RevisedJanuary26,

Transcription

1 EarliestEligibleVirtualDeadlineFirst:AFlexible andaccuratemechanismforproportionalshare DepartmentofComputerScience,OldDominionUniversity ResourceAllocation IonStoica,HusseinAbdel-Wahab Norfolk,Virginia, TR tobegrantedinarbitrarilysmallintervalsoftime.mainly,weshowthatinsteadyconditionsour denethenotionoffairnessinthecontextofanidealizedsysteminwhichtheresourceisassumed Weassumethattheresourceisallocatedintimequantaofsizeq.Toeachclient,weassociatea weightwhichdeterminestherelativesharefromtheresourcethattheclientshouldreceive.we algorithmguaranteesthatthedierencebetweentheservicetimethataclientshouldreceiveinthe Weproposeandanalyzeanewproportionalshareallocationalgorithmfortimesharedresources. Abstract ofatimequantum.thealgorithmprovidessupportfordynamicoperations,suchasaclientjoining orleavingthecompetition(fortheresource),andchangingaclient'sweight.byusinganecient idealizedsystemandtheservicetimeitactuallyreceivesintherealsystemisboundedbythesizeq representsthenumberofclientscompetingfortheresource. augmentedbinarysearchtreedatastructureweimplementtheseoperationsino(logn),wheren RevisedJanuary26,

2 1Introduction Oneofthemostchallengingproblemsinmodernoperatingsystemsistodesignexibleandaccurate inapredictableandresponsiveway.inaddition,ageneral-purposeoperatingsystemshouldseamlessly algorithmstoallocateresourcesamongcompetingclients.thisissuehasbecomemoreimportantwith constraints.inordertomeettheseconstraintstheunderlyingoperatingsystemshouldallocateresources integratethesenewtypesofapplicationswithconventionalinteractiveandbatchapplications. theemergenceofnewtypesofreal-timeapplicationssuchasmultimediawhichhavewelldenedtime thesecondclientshouldreceive75%oftheresource.real-timeschedulersarebasedonanevent-driven theshareoftheresourcethattheclientshouldreceive.theschedulertriestoallocatetheresource ulers[20,21,22].inproportionalsharealgorithmseachclienthasassociatedaweightwhichdetermines theseschedulersfallintwocategories:proportionalshare[2,26,28,29,30],andreal-time1basedsched- and3,respectivelythatcompeteforthesameresource.thentherstclientshouldreceive25%,while amongcompetingclientsinproportiontotheirshare.forexample,considertwoclientswithweights1 Manyschedulershavetried,andinparthavesucceeded,toaddresstheserequirements.Generally, policies,theseschedulersguaranteethatalleventsareprocessedbeforetheirdeadlines.whileingeneral loadsituations,real-timebasedschedulerstendtooerbetterguaranteesforapplicationswithtimeliness modelinwhichaclientischaracterizedbyasetofeventsarrivinginacertainpattern(usuallyperiodic), theproportionalshareschedulerstendtobemoreexibleandtoensureagracefuldegradationinover- andbyapredictedservicetimeandadeadlineassociatedtoeachevent.byimposingstrictadmission extendedtosupportbatchapplications.themainreasonisthatwhilemultimediaandinteractive constraints,suchasmultimedia. applicationsttheevent-drivenmodelimplicitlyassumedbytheseschedulers,batchapplicationsdonot. isthetimeatwhichthenextframeshouldarrive.similarly,aninteractiveapplication(suchasaneditor) Forexample,avideoapplicationcanbemodeledasaperiodicprocess,whereaneventrepresentsthe arrivalofaframe,itsservicetimeisthetimerequiredtoprocessanddisplaytheframe2,anditsdeadline canbemodeledasanaperiodicprocess,whereaneventisgeneratedasaresultoftheuserpressingakey, Althoughreal-timebasedschedulersprovidebettersupportformultimedia,theycannotbeeasily inmanysituationsitisdiculttoaccuratelydeterminethisservicetime.themainreasonisthat,evenfor thesameapplication,theservicetimemayhavelargevariations,andthesevariationsarehardtopredict. Forexample,theservicetimerequiredbyacompilermayvarysignicantlyevenforthesameprogram, isgivenbythelargestacceptabledelaybetweenthemomentwhenthekeyispressedandthemoment whenthecharacterisdisplayed.incontrasttobothmultimediaandinteractiveapplications,batch theservicetimeisthetimerequiredtoprocessanddisplaythecorrespondingcharacter,andthedeadline thatusereal-timeschemesforschedulingcontinuousmediaalsoemployconventionalalgorithms(e.g., dependingonhowmanymodulesarerecompiled.forthesereasonsmanygeneralpurposeschedulers applicationsareusuallycharacterizedbyonlyoneparameter,i.e.,therequestedservicetime.moreover, restrictive.thus,ausercanbeinthesituationinwhichhecannotrunanewapplication,althoughhe mightbewillingtoacceptadegradationinperformanceofotherapplicationsinordertoaccommodate in[18]:ratemonotonicandearlydeadlinerst. inordertosatisfystrongtimeconstraints,theyimposestrictadmissionpolicieswhichmakethemfairly round-robin)forschedulingbatchapplications.anotherdrawbackofreal-timebasedschedulersisthat, 1Usually,theseschedulersarebasedontwowell-knownreal-timealgorithmsproposedandanalyzedbyLiuandLayland 2Dependingonthedesiredqualitythiscanbeeithertheworstcase,orastatisticalaverageservicetime. 2

3 guaranteesfortheservicetimereceivedbyaclient.inthisway,ouralgorithmprovidesauniedapproach forschedulingcontinuousmedia,interactiveandbatchapplications.inpart,thealgorithmbuildson which,whileretainingalltheadvantagesoftheproportionalshareschedulers,providesstrongtimeliness forexample,whenanapplicationterminatesitisdiculttodistributerapidlyitsshareamongtheother applicationsthatarestillcompetingfortheresource. thenewone.finally,inahighlydynamicenvironmenttheseschedulersdonotprovideenoughexibility; tracktheworkprogressinanidealuid-owbasedsystem.toeachclientweassociateaweightwhich ideasfoundinpreviousnetworkfair-queueingalgorithms[23,31],andgeneral-purposeproportionalshare allocationalgorithms[29,30].asin[23,31],andsimilarto[29,30]weusethenotionofvirtualtimeto Inthispaperweproposeanewscheduler(calledEarliestEligibleVirtualDeadlineFirst{EEVDF) andavirtualdeadlinewhicharethecorrespondingstartingandnishingtimesofservicingtherequest determinestherelativeshareoftheresourcethattheclientshouldreceive.theclientrequirementsare uniformlytranslatedinasequenceofrequestsfortheresource.theserequestsareeitherissuedexplicitly bytheclients,orbythescheduleritselfonbehalfoftheclients.inthiswaybatchactivitiesaretreated thecurrentvirtualtime.thealgorithmsimplyallocatesanewtimequantumtotheclientthathasthe eligiblerequestwiththeearliestvirtualdeadline.wenotethatwhiletheconceptofvirtualdeadlineis alsoemployedbyotherproportional-sharealgorithms[23,31,29,30],theconceptofeligibletimeisa intheuid-owmodel.arequestissaidtobeeligibleifitsvirtualeligibletimeislessthanorequalto uniformlywithmultimediaandinteractiveactivities.basedontheclientshareandontheservicetime uniquefeatureofouralgorithm(which,aswewillshow,playsadecisiveroleinimprovingtheallocation thattheclienthasalreadyreceived,theschedulerassociatestoeachclient'srequestavirtualeligibletime accuracy). allocations,higherlevelresourceabstractionsareneededtospecifytheapplications'requirements.we noteherethatmanyoftheexistingabstractions,suchasticketsandcurrencies(developedbywaldspurger andweihl[28,30]),andprocessorcapacityreserves(proposedbymercer,savageandtokuda[20])are bedirectlytranslatedintotheweightassociatedtothatclientintheeevdfalgorithm.byeciently implementingdynamicoperations,theeevdfprovidesdirectsupportforhigherlevelmechanismssuch easilysupportedbyeevdf.forexample,inlotteryschedulingthenumberofticketsheldbyaclientcould WhileEEVDFimplementsaexibleandaccuratelow-levelmechanismforproportionalshareresource asticketstransferandinationemployedbylotteryscheduling[28,30].throughoutthispaperwewill notdiscussfurtherotherhigherlevelresourceabstractions;instead,wewillimplicitlyassumethatoneof concludesthepaper. thebasiceevdfalgorithm.section4discussestheconceptoffairnessindynamicsystems,whilesection 5presentsthreestrategiesforimplementingtheEEVDFalgorithm.InSection6wegivethefairness theexistingabstractionsisimplementedontopofeevdf.therefore,inthispaperwewillnotaddress problemsthatareusuallyhandledbythishigherlevel(e.g.,priorityinversion). analysisofthealgorithm.finally,insection7wegiveanoverviewoftherelatedwork,andsection8 Thispaperisorganizedasfollows.Thenextsectiondiscussesourassumptions,andSection3presents 2Assumptions Weconsiderasetofclientsthatcompeteforatimesharedresource(e.g.,processor,communication bandwidth).weassumethattheresourceisallocatedintimequantaofsizeatmostq.atthebeginning 3

4 asprocessorandcommunicationbandwidth.forexample,inmanypreemptiveoperatingsystems(e.g., simple,thismodelcapturesthebasicmechanismstraditionallyusedforsharingcommonresources,such higherprioritybecomesactive,oritmayvoluntarilyreleasethecpuwhileitiswaitingforaneventto useiteitherfortheentiretimequantum,oritmayreleaseitbeforethetimequantumexpires.although thesamefashion:aprocessusesthecpuuntilitstimequantumexpiresoranotherprocesswitha UNIX,Windows-NT),theCPUschedulerallocatestheprocessingtimeamongcompetingprocessesin ofeachtimequantumaclientisselectedtousetheresource.oncetheclientacquirestheresource,itmay outputlink.thus,inthiscase,thesizeqofatimequantumrepresentsthetimerequiredtosendapacket multiplexesasetofincomingsessionsonapacket-by-packetbasis.sinceusuallythetransmissionofa ofmaximumlength. packetcannotbepreempted,wetakeatimequantumtobethetimerequiredtosendapacketonthe itshouldreceive.theshareiscomputedastheratiobetweentheclient'sweightandthetotalsumover occur(e.g.,ani/ooperationtocomplete).asanotherexample,consideracommunicationswitchthat ofallclientsactiveattimet.thentheshareofclientiattimet,denotedfi(t),isdenedas: theweightsofallactiveclients.aclientissaidtobeactivewhileitiscompetingfortheresource,and passiveotherwise.moreformally,letwidenotetheweightassociatedtoclienti,andleta(t)betheset Further,weassociatetoeachclientaweightthatdeterminestherelativeshareoftheresourcethat usetheresourceforfi(t)ttimeunits.ingeneral,whentheclientsharevariesovertime,theservice Ideally,iftheclientshareremainsconstantduringatimeinterval[t;t+t],thenclientiisentitledto timethatclientishouldreceiveinaperfectfairsystemwhilebeingactiveduringatimeinterval[t0;t1] is Si(t0;t1)=Zt1 fi(t)=wi Pj2A(t)wj: (1) timeunits.theaboveequationcorrespondstoanidealuid-owsysteminwhichtheresourcecanbe quantacannotbetakenarbitrarilysmall.oneofthereasonsistheoverheadintroducedbythescheduling algorithmandtheoverheadinswitchingfromoneclienttoanother:takingtimequantaofthesameorder grantedinarbitrarilysmallintervalsoftime.inourcase,thisisequivalenttothesituationinwhich thesizeofatimequantumapproacheszero(q!0).3unfortunately,inmanypracticalsituations,time t0fi()d (2) ofmagnitudeastheseoverheadscoulddrasticallyreducetheresourceutilization.forexample,itwould beunacceptableforacputospendmoretimeinschedulinganewprocess,andcontextswitching betweentheprocesses,thandoingusefulcomputation.anotherreasonisthatsomeoperationscannot communicationswitchbeginstosendapacketforonesession,itcannotserveanyothersessionuntilthe beinterrupted,i.e.,oncestartedtheymustcompleteinthesametimequanta.forexample,oncea entirepacketissent. inours),itisnotpossibleforaclienttoalwaysreceiveexactlytheservicetimeitisentitledto.the dierencebetweentheservicetimethataclientshouldreceiveatatimet,andtheservicetimeitactually receivesiscalledservicetimelag.moreprecisely,letti0beatimeatwhichclientibecomesactive,and letsi(ti0;t)betheservicetimetheclientreceivesintheinterval[ti0;t](here,weassumethatclientiis activeintheentireinterval[ti0;t]).thentheservicetimelagofclientiattimetis 3AsimilarmodelwasusedbyDemersetalinstudyingfair-queuingalgorithmsincommunicationnetworks[9]. Duetoquantization,inasysteminwhichtheresourceisallocatedindiscretetimequanta(asitis 4

5 useitasthemainparameterincharacterizingourproportionalresourceallocationalgorithm. 3TheEEVDFAlgorithm Sincetheservicetimelagdeterminesboththethroughputaccuracyandthesystempredictability,we lagi(t)=si(ti0;t)?si(ti0;t): (3) case,therequestduration(length)iseitherspeciedbytheclient,orotherwisetheschedulerassumes processorcase,thescheduleritselfcouldbetheonetoissuetherequestsonbehalfoftheclient.inthis Inordertoobtainaccesstotheresource,aclientmustissuearequestwhichspeciesthedurationofthe a\default"duration.thisallowsustotreatallcontinuousmedia,interactive,andbatchactivitiesina request,andtobepassiveotherwise.foruniformity,throughoutthispaperweassumethattheclientis servicetimeitneeds.onceaclient'srequestisfullled,itmayeitherissueanewrequest,orotherwisethe consistentway. clientbecomespassive.noticethatwecanalternatelydeneaclienttobeactivewhileithasapending thesoleinitiatoroftherequests.however,inpracticethisisnotnecessarilytrue.forexample,inthe fewerlongerrequests,ormanyshorterones.forexample,aclientmayaskfor1mincomputationtime, itstimeexpires.noticethataclientmayrequestthesameamountofservicetimebygeneratingeither eitherbyissuing60requestswithadurationof1seceach,orbyissuing600requestswithadurationof expires,theclientispreemptedandthenexttimequantumcanbeallocatedtoanotherclient).whena atimequantum,thentheservicetimemightnotbeallocatedcontinuously(i.e.,whenatimequantum clientrequeststheresourceforlessthanonetimequantum,theschedulersimplypreemptstheclientonce Forexibilityweallowrequeststohaveanyduration.Ifthedurationoftherequestislargerthan 100mseceach.AswewillshowinSection6,shorterrequestsguaranteebetterallocationaccuracy,while longerrequestsdecreasethesystemoverhead.inthiswayaclientcouldtradebetweentheallocation byaxedintervaloftimebetweentwoconsecutiveevents,calledperiodanddenotedbyt,andby themaximumservicetimerrequiredtoprocessanevent.wheneveraneventoccurs,thetasksimply accuracyandtheschedulingoverhead. knownproblemofschedulingperiodictasksinareal-timesystem[18].aperiodictaskischaracterized thatthecurrenteventshouldbeprocessedbeforethenexteventoccurs.noticethatthisrequirement guaranteesthatthetaskreceivesasmuchasrservicetimeunitsduringeveryperiodt.consequently, requestsrservicetimeunitsforprocessingthatevent.acentralrequirementinreal-timesystemsis Toclarifytheideaswebrieypointoutthesimilaritiesanddierencesbetweenourmodelandthewell- beexpressedast+rf.similarly,inourmodel,bygivingthetimetatwhicharequestismadeand isidenticaltotheexpressionoftherequest'sdeadlineinthecaseofschedulingperiodictasks.asamajor servicetimerandthetimetatwhichtheeventoccurs,thedeadlineofthecorrespondingrequestcan anidealsystembysolvingtheequationr=s(t;d).ifthesharefoftheclientdoesnotchangeinthe itsdurationr,weobtainthetimedbeforewhichtheclientshouldreceivetherequestedservicetimein interval[t;d),thenfromeq.(2)followsthats(t;d)=f(d?t),andfurtherweobtaind=t+rf,which inthiscasethetaskreceivesasharef=rtoftheresource.thus,bygivingthesharef,therequested assumedtobegeneratedasaresultofanexternalevent(e.g.,apacketarrival,atime-out),inourmodel dierencebetweenourmodelandareal-timesystem,wenotethatwhileinareal-timesystemarequestis 5

6 asaresultofaninternalevent(whentheclientgeneratesanewrequestafterthecurrentonehasbeen arequestiseithergeneratedasresultofanexternalevent(whenthecliententersthecompetition),or continuousmediaandinteractiveapplications,andforconventionalbatch-applications. fullled).inthiswayourmodelprovidesintegratedsupportforbotheventdrivenapplications,suchas Similarlyto[31]and[23]wedenethesystemvirtualtimeas intheinterval[t1;t2)as BycombiningEq.(1)and(2)wecanexpresstheservicetimethatanactiveclientishouldreceive Si(t1;t2)=wiZt2 V(t)=Zt 01 Pj2A()wjd: Pj2A()wjd: 1 (4) inthecorrespondinguid-owsystemeachactiveclientireceiveswireal-timeunitsduringonevirtual allactiveclientsinonevirtualtimeunit.thatis,thesizeofavirtualtimeunitismodiedsuchthat competitiondecreasesitaccelerates.intuitively,theowofthevirtualtimechangesto\accommodate" activeclients.noticethatwhenthecompetitionincreasesthevirtualtimeslowsdown,whilewhenthe Wenotethatthevirtualtimeincreasesatarateinverseproportionaltothesumoftheweightsofall (5) Thus,ineachvirtualtimeunitthetwoclientsshouldreceivew1=2,andw2=3timeunits.Next,from timeunit.forexample,considertwoclientswithweightsw1=2andw2=3.thentherateatwhich Eq.(4)and(5)itfollowsthatSi(t1;t2)=wi(V(t2)?V(t1)): thevirtualtimeincreasesis1 w1+w2=0:2,andthereforeavirtualtimeunitequalsvereal-timeunits. ofactiveclientsisconstantandthesumoftheirweightsisone(pi2awi=1),i.e.,theshareofa clientiisfi=wi.thentheservicetimethatclientishouldreceiveduringaninterval[t1;t2)issimply virtualtimesv(t1)andv(t2)wearriveateq.(6).thus,eq.(6)canbeviewedasadirectgeneralization Si(t1;t2)=wi(t2?t1).Next,noticethatbyreplacingtherealtimest1andt2withthecorresponding Tobetterinterprettheaboveequationitisusefultoconsideramuchsimplermodelinwhichthenumber (6) ofcomputingtheservicetimesi(t1;t2)inadynamicsystem. deadlined.arequestofanactiveclientbecomeseligibleattimeewhentheservicetimethattheclient shouldreceiveinthecorrespondinguid-owsystemequalstheservicetimethattheclienthasalready received(intherealsystem)beforeissuingthecurrentrequest.letti0bethetimeatwhichclienti servicetimethanitwassupposedtoreceive(i.e.,itslagisnegativeattimet),thentheclientshouldwait newrequestischosensuchthatsi(ti0;e)=si(ti0;t).noticethatifattimetclientihasreceivedmore becomesactive,andlettbethetimeatwhichitinitiatesanewrequest.thentheeligibletimeeofthe Thebasicideabehindouralgorithmissimple.Weassociatetoeachrequestaneligibletimeeanda untiltimeebeforethenewrequestbecomeseligible.inthiswayaclientthathasreceivedmoreservice timethanitsshareis\sloweddown",whilegivingtheotheractiveclientstheopportunityto\catchup". alwayspositive.byusingeq.(6)wecanexpressthevirtualeligibletimev(e)as Ontheotherhand,ifattimetclientihasreceivedlessservicetimethanitwassupposedtoreceive(i.e., timethattheclientshouldhavereceivedattimeeitfollowsthattheclient'slagattimee(lagi(e))is Sinceineithercasetheservicetimethattheclienthasreceivedattimeeisnogreaterthantheservice itslagispositive),thenwehavee<t,andthereforethenewrequestisimmediatelyeligibleattimet. 6

7 whererrepresentsthelengthofthenewrequest.further,byusingagaineq.(6),wederivethevirtual betweentheeligibletimeeandthedeadlinedequalstheservicetimeofthenewrequest,i.e.,si(e;d)=r, deadlinev(d)as Next,thedeadlineoftherequestischosensuchthattheservicetimethattheclientshouldreceive V(e)=V(ti0)+si(ti0;t) wi: (7) inwhicheislargerthanthecurrenttimet.thenecannotbecomputedexactlyfromeq.(5)and(7), V(d),theydonotnecessarilygiveusthevaluesoftherealtimeseandd!Toseewhy,considerthecase havinganemptyreservoirinwhichwecollectwaterfromaspring.althoughwecandeneamarkto sincewedonotknowhowtheslopeofvwillvaryinthefuture.intuitively,thesituationissimilarto NoticethatalthoughEq.(7)and(8)giveusthevirtualeligibletimeV(e)andthevirtualdeadline V(d)=V(e)+rwi: (8) springmayvaryinthefuture.thereforewewillformulateouralgorithmintermsofvirtualeligibletimes indicatewhenthereservoirisfull,wecannotsayexactlywhenthiswillhappensincetheow-rateofthe earliestvirtualdeadline. anddeadlinesandnotoftherealtimes.withthistheearliesteligiblevirtualdeadlinefirst(eevdf) algorithmcanbesimplystatedasfollows: EEVDFAlgorithm.Anewquantumisallocatedtotheclientthathastheeligiblerequestwiththe doesnotcompromisethefairness.intuitively,thisresultsfromthedenitionofthevirtualeligibleeligible exactlytheamountofservicetimeaclientwillactuallyuse.however,aswewillshowinsection6,this thisisarealisticassumption4,forprocessorschedulingitismuchharder(andoftenimpossible)topredict allocatingbandwidthincommunicationnetworks[9,12,23],whichassumethatthelengthofapacket requested.thisisanimportantfeaturethatdierentiatesitfromthefairqueueingalgorithmsusedfor (andthereforetheservicetime)isknownwhenthepacketarrives.althoughincommunicationnetworks TheEEVDFalgorithmdoesnotassumethataclientwillalwaysusealltheservicetimeithas eligibletimeandthedeadlinearenotgiven.letr(k)denotethelengthofthekthrequestmadebyclienti, vdtodenotethevirtualeligibletimeandvirtualdeadlinerespectively,wheneverthecorrespondingreal andletve(k)andvd(k)denotethevirtualeligibletimeandthevirtualdeadlineassociatedtothisrequest. fornon-uniformquanta.5 time:wheneveraclientuseslessservicetimethanithasrequested,thevirtualeligibletimeofthenext Iftheclientuseseachtimetheentireservicetimeithasrequested,thenbyusingEq.(7)and(8)we requestispushedbackwards(suchthatthelagtobezero).inthiswayeevdfprovidesdirectsupport eachrequest: obtainthefollowingreccurencewhichcomputesboththevirtualeligibletimeandthevirtualdeadlineof SinceEEVDFisformulatedintermsofvirtualtimes,intheremainingofthispaperweuseveand received. rstcasethefractionfromthetimequantum(thattheclientwillactuallyuse)isassumedtobeknowninadvance,inthe tothedesiredfractionofatimequantum.thedierencebetweenfractionalandnon-uniformquantaisthatwhileinthe non-uniformquantacasethisfractionisnotknown. 5NoticethatEEVDFalsoprovidessupportforfractionalquantabysimplytakingthetimeoftherequesttobeequal 4Usually,thesealgorithmstakethepacketarrivaltimetobethetimeatwhichthelastbitfromthepackethasbeen 7

8 Figure1:AnexampleofEEVDFschedulinginvolvingtwoclientswithequalweightsw1=w2=2.All 1 time 0:5).Thearrowsrepresentthetimeswhentherequestsareinitiated(thepairassociatedtoeacharrow therequestsgeneratedbyclient1havelength2,andalloftherequestsgeneratedbyclient2areoflength (0, 1) (1, 2) (2, 3) representsthevirtualeligibletimeandthevirtualdeadlineofthecorrespondingrequest). 1.Client1becomesactiveattime0(virtualtime0),whileclient2becomesactiveattime1(virtualtime 2 time (0.5, 1) (1, 1.5) (1.5, 2) (2, 2.5) virtual time vd(k)=ve(k)+r(k) wi; (10) time onlythecasewhentheclientusestheresourceforlesstimethanrequested.letu(k)denotetheservice ithasrequested.sinceaclientneverreceivesmoreservicetimethanrequested,weneedtoconsider timethatclientiactuallyreceivesduringkthrequest.thentheonlychangeineq.(9){(11),willbein computingtheeligibletimeofanewrequest.specically,eq.(11)isreplacedby Next,letusconsiderthemoregeneralcaseinwhichtheclientdoesnotusetheentireservicetime t0=0.thus,accordingtoeq.(9)and(10)thevirtualeligibletimefortherstrequestofclient1 timequantumisofunitsize(q=1)andthatclient1istherstonewhichenterscompetitionattime w1=w2=2thatissuerequestswithlengthsr1=2,andr2=1,respectively.weassumethatthe Toclarifytheideas,letustakeasimpleexample(seeFigure1).Considertwoclientswithweights ve(k+1)=ve(k)+u(k) isve=0,whileitsvirtualdeadlineisvd=1.beingthesingleclientthathasanoutstandingeligible wi: (12) functionbecomes1 virtualdeadline1(whichwaitsforanothertimequantumtofulllitsrequest),andonefromclient2 whichhasthesamevirtualdeadlinei.e.,1.inthissituationwearbitrarilybreakthetieinthefavor secondquantumisallocated.thenatt=1therearetwopendingrequests:onefromclient1withthe isv(1)=r101 request,client1receivestherstquantum.attimet=1,client2entersthecompetition.sincethe virtualtimeincreasesataconstantrateduringtheinterval[0;1)(i.e.,1 ofclient2,whichthereforereceivesthesecondquantumupontermination.sincethisquantumfullls w1d=0:5.afterthesecondcliententersthecompetitiontheslopeofthevirtualtime w1+w2=0:25.next,letusassumethatclient2makesitsrstrequestbeforethe w1=0:5),itsvalueattimet=1 thecurrentrequestofclient2,client2issuesanewonewiththevirtualeligibletime1andthevirtual 8 ve(k+1)=vd(k): ve(1)=v(ti0); (11) (9)

9 deadline1:5.thus,attimet=2thesingleeligiblerequestistheoneofclient1,whichthereforereceives thenextquantum.further,att=3thereareagaintwoeligiblerequests:theoneofclient2thathas justbecomeeligible,andthenewrequestissuedbyclient1.sincethedeadlineofthesecondclient's request(1.5)isearlierthantheoneoftherstclient(2),thefourthquantumisallocatedtotheclient2. tion,clientleavingthecompetition,andchangingtheclient'sweight.inanidealizeduid-owsystem, thatadynamicsystemprovidessupportforthefollowingthreeoperations:clientjoiningthecompeti- 4FairnessinDynamicSystems Inthissectionweaddresstheissueoffairnessindynamicsystems.Throughoutthispaper,weassume Further,Figure1showshowthenextthreequantaareallocated. considerthefollowingtwoquestions: supportingdynamicoperationsistrivialsinceatanymomentoftimethelagofanyactiveclientiszero. true.intheremainingofthissectionwediscusshowthisaectsthefairnessinadynamicsystem6.we Unfortunately,inasysteminwhichtheservicetimeisallocatedindiscretetimequanta,thisisnolonger competitionwithanegativelag,i.e.,afterithasreceivedmoreservicetimethanitwasentitledto.as 1.Whatistheimpactofaclientwithnon-zerolagleaving,joining,orchangingitsweightonthe 2.Whenaclientwithnon-zerolagleavesthecompetition,whatshouldbeitslagwhenitrejoinsthe Inansweringtherstquestion,westartwithasimpleexample.Letusassumethataclientleavesthe otherclients? isequaltotheservicetimethattheclientsshouldreceive.therefore,ifaclientleavesthecompetition wewillshowinsection6,duringanytimeinterval,thetotalservicetimeallocatedtoallactiveclients withanegativelag,theremainingclientsshouldhavereceivedlessservicetimethantheywereentitled competition? to.inshort,againforoneclienttranslatesintoalossfortheotheractiveclients.inthiscase,the WaldspurgerandWeihlintheirstrideschedulingalgorithm[29,30],thetwopoliciesarenotequivalent. simplyupdatingthevirtualtime.wenotethatalthoughthispolicyissimilartotheoneemployedby activeclients.inotherwords,eachactiveclientwillinheritagain/lossproportionaltoitsweight.besides itsintuitiveappeal,aswewillshow,thispolicyhasanotheradvantage:itcanbeeasilyimplementedby operationtakesplace,theeect(i.e.,theresultinggainorloss)isproportionallydistributedamongall activeclientsinordertoattainfairness?weanswerthisquestionbyassumingthatwheneveradynamic questionweneedtoansweristhefollowing:howshouldthelossbedistributedamongtheremaining othertwoclients. competition(seefigure2).further,wediscusstheimpactofclient3leavingthecompetitiononthe Thedierenceconsistsinthewayinwhichthevirtualtimeisupdatedwhenaclientleavesorjoinsthe Supposethatthreeclientswithzerolagsbecomeactiveattimet0,andattimetclient3leavesthe competition:whileinstrideschedulingonlytheslopeofthevirtualtime7isupdated,inouralgorithm thevalueofthevirtualtimeisupdatedaswell(seeeq.18,19,20inthissection). 7Insteadvirtualtime,strideschedulingusesanequivalentconcept,calledglobalpass. 6TheseissueswerealsoaddressedbyWaldspurgerandWeihlinthecontextoftheirstrideschedulingalgorithm[30]. Tobetterunderstandourreasonsinconsideringtheabovepolicy,letusconsiderthefollowingexample. 9

10 Figure2:Thethreeclientsbecomeactiveattimet0.Attimet,client3leavesthecompetition. 1 2 SinceEEVDFisawork-conserving8algorithm,thetotalservicetimereceivedbyallactiveclientsduring Next,weturnourattentiontotheclients1and2whichremainactiveafterclient3leavesthecompetition. ofthevirtualtimeduringthisintervalisconstantandequalto1 thelagofeachclientattimetis Sincethenumberofactiveclientsandtheirsharesdonotchangeduringtheinterval[t0;t),theslope lagi(t)=wit?t0 w1+w2+w3.thenfromeq.(3)and(6), 3 t t thisintervalcanbeexpressedast?t0?s3(t0;t).lett+bethetimeimmediatelyafterclient3leaves theinterval[t0;t)isequaltot?t0.fromhere,theservicetimeallocatedtothersttwoclientsduring thecompetition,wherebyneglectingtheleavingoperationoverhead,wehavet+!t. 0 w1+w2+w3?si(t0;t);i=1;2;3: (13) Wenotethatiflag3(t)6=0,thenthisresultisdierentfromtheservicetimeeachclientshouldhave receivedjustbeforethedepartureofclient3,i.e.,(t?t0)wi t?t0?s3(t0;t))proportionaltotheclients'weights,i.e., andintuitiveapproachwouldbetosimplydividetheentireservicetimereceivedbybothclients(i.e., Now,whatistheservicetimethatclients1and2shouldhavereceivedattimet+?Anatural client3leaves,theremainingtwoclientswillproportionallysupporttheeventuallossorgaininthe thevirtualtime.byreplacings3(t0;t)fromeq.(13)intoeq.(14),weobtain servicetime.next,weshowthatintheeevdfalgorithmthisisequivalenttoasimpletranslationof Si(t0;t+)=(t?t0?s3(t0;t))wi w1+w2;i=1;2; w1+w2+w3(i=1;2;3).thisisbecauseonce (14) Finally,fromEq.(6)and(15)itfollowsthat Si(t0;t+)=(t?t0)wi V(t+)=V(t)+lag3(t) =wi(v(t)?v(t0))+wilag3(t) w1+w2+w3+wilag3(t) w1+w2;i=1;2: w1+w2;i=1;2: (16) (15) Thus,tomaintainthefairnessamongtheremainingclients,thevirtualtimeshouldbeupdatedaccording tothelagoftheclientwhichleavesthecompetition(asshownbytheaboveequation).sincet+is client(seesection6fordetails). becomputedas ithasreceivedattimet(i.e.,si(t0;t)=si(t0;t+)).fromhere,thelagsofthersttwoclientsatt+can asymptoticallyclosetot,theservicetimereceivedbyanyclientattimet+isequaltotheservicetime 8Aschedulingalgorithmissaidtobework-conservingiftheresourcecannotbeidlewhilethereisatleastoneactive lagi(t+)=wi(v(t+)?v(t0))?si(t0;t+)=lagi(t)+wilag3(t) 10 w1+w2;i=1;2: (17)

11 Thereforewhenclient3leaves,itslagisproportionallydistributedamongtheremainingclients,which Correspondingly,whenaclientjjoinsthecompetitionattimet,thevirtualtimeisupdatedasfollows updatingruleforthevirtualtimewhenaclientjleavesthecompetitionattimet isinaccordancewithourinterpretationoffairness.bygeneralizingeq.(16),wederivethefollowing V(t)=V(t)?lagj(t) V(t)=V(t)+lagj(t) Pi2A(t+)wi: Pi2A(t+)wi; (18) byupdatingthevirtualtimeaccordingtoeq.(18)and(19)weensurethatthesumoverthelagsofall activeclientsisalwayszero.thiscanbeviewedasaconservationpropertyoftheservicetime,i.e.,any representsthelagwithwhichclientjjoinsthecompetition.althoughitmightnotbeclearatthispoint, timeaclientreceivesmoreservicetimethanitsshare,thereisatleastanotherclientthatreceivesless. wherea(t+)containsalltheactiveclientsimmediatelyafterclientjjoinsthecompetition,andlagj(t) (19) weobtain timet,andrejoinsitimmediately(atthesametimet)havingweightw0j.byaddingeq.(18)and(19), fromwjtow0j.thenthisisequivalenttothefollowingtwooperations:clientjleavesthecompetitionat (18)and(19)thevirtualtimedoesnotchange.9 Wenotethatifthelagoftheclientthatleavesorjoinsthecompetitioniszero,thenaccordingtoEq. thattakeplaceatthesametime.tobespecic,supposethatattimettheweightofclientjischanged Wenotethatchangingtheweightofanactiveclientisequivalenttoaleaveandarejoinoperation clientwithzerolagismodied.thus,inasysteminwhichanyclientisallowedtojoin,leave,orchange Asforjoinandleaveoperations,noticethatthevirtualtimedoesnotchangewhentheweightofa V(t)=V(t)+lagj(t) (Pi2A(t)wi)?wj?(Pi2A(t)wi)?wj+w0j: lagj(t) becomespassivewithoutusingitsentiresharecoulduseitwhenitagainbecomesactivenexttime,and itsweightonlywhenitslagiszero,thevariationofthevirtualtimeiscontinuous. Nowletusturnourattentiontothesecondquestion.Weneedtodecidewhetheraclientthat (20) whetheraclientthatleavesthecompetitionafterithasusedmoreservicetimethanitsshareshouldbe penalizedwhenitrejoinsthecompetition.tobespecic,consideraclientthatleavesthecompetition considerthefollowingexample.supposethatbeforetimettherearetwoactiveclients1and2,andat aclientforthelostservicewhenitrejoinsthecompetition,thismighthamperotherclients.toseewhy, accumulateovermultipleperiodsofactivity,andconsequently,overlargeintervalsoftimetheclientmay iswhethertheclientshouldreceiveanycompensationwhenitrejoinsthecompetition.unfortunately, receivesignicantlylessservicetimethanitisentitledto.ontheotherhand,ifwedecidetocompensate thereisnosimpleanswertothisquestion.ifwedecidenottocompensate,thenthelostservicetimemay withapositivelag(i.e.,afterithasreceivedlessservicetimethanitwasentitledto).thenthequestion timetclient2becomespassivewithapositivelag,lag2(t)>0.next,assumethatatasubsequenttime t0client2rejoinsthecompetition,whileanotherclient3isactive(weassumethatatthistimeclient1 isnolongeractive).thenclient2willhavetorecovertheservicetimethatithaslosttoclient1,atthe expenseofclient3!consequently,client3willindirectlylosesomeservicetimebecauseclient2hasnot useditsentireservicetimewhileitwaspreviouslyactive,whichisnotfair. 9However,noticethattheslopeofthevirtualtimechanges. 11

12 ofaclientwhenitrejoinsthecompetition,inthissectionwepresentthreestrategiesforimplementing clientreceivescompensationoritispenalizedwhenitrejoinsthecompetition. theeevdfalgorithm.thecharacteristicsofthesestrategiesaredictatedbythedecisiononwhethera clientcouldleave,join,orchangeitsweightwhenitslagisnon-zero,andbythedecisiononwhethera 5AlgorithmImplementation Strategy1.Inthisstrategyaclientmayleaveorjointhecompetitionatanytime,anddependingon Since,aswehaveseenintheprevioussection,thereisnoclearansweronwhatisfairtodowiththelag occurs(e.g.,aclientjoining,leaving,orchangingtheweightofaclient),thevirtualtimeisupdated itslagitiseitherpenalizedoritreceivescompensationwhenitrejoinsthecompetition.moreprecisely, accordingtoeq.(18),(19),and(20)respectively.finally,wenotethatthisstrategyisappropriatefor iftheclientleavesthecompetitionattimet,andrejoinsatt0,thenlag(t)=lag(t0).eachtimeanevent notpreservedafteraclientleavesthecompetition,i.e.,anyclientthat(re)joinsthecompetitionhaszero lag.thisstrategyisappropriateforthosesystemsinwhichtheeventsthatcausetheclientstobecome activeareindependent.thisissimilartoareal-timesysteminwhichtheprocessingtimerequiredbyan wegiveanexampleofhowthisstrategymightbeactuallyimplemented. Strategy2.Thisstrategyissimilartothepreviousonewiththeonlydierencebeingthatthelagis systemswhereitisdesirabletomaintainfairnessovermultipleperiodsofclientactivity.inappendixa iszero.thus,inthiscase,thereisnoneedtoupdatethevirtualtimewhenadynamicoperationtakes Strategy3.Inthisstrategyaclientisallowedtoleave,join,orchangeitsweight,onlywhenitslag place.ontheotherhand,somecomplexityisaddedinensuringthatwhentheseeventsoccurthelag eventisassumedtobeindependentoftheprocessingtimerequiredbyanyotherevent. isindeedzero.inordertoensurethatalleventsinvolveonlyclientswithzerolag,weneedtoupdate implementingthisstrategyistoupdatetheslopeofthevirtualtimewhenaclientleavesthecompetition theslopeofthevirtualtimeatthecorrespondingtimesintheuid-owsystem.themainproblemin mayinvolveahighoverhead.insolvingthisproblem,weassumethatnoeventoccursduringanytime veryexpensivetoimplement;itrequirestostoretheeventhistoryand,inaddition,the\undo"operation allthemodicationsinthesystemthatoccurredbetweenthetimewhentheclientshouldleavethe owsystemissmallerthanthecorrespondingtimeintherealsystem.asolutionwouldbeto\undo" withapositivelag.10inthiscasethetimeatwhichtheclientshouldleavethecompetitionintheuidcase,thisisarealisticassumptionsince,ingeneral,theschedulingalgorithmexecutesonlybetweentime theservicetime(i.e.,transmissiontime)isassumedtobeknown,beforetherequestisinitiated.the quantum.wenotethatthisassumptionisnotasrestrictiveasitappears.forexample,intheprocessor quanta.ontheotherhand,incommunicationnetworkswedonotneedtoenforcethisassumptionsince competitionintheuid-owsystemandthetimewhenitactuallyleaves.unfortunately,thissolutionis wecancomputethevirtualtimewhentheclientshouldleaveintheuid-owsystemasthevirtualdeadlineoftheclient's timequantumisgivenbelow. basicmechanismstoimplementthisstrategy,undertheassumptionthatnoeventcanoccurduringa algorithm[23]. lastrequest.thisistheapproachusedbyparekhandgallagerintheirpacket-by-packetgeneralizedprocessorshare ideaissimplytodelaytheclientuntilitslagbecomeszero.thisisdonebyissuingadummyrequestof 10Wenotethatinasysteminwhichtheclientalwaysusestheentireservicetimeithasrequested,byusingEq.(9){(11) First,assumethataclientwantstoleavethecompetitionwhenitslagisnegative.Inthiscase,the 12

13 zerolength.thisapproachismotivatedbythefactthat,inthiscase,theeligibletimeisalwayschosen nolatterthanatimequantumafteritsdeadline.thus,betweenthemomentwhenthelagoftheclient suchthatthelagiszero.sincearequestcannotbeprocessedbeforeitbecomeseligible,andsincethe becomeszero,andthemomentwhentherequestisfullled,noothertimequantaareallocated.since lagandthereforewefurtherconsideronlythelatercase.aswewillshowinsection6,inasystemin whichthevirtualtimevariescontinuously(suchasinoursystem),arequestisguaranteedtobefullled caseofaclientwhichleaveswithanegativelagtothecaseofaclientwhichleaveswithanonnegative virtualeligibletimeofthedummyrequestisequaltoitsdeadline(seeeq.(10)),itfollowsthatthis noeventoccursduringthistimequantum,itwillnotmakeanydierencewhetherweupdatethevirtual requestwillbeprocessedafteritsdeadline.thus,byusingarequestofzerolength,wehavereducedthe timeafterthetimequantumexpires,insteadofexactlywhenthelagbecomeszero. clientleavesbeforeitslastrequesthasbeenfullled.herewetakethesimplestapproach:theextra timequantathatarenolongerusedbytheclientarerandomlyallocatedtootheractiveclientswithout chargingthem,i.e.,theirreceivedservicetimesandtheirslagsarenotupdated.althoughmorecomplex allocationschemesmightbedevised,theyarenotnecessarytoachievethegoalofthisstrategywhichis toguaranteethataslongasaclientcompetesfortheresourceitwillreceiveatleastitsshare. Asecondquestionregardingthisstrategyiswhattodowiththeremainingservicetimewhenthe 6FairnessAnalysisoftheEEVDFAlgorithm Inthissectionwedetermineboundsfortheservicetimelag.Firstweshowthatduringanytimeinterval inwhichthereisatleastoneactiveclient,thereisalsoatleastoneeligiblependingrequest(lemma thereisatleastoneactiveclienttheresourcecannotbeidle.byusingthisresult,intheorem1wegive 2).AdirectconsequenceofthisresultisthattheEEVDFalgorithmiswork-conserving,i.e.,aslongas tightboundsforthelagofanyclientinasteadysystem(seedenitions1and2below).finally,we showthatintheparticularcasewhenalltherequestshavedurationsnogreaterthanatimequantumq, algorithm(lemma5). ouralgorithmachievestightboundswhichareoptimalwithrespecttoanyproportionalshareallocation joiningorleavingthecompetition,andchangingtheclient'sweight,simplyasevent.weintroducenow withzerolag. somedenitionstohelpusinouranalysis. Denition1Asystemissaidtobesteadyifalltheeventsoccurringinthatsysteminvolveonlyclients Throughoutthissectionwerefertoanyeventthatcanchangethestateofthesystem,i.e.,aclient continuous.aswewillsee,thisisthebasicpropertyweusetodeterminetightboundsfortheclientlag. Recallthatinasysteminwhichalleventsinvolveonlyclientshavingzerolagsthevirtualtimeis Thefollowingdenitionrestrictsthenotionofsteadinesstoaninterval. clientswithzerolag. Denition2Anintervalissaidtobesteadyifalltheeventsoccurringinthatintervalinvolveonly Thus,inasteadysystemthelagofanyclientthatjoins,leaves,orhasitsweightchanged,iszero. issteady.thenextlemmagivestheconditionforaclientrequesttobeeligible. Wenotethatasteadysystemcouldbealternativelydenedasasystemforwhichanytimeinterval 13

14 Proof.Letrbethelengthofthependingrequestofclientkattimet(recallthatanactiveclienthas Lemma1Consideranactiveclientkwithapositivelagattimet,i.e., therequest.forthesakeofcontradiction,assumetherequestisnoteligibleattimet,i.e., Thenclientkhasapendingeligiblerequestattimet. alwaysapendingrequest),andletveandvddenotethevirtualeligibletimeandthevirtualdeadlineof lagk(t)0; (21) Lett0bethetimewhentherequestwasinitiated.ThenfromEq.(7)wehave Sincebetweent0andttherequestwasnoteligible,itfollowsthattheclienthasnotreceivedanyservice timeintheinterval[t0;t),andthereforesk(tk0;t0)=sk(tk0;t).bysubstitutingsk(tk0;t0)tosk(tk0;t)ineq. ve=v(tk0)+sk(tk0;t0) ve>v(t) wk: (22) (23)andbyusingEq.(3)and(6)weobtain lagk(t)=wk(v(t)?v(tk0))?sk(tk0;t) (24) (23) Finally,fromIneq.(22)itfollowsthatlagk(t)<0,whichcontradictsthehypothesisandthereforeproves thelemma. =wk(v(t)?v(tk0))?wk(ve?v(tk0)) followingcorollary. FromLemma1andfromthefactthatanyzerosumhasatleastanonnegativeterm,wehavethe =wk(v(t)?ve): Thenthereisatleastoneeligiblerequestattimet. Corollary1LetA(t)bethesetofallactiveclientsattimet,suchthat immediateconsequenceofthisresultandtheabovecorollaryisthatatanytimetatwhichthereisat Thenextlemmashowsthatatanytimetthesumofthelagsoverallactiveclientsiszero.An i2a(t)lagi(t)=0: X leastoneactiveclient,thereisalsoatleastonependingeligiblerequestinthesystem.thus,inthesense (25) ofthedenitiongivenin[23],theeevdfalgorithmiswork-conserving,i.e.,aslongasthereareactive clientstheresourceisbusy. Lemma2Atanymomentoftimet,thesumofthelagsofallactiveclientsiszero,i.e, i2a(t)lagi(t)=0: X14 (26)

15 Proof.Theproofgoesbyinduction.First,weassumethatattimet=0thereisnoactiveclientand thereforeeq.(26)istriviallytrue.next,fortheinductionstep,weshowthateq.(26)remainstrue aftereachoneofthefollowingeventsoccurs:(i)aclientjoinsthecompetition,(ii)aclientleavesthe competition,(iii)aclientchangesitsweight.finally,weshowthat(iv)duringanyinterval[t;t0)inwhich noneoftheaboveeventsoccursifeq.(26)holdsattimet,thenitalsoholdsattimet0. Case(i).Assumethatclientjjoinsthecompetitionattimetwithlaglagj(t).Lett?denotethetime immediatelybefore,andlett+denotethetimeimmediatelyafterclientjjoinsthecompetition,where t+andt?areasymptoticallyclosetot.next,letw(t)denotethetotalsumovertheweightsofallactive clientsattimet,i.e.,w(t)=pi2a(t)wi(t),andbyconvenienceletustakelagj(t?)=lagj(t).since t?!t+wehavesi(ti0;t?)=si(ti0;t+).thenfromeq.(3)weobtain: lagi(t+)=lagi(t?)+si(ti0;t+)?si(ti0;t?) Further,byusingEq.(6)and(19),thelagofanyactiveclientiattimet+(includingclientj)is lagi(t+)=lagi(t?)?lagj(t)wi W(t+): (27) SinceA(t+)=A(t?)[fjg,andsincefromtheinductionhypothesiswehavePi2A(t?)lagi(t?)=0,by usingeq.(27),weobtainx i2a(t+)lagi(t+)=x i2a(t+)(lagi(t?)?lagj(t)wi W(t+)) (28) =X i2a(t+)lagi(t?)?lagj(t)pi2a(t+)wi W(t+) =X i2a(t?)lagi(t?)+lagj(t?)?lagj(t)=0: Case(ii).Theproofofthiscaseisverysimilartotheoneofthepreviouscase;thereforeweomitithere. Case(iii).Changingtheweightofaclientjfromwjtow0jattimetcanbeviewedasasequenceoftwo events:rst,clientjleavesthecompetitionattimet;second,itjoinsthecompetitionatthesametime t,butwithweightw0j.thus,theproofofthiscasereducestotheprevioustwocases. Case(iv).Consideraninterval[t;t0)inwhichnoeventoccurs,i.e.,noclientleavesorjoinsthecompetition andnoweightischangedduringtheinterval[t;t0).next,assumethatpi2a(t)lagi(t)=0.thenweshall provethatpi2a(t0)lagi(t0)=0.byusingeq.(3)and(6)weobtain X i2a(t0)lagi(t0)=x i2a(t0)(si(ti0;t0)?si(ti0;t0)) (29) =X i2a(t)(si(ti0;t)?si(ti0;t))+x i2a(t)(si(t;t0)?si(t;t0)) =X i2a(t)lagi(t)+x i2a(t)si(t;t0)?x i2a(t)si(t;t0) =X i2a(t)wi(v(t0)?v(t))?x i2a(t)si(t;t0) =(t0?t)?x i2a(t)si(t;t0): Nextweshowthattheresourceisbusyduringtheentireinterval[t;t0).Forcontradictionassumethisis nottrue.letldenotetheearliesttimeintheinterval[t;t0)whentheresourceisidle.similarlytoeq. (29)wehave: 15

16 Sincetheresourceisnotidleatanytimebetweentandl,itfollowsthatthetotalservicetimeallocated toallactiveclientsduringtheinterval[t;t0)(i.e.,pi2a(t)si(t;l))isequaltol?t.further,fromthe aboveequationwehavepi2a(l)lagi(l)=0.butthenfromlemma1itfollowsthatthereisatleastone eligiblerequestattimel,andthereforetheresourcecannotbeidleattimel,whichprovesourclaim. i2a(l)lagi(l)=(l?t)?x X Further,withasimilarargument,itiseasytoshowthatPi2A(t0)lagi(t0)=0whichcompletestheproof i2a(t)si(t;l): system.wenotethatthisresultissimilartotheoneobtainedbyparekhandgallager[23]fortheir GeneralizedProcessorSharingalgorithm,i.e.,inacommunicationnetwork,apacketisguaranteednot ofthiscase. tomissitsdeadlinebymorethanthetimerequiredtosendapacketofmaximumlength. lemmafollows. Thefollowinglemmagivestheupperboundforthemaximumdelayoffulllingarequestinasteady Sincethesearetheonlycasesinwhichthelagsoftheactiveclientsmaychange,theproofofthe partitionofalltheactiveclientsattimed,intotwosetsbandc,wheresetbcontainsalltheclients Proof.Letebetheeligibletimeassociatedtotherequest(withdeadlined)ofclientk.Considerthe Lemma3Inasteadysystemanyrequestofanyactiveclientkisfulllednolaterthand+q,whered thathaveatleastadeadlineintheinterval[e;d],andsetccontainsalltheotheractiveclients(see istherequest'sdeadline,andqisthesizeofatimequantum. Case1(texists).Hereweconsidertwosub-caseswhethert2[e;d),ort<e.Firstassumethataclient any.furtherweconsidertwocaseswhethersuchatexistsornot. Figure3).LettbethelatesttimenogreaterthandatwhichaclientinCreceivesatimequantum,if increceivesatimequantumatatimet2[e;d).sinceallthedeadlinesofthependingrequestsissued byclientsincarelargerthand,thismeansthatattimetthependingrequestofclientkisalready fullled.consequently,intherstsub-casetherequestofclientkisfullledbeforetimed. noteligibleattimet,i.e.,t<ej.noticethatbesidestheclientsind(t),theotherclientsthatbelong (andthereforesmallerthananydeadlineofanyclientinc),itfollowsthatallthesependingrequestsare withthedeadlineintheinterval[t;d)(seefigure3).further,letd()denotethesubsetofdcontaining eligibletimeofitspendingrequestattimet.sincethedeadlinesoftheserequestsarenogreaterthand nootherclientwithanearlierdeadlineiseligibleatt.foranyclientjbelongingtod(t),letejbethe theactiveclientsattime.sinceatimequantumisallocatedtoaclientincattimet,itfollowsthat Forthesecondsub-case,letusDdenotealltheactiveclientsthathaveatleastoneeligiblerequest todarethosethateventuallyjointhecompetitionaftertimet.foranyclientjindthatjoinsthe itsrequests(noticethattheeligibletimeejandthedeadlinedjmightnotbeassociatedtothesame competitionaftertimet,wetakeejtobetheeligibletimeofitsrstrequest. request).fromeq.(10)iteasytoseethatafterclientjreceivessj(ej;dj)timeunits,allitsrequests intheinterval[ej;dj)arefullled.thus,theservicetimeneededtofulllalltherequestswhichhave deadlinesintheinterval[t;d)is Next,foranyclientjbelongingtoD,letdjdenotethelargestdeadlinenogreaterthandofanyof 16

17 ] ] ] ] ] client [ [ k t e B C d D(t) d + q [ [ [ ] [ Figure3:Thecurrentpendingrequestofclientkhastheeligibletimeeanddeadlined.SetBcontains alltheactiveclientsthathaveatleastonerequestwiththedeadlineintheinterval[e;d],whilesetc containsalltheotheractiveclients.timetrepresentsthelargesttimenogreaterthandatwhichaclient fromcreceivesatimequantum.finally,setd(t)containsalltheactiveclientsattimetthathaveat leastoneeligiblerequestwiththedeadlineintheinterval[t;d). Xj2DSj(ej;dj)=Xj2D(Zdj ejwj Pi2A()wid): (30) BydecomposingtheabovesumoverasetofdisjointintervalsJl=[al;bl)(1lm)covering[t;d), suchthatnointervalcontainsanyeligibletimeordeadlineofanyclientbelongingtod,wecanrewrite Eq.(30)asXj2DSj(ej;dj)=mXl=1(Zbl alpi2d(al)wi Pi2A(al)wid)<mXl=1(Zbl ald)=mxl=1(bl?al)=d?t; (31) TheaboveinequalityresultsfromthefactthatD()isapropersubsetofA()atleastforsomesubintervalsJi(otherwise,ifA()isidenticaltoD()overtheentireinterval[t;d),setsCandC0wouldbe empty). Assumethatattimed+qtherequestofclientk(havingthedeadlined)isnotfullledyet.Sinceno clientinccanbeservedbeforetherequestofclientkisfullled,itfollowsthattheservicetimebetween t+qandd+qisallocatedonlytotheclientsind.consequently,duringtheentireinterval[t+q;d+q), thereared?tservicetimeunitstobeallocatedtoallclientsind.next,recallthatanyclientjbelonging todwillnotreceiveanyothertimequantumafteritsrequesthavingdeadlinedjiseventuallyfullled,as longastherequestofclientkisnotfullled.thisissimplybecausethenextrequestofclientjwillhave adeadlinegreaterthand.butaccordingtoeq.(31)theservicetimerequiredtofulllalltherequests havingthedeadlinesintheinterval[t;d)islessthand?t,whichmeansthatatsomepointtheresource isidleduringtheinterval[t+q;d+q).butthiscontradictsthefactthateevdfiswork-conserving, andthereforeprovesthiscase. Case2.(tdoesnotexist)Inthiscasewetakettobethetimewhentherstclientjoinsthecompetition. Fromheretheproofissimilartotheonefortherstcase,withthefollowingdierence.SincesetCis empty,allthetimequantabetweentanddareallocatedtotheclientsind,andtherefore,inthiscase, weshowthatinfactclientkdoesnotmissthedeadlined. 17

18 requestsoftheclientswithnegativelagswhichareactiveatt1.thenanyrequestofanyactiveclientk isfulllednolaterthand+q,ifd2[dm;t2). Lemma4LetI=[t1;t2)beasteadyinterval,andletdmbethelargestdeadlineamongallthepending lagtojoin,leave,ortochangetheirweight,willeventuallyreachasteadystate. ofasteadyintervalthesameboundholds.thisshowsthatasystemwhichallowsclientswithnon-zero Followingwegiveasimilarresultforasteadyinterval.Mainly,weshowthatforcertainsubintervals Proof.SimilarlytotheproofofLemma3,weconsiderthepartitionofalltheactiveclientsattimed, intotwosetbandc,wheresetbcontainsalltheactiveclientsthathaveatleastadeadlineinthe interval[e;d],andsetccontainsalltheotherclients.similarly,welettdenotethelatesttimeinthe interval[t1;d)whenaclientincreceivesatimequantum,ifany.further,weconsidertwocaseswhether Case1.(texists)TheproofproceedssimilarlytotheoneforCase1inLemma3. Case2.(tdoesnotexist)Inthiscaseweconsidertwosub-setsofC:C?containingallclientsinC suchtexistsornot. allclientsinc+hadnonnegativelagsattimet1,andsincetheydonotreceiveanytimequantabetween dm)andthereforeallclientsinc?willhavenonnegativelagsattimedm.ontheotherhand,since toc?receivesanytimequantumbeforedmitfollowsthatnopendingrequestofanyclientinc?is fullledbeforeitsdeadline(recallthatthedeadlinesofalltheotherclientswithnegativelagsatt1are t1anddm,allofthemwillhavepositivelagsatdm.thus,wehave thathadnegativelagsattimet1,andc+containingalltheotherclientsinc.sincenoclientbelonging Ontheotherhand,wenotethatiftherequestofclientkisnotfullledbeforeitsdeadline,thenno karepositiveattimed,i.e., nogreaterthandisfullled.butthenfromeq.(3)itfollowsthattheirlagsaswellasthelagofclient otherclientbelongingtobwillreceiveanyothertimequantumafteritslastrequestwiththedeadline Xi2Clagi(d)>0 Xi2Blagi(d)0 (32) Further,byaddingEq.(32)and(33),weobtain i2a(d)lagi(d)>0 X (33) Theorem1Thelagofanyactiveclientkinasteadysystemisboundedasfollows, whichcontradictslemma2,andthereforecompletestheproof. Thenexttheoremgivestightboundsforaclient'slaginasteadysystem. (34) wherermaxrepresentsthemaximumdurationofanyrequestissuedbyclientk.moreover,thesebounds areasymptoticallytight.?rmax<lagk(d)<max(rmax;q); 18 (35)

19 have SinceSkincreasesmonotonicallywithaslopenogreaterthanone(seeEq.(4)),fromEq.(3)itfollows Proof.Leteanddbetheeligibletimeandthedeadlineofarequestwithdurationrissuedbyclientk. sincearequestisnotservicedbeforeitiseligible,itiseasytoseethattheminimumlagisachievedwhen minimumlagoccursattimee+r,iftherequestisfullledbythattime.further,byusingeq(3)we thatthelagofclientkdecreasesaslongasitreceivesservicetime,andincreasesotherwise.further, theclientreceivestheentirelyservicetimeassoonastherequestbecomeseligible.inotherwords,the Fromthedenitionoftheeligibletime(seeSection2)wehavelagk(e)0,andthusfromtheabove lagk(e+r)=sk(tk0;e+r)?sk(tk0;e+r) =Sk(tk0;e)+Sk(e;e+r)?(sk(tk0;e)+sk(e;e+r)) =lagk(e)+sk(e;e+r)?sk(e;e+r) (36) activewehave Sincethisisthelowerboundfortheclient'slagduringarequestwithdurationr,andsincermaxrepresents equationweobtainlagk(e+r)sk(e;e+r)?sk(e;e+r)>?sk(e;e+r)?r: themaximumdurationofanyrequestissuedbyclientk,itfollowsthatataanytimetwhileclientkis lagk(t)?rmax: (37) aslateaspossible.sinceaccordingtolemma3,therequestisfulllednolaterthand+q,itfollows thatthelatesttimewhenclientkshouldreceivetherstquantumisd+q?r.weconsidertwocases: rqandr<q.intherstcased+q?rd,andthereforeweobtainsk(e;d+q?r)<sk(e;d)=r. Lett1bethetimeatwhichtherequestisissued.Further,fromthedenitionoftheeligibletime,and Similarly,themaximumlagintheinterval[e;d)isobtainedwhentheentireservicetimeisallocated (38) fromthefactthattheclientisassumedthatitdoesnotreceiveanytimequantumduringtheinterval [t1;d+q?r),wehaveforanytimetwhiletherequestispending SincetheslopeofSkisalwaysnogreaterthanone,inthesecondcasewehaveSk(e;d+q?r)= lagk(t)sk(tk0;d+q?r)?sk(tk0;d+q?r) =(Sk(tk0;e)?sk(tk0;t1))+Sk(e;d+q?r)?sk(t1;d+q?r) =Sk(tk0;e)+Sk(e;d+q?r)?sk(tk0;t1)?sk(t1;d+q?r) =Sk(e;d+q?r)<r: (39) Finally,bycombiningEq.(39)and(40)weobtainlagk(t)<max(q;r).Thus,atanytimetwhilethe Sk(e;d)+Sk(d;d+q?r)<r+q?r=q,andfromhereweobtain clientisactivewehave lagk(t)sk(e;d+q?r)<q: lagk(t)<max(q;rmax): (40) 19 (41)

20 virtualdeadlineofthesecondclient'srequest,i.e.,t0+rmax rmaxandr0maxarechosensuchthatthevirtualdeadlineoftherstclient'srequestissmallerthanthe activeattimet0andtheirrstrequestshavethelengthsrmaxandr0max,respectively.weassumethat w1,w2betheweightsoftwoactiveclients,suchthatw1w2.next,supposethatbothclientsbecome Toshowthattheboundlagk(t)>?rmaxisasymptoticallytight,considerthefollowingexample.Let w1<t0+r0max thevirtualdeadlineoftherstrequestofclient2,suchthatclient1receivesitsentireservicetimejust consequentlylag1(rmax)approaches?rmax. However,inthiscaseweassumethatthevirtualdeadlineoftherstrequestofclient1isearlierthan entireservicetimebeforeclient2,andthusfromeq.(3)wehavelag1(rmax)=s1(t0;t0+rmax)?rmax. Next,byusingEq.(4)weobtainS1(t0;t0+rmax)=w1 Toshowthattheboundlagk(t)<max(rmax;q)isasymptoticallytight,weusethesameexample. w1+w2,whichapproacheszerowhenw1 w2.thenclient1receivesthe priortoitsdeadline.sincethedetailsoftheproofaresimilarwiththepreviouscasewedonotshow themhere. w2!1,and oftherequesttobeintheorderofseconds.ontheotherhand,inthecaseofamultimediaapplication weneedtotakethelengthofarequestnogreaterthanseveraltensofmilliseconds,duetothedelay requirements.forexample,foranintensivecomputationtaskitwouldbeacceptabletotakethelength thelengthoftheirrequests.whileshorterrequestsoerabetterallocationaccuracy,thelongerones reducethesystemoverheadsinceforthesametotalservicetimefewerrequestsneedtobegenerated. Itisthereforepossibletotradebetweentheaccuracyandthesystemoverhead,dependingontheclient NoticethattheboundsgivenbyTheorem1applyindependentlytoeachclientanddependonlyon constraints.theorem1showsthateevdfcanaccommodateclientswithdierentrequirements,while guaranteeingtightboundsforthelagofeachclientduringasteadyinterval.thefollowingcorollary followsdirectlyfromtheorem1. Corollary2Considerasteadysystemandaclientksuchthatnorequestofclientkislargerthana timequantum.thenatanytimet,thelagofclientkisboundedasfollows: holdforanyproportionalsharealgorithm. Lemma5Givenanysteadysystemwithtimequantaofsizeqandanyproportionalsharealgorithm,the lagofanyclientisboundedby?qandq. NextwegiveasimplelemmawhichshowsthattheboundsgiveninCorollary3areoptimal,i.e.,they?q<lagk(t)<q: (42) Proof.Considernclientswithequalweightsthatbecomeactiveattime0.Weconsidertwocases:(i) Similarly,thelagoftheclientwhichreceivesthenthtimequantumis(attimen?1,immediatelybefore eachclientreceivesexactlyonetimequantumoutoftherstnquanta,and(ii)thereisaclientkwhich itreceivesthetimequantum) receivesmorethanatimequanta.fromeq.(3),itiseasytoseethat,attimeq,thelagoftheclient thatreceivestherstquantumislag(q(n?1))=q?qn: lag(q)=qn?q: (43) 20 (44)