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IEEE T ansac ions on Mobile Compu ing.
Ci a ion in o ma ion: DOI 10.1109/TMC.2018.2890235
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1
A Comple e LTE Ma hema ical F amewo k o he
Ne wo k Slice Planning o he EPC
Jona han P ados-Ga zon, Abdelquoddouss Lagh issi, Miloud Bagaa, Ta ik Taleb, and Juan M. Lopez-Sole
Abs ac —5G is he nex elecommunica ions s anda ds ha
will enable he sha ing o physical in as uc u es o p o i-
sion ul a sho -la ency applica ions, mobile b oadband se ices,
In e ne o Things, e c. Ne wo k slicing is he i ualiza ion
echnique ha is expec ed o achie e ha , as i can allow logical
ne wo ks o un on op o a common physical in as uc u e
and ensu e se ice le el ag eemen equi emen s o di e en
se ices and applica ions. In his ein, ou pape p oposes a no el
and comple e solu ion o planning ne wo k slices o he LTE
EPC, ailo ed o he enhanced Mobile B oadBand use case. The
solu ion de ines a amewo k which consis s o : i) an abs ac ion
o he LTE wo kload gene a ion p ocess, ii) a compound a ic
model, iii) pe o mance models o he whole LTE ne wo k, and
i ) an algo i hm o join ly pe o m he esou ce dimensioning
and ne wo k embedding. Ou esul s show ha he agg ega ed
signaling gene a ion is a Poisson p ocess and he da a a ic
exhibi s sel -simila i y and long- ange-dependence ea u es. The
p oposed pe o mance models o he LTE ne wo k ely on hese
esul s. We o mula e he join op imiza ion p oblem o esou ces
dimensioning and embedding o a i ualized EPC and p opose
a heu is ic o sol e i . By using simula ion ools, we alida e he
p ope ope a ion o ou solu ion.
Index Te ms—LTE, EPC, Ne wo k Slicing, NFV, So wa ized
Ne wo ks, Mobile Ne wo ks, T a ic cha ac e iza ion, Resou ces
dimensioning, and Ne wo k embedding.
I. INTRODUCTION
FIFTH Gene a ion (5G) mobile ne wo ks play a pa amoun
ole in he o hcoming global indus ial digi aliza ion. 5G
will co e all he e ical ma ke needs in a cos e ec i e
manne . Compa ed o i s p edecesso (i.e., he Long-Te m
E olu ion (LTE) echnology), he equi emen s o 5G sys ems
include, among many o he s, highe ne wo k lexibili y and
scalabili y, as well as x100 inc ease in cos e ec i eness [1]–
[4]. To mee hese challenging goals, ne wo k so wa iza-
ion (NS) is en isaged as he co ne s one o build he 5G
echnology [5], [6]. The concep o NS is mainly based on
i) Ne wo k Func ion Vi ualiza ion (NFV), which decouples
ne wo k unc ions om p op ie a y ha dwa e enabling hem
o un as so wa e on i ualiza ion con aine s such as i ual
machines (VMs) [7], and ii) So wa e De ined Ne wo king
Jona han P ados-Ga zon and Juan M. Lopez-Sole a e wi h he Resea ch
Cen e o In o ma ion and Communica ions Technologies o he Uni e si y
o G anada (CITIC-UGR); and he Depa men o Signal Theo y, Telema ics
and Communica ions o he Uni e si y o G anada, G anada, 18071 Spain
(email: jpg@ug .es, juanma@ug .es).
Abdelquoddouss Lagh issi, Miloud Bagaa, and Ta ik Taleb a e wi h he
Depa men o Communica ions and Ne wo king, School o Elec ical En-
ginee ing, Aal o Uni e si y, Espoo, Finland. Ta ik Taleb is also wi h he
Cen e o Wi eless Communica ions (CWC), Uni e si y o Oulu, 90014 Oulu,
Finland, and also wi h he Compu e and In o ma ion Secu i y Depa men ,
Sejong Uni e si y, 143-747 Seoul, AQ3 Sou h Ko ea. (emails: abdelquod-
[email p o ec ed], [email p o ec ed], [email p o ec ed]).
(SDN), which ully sepa a es con ol and da a planes in
ne wo k nodes allowing ne wo k p og ammabili y.
Unde he NS app oach, isola ed, ully au oma ed, p o-
g ammable, lexible, and se ice-cus omized ne wo ks known
as ne wo k slices can be deployed on op o a common physical
in as uc u e [8]–[10]. This app oach is e e ed o as ne wo k
slicing. I will allow he mobile ope a o s o co e he di e en
ma ke scena ios and use cases ha demand he e ogeneous,
di e se and possibly mu ually incompa ible equi emen s [5].
The adop ion o ne wo k slicing in 5G mobile ne wo ks
equi es op imal solu ions o planning he slices acco ding
o he di e en use cases equi emen s. This mainly in ol es
he dimensioning o he esou ces and i s embedding in a
gi en in as uc u e. Fu he mo e, hese p ocesses ha e o
be done in a manne ha ensu es he Quali y o Se ice
(QoS) equi emen s o each use case. Likewise, aced wi h a
dec easing A e age Re enue Pe Use (ARPU), ope a o s a e
challenged o educe, o e en op imize, i) he acqui emen and
main enance o he physical in as uc u e (i.e., capi al expen-
di u es -CAPEX-), and ii) he ongoing expenses o p ope ly
ope a e he ne wo k equipmen (i.e., ope a ing expendi u es -
OPEX-). Many echno-economic models ha e been p oposed
o educe he CAPEX and OPEX such as in [11], [12].
Ou wo k aims o design a comple e solu ion o ne wo k
slices planning o he LTE E ol ed Packe Co e (EPC), which
is ailo ed o he enhanced Mobile B oadBand (eMBB) use
case [7], [13]. To ha end, we p opose a amewo k consis ing
o he ollowing componen s:
•An abs ac ion o he LTE wo kload gene a ion p o-
cess, o bo h Con ol Plane (CP) and Da a Plane (DP),
along wi h a compound a ic model ha includes he
mos ep esen a i e se ices consumed in cu en cellu-
la ne wo ks. This is equi ed o es ima e he se ice
consump ion when he e is no p e ious knowledge o
he wo kload demand. This componen is also use ul o
gene a e syn he ic wo kloads o expe imen a ion (e.g., o
s ess a i ualized LTE ne wo k).
•Holis ic analy ical models o p edic he pe o mance
(e.g., packe loss p obabili y and esponse ime) o a
i ualized EPC ( EPC). We apply queuing heo y and
s ochas ic ne wo k calculus o de elop he CP and DP
models, espec i ely. Fo a gi en wo kload and a se
o QoS equi emen s, ou models acili a e esou ces
dimensioning.
•The co esponding o mula ion and heu is ic o sol e he
join op imiza ion p oblem o esou ces dimensioning and
embedding o he EPC. We ha e sugges ed a mul i-
objec i e op imiza ion p oblem ha minimizes he wo k-
2
load imbalances among a se o candida e Edge Clouds
(ECs) (i.e., Da a Cen e s (DCs) deployed close o end
use s) and maximizes he esou ces u iliza ion, on he
ne wo k side, and Quali y o Expe ience (QoE), on he
end use ’s side. These objec i es a e subjec o mee a se
o QoS equi emen s. Fo he CP, he QoS equi emen s
a e de ined as an uppe bound on he a e age elapsed ime
o mo e a Use Equipmen (UE) om IDLE o ACTIVE
s a es. Fo he DP, he QoS equi emen s conside ed a e
he limi on he maximum one-way ne wo k delay and a
maximum packe loss p obabili y a he EPC. Addi ion-
ally, we impose a condi ion o limi he maximum numbe
o Cen al P ocessing Uni (CPU) co es o be assigned o
a single Vi ual Ne wo k Func ion Componen (VNFC)
ins ance. Tha is o ake in o accoun he ac ual limi a ion
on he numbe o CPU co es o he Physical Machines.
Sha ing ne wo k esou ces be ween di e en use s has
p o en o educe CAPEX and OPEX [14]–[16]. The abo e-
men ioned ea u es, namely he pe o mance-p edic i e mod-
els, he load balancing among ECs, and he maximiza ion
o esou ce u iliza ion will ce ainly induce conside able cos
sa ings. Also, he maximum numbe o CPU co es cons ain
will ha e an impac on educing he cos s due o OPEX
[17]. Las , al hough he NS pa adigm enables ope a o s o
dynamically adap he esou ces alloca ed o each ne wo k
slice and se ices [18], he on-demand plans o e ed by in-
as uc u e p o ide s a e mo e expensi e han he ese a ion
plans. Speci ically, esou ces can be pu chased as a ese a ion
o up o 70% o he on-demand p ice [19]. Thus, he ne wo k
slices planning is c ucial o ope a o s o sa e money.
As a s a ing poin , his wo k is mean o enhance he
“Ne wo k Slice Planne ” (NSP) [20]. NSP is a simula ion ool
ha implemen s accu a e models o he use s’ beha io , mo-
bili y, and da a consump ion in cellula ne wo ks. Speci ically,
we ex end i s da a consump ion model o include he mos
ep esen a i e se ices consumed in cu en mobile ne wo ks.
Then, by using NSP we cha ac e ize s ochas ically he ag-
g ega ed wo kload gene a ion p ocesses o he CP and DP.
Unde ou wo kload gene a ion model, he esul s show ha
he agg ega ed signaling gene a ion p ocess ollows a Poisson
dis ibu ion and he agg ega ed DP wo kload exhibi s Sel -
Simila i y (SS) and Long-Range Dependence (LRD) ea u es.
Based on he a o esaid esul s, we de elop holis ic pe -
o mance models o a i ualized LTE ne wo k. The CP is
modeled ollowing he same echnique as in [21] o chains
o Vi ual Ne wo k Func ions (VNFs). The model includes
he main LTE en i ies and hei messages exchange. The DP
is modeled as a queue ed by a ac ional B ownian Mo ion
( Bm) p ocess [22]. These comp ehensi e models allow us o
de ine e icien esou ces dimensioning algo i hms.
Finally, he heu is ic p oposed in his wo k o sol e he
planning o EPC elies on he a o emen ioned pe o mance
models. The algo i hm is dubbed “Planne o he EPC as a
Se ice” (PES). By using a sys em-le el LTE simula o , we
alida e he co ec ope a ion o PES. We also show ha PES
embedding algo i hm educes he wo kload imbalances among
candida e ECs in con as o o he baseline echniques.
The emainde o he pape is o ganized as ollows. Sec ion
II b ie ly e iews he ela ed li e a u e. Sec ion III desc ibes
he sys em model. Sec ion IV includes he o mula ion o he
join op imiza ion p oblem o esou ce dimensioning and em-
bedding o he EPC. In Sec ion V, he modeling and analysis
o es ima e he pe o mance o he CP and DP a e p esen ed.
Nex , in Sec ion VI, we in oduce he p oposed heu is ic o
pe o m he planning o he EPC. Sec ion VII explains he
expe imen al se up. Sec ion VIII p o ides nume ical esul s
ha show he p ope ope a ion o ou solu ion. Finally, Sec ion
IX summa izes he main conclusions.
II. RELATED WORKS
This sec ion b ie ly e iews he ela ed li e a u e. In pa -
icula , we ocus on pe o mance models and embedding
algo i hms (i.e., on how o map VNFC ins ances o physical
in as uc u es) o he EPC.
A. Modeling o he EPC
Analy ical models cons i u e an agile way o p edic he pe -
o mance o a sys em in ad ance. The e a e se e al p oposals
in he li e a u e ackling he analy ical modeling o pa s o he
en i e EPC [21], [23]–[27]. In a iably, hese wo ks employ
queuing heo y.
In [24], Rajan e al. model he EPC as a D/D/m node. They
conclude ha when simply eplacing exis ing EPC elemen s
wi h i ualized equi alen s, se e e pe o mance bo lenecks
occu . In [26], [27], P ados e al. analyze he pe o mance o a
i ualized Mobili y Managemen En i y ( MME) wi h a h ee-
ie design, inspi ed by web se ices, and using a Jackson’s
ne wo k (i.e., a ne wo k o M/M/m queues). Each queue ep e-
sen s a ie o VNFC o he MME. The au ho s show ha he
p oposed model p o ides ai ly good esul s o compu a ional
esou ces dimensioning. In [21], he same au ho s enhance
he p e ious model by ex ending i s applicabili y domain o
any chain o VNFs, inc easing i s lexibili y, and using a
mo e accu a e echnique o analysis. Speci ically, each VNFC
ins ance is modeled as a G/G/m queue. The esul ing ne wo k
o queues is sol ed by using he app oxima ed echnique
p oposed by Whi e al. in [28] o he Queuing Ne wo k
Analyze e e ed o, he eina e , as he QNA me hod. Fo he
abo emen ioned use case (a h ee- ie ed MME), he au ho s
show he QNA me hod ou pe o ms Jackson’s ne wo ks and
Mean Value Analysis echniques in e ms o he esponse
ime es ima ion e o . Tanabe e al. p opose in [23] a bi-
class (i.e., Machine- o-Machine and Mobile B oadband -MBB-
communica ions) queuing model o he EPC. The CP and
DP o he EPC a e modeled as M/M/m/m and M/D/1 nodes,
espec i ely. This model cons i u es he co e o he EPC-ORA
me hod which aims o op imize he esou ce assignmen o
he CP and DP o he EPC. Finally, in [25], Ren e al. p opose
a dynamic esou ce p o isioning algo i hm o he EPC
conside ing he capaci y o legacy ne wo k equipmen al eady
deployed. To e alua e he pe o mance o hei solu ion, hey
model each EPC elemen as a M/M/m/K queue and assume
ha he VNF ins an ia ion ime is exponen ially dis ibu ed.
The a o emen ioned wo ks only model pa s o he EPC
and/o do no cap u e he in e ac ions among i s elemen s.
3
In his pape , his gap is co e ed. We conside he main
elemen s o he LTE ne wo k CP (i.e., UE, e ol ed Node B
-eNB-, MME, Se ing Ga eway -SGW-, Packe Da a Ne wo k
Ga eway -PGW-, Home Subsc ibe Se e -HSS-, and Policy
and Cha ging Rules Func ion -PCRF-) as well as hei in e ac-
ions. In his way, i is possible o p edic he pe o mance o
he whole LTE CP om he agg ega ed signaling gene a ion
p ocess. In [23], [25], he esou ces dimensioning o he
EPC is also isi ed. Ne e heless, hese wo ks add ess he
dimensioning o each componen in an isola ed way. Only
hen, i is necessa y o de ine a p ocessing delay budge o
each en i y o be dimensioned in ad ance. Ou holis ic model
o an LTE ne wo k o e comes his limi a ion by enabling
he esou ces dimensioning algo i hm o conside an o e all
p ocessing delay budge o he whole EPC. This leads o
esou ces sa ings.
Fo he DP, we le e age he esul s ob ained o he analysis
o he LTE da a a ic aces o de i e i s pe o mance me ics.
Speci ically, he EPC DP is modeled as a single queue ed
by a Bm p ocess. To he bes knowledge o he au ho s, his
is he i s wo k ha uses s ochas ic ne wo k calculus esul s
o analyzing he pe o mance o a EPC.
B. Algo i hms o he EPC embedding
The e is a ich li e a u e p oposing algo i hms o embed he
whole EPC o some o i s en i ies in a physical in as uc u e
[29]–[38]. In [29], Taleb e al. p opose a heu is ic algo i hm o
i ualized SGWs ( SGWs) embedding. The algo i hm ies o
minimize he equency o mobili y ga eway eloca ions while
ensu ing ha a maximum capaci y o each SGW, which
handles he a ic load o a se ing a ea, is no exceeded. This
wo k is ex ended in [32] whe e some addi ional objec i es and
es ic ions a e conside ed. Rega ding he objec i es, he pa h
be ween UEs and PGWs is minimized, and he o e all ne wo k
esou ce u iliza ion is op imized. Conce ning he es ic ions,
his wo k was a pionee in conside ing some ele an hi d
gene a ion pa ne ship p ojec (3GPP) cons ain s.
In [30], Bagaa e al. add ess he embedding o he i u-
alized PGW ( PGW). The embedding p oblem is o mula ed
as a mul i-objec i e non-linea op imiza ion p oblem which
minimizes he cos s o he ne wo k ope a o s, maximizes he
ne wo k pe o mance, and balances he load equally among
he PGW ins ances. To sol e he p oblem, h ee heu is ic
algo i hms a e p oposed o achie e nea -op imal solu ions. In
[31], Bas a e al. in es iga e di e en app oaches o deploy he
co e ga eways (i.e., SGW and PGW) in he DCs. Speci ically,
hey conside a ully and pa ially i ualiza ion app oaches
o he ga eways. The o me consis s in mo ing he CP and
DP unc ionali ies o each ga eway o a DC. The la e decou-
ples CP and DP unc ionali ies by using he SDN pa adigm
and only he CP pa is hos ed wi hin a DC. In he same
con ex ; elying on an SDN amewo k ha decouples he
CP om he DP, Da sika e al. p opose in [39] a Ma ching
Theo e ic Flow P io i iza ion algo i hm ha aims o imp o e
he g ade o se ice le el and delay induced by he co e
ne wo k conges ion. This app oach allows o e he op se ice
p o ide s o in e ene in he i ual slices alloca ion p ocess.
TABLE I: No a ion.
No a ion Desc ip ion
m(c)
l
Numbe o dedica ed physical CPU co es alloca ed o ins ance
lo he VNFC c∈C.
mmax Maximum numbe o dedica ed physical CPU co es o be
alloca ed o a single i ualiza ion con aine .
TeAc ual mean esponse ime o he CP en i y e∈E.
Ti Ac ual mean esponse ime o he LTE in e ace i ∈IF .
T(SR)Ac ual mean delay o he CP o ca y ou an SR p ocedu e.
T(CP )
budge Mean delay budge o he CP.
T(max)
uAc ual maximum esponse ime o he DP en i y u∈U=
{UE, eNB, DP GW}.
T(max)
i
Ac ual maximum esponse ime o he LTE DP in e ace
i ∈IF U ={Uu, S1−U}.
T(DP )
max Ac ual maximum delay o he DP.
T(DP )
budge Maximum delay budge o he DP.
P(EP C)Ac ual EPC packe loss p obabili y.
P(EP C)
budge EPC packe loss p obabili y budge .
I has pe mi ed o achie e e icien lows p io i iza ion wi h
espec o he se ice p o ide s’ policies and QoS demands.
In [33], Ma ini e al. o mula e he p oblem o choosing he
VNF ins ances p o ided by a dis ibu ed se o DCs o se e
a gi en se ice chain eques . The objec i e is o minimize
he o e all la ency o he chain. This op imiza ion p oblem
can be o mula ed as a esou ce cons ained sho es pa h
p oblem. In [34] [35], Baumga ne e al. o mula e he join
op imiza ion p oblem o he i ual mobile co e ne wo k opol-
ogy composi ion and embedding. The o mula ion gua an ees
a maximum end- o-end la ency and akes in o accoun he
p ocessing, queuing, and p opaga ion delays. In [36], Bagaa
e al. add ess he placemen o i ual ins ances o 4G (MME,
SGW, PGW) and 5G (AMF, SMF and AUSF) co e ne wo k
elemen s o e a ede a ed cloud based on Mixed In ege Linea
P og amming and coali ional o ma ion game. Finally, Die ich
e al. [37] o mula e a mixed-in ege linea p og am o he
SGW and MME embedding. To educe i s ime complexi y,
hey ans o m i in o a linea p og am by employing elaxa ion
and ounding echniques. Thei p oposal mi iga es he load
imbalance in oday’s mobile ne wo ks, which imp o es eques
accep ance and esou ce u iliza ion.
The esou ces dimensioning and embedding a e ea ed
h oughou he li e a u e as sepa a e p oblems. These wo
s ages o esou ces alloca ion a e closely ela ed and pe o m-
ing hem in a coo dina ed way b ings bene i s. Fo ins ance,
he e is a ade-o be ween he wo kload balance among a se
o candida e DCs (i.e., p opaga ion delays) and he esou ces
u iliza ion (i.e., p ocessing delays) when an o e all delay
budge o be me is pa i ioned among hese wo s ages. He ein,
we o mula e he join op imiza ion p oblem o planning he
EPC o add ess his ade-o .
III. SYSTEM MODEL
A. Sys em A chi ec u e
Le us assume an e ol ed uni e sal e es ial adio access
ne wo k (E-UTRAN), al eady deployed wi h IeNBs, which
p o ides connec i i y o a se o JUEs o he LTE EPC (see
Fig. 1). Each UE jis a ached o an eNB i.
4
Fig. 1: E-UTRAN deploymen and ECs si es.
Fig. 2: Assumed LTE ne wo k a chi ec u e.
Le uji be a bina y a iable indica ing whe he he UE jis
a ached o he eNB i(uji = 1) o no (uji = 0). We conside
he co e age map o his E-UTRAN as a ec angula a ea A
wi h heigh hand wid h w.
Wi hin A, he e a e al eady deployed KECs (see Fig.
1). Le (eNB)
i= (x(eNB)
i, y(eNB)
i)∀i∈N∩ {1, .., I},
(UE)
j= (x(UE)
j, y(UE)
j)∀j∈N∩{1, .., J}, and (EC)
k=
(x(EC)
k, y(EC)
k)∀k∈N∩{1, .., K}deno e wo dimensional
ec o s ep esen ing he posi ions o eNBs, UEs, and ECs
wi hin A, espec i ely.
The MME, SGW, and PGW o he EPC will be implemen ed
as a se o VNFs ha makes up a ne wo k se ice [18],
he ea e e e ed o as EPC, and deployed on he candida e
ECs. We disca d he op ion o deploying he EPC as a single
VNF wi h se e al componen s (VNFCs), since, in his wo k,
we will assume ha he LTE EPC in e nal in e aces such as
S11 and S5 will emain unchanged. O he EPC en i ies, such
as he HSS and he PCRF, migh be loca ed ou side o he
ECs and implemen ed ei he as VNFs o physical ne wo k
unc ions (PNFs).
The agg ega ed wo kload gene a ed by he JUEs a ached
o he E-UTRAN is dis ibu ed among he Kcandida e ECs.
This wo kload dis ibu ion is pe o med a he g anula i y o
eNBs (i.e., each eNB iis assigned o a candida e EC k). Le ik
be a bina y a iable indica ing whe he he eNB iis assigned
o he EC k(i.e., ik = 1) o no (i.e., ik = 0). To se e i s
co esponding wo kload, a EPC is ins an ia ed on each EC.
The LTE ne wo k a chi ec u e deemed in his wo k is
depic ed in Fig. 2. We conside ha he CP and DP o he
EPC a e ully decoupled. Also, we assume he in e aces,
be ween he CP unc ional en i ies, as he ones de ined in
he 3GPP LTE s anda ds. Consequen ly, each CP en i y (e.g.,
Fig. 3: Wo kload gene a ion model.
he MME, and he con ol unc ionali ies o he SGW and
PGW -cSGW and cPGW-) a e implemen ed sepa a ely as
a single VNF wi h a single componen (VNFC). The DP
unc ionali ies o he SGW and PGW a e in eg a ed on a
single VNF, wi h only one VNFC, ha exposes he LTE S1-U
and SGi in e aces. We assume ha all VNFCs o he EPC
execu e CPU-in ensi e asks. Each VNFC migh ha e mul iple
ins ances. Conside ing he ETSI NFV a chi ec u al amewo k
and e minology [40][18] and wi hou loss o gene ali y, each
VNFC ins ance is supposedly unning on an isola ed i ual-
iza ion con aine such as a VM. Le m(c)
ldeno e he numbe
o dedica ed physical CPU co es alloca ed o he ins ance l
o he VNFC c∈C={MME, cSGW, cPGW, DP GW}.
Since he numbe o CPU co es o a physical se e is ini e
and he la e a e sha ed among se e al VMs, we conside ha
m(c)
lis limi ed o mmax (i.e., m(c)
l≤mmax).
B. Wo kload gene a ion model
In his pape , we add ess he eMBB use case. In his con ex ,
he UEs un applica ions ha gene a e and consume DP a ic.
We conside he abs ac ion p esen ed in [27] o such a
p ocess (see Fig. 3).
A session wi h du a ion Tsd is de ined as he use ’s ac i i y
beginning om he ime an applica ion is launched o he ime
i closes. A session consis s o Napplica ion ac i i y pe iods
(AAPs) o leng h Ton sepa a ed by N−1 eading imes o
du a ion D. An AAP is a ime pe iod in which he applica ion
gene a es o consumes all necessa y ne wo k a ic o pe o m
a gi en ask (e.g., download he p o ile o a iend, o send
an ins an message). A eading ime is he empo al in e al
du ing which he use pe o ms any ac ion ha does no equi e
o gene a e ne wo k a ic such as deciding which iend’s
p o ile o isi nex o eading a message.
Rega ding he signaling wo kload, he use s’ ac i i y and
mobili y igge he LTE CP p ocedu es. In his wo k, we only
conside he UE- igge ed se ice eques (SR), S1-Release
(S1R), X2-based Hando e (HO), and acking a ea upda e
(TAU) p ocedu es. Al hough o he p ocedu es such as a ach
and S1-based hando e a e hea ie in e ms o compu a ional
esou ces consump ion, hey do no occu equen ly in LTE
ne wo ks [41].
Once he UE is egis e ed in he ne wo k, an SR p ocedu e is
igge ed du ing i s idle- o-connec ed (i.e., IDLE o ACTIVE)
ansi ions. Then, whene e an AAP s a s while he UE is
in idle mode, an SR p ocedu e akes place (see Fig. 3).

5
Con e sely, an S1R p ocedu e occu s du ing UE’s connec ed-
o-idle ansi ions du ing which he ne wo k eleases he UE’s
esou ces. We also ake in o accoun he e ec s o an inac i i y
ime . I s alue is deno ed as I. The ne wo k wai s Iuni s
o ime a e ha an AAP inishes be o e igge ing an S1R
(see Fig. 3). A HO p ocedu e is igge ed when a UE is in
connec ed mode and pe o ms a cell change, bu he a ge
cell is a ached o he same MME as he sou ce cell’s. Finally,
we assume ha a TAU p ocedu e is igge ed whene e a
UE ca ies ou a T acking A ea (TA) change. These TAs a e
p ede ined and a e he same o any UE.
C. Pe o mance Requi emen s
The LTE ne wo k has o mee a se o pe o mance e-
qui emen s in e ms o la ency and packe loss p obabili y
[42]. Fo he CP, he conside ed pe o mance equi emen is
an uppe bound on he mean CP la ency T(CP )
budge de ined by
he 3GPP, i.e., he a e age elapsed ime o mo e an UE om
IDLE s a e o ACTIVE s a e [42]. In his wo k, we ansla e
his speci ica ion as he equi ed a e age ime o ca y ou a
se ice eques p ocedu e. Mo eo e , we conside he wo s -
case scena io o he se ice eques p ocedu e, whe e he UE
au hen ica ion, NAS (Non-Access S a um) secu i y se up, and
he EPS (E ol ed Packe Sys em) session modi ica ion s eps
occu du ing he SR.
Le Teand Ti deno e, espec i ely, he mean
esponse imes o he CP en i y e∈E=
{UE, eNB, MME, cSGW, cPGW, HSS, PCRF}and he
LTE in e ace i ∈IF ={Uu, S1−C, S11, S6a, S5, Gx}.
The mean ime equi ed o ca y ou an SR, T(SR), in he
wo s -case scena io can be compu ed as:
T(SR)= 5 ·TUE + 8 ·TeNB + 5 ·TMME + 2 ·TcSGW
+ 2 ·TcP GW +THSS +TP CRF + 8 ·TUu + 7 ·TS1−C
+ 2 ·TS11 + 2 ·TS6a+ 2 ·TS5+ 2 ·TGx
(1)
The abo e equa ion means ha du ing an SR call low in he
wo s case scena io he UE, eNB, MME, cSGW, cPGW, HSS,
and PCRF en i ies ha e o p ocess, espec i ely, 5, 8, 5, 2, 2,
1, and 1 con ol messages. Also, 8, 7, 2, 2, 2, and 2 con ol
messages ha e o a e se, espec i ely, he LTE Uu, S1-C,
S11, S6a, S5, and Gx in e aces [43]. Then, he CP delay
equi emen can be exp essed as T(SR)≤T(CP )
budge .
Fo he DP, he pe o mance equi emen s conside ed a e
he maximum DP delay budge T(DP )
budge and he packe loss
p obabili y a he EPC P(EP C)
budge . We conside T(DP )
budge as he
maximum ime i akes o a packe o a el om he SGi
in e ace a he SGW/PGW VNFC o he UE applica ion. The
P(EP C)
budge is he maximum allowable packe loss a he DPGW
VNFC ecei e bu e .
Le T(DP )
max and P(EP C)deno e he ac ual maximum delay o
he DP and he packe loss p obabili y o he EPC, espec i ely.
We can compu e T(DP )
max as:
T(DP )
max =T(max)
UE +T(max)
eNB +T(max)
DP GW +T(max)
Uu +T(max)
S1−U
(2)
whe e: T(max)
UE ,T(max)
eNB , and T(max)
DP GW a e espec i ely he
ac ual maximum DP packe p ocessing delay a he UE,
eNB, and DPGW. And T(max)
Uu and T(max)
S1−Ua e he ac ual
maximum delays o he DP adio and backhaul in e aces,
espec i ely. Then, he DP equi emen s can be exp essed as
T(DP )
max ≤T(DP )
budge and P(EP C)≤P(EP C)
budge .
IV. PROBLEM FORMULATION
In his sec ion, we o mula e he join op imiza ion p oblem
o dis ibu e he agg ega ed wo kload gene a ed by he E-
UTRAN among he candida e ECs and o pe o m he di-
mensioning o he equi ed esou ces o each EPC ins ance.
Taking in o accoun he de ined sys em model, i can be
o mula ed as ollows:
Objec i es :
minimize 

|K|
X
k=1 
|I|
X
i=1
|J|
X
j=1
ikuij −|J|
|K|
(3a)
minimize 

|K|
X
k=1
|I|
X
i=1
ik ·dik
(3b)
minimize 

|K|
X
k=1 X
c∈CX
l
m(c)
l
m(c)
l∈N(3c)
whe e dik =|| (eNB)
i− (EC)
k|| is he Euclidean dis ance
be ween eNB iand EC k.
Cons ain s :
CP :
C1 : T(SR)
k≤T(CP )
budge ,(3d)
DP :
C2 : max T(DP )≤T(DP )
budge ,(3e)
C3 : P(EP C)≤P(EP C)
budge ,(3 )
O he s
C4 : m(c)
l≤mmax ∀k∈[1,|K|]∩N(3g)
C5 :
|K|
X
k=1
|I|
X
i=1
ik =|I|, ik ∈ {0,1}(3h)
The decision a iables o he op imiza ion p oblem a e ik
and m(c)
l. Objec i e (3a) aims o dis ibu e he wo kload as
equally as possible o o minimize he wo kload imbalances
ac oss he candida e ECs. The goal is op imally achie ed when
he same numbe o use s (|J|/|K|) is assigned o e e y EC
k∈K. Objec i e (3b) aims o minimize he p opaga ion
delays. The co esponding objec i e unc ion is minimized
when e e y eNB i∈Iis assigned o he nea es EC k∗∈K,
whe e k∗=a gmink∈K(dik). Las , objec i e (3c) in ends
o minimize he o al numbe o CPU ins ances alloca ed o
he EPC o , equi alen ly, o maximize he u iliza ion o he
compu a ional esou ces.
Cons ain s (3d), (3e), and (3 ) gua an ee ha he QoS
equi emen s a e ul illed. Speci ically, Cons ain (3d) ensu es
6
Fig. 4: LTE con ol plane model.
ha he ac ual mean delay o ca y ou a se ice eques o
he EPC k(i.e., EPC ins ance unning on EC k) is lowe o
equal han he mean CP la ency T(CP )
budge . Cons ain (3e) and
(3 ) ensu e ha he maximum DP delay budge and he packe
loss p obabili y a he EPC a e me , espec i ely. Cons ain
(3g) limi s he maximum numbe o physical co es eques ed
o a single VNFC ins ance. Ha ing a single VNFC ins ance
would be op imal o minimizing he amoun o equi ed
esou ces (s a is ical mul iplexing). Howe e , each physical
se e has a maximum numbe o physical co es, i.e., he
numbe o physical co es we can eques pe VNFC ins ance
is limi ed. Mo eo e , in gene al, he highe is he numbe o
physical co es eques ed o a VNFC ins ance, he lowe is i s
a ailabili y. Finally, Cons ain (3h) gua an ees ha all eNBs
a e assigned o a candida e EC k(o EPC ins ance k).
V. ANALYSIS AND MODELING
A. LTE CP modeling
We model he CP o he LTE as an open ne wo k o G/G/m1
queues (see Fig. 4), whe e each queuing node ep esen s an
ins ance o a gi en en i y o he LTE ne wo k. The MME,
cSGW, and cPGW migh ha e se e al ins ances, each o which
is modeled as a G/G/m queuing node wi h m(c)
lse e s.
The se e s o a queuing node ep esen he CPU ins ances,
alloca ed o he en i y ins ance, p ocessing con ol messages
in pa allel. As s a ed in Sec ion III-A, m(C)
l≤mmax. Fo
he sake o simplici y, only one ins ance is conside ed o he
es o LTE CP en i ies (e.g., UE, eNB, HSS, and PCRF). The
co esponding G/G/m queuing node ha models he ins ance
o hese en i ies migh ha e an a bi a y numbe o se e s as
hey migh be deployed as PNFs.
The a ic sou ces a e loca ed a he eNB and he UE,
since he LTE signaling p ocedu es conside ed in his wo k
(e.g., SR, S1R, HO, and TAU) a e igge ed by hese en i ies.
Speci ically, he TAU and SR p ocedu es a e igge ed by he
UE and he S1R and HO p ocedu es a e igge ed by he eNB.
In he same way, he a ic sinks a e placed a he MME
ins ances.
To sol e he ne wo k o queues, we employ he QNA
me hod [28] which is desc ibed in Appendix A. This echnique
1In Kendall’s no a ion, a G/G/m queue is a queuing node wi h mse e s,
a bi a y a i al and se ice p ocesses, FCFS (Fi s -Come, Fi s -Se ed)
discipline, and in ini e capaci y and calling popula ion.
was applied and alida ed in [21] o es ima e he mean
esponse ime o a VNF wi h se e al VNFCs. In his wo k, we
use he QNA me hod o es ima e he mean esponse imes o
he LTE CP en i ies Te∀e∈E. To ha end, he QNA me hod
uses a educed se o he ollowing inpu pa ame e s:
•The s eady s a e ansi ion p obabili ies ma ix P= [pki],
whe e pki deno es he p obabili y o a packe o lea e
node k o node iand p0k= 1 −Pipki deno es he
p obabili y o a packe a node k o lea e he ne wo k. In
his wo k, we p o ide he exp essions o compu e P o
he LTE CP ( e e o Appendix B).
•The mean and squa ed coe icien o a ia ion (SCV) o
he ex e nal a i al p ocesses a node k,λ0k, and c2
0k.
Please no e ha only he UE and he eNB ha e ex e nal
a i al p ocesses in ou model (see Fig. 4). Conside ing
he abs ac ion desc ibed in sec ion III-B o he signaling
gene a ion p ocess, we ound ha hese a i al p ocesses
a e Poissonian (see Sec ion VIII-A). Then, c2
0k= 1 ∀k.
•The mean and he SCV o he se ice p ocesses a each
queue k,µkand c2
sk.
B. LTE DP modeling
Fo he conside ed a chi ec u e, he LTE DP consis s o
h ee ne wo k en i ies namely, UE, eNB, and DPGW, which
a e connec ed in andem. Since he ocus is on he EPC
dimensioning, we assume ha he UE and eNB en i ies ha e
cons an maximum delays.
The same me hodology as applied o model he LTE CP
canno be used o model he EPC DP as i can only p o ide
o e all mean pe o mance me ics o a queuing ne wo k, bu
no he pe o mance bounds such as hose de ined in Sec ion
III-C (e.g., T(DP )
max and P(EP C)) o he DP. Mo eo e , he
s ochas ic cha ac e iza ion o he agg ega ed DP a ic ca ied
ou in his wo k (see Sec ion VIII-A) shows ha he EPC
DP wo kload a i al p ocess exhibi s SS and LRD ea u es.
Con en ional queuing heo y does no comp ise such kind o
a i al p ocess [44]. Then, we model he DPGW as a single
queue ed by a Bm p ocess. Mo e p ecisely, we use he model
ha was i s epo ed in [22] and also de i ed in [44] om
s ochas ic ne wo k calculus esul s. This model can p o ide
he pe o mance bounds o a andem o queues wi h SS and
LRD inpu in an e ec i e and simple way.
To cha ac e ize he a i al p ocess, we adop he model
p oposed in [22]. Le A deno e he cumula ing a i al p ocess
o he DPGW queue, i.e., he cumula i e amoun o a ic
(i.e., in numbe o packe s) a i ing a he DPGW in he ime
in e al [0, ]. The ollowing model is conside ed o A [22]:
A =λ· +√λ·α·Z (4)
whe e Z is a no malized Bm pa ame e wi h Hu s pa ame e
H∈(1/2,1],λ > 0is he mean inpu a e, and α > 0is a
a iance coe icien .
Unde he abo e packe a i al model and conside ing a
cons an a e se e wi h capaci y C, he iola ion p obabili y
=P[B > b]o a backlog bound bcan be app oxima ed as
[22], [44]:
≈exp −(C−λ)2H
2·κ(H)2·λ·αb2−2H(5)
7
whe e κ(H) = HH(1 −H)1−H. The abo e equa ion gi es us
an app oxima ion o he p obabili y o sa u a ion o a bu e
o size bpacke s o equi alen ly he packe loss p obabili y a
a queue ed wi h a Bm a i al p ocess.
Finally, he maximum esponse ime o a queuing node wi h
bu e size band cons an a e se e wi h capaci y Ccan be
compu ed as:
T(max)=b+ 1
C(6)
By using (5) and (6), we can pe o m he dimensioning o
he equi ed capaci y o he DPGW.
VI. PES: PLANNER FOR THE EPC AS A SERVICE
Algo i hm 1 PES Algo i hm
Inpu : eNBs posi ions (eNB)
ialong wi h he numbe o UEs
hey se e NUE
eNB(i) = Pjuji, and he QoS specs T(DP )
budge ,
P(EP C)
budge , and T(CP )
budge .
Ou pu : eNBs assigmen (i.e., ik), and o al numbe o
p ocessing ins ances alloca ed o each EPC en i y pe
EC (e.g., mMME,mcSGW ,mcP GW , and mDP GW ).
1: [NUE
EC, ik]⇐Pa i ioning( (eNB),NUE
eNB)
2: o each k∈Kdo
3: Compu e he p ocessing delay budge s o he EPC CP
and DP, T(CP )
p oc−budge and T(DP )
p oc−budge , using (7) and (8).
4: Fo NU=NUE
EC(k), es ima e he ex e nal a i al p o-
cesses (λ(CP ),λ(DP ),α(DP ), and H(DP )) using (9)-(15)
5: [mMME(k),mcSGW (k),mcP GW (k),mDP GW (k)]
⇐Dimensioning(λ(CP ),λ(DP ),α(DP ),H(DP ),
T(CP )
p oc−budge ,T(DP )
p oc−budge ,P(EP C)
budge )
6: end o
In his sec ion, we p opose a heu is ic me hod o ind a
sub-op imal solu ion o he p oblem o mula ed in Sec ion IV.
To achie e a me hod wi h low-complexi y, we decouple he
p ocess o wo kload dis ibu ion among he candida e ECs
and he esou ces dimensioning o he EPC a each EC.
The heu is ic me hod, depic ed in Algo i hm 1, p oceeds as
ollows. Ini ially, he pa i ioning algo i hm assigns each eNB
o a candida e EC (see Algo i hm 2). The idea in his algo i hm
is o dis ibu e he wo kload as equally as possible among he
candida e ECs, while gua an eeing a maximum p opaga ion
delay o he backhaul ne wo k (max)
p op−backhaul. The algo i hm
ini ializes he wo kload assigned o each EC kNUE
EC(k), which
is measu ed as he numbe o assigned UEs, o ze o. Then, i
i e a i ely inds he candida e EC k∗wi h he lowes wo kload
alloca ed and i s nea es eNB i∗being no assigned ye . I
he p opaga ion delay limi be ween he EC k∗and he eNB
i∗is no iola ed, hen, he eNB i∗is a ached o he EC
k∗( i∗k∗= 1). O he wise, he EC k∗is excluded om he
se o candida e ECs K. The algo i hm ends when all eNBs
a e alloca ed. Obse e ha , in he wo s case scena io, he
algo i hm equi es NeNB +NEC i e a ions o assign all eNBs.
Please no e ha he numbe o UEs a ached o each eNB
is assumed o be known. On he one hand, i he E-UTRAN
is in he ope a ion phase, he ope a o can know accu a ely
he a e age numbe o UEs a ached o each eNB. On he
o he hand, i he E-UTRAN is no in he ope a ion phase, he
ope a o can es ima e he a e age numbe o UEs a ached o
each eNB om he popula ion densi y map o he co e age
geog aphical a ea and he expec ed ma ke sha es.
Algo i hm 2 E-UTRAN Pa i ioning Algo i hm
Requi e: All eNBs o he se Iha e o be assigned o an EC
o he se K.
Inpu : eNBs posi ions (eNB)
ialong wi h he numbe o UEs
hey se e NUE
eNB(i) = Pjuji, he ECs posi ions (EC)
k,
and he maximum p opaga ion ime o he backhaul
ne wo k (max)
p op−backhaul.
Ou pu : eNBs assigmen , i.e., ik
1: Ini ializa ion NUE
EC =−→
0, ik = 0
2: while I6=∅do
3: k∗= a g min
k∈K
(NUE
EC(k))
4: i∗= a g min
i∈I|| (EC)
k∗− (eNB)
i||
5: i || (EC)
k∗− (eNB)
i∗|| ≤ (max)
p op−backhaul ·c hen
6: I⇐I i∗, i∗k∗= 1
7: NUE
EC(k∗)⇐NUE
EC(k∗) + NUE
eNB(i∗)
8: else
K⇐K k∗
9: end i
10: end while
Once he eNBs assignmen is ca ied ou , he p ocess-
ing ime budge s o he EPC CP T(CP )
p oc−budge and DP
T(DP )
p oc−budge can be compu ed. To ha end, we can e alua e
T(SR)and T(DP )
max , in (1) and (2), o TMME,TcSGW ,
TcP GW , and T(max)
DP GW , equal o ze o, espec i ely. Fo mally,
T(SR)
0=T(SR)(TMME = 0, TcSGW = 0, TcP GW = 0) and
T(DP )
max0=T(DP )
max (T(max)
DP GW = 0). Then,
T(CP )
p oc−budge =T(CP )
budge −T(SR)
0(7)
T(DP )
p oc−budge =T(DP )
bugde −T(DP )
max0(8)
Then, once he e is an es ima ion o he numbe o UEs o
be se ed by each EC, we can also es ima e he agg ega ed
ex e nal a i al p ocesses, o bo h he LTE CP and DP, which
a e inpu s o he esou ces dimensioning algo i hm. We use
an abs ac ion o he LTE wo kload gene a ion p ocess, along
wi h a compound a ic model, o pe o m such an es ima ion.
We cha ac e ize s ochas ically hese a i al p ocesses in Sec-
ion VIII-A, whe e he cu e i ings a e p o ided o es ima e
he main pa ame e s o model hem as a unc ion o he use s’
numbe .
Finally, he esou ces dimensioning is ca ied ou (see
Algo i hm 3). The dimensioning o he EPC CP and DP
is pe o med sepa a ely. Since we a e conside ing only one
VNFC o he EPC DP, i s dimensioning simply equi es
sol ing nume ically (5). Fo he CP, we p opose a no el algo-
i hm which sea ches o he minimum numbe o p ocessing
ins ances o be alloca ed o he EPC CP o a gi en EC so ha
a p ocessing delay budge T(CP )
p oc−budge is me . The algo i hm
8
Algo i hm 3 Dimensioning Algo i hm
Inpu : P ocessing delay budge s o he EPC CP
T(CP )
p oc−budge and DP T(DP )
p oc−budge ;P(EP C)
budge ; Ex e nal
a i al p ocesses cha ac e iza ion o CP and DP (λ(CP ),
λ(DP ),α(DP ), and H(DP )).
Ou pu : numbe o physical co es alloca ed o each EPC
en i y mMME,mcSGW ,mcP GW , and mDP GW
1: {DATA PLANE:}
2: Sol e (5) nume ically o b≤T(DP )
p oc−budge ·C−1and
≈P(EP C)
budge o ob ain he equi ed DP p ocessing capaci y
C. Then, mDP GW =dC/µDP GW e.
3: {CONTROL PLANE:}
4: Ini ializa ion mMME =dλMME/µMMEe,mcSGW =
dλcSGW /µcSGW e,mcP GW =dλcP GW /µcP GW e,
MCP =mMME +mcSGW +mcP GW ,T(CP )
p oc =
8·TMME(mMME)+3·TcSGW (mcSGW )+2·
TcP GW (mcP GW );
5: while T(CP )
p oc > T(CP )
p oc−budge do
6: MCP ⇐MCP + 1
7: o each m∈ {mMME, ..., MCP −mcSGW −
mcP GW }∩Ndo
8: o each n∈ {mcSGW , ..., MCP −mMME −
mcP GW }∩Ndo
9: l=MCP −m−n
10: Taux = 8·TMME(m)+3·TcSGW (n)+2·TcP GW (l)
11: i T(CP )
p oc > Taux hen
12: T(CP )
p oc ⇐Taux,mMME ⇐m,mcSGW ⇐n,
mcP GW ⇐l
13: end i
14: end o
15: end o
16: end while
i e a es un il he p ocessing delay budge is ul illed. A each
i e a ion, i inc emen s by one he numbe o p ocessing
ins ances MCP alloca ed o he EPC CP. Fo a gi en MCP ,
he algo i hm explo es di e en combina ions o dis ibu e
hese ins ances among he di e en VNFCs o be dimensioned
(e.g., MME, cSGW, cPGW), and choose he one p o iding
he lowes p ocessing delay. To achie e he linea complexi y,
he sea ch space is limi ed a each i e a ion (see line 12 o
Algo i hm 3). In he algo i hm, Tmme(m),TcSGW (n), and
TcP GW (l)deno e, espec i ely, he mean esponse imes o
he MME, cSGW, and cPGW o a gi en numbe o alloca ed
p ocessing ins ances m,n, and l. These mean esponse imes
a e es ima ed by using he QNA me hod ( e e o Appendix
A). Please no e ha , al hough i is no explici ly included in
Algo i hm 3, o each ‘p ocessing ins ances alloca ion (m,
n,l), i is necessa y o e-es ima e bo h he in e nal low
pa ame e s a each queue, using (16)-(22), and he ansi ion
p obabili y ma ix, using (31)-(41).
The numbe o ins ances o , equi alen ly, he num-
be o i ualiza ion con aine s o each EPC en i y
a a gi en EC can be simply compu ed as ollows:
dmMME/mmaxe,dmcSGW /mmaxe, and dmcP GW /mmaxe,
and dmDP GW /mmaxe.
Fig. 5: Ma ko chain based model o social ne wo king.
Fig. 6: Scena io ealiza ion wi h a popula ion densi y o 1000
use s pe km2.
VII. EXPERIMENTAL SETUP
To alida e he models de eloped in his wo k and o
assess ou solu ion o EPC slices planning, we employed wo
so wa e ools: i) he NSP [20], and ii) a sys em-le el simula o
o an LTE ne wo k.
A. Ne wo k Slice Planne
We used he NSP [20] o gene a e he syn he ic signaling
and da a a ic in an LTE ne wo k. We ex ended he compound
a ic model o his ool by including he a ic models
employed in [27]. The se up o each se ice ype (see Table
II and Fig. 5) elies on models aken om he li e a u e,
which a e de i ed om eal aces. Speci ically, he main
e e ences used o he di e en se ices se up a e [46] o
social ne wo king; [47] o Mobile Ins an Messaging; [48] o
web b owsing; [49] and [45] o ideo s eaming; and [50] and
[51] o ideo calls. Acco ding o [52], he se ices conside ed
accoun o mo e han 70% o he peak agg ega e a ic in he
Ame ican mobile access ne wo ks.
The ou pu aces o he NSP we e used o cha ac e ize
he agg ega e packe a i al p ocesses a he LTE CP and DP.
These aces a e also used as inpu s o ou sys em-le el LTE
ne wo k simula o .
B. LTE ne wo k simula o
The sys em-le el LTE ne wo k simula o was de eloped
wi hin he NS3 en i onmen . I implemen s he messages
exchange be ween he main LTE ne wo k en i ies. The aces
gene a ed om he NSP a e used as inpu s o he simula o
15
APPENDIX A
QNA METHOD
This appendix desc ibes he main s eps ollowed by he
QNA me hod o es ima e he mean esponse ime o each
indi idual queue in a ne wo k o G/G/m queues.
A. In e nal lows pa ame e s es ima ion
As in he case o Jackson’s ne wo ks, he mean a i al a e
o each queue λkcan be compu ed by sol ing he low balance
equa ions:
λk=λ0k+
K
X
i=1
λi·pik (16)
The mos in e es ing aspec o he QNA me hod is ha
i es ima es he Squa ed Coe icien o Va ia ion (SCV) o
he agg ega ed a i al p ocess o each queue c2
ak om he
ollowing se o linea equa ions:
c2
ak =ak+
K
X
i=1
c2
aibik,1≤k≤K(17)
ak= 1 + ωk(q0kc2
0k−1)
+
K
X
i=1
qik[(1 −pik) + pikρ2
ixi](18)
bik =ωkqikpik(1 −ρ2
i)(19)
xi= 1 + m−0.5
i(max{c2
si,0.2}−1) (20)
ωk=1 + 4(1 −ρk)2(γk−1)−1(21)
γk= K
X
i=0
q2
ik!−1
(22)
whe e q0k=λ0k/λkand qik = (λi··pik)/λka e espec i ely
he p opo ion o a i als o he node kcoming om i s
ex e nal a i al p ocess and node i, and ρk=λk/(µk·mk)is
he u iliza ion o he node k.
B. Mean esponse ime compu a ion pe node
Once he λkand c2
ak o he agg ega ed a i al p ocess o
each node ka e es ima ed, we can compu e he mean esponse
ime o each node k. I node khas only one se e (mk= 1),
Tkcan be es ima ed as:
Tk=ρk·(c2
ak +c2
sk)·β
2·µk(1 −ρk)+1
µk
(23)
wi h
β=(exp(−2·(1−ρk)·(1−c2
ak )2
3·ρk·(c2
ak +c2
sk ))c2
ak <1
β= 1 c2
ak ≥1(24)
I , by con as , he node kis a GI/G/m queue (mk=m),
Tkcan be es ima ed as:
Tk= 0.5·c2
ai +c2
si·WM/M/m
k+1
µk
(25)
whe e WM/M/m
kis he mean wai ing ime o a M/M/m queue,
and can be compu ed as:
WM/M/m
k=C(mk,λk
µk)
mkµk−λk
(26)
and C(m, ρ) ep esen s he E lang’s C o mula which has he
ollowing o mula ion:
C(m, ρ) = (m·ρ)m
m!·1
1−ρ
Pm−1
k=0
(m·ρ)k
k!+(m·ρ)m
m!·1
1−ρ(27)
APPENDIX B
TRANSITION PROBABILITIES FOR THE LTE CP QUEUING
MODEL
This appendix includes exp essions o compu e he ansi-
ion p obabili ies o he p oposed LTE CP queuing model.
Le VEdeno e he isi a io o he CP en i y E∈
{UE, eNB, MME, cSGW, cP GW, HSS, PCRF}which is
de ined as he a e age numbe o isi s o en i y Eby a sig-
naling p ocedu e du ing i s li e ime in he ne wo k. Fo mally,
VE=λE/PEλ0E=λE/(λ0UE +λ0eNB). Please no e ha
VEis equal o he a e age numbe o packe s o be p ocessed
by he LTE CP en i y Epe con ol p ocedu e. Then,
VE=PCP λCP ·n(E)
CP
PCP λCP
(28)
whe e n(E)
CP is he numbe o packe s o be p ocessed by
he LTE CP en i y E o he con ol p ocedu e CP ∈
{SR, S1R, HO, TAU}.
The isi a ios and he ansi ion p obabili ies a e ela ed
h ough (16) ( low balance equa ions):
VE=λ0E
PEλ0E
+X
E
VE·pE1→E2(29)
The ansi ion p obabili ies also sa is y
p0E1+X
E2
pE1→E2= 1 (30)
Assuming ha he wo kload is dis ibu ed among he ins ances
o he VNFCs (e.g., MME, cSGW, and cPGW), acco ding o
hei capaci ies, i.e., VEl=m(E)
l/(Plm(E)
l)·VE, and using
(29) and (30), we can compu e he ansi ion p obabili ies o
ou LTE CP queuing model by he ollowing equa ions:
peNB
UE =
VUE −λ(U E)
0
PEλ(E)
0
VeNB
(31)
peNB
MMEl=m(MME)
l
Plm(MME)
l·(1 −peNB
UE )(32)
pMMEl
eNB =
VeNB −λ(eN B)
0
PEλ(E)
0−VUE
VMME
(33)
pMMEl
cSGWl=
m(cSGW )
l
Plm(cSGW )
n·1−pMMEl
eNB −pMMEl
HSS −1
VMME (34)

16
pMMEl
HSS =VHSS
VMME
(35)
pcSGWl
MMEl=m(MME)
l
Pmm(MME)
m· 1−X
l
pcSGWl
cP GWl!(36)
pcSGWl
cP GWl=m(cP GW )
l
Pmm(cP GW )
m·(VP GW −VP CRF )
VSGW
(37)
pcP GWl
cSGWl=m(cSGW )
l
Pmm(cSGW )
m·1−VP CRF
VP GW (38)
pP GWl
P CRF =VP CRF
VP GW
(39)
pHSS
MMEl=m(MME)
l
Pmmm
(40)
pP CRF
cP GWl=m(cP GW )
l
Pnm(cP GW )
n
(41)
Please no e ha he ansi ion p obabili ies depend on he
a e age numbe o packe s o be p ocessed o each LTE CP
en i y pe con ol p ocedu e, which is equal o he isi a io o
he en i y; he ex e nal a i al p ocesses λ0UE and λ0eNB; and
he numbe o p ocessing ins ances assigned o each VNFC
ins ance m(C)
l.