In Proceedings of Performance Tools 98 Lecture Notes in Computer Science, Vol. 1468, pp , September 1998

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1 In Proceedings of Performance Tools 98 OnChoosingaTaskAssignmentPolicyfora Lecture Notes in Computer Science, Vol. 1468, pp , September 1998 MorHarchol-Balter?;1,MarkE.Crovella??;2,andCristinaD.Murta???;2 DistributedServerSystem 2DepartmentofComputerScience,BostonUniversity 1LaboratoryforComputerScience,MIT Abstract.Weconsideradistributedserversystemmodelandaskwhich policiescommonlyproposedforsuchdistributedserversystems:round- vicedemandisknowninadvance.weconsiderfourtaskassignment hostprocessestasksinfirst-come-first-serveorderandthetask'sser- Robin,Random,Size-Based,inwhichalltaskswithinagivesizerange policyshouldbeusedforassigningtaskstohosts.inourmodeleach areassignedtoaparticularhost,anddynamic-least-work-remaining, criticallyonthevariabilityinthetasksizedistribution.inparticular decisionofwhichtaskassignmentpolicyisbest.wendthatnooneof inwhichataskisassignedtothehostwiththeleastoutstandingwork. wendthatwhenthetasksizesarenothighlyvariable,thedynamic theabovetaskassignmentpoliciesisbestandthattheanswerdepends Ourgoalistounderstandtheinuenceoftasksizevariabilityonthe policyispreferable.howeverwhentasksizesshowthedegreeofvariabilitymorecharacteristicofempiricallymeasuredcomputerworkloads,the Size-Basedpolicyisthebestchoice.Weusetheresultingobservations performthedynamicpolicybyalmost2ordersofmagnitudeandcan outperformothertaskassignmentpoliciesbymanyordersofmagnitude, toargueinfavorofaspecicsize-basedpolicy,sita-e,thatcanout- 1Introduction underarealistictasksizedistribution. ofthistrendincludedistributedwebservers,distributeddatabaseservers,and distributeddesignsbecauseoftheirscalabilityandcost-eectiveness.examples Tobuildhigh-capacityserversystems,developersareincreasinglyturningto highperformancecomputingclusters.insuchasystem,requestsforservicearrive assigningtaskstohostmachinesisknownasthetaskassignmentpolicy. andmustbeassignedtooneofthehostmachinesforprocessing.therulefor???supportedbyagrantfromcapes,brazil.permanentaddress:depto.deinformatica,universidadefederaldoparana,curitiba,pr81531,brazil.??supportedinpartbynsfgrantsccr andccr ?supportedbythensfpostdoctoralfellowshipinthemathematicalsciences.

2 systeminwhicheachincomingtaskisimmediatelyassignedtoahostmachine, andeachhostmachineprocessesitsassignedtasksinrst-come-rst-served (FCFS)order.Wealsoassumethatthetask'sservicedemandisknowninadvance.Ourmotivationforconsideringthismodelisthatitisanabstractionof Inthispaperweconcentrateontheparticularmodelofadistributedserver someexistingdistributedservers,describedinsection3. tributedserversystems:round-robin,inwhichtasksareassignedtohostsina cyclicalfashion;random,inwhicheachtaskisassignedtoeachhostwithequal probability;size-based,inwhichalltaskswithinacertainsizerangearesent toanindividualhost;anddynamic(alsoknownasleast-work-remaining)in Weconsiderfourtaskassignmentpoliciescommonlyproposedforsuchdis- whichanincomingtaskisassignedtothehostwiththeleastamountofoutstandingworklefttodo(basedonthesumofthesizesofthosetasksinthe queue). ofwhichtaskassignmentpolicyisbest.wearemotivatedinthisrespectby inmanymeasurementsofcomputerworkloads.inparticular,measurementsof theincreasingevidenceforhighvariabilityintasksizedistributions,witnessed manycomputerworkloadshavebeenshowntotaheavy-taileddistributions Ourgoalistostudytheinuenceoftasksizevariabilityonthedecision analysisoranalyticapproximations.weshowthatthevariabilityofthetasksize withveryhighvariance,asdescribedinsection2.2. distributionmakesacrucialdierenceinchoosingataskassignmentpolicy,and thatworkswellunderconditionsofhightasksizevariance. weusetheresultingobservationstoargueforaspecictaskassignmentpolicy Incomparingtaskassignmentpolicies,wemakeuseofsimulationsandalso 2BackgroundandPreviousWork ied,butmanybasicquestionsremainopen.inthecasewheretasksizesare 2.1FundamentalResultsinTaskAssignment Theproblemoftaskassignmentinamodellikeourshasbeenextensivelystudtion,theoptimalityofShortest-Linetaskassignmentpolicy(sendthetaskto unknown,thefollowingresultsexist:underanexponentialtasksizedistribu- thehostwiththeshortestqueue)wasprovenbywinston[14]andextended byweber[12]toincludetasksizedistributionswithnondecreasingfailurerate. TheactualperformanceoftheShortest-Linepolicyisnotknownexactly,butis approximatedbynelsonandphillips[9].infactasthevariabilityofthetask alentoptimalityandperformanceresultshavenotbeendevelopedforthetask sizedistributiongrows,theshortest-linepolicyisnolongeroptimal,whitt[13]. assignmentproblem,tothebestofourknowledge.forthescenarioinwhichthe agesofthetaskscurrentlyservingareknown,weber[12]hasshownthatthe Shortest-Expected-Delayruleisoptimalfortasksizedistributionswithincreasingfailurerate,andWhitt[13]hasshownthatthereexisttasksizedistributions Inthecasewheretheindividualtasksizesareknown,asinourmodel,equiv- forwhichtheshortest-expected-delayruleisnotoptimal.

3 1 Distribution of process lifetimes (log plot) (fraction of processes with duration > T) 1/2 1/4 1/8 Fig.1.MeasureddistributionofUNIXprocessCPUlifetimes,from[5].Dataindicates 1/16 fractionofjobswhosecpusevicedemandsexceedtseconds,asafunctionoft. 1/32 1/64 2.2Measurementsoftasksizedistributionsincomputer applications Duration (T secs.) Manyapplicationenvironmentsshowamixtureoftasksizesspanningmanyordersofmagnitude.Insuchenvironmentstherearetypicallymanysmalltasks, andfewerlargetasks.muchpreviousworkhasusedtheexponentialdistribution poormodelandthataheavy-taileddistributionismoreaccurate.ingenerala heavy-taileddistributionisoneforwhichprfx>xgx?;where0<<2. surementsindicatethatformanyapplicationstheexponentialdistributionisa tocapturethisvariability,asdescribedinsection2.1.however,recentmea- 1.Decreasingfailurerate:Inparticular,thelongerataskhasrun,thelonger Tasksizesfollowingaheavy-taileddistributionshowthefollowingproperties: 3.Thepropertythataverysmallfraction(<1%)oftheverylargesttasks 2.Innitevariance(andif1,innitemean). makeupalargefraction(half)oftheload.wewillrefertothisimportant itisexpectedtocontinuerunning. Thelowertheparameter,themorevariablethedistribution,andthemore tasksthatcomprisehalftheload. pronouncedistheheavy-tailedproperty,i.e.thesmallerthefactionoflarge propertythroughoutthepaperastheheavy-tailedproperty. measureddistributionofcpurequirementsofoveramillionunixprocesses, takenfrompaper[5].thisdistributioncloselytsthecurve Asaconcreteexample,Figure1depictsgraphicallyonalog-logplotthe ronments,includinginstructional,reasearch,andadministrativeenvironments. In[5]itisshownthatthisdistributionispresentinavarietyofcomputingenvi- PrfProcessLifetime>Tg=1=T:

4 Infact,heavy-taileddistributionsappeartotmanyrecentmeasurementsof computingsystems.theseinclude,forexample: {UnixprocessCPUrequirements,measuredatUCBerkeley:1[5]. {UnixprocessCPUrequirementsmeasuredatBellcore:11:25[8]. {SizesoflesstoredinUnixlesystems:[7]. {SizesoflestransferredthroughtheWeb:1:11:3[1,3]. {I/Otimes:[11]. servicerequirements. typicallytendstobecloseto1,whichrepresentsveryhighvariabilityintask Inmostofthesecaseswhereestimatesofweremade,12.Infact, {SizesofFTPtransfersintheInternet::91:1[10]. 3ModelandProblemFormulation Weareconcernedwiththefollowingmodelofadistributedserver.Theserver iscomposedofhhosts,eachwithequalprocessingpower.tasksarrivetothe systemaccordingtoapoissonprocesswithrate.whenataskarrivestothe system,itisinspectedbyadispatcherfacilitywhichassignsittooneofthe tasks. hostsforservice.weassumethedispatcherfacilityknowsthesizeofthetask. ThetasksassignedtoeachhostareservedinFCFSorder,andtasksarenot preemptible.weassumethatprocessingpoweristheonlyresourceusedby ontheirjob'sprocessingdemand.ifthejobexceedsthatdemand,itiskilled. batchdistributedcomputingfacilityatmit'slaboratoryforcomputerscience. Thexolasfacilityhasadispatcherfrontendwhichassignseachjobtooneof Xolasconsistsof4identicalmultiprocessorhosts.Usersspecifyanupperbound Theabovemodelforadistributedserverwasinitiallyinspiredbythexolas thehostsforservice.theuserisgivenanupperboundonthetimetheirjob willhavetowaitinthequeue,basedonthesumofthesizesofthejobsinthat queue.thejobsqueuedateachhostareeachruntocompletioninfcfsorder. terizedbythreeparameters:,theexponentofthepowerlaw;k,thesmallest sult,wemodeltasksizesusingadistributionthatfollowsapowerlaw,buthas possibleobservation;andp,thelargestpossibleobservation.theprobability anupperbound.werefertothisdistributionasaboundedpareto.itischarac- Weassumethattasksizesshowsomemaximum(butlarge)value.Asare- massfunctionfortheboundedparetob(k;p;)isdenedas: f(x)= 1?(k=p)x??1kxp: k andvaryovertherange0to2inordertoobservetheeectofchanging ThroughoutthispaperwemodeltasksizesusingaB(k;p;)distribution, (1) variabilityofthedistribution.tofocusontheeectofchangingvariance,we keepthedistributionalmeanxed(at3000)andthemaximumvaluexed(at

5 10 14 Second Moment of Bounded Pareto Distribution 10 power law f(x) w/ exponent α Fig.2.ParametersoftheBoundedParetoDistribution(left);SecondMomentof B(k;1010;)asafunctionof,whenEfXg=3000(right). 9 Numberofhosts Systemload 8 Meanservicetime h=8. =: k p Taskarrivalprocess EfXg=3000timeunits Poissonwithrate=1=EfXgh=:0021 Maximumtaskservicetime parameter Minimumtaskservicetime p=1010timeunits chosensothatmeantaskservicetimestays 0<2 constantasvaries(0<k1500) tasks/unittime p=1010).inordertokeepthemeanconstant,weadjustkslightlyaschanges Table1.Parametersusedinevaluatingtaskassignmentpolicies thisdistributionwillstillshowveryhighvariabilityifkp.forexample, (0<k1500).TheaboveparametersaresummarizedinTable1. isnotaheavy-taileddistributioninthesensewehavedenedabove.however, Figure2(right)showsthesecondmomentEX2 functionofforp=1010,wherekischosentokeepefxgconstantat3000,(0< NotethattheBoundedParetodistributionhasallitsmomentsnite.Thus,it declines.furthermore,theboundedparetodistributionalsostillexhibitsthe k1500).thegureshowsthatthesecondmomentexplodesexponentiallyas heavy-tailedpropertyand(tosomeextent)thedecreasingfailurerateproperty ofthisdistributionasa oftheunboundedparetodistribution. thebesttaskassignmentpolicy.thefollowingfourarecommonchoices: Random:anincomingtaskissenttohostiwithprobability1=h.Thispolicy Giventheabovemodelofadistributedserversystem,weaskhowtoselect Round-Robin:tasksareassignedtohostsincyclicalfashionwiththeith taskbeingassignedtohostimodh.thispolicyalsoequalizestheexpected equalizestheexpectednumberoftasksateachhost.

6 Size-Based:Eachhostservestaskswhoseservicedemandfallsinadesignated largetasks. range.thispolicyattemptstokeepsmalltasksfromgetting\stuck"behind timesthanrandom. numberoftasksateachhost,andtypicallyhaslessvariabilityininterarrival Dynamic:Eachincomingtaskisassignedtothehostwiththesmallestamount ofoutstandingwork,whichisthesumofthesizesofthetasksinthehost's queueplustheworkremainingonthattaskcurrentlybeingserved.this tasksizes.theeectivenessofthesetaskassignmentschemeswillbemeasured Inthispaperwecomparethesepoliciesasafunctionofthevariabilityof systemstandpointattemptstoachieveinstantaneousloadbalance. policyisoptimalfromthestandpointofanindividualtask,andfroma 3.1ANewSize-BasedTaskAssignmentPolicy:SITA-E intermsofmeanwaitingtimeandmeanslowdown,whereatask'sslowdownis Beforedelvingintosimulationandanalyticresults,weneedtospecifyafew itswaitingtimedividedbyitsservicedemand.allmeansareper-taskaverages. moreparametersofthesize-basedpolicy. taskissenttotheappropriatehostbasedonitssize.inpracticethesizeranges fortasksofsizebetween15minutesand3hours,a6-hourqueue,a12-hour associatedwiththehostsareoftenchosensomewhatarbitrarily.theremightbe a15-minutequeuefortasksofsizebetween0and15minutes,a3-hourqueue Insize-basedtaskassignment,asizerangeisassociatedwitheachhostanda queueandan18-hourqueue,forexample.(thisexampleisusedinpracticeat thecornelltheorycenteribmsp2jobscheduler[6].) ment,whichwerefertoassita-e SizeIntervalTaskAssignmentwithEqual Load.Theideaissimple:denethesizerangeassociatedwitheachhostsuch thatthetotalwork(load)directedtoeachhostisthesame.themotivationfor doingthisisthatbalancingtheloadminimizesmeanwaitingtime. Inthispaperwechooseamoreformalalgorithmforsize-basedtaskassign- iseasytoobtainbymaintainingahistogram(inthedispatcherunit)ofalltask theexpectedworkdirectedtoeachhostisthesame.thetasksizedistribution sizeswitnessedoveraperiodoftime. thetasksizedistributiontodenethecutopoints(deningtheranges)sothat Themechanismforachievingbalancedexpectedloadatthehostsistouse ofhosts.thenwedetermine\cutopoints"xi,i=0:::hwherek=x0<x1< p(possiblyequaltoinnity)denotethelargesttasksize,andhbethenumber functionoftasksizeswithnitemeanm.letkdenotethesmallesttasksize, x2<:::<xh?1<xh=p,suchthat Moreprecisely,letF(x)=PrfXxgdenotethecumulativedistribution andassigntotheithhostalltasksranginginsizefromxi?1toxi. Zx1 x0=kxdf(x)=zx2 x1xdf(x)==zxh=p xh?1xdf(x)=mh=rpkxdf(x) h

7 10 8 Simulated mean waiting time 10 5 Simulated mean slowdown 10 7 Random. Round Robin 10 4 Random... Dynamic. Round Robin 10 6 SITA E Dynamic SITA E mean.intheremainderofthepaperwewillalwaysassumethetasksizedistributionistheboundedparetodistribution,b(k;p;). SITA-Easdenedcanbeappliedtoanytasksizedistributionwithnite policiesviasimulation.simulationparametersareasshownintable1. 4SimulationResults InthissectionwecomparetheRandom,Round-Robin,SITA-E,andDynamic outsetwellbelowthetruemean;thetruemeanisn'tachieveduntilenoughlarge below[2].thisoccursbecausetherunningaverageoftasksizesistypicallyatthe tasksarrive.theconsequenceforasystemlikeourownisthatsimulationoutputs dicultbecausethesystemapproachessteadystateveryslowlyandusuallyfrom Simulatingaserversystemwithheavy-tailed,highlyvariableservicetimesis appearmoreoptimisticthantheywouldinsteady-state.tomakeoursimulation only.eachdatapointshowninourplotsistheaverageof400independentruns, measurementslesssensitivetothestartuptransient,werunoursimulationfor 4105arrivalsandthencapturedatafromthenextsinglearrivaltothesystem eachofwhichstartedfromanemptysystem. ability).asdescribedinsection2.2,valuesintherange1.0to1.3tendtobe commoninempiricalmeasurementsofcomputingsystems. Weconsidervaluesintherange1.1(highvariability)to1.9(lowervari Fig.3.MeanWaitingTime(a)andMeanSlowdown(b)underSimulationofFour 10 1 TaskAssignmentPoliciesasaFunctionof. (b) Figure3showstheperformanceofthesystemforallfourpolicies,asafunctionof(notethelogarithmicscaleontheyaxis).Figure3(a)showsmean waitingtimeand3(b)showsmeanslowdown.belowwesimplysummarizethese results;inthenextsection,wewilluseanalysistoexplaintheseresults.

8 androundrobinpoliciesissimilar,andthatbothcasesperformmuchmore poorlythantheothertwo(sita-eanddynamic).asdeclines,bothofthe performancemetricsundertherandomandround-robinpoliciesexplodeapproximatelyexponentially.thisgivesanindicationofthesevereimpactsthat Firstofall,observethattheperformanceofthesystemundertheRandom heavy-tailedworkloadscanhaveinsystemswithnaivetaskassignmentpolicies. namicisontheorderof100timesbetterforbothmetricswhencomparedto RandomandRoundRobin.Forlarge,thismeansthatDynamicperformsquite well withmeanslowdownlessthan1.howeverasthevariabilityintasksize increases(as!1),dynamicisunabletomaintaingoodperformance.ittoo TheDynamicpolicyshowsthebenetsofinstantaneousloadbalancing.Dy- suersfromroughlyexponentialexplosioninperformancemetricsasdeclines. undersita-eisrelativelyunchanged,withmeanslowdownalwaysbetween2 formancemetricsforrandom,roundrobin,anddynamicallexplode,sita-e's three.overtheentirerangeofvaluesstudied,theperformanceofthesystem and3.thisisthemoststrikingaspectofourdata:inarangeofinwhichper- Incontrast,thebehaviorofSITA-Eisquitedierentfromthatoftheother performanceremainsremarkablyinsensitivetoincreaseintasksizevariability. canbeontheorderof100timesbetterthanthatofdynamic. morecharacteristicofempiricalmeasurements(1:1),sita-e'sperformance mentexhibitsbetterperformance;butwhentasksizesshowthevariabilitythatis Asaresultwendthatwhentasksizeislessvariable,Dynamictaskassign- SITA-EbecomespreferabletoDynamicoveralargerrangeof. acrosstherangeoffrom1.1to1.9isdiculttounderstandusingthetoolsof In[4]wesimulatearangeofloads()andshowthatasloadincreases, simulationalone.forthatreasonthenextsectiondevelopsanalysisofsita-e andtheotherpolicies,andusesthatanalysistoexplainsita-e'sperformance. TheremarkableconsistencyofsystemperformanceundertheSITA-Epolicy Tounderstandthedierencesbetweentheperformanceofthefourtaskassignmentpolicies,weprovideafullanalysisoftheRound-Robin,Random,and 5AnalysisofTaskAssignmentPolicies formulabelowwhichanalyzesthem/g/1fcfsqueue: SITA-Epolicies,andanapproximationoftheDynamicpolicy. IntheanalysisbelowwewillrepeatedlymakeuseofthePollaczek-Kinchin wheredenotestherateofthearrivalprocess,xdenotestheservicetime EfWaitingTimeg=EX2 EfSlowdowng=EfW=Xg=EfWgEX?1 =2(1?) [Pollaczek-Kinchinformula] distribution,anddenotestheutilization(=efxg).theslowdownformulas followfromthefactthatwandxareindependentforafcfsqueue.

9 tailed,thesecondmomentoftheservicetimeexplodes,asshowninfigure2. RandomTaskAssignment.TheRandompolicysimplyperformsBernoullisplittingontheinputstream,withtheresultthateachhostbecomesanindependenerallypoorbecausethesecondmomentoftheB(k;p;)ishigh. metricsareproportionaltothesecondmomentofb(k;p;).performanceisgen- is,i=.sothepollaczek-kinchinformulaappliesdirectly,andallperformance M=B(k;p;)=1queue.Theloadattheithhost,isequaltothesystemload,that thesecondmomentoftheservicetime.recallthatiftheworkloadisheavy- ObservethateverymetricforthesimpleFCFSqueueisdependentonEX2, RoundRobin.TheRoundRobinpolicysplitstheincomingstreamsoeachhost formanceclosetotherandomcasesinceitstillseeshighvariabilityinservice seesaneh=b(k;p;)=1queue,withutilizationi=.thissystemhasper- SITA-E.TheSITA-EpolicyalsoperformsBernoullisplittingonthearrival times,whichdominatesperformance. stream(whichfollowsfromourassumptionthattasksizesareindependent).by thedenitionofsita-e,i=.howeverthetasksizesateachqueuearedeterminedbytheparticularvaluesoftheintervalcutos,fxig;i=0;:::;h.infact, hostiseesam=b(xi?1;xi;)=1queue.thereasonforthisisthatpartitioning oftheresultingregionstounitprobabilityyieldsanewsetofboundedpareto theboundedparetodistributionintocontiguousregionsandrenormalizingeach distributions.in[4]weshowhowtocalculatethesetofxisfortheb(k;p;) distribution,andwepresenttheresultingformulasthatprovidefullanalysisof thesystemunderthesita-epolicyforalltheperformancemetrics. knownapproximationsfortheperformancemetricsofthem/g/hqueue[15]: performedthesimulationstudy.however,in[4]weprovethatadistributed assignmentisactuallyequivalenttoanm/g/hqueue.fortunately,thereexist Dynamic.TheDynamicpolicyisnotanalyticallytractable,whichiswhywe systemofthetypeinthispaperwithhhostswhichperformsdynamictask wherexdenotestheservicetimedistributionandqdenotesthenumberin queue.what'simportanttoobservehereisthatthemeanqueuelength,and thereforethemeanwaitingtimeandmeanslowdown,areallproportionaltothe EQM=G=h =EQM=M=h EX2 =EfXg2; andround-robintaskassignmentpolicies. secondmomentoftheservicetimedistribution,aswasthecasefortherandom icyoverthewholerangeof.figure5againshowstheseanalytically-derived assignmentpoliciesoverarangeofvalues.figure4showstheanalyticallyderivedmeanwaitingtimeandmeanslowdownofthesystemundereachpol- metrics,butonlyovertherangeof12,whichistherangeofcorrespondingtomostempiricalmeasurementsofprocesslifetimesandlesizes(see Section2.2).(Notethat,becauseofslowsimulationconvergenceasdescribedat thebeginningofsection4,simulationvaluesaregenerallylowerthananalytic Usingtheaboveanalysiswecancomputetheperformanceoftheabovetask predictions;howeverallsimulationtrendsagreewithanalysis).

10 boththeseguresgrowsworseasdecreases,wheretheperformancecurves followthesameshapeasthesecondmomentoftheboundedparetodistribution,showninfigure2.thisisexpectedsincetheperformanceofrandomand Dynamicisdirectlyproportionaltothesecondmomentoftheservicetimedistribution.Bycontast,lookingatFigure5weseethatintherange1<<2, themeanwaitingtimeandespeciallymeanslowdownunderthesita-epolicy isremarkablyconstant,withmeanslowdownsaround3,whereasrandomand Dynamicexplodeinthisrange.TheinsensitivityofSITA-E'sperformanceto FirstobservethattheperformanceoftheRandomandDynamicpoliciesin inthisrangeisthemoststrikingpropertyofoursimulationsandanalysis. Analytically derived mean waiting time Random 10 9 SITA E... Dynamic Approximation 10 8 Analytically derived mean slowdown Random SITA E Dynamic Approximation <2. Fig.4.Analysisofmeanwaitingtimeandmeanslowdownoverwholerangeof, Analytically derived mean waiting time: Upclose Random SITA E... Dynamic Approximation Analytically derived mean slowdown: Upclose 10 7 Random 10 6 SITA E... Dynamic Approximation rangeof,12. Fig.5.Analysisofmeanwaitingtimeandmeanslowdownoverempiricallyrelevant

11 adynamicpolicyexplodes?acarefulanalysisoftheperformanceofsita-eat eachqueueofthesystem(see[4])leadsustothefollowinganswers: 1.Bylimitingtherangeoftasksizesateachhost,SITA-Egreatlyreduces WhydoesSITA-Eperformsowellinaregionoftasksizevariabilitywherein 2.Whenloadisbalanced,themajorityoftasksareassignedtothelow-numbered thevarianceofthetasksizedistributionwitnessedbythelowered-numbered hosts,therebyimprovingperformanceatthesehosts.infacttheperformance atmosthostsissuperiortothatofanm/m/1queuewithutilization. 3.Furthermore,meanslowdownisimprovedbecausesmalltasksobserveproportionatelylowerwaitingtimes. hosts,whicharethehostswiththebestperformance.thisisintensiedby theheavy-tailedpropertywhichimpliesthatveryfewtasksareassignedto highnumberedhosts. systemperformanceeventuallydeterioratesbadly.thereasonisthatasoverallvariabilityintasksizesincreases,eventuallyevenhost1willwitnesshigh Forthecaseof1,showninFigure4,evenundertheSITA-Epolicy, overwhichsita-eshowsgoodperformance.forexample,whenthenumberof hostsis32,sita-e'sperformancedoesnotdeteriorateuntil:8. variability.furtheranalysis[4]indicatesthataddinghostscanextendtherange 6Conclusion Dynamic(sendingthetasktothehostwiththeleastremainingwork). inuenceswhichtaskassignmentpolicyisbestinadistributedsystem.we Inthispaperwehavestudiedhowthevariabilityofthetasksizedistribution onthevariabilityoftasksizedistribution.whenthetasksizesarenothighly considerfourpolicies:random,round-robin,sita-e(asize-basedpolicy),and variable,thedynamicpolicyispreferable.however,whentasksizesshowthe SITA-Eisbest. degreeofvariabilitymorecharacteristicofempiricalmeasurements(1), Wendthatthebestchoiceoftaskassignmentpolicydependscritically teristicofempiricalmeasurements,sita-eoutperformsdynamicbycloseto2 severalordersofmagnitude.andintherangeoftasksizevariabilitycharac- large:randomandround-robinareinferiortobothsita-eanddynamicby Themagnitudeofthedierenceinperformanceofthesepoliciescanbequite fourpoliciesgleanedfromouranalysis: theirperformanceisdirectlyproportionaltothesecondmomentofthetask OuranalysisoftheRandom,Round-RobinandDynamicpoliciesshowsthat Moreimportantthantheaboveresults,though,istheinsightsaboutthese sizedistribution,whichexplainswhytheirperformancedeterioratesasthetask sizevariabilityincreases.thus,eventhedynamicpolicy,whichcomesclosesto achievinginstantaneousloadbalanceanddirectseachtasktothehostwhere

12 itwaitstheleast,isnotcapableofcompensatingfortheeectofincreasing varianceinthetasksizedistribution. SITA-Epolicywhichisasimpleformalizationofsize-basedpolicies,denedto afullanalysisofthesita-epolicy,leadingtoa3-foldcharacterizationofits equalizetheexpectedloadateachhost.thisformalizationallowsustoobtain power:(i)bylimitingtherangeoftasksizesateachhost,sita-egreatlyreduces Tounderstandwhysize-basedpoliciesaresopowerful,weintroducethe aresenttothesubsetthehostshavingthebestperformance.(iii)meanslowdown thevariabilityofthetasksizedistributionwitnessedbyeachhost{thereby isimprovedbecausesmalltasksobserveproportionatelylowerwaitingtimes. These3propertiesallowSITA-Etoperformverywellinaregionoftasksize improvingtheperformanceatthehost.(ii)whenloadisbalanced,mosttasks variabilityinwhichthedynamicpolicybreaksdown. References 1.M.E.CrovellaandA.Bestavros.Self-similarityinWorldWideWebtrac:Evidenceandpossiblecauses.IEEE/ACMTransactionsonNetworking,5(6):835{846, 3.M.E.Crovella,M.S.Taqqu,andA.Bestavros.Heavy-tailedprobabilitydistributionsintheworldwideweb.InAPracticalGuideToHeavyTails,pages1{23. December M.E.CrovellaandL.Lipsky.Long-lastingtransientconditionsinsimulationswith heavy-tailedworkloads.in1997wintersimulationconference, M.Harchol-Balter,M.E.Crovella,andC.D.Murta.Onchoosingataskassignmentpolicyforadistributedserversystem.TechnicalReportMIT-LCS-TR-757, Chapman&Hall,NewYork, S.Hotovy,D.Schneider,andT.O'Donnell.Analysisoftheearlyworkloadonthe 5.M.Harchol-BalterandA.Downey.Exploitingprocesslifetimedistributionsfor CornellTheoryCenterIBMSP2.TechnicalReport96TR234,CTC,Jan dynamicloadbalancing.acmtransactionsoncomputersystems,15(3),1997. MITLaboratoryforComputerScience, G.Irlam.Unixlesizesurvey. 8.W.E.LelandandT.J.Ott.Load-balancingheuristicsandprocessbehavior.In 10.V.PaxsonandS.Floyd.Wide-areatrac:ThefailureofPoissonmodeling. 9.R.D.NelsonandT.K.Philips.Anapproximationforthemeanresponsetimefor ProceedingsofPerformanceandACMSigmetrics,pages54{69, D.L.PetersonandD.B.Adams.FractalpatternsinDASDI/Otrac.InCMG IEEE/ACMTransactionsonNetworking,pages226{244,June1995. Evaluation,17:123{139,1998. shortestqueueroutingwithgeneralinterarrivalandservicetimes.performance 12.R.W.Weber.Ontheoptimalassignmentofcustomerstoparallelservers.Journal 13.WardWhitt.Decidingwhichqueuetojoin:Somecounterexamples.Operations Proceedings,December1996. ofappliedprobability,15:406{413, R.W.Wol.StochasticModelingandtheTheoryofQueues.PrenticeHall, W.Winston.Optimalityoftheshortestlinediscipline.JournalofAppliedProbability,14:181{189,1977. Research,34(1):226{244,January1986.