Simulating Bandwidth Sharing with Pareto distributed File Sizes

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File Sizes

Eitan Altman, Julio Rojas-Mora, Tania Jimenez

To cite this version:

Eitan Altman, Julio Rojas-Mora, Tania Jimenez. Simulating Bandwidth Sharing with Pareto distributed File Sizes. [Research Report] RR-6926, INRIA. 2009. �inria-00383079�

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a p p o r t

d e r e c h e r c h e

0249-6399ISRNINRIA/RR--6926--FR+ENG

Thème COM

Simulating Bandwidth Sharing with Pareto distributed File Sizes

Julio Rojas-Mora — Tania Jiménez — Eitan Altman

N° 6926

May 2009

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Centre de recherche INRIA Sophia Antipolis – Méditerranée

JulioRojas-Mora

∗

,Tania Jiménez, Eitan Altman

†

ThèmeCOMSystèmesommuniants

Équipe-ProjetMaestro

Rapportdereherhe n° 6926May200920pages

Abstrat: Thetraontheinternethasknowntobeheavytailed: thesizeof

letransfersthroughFTPorHTTPappliations,aswellasthosetransferredby

P2Pappliations hasbeenobservedtohaveaveryheavytail. Typiallymod-

eledasParetodistributed withparameterbetween1.05to1.5, thelesizehas

innitevariane. Thisisthesoureofmanydiultiesin simulatingdatatraf-

: onvergeneisveryslow,simulationshavetobeverylong,andthestandard

methodsforderivingondeneintervals,basedontheCLT,arenotappliable

here. Weillustratethese wellknownproblemsthroughthesimulationstudyof

a proessorsharingqueue, whih is oftenused to model session levelresoure

sharingin theinternet. Wetest bootstrapmethods to aelerate onvergene

and improvethepreisionofsimulations,and testadiret approah toobtain

ondene intervalbasedonthe histogramoftheempirial distributions. The

onlusiondrawnarethenompared tothoseobtainedwhensimulatingin ns2

datatransferusingTCP.

Key-words: M/G/1 queue. Proessor Sharing. Simulations. Condene

interval. Bootstrap.

∗

LIA, University of Avignon, 339, hemindes Meinajaries, Agropar BP 1228, 84911

AvignonCedex9,Frane. {julio.rojas,tania,jimenez}univ-avignon.fr

†

MaestrogroupINRIA,2004 RoutedesLuioles06902 SophiaAntipolisCedex,Frane.

eitan.altmansophia.inria.fr

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Résumé: Lataille des hierstransférés surl'Internet parFTPouHTTP,

ainsi que eux transférés par les appliations P2P, a souvent été observée et

mesurée. La distribution observée a été à queue lourde et elle a souvent été

modélisée par une une distribution de Pareto ave un paramètre entre 1,05

et 1,5; ela signie que la taille des hiers a une variane innie. Cei est

la soure de nombreuses diultés dans la simulation du tra de données:

la onvergene est très lente, les simulations doivent être très longues, et les

méthodeshabituellesdealuld'intervallesdeonane,baséessurleThéorème

Limite Centrale, ne sont pas appliables. Nous illustrons es problèmes bien

onnus à traversla simulationd'une le "Proessor Sharing",qui est souvent

utiliséepourmodéliserlepartagedesressouresauniveausessiondansl'Internet.

Lesontributionprinipaledenotretravailsont(i)l'appliationdelaméthode

deBootstrapquinouspermetd'aélérerlaonvergeneetd'améliorerlapréision

des simulations, et (ii) l'étude de l'approhe pour obtenir des intervales de

onanebaséssurladistributionempirique. Puisnousomparonslesonlusions

à elles obtenues lors de la simulation en NS2 des transfers de hiers sur

l'Internet.

Mots-lés: FileM/G/1,ProessorSharing,Simulatiojs,IntervaldeConane,

Bootstrap

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1 Introdution

The proessor sharing queue has been perhaps the most ommon model for

bandwidthsharingofsessions(havingthesameRTT)ofdatatransferoverthe

Internet [4℄, along with the Disriminatory Proessor sharing Sharing queue

[7℄ adapted to the ase of unequal round trip times. Several researh groups

[17, 14, 11, 13℄ haveexamined itsvalidity throughsimulations. Theexpeted

sojourntimeofaustomerin aproessorsharingqueueisknowntobeinsensi-

tivetothelesizedistribution(itdependsonthedistributiononlythroughthe

expetation). Its variane,however,isnotinsensitiveanymore,andinpartiu-

lar,itisinnitewhentheservietimeofalient(orequivalentlyinoursetting

-thesizeofalethatistransferred)hasaninniteseondmoment[3℄.

The size of a data transfer overthe Internet is known to be heavy tailed

[12℄. (This is thease both for FTPtransfers aswell asfor the "on"periods

of HTTP transfers). Among several andidates for modeling the distribution

ofthesetransfers,theParetodistributionwithparameterbetween1.05and1.5

has been the one that gave the best t with experiments for the last twenty

years,see[8,2,6,15℄.

Thisveryheavytail hasbeenausing various serious problemsfor simula-

tionsofdatatransfers:

Theauthorsof[11℄write: "sineasubstantialpartof thedistribution is

in the tail, if the simulation is not runfor very longthe average of the

sampledlesizeswouldbelessthanthenominalaverage,thusleadingto

aloweroeredloadandheneoverestimationofthethroughput."

Centrallimitbasedondeneintervalsarenotavailablesinetheseond

momentof the number of sessions orof the workload in thesystem are

innite.

Duetothelastpointsonemayhaveproblemsininterpretationofsimula-

tionresults. Indeed,variouspapersin whih sharingbandwidthismod-

eledbyproessorsharingreportdierenesbetweentheexpetedtheoret-

ialvalueandthesimulationvalue[13,14,11,17℄that varyfromaround

10%in [11℄ andgoupto afatorof10in somesituationsin [13℄. When

suh deviations our,it is importantto know whether theyanbedue

totheimpreisionsin thesimulationortorealphenomena.

Thewarm-uptimeisextremelylong,see[9℄.

Other approahes to simulating networks with Pareto dis-

tributed le size

Onewayto avoidheavytailsisto trunatethem. Thisis in spiritof thesug-

gestioninthepaper"DiultiesinsimulatingqueueswithParetoservie"[10℄

that says: "Sinefor any nite simulation runlength, there isalways amaxi-

mum value of the random variables generated, we, inatuality, are simulating

atrunatedPareto servie distribution. Ithas alsobeen arguedthat thereisal-

waysamaximumlesizeorlaim amountso,inreality,wearealways dealing

with trunated distributions." Butwhere should we trunate thedistribution?

TrunationatsomesizeM ^would^be^valid^if^the^dierene^between^performane

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with trunation at anyother valueL ^that ^is ^greater ^than M ^has^a ^negligible

impatonperformane. Simulating withatrunated Pareto distribution may

notbesuientandseveralothertestsoftrunationatlargerthresholdvalues

maybeneeded.

Our ontributions

Our main ontribution is in introduing to the networking ontext statistial

methods that, to the best of our knowledge have notbeenused before in the

networking ontext. Inpartiular,

Weobtainanimpressiveimprovementinthepreisionofsimulationsusing

thebootstrapapproah;equivalently,thisallowsusto dereaseonsider-

ably the simulation run times or the number of simulations needed in

order toahieveagivenpreisionlevel. Weshall nallypresentdetailed

experimentationswithsomeofthem.

Condene Intervals Weusethequantilebasedapproahasanalter-

native to the entral limit based approah for obtaining the ondene

intervals. Theentrallimitapproahis notdiretly appliablewhenever

thevarianeisinnite,whihistheasewiththestationarysojourntime

of adata transfersession(having aPareto distribution withashapepa-

rameterK^lower^than^2).

We then apply these ideasdiretly to the simulationof ompeting non-

persistent TCP transfers of les that have heavy tail distribution. We

omparethesimulationresultstothoseobtainedfortheproessorsharing

queue. Weidentify various reasons for the deviation of the behaviorof

TCPfromtheidealproessorsharingmodelandmanageto quantifythe

impatofsomeofthese.

Struture of the paper

We begin by presenting in the next setion a bakground on the bootstrap

methodandonthequantilemethodforobtainingondeneinterval. InSetion

3 we present our approah onerning the use of simulation to estimate the

average number of pakets in the proessor sharing queue. In Setion 4 we

study someissues onerning the simulation of the workload of the proessor

sharingqueue. Wethen presentin Setion 5simulationsof TCP onnetions

that share a ommon bottlenek link and ompare the preision that an be

obtained(with andwithoutthebootstrapapproah)tothepreisionobtained

in simulating the proessor sharing queue. We provide explanations for the

dierenesbetween theTCPsenario andits orresponding proessorsharing

model. Aonludingsetionsummariesthepaper. An appendix lariessome

basiquestionsonerningondeneintervals.

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Algorithm1Bootstrapalgorithmforestimatingthemeanqueuesize

1. Makensimulationsofthe queuequeuesizeand letXi,∀i= 1, . . . , n,^be

theestimationoftheparameterofinterestobtainedfromeahsimulation.

2. Forj = 1, . . . , k^do:

(a) Let X_1,j^∗ , . . . , X_n,j^∗ ^be ^a ^resample, ^with replaement, taken from

X1, . . . , Xn^.

(b) Letθˆ_j^∗=n⁻¹Pn i=1X_i,j^∗ ^.

3. Calulateθˆ^∗=k⁻¹Pk

j=1θˆ_j^∗^,^the^Monte^Carloapproximationoftheboot- strapestimationofθ^.

4. Calulateondeneintervalsforθˆ^∗ ^using^the^quantile^method.

2 Bakground

Bootstrap

BootstrapisamethodreatedbyEfron[20℄fornon-parametrialestimation. Let

θ^beâ^parameterôfâômpletelyûnspeieddistributionF^,^for^whih^we^have

asampleofi.i.d. observationsX1, . . . , Xn^,ând^letθˆ^be^theêstimation^madeôf

theparameter. Fromthesample, wewill makek ^resamples^withreplaement,

X_1,j^∗ , . . . , X_n,j^∗ ∀j= 1, . . . , k^,^withêahêlement^havingprobability1/nôf^being

seleted, and foreah resamplean estimatorθˆ^∗_j ^will ^be^alulated. ^By^Monte

Carloapproximation,thedistributionofθˆîs^thenêstimated^by^thedistribution of θˆ^∗^. ^When k → ∞^, ^the êstimation ôf θˆ ^will ^be ^better ând, ⁱⁿ ^turn, ^the

real distribution of θ ^will ^also ^be ^better ^estimated. ^We ^note ^that ^the ^time

and memoryoverheadfor resamplingand performing thebootstrapalgorithm

are oftenmuhsmallerthan theones neededto reatemoresamples, andan

beperformedwithin averyshort amount oftime. Singh [21℄, and Bikel and

Freedman[19℄aregoodreferenestounderstandtheasymptotiharateristis

ofthebootstrap.

Remark. An alternativeway to aelerate the simulationsis the important

samplingormoregenerally,varianeredutiontehniquesseee.g. [22,Chap4℄

for ageneralintrodution. Theyare dierentthan thebootstrapapproahin

thattheyarebasedonsimulatinganothermodel(e.g. usealargerloadinorder

to obtainbetterestimateof arareeventof reahingalargequeuesize). Then

some knowledge of the system is needed in order to transform the simulated

results of the new model to that of the original one. The bootstrap method

that westudyis apost-simulationapproah: it onernsstatistial proessing

ofsimulatedtraes. Itanbeusedontopofvarianeredutiontehniqueswhen

theyareavailable.

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Quantile-based ondene interval

Assumewewishtoobtaintheondeneintervaloftheestimationofsomepa-

rameterof asimulatedproessXt^. ^The^quantile^approah^to^derive^ondene

intervalsisbasedonrunninganumberN ^of^i.i.d. simulations(eahsimulation orrespondsinourasetothequeuelengthproessortofuntions ofthispro-

ess). Wethenusethese toomputetheempirialdistributionof thefuntion

oftherandomvariable.

For(1−α)·100% ôndene ^level, ^the (1−α)·100% ôndene înterval

by the quantile method is the interval between the (α/2)·100^- ^th ^and ^the (1−α/2)·100^-th^points^of^the^sorted^sample.

3 Simulating the queue size of a proessor shar-

ing queue

ThroughaseriesofsimulationsperformedinJAVA(availablefromtheauthors

byrequest) we exhibit thepower ofthe bootstrapapproah: itsability to in-

reasethepreisionofthesimulationofInternettrathatisthrottledbysome

bottleneklinkandatthesametimedereasetherequireddurationofthesimu-

lation. Inthissetionwerestrittostudyoftheproessorsharingqueuewhih

hadoftenbeenproposedasamodelforTCPtransferssharingabottleneklink

intheInternet. A TCPsession isthenrepresentedbyoneustomerin thePS

queue.

Belowweused the PS queue with aPoisson arrivalproesswith a rateof

λ= 1ûstomers ^per^seond ^(A ûstomer ^representsâ^le ^when ^the ^proessor

sharingqueueis used to model le transfers). We onsider three servietime

distributions: Pareto with shape parameter 1.5, Pareto with shape 2.5, and

exponentially distributed. Wevary the average servietime σ ^so^as^to ^obtain

anaverageloadρ=λσ ^that^takes^the^values^0.6^and^0.7.

Eahoneoftheguresbeloworrespondto thequeuesize oftheproessor

sharing averaging over 100 independent samples. When onsidering the PS

queueasamodel forbandwidthsharingintheInternet,thequeuesize should

beinterpretedasthenumberofongoingsessions.

Thedurationofeahsimulationis6·10⁶^seând^thereîsâ^warmûp^timeôf

300000se. Ineahoneofthesenariosdesribedin Figure1,wepresent:

1. thetheoretialsteadystateexpetedqueuesize

2. thevalueobtainedbythesimulations

3. theondeneintervalorrespondingtoa95%perentileofobtainedusing

thequantilemethod

4. the ondene intervalorrespondingto a 95%perentile obtainedafter

applyingthebootstrapmethod.

Wetook100000resamplesoutofour100originalsamplesforthebootstrap.

Figure1displaysthepreisionofthesimulationswithandwithouttheboot-

strap approahfor ρ = 0.6 ^(up) ^and ρ = 0.7 ^(down). ^The ^left ^subgures ^are

foranexponentialdistributed servietime,themiddleand rightonesarefora

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Pareto distributed servietime withparameterK = 2.5 ^and K = 1.5^,^respe-

tively. Theaverageservietimeis thesamein theallthree sub-guresgures

orrespondingto thesameρ^. ^(It^was^hosen^so^that ^indeedρ=λ·σ^will^have

thevalues0.6 ^and0.7^,respetively.)

Weobservethefollowingpointsfrom thesimulations:

ThesimulationsshowwelltheinsensitivityofthePSregimetotheservie

time distribution. Indeed, in eah set of simulations having the same

loadρ^, ^we ^see ônvergene ^to ^the ^sameâverage ^rate^for ^the exponential ase,thePareto distributionwithparameterK= 2.5ând ^for^the^Pareto

distributionwithK= 1.5^.

The95%ondene intervalwithoutbootstrapisaround10timeslarger

than that of the bootstrap (whih establishes learly the advantage of

usingit). Thisisseentoholdforanydurationofthesimulation.

Weseethattheondeneintervalarearound10timessmallerinthease

ofexponentialand Pareto with shapeparameterof 2.5 than in the ase

ofParetoshapeparameter1.5. Thisanbeexpetedsinethetailofthe

latterdistributionismuh heavier.

Durationofthesimulation:Forallthreedistributions,andforthedierent

loads, we see that the preision obtained with bootstrap after already

400000se is morethan three timesbetterthan that without bootstrap

after wesee that even after 6millionseonds. It thus seemsthat to get

thesamepreisionwithoutbootstrap,onewouldneedtousesimulations

muhlongerthan15timesasmuhaswithbootstrap.

4 On the simulation of the workload in a queue

with heavy tailed servie time

Thestatespaeneededtorepresenttheevolutionofthenumberofpaketsina

proessorsharingisquiteomplex: weneedtheresidualtimeofeahpaketthat

is in the system. Inontrast, the evolutionworkload in the systemis easyto

desribeasitanbewritten asaonedimensionalreursiveequation. Wethus

hose to illustrate some of the diulties in simulating the proessorsharing

queuebyashortdisussiononerningthesimulationoftheworkloadproess.

We onsider in this Setion the workload of an M/G/1 proessorsharing

queue withaPoisson arrivalproess withrate λ^and ^with^i.i.d. ^servie^times.

Theworkloadisthetimeittakestoompletetransmissionofalltheonnetions

thatarepresentin thesystem. Assume throughoutthatρ <1^so^the^workload

proessisergodi.

Wenote thattheworkloadproess isthesameastheoneofaFIFOqueue

withthesamearrivalsandservietimes.

In ase servie times do not have a nite seond moment, the expeted

workloadat steady stateis innite. (Indeed, the workloadis invariantunder

theservie disipline,and is thus equalto theoneof FIFOdisipline. Dueto

thePASTAproperty,theexpetedstationaryworkloadequalstothatatarrival

epohs whih equals the expeted waiting time. The latter is innite in the

M/G/1FIFOqueuesine sinethearrivalhasto wait morethan theresidual

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servie time of the ustomer in servie, and the latter is proportional to the

seondmomentoftheservietime.)

Considernow the asewhere the servie time has anite seond moment

but innite third moment. In that ase the expeted stationary workload is

nitebutitsvarianeisinnite.

Assumethat wewishto estimatetheexpetedstationaryworkloadE[V]ⁱⁿ

thesystem. Theworkloadsatisesthereursion

V0(v) =v, Vn+1(v) = (Vn(v) + Θn−τn)⁺, n≥0

whereVnîs^the^workloadâs^seen^by^then^thârrival,Θn îs^the^workload^brought

bythen^thârrival,ândτn îs^the^time^between^then^thârrivalêpohTn ând^the

(n+ 1)thârrivalînstantTn+1^. E[V]ân^then^beêstimated^by Vbn=

Pn i=1Vi

n

forn ^large^(due ^to ^the ^PÂSTA ^property^, êstimation ât ârrivalînstantsîndeed

onvergestothestationaryrandomvariable). Denethebiasfuntion:

Rn(v) =E[Sn(v)]−nE[V] where Sn(v) = Xn

i=1

Vi(v).

WenotethatRn(0)îs^monotone^dereasingⁱⁿn^. Ît^hasâ^limitâsn→ ∞^whih

wedenote by R^. Rn(0) îs^known âs^the^bias; îfît ^were^zero ^then Vbn =E[V]^.

OtherwizethetotalexpetedestimationerrorattimeTn ^is^given^byRn(v)/n^.

Toobtainanestimationerrorsmallerthanǫ^,^we^need^to^hoosen^suh^that Rn/n < ǫ^. Ît îs âdvoated ⁱⁿ ^[18℄ ^to ^hoose n^suh ^that R/n < ǫ^. Ît ênsures

that|Rn|/n < ǫ^. ^Following^this^approah,^we^disover,unfortunately, thatthe simulationlength should betaken to be innitesine |R| ^turns ^to ^be^innite.

Wenextestablishthisonlusion.

LetVn^∗ ^be^the^stationary ^workload^proess ^Weâssume^thatVn^∗ ândVn âre

dened on ajoint probability spae: the arrivalproess and servietimes are

thesameinboth. Theonlydiereneisintheinitialstate.

Deneη(v)^to^be^the^rst^timen^that Vn ^ouples^withV_n^∗^. ^Letθ(v)^be^the

rsttimethatithits0. Then

Rn(v) = Xn

i=0

E

1{η(v)> i}[Vi(v)−Vi^∗]

Wemakethefollowingkeyobservation: Whenv > V0^∗^thenη(v) =θ(v)^. ^Indeed,

aslongasV_n^∗ândVn âre^both^positive^then^the^dierene|Vn−V_n^∗|îsônstant.

Itstart dereasingwhen thesmallerof thetwohitszeroand itvanisheswhen

theyarebothzero.

LetQ0(v) =v ^and^dene ^forn >0^: Qn(v) =

Xn

i=0

1{θ(v)> i}Vi(v)

NotethatRn(v)≥P(V^∗ < v)Qn(v)^.