A simple stability condition for RED using TCP mean-field modeling

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mean-field modeling

Julien Reynier

To cite this version:

Julien Reynier. A simple stability condition for RED using TCP mean-field modeling. [Research

Report] 2006, pp.24. �inria-00091036�

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inria-00091036, version 1 - 5 Sep 2006

A simple stability condition for RED using TCP mean field modeling

Julien Reynier

N° ????

Août 2006

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Julien Reynier

∗

ThèmeCOM Systèmesommuniants

ProjetsTREC

Rapportdereherhe n°???? Août200621pages

Abstrat: Congestion on the Internet is an old problem but still a subjet of intensive researh. The

TCPprotoolwith itsAIMD(AdditiveInreaseandMultipliativeDerease) behaviorhidesveryhallenging

problems;oneofthemistounderstandtheinterationbetweenalargenumberofuserswithdelayedfeedbak.

This artilewill fouson twomodeling issues ofTCP whih appeared to be importantto takleonrete

senarioswhenimplementingthemodelproposedin[7℄;rstlythemodelingofthemaximumTCPwindowsize:

thismaximumanbereahedquiklyinmanypratialases;seondlythedelaystruture: theusualLittle-like

formula behavesreally poorly when queuing delays are variable, and may hange dramatially the evolution

of the predited queue size, whih makes it useless to study drop-tail or RED (Random Early Detetion)

mehanisms.

Within proposed TCPmodeling improvements,weareenabled to lookat aonrete examplewhere RED

should beusedinFIFOroutersinsteadoflettingthedefaultdrop-tailhappen. Westudymathematiallyxed

pointsofthewindowsize distributionandloalstabilityofRED.AninterestingaseiswhenRED operatesat

thelimitwhentheongestionstarts,itavoidsunwantedlossofbandwidthanddelayvariations.

Key-words: TCP,AQM,drop-tail,RED,ongestion

TREC

∗

ÉoleNormaleSupérieure,45,rued'Ulm,75005Paris

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Résumé : Le ontrle de ongestion dans Internet est depuis longtemps le sujet de reherhes poussées.

Le protooleTCPavesonomportementAIMD(pouraroissementslinéaire,déroissanemultipliativeen

anglais)ahedesproblèmesexessivementompliqués. L'und'entreeuxestdeomprendrel'interationentre

denombreuxutilisateursaveundélaideréponsedusystème.

CerapportvasefoalisersurdeuxpointsdanslamodélisationdeTCP.Cespointssontapparusimportant

lorsque nous avons voulu onfronter à des sénarii onrets le modèle proposé dans [7℄; Tout d'abord la

modélisation delafenêtre maximale deTCP: ette valeurpeur être atteinte très failementdansla pratique;

ensuite, lastruture des délais: la formuletype Littlehabituellement employée donnedes résultatsloin de la

réalité quand les délaissont variables. Cette hypothèse de modélisation a un impat important sur la taille

prédite de lale d'attente e qui rend vaineslestentativesde omparaisonentre lesméanismes drop-tailet

RED.

Grâeàesaméliorationsapportéesaumodèle,noussommesapablesdansunaspréisd'étudieromment

paramétrer RED dansdes routeurs FIFOpourqu'ilaméliore lesperformanespar rapport auas pardéfaut

de drop-tail. Nousétudions mathématiquementlespointsxes etladistribution delataille desfenêtres et la

stabilité loaledeRED. Unas intéressantestquand RED setrouvedans sondomaine defontionnementau

début delaongestion,iléviteunemauvaise utilisationdelabandepassanteetdesvariationsdanslesdélais.

Mots-lés : TCP,AQM,drop-tail,RED,ongestion

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Contents

1 Introdution 4

1.1 TCProuterontrolissue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Ourpreviousworksandmotivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Newresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Model and equations 4 2.1 Evolutionof ongestionwindowsizes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Delayin thesystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 LimitsofLittle-likeformula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.2 Howtoimprovedelaymodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.3 Delayequations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.4 Bandwidthevolutionwhenrossingthereeivinguser . . . . . . . . . . . . . . . . . . . . 6

2.2.5 Generationoflosses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Fixed point 7 3.1 Fixedpointequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2 Fixedpointresolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.2.1 Firstiteration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2.2 Seonditeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3 Normalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.3.1 Regularityproperties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.3.2 Computationoftheintegral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.4 Equilibriumvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Stabilityanalysis 12 4.1 Arstremark. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.2 Stabilityequations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.3 Dierentialequationswithtimedelay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Simulation results 15 5.1 Resultswithdrop-tail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.1.1 Beforeongestionhappens. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.1.2 Theearlyongestionphase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.1.3 Strongongestionphase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.2 ResultswithRED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.2.1 Conguration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.2.2 RED initsworkingregime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.2.3 RED workinglikeadrop-tail . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.3 Inreasingthelateny . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

5.4 Mixinglatenies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

6 Conlusion 18 7 Related Works 18 7.1 TCPmodelingarea. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

7.2 ControltheoryappliedtoTCP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

7.3 BuersizingforIProuters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

8 Further Works 19

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1 Introdution

1.1 TCP router ontrol issue

TCPahievesadistributedongestionontroloftheInternet. ThisartileprovesausablelosedformulaRED

stability. RED (RandomEarlyDetetion)wasintroduedbyFloyd in[14℄;it istobedeployedat arouterto

send ongestioninformationtoTCPRenoend users. TheideabehindREDisthat therstsignofongestion

is whentherouterqueuestartstobeusedmorethanto buernormaltrautuations;thenthebuergets

full andthedrop-tailmehanismdestroyspaketsarrivingwithoutanyroomleft inthequeueto tin.

Drop-tailleadstotwoissues: rstly,thequeue sizeosillationsprovokedelayjitters- thishasdetrimental

eets for appliations using TCPfor realtime ontent -seondly, drop-tail synhronizes soures, resultingin

bandwidthunder-utilizationoftheongestedlink (thisideawasrstintrodued in[41℄for TCPTahoe). This

serves as leverage beause at the time bandwidth demand reahes apaity, the goodput diminishes by the

synhronizationeet,worseningthestartingongestion(thisassertionwillbeexplainedlearlylater).

Thesetwomain reasonsexplaintheinterestforRED andotherAQM(AtiveQueueManagement)to deal

with TCP ongestion at therouter level. RED often works in an admirableway, leadingto reduedqueuing

delays,avoidingjittersandreahingoptimalbandwidthutilization... butsometimesREDperformsworsethan

doingnothingatall(drop-tail). Thisis thereasonwhymanysystemadministratorsarerelutanttouseRED

althoughitisdeployedinalmosteveryrouteroftheInternet. ThispaperwillshowhowtotuneRED inaway

itissometimesoptimalandalwaysbetterthandrop-tail.

1.2 Our previous works and motivation

In[7,29℄and[40℄weinvestigatedmeaneldTCPmodelingbyontinuingtheuidTCPmodelintroduedand

studiedin[15,25,31℄. Despitetheinterestingresultsarisingfromthemodels,therewerestillsomediultiesin

understandingtheoriginalproblemoftuningREDandomparingitauratelytodrop-tail. Twopointsneeded

to beaddressed. Firstly,withthedevelopmentof highspeedaess,itbeomesdiulttosupposethat TCP

alwaysworkswithinitsongestionavoidanemodeinaAIMDmanner. Thesizeofpaketsoftheorderof1kB

makesthemaximumTCPwindowsize relativelysmall(most ommonpaketsize isaround1.4kB^). ^We^shall

sayWmax= 64 ^paketsêvenîf^the^reeiver^does^notîmposeâny^reeption^windowlimitation. Thisfat isdue to theodingofthewindowsize on16^bitsâddressing^window^by^Bytes⁽2¹⁶B = 64kB^).

Seondly, another limiting modeling assumption is afat notied by Hongin [16℄: when thequeue is not

empty,aknowledgementsarriveobviouslyattheongested routerbandwidth. This remarkisruial beause

TCPdynamiisverysensitivetothedelayedfeedbak.

1.3 Outline

Setion 2explainsanddenes ourmodel;thenwestudy thesteadystatewindowdistributionwith amaximal

windowsizeinsetion3. NextsteponsistsinseekingastabilityregionfortheRED algorithm,whih isdone

in setion4. Wenishwith showingsimulationresultsonaonreteexamplein setion5. Thislast example

showshowtousepreviousresultstoongureRED inarouterinordertoavoidollapseattheearlystagesof

ongestion.

1.4 New results

Whereas modelingWmax ând ÂCK ^bandwidthâre ^not ^newîdeas^(see ^[34℄ ând ^[16℄), âdapting ^them ⁱⁿ ^mean

eld equationsto obtainaurateevolutionequationstogetherwiththewindowdistribution onstitutesastep

forward. Thesteady statesolutionforthewindowdistribution takinginto aounttheWmax ^phenomenon ⁱⁿ

setion3isanextensionof[7℄whihisimportantfromapratialpointofview. Insetion4,thestabilityresult

obtainedforREDwiththeorem4isaverysimplelosedformula. Finallytheexampleinsetion5explainshow

REDshouldbetunedtoinreaseroutereieny,thisisanimportantresultbeause,aswesaid,thesuggested

tuning anbeapplied withoutanyhardwaremodiationinalmosteveryrouterbyenablingRED.

2 Model and equations

A number N^, ^relatively ^large, ôf ûsers ^share â ômmon ^bottlenek ^router ^(gure ¹ ^to ^see ^the ^modeled ^net-

work topology). We an onsider the histogram of users' ongestion windowsizes; in [29℄, we sawthat this

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histogram onverges"gently"to adeterministiwindowdistribution asN ^tends ^to^innity^. ^In^[7℄, ^we^studied

this asymptoti distributionwhih satises apartial dierentialequation the resultswere appliable evenfor

small numbers (N = 25 ^for ^RED, ^and N = 10 ^for ^adrop-tail). Hereinafter weadapt thepartial dierential equationsrstpresentedin [7℄;thisisdoneinawayweouldprovethemeaneld limitaswedidin[29℄,but

weshallnotenter insuhdevelopmentsinthisartile.

Queue, 235 packets User 1, W

¹

User 2, W

²

… User N, W

^N

1Gb/s, 4 ms

100 Mb/s, 1 ms

TCP sink 1 ms

1 ms

Queue, 235 packets User 1, W

¹

User 2, W

²

… User N, W

^N

1Gb/s, 4 ms

100 Mb/s, 1 ms

TCP sink 1 ms

1 ms

Figure1: Modeled topologyandsimulatedsenario,N ^is^variable.

2.1 Evolution of ongestion window sizes

Imagineusershaveanotiationoflossesoftheformκ(t)ⁱⁿ^proportion^of^the^inomingaknowledgementow.

Denote byA(t) â ^funtion îndiating ^theôwêvolutionôf ^windows^(for êxampleA(t) = _rtt(t)¹ ⁱⁿ ûsual ^TCP

models). ThequestionofdelaysandhowthefuntionsAândκêvolveôme^laterⁱⁿ^theârtile;^these^questions

arenotrelevanttostudytheintrinsiuserongestionwindowsizeevolution. Then,thedistributionofwindow

sizes isoftheform

D(t, w) =p(t, w)dw+M(t)δWmax.

Whih leadsto twoequations,thePDE:

1 A(t)

∂p

∂t(t, w) + ∂p

∂w(t, w) = ⁽¹⁾

κ(t) 4wp(t,2w)χw<Wmax/2−wp(t, w)χw<Wmax

+δ^Wmax

2 M(t)κ(t)Wmax

and

1 A(t)

dM

dt (t) =p(t, Wmax)−M(t)κ(t)Wmax. ⁽²⁾

Intuitively, theoeientA(t) îs^the înoming^bandwidth; ^when îtîs ^small, ^the^window^sizes ^haveâ^slow

reation,when itislarge,theyreatinafasterway. Theoeientsχw<Wmax ^only ^indiate^that ^the^window

size annotbelargerthanWmax^. ^When^no^loss^ours,^the^oeient ∂p

∂w(t, w)^indiates^that ^the^window^size

inreaseslinearly. Whenlossesarise,theoeientκênables−κw(p(w))^,^whih^means^thatâêrtain^proportion

ofusersthatwereatwindoww^hange^to ânother^valueôf^the^window^size;îtâlsoênables4κwp(t,2w)^,^whih

meansthat usersthat wereatwindow2w^and2w+ 1 ^(or2w−1⁾^move^to ^windoww^.

2.2 Delay in the system

2.2.1 Limitsof Little-likeformula

Asnotiedin [16℄andin[40℄,theLittle-likeapproximationmadeinusualTCPmodels(forexample[7,15,35℄)

laks realism and strongly limits the way models an explain reality. This approximationonsists in saying

that attimet^,^the^bandwidthB(t)ôfâûserîs^related^to ^the^RTT,R(t)(Round-TripTime)anditsongestion

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windowsizeW(t)ând^byB(t) =W(t)/R(t)^. Îf^the^RTTîsâlmostônstant^(forînstane^lose^to^thepropagation delay),itisaratheraeptablesimpliation,whereaswhenR(t)îs^variable,^the^modelân^lead^toûnaeptable

onsequenes: itis easytounderstand that whenonewantstostudythe stabilityofRED (withanon empty

queue),sayingB(t) =W(t)/R(t)ôrB(t)îsônstantêntails^dierentônlusions.

2.2.2 How to improve delaymodel

Theideaomesfrom[8℄whereasimpledelaylineisintroduedtostudythelimitbehavior(whenthebandwidth

tendstoinnity)ofoneuserimplementingMulTCPorsalableTCP([12,20,21℄). Althoughtheuseofadelay

line ompliates equations,the model isstill easy to simulate. Furthermore loal stability ofxed pointsan

bestudiedmathematially. Herewewilladaptdelayline modelingtolargenumberofTCPRenousers.

2.2.3 Delayequations

Delayandqueuesize LetusintrodueQ(t)^the^queue^size^mesuredⁱⁿ^seonds,^the^router^is^supposed^FIFO

(in otherwords,Q(t)îs^the^queuing^delay). ^Denote^byK(t)^,^the^destrution probabilityforapaket entering the queue; allBi(t)^theînoming ^bandwidth ^to^the ^queueând Bo(t) ^theôutgoing ^bandwidth,^saled ^by ^the

numberofusers. C îs^the^routerâpaity^perûser. ^Then:

Bo(t) =

min(C, Bi(t))^ifQ(t) = 0

C ^else. ⁽³⁾

RTT CallR(t) = T+Q(τ(t))^, ^the ^R^TT ^virtually ^written ^by^the ^queue ^on^paket ^arriving ^the ^router τ(t)^.

This paket beomesanACKthat generatesnewpaketswhere thevalueisopied. Bydenition wesaythat

this valueomesbakattherouterrouterattimet^,^Whih^makest=τ(t) +R(t)^leading^to^the^relation:

R(t) =T+Q(t−R(t)). ⁽⁴⁾

Wedisussedin[7℄thefat thatthisimpliitequationanalsobewritten: R(t+T+Q(t)) =T+Q(t),^whih

doesnotraiseanydenitionissuesandiseasierfornumerialomputations.

Advane A(t)^of^window^sizes ^The^window^sizeapproximatelyinreasesbyoneeveryW(t)^arrived^pakets.

Theinomingbandwidthforagivenuseristheprobabilitythatthepaketisoneofhismultipliedbythetotal

bandwidth:

Wj(t−R(t)) PN

i Wi(t−R(t))Bo(t−T).

Infat wewantto omputetheadvane ofwindowsize,whihmeans thatdestroyednon-arrivingpakets

arryinformation. ThusthemodiedACKbandwidthis:

1 1−K(t−R(t))

Wj(t−R(t)) PN

i Wi(t−R(t))Bo(t−T).

Thefatorofadvaneweshalluseinwindowsizes evolutionequationsis:

A(t) = 1 1−K(t−R(t))

1

F(t−R(t))Bo(t−T), ⁽⁵⁾

where F(t) = _N¹ P

iWi(t) =R

wp(t, w)dw ^represents^the^number^of ^paketson-the-ight.

Loss rateindiator Itisgivenby:

κ(t) =K(t−R(t)). ⁽⁶⁾

2.2.4 Bandwidthevolutionwhen rossing thereeiving user

Togofullirle 1

weneedtosaywhat isthevalueofthebandwidthBi(t)^knowing^the^window^size ^evolutions

andtheACKbandwidthBo(t)^. ^Theêvolutionôf^this^numberônlyômes^from^new^pakets^being^sentôrÂCK

beingreeived(ountingasACKtheindiationofalost paket);thus:

dF

dt (t) =Bi(t)− 1

1−K(t−R(t))Bo(t−T). ⁽⁷⁾

1

AKA:givethelastequation