Prasad and Constantine Do vrolis Colle ge of Computing Geor gia Institute of echnology ra vido vrolis ccgatechedu Abstract It is common for simulation and analytical studies to model Internet tr af 57346c as an gr gation of mostly per sis tent TCP 5 ID: 28576
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BeyondtheModelofPersistentTCPFlows:Open-LoopvsClosed-LoopArrivalsofNon-PersistentFlowsRaviS.PrasadandConstantineDovrolisCollegeofComputing,GeorgiaInstituteofTechnologyfravi,dovrolisg@cc.gatech.eduAbstractItiscommonforsimulationandanalyticalstudiestomodelInternettrafcasanaggregationofmostlypersis-tentTCPows.Inpractice,however,owsfollowaheavy-tailedsizedistributionandtheirnumbeructuatessigni-cantlywithtime.Oneimportantissuethathasbeenlargelyignorediswhethersuchnon-persistentowsarriveinthenetworkinanopen-loop(sayPoisson)orclosed-loop(in-teractive)manner.ThispaperfocusesonthedifferencesthattheTCPowarrivalprocessintroducesinthegener-atedaggregatetrafc.WerstreviewtheProcessorSharingmodelsforsuchowarrivalprocessesaswellasthecorre-spondingTCPpacket-levelmodels.Then,wefocusonthequeueingperformancethatresultsfromeachmodel,andshowthattheclosed-loopmodelproduceslowerlossrateandqueueingdelaysthantheopen-loopmodel.Weexplainthisdifferenceintermsoftheincreasedtrafcvariabilitythattheopen-loopmodelproduces.Thecauseofthelat-teristhattheowarrivalrateintheopen-loopmodeldoesnotreduceuponcongestion.Wealsostudythetransientef-fectofcongestioneventsonthetwomodelsandshowthattheclosed-loopmodelresultsincongestion-responsivetraf-cwhiletheopen-loopmodeldoesnot.Finally,wediscussimplicationsofthedifferencesbetweenthetwomodelsinseveralnetworkingproblems.1IntroductionModelsofaggregateTCPtrafcarevaluableinnet-workingresearchandpractice.MuchofthepreviousworkinthisareahasbeenfocusingonthemodelofpersistentTCPows,i.e.,onowsthathaveunlimiteddatatosendandthatarenotlimitedbythereceiveradvertisedwindow.Thismodelismathematicallytractableanditiseasiertosimulate,butatthesametimeitfailstocapturekeyas-pectsofrealInternettrafc[10].Specically,itignorestheheavy-tailednatureoftheowsizedistribution(thatcanproduceLong-RangeDependency),thesignicantvari-ationsinthenumberofactiveowswithtime,andthere-lationbetweencongestionandtheowarrivalprocess.Ontheotherhand,somepreviousworkhasconsiderednon-persistentTCPows,followingaheavy-tailedsizedistri-bution.TheopenissuethereiswhetherthearrivalprocessoftheTCPowsshouldbemodeledinanopen-loop(OL)manner(say,accordingtoaPoissonprocess),orinaclosed-loop(CL)manner(say,fromanumberofinteractiveusers).Thispaperfocusesonthedifferencesthattheowarrivalprocess,OLversusCL,causesinthegeneratedaggregatetrafc.Therelatedissueofwhichmodelismorerealistichasbeenthefocusofarecentmeasurementstudy[16].WestartwiththeuidProcessorSharing(PS)mod-elsfortheOLandCLowarrivalprocesses.ThePSmodelsprovideanaccurateestimateoftheofferedloadinlight/moderateloadconditions.Ontheotherhand,whentheloadapproachesthecapacity,thePSmodelscanleadtosignicantunderestimationoftheofferedload.ThemainproblemisthatthePSmodelsignorepacketlossesandTCPretransmissions,whichareasignicantcontributionofad-ditionalloadincongestedlinks.Nevertheless,thePSmod-elsshowclearlythattheopen-loopmodelcanbeunstable,whiletheclosed-loopmodelisalwaysstable,asthenumberofactiveowsisbounded.Wethencomparethequeueingperformanceofthe(packetlevel)OLandCLmodels,examiningthelossrateandqueueingdelaydistributionthatthetwomodelspro-duceunderthesameofferedload.TheOLmodelproduceshigherlossrateandqueueingdelaysthantheCLmodel.Toexplainthisdifference,weexaminethetrafcvariabil-ityproducedbythetwomodelsinarangeoftimescales(10msec-1sec).WendoutthattheOLmodelresultsinhighervariancethantheCLmodel,especiallywhenthetimescaleexceedstheTCPRound-TripTime(RTT).ThecauseoftheincreasedtrafcvariabilityintheOLmodelisthatthelatterdoesnotreducetheowarrivalrateuponcongestion.Thisleadstomoresignicantoverloadevents,inmagnitudeandduration,thantheCLmodel,generatinghighertrafcvariability.TheCLmodelrespondstoconges-1 tionroughlyoneRTTafteritsoccurrence,whichexplainswhythevariabilitydifferencebecomessignicantwhenthetimescaleislargerthantheRTT.Wealsoexaminethedistributionofthenumberofac-tiveowswitheachowarrivalmodel.Here,wendthattheOLmodelresultsinhighervariabilityinthenumberofowsthantheCLmodelwhentheofferedloadissig-nicant.TherearetimeperiodsinwhichthenumberofongoingowswiththeOLmodelismuchhigherthantheaverage.ThisobservationisrelatedtoanearlierstudybySchroederetal.whichshowedthatjobschedulingiscru-cialmostlywiththeOLmodel,astheformergivesawiderleewaytotheschedulerthantheCLmodel[18].Finally,wefocusonthetransientresponseofthetwomodelsintermsofthecongestionresponsivenessoftheag-gregatetrafc.WithOLowarrivals,theresultingtrafcisnotcongestionresponsive,meaningthattheofferedloaddoesnotfollowtheavailablecapacityinthenetwork.WithCLowarrivals,ontheotherhand,thetrafciscongestionresponsive.Therestofthepaperisorganizedasfollows.Sections2and3reviewpreviousworkandthelimitationsoftheper-sistentowsmodel,respectively.Section4describetheOLandCLmodelsandreviewbasicresultsaboutthecorre-spondingPSmodels.OursimulationsetupispresentedinSection5,whilethequeueingandofferedloaddifferencesbetweenthetwomodelsarepresentedinSections6and7.Section8examinesthevariabilityintheofferedloadatdif-ferenttimescales,whileSection9showsthevariationinthenumberofactiveows.Section10focusesontheconges-tionresponsivenessofthetwomodels.Weconcludethepa-perinsection11,alsodiscussingsomeimplicationsofthisworkinvariousareasofnetworkingresearchandpractice.2RelatedWorkOverthelastfewyears,andespeciallyafterthesemi-nalworkbyKellyetal.[11],severalresearchersappliedcontroltheorytoexaminethestabilityofTCPcongestioncontrol[14,20,15].AkeypointaboutthatlineofworkisthatitassumespersistentTCPows,anditfocusesontheasymptoticstabilityofthequeuesizeatthenetworkbottle-neck.Theassumptionofpersistentowsremovesfromtheproblemtheimportanceoftheowarrivalprocess.Somepreviousworkusesnon-persistentowmodels,butoftenwithoutdiscussingwhethertheOLortheCLmodelismoreappropriate.BenFredjetal.[7]consideredtheOLmodel.Theynotedthattheonlyreductionintheof-feredloaduponacongestioneventisduetoabortedtrans-fers.Suchtransfers,however,resultinwastedthroughputanduserdissatisfaction.Forthisreason,theauthorspro-posedadmissioncontrolastheonlyefcientwaytopreventpersistentoverload.Vecianaetal.[4]consideredtheOLmodelandconcludedthattheInternettrafcmaybecomeunstableundercertainconditions.Heymanetal.[8]usedaCLmodeltoanalyzetheper-formanceofWeb-liketrafcoverTCP.Theyshowedthatthesessiongoodputandthefractionoftimethesystemhasagivennumberofactivesessionsareinsensitivetothedis-tributionofsessionsizesandthinktimes,andtheyonlydependonthemeanofthesedistributions.BergerandKo-gan[2],aswellasBonaldetal.[3],usedasimilarCLmodeltodesignbandwidthprovisioningrulesformeetingcertainthroughput-relatedQoSobjectives.BondiandWhitt[1]examinethedifferencesbetweentheOLandCLmodelsinthecontextofnetworksofqueues,focusingontherela-tionbetweentheaveragequeuesizeatthebottleneckqueueandthevariabilityinthejobservice-timedistribution.MostofthepreviousworkwiththeOLorCLmodelsassumesthatTCPcongestioncontrolcansharethecapacityofalinkasauidPSserver[17].KheraniandKumar[12]showedthatthePSmodelisnotalwaysaccurate,mostlybecauseTCPtransfersdonotmanagetokeepthelinkfullyutilizedundercertainconditions.Inthispaper,weusethePSmodeljusttogainsomebasicanalyticinsight.MostofourconclusionsarebasedonNS2simulationswithTCPtransfers.Inapaperthatiscloselyrelatedtoourwork,Schroederetal.[18]comparetheOLandCLmodelsinageneralcon-textofjobarrivalsataserver.Theyhighlightthedifferencesbetweenthetwomodelsintermsofthemeanjobcomple-tiontime,andtheyfocusontheeffectivenessofdifferentjobschedulingpolicieswitheachmodel.Morerecently,wehaveanalyzedseveraltrafctracescollectedatadozenofInternetlinksinordertoestimatethefractionoftrafcthatcanbemappedtoeithertheOLorCLmodel[16].Thatmeasurementstudyshowsthatabout60-80%oftrafcto/fromwell-knownports(mostlyHTTP)followtheCLmodel.Nevertheless,thepercentageofOLtrafcissignicantinsomelinks.Additionally,wecouldnotclassifyreliablyupto70%ofthetrafcincertaintraces.3CritiqueofthePersistentFlowsModelItiscommonforanalyticalandsimulationstudiestomodelmostofthetrafcwithpersistentTCPconnections.Acommonargumenttojustifythismodelisthatmosttraf-cinanInternetlinkiscarriedbyafewlargeTCPowselephantsandsothoseowscanbemodeledaspersistent.Thesmallerows,referredtoasmice,donotcontributeasignicantamountoftrafcandsotheyareoftenignored,ortheyareviewedasasourceofstochasticnoiseinsim-ulationstudies.Thepreviousargumentisanoversimpli-cationanditignorestwokeycharacteristicsofrealInternettrafc.First,thesizeofTCPowsfollowsacontinuousandheavy-taileddistributioninpractice,ratherthanabimodal distributioninwhichowsareeitherveryshort(mice)orverylong(elephants).Inotherwords,thepreviousargu-mentignorestheowsofsignicant,butnotextreme,size.Second,owswithverylargesize(relativetootherowsintheaggregate)donotalwayshaveverylongduration.Somelargeowsgethigherthroughput,andsotheirdurationcanbecomparabletothatofshortows.Suchowscannotbemodeledaspersistent,especiallywhenthetimescaleofin-terest(forexample,thedurationofthesimulationstudy)islongerthantheirduration.Toillustratetheseissues,weanalyzeapackettracethatwascollectedattheborderrouterofGeorgiaTechinJan-uary2005.Thetracedurationistwohoursandthemoni-toredlinkcarriestheinboundtrafcinaGigabitEthernetsegmentthatconnectsthecampusnetworktotheSoXGi-gaPoP.Theobjectiveofthistrafcanalysisistoexaminetheassumptionsbehindthepersistentowsmodel.Notethatsimilarstudieshavebeenconductedseveraltimesinthepast(forinstance,see[6]),usingtracesfrommanylinksandunderdiverseloadconditions.0102030405060708090100110120Time interval T (min.)0.20.30.40.50.60.7Fraction of bytes from Figure1.ThefractionofbytesfgeneratedbyowsthatareactivefortheentiredurationofagiventimeintervalTasafunctionofT.Theerrorbarsdepicttheminimumandthemaximumvaluesofthefractionf.Werstlookedattheowsizedistribution.WendthattheC-CDFofthatdistributionshowsclearlineardecreaseinalog-logplot,pointingtotheheavy-tailedParetodistri-bution(withshapeparameterabout1.3).Wealsoexaminedthedistributionofowinterarrivals.Whentheinterarrivalsarelargerthan100msecorso,theycanbemodeledasex-ponentialandindependent(pointingtoaPoissonowar-rivalprocess).However,therearesignicantcorrelationsinlowerinterarrivals,probablyduetothegenerationofsimul-taneousowsbythesameapplicationsession.Next,wemeasuredthenumberofactiveowsasafunc-tionoftimefordifferentowsizethresholds.Ifweonlyconsiderowsthatarelargerthan1.5MB(or1000pkts),thenumberofactiveowsremainsalmostconstantwithtime.Thisobservation,however,shouldnotbeinterpretedasvalidationofthepersistentowsmodel.Thereasonisthateventhoughthenumberof(sufcientlylarge)activeowsremainsroughlyconstantwithtime,thesetofactiveowschangessignicantlywithtime.ToillustratethispointweexaminedthefractionofbytesfthatisgeneratedbyowsthatremainedactivethroughoutagiventimeintervaloflengthT.Withthepersistentmodel,allowsareactivethroughoutTandthisfractionshouldbecloseto100%.IfallowslastedforlessthanTseconds,thenthisfrac-tionshouldbezero.Wemeasuredfforthefollowingval-uesofT:7.5,15,30,60and120minutes.ForeachvalueofT(exceptfor120minutes),weobtained30samplesofthefractionf,ignoringthersttwominutesofthetrace,andconsideringowsthatlastlongerthan0:95TasactivethroughoutthedurationT.Figure1showsthemean,theminimumandthemaximumvalueoffasafunctionofT.Thekeyobservationhereisthatevenfortimeintervalsthatlastonly5-10mins,thefractionoftrafcfrompersistentowsisonly40-70%.So,theassumptionthatthesamesetofowscarriesalmostalltrafcignoresthevariabilityduetothedynamicowarrivalandcompletionprocesses.4TwoModelsofNon-PersistentFlowAr-rivalsInthissection,wedescribetwobasicmodelsofnon-persistentowarrivals:theOLandCLmodels.Bothmod-elsaresimpleandwell-studiedintheperformanceevalua-tionliterature.Notethatthetermsopen-loopandclosed-loophavebeenpreviouslyusedtodistinguishbetweennon-TCPtraf-c(viewedasopen-loopbecausepacketsarriverandomlybasedonanexogenousprocess)andTCPtrafc(viewedasclosed-loopbecausetheowisregulatedbyTCPcon-gestioncontrol).Inthispaper,bothOLandCLmodelsdescribeanaggregateofTCPows.Theydiffer,however,inthehigherlevelprocess,operatingatthesessionorap-plicationlayer,thatgeneratestheseows.Figure2showsaschematicdiagramoftheowgenerationprocess.Ifthesessionlayerusessomefeedbackfromthenetwork,sothatitslowsdownthegenerationofnewowsuponcongestion,theresultingtrafcwillbeclosertotheCLmodel.Oth-erwise,intheabsenceofsuchfeedback,theOLmodelismoreappropriate.4.1Open-LoopmodelIntheOLmodel,usersorapplicationsgenerateowsindependentofanypreviousowstheymayhavegenerated.Tomotivatethismodel,considertheaccesslinkofaWebserver.Intheoutbounddirection,theserversendslestoalargepopulationofuserslocatedanywhereintheInternet. NetworkTCP feedback loopSession feedback loop ?TransportSessionClientServerFigure2.Theowarrivalprocessiscon-trolledbythesession/applicationlayer.Isthatlayerresponsivetonetworkcongestion?Assumethatauserdoesnotreturntothisserver,atleastforalongtime,aftercompletingaletransfer.Consequently,theserver'ssessionsarealwayswithnewusers.Ifthelinkbecomescongested,thearrivalrateofnewsessionswillnotbeaffected,asInternetusersaretypicallyunawareofthenetworkstateinagivenpath.ConsideraPSserverwithcapacityC(bytes/sec),aver-ageowarrivalrate(ows/sec),andaverageowsizeS(bytes).WerefertothismodelasPS-OL.Theaverageof-feredloadintheserverisgivenbySandthenormalizedofferedloadiso=S=C:(1)Ifo1,theserverisstableandoistheaverageutiliza-tion.ForaPoissonowarrivalprocess,itcanbeshownthattheaveragenumberofactiveowsisgivenby[13]No=1(1 o):(2)Otherwise,ifo�1,theserverisunstable(aslongasowsareneveraborted).Sinceboththeowarrivalrateandtheaverageowsizeareindependentofthenetworkstate,theaverageofferedloadremainsconstanteveninthepresenceofcongestion.Further,theexpectedthroughputofanewowinthePS-OLmodelisgivenbytheavailablecapacityintheserver,R=C(1 o):(3)DeviatingfromthePSmodel,wecanconsiderapacket-levelmodelofaFirst-Come-First-Served(FCFS)queuewithanitebufferandwithowsthatarecontrolledbyTCPcongestioncontrol(TCP-OL).NoticetwoimportantdifferencesbetweentheTCP-OLmodelandthePS-OLmodel.Intheformer,wecanhavepacketdrops.TCPreactstothemwithretransmissions,whicheffectivelyincreasethesizeoftheaffectedows.Further,itiswellknownthatTCPcangenerateredundantretransmission.ThismeansthattheactualofferedloadbyasetofTCPowsintheOLmodelcanbehigherthanwhatthePSmodelpredictsinEqua-tion(1).Second,aTCPowcanbeactiveevenwhenitdoesnotcompeteforavailablecapacity,becauseofwindowlimitationsduetoslow-start,retransmissiontimeouts,lim-itedadvertisedwindow,etc.ThismeansthattheaveragenumberofactiveTCPowscanbemuchlargerthanEqua-tion(2).4.2Closed-LoopmodelToillustratetheCLmodel,considertheaccesslinkofasmallenterprisewith,sayN,users.Intheinbounddi-rection,mostofthetrafcatthelinkisdownloadsthataregeneratedbytheactivityoftheseNusers.Inthesimplestmodel,eachusercanbeintheActivestatedownloadingale,thenspendingsometimeintheIdle(orThinking)state,andtheneitherdownloadinganotherle,orleavingthesystemforalongertimeperiod(Inactivestate).ThislinkwouldnotcarrymorethanNactiveowsatanytime.Furthermore,ifthelinkbecomescongested,thenthedown-loadlatenciesofallactiveowswillincrease,reducingtheratewithwhichnewowsaregenerated.InthePSversionoftheCLmodel,wehaveaxednum-berofusersN.Eachusergoesthroughcyclesofactivity,withowsofaveragesizeS,followedbyidleperiodsofaveragelengthTi.TheaveragesessionarrivalrateintheCLmodelisgivenbyc=NTt+Ti(4)whereTtistheaverageowtransferlatency.Thelatterde-pendsontheloadatthePSserver.ThustheaverageserverutilizationatthePS-CLmodelisgivenbyc=NSC(Ti+Tt):(5)TheaveragenumberofactiveowsinthePS-CLmodelisgivenby(see[2])Nc=a1 afora 1=N 1 a 1=N CTiSfora�1(6)wherethenormalizedofferedloadisgivenbya=NS=CTi:(7)Notethattheexpectednumberofactiveowsfora 1issamewiththeOLmodel.Ontheotherhand,when�a1,NcincreasesslowlywithaandremainsboundedbyN.SimilartotheTCP-OLmodel,theCLmodelwithaFCFSqueueandTCPows(TCP-CL)candeviatesigni-cantlyfromitsPS-CLcounterpart.First,asinTCP-OL,weneedtoconsidertheextraloadduetorequiredorredundantretransmissions.Second,asintheTCP-OLmodel,TCPisnotabletoalwaysusetheavailablecapacity. 5SimulationSetupTheprevioussectionreviewedwell-knownresultsfortheOLandCLmodels,basedonthePSmodel.Inthiswork,wearemoreinterestedinTCP-speciceffectsthatcannotbecapturedbythePSmodel,aswellasonthevarianceoftheresultingaggregatetrafc.Forthesereasons,werelymostlyonsimulation.Figure3showsourNS-2simulationsetup.Thereare 1Gbps1Gbps50MbpsB=250pkts1105ms45ms5ms5msServersClientsFigure3.Simulationsetup10inputlinks,eachwithcapacity1Gbps,connectedtoanoutputlinkwithcapacityC=50MbpsandbuffersizeB.Thistopologydescribesascenarioinwhichthebottleneckistheingresslinkofanenterprisenetwork,andwheretheserver,backboneandclientlinksareover-provisioned.Inthissetup,wehave20serversthatareconnectedtothebot-tleneckwith1Gbpslinksandwithpropagationdelaysthatvarybetween5msecand45msec.Theround-trippropaga-tiondelayinthissetupvariesfrom30msecto110msec,withaharmonicmeanofaboutT0=60msec1.InallsimulationsweusetheSACK-enabledNS-2TCPmodulesack1.ThebuffersizeBissettothebandwidth-delayproductofthepath(250packets),consideringT0astherepresentativedelay.Themaximumadvertisedwindowissetto256packets.IntheCLsimulations,thereareNclientsthatinitiateTCPtransfers.Theseusersarriveforthersttimeatthenetworkatarandominstantduringtherstfewsecondsofthesimulation.Afterarriving,eachuserfol-lowstheCLowgenerationprocessselectingaserverforeachtransferrandomlyfromthesetof20servers.IntheOLsimulations,theowarrivalprocessisPoissonwithar-rivalrate.Inallsimulations,theowsizefollowsaParetodistributionwithameanof25packetsandshapeparameter1.5.ThethinktimeTifortheCLmodelfollowsanexpo-nentialdistributionwithameanof2seconds.ThevaluesofandNarevariedtoobtaindifferentofferedloadsintheOLandCLmodels,respectively.Eachsimulationrunsfor1000secondsandwereportresultsfortheperiodfrom200to950seconds.6ControllingtheOfferedLoadTocomparethetrafccharacteristicsandqueueingper-formanceoftheOLandCLmodels,werstneedtomake1Theuseoftheharmonicmeanhasbeenrecommendedin[5].surethattheirparametersareselectedsothatbothmodelsproduceequalaverageofferedload.Theofferedloadisde-nedastheamountoftrafcthatarrivesatthebottlenecklinkperunitoftime,anditincludestrafcthatmaygetdroppedduetocongestion.ControllingtheofferedloadintheOLandCLmodels,however,isnottrivial.SupposethatwewanttogenerateacertainofferedloadXatthebottlenecklinkoftheprevioussimulationsetup.GiventheaverageowsizeS(andtheav-eragethinktimeTiinthecaseoftheCLmodel),acommonapproachistorelyonthePSmodel.FortheOLmodel,wecancalculatetherequiredowarrivalrateas=X=S.FortheCLmodel,however,thetermTtdependsonthegivenloadconditions.Acrudeapproximationistoassumelightloadconditions(a 1),andthusTt Ti.Then,therequirednumberofusersisN=XTi=S.010203040506070PS Model Offered Load (Mbps)020406080100Offered Load (Mbps)PS-OLPS-CL-TPS-CLCapacityFigure4.TheofferedloadwiththeTCP-OLandTCP-CLmodels(simulated,y-axis)asafunctionoftheofferedloadXthatispre-dictedbythecorrespondingPSmodels(cal-culated,x-axis).Next,weexaminetherelationbetweentheaverageof-feredloadXpredictedbythetwoPSmodels,aspreviouslydescribed,andtheactualofferedloadthatweobserveinsimulationswithTCPtrafc(TCP-OLandTCP-CLmod-els).Figure4showstheresultsofthiscomparison.ThecapacitylinesX=CandY=Careshownforreference.WeobservethattheofferedloadwiththeTCP-OLmodelisveryclosetotheloadXpredictedbythePS-OLmodel,aslongasXremainsbelowthecapacityC.AsXapproachesCtheTCP-OLofferedloadstartsdeviatingfromX,andwhenX�C(overload)theTCP-OLofferedloadissignif-icantlyhigherthanX.ThereasonisthattheTCP-OLof-feredloadincludesretransmissions(requiredorredundant)ofdroppedpackets.ThefactthattheincreaserateoftheTCP-OLofferedloaddropsasXgoesmoredeeplyintooverloadisduetotheincreasingfrequencyofretransmis-siontimeoutsthattheTCPconnectionsexperience.Nev-ertheless,theimportantobservationhereisthatwecanuse theofferedloadpredictedbyPS-OLmodelasareasonableapproximation,aslongasXC.InthecaseoftheCLmodel,theofferedloadpredictedbythePS-CLmodelislowerthanthatwithTCP-CL,eveninlight/moderateloadconditions.Thereason,ofcourse,isthatwehaveignoredtheload-dependenttransfertimeTt,assumingthatitismuchlessthanTi.EspeciallyforTCPows,however,wecannotignorethatforaowofsizeSthereisaminimumtransferlatencyofseveralRTTsdueslow-start,eveniftherearenoqueueingdelaysorpacketlosses.Thus,wenextconsiderthefollowingimprovedap-proximationoftheofferedloadwiththePS-CLmodel,X=NSTi+Tt;min(S)(8)whereTt;min(S)istheminimumlatencyrequiredbyTCPtotransferaowofsizeSusingslow-start.ItissimpletoestimatethisparameteraslongastheRTTandtheTCPvariantusedareknown.WerefertothisapproximationasthePS-CLmodelwithaconstanttermfortheminimumtransfertimeoftheaverageowsize,orPS-CL-Tforshort.Figure4showstherelationbetweentheofferedloadoftheTCP-CLandPS-CL-Tmodels(withTt;min(S)=0.36secinoursimulations).Notethatthelatterisareasonablygoodapproximationbothwhenthelinkisnotcongested(XC)andinoverload(X�C).ThereasontheofferedloadisslightlyabovethecapacityinoverloadisagainthepresenceofsomeTCPretransmissions.Insummary,thePSmodelcanprovideareasonableapproximationoftheofferedloadintheTCP-CLmodel,aslongasweconsidertheminimumtransfertimewithTCPslow-startfortheaverageowsize.010203040506070PS Model Offered Load (Mbps)020406080100Offered Load (Mbps)PS-OLPS-CL-T (TI=20sec)PS-CL-T (TI=2sec)CapacityN=3200N=390Figure5.TheofferedloadfromtheCLmodeltendstothatoftheOLmodelasNandTiincrease.NoticethattheOLmodelcanbeviewedastheasymp-toticlimitoftheCLmodel,ifweletthenumberofusersNandtheaverageidletimeTigotoinnity,whiletheinitialtransferofeachuserisrandomlyplacedonthetimeaxis.Indeed,wemaywonderwhethertheofferedloadwiththeTCP-CLmodelapproachesthatoftheTCP-OLmodelasweincreaseNandTi.Figure5showstheofferedloadfromtheTCP-CLmodelfortwovaluesofTi,2and20seconds.Notethatanincreaseintheidletimealsorequiresanincreaseinthenumberofusersinordertoattainthesameofferedload.Forexample,withTi=2secweneed400userstoget46Mbpsofofferedload,whilewithTi=20secweneed3200users.Weseethattheofferedloadbetweenthethreecurvesdiffersmostlyinoverload,asexpected.AsweincreaseTiandN,theTCP-CLcurveapproachestheTCP-OLcurve,implyingthegradualconvergenceoftheCLmodeltotheOLmodel.NoticehoweverthatthisconvergenceisveryslowandinpracticewewouldneedaverylargenumberofusersbeforewecanclaimthattheaclosedpopulationofuserscanbemodeledwiththeOLmodel,inoverloadconditions.Intherestofthepaper,weusetheofferedloadthatiscalculatedfromns-2simulations.7QueueingPerformanceNext,wecomparethequeueingperformanceoftheTCP-OLandTCP-CLmodels.Themainobservationisthat,un-derthesameofferedload,theTCP-OLmodelresultsinhigherqueueingdelaysthantheTCP-CLmodel.Iftherearepacketlosses,thenthelossratewithTCP-OLisalsohigherthanwithTCP-CL.01020304050Offered Load (Mbps)00.511.52Loss Rate (%)TCP-OLTCP-CLFigure6.Thelossrateasafunctionoftheof-feredloadfortheTCP-OLandTCP-CLmod-els.6and7showthelossrateandthequeueingde-laysfortheTCP-OLandTCP-CLmodelsasafunctionoftheofferedload.Forqueueingdelays,wereporttheme-dianandthe90-thpercentileoftheper-packetdelaydistri-bution.Thedifferencesareofcourseminorforlightloadconditions,whentheofferedloadis,say,below50%ofthecapacity.Inheavierloadconditions,however,thediffer- 01020304050Offered Load (Mbps)00.010.020.030.040.050.06Queueing Delay (sec.)TCP-OL: 90thPercentileTCP-CL: 90thPercentileTCP-OL: MedianTCP-CL: MedianFigure7.Themedianand90-thpercentileofthequeueingdelaydistributionasafunctionoftheofferedloadfortheTCP-OLandTCP-CLmodels.encesaresignicantandcannotbeignored.Inthenextsec-tionweexplainthesedifferencesexaminingthestatisticalvariabilityoftheaggregatetrafcindifferenttimescales.8TrafcVariabilityatDifferentTimescalesTheresultsoftheprevioussectionsuggestthattheTCP-OLmodelproduceslargertrafcburstinessthantheTCP-CLmodel.Inthissectionweaimtofurtherunderstandwhatcausesthisdifferenceandtoidentifytheloadconditionsandtimescalesinwhichthisismoreevident.Figure8showsthevarianceoftheofferedloadforanaveragingtimescaleof10msec,100msecand1sec.First,noticehowthevariancedependsontheofferedload.Thevarianceincreasesuptoacertainpoint(20-45Mbps,de-pendingonthetimescaleandthemodel).Afterthatpointthevariancedecreaseswiththeofferedload.Foranex-planationofthiswell-understoodtrendwereferthereaderto[9,19].WhatismorerelevanthereisthattheTCP-OLmodelproduceshighervariancethantheTCP-CLmodelinmoderate/heavyloadconditions.Sincetheround-tripprop-agationdelaysinoursimulationtopologyvaryfrom30msecto110msec,weviewthetimescaleof10msecasbelowthetypicalRTT,100msecasroughlyequaltotheRTT,and1secaslargerthantheRTT.TheresultsofFigure8alsosuggestthatthedifferenceinthevarianceofthetwomodelsismoresignicantwhenthetimescaleisaroundtheRTTorhigher.Inlightloadconditionsthetwomodelsarepracticallyequivalent,asthereisnosignicantqueueingorpacketlossesandtransfersareonlylimitedbyTCP'sslow-start.Astheofferedloadincreasesbeyondroughly50%ofthecapacity,congestionepisodesstarttooccur.IntheTCP-OLmodel,newowsarriveindependentofwhetherthebot-tleneckiscongestedornot.IntheTCP-CLmodel,whenaowslowsdownbecauseofcongestionitalsodelaysthegenerationofthenextowfromthesameuser.Thisreducesthedurationandmagnitudeofcongestionevents,leadingtolowertrafcvariabilitythanintheTCP-OLmodel.TheresponselatencyoftheTCP-CLmodelcannotbefasterthanTCP'sRTThowever;thisexplainswhythetwomodelslookthesameinsub-RTTtimescales.0100200300400500050100150VarianceTCP-CLTCP-OL01020304050Offered Load (Mbps)0255075Time Scale = 10 msecFigure8.VarianceoftheofferedloadwiththeTCP-OLandTCP-CLmodelsforthreeaverag-ingtimescales.0102030405000.10.20.30.40.5TCP-CLTCP-OL0102030405000.10.20.30.40.501020304050Offered Load (Mbps)00.10.20.30.40.5�Fracion of time offered load Capacity01020304050Offered Load (Mbps)00.10.20.30.40.5Time Scale = 10msecTime Scale = 100msecTime Scale = 1 secTime Scale = 10 secFigure9.Fractionoftimetheofferedloadisgreaterthanthecapacityforfouraveragingtimescales.ofurtherillustratethepreviousexplanation,Figure9showsthefractionoftimetheofferedloadexceedsthelinkcapacityinfouraveragingtimescales.Hereweseethatinthesub-RTTtimescaleof10msec,bothmodelsexperienceoverloadforpracticallythesamefractionoftime.WhenweexaminethetrafcathighertimescalesthantheRTT,however,weconrmthattheTCP-OLisoverloadedmorefrequently.TheTCP-CLmodelexperiencesoverloadlessoftenbecauseitsowarrivalratereducesupontheoccur-renceofpacketlosses.Sincethetwomodelshavethesameaverageofferedload,thehigheroverloadfrequencyinTCP-OLiscompensatedwithtimeperiodsinwhichtheTCP-OL offeredloadislessthanthatinTCP-CL.Thesewideruc-tuationsmakethevarianceofTCP-OLhigher,aslongasthetheofferedloadandtimescalearesufcientlylarge.02468101200.20.40.60.81CDF02468101200.20.40.60.8102468101200.20.40.60.81CDFTCP-OLTCP-CL024681012Number of Consecutive Congestion Periods00.20.40.60.81Time Scale = 10 msecTime Scale = 100 msecTime Scale = 10 secTime Scale = 1 secFigure10.TheCDFofthelengthofcon-secutiveoverloadperiodsforfouraveragingtimescales.Theaverageofferedloadis95%ofthecapacity.Itisnotjustthefrequencyofoverloadeventsthatdif-fersbetweenthetwomodels,butalsotheirduration.ThisisshowninFigure10,whereweplottheCDFofthedurationofoverloadeventsatdifferenttimescalesforanaverageof-feredloadof47.5Mbps.Thisdurationismeasuredasthenumberofconsecutivetimeperiods(withlengthequaltotheaveragingtimescale)inwhichtheofferedloadishigherthanthecapacity.Inthesub-RTTtimescalebothmodelshavethesamedistribution.Asthetimescaleincreases,how-ever,thegapbetweenthetwodistributionsincreases,astheTCP-OLmodelisunabletoself-regulateitsofferedloadbe-lowthecapacity.Forinstance,whenwelookatthetrafcinsuccessiveintervalsof10seconds,about85%oftheover-loadeventsinTCP-CLlastforonlyoneinterval.Thecor-respondingpercentageisonly40%intheTCP-OLmodel.9NumberofActiveFlowsInthissection,weexaminethenumberofactiveowscreatedbytheTCP-OLandTCP-CLmodels.WeshowthatthenumberofactiveowsinthesetwoTCPmodelsismuchlargerthanthatpredictedbythePSmodel,andthatTCP-OLproduceshighervariabilityinthenumberofactiveowsthantheTCP-CLmodel,inheavyloadcondi-tions.Thelatterimpliesthattheper-owthroughputintheTCP-OLmodelisalsolesspredictable.Figure11showstheCDFoftheaveragenumberofac-tiveowswhentheofferedloadis70%and95%ofthecapacity.Thenumberofactiveowsisaveragedover1-secintervals.Werstnotethatthenumberofactiveowsin5010015020025000.20.40.60.81CDFTCP-CLTCP-OL50100150200250Average Number of Active Flows00.20.40.60.81CDF70% Offered Load 95% Offered Load Figure11.TheCDFoftheaveragenumberofactiveows,measuredat1-secintervals,fromtheTCP-OLandTCP-CLmodelswhentheofferedloadis70%and95%oftheca-pacity.bothmodelsismuchhigherthanthatpredictedbythepro-cessorsharingmodel(seeEquations2and6).Specically,thePS-OLmodelpredictsabout3and20owsforofferedload70%and95%,respectively.Thecorrespondingaver-agesfromtheTCP-OLsimulationsare70and131.ForthePS-CLmodel,ontheotherhand,Equation6predictsanav-erageof102activeowsfor95%offeredload,whiletheaveragefromtheTCP-CLsimulationsis115.Thesediffer-encescanbeattributedtothefactthat,withTCP,thereisalargenumberofsmallowsthatarenotalwayscompetingforavailablecapacitybecauseofslow-start,retransmissiontimeouts,orotherlimitations.AlsonoticethattheTCP-OLmodelresultsinmuchhighervariabilityinthenumberofactiveowsinheavyloadconditions.Again,thisisbecausetheTCP-OLmodeldoesnotreducetheowarrivalrateuponcongestion.ThenumberofactiveowsintheTCP-CLmodel,ontheotherhand,isalwaysboundedbyN.Theincreasedvariabil-ityinthenumberofactiveowswiththeTCP-OLmodelmeansthattheper-owthroughputwiththatmodelislesspredictablethanwithTCP-CL.10CongestionResponsivenessSofarwehavefocusedonthesteady-statebehaviorofthetwomodels.Inthissection,weexaminetheirtransientresponsetoindividualcongestionevents.Werefertoatrafcaggregateascongestionresponsiveifitreducesitsofferedloaduponoverloadtoapointthatthereisnolongercongestion.ThespeciccongestioneventthatweconsiderhereisaperiodicUDPstreamwithratethatishigherthantheavailablecapacityinthebottleneck.GiventhattheUDPstreamdoesnotreacttocongestion, theeventthatwesimulaterepresentsasuddenreductionoftheavailablecapacityfortheTCPaggregatefromCtoC0=(1 f)C,wherefCistherateoftheUDPstream.Inthefollowing,wemaketheofferedloadCbeforethecongestioneventtobeatthesamelevelintheTCP-OLandTCP-CLmodels.Weset1 f1,sothatthebottle-neckbecomescongestedwhentheUDPstreamstarts.100200300400Time (sec.)3040506070Offered Load (Mbps)TCP-OLTCP-CL15Mbps UDP Stream Figure12.Theresponseofthetrafcag-gregateintheTCP-OLandTCP-CLmodels,whenacongestioneventiscausedbyaUDPstreamofrate15Mbps.Thecapacityis50Mbpsandtheofferedload(beforethecon-gestionevent)is47.5Mbps.Figure12showstheofferedloadfromthetwotrafcmodelsin1-secintervals.Thecongestioneventiscausedbya15MbpsCBRUDPstreamanditlastsfrom200secto275sec.Theeffectsofthecongestioneventcanbeex-aminedinthreestages:rst,justafterthecongestioneventstarts,second,duringthecongestionevent,andthird,afterthecongestioneventnishes.Beforethestartofthecongestionevent,bothTCP-OLandTCP-CLhavethesameaverageofferedload.TheirresponsewhentheUDPstreamstartsisthat,becauseofTCP'scongestioncontrol,thetrafcfrombothmodelsdropsatalevelthatisclosetothenewavailablecapacity(35Mbps).Thesimilaritybetweenthetwomodels,how-ever,endsthere.AfewsecondslatertheofferedloadintheTCP-OLmodelstartsincreasing,asmoreandmorenewowsarriveandcompeteforthroughput.TheofferedloadintheTCP-CLmodel,ontheotherhand,isself-regulatedattheleveloftheavailablecapacity,becauseanewowcan-notstartunlessanexistingowhascompleted.Thus,thenumberofactiveowsintheTCP-OLmodelkeepsincreas-ing,whilethecorrespondingnumberintheTCP-CLmodelstaysroughlythesame(alsoseeFigure13).Finally,afterthecongestioneventends,theofferedloadfrombothmodelsincreasestocapturetheavailablecapacitythathasbeenreleasedbytheUDPstream.IntheTCP-CLmodel,thisprocessiscompletedwithinafewseconds.IntheTCP-OLmodel,however,thereisalargebacklogofac-tiveowsthatneedstobeclearedbeforetheofferedloadreturnsatitspre-congestionlevel.AsFigure13shows,thiseffectlastsforhundredsofseconds(thisdependsofcourseonthedurationandmagnitudeofthecongestioneventandontheTCPofferedloadbeforecongestion).Figure13fur-thershowsthequeueingdelayinthebottleneckwitheachmodel.Noticethateventhoughthecongestioneventendsatt=275sec,thequeueremainsalmostfullforhundredsofsecondswiththeTCP-OLmodel.1001502002503003504000500100015002000Num. of Active FlowsTCP-OLTCP-CL100150200250300350400Time (sec)00.020.040.060.08Queuing Delay (sec.)15Mbps UDP streamFigure13.ThetimeseriesofthenumberofactiveowsandofthequeueingdelaywiththeTCP-OLandTCP-CLmodelswhenacongestioneventiscausedbyaCBRUDPstream.enifthelong-termofferedloadatalinkisbelowthecapacity,therecanbeoverloadeventsthatlastforafewtensofseconds.Theimportantlessonfromthepreviousdiscussionisthatduringsucheventsanopen-looptrafcaggregateiseffectivelycongestionunresponsivedespitethefactthatitconsistsofTCPows.Further,theconsequencesofanexternallyimposedcongestionevent(sayalargeUDPstreamoraDOSattack)canlastformuchlongerthanthedurationoftheeventitself,ifthetrafcisopen-loop.11DiscussionInthispaper,weexaminedtwobasicmodelsofnon-persistentowarrivals,andexplainedhowtheyleadtodif-ferenttrafccharacteristics,intermsofofferedload,vari-abilityindifferenttimescales,queueingperformance,num-berofactiveows,congestionresponsivenessandelastic-ity.Inthefollowing,wediscusssomemoreimplicationsofthisworkinotherareasofnetworkingresearchandpractice.AQMandnetworkstability:Activequeuemanage-ment(AQM)mechanisms,suchasRED,REM,PIcon-trollers,etc.,havebeenproposedasawaytostabilizecon-gestioncontrol.Itisimportanttonotethatsuchstudiesas-sumepersistentTCPconnections.Withthatmodel,AQM mechanismscancontrolthequeuelengthandthebottle-necklinkutilization.TheeffectivenessofAQMmecha-nismswithnon-persistenttrafc,however,ismuchlessun-derstood.TheofferedloadofTCP-OLtrafcdoesnotde-pendonnetworkstate.AQMmechanismscannotregulatesuchanaggregate,andtheyareunabletoavoidpersistentoverloadiftheofferedloadexceedsthecapacity.Isadmissioncontrolnecessary?Severalresearchersadvocatetheuseofadmissioncontrolastheonlywaytoregulatetheofferedloadandavoidcongestioncollapse.Weagree,ifthetrafcismostlyOL.Withoutadmissioncontrol,theonlywaytoavoidcongestioncollapseistoexpectthatuserswillbeimpatientandabandonslowongoingtransfers.Admissioncontrolcanlimitthenumberofactivesessionsorowsinthenetwork.Admissioncontrolmaynotbenec-essary,however,ifmostofthetrafcfollowstheCLmodel.TCP-friendlycongestioncontrol:TheuseofTCP-friendlycongestioncontrolhasbeenencouragedinallnon-TCPprotocolsandapplications.ThebasicmotivationforsuchproposalsisthatTCP-friendlytransferscanavoidcon-gestioncollapse.Itshouldbeclearhowever,thatevenifatrafcaggregateconsistsentirelyofTCPows,itcanstillcausecongestioncollapseorpersistentoverloadifitisOL.ThesameisobviouslytrueforTCP-friendlytrafc.There-fore,theuseofTCP-friendlycongestioncontrolisnotsuf-cienttoguaranteestability.Trafcengineeringandnetworkprovisioning:Traf-cengineering,aswellasotherprovisioningmecha-nisms,requireanestimatefortheofferedloadbetweenanyingress/egresspair.Furthermore,suchmechanismsassumethatifagiventrafcaggregateisswitchedfromoneroutetoanother,thenthethroughputofthataggregatewillnotchange.ThisassumptionisnottrueforTCP-CLtrafc.TheofferedloadfromsuchaggregatesdependsontheRTTandlossrateintheunderlyingpath.Ontheotherhand,theof-feredloadfromTCP-OLtrafcdoesnotdependontheun-derlyingpath(ignoringretransmissions),makingsuchtraf-cconsistentwithcommonassumptionsintrafcengineer-ing.Sessionlayercongestioncontrol:Atthemorepracti-calside,werecommendthatallnetworkapplicationsusesomeformofcongestioncontrolatthesessionlayer.Thiscanbeassimpleasadoptingoneofthefollowingrules:donotgenerateanewsessionuntiltheprevioussessionhascompleted,slowdownthegenerationofnewsessionsifthenetworkiscongested,ordonotkeepmorethanacertainnumberofsessionsactive.Itisalsoimportantthatsessionlayercongestioncontrolisimplementedinapplicationsthatgeneratetransfersautomatically,withoutuserintervention.Forexample,NNTPserverstransfernewstotheirpeerspe-riodically,independentofwhethertheunderlyingnetworkiscongestedornot.Effectively,suchapplicationsgenerateTCP-OLtrafc.References[1]A.BondiandW.Whitt.TheInuenceofService-TimeVari-abilityinaClosedNetworkofQueues.PerformanceEvalu-ation,6:219234,1986.[2]A.BergerandY.Kogan.DimensioningBandwidthforElas-ticTrafcinHigh-SpeedDataNetworks.IEEE/ACMTrans.onNetworking,8(5):643654,2000.[3]T.Bonald,P.Olivier,andJ.Roberts.DimensioningHighSpeedIPAccessNetworks.In18thITC,2003.[4]G.deVeciana,T.Lee,andT.Konstantopoulos.StabilyandPerformanceAnalysisofNetworksSupportingServices.IEEE/ACMTrans.onNetworking,9(1),2001.[5]A.DhamdhereandC.Dovrolis.BufferSizingforCongestedInternetLinks.InIEEEInfocom,2005.[6]A.Feldmann.CharacteristicsofTCPConnectionArrivals.TechnicalReport,AT&TLabsResearch,1998.[7]S.B.Fredj,T.Bonald,A.Proutiere,G.Regnie,andJ.W.Roberts.StatisticalBandwidthSharing:AStudyofCon-gestionatFlowLevel.InACMSigcomm,2001.[8]D.Heyman,T.V.Lakshman,andA.L.Neidhardt.ANewMethodforAnalysisFeedback-BasedProtocolswithAp-plicationstoEngineeringWebTrafcovertheInternet.InACMSigmetrics,1997.[9]M.JainandC.Dovrolis.End-to-endEstimationoftheAvail-ableBandwidthVariationRange.InACMSigmetrics,2005.[10]Y.Joo,V.Riberio,A.Feldmann,A.Gilbert,andW.Will-inger.TCP/IPtrafcdynamicsandnetworkperformance:Alessoninworkloadmodeling,owcontrol,andtracedrivensimulations.ACMCCR,Apr2001.[11]F.P.Kelly,A.Maulloo,andD.Tan.RateControlinCom-municationNetworks:ShadowPrices,ProportionalFairnessandStability.JournaloftheOperationalResearchSociety,49:237252,1998.[12]A.A.KheraniandA.Kumar.StochasticModelsforThroughputAnalysisofRandomlyArrivingElasticFlowsintheInternet.InIEEEInfocom,2002.[13]L.Kleinrock.Time-sharedSystems:ATheoreticalTreat-ment.JournaloftheACM,14(2):242261,1967.[14]S.KunniyurandR.Srikant.Stable,Scalable,FairCon-gestionControlandAQMSchemesthatAchieveHighUti-lizationintheInternet.IEEETrans.onAutomaticControl,49:20242029,2004.[15]S.H.Low,F.Paganini,J.Wang,S.Adlakha,andJ.Doyle.DynamicsofTCP/REDandaScalableControl.InIEEEInfocom,2002.[16]R.S.PrasadandC.Dovrolis.MeasuringtheCongestionResponsivenessofInternetTrafc.InPAM,2007.[17]J.Roberts.ASurveyonStatisticalBandwidthSharing.ComputerNetworks,45:319332,2004.[18]B.Schroeder,A.Wierman,andM.Harchol-Balter.ClosedVersusOpen:ACautionaryTale.InNSDI,2006.[19]X.Tian,J.Wu,andC.Ji.AUniedViewofHeterogeneousNetworkTrafc:ImpactofNetworkLoad.InIEEEInfo-com,2002.[20]Y.Zhang,S.Kang,andD.Loguinov.DelayedStabilityandPerformanceofDistributedCongestionControl.InACMSigcomm,2004.