served by Bit will deactivate again immediately. The deactivation is
delayed if the BS can support another BS via cooperative transmission.
Generalizing the greedy policy leads to the accumulate and fire policy AP(k) [CX07].
Using this policy, the pico BS only activates if the on threshold noted as kof UEs which
would be assigned to it if it were active is exceeded. Once there is no UE connected to
a pico BS it will deactivate. I denote it by AP(k). The accumulate and fire policy can
cause cascading effects: when a BS finishes its transmission, the interference of other
channels changes. Other UEs are reassigned and other BSs are (de-)activated. Note that
the greedy policy, described earlier, is just the special case AP(1). Both the greedy policy
and the accumulate and fire policy are direct applications from policies from chapter 2.
In comparison to the always on/off strategies, the accumulate and fire policy tries
to make better use of the knowledge about active UEs but coordinate the activity of
neighboring BSs. Hence, a BS can be activated only to discover that the UE which it
was activated to serve has already been served by another BS. This can lead to increased
energy consumption.
7.4.3. Set cover
Another approach to decide on (de-)activation decisions is to understand it as a set cover
problem. That is, I look for the smallest subset of all BSs that is able to serve all UEs.
I use the standard notation of set cover problems as used by Vazirani [Vaz01]. I have
a universe Uof UEs as well as a set of covering sets Sof pico BSs. While Usimply
contains all active UEs, the set for a BS in Sis built by analyzing which of the UEs in U
can be served by which BS, so that the data rate requirement of the UE can be fulfilled.
In terms of activity of BSs that means, I search for the smallest set of active BSs such
that each UE achieves at least an given SINR.
To keep the set cover approach as simple as possible, I define a threshold of SINR
over which I consider the connection as valid. During this calculation I do not update
the interference calculation because this would change the possible associations and has
undetermined effects for the set cover algorithms. While the activation algorithm ignores
these changes in interference they are of course considered during simulation to compute
the resulting download times.
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I determine a solution to the set cover problem using a greedy approach [Vaz01]. Since
the macro BSs are able to serve UEs as well, it is unnecessary to cover all UEs with pico
BSs. Hence, I stop the set cover approach once the first 90% of all UEs are connected.
I selected 90% as the remaining 10% can be served by the macro BSs in my model. For
other scenarios this value has to be newly estimated. Similar to the accumulate and fire
policy, a UE is assigned to the best active BS during state changes of BSs. I use SP(C)
to describe the set cover algorithm based on the currently active UEs. The SP(C) policy
one possibility to solve the problem of covering all UEs, which I analyzed in chapter 5.
As the set cover approach has no notion of activation times, the solution it determines
may be outdated when the BSs reach their states. In addition, the set cover approach
ignores the fact that the data rates of UEs share BSs decreases. These effects may lead
to higher energy consumption and download times, which I quantify in my simulation.
Variants The set cover policy can be varied by changing the algorithm that determines
the solution to the set cover problem (only greedy in my case) and the elements of the
sets to cover (i.e., the contents of the sets U, S). The algorithm described above considers
currently active UEs. Because this can change very fast, while the BSs might not be
able to react that fast, it can be useful to use the set cover to cover the average number
of UEs in a given area. In effect, this approach tries to cover areas where many requests
are generated and does not try to adapt to the currently active UEs. The idea behind
this variant is that it results in fewer power cycles and, thus, can better cope with long
power cycles.
In this variation I split up the whole coverage area into smaller parts, each covered by
a subset of BSs. Uthen represents the average number of UE arrivals in one of these
areas. The subsets of BSs, able to serve UEs in each particular area, are defined as S.
I use SP(A) to refer to the set cover algorithm using the average UE arrival to select
the active BSs.
7.5. Results
In this section, I present the results of simulating the algorithms using the model de-
scribed earlier. To do so, I implemented an event-based simulation in OMNeT++.
I assume the state change durations Aand Zare 1 second unless further specified. I
arbitrarily selected this value but determine the results of my simulation for other values
as well.
The following figures show the mean power consumption of all macro and pico BSs
in watts on the X-Axis in relation to the mean download duration in seconds on the Y-
Axis. Each point in the plot, stating power consumption and download duration, consists
of two 95% confidence intervals for power and download duration (without Bonferroni
correction). I consider a policy to be good if it results in low download times and low
power consumption.
Figure 7.3 shows how the different strategies trade off the mean download duration
and total power consumption. As expected, the always-on policy OnP results in the
122
0
0.5
1
1.5
2
2.5
3
3.5
1300 1400 1500 1600 1700 1800 1900 2000 2100
Mean download duration L (s)
Mean power P [macro+pico] (W)
OffP
OnP
GP
AP(5)
SP(C)
SP(A)
Min. power
7 macros
Figure 7.3.: Comparing policy performance for 20% of the average load, dense urban
(DU) deployment, 42 pico BSs, 7 macro BSs, pico BSs in hotspots and
random deployment, and 2 hotspots.
0
1
2
3
4
5
1300 1400 1500 1600 1700 1800 1900 2000 2100
Mean download duration L (s)
Mean power consumption P [macro+pico] (W)
GP = AP(1)
AP(20)
Min. power
7 macros
Figure 7.4.: Comparing accumulate and fire thresholds kfor DU deployment, 42 pico
BSs, 7 macro BSs, pico BSs deployed in hotspot and random deployment, 2
hotspots.
123
lowest download duration. However, the always-on policy consumes less power than the
always-off policy. The reason for this is that the pico BSs can transmit the requested data
to the UEs more efficiently than the macro BSs. In this scenario the greedy policy GP
and the always-on policy OnP provide the best trade-offs between power consumption
and download duration. The always-off policy OffP, on the contrary, has the highest
download duration and highest power consumption. This shows that with these power
consumption parameters it is not generally a good idea to disable as many pico BSs as
possible and off-load the traffic onto the macro BSs.
As already determined analytically in section 2.6, figure 7.4 illustrates that higher acti-
vation thresholds in the accumulate and fire policy AP(k) increase the latency. However,
here a higher threshold also increases power consumption as requests are not waiting,
but assigned to a macro BS which serves the requests and consumes more power than the
pico BS. Therefore, I will compare different ideas how to reduce the power consumption
of macro BSs next.
0
1
2
3
4
5
6
7
0 500 1000 1500 2000 2500
Mean download duration L (s)
Mean power per unit area Q [macro+pico] (W)
OffP
OnP
GP
SP(C)
OnP
GP
SP(C)
SDU
DU
Min. power
SDU
Min. power DU
Figure 7.5.: Comparing different strategies in SDU and DU deployment, 42/168 pico
BSs, 7 macro BSs, pico BSs in hotspots and random deployment, and 2/8
hotspots.
Figure 7.5 shows how the algorithms perform if the distance between macro BSs is
doubled. Using the sparse deployment SDU, the download durations using the always-off
policy and set cover policy SP(C) are higher but the greedy policy GP and the always-on
policy OnP fulfill the demands of the UEs with the pico BSs nearly as good as the using
the normal spacing. Important here is that all policies consume less power in the sparse
SDU placement than in the normal placement DU. However, as it would not be possible
to change the spacing at after construction of the RAN, it is necessary to determine if
it also can support high load. Figure 7.6 shows that the sparse SDU deployment has
longer download durations than the normal DU deployment. Next, I will explain why
cooperation does not reduce the data rate, but increases the energy consumption in this
124
0
10
20
30
40
50
0 500 1000 1500 2000 2500 3000 3500 4000
Mean download duration L (s)
Mean power per unit area Q [macro+pico] (W)
NoCoop Coop
NoCoop Coop
SDU
DU
Min. power
SDU Min. power
DU
Figure 7.6.: Comparing the always-on policy OnP for 140% of the average load in SDU
and DU deployment, 42/168 pico BSs, 7 macro BSs, pico BSs in hotspot
and random deployment, and 2/8 hotspots.
scenario.
In contrast to the analysis of chapter 5 I assume the request size to be constant
instead of the duration (this is a good model for bulk transfers, while the other is a good
model for phone calls). Also cooperation has the following effects: (1) the SINR the UE
receives is higher and thus the download duration is lower and (2) as the second BS that
cooperates has a lower signal its energy-efficiency is lower than the first cooperating BSs.
Thus, data is transmitted at a lower mean efficiency and the total energy consumption
is higher. The time gained by faster downloads does not allow conserving the energy
additionally spent using cooperation. Figure 7.6 shows that when BSs cooperate the
power consumption increases from the increased amount of transmitted data, but the
download durations do not decrease as the cooperation partners cannot increase the data
rate significantly.
I also simulate scenarios in which I deactivate either 1 or 6 of the macro BSs and keep
them in place in figure 7.7. This allows turning them on again when necessary during
high load and still conserves power during times of low load. Deactivating macro BS
is one of the methods to conserve power which I describe in chapter 4. While it does
not recreate a hexagonal deployment it is also an example for re-tiling (see figures 5.9
and 5.10). Using the exact method described earlieer is not possible in a scenario with
only 7 BSs. Note that I do not use the activation and deactivation strategies for the
macro BSs but keep them in a single state during a simulation run. Figure 7.7 shows
that deactivating macro BSs conserves large amounts of power while the pico BSs can
help to provide data to the UEs.
I compare the results for different power-cycle durations in figure 7.8. It shows that
faster power cycles lead to both lower power consumption and lower download durations
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0
1
2
3
4
5
6
0 500 1000 1500 2000 2500
Mean download duration L (s)
Mean power P [macro+pico] (W)
OnP
GP
OffP
OnP
GP
SP(A)
OffP
OnP
GP
SP(A)
Min. power
1 macro
Min. power
6 macro
Min. pwr.
7 macro
1 Macro
6 Macro
7 Macro
Figure 7.7.: Comparing performance when varying the number of macro BSs; 20% of the
average load, DU deployment, 42 pico BSs, pico BSs in hotspot and random
deployment, and 1 hotspot per sector. Always-off policy with one macro BS
has a mean download duration of about 5 s and is clipped from the plot.
0
0.5
1
1.5
2
2.5
3
3.5
0 10 20 30 40 50 60 1300
1400
1500
1600
1700
1800
1900
2000
2100
2200
2300
Mean download duration L (s)
Mean power P [macro+pico] (W)
Startup and shutdown time A=Z(s)
Min. power
7 macro
Figure 7.8.: Deactivation times for 20% of the average load in DU deployment, 42 pico
BSs, 1 macro BSs, pico BSs in hotspot and random deployment, 2 hotspots,
with the greedy policy GP.
126
0
0.5
1
1.5
2
2.5
3
3.5
1300 1500 1700 1900 2100 2300
Mean download duration L (s)
Mean power P [macro+pico] (W)
0.01s
1s
10s
60s
OffP
OnP
GP
SP(A)
Min. power 7 macros
Figure 7.9.: Deactivation durations for 20% of average load, DU, 42 pico BSs, 1 macro
BSs, pico BSs in hotspot and random deployment, and 2 hotspots. Durations
are set for both activation Aand deactivation durations Z.
0
0.5
1
1.5
2
2.5
3
3.5
1300 1400 1500 1600 1700 1800 1900 2000 2100
Mean download duration L (s)
Mean power P [macro+pico] (W)
0
20
40
80
160
GP
SP(A)
Min. power
7 macros
Figure 7.10.: Comparing different number of pico BSs for DU deployment, 1 macro BS,
pico BSs deployed in hotspot and random deployment, and 2 hotspots.
127
with the greedy policy. Figure 7.9 shows the same information, and additionally adds
the SP(A), the always-on, and always-off policies as a comparison. The set cover policy
based on average arrival rates SP(A) is better suited to cope with high power cycle
times than the greedy policy GP, but only for low power-cycle durations does the greedy
policy consume less power than the always-on policy. This shows that it is important
to determine and reduce the power-cycle durations of future BSs.
Figure 7.10 shows that increasing the number of pico BSs can reduce both the power
consumption as well as the latency. This is an interesting effect, but it is necessary to
determine if the modeling assumptions still hold in scenarios with a high number of pico
BSs and how it interacts with fewer macro BSs.
7.6. Conclusion
I described different theoretically analyzed algorithms to deactivate BSs in RANs and
simulated them in a realistic scenario. My results show that statically deactivating and
sparser deployment of macro BSs can conserve power and still provide high data rates.
An important parameter which describes the trade-off between power and data rate is the
power-cycle duration of a BS. Low power cycle times reduce both power consumption
and download durations. In contrast to the previous chapter, cooperation does not
reduce the power consumption in this chapter as I compare them with strategies with
the same deployment of BSs just without cooperation.
I conclude that activating and deactivating BSs can conserve power in realistic sce-
narios. Deactivating macro BSs reduces the power consumption more than any strategy
to deactivate pico BSs. Future work will need to include other scenarios as well as other
deployment strategies for pico BSs. Especially interesting are scenarios with a higher
number of pico BSs. Assigning UEs to BSs not based on signal-to-noise ratio (SNR)
but on energy efficiency is a potential next step. It is also necessary to determine the
power-cycle duration and power profiles of future BSs.
128
8. Final thoughts
In this chapter, I first summarize the content of this dissertation. I then outline which
future work is necessary to continue my work to reduce the energy consumption of radio
access networks (RANs). In the end, I present a final conclusion.
8.1. Summary
In my dissertation I compared ways to reduce the energy consumption of RANs on the
network level. To conserve energy, I considered using fewer base stations (BSs) and
deactivating idle BSs. Deactivating idle BSs reduces energy consumption because their
power consumption is still considerable when idle. To deactivate BSs, I assumed BSs
can be put into a low-power sleep mode.
The large difference in required radio resources between signaling and data traffic
motivated the idea of a RAN in which signaling and data traffic are split. Splitting the
traffic allows large-range macro BSs to detect the requests of user equipments (UEs) and
to activate, on demand, low-range pico BSs, which serve the requests. Hence, the split
of signaling and data traffic allows the energy consumption of RANs to be adapted to
the load.
In chapter 2 I analyzed an abstract queuing system of a single server with a sleep mode
on its own. I quantified the effect of power-cycle durations on energy consumption and
latency. Moreover, I analyzed the trade-off between energy consumption and latency for
Poisson arrivals and the competitive ratio for worst-case arrivals.
When considering a single server, the latency increases approximately linearly with the
power-cycle durations. The energy consumption asymptotically approaches its maximum
energy consumption when the power-cycle durations increase. While this is intuitively
clear, I provided an analytic derivation for all power-cycle durations. It shows that
considering each individual BS with a sleep mode only reduces the energy consumption
if the power-cycle durations are low.
In chapter 3 I showed that considering a network as a whole (instead of each device
individually) allows the energy consumption to be reduced even if power-cycle durations
are high. To illustrate the idea, I used an example of wired networks to express the
general idea of network-level energy-saving methods without having to consider the
complex effects of wireless transmissions. For the rest of the dissertation, I applied
the idea of network-level energy saving to RANs.
Both for signaling and data BSs, increasing the range allows having fewer active BSs
per area and thereby conserving energy when the load is low. One technique to increase
the range is to use cooperative transmissions from several BSs to a single UE. This allows
129
the BSs to reach a UE that is not in range of any individual BS. This requires more
energy per transmission but allows deactivating of other BSs. Because the energy offset
for deactivating a BS under low load is higher than the additionally consumed energy,
this will in total conserve energy.
I described how much energy can be conserved in RANs when BSs cooperatively
transmit to extend their signaling range in chapter 4. I analyzed the effect for different
path-loss exponents and a varying number of cooperating BSs. The analytical results
show that the area a BS covers can be significantly increased when cooperative transmis-
sions are introduced. Moreover, scenarios with a low path-loss exponent are best suited
for cooperation. I concluded that it is reasonable to let a few BSs cooperate to detect
requests of UEs to reduce the overall energy consumption of RANs.
Cooperative transmissions can also be used to reduce the number of active pico BSs
which serve the data traffic. I analytically quantified the reduction in activity when
pico BSs can cooperate in chapter 5. Reducing the fraction of time BSs are active
reduces energy consumption. Placing the BSs at the optimal distance for cooperation
additionally reduces energy consumption.
When BSs cooperate to serve data to a UE, more BSs will be transmitting and, thus,
create more interference for other UEs. The interference can be dedreased by selecting
which BSs actually transmit data to a UE based on instantaneous channel knowledge
instead of average channel knowledge. This allows the effective radiated power (ERP)
to be at nearly non-cooperative levels and have the outage probability of cooperative
transmissions. The greatest gain (both in terms of outage probability and ERP) from
using instantaneous channel knowledge is achieved when the average channel gains of all
possibly cooperating BSs are the same. From chapter 6 I conclude that using cooperative
transmission based on instantaneous channel knowledge nearly removes the drawbacks
that cooperative transmissions introduce.
In chapter 7 I developed an event-based simulation of a RAN based on the specifi-
cations agreed upon within the mobile working group of the GreenTouch consortium.
I used this simulation to analyze the behavior of a RAN when all the effects, which I
studied analytically earlier, interact. The policies to conserve energy have no notion of
some effects (e.g., power-cycle durations or BS interaction), but have to cope with them
in the simulation. The results show that they are able to conserve significant amounts
of energy while keeping the quality of service high.
8.2. Future work
The simulation I described in chapter 7 only describes a single scenario (a dense urban
environment under low load). While this is the scenario with the greatest potential to
conserve energy, it is necessary to consider other scenarios as well. For example in rural
areas, where most BSs are needed to cover the area, conserving energy by deactivating
BSs is harder. One possibility is to use the cooperative techniques, which I described
in chapter 4, to increase the spacing and reduce the energy consumption. My work
only showed the general applicability and potential gains, but this needs to be tested in
130
realistic environments. In general, the simulation should be extended to include signaling
traffic and movement of users.
I did not analyze uplink transmissions and interactive traffic. While both of them are
not as prevalent as video streaming, RANs have to support them. Because they have
stricter requirements a detailed analysis for these types of traffic is needed.
Traffic with less strict requirements (e.g., updates) can be processed when this is
possible with less energy. This introduces additional complexity because the UEs and
the RANs have different interests but need to coordinate the transmissions anyway.
As both the theoretical analysis (chapter 2) and the simulation (chapter 7) showed, the
power-cycle duration is an important parameter to determine the energy consumption of
a RAN. Therefore, it is important to have good predictions of the power-cycle durations
of future BSs. Also reducing the power-cycle durations will reduce the energy consump-
tion and increase data rates of the RANs. In addition to determining the power-cycle
durations, the power profile of future BSs are important. Power profiles which are closer
to the linear profile can be combined with BS deactivation. Hence, it is necessary for
future work to determine the power profile of future BSs and also reduce it.
To make the simulations more realistic a more detailed user model is needed. Because
user distributions are usually only approximated by a Poisson process, a deeper analysis
of real user locations is needed. This can for example be achieved by a more detailed
model of the hotspots and studies to determine their sizes and traffic in reality. This is
also valid for the very simplified model of demands that I used in chapter 7.
Moreover, it is necessary to define the interfaces and protocols which BSs can use to
(de-)activate each other. Because potentially many BSs will be deactivated most of the
time it is necessary that the activation methods do not consume much energy. This is
also valid for the backbone network which the BSs might use to transmit the activation
signals.
While cooperative transmission from BSs are part of the Long Term Evolution Ad-
vanced (LTE-A) standard, it was not designed to reduce the energy consumption. To
use cooperation to reduce the energy consumption it is necessary to allow BSs to go into
sleep modes. The protocols to allow this need to be developed.
As I showed in chapter 6, selecting the cooperating BSs based on instantaneous channel
conditions enables a reduction of the radiated power. Radiated power becomes inter-
ference at other receivers. With less interference, transmissions are finished faster and
thus consume less energy. Hence, future work is needed to make sure the transmissions
of BSs can quickly adapt to changing channel conditions.
8.3. Conclusion
There is no single method to reduce the energy consumption of RANs to the minimum.
A mix of methods to conserve energy on different levels is needed. These include reducing
the maximum power consumption of BSs, making the power profile of BSs more linear,
adapting the activity of BSs to the load in the network and implementing low-power
signaling.
131
A RAN that is based on a new technology can be both more energy-efficient and
provide a higher quality of service (QoS) to the end user. But in a given system there
is usually a trade-off between energy consumption and QoS. Also, a system that is very
energy-efficient but cannot provide the needed service to the end users is not very useful.
Hence, it is not only necessary to understand the behavior of RANs but also what the
users demand. Examples are: daily traffic patterns, locations of hotspots, and the mix
of voice and data traffic.
A very important factor that will influence both the QoS and energy consumption of
RANs is the power-cycle duration. For low power-cycle durations a simple greedy on/off
policy provides good results. Deactivating macro BSs during low load is more important
than deactivating pico BSs. However, for future BSs neither their power-cycle durations
nor their power profiles are known. This information is necessary, if one wants to make
informed decisions about methods to conserve power.
132
A. Proofs for the stretch metrics
In this chapter, I describe the technical details of the proofs referenced in chapter 3.
Because all sums, products, maxima, minima, and averages in this chapter are over
“d∈D” I will drop them for readability. To further improve readability I drop the
“(d)” after x,yand φ. I use the following short-hand notations to reduce the number of
indexes used in the proofs:
•x(d) := LC(d),
•y(d) := LCL(d), and
•max(x) := max
d∈D(x(d)).
To follow the proofs it is helpful to consider xand yas vectors containing the latencies
of the demands in the two considered configurations.
A.1. Equality of stretch metrics for the geometric mean
The two metrics SGS and SSG are equal (referred to in section 3.3.3):
SGS = geo x
y
=Y(x/y)φ1/Pφ
=Qxφ
Qyφ1/Pφ
=Qxφ1/Pφ
(Qyφ)1/Pφ
=geo(x)
geo(y)=SSG.(A.1)
A.2. Possible orders of metrics
In this section, I present examples using which each possible order of the stretch metrics
is achieved. I refer to them in section 3.4.1. I denote the examples for the latencies using
the latency-minimizing configuration L(CL) and the latencies using energy-conserving
configuration L(C) as [L(CL)] →[L(C)].
Possible orders with two demands:
133
•SSM > SAS > SSA: [3,2] →[3,7],
•SSM > SSA > SAS: [2,1] →[3,1],
•SAS > SSM > SSA: [2,1] →[2,5], and
•SAS > SSA > SSM: [2,1] →[2,2].
Orders in which SSA > SSM and SSA > SAS (which include SSA > SSM > SAS and
SSA > SAS > SSM) are not possible with only two demands. I prove this in the next
section.
The last two possible orderings are (with 3 demands):
•SSA > SSM > SAS: [1,2,7] →[1,3,9] and
•SSA > SAS > SSM: [1,2,5] →[1,3,6].
A.3. Impossible orders
There is no configurations in which SSA > SAS and SSA > SSM hold at the same time
with exactly two unweighted demands when all values are positive. I refer to this in
section 3.4.1. This includes the two orders SSA > SSM > SAS and SSA > SAS > SSM.
Proof. Assume a configurations for two demands exists in which SSA > SAS and SSA >
SSM holds. Thus, the two equations
SSA > SAS ⇔x1φ1+x2φ2
y1φ1+y2φ2
>x1φ1
y1(φ1+φ2)+x2φ2
y2(φ1+φ2)(A.2)
and
SSA > SSM ⇔x1φ1+x2φ2
y1φ1+y2φ2
>max(x1, x2)
max(y1, y2)(A.3)
must hold. Without loss of generality I assume x1≥x2and look at the three cases:
1. y1=y2: Simplifying SSA > SAS gives 0 >0
2. y1< y2: Simplifying SSA > SAS gives x2y1
y2> x1, which leads to
x2≥x2
y1
y2
> x1≥x2 (A.4)
3. y1> y2:
a) Simplifying SSA > SAS gives x2y1
y2< x1and
b) Simplifying SSA > SSM gives x2y1
y2> x1
134
A.4. Bounds between metrics
In this section, I prove that the bound
A≤skew(C) ·skew(CL)·B(A.5)
holds for all combinations of Aand Bfrom the five metrics. I improve the bound where
possible. I refer to the bounds in section 3.4.1. The following bounds hold for strictly
positive values (which is a useful assumption for latencies):
max(x)
max(y)≤max x
y(A.6)
min(x)
min(y)≤max x
y(A.7)
max x
y≤max(x)
min(y)(A.8)
min(x)
max(y)≤min x
y.(A.9)
The following proofs either use transitivity to directly show the result or are proofs
by contradiction. The proofs of contradiction start by assuming the opposite is true and
construct an inequality chain from it which leads to a contradiction.
A.4.1. Maximum of stretches
I start by showing that the maximum of the stretches SMS is always higher than the
other four metrics.
Proof of SSM ≤SMS,Assume: SMS < SSM
max(x)
max(y)
A.6
z}|{
≤max x
y<max(x)
max(y) (A.10)
Proof of SAS ≤SMS,Assume: SMS < SAS and define z:= max x
y
z= max x
y<Pxφ
y
Pφ≤Pzφ
Pφ=zPφ
Pφ=z (A.11)
Proof of SSA ≤SMS,Assume: SMS < SSA and define z:= max x
y
z= max x
y<Pxφ
Pyφ
x≤zy
z}|{
≤Pzyφ
Pyφ =zPyφ
Pyφ =z (A.12)
Note that this one half of Cauchy’s Third Inequality [Ste04].
135
Proof of SGS ≤SMS,Transitivity of SGS ≤SAS ≤SMS
A.4.2. Stretch of maximum
Now I show that A≤skew(y)SSM holds for any of the metrics AI described.
Proof of SMS ≤skew(y)SSM,Assume: skew(y)SSM < SMS
max(x)
min(y)=
skew(y)
z }| {
max(y)
min(y)
SSM
z }| {
max(x)
max(y)<max x
yA.8
z}|{
≤max(x)
min(y) (A.13)
Proof of SSA ≤skew(y)SSM,Transitivity of SSA ≤SMS ≤skew(y)SSM
Proof of SAS ≤skew(y)SSM,Transitivity of SAS ≤SMS ≤skew(y)SSM
Proof of SGS ≤skew(y)SSM,Transitivity of SGS ≤SAS ≤skew(y)SSM
A.4.3. Stretch of average
Now I show the inequalities with SSA on the right side.
Proof of SMS ≤skew(x)skew(y)SSA,Assume: skew(x) skew(y)SSA < SMS
max(x)
min(y)=max(x)
min(x)
max(y)
min(y)Pmin(x)φ
Pmax(y)φ≤max(x)
min(x)
max(y)
min(y)Pxφ
Pyφ
<max x
yA.8
z}|{
≤max(x)
min(y) (A.14)
Proof of SSM ≤skew(x)SSA,Assume: skew(x)SSA < SSM
max(x)
max(y)=max(x)
min(x)Pφmin(x)
Pφmax(y)≤max(x)
min(x)Pφx
Pφy <max(x)
max(y) (A.15)
Proof of SAS ≤skew(x)skew(y)SSA,Transitivity of SAS ≤SMS ≤skew(x) skew(y)SSA
Proof of SGS ≤skew(x)skew(y)SSA,Transitivity of SGS ≤SAS ≤skew(x) skew(y)SSA
A.4.4. Average of stretches
Now I show the inequalities with SAS on the right side.
136
Proof of SMS ≤skew(x)skew(y)SAS,Assume: skew(x) skew(y)SAS < SMS
max(x)
min(y)
A.9
z}|{
≤max(x)
min(x)
max(y)
min(y)min x
y≤max(x)
min(x)
max(y)
min(y)Pxφ/y
Pφ
<max x
yA.8
z}|{
≤max(x)
min(y) (A.16)
Proof of SSM ≤skew(x)SAS,Assume: skew(x)SAS < SSM
max(x)
max(y)
A.9
z}|{
≤max(x)
min(x)min x
y≤max(x)
min(x)Pφx/y
Pφ<max(x)
max(y) (A.17)
Proof of SSA ≤skew(x)skew(y)SAS,Transitivity of SSA ≤SMS ≤skew(x) skew(y)SAS
Proof of SGS ≤SAS,Generalized inequality of arithmetic and geometric means to
weighted means [Ste04].
A.4.5. Geometric mean of stretches
Now I show the inequalities with SGS on the right side.
Proof of SMS ≤skew(x)skew(y)SGS,Assume: skew(x) skew(y)SGS < SMS
max(x)
min(y)
A.9
z}|{
≤max(x)
min(x)
max(y)
min(y)min x
y≤max(x)
min(x)
max(y)
min(y) Yx
yφ!1
Pφ
<max x
yA.8
z}|{
≤max(x)
min(y)(A.18)
Proof of SSM ≤skew(x)skew(y)SGS,Transitivity of SSM ≤SMS ≤skew(x) skew(y)SGS
Proof of SAS ≤skew(x)skew(y)SGS,Transitivity of SAS ≤SMS ≤skew(x) skew(y)SGS
Proof of SSA ≤skew(x)skew(y)SGS,Transitivity of SSA ≤SMS ≤skew(x) skew(y)SGS
A.5. Latency in rings
The following holds in an n-circle when an edge is deactivated:
lim
n→∞SGS =2
√e.(A.19)
This is approximately 1.21. I refer to this in section 3.3.3.
137
A.5.1. Limit in odd-length rings
Proof. First consider the case that nis odd. In the latency-minimizing configurations
CL(the circle) the geometric mean of the latencies is:
geo(LCL) = n(n−1)
v
u
u
u
t
(n−1)/2
Y
i=1
i
2n
.(A.20)
And in the energy-minimizing configurations C (the path) it is:
geo(LC) = n(n−1)
v
u
u
t
n
Y
s=1
n
Y
d=1,d6=s|d−s|.(A.21)
Thus, I need to determine:
lim
n→∞SGS = lim
n→∞ n(n−1)
v
u
u
u
u
u
u
u
t
n
Q
s=1
n
Q
d=1,d6=s|d−s|
(n−1)/2
Q
i=1
i!2n= lim
n→∞
n−1
Q
s=1
s!
(((n−1)/2)!)n
2
n(n−1)
.(A.22)
I replace nwith 2m+ 1 and use Stirling’s approximation n!≈n
en. I only need to
consider terms that do not approach 1 after the exponentiation with 2/(2m(2m+ 1)).
lim
m→∞
2m
Q
s=1
s!
(m!)2m+1
2
2m(2m+1)
= lim
m→∞
2m
Q
s=1 s
es
m
em(2m+1)
2
2m(2m+1)
(A.23)
Using Q2m
s=1 e−s= e−m(2m+1) I get
lim
m→∞ m−m(2m+1)em(2m+1)e−m(2m+1)
2m
Y
s=1
ss!2
2m(2m+1)
.(A.24)
A Stirling-like series for the hyperfactorial [Weib] is Qn
i=1 ii≈e−n2/4nn(n+1)/2, where
again all terms have been dropped that approach 1 in the end result.
lim
m→∞m−m(2m+1)e−(2m)2/4(2m)2m(2m+1)/22
2m(2m+1) =2
√e(A.25)
138
A.5.2. Limit in even-length rings
Proof. In case of even length, the formula for the path is the same, but the formula for
the circle is
n(n−1)
v
u
u
u
t
n/2−1
Y
i=1
i
2
n
2
n
.(A.26)
Replacing nwith 2mand inserting Sterling’s approximation I get
lim
m→∞
2m−1
Q
s=1
s!2
(m−1)!2m2m
1
2m(2m−1)
= lim
m→∞
2m−1
Q
s=1 s
es
m−1
e(m−1)2m
2
2m(2m−1)
.(A.27)
Using Q2m−1
s=1 e−s= em−2m2I get
lim
m→∞ (m−1)−(m−1)2me(m−1)2mem−2m22m−1
Y
s=1
ss!2
2m(2m−1)
.(A.28)
Using the same Stirling-like approximation of the hyperfactorial the result is:
lim
m→∞(m−1)−(m−1)2me(m−1)2mem−2m2e−(2m−1)2/4(2m−1)(2m−1)2m/22
2m(2m−1) =2
√e.
(A.29)
139
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C. Glossary
activation A transient state in which a base station (BS) changes
from sleep to active denoted as SU. xii, xiii, xvi, 10–
12, 20–28, 31, 33–35, 37, 38, 40, 42–44, 47, 60, 77–79,
91, 94, 115, 118, 120–122, 124, 125, 127, 129, 131, 159
active A state in which a BS is ready to perform work (but
does not necessarily perform work) denoted as SA.
iii, xii, xv, 3, 20, 23, 25, 28, 32, 34, 35, 40, 47, 48, 53,
72, 73, 78–81, 84–86, 90–94, 114, 116–118, 120–122,
124, 129, 130, 157, 158
cell The area a BS provides coverage to (either data or
signaling traffic). 62, 64–66, 68, 70, 80, 90, 107, 108,
120, 159
configuration A state of a radio access network (RAN) which de-
scribes the activity state of each BSs denoted as C.
xii, 45–49, 51, 55, 57, 78, 86, 115, 133, 134, 138
cooperation Multiple BSs transmitting to the same user equip-
ment (UE) to increase its signal strength. iii, xii–xiv,
xvi, xix, xx, 2, 12–15, 17–19, 60–73, 75–81, 85–89,
91, 93–98, 100, 101, 104, 108–114, 118–121, 124, 125,
127–131
data traffic The traffic that a UE wants to transmit (compare
signaling traffic). xv, 2, 4, 5, 11, 12, 17, 78, 80, 129,
157–159
deactivation A transient state in which a BS changes from active
to sleep denoted as SD. iii, xiv, xvi, xvii, 10, 11, 19,
21–24, 28, 29, 31, 35, 37, 40, 43, 44, 46, 56, 58, 91,
115, 118, 120, 121, 125, 127, 131, 159
deployment The distribution of BSs in an area or plane. xii, xvi,
xix, 14, 61, 64–67, 72, 78, 80, 86–91, 94, 116–118,
123–128, 158
157
hexagonal A deployment in which each BSs has six neighboring
BSs which are all at the same distance to it and each
other. 64, 66, 69, 72, 73, 78–80, 84, 90, 91, 94, 117,
118, 125, 159
hotspot A (small) area where users generate above-mean traf-
fic (e.g., in a hotel or restaurant). 11, 12, 16, 117,
123–127, 131, 132
idle A state in which a BS is active, but is not performing
any work. 3, 4, 10, 12, 26, 28, 29, 32, 46, 48, 57, 58,
118, 120, 124, 129, 159
interference Signals from other transmissions that alter the re-
ceived signal. iii, x, 5, 17–19, 63, 95, 101, 117, 120,
121, 130, 131
latency The time it takes the RANs to fully process a request
from a UE (e.g., a file transmission) denoted as L. iii,
xii–xiv, xviii, 5, 8, 10, 13, 16, 17, 20–33, 35–55, 57,
114, 124, 129, 133
load The ratio of work a system performs to the work in
could maximally perform (depending on the context
this can be averaged over different time scales). 4–10,
13–16, 19, 23, 38–40, 45, 46, 55–58, 61, 62, 78, 80,
115–118, 121, 123–127, 129–131, 159
macro A size category of BSs with long range and usually
multiple sectors which I consider to serve only or
mostly signaling traffic. xx, 11–14, 62, 79, 80, 114–
118, 120, 122–129
noise Random fluctuation which of a signal denoted as N.
x, xiv–xvi, 4, 5, 63, 97, 101, 117, 158
outage The state of a channel in which the signal-to-noise
ratio (SNR) is too low to transmit at the outage ca-
pacity. xiv, xix, 5, 61, 62, 75, 76, 79, 95–105, 107–113,
130, 158
path loss The reduction of signal strength over traveled dis-
tance. xiii, 3, 5, 19, 60, 63–66, 68–73, 75–77, 80, 107,
108, 117, 130
pico A size category of BSs with short range which I con-
sider to serve only data traffic. 11–13, 79, 80, 115–
118, 120–130
158
policy A procedure that describes how to determine which
BS are activated and which are deactivated at which
time ((de-)activation policy). xii–xv, xviii, 10, 20–44,
48, 120–126, 132, 159
power cycle The activation and deactivation of a BS or the other
way around. iii, xviii, xx, 11, 16, 20, 23, 24, 27–
29, 32–35, 37, 40, 42–45, 59, 78, 114, 120, 122, 125,
128–132
power profile The function from load to power consumption of BS
or the complete RAN. iii, 6–9, 11, 12, 45, 52, 59, 101,
118, 131, 132
range The distance over which a BS can reliably transmit
(data or signaling traffic) denoted as rSfor signaling
traffic and rDfor data traffic. iii, 4, 5, 11, 12, 16,
21, 60–66, 71–73, 75, 78, 80, 81, 84, 85, 94, 112, 129,
130, 158
scheme A procedure that describes how to determine which
BS serves the data traffic of which UE (association
scheme). xiii–xvi, xix, 9, 12, 21, 79, 85, 86, 88, 91,
93, 94, 96, 100, 101, 106, 108–113, 120, 159
sector The area of a cell that is covered by a single antenna.
19, 116, 117, 158
signaling traffic Traffic that is needed to control the flow of data traf-
fic. xv, xvi, 2, 4, 5, 11, 12, 17, 60, 63, 72, 73, 77, 78,
117, 120, 131, 157–159
sleep A state in which a BS is not ready to perform work
and consumes less power than being idle denoted as
SS. iii, 1, 6, 8, 11, 12, 14–16, 20, 24, 25, 29, 44, 45,
47, 51, 79, 90, 114–116, 118, 120, 121, 129, 131, 157
spacing The distance of neighboring BSs in a hexagonal de-
ployment, also known as inter-site distance (ISD) de-
noted as ξ. xvi, 60, 64–66, 68–70, 72, 81–94
traffic The flow of data between UEs and BSs. xviii, 3, 4,
8–12, 15, 16, 20, 23, 45, 52, 80, 91, 115, 117, 129,
131, 132, 157–159
159
