Ci a ion: Zabala, L.; Doncel, J.; Fe o,
A. Op imali y o a Ne wo k
Moni o ing Agen and Valida ion in a
Real P obe. Ma hema ics 2023,11, 610.
h ps://doi.o g/10.3390/ma h11030610
Academic Edi o : Kagan Eugene
Recei ed: 20 Decembe 2022
Re ised: 13 Janua y 2023
Accep ed: 20 Janua y 2023
Published: 26 Janua y 2023
Copy igh : © 2023 by he au ho s.
Licensee MDPI, Basel, Swi ze land.
This a icle is an open access a icle
dis ibu ed unde he e ms and
condi ions o he C ea i e Commons
A ibu ion (CC BY) license (h ps://
c ea i ecommons.o g/licenses/by/
4.0/).
ma hema ics
A icle
Op imali y o a Ne wo k Moni o ing Agen and Valida ion in a
Real P obe
Luis Zabala 1, Josu Doncel 2,* and A mando Fe o 1
1Depa men o Communica ions Enginee ing, Uni e si y o he Basque Coun y, UPV/EHU,
48013 Bilbao, Spain
2Depa men o Ma hema ics, Uni e si y o he Basque Coun y, UPV/EHU,
48940 Leioa, Spain
*Co espondence: [email p o ec ed]
Abs ac :
The e olu ion o commodi y ha dwa e makes i possible o use his ype o equipmen
o implemen a ic moni o ing sys ems. A p elimina y empi ical e alua ion o a ne wo k a ic
p obe based on Linux indica es ha he sys em pe o mance has signi ican losses as he ne wo k a e
inc eases. To assess his issue, we conside a model wi h wo andem queues and a mo ing se e . In
his sys em, we o mula e a h ee-dimensional Ma ko Decision P ocess in con inuous ime. The goal
o he p oposed model is o de e mine he posi ion o he se e in each ime slo so as o op imize
he sys em pe o mance which is measu ed in e ms o h oughpu . We i s o mula e an equi alen
disc e e- ime Ma ko Decision P ocess and we p opose a nume ical me hod o cha ac e ize he
solu ion o ou p oblem in a gene al se ing. The solu ion we ob ain in his p oblem has been es ed
o a wide ange o scena ios and, in all he ins ances, we obse e ha he op imali y is close o
a h eshold ype policy. We also conside a eal p obe and we alida e he good pe o mance o
h eshold policies in eal applica ions.
Keywo ds:
ne wo k moni o ing; pe o mance e alua ion; Ma ko decision p ocess; Linux ne wo k
subsys em
MSC: 90B15; 60K30; 93E20
1. In oduc ion
The complexi y o mode n communica ion ne wo ks has d ama ically inc eased in
he las yea s. As a esul , a ic moni o ing o such complex sys ems, which is equi ed
by ne wo k ope a o s o cope wi h hei use s’ needs o gua an ee hei sa is ac ion, has
become an ex emely challenging ask.
Di e en applica ions can be ela ed o a ic moni o ing sys ems, anging om eal-
ime mul imedia a ic moni o ing [
1
] o in usion de ec ion sys ems [
2
] and ne wo k
an i i uses [
3
], o men ion a ew. Fu he mo e, nowadays, wi h he p oli e a ion o cloud
se ices, adi ional ne wo k and da a cen e en i onmen s mus be complemen ed wi h
cloud ne wo k moni o ing ools in o de o keep ack o pe o mance indica o s. This
is accomplished by deploying some ligh weigh moni o ing so wa e agen s ha collec
s a is ics on physical o i ual se e s and physical o i ual ne wo k de ices [
4
]. Then,
each o hese agen s will send hei da a o a eposi o y ha ypically p o ides big da a-
based analy ics o ale ing, diagnos ics and displaying summa ized in o ma ion.
I has been shown ha he ad ances o commodi y ha dwa e allow so wa e o a ain
good pe o mance while he main ad an ages o lexibili y and modula i y s ill hold,
see [5,6]. This can be achie ed, o ins ance, wi h an e icien use o he ke nel modules.
Wi hin he da a moni o ing sys ems based on commodi y ha dwa e, Linux-based
ne wo k end sys ems ha e been widely deployed [
7
,
8
]. F om a ne wo k pe o mance
pe spec i e, Linux ep esen s an oppo uni y since i is amenable o op imiza ion and
Ma hema ics 2023,11, 610. h ps://doi.o g/10.3390/ma h11030610 h ps://www.mdpi.com/jou nal/ma hema ics
Ma hema ics 2023,11, 610 2 o 23
uning ne wo k s ack, due o i s open sou ce suppo . Whene e we wan o know how
ou ne wo k beha es, i is impo an o ha e a packe cap u ing de ice able o achie e a es
simila o hose on he ne wo k, o ob ain he mos comp ehensi e iew.
We ep esen in Figu e 1 he e olu ion o packe s since en e ing he Linux end sys-
em [
9
] o hei deli e y o he moni o ing applica ion. This is achie ed in h ee di e en
s ages:
•
The packe is ans e ed om Ne wo k In e ace Ca d (NIC) o he ing bu e . Since
his is a Di ec Memo y Access (DMA) ans e , all o he ac ions ela ed o his packe
mo emen a e comple ed wi hou consuming any CPU’s esou ces.
•
The packe is ans e ed om he ing bu e o he analysis bu e , d i en by a
so wa e in e up eques (so i q) [10]. This is he cap u ing s age in Figu e 1.
•
Finally, he packe s a e ea ed by he moni o ing applica ion and hey a e pulled ou
om he analysis bu e . This is he analyzing p ocess in Figu e 1.
Figu e 1. Packe ecei ing p ocess wi h Linux ne wo king subsys em.
I is ema kable ha , while he DMA ans e does no ha e any CPU consump ion,
bo h he cap u ing p ocess and he analyzing p ocess do ha e i . Addi ionally, i he e is only
one p ocesso , bo h p ocesses will no be unning a he same ime. As a consequence, in
o de o design a ic moni o ing sys ems op imally, i is in e es ing o build ma hema ical
models which allow us o s udy hei pe o mance heo e ically. Speci ically, he e is an
inc easing demand o analy ical models o aid p obe designe s in p edic ing how e ec i e
and e icien is he ne wo k p obe when subjec ed o high-speed a ic. In addi ion,
modeling and analyzing he pe o mance o ne wo k p obes can be ex emely use ul in
gaining a deepe unde s anding o he p obe’s beha io and cha ac e is ics. P obe designe s
and sys em adminis a o s can iden i y bo lenecks and key pa ame e s ha impac i s
pe o mance, and hen pe o m he necessa y uning o op imal pe o mance. Analysis
can p o ide quick answe s o nume ous design and ope a ional ques ions.
Ou objec i e is o de elop a ma hema ical model ha desc ibes he beha io o a
ne wo k a ic moni o ing agen and o gain insigh s abou how o op imize i s pe o -
mance. Fo his pu pose, we conside an analy ical queuing model de eloped o he
s udy and analysis o he pe o mance o a single-p ocesso ne wo k a ic p obe buil on
o - he-shel ha dwa e. We ep esen he wo main s ages (cap u ing and analysis) o he
a ic moni o ing sys em using a single-se e andem queue. Howe e , as we ep esen a
single-p ocesso sys em, he wo se e s o he andem queue canno be ac i e simul ane-
ously and, o his eason, we p opose a Ma ko Decision P ocess (MDP) o choose which
se e has o be ac i e a each ime and ob ain he op imum scheduling policy ha allows
us o achie e a be e pe o mance in e ms o ne wo k analysis h oughpu . The model
p oposal is e alua ed and, inally, he esul s ob ained om he model a e compa ed o
hose ha we ob ain in a eal Linux-based ne wo k p obe.
The pe o mance o a ic analysis solu ions has been s udied expe imen ally unde
mul i-Gigabi ne wo k en i onmen s [
11
–
13
]. Howe e , ew o hese solu ions ha e been
modeled and s udied analy ically in o de o assess he pe o mance and beha io o sys em
pe o mance unde hea y ne wo k loads. Fo ins ance, in [
14
], he Linux sys em’s packe
ecei e p ocess is ep esen ed by he oken bucke algo i hm and queuing p ocesses. The
au ho s o [
15
] p esen a ini e queuing sys em wi h mul iple s ages o se ice o ep esen
he beha io o a ule-based ne wo k i ewall. Ano he app oach is he one p esen ed in [
16
]
Ma hema ics 2023,11, 610 3 o 23
which desc ibes an
Mx/GB/
1
/K
queue wi h aca ions o o mula e high-speed packe I/O
amewo ks. In [
17
,
18
], he analy ical models, based on
M/M/
1 and
M/M/c/K
queues,
es ima e he compu ing esou ces equi ed o sa is y he Quali y o Se ice (QoS) a ge s o
some cloud moni o ing applica ions.
The e a e some esea ches o measu e powe consump ion o packe p ocessing en-
gines [
19
,
20
]. Among hem, we would like o men ion [
19
]. This wo k p oposes an
Mx/D/
1
queue wi h a se up pe iod and i adds he esolu ion o an op imiza ion p oblem o s udy
he ade-o be ween ene gy consump ion and ne wo k pe o mance indexes.
To inish he ela ed wo k e iew, we would like o highligh he applica ion o queuing
models in he ield o Fi h-Gene a ion (5G) ne wo ks. He e, mul iple so wa e componen s,
e e ed o as Vi ual Ne wo k Func ions (VNF), a e connec ed and hey p ocess packe s o
p o ide a se ice chain. Ne wo k moni o ing can be one o hose se ice chains. This ype
o model ep esen s each VNF wi h a queue. Fo ins ance, in [
21
], he au ho s o mula e an
open queuing ne wo k o G/G/m queues o p edic he pe o mance o gene ic so wa ized
ne wo k se ices in e ms o esponse ime. O he examples o his esea ch line can be [
22
],
which add esses a p oblem o VNF placemen by using an open queuing ne wo k o
M/M/1 queues, and [23], which explains how o model he building block o a 5G node.
As a as we know, mos o he p e ious s udies p esen ed in he li e a u e do no ake
accoun o he ela ion be ween he cap u ing s age and he analysis s age o a ne wo k
moni o ing sys em. Fo his eason, he main con ibu ion o his a icle is o s udy he
applica ion o a no el analy ical model based on classical concep s o MDPs, conside ing
p ocessing and bu e ing limi a ions o he cap u e-analysis sys em and op imizing he
pe o mance in e ms o analysis h oughpu .
The emainde o he pape is o ganized as ollows. The queuing model o he
cap u ing and analysis sys em is p esen ed in Sec ion 2. Then, in Sec ion 3, he op imiza ion
p oblem based on MDP is de ailed including i s pa ame e s and he solu ion. The model
alida ion and he ob ained esul s a e discussed in Sec ions 4and 5. Finally, we conclude
he pape in Sec ion 6.
2. Model Desc ip ion
In his sec ion, we ep esen he packe ecei ing p ocess o a a ic moni o ing sys em
(see Figu e 1) using an open queuing ne wo k, as d awn in Figu e 2. The sys em desc ibed
in Sec ion 1is modeled by a andem queue, and compu e consump ions a e ela ed o
se ice capaci ies o he queuing ne wo k.
Figu e 2. Queuing sys em ep esen a ion o he case o one single p ocesso .
As can be seen in Figu e 2, packe s a i e a he i s queue o he model. This packe
a i al co esponds o he DMA ans e om he NIC o he ing bu e wi hou any CPU
consump ion. Thus, he i s queue ep esen s he ing bu e o he cap u ing s age whe e
he packe s a e s o ed be o e being p ocessed by he so i q. The model assumes ha he
i s queue, he cap u ing queue hence o h, has go a bounded capaci y o M packe s. I a
packe a i es when he cap u ing queue is ull, hen he packe is d opped ou o he sys em
wi hou being p ocessed. Nex , he i s se e ea s he packe s o he cap u ing queue.
This ep esen s he so i q p ocessing. Fo his eason, he se ice ime o he i s se e is
ela ed o he p ocessing ime equi ed by he CPU o he so i q.
Ma hema ics 2023,11, 610 4 o 23
Once a packe is se ed by he i s se e , i.e., he so i q p ocessing o he packe is
comple ed, i mo es o he second s age o he queuing ne wo k. This is ela ed o he
analysis s age whe e packe s a e wai ing in he analysis bu e un il hey a e p ocessed by
he moni o ing applica ion o he ne wo k a ic moni o ing sys em (see Figu e 1). In he
model, he second wai ing queue and he second se e ep esen he analysis bu e and
he analysis p ocessing, espec i ely. Jus as he cap u ing queue, he second queue, e e ed
o as he analysis queue hence o h, is also conside ed a ini e queue. I has go a capaci y o
N packe s and, when i is ull, he a i ing packe s canno en e i . Finally, as soon as he
second se e comple es he analysis p ocessing o a packe , his packe lea es he ne wo k,
meaning ha he moni o ing applica ion has comple ed he ask ela ed o his packe .
In o de ha he analy ical solu ion does no become unmanageable, we assume ha
packe s a i e a he cap u ing queue ollowing a Poisson p ocess wi h pa ame e
˜
λ
. We also
suppose ha bo h he ime needed by he p ocesso o comple e he cap u ing p ocess o
each packe and he analysis ime o each packe in he second s age ollow an exponen ial
andom a iable wi h pa ame e s
˜
µ1
and
˜
µ2
, espec i ely. I is p o en ha packe a i als
do no ollow a Poisson p ocess o some ypes o a ic such as E he ne ne wo k and ha
hey a e a he bu s y [
24
], bu he case we a e dealing wi h is sligh ly di e en . In ou case,
he packe a i al is no di ec ly he a ic o he E he ne ne wo k, bu i is he incoming
packe s om he ne wo k ca d’s bu e o he ke nel memo y a ea ia DMA. Rega ding
se ice a e modeling, al hough he p og am’s code has a qui e de e minis ic beha io ,
some andomness is in oduced by Poisson incoming a ic, a iable leng h o packe s and
ke nel schedule unce ain y. Despi e o hese limi a ions, we will keep wo king wi h hese
assump ions o simplici y o he analysis and, as will be demons a ed la e , he esul s
ob ained om ou model we e closely ma ched o esul s ob ained om eal expe imen al
measu emen s.
The queuing ne wo k consis s o wo se e s, howe e , as we a e going o conside
he case o a single-p ocesso sys em, his single CPU has o di ide i s e o s among he
wo asks (cap u ing and analysis). Fo his eason, in he model, he wo se e s o he
queuing ne wo k (bo h o hem named as #CPU1 in Figu e 2) canno be ac i e a he same
ime. The e o e, he sys em consis s o a wo-s age andem queue a ended by a mo ing
se e . Acco ding o he p e e ences o he sys em schedule , labeled as he decision make
hence o h, he p ocesso is alloca ed o he cap u ing queue o he analysis queue.
The way he decision make chooses he igh ac ion o he dynamic p ocesso a any
ime has a big impac in he pe o mance o he ne wo k, measu ed in e ms o analysis
h oughpu . Alloca ing i oo much ime o he analysis queue may lead o packe loss in he
cap u ing queue, whe eas alloca ing i oo much o he cap u ing queue may dec ease he o al
capaci y o packe analysis. In his pape , we aim o p o ide insigh s in o how he decision
make should p oceed wi h he goal o op imizing he pe o mance o he sys em. Ou
analysis is based on a h ee-dimensional MDP. This allows us o ob ain a dynamic policy
ha ells he decision make whe e o alloca e he p ocesso a any ime. This decision will
depend on he numbe o packe s wai ing in each queue and he posi ion o he p ocesso
when he decision is made.
An Equi alen Disc e e Time Model
Nex , we build a disc e e ime model, which desc ibes he dynamics o he con inuous
ime model in oduced abo e. This is done ia uni o miza ion echniques. The disc e e
ime model allows us o es i s accu acy when explaining he pe o mance o he a ic
moni o ing sys em. On he o he hand, i p o ides an easie op imiza ion amewo k,
which helps o ind he p o ocol ha op imizes he analysis h oughpu .
Using uni o miza ion a gumen s, we de ine he ollowing quan i ies:
λ=˜
λ
˜
λ+˜
µ1+˜
µ2
. (1)
Ma hema ics 2023,11, 610 5 o 23
µ1=˜
µ1
˜
λ+˜
µ1+˜
µ2
. (2)
µ2=˜
µ2
˜
λ+˜
µ1+˜
µ2
. (3)
Thus, we mo e om a con inuous o a disc e e sys em whe e:
•
A any ime slo , and gi en ha he cap u ing queue is no ull (wi h
M
packe s in i s
bu e ), a new packe a i es o he ne wo k wi h p obabili y λ;
•
I he p ocesso is wo king in he cap u ing queue, and he queue is no emp y, he
p ocesso cap u es a packe wi hin one ime slo wi h p obabili y µ1;
•
I he p ocesso is wo king in he analysis queue, and his queue is no emp y, he p o-
cesso inishes he analysis s age o a packe wi hin one ime slo wi h p obabili y µ2.
The p obabili ies gi en in Equa ions (1)–(3) a e independen om one ime slo o
ano he . The leng h o each ime slo , in uni s o ime, is gi en by
s=1
˜
λ+˜
µ1+˜
µ2
. (4)
3. MDP Fo mula ion
In his sec ion, we p esen an op imiza ion p oblem which leads o he maximiza ion o
he analysis h oughpu in ou disc e e ime model desc ibed in Sec ion 2. This op imiza ion
p oblem alls in he amewo k o he MDPs. An MDP consis s o a s a e space, an ac ion
space, and a ewa d s uc u e. Whene e he decision make chooses among he a ailable
ac ions, i ea ns a ewa d which depends on he cu en s a e o he sys em, and a e
ecei ing i , he sys em e ol es i s s a e o a new one. The goal o he decision make is o
ind he igh sequence o ac ions, maximizing he expec ed o al ewa d ea ned om he
sys em.
I is easy o see ha he model can be seen as a a h ee-dimensional disc e e ime MDP.
We conside a ime-slo ed sys em
∈ T :={
0, 1, 2,
. . . }
, a which he sys em can make
decisions. The ime epoch
co esponds o he beginning o he ime pe iod
. A any
,
he decision make has o choose among wo ac ions: ei he alloca e he p ocesso o he
cap u ing queue o he analysis queue. The s a e o he sys em is ep esen ed by he numbe
o packe s wai ing in bo h queues and he posi ion o he p ocesso be o e he decision is
made.
We p o ide below he elemen s ha desc ibe he e olu ion o he sys em:
•A:={C
,
A}
is he ac ion se o he sys em. A any ime, selec ing ac ion
C
ep esen s
alloca ing he p ocesso o he i s queue o packe cap u ing, whe eas ac ion
A
ep esen s alloca ing he p ocesso o he second queue o packe analysis;
•S:=M × N × A
is he s a e space o he sys em. I he s a e o he sys em is
s=
(m
,
n
,
a)
, wi h
m∈ M :={
0, 1,
. . .
,
M}
,
n∈ N :={
0, 1,
. . .
,
N}
and
p∈ A
, his means
ha he e a e
m
packe s queuing in he cap u ing queue,
n
packe s queuing in he analysis
queue, and ha he p e ious ac ion aken by he sys em has been
p
, ep esen ing he
cu en posi ion o he p ocesso ;
•Ra
s
is he one-pe iod expec ed ewa d ea ned by he sys em i ac ion
a∈ A
is chosen when
he sys em is in s a e s∈ S;
•Pa:=pa
s,s0
is he s a e- ansi ion ma ix i ac ion
a∈ A
is decided a he beginning
o a pe iod. I
s= (m
,
n
,
p)
and
s0= (m0
,
n0
,
p0)
, hen he ansi ion p obabili y o
mo ing om s a e s o s a e s0is pa
s,s0.
3.1. Rewa d S uc u e
Since he goal o he decision make is o maximize he analysis h oughpu , we es ablish
ha i ea ns 1 uni o ewa d whene e a packe lea es he analysis queue due o i s comple-
Ma hema ics 2023,11, 610 6 o 23
ion. Mo eo e , we also conside ha he decision make incu s a cos whene e i decides o
change he posi ion o he p ocesso . This cos ep esen s he ime ha he p ocesso spends
changing i s posi ion, e ec i ely a ec ing he pe o mance o he sys em. We deno e his
swi ching cos by
SC
. The one-pe iod expec ed ewa ds
Ra
s
, depending on he cu en s a e
s= (m,n,p)and he ac ion achosen by he decision make , can be exp essed as ollows:
Ra
s=(m,n,p)=
0, p=C,a=C, 0 ≤n≤N,
−SC,p=C,a=A,n=0,
µ2−SC,p=C,a=A, 1 ≤n≤N,
−SC,p=A,a=C, 0 ≤n≤N,
0, p=A,a=A,n=0,
µ2,p=A,a=A, 1 ≤n≤N,
0≤m≤M(5)
The ewa d is ze o when (i) he p ocesso s ays in he cap u ing queue o (ii) he
p ocesso s ays in analysis queue and he analysis queue is emp y. The ewa d is minus
he swi ching cos when (i) he p ocesso mo es om he cap u ing queue o he analysis
queue and he analysis queue is emp y o (ii) he p ocesso mo es om he analysis queue
o he cap u ing queue. The ewa d is equal o
µ2
when he p ocesso s ays a he analysis
queue and his queue is no emp y. Finally, he ewa d is
µ2
minus he swi ching cos when
he p ocesso mo es om he cap u ing queue o he analysis queue and he analysis queue
is no emp y.
Obse e ha his p oblem o minimizing cos s is no equi alen o maximizing he
h oughpu , bu , as we will see la e , he ob ained esul s ul ill he goal o his wo k.
3.2. T ansi ion P obabili ies
As men ioned be o e,
pa
s,s0
is he p obabili y o mo ing om he s a e
s= (m
,
n
,
p)
o
he s a e
s0= (m0
,
n0
,
p0)
when ac ion
a
is aken a he beginning o he ansi ion pe iod.
Thus, he alue o
p
, he posi ion o he p ocesso a he s a e
s= (m
,
n
,
p)
, coincides wi h
he ac ion
a
, i.e.,
p=a
a he s a e
s
. On he o he hand, he posi ion
p0
o he s a e
s0
will
be he alue o he ac ion decided a he beginning o he nex ansi ion pe iod. The alue
o p0does no a ec he ansi ion p obabili y.
The ansi ion p obabili ies
pa
s,s0
can be sepa a ely ob ained by using he wo cases o
a
.
Fi s ly, o ac ion a=C, s a e s= (m,n,p)wi h p=a=C, and s a e s0= (m0,n0,p0),
pa=C
s,s0=
1−λm=0, 0 ≤n≤N,m0=0, n0=n
λ0≤m≤M−1, 0 ≤n≤N,m0=m+1, n0=n
1−λ−µ11≤m≤M−1, 0 ≤n≤N,m0=m,n0=n
µ11≤m≤M, 0 ≤n≤N−1, m0=m−1, n0=n+1
µ11≤m≤M,n=N,m0=m−1, n0=N
1−µ1m=M,n=0, m0=M,n0=0
1−λ−µ1m=M, 1 ≤n≤N,m0=M,n0=n
λm=M, 1 ≤n≤N,m0=M,n0=n
(6)
In Equa ion (6), he p obabili ies assume ha
λ(1−µ1)≈λ(7)
(1−λ)µ1≈µ1(8)
(1−λ)(1−µ1)≈1−λ−µ1(9)
The abo e exp essions ep esen he ansi ion p obabili ies when he p ocesso is in
he cap u e node. We desc ibe a ew o hem he e. When he cap u e queue is no ull, i
ecei es a new packe wi h p obabili y
λ
. When his queue is emp y, he p obabili y ha he
s a e is no modi ied is 1
−λ
, whe eas when i is no emp y, his p obabili y is 1
−λ−µ1
.
When he cap u ing queue is ull, he p obabili y ha he s a e does no change is λ.
Ma hema ics 2023,11, 610 7 o 23
Secondly, o ac ion
a=A
, s a e
s= (m
,
n
,
p)
wi h
p=a=A
, and s a e
s0=
(m0,n0,p0),
pa=A
s,s0=
1−λ0≤m≤M−1, n=0, m0=m,n0=0
λ0≤m≤M−1, 0 ≤n≤N,m0=m+1, n0=n
1−λ−µ20≤m≤M, 1 ≤n≤N,m0=m,n0=n
µ20≤m≤M, 1 ≤n≤N,m0=m,n0=n−1
1m=M,n=0, m0=M,n0=0
λm=M, 1 ≤n≤N,m0=M,n0=n
(10)
Jus as he case o ac ion a=C, in Equa ion (10), he p obabili ies assume ha
λ(1−µ2)≈λ(11)
(1−λ)µ2≈µ2(12)
(1−λ)(1−µ2)≈1−λ−µ2(13)
The abo e exp essions ep esen he ansi ion p obabili ies when he p ocesso is in
he analysis node. As i can be seen, he ansi ion p obabili ies a e analogous o hose o
he case whe e he p ocesso is in he cap u ing node.
Thus, he dynamics o he sys em is cap u ed by he s a e p ocess
s(˙
)
and he ac ion
p ocess
a(˙
)
, which co espond o s a e
s( )∈ S
and ac ion
a( )∈ A
a all ime epoch
∈ T
.
As a esul o choosing ac ion
a( )
a he beginning o ime slo
∈ T
while he sys em is in
s a e
s( )
, he decision make ea ns a ewa d, and he sys em e ol es i s s a e o he nex
ime slo , +1.
3.3. Op imiza ion P oblem
Now, we desc ibe, in mo e de ail, he p oblem we conside in his pape , and we
gi e i s ma hema ical o mula ion. We deno e by
Πs,a
he se o andomized and non-
an icipa ing policies, a ailable o he decision make . These policies assume ha , a ime
∈ T
, he decision make has exac in o ma ion abou he s a e p ocess un il
, i.e., i knows
s(
0
)
,
s(
1
)
,
. . .
,
s( )
, and i also knows all he decisions ha ha e been made be o e, i.e,
a(0),a(1), . . . , a( −1).
The p oblem is o ind a policy
π∈Πs,a
ha maximizes he expec ed o al u u e
ewa ds conside ing an in ini e ime ho izon, i.e.,
max
π∈Πs,a
Eπ
0 ∞
∑
=0
Ra( )
s( )!, (14)
whe e
E0
deno es he expec a ion o e he e olu ion o he sys em, condi ioned o he
in o ma ion a ailable a =0.
3.4. Nume ical Solu ion
P oblem (14) is a s anda d MDP, o which i is well known ha he e exis s an op imal
policy which is de e minis ic (non- andomized), s a iona y (Ma ko ian) and independen
o he ini ial s a e. In pa icula , his implies he exis ence o an op imal policy which, a
any ime
, only depends on he s a e o he sys em
s( )
. An op imal Ma ko ian solu ion
o he MDP can be ob ained by sol ing he Bellman equa ion (see [
25
,
26
]). Howe e , he
ac ha ou MDP lies on a h ee-dimensional s a e space, oge he wi h he exis ence o
swi ching cos s, makes he Bellman equa ion in ac able om an analy ic poin o iew, and
hus, we a e o ced o sol e he MDP using nume ical me hods. In pa icula , ou solu ions
a e ob ained ollowing he alue i e a ion algo i hm, which we b ie ly desc ibe below:
Ma hema ics 2023,11, 610 8 o 23
Le us deno e by
Vπ
s
he alue unc ion o he MDP s a ing om s a e
s∈ S
a ime
=0 and ollowing policy π∈Πs,a, i.e.,
Vπ
s=Eπ ∞
∑
=0
Ra( )
s( )|s(0) = s!. (15)
The alue i e a ion algo i hm p oceeds as ollows:
• Fix an a bi a y ini ial alue V0
s o all s∈ S.
• Fo i=1, 2, . . . , n, calcula e ecu si ely:
–Vi
s=max
a∈A {Ra
s+∑
s0∈S
pa
s,s0Vi−1
s0}
–ai
s=a g max
a∈A
{Ra
s+∑
s0∈S
pa
s,s0Vi−1
s0}
• Le ng ow
This i e a i e algo i hm con e ges o he op imal solu ion o he MDP, p o iding in
he limi wo impo an alues:
• lim
i→∞
Vi
s=Vπ∗
s, and
• lim
i→∞ai
s=aπ∗
s,
whe e
Vπ∗
s
is he alue unc ion ollowing an op imal policy and s a ing om s a e
s
, and
aπ∗
s ep esen s he ac ion chosen by he op imal policy whene e he sys em is in s a e s.
The goal is, he e o e, o ob ain nume ically he op imal policy o he h ee-dimensional
s a e space MDP, by using a se o pa ame e s which e lec he expe imen s ca ied ou in
a eal a ic moni o ing sys em. Once we ob ain his policy, we will be able o compa e
he pe o mance o he eal sys em and he one ob ained by he Ma ko ian model which
ollows he op imal policy. This imp o emen migh be seen as he po en ial imp o emen
o he pe o mance in he a ic moni o ing sys em when using a dynamic alloca ion policy
which depends on he amoun o packe s in cap u ing and analysis bu e s.
4. Pe o mance E alua ion o he Model
In his sec ion, we sol e he op imiza ion p oblem desc ibed abo e o a gi en se o
pa ame e s. To do so, i s o all, we explain how o ob ain hose inpu pa ame e s. Nex ,
we desc ibe he e alua ion o he MDP model. To conclude he sec ion, we p esen he
esul s o he simula ion ha ollows he op imal policy ob ained om he MDP model.
4.1. Es ima ion o Pa ame e s
In o de o se he model’s inpu pa ame e s (
˜
λ
,
˜
µ1
,
˜
µ2
,
M
and
N
), we pe o m some
es s on a Linux-based ne wo k p obe, which is a p o o ype called Ksenso [
27
]. This eal
p obe is ins alled inside a es bed pla o m [28], as can be seen in Figu e 3.
Inside he es ing pla o m, he a ic gene a o is in cha ge o sending da a packe s
o he ne wo k (o o a po en ial ecei e ). The p obe has o cap u e and p ocess as many
ne wo k packe s as possible. The es manage , wi h he help o so wa e agen s ins alled
in he gene a o and p obe, measu es he me ics o in e es . The e o e, he es manage
p o ides us wi h some s a is ical esul s o he expe imen al es s om which we can
es ima e alues o he inpu pa ame e s o he ma hema ical model. Such expe imen al
esul s a e, o example, he numbe o cap u ed packe s, CPU consump ion in cap u ing
and analysis asks, he es ’s du a ion, e c.
Ma hema ics 2023,11, 610 9 o 23
Figu e 3. Tes bed pla o m o expe imen al measu emen s.
Rega ding he ha dwa e and so wa e de ails o he expe imen al se up, on he one
hand, he a ic gene a o , he es manage and he ecei e a e ins alled on he same
kind o physical machine: a compu e wi h wo In el Xeon 5110 CPUs a 1.66 GHz, 2 GB
o RAM and unning Debian 7 GNU/Linux. In o de o injec syn he ic ne wo k a ic,
he a ic gene a o has an Endace DAG 4.3GE ca d ins alled [
29
]. The packe s gene a ed
a e minimum sized IP packe s (40 by es) o 1 Gbps E he ne ne wo k. This means ha
he p obe will ecei e he maximum numbe o packe s possible. On he o he hand, he
p obe is ins alled on a machine wi h wo In el Quad Xeon 5420 CPUs a 2.5 GHz, wi h
4 co es each, 4 GB o RAM and unning a modi ica ion o he ke nel Linux 2.6.23 wi h a
module ha implemen s he analysis ask. In ou MDP model-o ien ed expe imen s, he
p obe is con igu ed o use a single CPU co e. Las ly, as Figu e 3shows, all he machines a e
connec ed o he same 1 Gbps E he ne swi ch.
Some expe imen al esul s allow us o se he alues o he inpu pa ame e s o he
model as ollows:
•˜
λ
is ela ed o he ne wo k packe a e. In he es ing scena io, he a ic gene a o
allows us o se he packe injec ion a e in o he ne wo k and we choose 21 a es
a ying om 50,000 o 1,500,000 pps. In addi ion, he es manage ga he s he packe
a e om he ne wo k swi ch o which he p obe is connec ed. These alues o ne wo k
a ic a e a e di ec ly used as he inpu pa ame e
˜
λ
o he model. As we keep he
same alues o he expe imen al es s, i will be possible o compa e he heo e ical
esul s o he nume ical e alua ion o he model wi h he expe imen al ones. Due o
he ac ha he es s a e pe o med o each one o he a es p oduced by he a ic
gene a o , he e a e expe imen al esul s o each one o hose ne wo k a ic a es;
•˜
µ1
is he packe cap u ing a e. We es ima e i by conside ing he o al amoun o
packe s cap u ed by he so i q p ocessing du ing he es (which we deno e as
ncap
)
and he ime spen by he CPU in he so i q (deno ed as
Tcap
). Bo h
ncap
and
Tcap
a e
p o ided by he es manage since, du ing he es , he numbe o packe s cap u ed
by he polling p ocess o he so i q and he numbe o cycles consumed by he CPU
in he cap u ing s age a e coun ed. A simple con e sion om CPU cycles o seconds
allows us o ob ain Tcap. Thus, ˜
µ1is es ima ed as ollows:
˜
µ1=ncap
Tcap . (16)
Ma hema ics 2023,11, 610 16 o 23
op imal policy. The conclusion ex ac ed om ha s udy (see Sec ion 4.2) is ha he mo e
he ne wo k da a a e inc eases, he mo e he op imal policy app oaches a h eshold alue.
A his poin , we can de ine se e al h eshold based policies o he case when he
p ocesso is in he analysis s age, such as:
• Policy based on a h eshold Tmwhich depends on m. I wo ks as ollows:
i m≥Tm hen
ac ion ←CAPTURE
else
ac ion ←ANALYSIS
end i
• Policy based on a h eshold Tnwhich depends on n. I wo ks as ollows:
i n≤Tn hen
ac ion ←CAPTURE
else
ac ion ←ANALYSIS
end i
• S ep-wise policy which depends on mand n. I wo ks as ollows:
i (n=0)o (m≥Tmand n≤Tn) hen
ac ion ←CAPTURE
else
ac ion ←ANALYSIS
end i
The simula ion p og am used o e alua e he pe o mance o he op imal policy can
also o esee he esul s o he h eshold based policies. Fo example, Figu e 9shows he
analysis h oughpu ob ained when an m-based h eshold policy wi h alue
Tm=
196 is
used. I is isible in he g aph ha he policy wi h
Tm
is no bene icial, on he con a y, i is
de imen al, since, om a ne wo k a e o a ound 500,000 pps, he h oughpu is e y low.
Figu e 10 shows an example o an n-based h eshold policy wi h
Tn=
3000. In con as
wi h he m-based h eshold policy, his policy achie es a conside able h oughpu imp o e-
men whose o m is simila o he op imal policy’s one, i.e., i p e en s he h oughpu
dec ease in he ange o in e media e a es, and i manages o inc ease he h oughpu a
he highes a es.
I is also possible o simula e he beha io o a s ep-wise policy. Figu e 11 p esen s an
example o his ype o policy wi h
Tm=
198 and
Tn=
2048. The esul s a e posi i e oo,
bu i s implemen a ion will be mo e di icul compa ed o he n-based policy.
F om he poin o iew o he implemen a ion, he policies based on a h eshold alue
a e easible, in pa icula , hose ones whose h eshold alue depends on he pa ame e
n
. I
he p ocesso is alloca ed o he cap u ing ask, he e is no swi ch un il he analysis queue is
ull o he cap u ing queue is emp y. This happens o all he alues o ne wo k da a a e,
hus, he e, we iden i y a h eshold policy o he alues
n=N
and
m=
0. Howe e , i
he p ocesso is alloca ed o he analysis p ocess, he momen when he p ocesso swi ches
o he cap u ing ask is di e en depending on he ne wo k da a a e and he numbe o
packe in bo h bu e s (
m
and
n
). Le us see i in he con ex o h ee anges o ne wo k
a ic a e.
Ma hema ics 2023,11, 610 17 o 23
Figu e 9. Pe o mance simula ion o m-based h eshold policy.
Figu e 10. Pe o mance simula ion o n-based h eshold policy.
Ma hema ics 2023,11, 610 18 o 23
Figu e 11. Pe o mance simula ion o s ep-wise policy.
Fi s , wi h low a ic a es, he op imal policy ells us ha while he analysis bu e is
no emp y, he e is no swi ch om analysis o cap u e un il he cap u ing bu e is almos
ull and he numbe o packe s in he analysis bu e is less han a gi en alue. As seen in
Figu e 4, o he ne wo k da a a e o 100,000 pps and null analysis load, he p obabili y
o
m>
197 is e y low and, as a esul , i is possible o se he h eshold alue o
n=
0
which is he common alue p oposed by he op imal policy o e e y
m<
197. Howe e ,
as da a ne wo k a e inc eases, he p obabili y o ha ing ull o almos ull cap u e bu e
also g ows. In hese cases, he p e ious h eshold alue o
n=
0 could no be app op ia e
because i would no a oid he isk o d opping packe s a he en ance o he cap u e bu e .
Thus, secondly, wi h in e media e a ic a es, i is necessa y o conside he case
when he p ocesso is analyzing packe s, he cap u e bu e is almos ull and he p ocesso
mus change om analysis o cap u e in o de o analyze he 100% o he incoming a ic
wi hou d opping any packe a he cap u ing en ance. Fo example, o he ne wo k
da a a e o 525,000 pps and null analysis load, he ob ained policy indica es ha he
p ocesso has o change om analysis o cap u e when
m
is g ea e han o equal o 194;
o he wise, he analysis ask does no inish un il he analysis bu e is emp y. I
m≥
194,
he n- alue o he swi ching poin
(m
,
n)
di e s om one case o ano he . Speci ically, we
ha e he ollowing pai s:
(m=
194,
n=
3609
)
,
(m=
195,
n=
3840
)
,
(m=
196,
n=
3910
)
,
(m=
197,
n=
3951
)
,
(m=
198,
n=
3978
)
,
(m=
199,
n=
3996
)
,
(m=
200,
n=
4010
)
.
As seems logical, he g ea e he occupa ion o he cap u e bu e , he soone he swi ch
should ake place ( ha is, wi h a g ea e alue o
n
). Howe e , as men ioned p e iously,
he implemen a ion o a policy based on bo h pa ame e s,
m
and
n
, is di icul . Fo his
eason, we op o ixing a policy based on a h eshold whose alue depends only on n.
Then, wi h highe in e media e a ic a es, o example, 620,000 pps, as can be seen
in Figu e 5, he op imal policy is cha ac e ized by a h eshold alue because he p ocesso
only swi ches om he analysis o he cap u e when he cap u e bu e is ull. Thus, he
h eshold alue is
m=M
, bu , as be o e, gi en he di icul ies o ha e moni o ed he
occupa ion o he cap u e bu e , i can be ansla ed o an n-based h eshold alue.
Finally, wi h he highes da a ne wo k a es, he op imal policy is a non-idling policy
ha can be o ally exp essed in e ms o n-based h eshold alues. Fo example, as can be
seen in Figu e 6 o he ne wo k da a a e o 1,100,000 pps, he p ocesso swi ches om
Ma hema ics 2023,11, 610 19 o 23
analysis o cap u e when he analysis bu e is emp ied. Thus, in his case, he h eshold
alue is Tn=0 o all he alues o mwhich a e in he ange 0 ≤m≤M.
5.2. Compa ing Op imal and Real Analysis Th oughpu s
Gi en wha we ha e jus seen, a con ol mechanism is in oduced in o he eal p obe.
I wo ks as ollows: e e y ime a new packe is in oduced in o he analysis queue ( he bu e
o he moni o ing applica ion), he sys em checks he numbe o packe s in his queue and i
does no swi ch o analyze packe s as long as he analysis queue is no ull and he cap u ing
bu e is no emp y. This ollows exac ly he esul o he op imal policy con ained in he
ma ix C. When he analysis queue is ull, his means ha he cap u e h eshold is eached
and he sys em disables he packe cap u ing. To do ha , ha di qs and NIC polling a e
disabled. Hence, in ha momen , he analyzing ins ance will p ocess he packe s in he
queue. This will happen un il he packe amoun eaches he analysis h eshold ha has
been p e iously es ima ed as
Tn
. In ha momen , he packe cap u e ask is enabled again
so ha i can s a again.
When implemen ing his con ol mechanism, we can s a e ha wo h esholds ha e o
be se : he maximum h eshold,
Tmax
, as he maximum size o he analysis queue (
Tmax =N
)
and he minimum h eshold as Tmin =Tn.
Ou es se up (see Figu e 3) allows us o implemen he h eshold policy based on
Tmax =N
and
Tmin =Tn
due o he ac ha we can con ol he ne wo k da a a e managing
p ope ly he a ic gene a o . Figu e 12 p esen s he analysis h oughpu o he eal p obe
wi h he implemen ed con ol mechanism o di e en ne wo k da a a es and analysis
loads. We can see ha he ob ained esul is e y close o he simula ed pe o mance o he
op imal policy (Figu e 8).
In Figu e 13, we show he di e ence be ween he op imal policy and he implemen ed
one in e ms o h oughpu e iciency, assuming ha he op imal policy achie es 1.0 (100%).
The esul ells us ha e en i he implemen ed policy does no ollow he op imal one so
closely, he di e ence be ween bo h o hem is no bigge han 2% in all o he ange o
ne wo k da a a es. Figu e 13 also shows esul s o s ep-wise,
Tn
- and
Tm
-based policies
assessed by he simula ion ool. In he case o
Tn
-based policy, i is also e y close o he
eal p obe’s one. Thus, we can conclude ha he eal implemen a ion is accep able, as well
as ha he simula ion ool ep oduces he beha io o he eal sys em.
Ano he aspec conside ed o he implemen a ion in he eal p obe is he a iabili y
o he p oposed h eshold,
Tn
, wi h he ne wo k da a a e. As men ioned p e iously, in
ou expe imen al scena io wi h a sou ce o syn he ic a ic, i is easible o conduc es s
a ying he pa ame e
Tn
acco ding o a con olled ne wo k da a a e. Howe e , in a eal
en i onmen , i can be complex o con ol he da a low speed and adap he n- h eshold
policy simul aneously. Fo ha eason, we implemen a h eshold policy wi h he same
alue o
Tn
o all he ne wo k da a a es and analysis loads. In his way, we es di e en
alues o
Tn
such as 512, 1024, 2048 and 3000. Figu e 14 p esen s a compa ison in e ms
o h oughpu e iciency wi h espec o he op imal policy o hose se ings. The mos
signi ican esul is he pe o mance imp o emen achie ed when in oducing he con ol
mechanism o e ne wo k da a a es ha o e load he sys em. Ob iously, he op imal
policy is he bes and, al hough he implemen ed policy in he eal sys em does no each i ,
he ob ained esul is e y accep able. The pe o mance is imp o ed un il 80% o some
ne wo k da a a es. In addi ion, he eliabili y o he de eloped simula ion ool o he cases
which ha e been able o achie e a eal implemen a ion is ema kable.
Ma hema ics 2023,11, 610 20 o 23
Figu e 12. Pe o mance o he eal p obe wi h he con ol mechanism.
Figu e 13. Compa ison in e ms o h oughpu e iciency wi h espec o he op imal policy.
Ma hema ics 2023,11, 610 21 o 23
Figu e 14.
Pe o mance compa ison be ween implemen ed scena ios on he eal p obe wi h n-based
h eshold policies.
6. Conclusions
This pape p esen s an MDP model o a ne wo k a ic moni o ing sys em based on
commodi y ha dwa e in o de o imp o e he pe o mance in e ms o analysis h oughpu .
Fi s , we iden i ied he pe o mance p oblem obse ed in a eal ne wo k a ic p obe when
he ne wo k da a a e inc eases. I happened o he di e en analysis loads ha he p obe
suppo s in he p elimina y empi ical pe o mance e alua ion (see Figu e 7).
Then, we p opose a model ha consis s o a wo-s age andem queue a ended by
a mo ing se e . In his amewo k, a h ee-dimensional MDP is o mula ed in which
he objec i e is o de e mine he posi ion o he se e in each ime slo o imp o e he
h oughpu o he queuing sys em. We ha e analyzed he s uc u e o he op imal policy
o a wide ange o scena ios whe e we show ha i has a h eshold- ype beha io .
We ha e also conside ed a eal p obe so as o alida e ou heo e ical indings. Fi s ,
we analyzed se e al h eshold ype policies and we also simula ed he op imal policy using
pa ame e s om he eal p obe sys em. These expe imen s show he imp o emen o he
op imal policy in e ms o h oughpu . Fu he mo e, we conside ed he eal sys em in which
we in es iga e he pe o mance o he h eshold-based policy, which is close o he op imal
one, and allows o demons a e he imp o emen de i ed om ou ma hema ical model.
As u u e wo k, we aim o adap some o he undamen als applied in he modeling
o his wo k o 5G ne wo k en i onmen s. We belie e ha i is app op ia e o ep esen
wi h queuing models he compu a ional consump ion o he di e en p ocessing-s ages o
VNFs. In addi ion, i is common o app oach limi ed- esou ce applica ions in eal in o -
ma ion sys ems on cu en 5G mobile ne wo ks. Fo ins ance, mo e and mo e mul imedia
applica ions equi e in ensi e edge compu ing in ne wo k nodes whe e he e a e limi ed
esou ces. The p oposal p esen ed in his wo k iden i ies which s a egy could be op imal
o adap he a ailable esou ces o he s a e o he a ic cap u e ask and he analysis needs.
Addi ionally, he e is a cu en end owa ds dis ibu ed compu ing esou ces ha is
applied o ne wo k moni o ing sys ems, he in as uc u e o ne wo ks hemsel es and new
i ualized se ices. The e o e, wi hin hese con ex s, he e a e p oposals ha de e mine
how o conduc esou ce placemen in a dynamic and e icien way (see, o ins ance, [
22
]).
In his sense, we would like o ex end his wo k based on an MDP model in ha esea ch
Ma hema ics 2023,11, 610 22 o 23
line, by conside ing ubiqui y issues on dis ibu ed compu ing sys ems, i ualized ne wo k
se ices and edge compu ing on cu en elecommunica ion in as uc u e.
Au ho Con ibu ions:
Concep ualiza ion, L.Z., J.D. and A.F.; me hodology, L.Z., J.D. and A.F.;
so wa e, L.Z. and A.F.; alida ion, L.Z. and A.F.; o mal analysis, L.Z., J.D. and A.F.; in es iga ion,
L.Z., J.D. and A.F.; esou ces, L.Z., J.D. and A.F.; da a cu a ion, L.Z., J.D. and A.F.; w i ing—o iginal
d a p epa a ion, L.Z., J.D. and A.F.; w i ing— e iew and edi ing, L.Z., J.D. and A.F.; isualiza ion,
L.Z., J.D. and A.F.; supe ision, L.Z., J.D. and A.F.; p ojec adminis a ion, A.F.; unding acquisi ion,
A.F. All au ho s ha e ead and ag eed o he published e sion o he manusc ip .
Funding:
This esea ch was pa ially suppo ed by he Depa men o Educa ion o he Basque
Go e nmen , Spain h ough he Consolida ed Resea ch G oups NQaS (IT1635-22) and MATHMODE
(IT1456-22), by he Ma ie Sklodowska-Cu ie, Spain g an ag eemen No 777778, by he Spanish
Minis y o Science and Inno a ion, Spain wi h e e ence PID2019-108111RB-I00 (FEDER/AEI), by
g an PID2020-117876RB-I00 unded by MCIN/AEI (10.13039/501100011033) and by G an KK-
2021/00026 unded by he Basque Go e nmen .
Ins i u ional Re iew Boa d S a emen : No applicable.
In o med Consen S a emen : No applicable.
Da a A ailabili y S a emen : No applicable.
Con lic s o In e es : The au ho s decla e no con lic o in e es .
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Disclaime /Publishe ’s No e:
The s a emen s, opinions and da a con ained in all publica ions a e solely hose o he indi idual
au ho (s) and con ibu o (s) and no o MDPI and/o he edi o (s). MDPI and/o he edi o (s) disclaim esponsibili y o any inju y o
people o p ope y esul ing om any ideas, me hods, ins uc ions o p oduc s e e ed o in he con en .
