134A.KesselmanandY.MansourDolevetal.[6]studyretransmissiontimeoutsetti - PDF document

134A.KesselmanandY.MansourDolevetal.[6]studyretransmissiontimeoutsettingforasimpletransmissionproto-colbymeansofcompetitiveanalysis.LudwigandKatz[17]proposetheEifelalgorithmtoeliminatetheunnecessaryretransmissionsthatcanresultfromaspuriousretransmis-siontimeout.GurtovandLudwig[11]presentanenhancedversionoftheEifelalgorithmandshowitsperformancebenetsonpathswithahighbandwidth-delayproduct.EkstrandLudwig[7]proposeanewalgorithmforcalculatingtheRTO,namedthePeak-Hopper-RTO(PH-RTO),whichimprovesupontheperformanceofTCPinhighlossenvironments.SignicanteffortshavebeenalsodevotedtomodelingsuchacomplexprotocolasTCP[16,15,18,20].InthispaperwestudyhowtondtheoptimalRTOmaximizingtheTCPthroughputusingthemodelof[20].OurmaincontributionistoshowthattheoptimalRTOneedtodependontheTCPwindowsize.WederivetheoptimalRTOasafunctionofRTTandtheTCPwindowsizeforageneralandsomespecicdistributionsofRTT.Intuitively,thelargertheTCPwindowsize,thelongertheoptimalRTO.Wenotethattheheavy-tailedParetodistributionhasbeenshowntoapproximatemostaccuratelytheactualRTTdistributionintheInternet[1,2,10,5].AnimportantadvantageofouralgorithmisthatitcanbeeasilyimplementedonthetopoftheexistingTCPtimeoutmechanism.RFC3649[8]proposesamodicationofTCPcongestioncontrolthatadaptstheincreasestrategyandmakesitmoreaggressiveforhighbandwidthlinks(i.e.forlargewindowsizes).InthisworkwedemonstratethatforsuchscenariosTCPthroughputcouldbefurtherincreasedbyselectingalargerRTO.Ourresultsarestrengthenedbysimulationsin[11],whichshowthatproperhandlingofspurioustimeoutsonpathswithahighbandwidth-delayproductcanincreaseTCPthroughputbyuptoTherestofthepaperisorganizedasfollows.SummaryofourresultsappearsinSection2.InSection3wedescribetheTCPmodel.Section4containsananalyticderivationoftheoptimalRTO.AgeneralRTTdistributionandsomespecicdistributionsareconsideredinSection5andSection6,respectively.2SummaryofResultsInthissectionwegiveanoverviewofourmainresultswhiletheformaldenitionsandproofsaredeferredtothefollowingsections.WeassumethatRTTbehaveslikearandomvariableandderivetheoptimalretransmissiontimeoutasafunctionofthemeanandthevarianceofRTTandtheTCPwindowsize.TheinputparameterstoouralgorithmaretheRTTmean,theRTTvarianceandtheTCPwindowsize.(Weassumethatbotharenite.)OurgoalistondtheoptimalRTOmaximizingtheTCPthroughput.WeshowthatitisanincreasingfunctiononFirstweobtainsomeupperboundsontheoptimalRTOforageneralRTTdistri-bution.Theseboundsmaybeconsideredasworst-caseboundssincetheyholdforanydistribution.TheresultsarepresentedinTable1.WeshowthatforanyRTTdistributiontheoptimalRTOisboundedfromaboveby timesthemeanofRTT.Pro-videdthathighermomentsofRTTexist,weestablishboundswhicharemostlydrivenbythosemomentswhiletheeffectofthewindowsizebecomesinsignicant.NoticethatwhenRTTisaxedconstant,weobtainanupperboundwhichtendstoRTT. OptimizingTCPRetransmissionTimeout135Table1.Generaldistribution Moment RTO–UpperBound Firstmoment 1
3W
RTT ’thmoment
W2 31 RTTTRTT k Table2.Specicdistributions RTTDistribution RTO–OptimalValue Normal µ+
·O +ln  Exponential µ·O(lnW) Pareto
W2 1Š1 NextwederivetheoptimalRTOforsomespecicdistributions.ThecorrespondingresultsarepresentedinTable2.Basically,wewouldliketheprobabilityofaprematureretransmissiontimeouttobeverysmall.Therationalisthatthethroughputdegradationduetoaprematureretransmissiontimeoutismuchhigherthanthatduetoalateretrans-missiontimeout.Ourmodelsetstheprobabilityofaprematureretransmissiontimeoutatabout,foroptimizingtheTCPthroughput.IncaseRTTisdistributedaccordingtotheNormaldistribution,onewouldexpecttheoptimalRTOtobeasumofthemeanplusthestandarddeviationtimessomefactor,asouranalysisindeedshows.ThefactorofisduetothefactthatwhenRTOtheexpectednumberofroundswastedasaresultofalateretransmissiontimeoutis
/µ.ThissettingissimilartotheRTOcalculationofJacobson[12]whilethemaindifferenceisthedependenceonthewindowsize.FortheExponentialRTTdistribution,weshowthattheoptimalRTOisproportionaltothemeanofRTTandthelogarithmofthewindowsize.Thelogarithmicfactorofthewindowsizefollowsfromtheformofthedensityfunction.Finally,weconsidertheheavy-tailedParetodistributionofRTTandestablishthattheoptimalRTOisthemeanofRTTmultipliedbyapowerofthewindowsize.Suchadependenceisduetotheheavy-tailpropertyoftheParetodistribution.3TCPModelWeadoptthemodelof[20]thatisbasedonRenoversionofTCP.TheTCP’scongestionavoidancebehaviorismodeledintermsof"rounds."Thedurationofaroundisequaltotheroundtriptimeandisassumedtobeindependentofthewindowsize.WedenearoundofTCPtobeatimeperiodstartingfromtransmittingawindowofpacketsback-to-backandendinguponreceivingtheacknowledgmentsforthesepackets.Wemakethefollowingsimplifyingassumptions.Thereisalwaysdatapendingatthesender,suchthatthesendercanalwaystransmitdataaspermittedbythecongestionwindowwhilethereceiver’sadvertisedwindowissufcientlylargetoneverconstrainthecongestionwindow.Everypacketisassumedtobeindividuallyacknowledged(the 136A.KesselmanandY.Mansourdelayedacknowledgmentalgorithmisnotineffect).Apacketislostinaroundindepen-dentlyofanypacketslostinotherrounds.However,packetlossesarecorrelatedamongtheback-to-backtransmissionswithinaround:ifapacketislost,alltheconsequentpacketstransmitteduntiltheendofthatroundarealsolost.Wedenepacketlossprob-tobetheprobabilitythatapacketislost,giventhateitheritistherstpacketinaroundortheprecedingpacketintheroundisnotlost.Wecallthecongestionavoidancephaseasteadystate.Weassumethattimeoutexpirationdoesnotoccurduringaslowstartphaseandconcentrateonthetimeoutsettinginasteadystate.WealsoassumethatthemeanandthevarianceofRTTareavailableorcouldbeestimated.WeapproximatethepacketlossprobabilityasafunctionoftheTCPwindowsizeinasteadystate,whichisasimplicationof[20],as .Wenotethatthemodelof[20]capturestheeffectofTCP’stimeoutmechanismonthroughput.4TCPTimeoutOptimizationInthissectionweconsideroptimizationoftheretransmissiontimeout.ThegoalistomaximizethethroughputofTCP.NoticethattheoptimalRTOistheactualRTT,whichisunknowntoouralgorithm.Thus,theonlinedecisionmustbebasedonlyontheavailableestimatesofthemeanandthevarianceofRTT.WetrytondthevalueofRTOthatbalancesthroughputdegradationbetweenaprematureretransmissiontimeoutandalateretransmissiontimeout,whicharesocalled“badevents”(thatwillbeformallydenedlater).Recallthatinourmodelbadeventsoccuronlyinasteadystate.Whenabadeventhappens,weconsidertheconvergenceperiodduringwhichTCPreachesasteadystate.WecomparethethroughputofTCPduringwiththatofanoptimalalgorithmthatusestheactualRTTasitsRTOandsendsinaveragepacketseveryround.Wecalltothenumberofextrapacketssentbytheoptimalalgorithmduringthroughputdegradation.Thegoalistominimizetheexpectedthroughputdegradationduetobadevents.Firstwewillderivetheexpecteddurationoftheconvergenceperiod.Inthecaseofaprematureretransmissiontimeout,ittakesexactlyroundsforTCPtoreachasteadystatesincetheTCPwindowgrowsexponentiallyduringaslowstartphase.Inthecaseofalateretransmissiontimeout,TCPisidleinsteadoftransmittingduringRTORTTtimeunits.Thus,theexpectationofthelengthofinroundsis:is:lengthRTO�RTT RTO�RTTRTORTORTTdRTT.WeapproximatetheexpectednumberofroundsusingtheLawofLargeNumbersasaslengththroundsinRTO�RTT RTO�RTTRTORTORTT dRTT. Suchsituationnaturallyoccurswhenthetail-droppolicyisdeployedbythebottleneckrouter. OptimizingTCPRetransmissionTimeout137Assumingthatthereisasequenceofoneormorelossesinagivenround,theproba-bilityofretransmissiontimeoutis [20].Inthesequel,weassumethatNextwewilldenethebadeventsmoreformally.Prematureretransmissiontimeout.Wesaythatatimeoutoccurredprematurelynopacketintheroundislostorthelosscanbecapturedbythefastretransmissionmechanism.NotethatRTOmustbesmallerthanRTT.Theprobabilityofthiseventis:is:RTORTT RTORTTThethroughputdegradationduetothiseventis:.Observethatduringtheslowstartphase,TCPsendsatmostpackets.Weobtainthattheexpectedthroughputdegradationasaresultofaprematureretransmissiontimeoutis:is:RTORTTLateretransmissiontimeout.Wesaythatatimeoutoccurredifsomepacketsintheroundarelostandthelosscannotbecapturedbythefastretransmissionmechanism.NotethatRTOmustbelargerthanRTT.Theprobabilityofthiseventis:is:RTO&#x-270;&#x.100;RTT RTO�RTT Thethroughputdegradationduetothiseventis: RTO�RTTRTORTORTT dRTT.Wegetthattheexpectedthroughputdegradationasaresultofalateretransmissiontimeoutis: RTORTORTT dRTT.TheoptimalRTO,RTO,minimizestheexpectedthroughputdegradation,thatis:RTORTORTORTOThus,giventheprobabilitydistributionofRTT,theoptimalRTOminimizes:minimizes:RTORTT RTORTORTT dRTT.Forsimplicity,wewillderiveanapproximationfortheoptimalRTO,thebalancedRTO,forwhichtheexpectedthroughputdegradationisthesameforbothofthebadevents:ents:RTORTT RTORTORTT dRTTNotethatintheworstcasetheexpectedthroughputdegradationforthebalancedRTOisatmosttwiceaslargeasthatfortheoptimalRTO. OptimizingTCPRetransmissionTimeout1396.2ExponentialDistributionInthissectionweconsidertheExponentialdistributionofRTTwiththerateparameterthemeanmeanx]=1/,thedensityfunctionandthedistributionfunction.WeshowthattheoptimalRTOisproportionaltothemeanofRTTandthelogarithmoftheTCPwindowsize.Substitutingto(1),(1),RTORTTRTOORTTRTTweget:RTO RTORTORTOThisgivesusthefollowingRTO:RTO ln W2 3 =1 .Theloga-rithmofachievestheeffectofsettingtheprematureretransmissiontimeoutprobabilitytobeorderof6.3ParetoDistributionInthissectionweconsidertheheavy-tailedParetodistributionofRTTwiththeshape,themean ,thedensityfunction andthedistributionfunction .WeshowthattheoptimalRTOisthemeanofRTTmultipliedbyapowerofthewindowsize,whichisduetotheheavy-tailpropertyofthePareto.Substitutingto(1),(1),RTORTT RTOORTT RTT givesus: RTO WaŠ1 a·1Š1 RTORTORTORTT SolvingthisequationderivesthefollowingRTO:RTO .AninterestingsettingisisRTT.InthiscasewegetthatRTO ,whichjustiestheformoftheboundwehaveforanarbitrarydistribution.References1.A.AcharyaandJ.Saltz,"AStudyofInternetRound-tripDelay,"TechnicalReportCS-TR-,UniversityofMaryland,December1996.2.M.AllmanandV.Paxson,“OnEstimatingEnd-to-EndNetworkPathProperties,”InProceed-ingsofSIGCOMM’99,pp.263-274.3.H.Balakrishnan,S.Seshan,M.Stemm,andR.H.Katz,“AnalyzingStabilityinWide-AreaNetworkPer-formance,”InProceedingsofSIGMETRICS’974.J.C.Bolot,“CharacterizingEnd-to-EndPacketDelayandLossintheInternet,”JournalofHighSpeedNetworks,2(3),September1993.5.C.J.Bovy,H.T.Mertodimedjo,G.Hooghiemstra,H.UijterwaalandP.VanMieghem,"Anal-ysisofEndtoendDelayMeasurementsinInternet,"InProceedingsofPAM2002,March