stchuang sundaes nickm stanfordedu Abstract This paper is about high capacity switches and routers that give guaranteed throughput rate and delay guaran tees Many routers are built using input queueing or combined input and output queueing CIOQ usi ID: 72480
Download Pdf The PPT/PDF document "Practical Algorithms for Performance Gua..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Inputs R 2 3 Outputs R R2 R3 Buffered CrossbarFig.1.Thearchitectureofthebufferedcrossbarwiththreeports.ResearchersÞrstnoticedviasimulationthatbufferedcross-barsprovidegoodthroughputforadmissibleuniformtrafÞcwithsimplealgorithms[9][10][11][12].Simulationsalsoin-dicatedthat,withmodestspeedup,abufferedcrossbarcancloselyapproximatefairqueueing[13].In[14],theauthorsde-scribedamechanismtoprovidefairallocationandconÞrmedthroughsimulationsthatabufferedcrossbarcanallocateserviceinaweightedmax-minfairmanner.Untilrecently,therewerenoanalyticalresultsonguaranteedthroughputtoexplainorconÞrmtheobservationsmadebysimulations.TheÞrstanalyticalresultscamein2001,whenJavidiprovedthat,withuniformtrafÞc,abufferedcrossbarwithnospeedupcanachieve100%throughput[15].Morerecently,etal.provedthatabufferedcrossbarwithaspeedupoftwocanmimicaÞrstcomeÞrstserveoutputqueued(FCFS-OQ)switchwithanyarrivaltrafÞcpattern[16].Magilletal.alsoshowedthatabufferedcrossbarwithcellspercrosspointcanmimicaFCFS-OQswitchwithstrictpriorities.Inthispaper,wedescribeaseriesofalgorithmswithabroadclassofperformanceguaranteesoverandaboveFCFSandstrictpriorityFCFSemulation.Weprovethatthesealgorithmscanachieve100%throughput,canmimicanOQswitchusingaweightedroundrobinscheduler(whichgivesrateguarantees),andcanalsoachievedelayguarantees.ThemainbeneÞtofthesealgorithmsisthateachinputandoutputmakessimpleschedulingdecisionsandinparalleleliminatingtheneedforacentralizedscheduler.OurresultsshowbufferedcrossbarscangreatlysimplifytheschedulingOfcourse,simplifyingtheschedulercomesattheexpenseofamorecomplicatedcrossbar;itnowhastoholdandpacketbuffers.InthepastthiswouldhavebeenTwodifferentswitchesaresaidtomimic[2][17]eachother,ifunderidenticalinputs,identicalpacketsdepartfromeachswitchatthesametime.prohibitivelycomplex:Thenumberofportsandcapacityofacrossbarswitchusedtobelimitedbythecrosspointsthatdominatedthechiparea(hencethedevelopmentofmulti-stageswitchfabrics,suchasClos,BanyanandOmegaswitchesbasedonsmallercrossbarelements).Butnowadays,crossbarswitchesarelimitedbythenumberofpinsrequiredtogetdataonandoffthechip[18].Improvementsinprocesstechnology,andreductionsingeometries,meansthatthelogicrequiredforcrosspointsissmallcomparedtothesizeofchipneededinputsandoutputs.Thechipsarepad-limited,withanunderutilizeddie.Abufferedcrossbarcanusetheunuseddieforpacketbuffers.Webelievethatincurrenttechnology,theunbufferedcrossbarswitchreportedin[18]couldcellbuffers.C.OrganizationofthispaperTherestofthepaperisorganizedasfollows.InSectionII,weshowthatabufferedcrossbarwithspeeduptwocangive100%throughputfornon-uniformtrafÞc.InSectionIII,webrießyreviewthecountingmethodintroducedin[2]usedtoshowhowaCIOQswitchcanmimicanOQswitch.InSectionIVweshowhowthecountingmethodcanbeappliedtoabufferedcrossbarandsoastomimicaclassofOQswitches.InSectionV,wedescribehowabufferedcrossbarcangiverateguaranteesbetweenaninput/outputpairforaweightedroundrobinscheduler.InSectionVI,weshowthatthebufferedcrossbarwithonecellpercrosspointcangivedelayguaranteeswhen,andintroduceanovelmechanismcalledheaderschedulingwhichsupportsdelayguaranteeswhenII.ATHROUGHPUTWITHANARBITRARYSCHEDULINGALGORITHMFigure2showstheschedulingphasesinabufferedcrossbarwithaspeedupoftwo.Thetwoschedulingphaseseach consistsoftwoparts:inputscheduling,andoutputscheduling.Intheinputschedulingphase,eachinput(independentlyandinparallel)picksacelltoplaceintoanemptycrosspoint.Intheoutputschedulingphase,eachoutput(independentlyandinparallel)picksacellfromanon-emptycrosspointtotakefrom.Thekeytocreatingaschedulingalgorithmisdeterminingtheinputandoutputschedulingpolicywhichdecideshowinputsandoutputspickcellsintheschedulingphases.Wewillseeanumberofdifferentpolicieseachofwhichprovidesadifferentschedulingalgorithm.TheÞrstalgorithmweÕllconsideristhemostgeneral.Ineachschedulingphase,theinputpicksanynon-emptyVOQ,andtheoutputpicksanynon-emptyWewilladoptthefollowingnotationanddeÞnitions.Theswitchhasports,andVOQholdscellsatinputforoutputistheoccupancyofVOQisthesumofthenumberofcellsintheVOQthecorrespondingcrosspoint.Wewillassumethatallarrivalstoinput,...,NareBernoullii.i.d.withrate,andaredestinedtoeachoutput,...Nwithprobability.Wewilldenotethearrivalmatrixas,s,ij],whereDeÞnition1:Anarrivalprocessissaidtobeadmissiblewhennoinputoroutputisoversubscribed,i.e.,whenDeÞnition2:100%ThroughputAnalgorithmissaidtogive100%throughputiffforanyadmissibletrafÞc,thequeuesizesareÞnite,i.e.,forevery,thereexistssuchthatInwhatfollows,wewillshowthatthebufferedcrossbarcangive100%throughput.Theresultisquitestronginthesensethatitholdsforanyarbitrarywork-conservinginputandoutputschedulingpolicywithaspeedupoftwo.Inotherwords,eachinputcanchoosetoserveanynon-emptyVOQforwhich,andeachoutputcanchoosetoserveanycrosspointforwhichTheorem1:(SufÞciency)Abufferedcrossbarcanachieve100%throughputwithspeeduptwoforanyBernoullii.i.d.admissibletrafÞc.Proof:Wedescribeanintuitionoftheproof.ThemainproofappearsinAppendixA.ForeachVOQ,letdenotethesumofthecellswaitingatinputandthecellswaitingatallinputsdestinedtooutput(includingcellsinthecrosspointsforoutputWeÕllseelaterthatotherqueueingstructuresareusefulanditisnotnecessarytoplacecellsinVOQsItiseasytoseethatwhenVOQisnon-empty(i.e.),thendecreasesineveryschedulingphase.Therearetwocases:Case1:.Outputwillreceiveonecellfromthebuffersdestinedtoitandwilldecreasebyone.Case2:.InputwillsendonecellfromitsVOQstoacrosspoint,andwilldecreasebyWithwilldecreasebytwopertimeslot.Whentheinputsandoutputsarenotoversubscribed,theexpectedincreaseinisstrictlylessthantwopertimeslot.Sotheexpectedchangeinisnegativeoverthetimeslot,andthismeansthattheexpectedvalueofisbounded.Thisinturnimpliesthattheexpectedvalueofisboundedandthebufferedcrossbarhas100%throughput. III.BACKGROUNDOFTHEAnOQswitchcanprovidethroughput,rateanddelayguarantees.In[2],theauthorsusedacountingmethodtoshowthataCIOQswitchusingatraditionalunbufferedcrossbarwithaspeedupoftwocanmimicanOQswitchandthereforealsoproviderateanddelayguarantees.Themethodin[2]wastoshowthataCIOQswitchwithspeedupoftwocanmimicaso-calledPush-InFirst-Out(PIFO)queueingpolicy.PIFOisageneralqueueingpolicythatincludesWFQ(weightedfairqueueing)asaspecialcase.IfaCIOQswitchcanimplementaPIFOpolicy,thenitcanimplementWFQwhichmeansitcansupportrateanddelayguarantees[19][20].InaPIFOqueue,thereisasinglequeueofallcellswaitingtodepartfromanoutput.Whenanewcellarrivesfortheoutput,itisÒpushed-inÓtoanarbitrarylocationinthequeue.Onceinthequeue,thecellÕsrelativeorderingwithcellsalreadyinthequeuedoesnÕtchange;cellscanÕtswitchplaces.Ofcourse,newcellscanarrivelaterandgetpushed-inaheadof(orbehind)thecell.CellscanonlydepartfromheadofInordertomimicaPIFO-OQswitch,acellmustbetransferredtotheoutputoftheCIOQswitchbeforethedeparturetimeofthecounterpartcellintheOQswitch.Ifthecellispreventedfromreachingitsoutputintime,thedeparturesfrombothswitchesisnotidentical,andweÕllfailtomimictheOQswitch.SelectingtheappropriateschedulingalgorithmisthekeytoensuringthatthecellwillreachtheoutputoftheCIOQswitchintime.Thegeneralstructureoftheschedulingalgorithmdescribedin[2]assumedthateachinputoftheCIOQswitchmaintainsaprioritylist,whichcanbethoughtofasanorderedsetofcellswaitingattheinputport.Acellcanbepreventedfromreachingitsoutputontimebyothercellsatitsinputwithahigherpriority.Themorecellsaheadofitintheprioritylist,IfacellfromVOQissenttocrosspoint,thenstaysthesameattheendoftheinputschedulingphasesincebyoneandincreasesbyone.Intheoutputschedule,Case1appliesandwillfurtherdecreasebyone.Asaresult,ifacellfromVOQissenttocrosspoint,thendecreasesbytwointhatschedulingphase. Scheduling Phase 1 Scheduling Phase 2 Arrival Input Schedule 1 Output Schedule 1 Input Schedule 2 Output Schedule 2 Depart Fig.2.Theschedulingphasesforthebufferedcrossbar.TheexactorderofthephasesdoesnÕtmatter;butwewillusethisordertosimplifyproofs.thelongeritmighttaketobetransferredtotheoutput.ManyorderingsofthecellsarepossibleÑeachorderingleadingtoadifferentswitchschedulingalgorithm.Inaddition,eachoutputalsomaintainsanoutputprioritylist:anorderedlistofcellsattheinputswaitingtobetransferredtoaparticularoutput.TheoutputprioritylistisconstructedbasedontheorderinwhichthecellswoulddepartfromtheOQswitch.ThisprioritylistwilldependonthequeueingpolicyfollowedbytheOQswitch(i.e.,WFQ,strictpriorities,etc.).ThefollowingdeÞnitionspreviouslydeÞnedin[2]isnec-essaryfortheunderstandingoftherestofthepaper.DeÞnition3:OutputCushionAtanytime,theoutputcushionofacell,isthenumberofcellsatÕsoutputportthathasanearlierdepartureorderthancellDeÞnition4:InputThreadAtanytime,theinputthreadofcell,isthenumberofcellsatÕsinputwithahigherpriority.DeÞnition5:SlacknessAtanytime,theslacknessof,equalsitsoutputcushionminusitsinputthread;Theslacknessreßectstheurgencywithwhichwemusttransferthecelltoitsoutput.ThekeytoidenticalbehavioristoÞndschedulingalgorithmsforwhichtheslacknessisalwaysnon-negative.Althoughnotstrictlynecessary,thiswillensurethatwhenacellistransferredtotheoutputitsoutputcushionisnon-negative,(orreacheszerointhetimeslotitistransferred).Theideaisthatwhentheoutputcushionofacellreacheszero,theinputthreadofthatcellmustalsoequalzero.Thismeansthateither:(1)thecellisalreadyatitsoutput,andwilldeparttheoutputontime,or(2)thecellisattheheadofitsprioritylist(becauseitsinputthreadiszero),andwillbetransferredtotheoutputimmediately,whichensuresthatthecellwilldeparttheoutputontime.Basedontheinputandoutputprioritylists,thecountingmethodrequiredthatineachschedulingphase,atleastoneofthefollowingconditionsforeachcellissatisÞed:(1)cellistransferredfromtheinputside,(2)acellthatisaheadofcellinitsinputprioritylististransferredfromtheinputside,or(3)acellthatisaheadofcellinitsoutputprioritylististransferredtotheoutputside.In[2],itwasprovedthatmeetingtheconditionsofthecountingmethodensuredthattheslacknessofacellin-creasedbyatleastoneineachschedulingphase,whichwasessentialinprovingthattheslacknessofanycellisalwaysnon-negative.However,[2]requiredastablemarriage[21]tomeettheconditionsofthecountingmethod.Thesolutionrequireduptoatleastiterationsofthestablemarriagealgorithmforaswitchwithports,makingthemtoocomplextoimplementforfastswitcheswithlargenumberofports.Wewillnowshowhowthebufferedcrossbarcanalsomeettheconditionsofthecountingmethodinasimpledistributedmannerwhereeachinputandoutputmakesdecisionsinde-pendentlyandinparallel.IV.CETHODWITHAUFFEREDROSSBARInordertoensurethattheslacknessofacellincreasesbyatleastoneineachschedulingphaseforabufferedcrossbar,theinputandoutputschedulingpoliciesmustcarefullybeselectedtoguaranteethattheconditionsofthecountingmethodaremet.Theinputschedulingpolicygivespreferencetocellsbasedontheinputprioritylist.Similarly,theoutputschedulingpolicygivespreferencetocellsbasedontheoutputprioritylist.Sincetheoutputprioritylistisorderedbasedondepartureorder,preferenceisgiventocellswithanearlierdepartureorder.However,thebufferedcrossbarhasanadditionalrequire-menttomeettheconditionsofthecountingmethod.Theinputprioritylistalsomustbearrangedsocellsdestinedtothesameoutputareorderedbasedondepartureorder.SpeciÞcally,cellstothesameoutputwithanearlierdepartureordermusthaveahigherpriority.Cellstodifferentoutputscanstillbeorderedinanyway.Thisrequirementisnecessarytoensurethat,intheoutputschedulingphase,thecellselectedhastheearliestdepartureorderofthecellsstoredintheinputqueuescorrespondingtothenon-emptycrosspoints,ascanbeseeninthefollowingexample.Letcells,andallbedestinedtooutput.Cellstoredininputqueue,cellisstoredininputqueue,cellisstoredincrosspoint,andnoothercellsaredestinedtoattime.Thedepartureorderisforcells,andrespectively.Intheinputschedulingphase,inputdoesnotselectcellsincecellisalreadyinthecrosspointandinputselectscell.Intheoutputschedulingphase,isselectedsinceithasanearlierdepartureorderthan.Asaresult,theconditionsofthecountingmethodisnotmetforcellsincecellwhichhasalaterdepartureorderdoesnothaveahigherprioritythancellintheoutputprioritylist.Thisoccurredbecauseatsomepointintimecellwasincorrectlygivenahigherprioritythancellintheinputprioritylist.ThismotivatesthefollowingÒGroupByVirtualOutputQueueÓinsertionpolicypreviouslydescribedin[2].GBVOQInsertionPolicy:1)Whenacellarrivestoanon-emptyVOQ,thecellisinsertedintheinputprioritylistjustbehindthelast cellbelongingtothesameVOQ.Thisensuresthatcellsdestinedtothesameoutputareorderedbasedondepartureorder.2)WhenacellarrivestoanemptyVOQ,thecellisinsertedattheheadoftheinputprioritylist.AtÞrstglance,itseemsunfairtoinsertacellwhicharrivestoanemptyVOQattheheadoftheinputprioritylist.However,itispossiblethattherearenoothercellsinthesystemdestinedtothatoutput.Therefore,thecellmayimmediatelyneedtobetransferredtotheoutputinordertokeepthatoutputbusy.Wewillnowproveinthefollowinglemmathatthebufferedcrossbarcansatisfytheconditionsofthecountingmethod.Lemma1:Theslacknessofacelldecreasesbyatleastoneineachschedulingphase.Proof:LetÕsassumethatcellbelongstoVOQ.Therearetwocasestoconsiderinaschedulingphase.Case1:,thenweknowthatintheinputschedulingphase,acellwillbetransferredfrominputtooneofthebuffers.Ifcellistransferredtothenwenolongerneedtoconsiderit.Ifadifferentcellistransferredtoitscrosspoint,thecellwouldbelongtoÕsinputthread,andwilldecreasebyone.Case2:,thenacellwillbetransferredfromoneofthecrosspointstooutputintheoutputschedulingphase.BydeÞnitionoftheGBVOQinsertionpolicythecellinthecrosspointhasanearlierdepartureorderthancell.Sincetheoutputschedulingpolicyselectsthenon-emptycrosspointthatcontainsthecellwiththeearliestdepartureorder,increasesbyincreasesbyatleastoneperscheduling ThecountingmethodusingtheGBVOQinsertionpolicycanbeappliedtriviallytoshowthatabufferedcrossbarcanmimicarestrictedPIFO-OQswitch,i.e.,aPIFO-OQswitchwiththerestrictionthatcellsfromaninput/outputpairdeparttheswitchintheordertheyarrive.Thisrestrictedpolicyincludesoutputlinkschedulerswhicharefairacrossallinputs,i.e.,providerateguaranteesbetweeneachinput/outputpair.Theorem2:(SufÞciency)AbufferedcrossbarwithaspeedupoftwocanmimictherestrictedPIFO-OQswitch,regardlessoftheincomingtrafÞcpattern.Proof:SeeAppendixB. V.RATEUARANTEESWITHAUFFEREDROSBAROurgoalistoÞndawayforabufferedcrossbartoprovideapre-determinedandguaranteedrateforeachßowpassingthroughtheswitch.InanOQswitch,thisisstraightforwardtodowith,forexample,aweightedroundrobin(WRR)scheduler.AWRRschedulerserveseachßowqueueinturninIfacellistransferredtothecrosspointthenitisavailableforselectionintheoutputschedulingphaseandcanbeplacedontheoutputlinewhenevernecessary.FCFSisanexampleofthisrestrictedpolicyandisextensivelyanalyzedin[16].round-robinorder,givingservicetoeachqueueinproportiontotheweightassignedtoit.Ifaqueueisempty,itisskippedandnotserved.ItiswellknownthatÑwhenpacketsareallofequallengthÑWRRgiveseachßowarateinproportiontoitsweight,andhencecangiveaminimumrateguaranteetoeachßow.Furthermore,ifthearrivalprocessesaresuitablyconstrained(e.g.byleakybuckets),thenthedelayofeachpacketthroughtheswitchcanbebounded[20].OneapproachtoprovingthatabufferedcrossbarcanproviderateguaranteeswouldbetoshowthatthebufferedcrossbarcanemulateaPIFO-OQswitch.Then,becauseWRRisaspecialcaseofPIFO,wecanconcludethatthebufferedcrossbarcansupportWRRandproviderateguaranteesjustasanOQswitchcan.ButasweÕllseeinthenextsection,itisharderforabufferedcrossbartoemulateaPIFO-OQswitch.Soinstead,weÕregoingtostartbysolvinganeasierproblembyconsideringarestrictedPIFO-OQswitch.Inwhatfollows,aßowisdeÞnedtobethosecellsbetweenaspeciÞcinput/outputpair.WeÕllconsideraWRR-OQswitchthatservestheseßowsinWRRorder;andtrytoemulatethesamebehaviorwithabufferedcrossbar.IntheWRR-OQswitch,eacharrivingcellisassignedavirtualÞnishingtimebytheWRRscheduler.ReviewingthewayaWRRschedulerworks,considerthecellsatjustoneoutput.ThecellfrominputisassignedvirtualÞnishing,where isthearrivaltimeofthecellfrominputtothatoutput,isthevirtualtime(roundnumber)attime,andtheweightassignedtoinput.Whentheoutputlineisfree,theWRRschedulerservesthecellwiththesmallestvirtualÞnishtime.Similarly,intheWRRbufferedcrossbar,avirtualÞnishingtimeneedstobeassignedtoeachcell,soastodeterminethecorrectdepartureorder.Theproblemisthatcellsarebufferedatbothinputsandoutputs(andinthecrossbar).CalculatingthevirtualÞnishingtimewhenthecellarriveswouldrequiretheinputtohaveinformationaboutthecellÕsoutput,andallthecellsatotherinputsdestinedtoit.Thisisimpractical.FortunatelywithourrestricteddeÞnitionofßows,cellsareheldintheinputprioritylistintheirdepartureorderandÑasweÕllshowbelowÑitissufÞcientfortheoutputtoassignavirtualÞnishtimeonlywhencellsreachthecrosspoint.Theoutputneedstoknowuponarrivalofthecelltotheswitchwhetherthecell(fromthisinputtothegivenoutput)hasdeparted.Ifithasdeparted,thenthecellistransferredtothecrosspointimmediatelyandisassignedthevirtualÞnishtimebasedonlyonthecurrentvirtualtime.Ifithasnotdeparted,thenthecellisnotbetransferredimmediatelyorthecellmustbeintheoutputqueue.Therefore,thecellisassignedthevirtualÞnishtimebasedonthevirtualÞnishtimeofthecell.Wewillformalizethesecasesintheproof. Theorem3:(SufÞciency)AbufferedcrossbarcanmimicanOQswitchusingaweightedroundrobinpolicywithspeeduptwo,regardlessoftheincomingtrafÞcpattern.Proof:SeeappendixC. AconsequenceoftheproofisthattheschedulingofcellstoemulateWRRserviceispractical,andcanbedoneinde-pendentlyandinparallelbyeachinputandoutput.Aninputindependentlygeneratesitsinputprioritylistwithonlylocalknowledge.Itpicksacelltoplaceinthecrosspointknowingonlywhichcrosspointsareempty.AnoutputcalculatesthevirtualÞnishtimeofacellwhenthecellarrivestothecrosspoint.Theoutputjustneedstoknowifthecellwastransferredtothecrosspointimmediatelyuponarrival,whichcanbecarriedinonebitofthecell.EachoutputpicksthecellinthecrosspointwiththesmallestvirtualÞnishtime.VI.DELAYUARANTEESINAUFFEREDROSSBARInpractice,aninput/outputpaircarriesmanyßows,notjustone.Forexample,itcarriesTCPßowsbetweensource/destinationpairs,andwemightwanttogiveeachßowadifferentrateordelayguarantee.Inordertodothis,weneedtorelaxourconstraintonthedeÞnitionoftheßow,anddeterminehowtoassignadifferentratetoeachßow.Thisiswhatwewilldonext;andweÕllseethatitincreasesthecomplexityofthebufferedcrossbarandrequiresmorespeedup.InaPIFO-OQswitch,anarrivingcellcanbepushedintoanylocationinthequeue.Itcould,forexample,bescheduledtodepartaheadofallcurrentlyqueuedcellsbetweenthesameinput/outputpair.Inordertomeettheconditionsofthecountingmethod,thecellinthecrosspointmusthavetheearliestdepartureorderofallcellsstoredintheinputqueuebelongingtoitsinput/outputpair.Thiscausesproblemsforthebufferedcrossbarswitch.Imaginethesituationinwhichacrosspointhasacellinit,andanarrivingcellhasandepartureorderthanthecellinthecrosspoint.ThiscauseswhatwecallÓcrosspointblockingÓsincethearrivingcellcannotovertakethecellinthecrosspoint.Ifeachcrosspointhadacellbufferforeachßow,crosspointblockingcouldbeavoided.However,thisdoesnÕtscaleforalargenumberofßows.Wenowshowhowabufferedcrossbarcanovercomecrosspointblockinginamannerwhichisindependentofthenumberofßowsbetweenaninput/outputpair.A.DelayguaranteeswithspeedupthreeWhenacellarrivestoaninputwithanearlierdepartureorderthanthecellinthecrosspointbuffer,wewillswapthecellinthecrosspointwiththenewlyarrivingcell.Logically,thecellthatwaspreviouslyinthecrosspointisrecalledtotheinputwhereitistreatedlikeanewlyarrivingcell.Bymodifyingthearrivalphasetoincludeswapping,crosspoint[16]showedabufferedcrossbarwithcellspercrosspointcouldmimicanOQswitchwithstrictpriorityßowsbetweeneachinput/outputpair.Notethatthecellwhicharrivedwithanearlierdepartureordermustbelongtoadifferentßowthanthecellinthecrosspoint.Asaconsequence,cellsfromthesameTCPßowwillneverbereordered. A2 B4 A3 A1 A2 B4 Fig.3.Theinsertionpolicyforachievingdelayguarantees.TheÞgureshowstheprioritylistforagiveninput.Theletterdenotestheoutputdestination,andthenumberdenotesthecellÕsdepartureorderforagivenoutput.(a)Arrivingisinsertedimmediatelyaftercell.(b)Arrivingcellisinsertedattheheadoftheprioritylistsincenoothercellhasadepartureorderlessthancelldestinedtooutputblockingcanbeavoided.Thisisattheexpenseofadditionalspeeduptoperformtheswap.ThemodiÞedarrivalphaserequiresanewinsertionpolicy.Thispolicyneedstomeettworequirements:(1)Topreventcrosspointblocking,cellsfromaninput/outputpairmustbeinsertedbasedontheirdepartureorder.(2)Theslacknessofacellmustbenon-negativewheninserted.TheInsertionPolicy:AsaconsequenceoftheÞrstrequire-ment,anarrivingcelldestinedtooutputportisinsertedbehindallcellsdestinedtooutputwithadepartureorderlessthancell.Tosatisfythesecondrequirement,cellbehindthecellthatdepartsbeforeit(ifitexists),destinedtothesameoutput.Ifnosuchcellexists,isinsertedattheheadoftheprioritylist.Thisensuresthattheslacknessofthecellisnon-negative.TheprioritylistdeÞnedbythisinsertionpolicyhasthepropertythatcellsfrominputtooutputareorderedbasedontheirPIFOdepartureorder.AnexampleisshowninFigureTheorem4:(SufÞciency)AbufferedcrossbarcanmimicaPIFO-OQswitch(andhencegivedelayguarantees)withspeedupthree,regardlessoftheincomingtrafÞcpattern.Proof:SeeAppendixD. B.DelayguaranteeswithspeeduptwoWeovercamecrosspointblockingbyswappingthecellinacrosspointwithanewlyarrivingcell.Thiswasnecessarybecauseweallowedcellstobetransferredtothebufferedcrossbarevenbeforetheywerescheduledtodepart.Thisearlytransferwasthecauseofcrosspointblocking,andthusrequiredswapping.Butwecouldeliminatetheneedforswappingifweavoidedtransferringacelltothecrosspointuntilitwasreallyreadytobetransferredtotheoutput.Forexample,wecouldputthecellheaderinthecrosspointbuffer,andonlytransferthecellacrossthecrossbarwhenchosenbytheoutput.SchedulingwouldnowbedoneintwoThisinsertionpolicyshouldbecontrastedwiththeCCFinsertionpolicyin[2],whichdoesnotmaintaincellsfrominputtooutputinthecorrectdepartureorder.CCFdoesnotneedtomaintainthisorderingbecausethestablemarriageproblemdescribedin[2]considersallcellsqueuedattheinputswhenscheduling. distinctphases.First,inputandoutputschedulingwouldbedonebasedonthecellheaders.Second,thecellbodieswouldbetransferredwhentheoutputchoosesacell.WecallthisheaderschedulingHowever,thiscreatesaproblem.Aninputcouldreceiveuptogrants(onefromeachoutput)inasingleoutputschedulingphase.Fortunately,overconsecutivephasesthenumberofgrantsreceivedbyaninputisboundedby.Thisisbecauseaninputcancommunicateatmostoneheaderperinputschedulingphase,andthereareatmostoutstandingheaders(oneforeachcrosspoint)perinput.Ontheotherhand,eachoutputgrantsatmostoneheaderperschedulingphase.Sothereareatmostgrantsforanoutputoveranyconsecutiveschedulingphases.Sinceinputssendatmostonecellperschedulingphaseandaninputcanreceiveuptograntsinconsecutiveschedulingphases,thecellforaheadergrantedinphasemightnotbetransferreduntilphase.Withonlyonecellpercrosspoint,acellcanpreventaninputfromsendinganothercelltothesameoutputforanotherphases.WethereforemodifythebufferedcrossbartohavecellsofbufferingperoutputasshowninFigure4.Therearebuffersinthecrossbar,butbuffersarededicatedtooutputs,ratherthaninput/outputpairs.Theorem5:(SufÞciency)Withspeeduptwoandcellsofbufferingperoutput,abufferedcrossbarcanmimicaPIFO-OQswitchwithaÞxeddelayoftimeslots.Proof:SeeappendixE. Theresultcomesattheexpenseofamorecomplicatedbufferingschemeinthecrossbarandrequirescellsbufferingperoutput.Sincethesecellscanarriveinthesameschedul-ingphase,thereisanadditionalimplementationcomplexity.Thiscanbeeliminatedbymodifyingthebufferedcrossbarsoithascellsforeach,foratotalofcells.Inthelattercase,nomorethanonecellcanarrivetoacrosspointineachschedulingphase.Whilerequiringmorestorage,itwillalsomimicaPIFO-OQswitchwithaÞxeddelayoftimeslotswithspeeduptwo.Thismightbepracticalforsmallvaluesof.InbothmodiÞedbufferedcrossbararchitectures,thenumberofcrosspointsisindependentofthenumberofßowsintheswitch.VII.CItishardtoscalecrossbar-basedroutersbecausetheschedulerforacrossbarmustresolvetheinputandoutputconstraintssimultaneously.Whereascentralizedschedulersgetverycomplicated,theschedulerforabufferedcrossbarallowsinputsandoutputstomakedecisionsindependentlyandinparallel.Withspeeduptwo,andschedulingalgorithmswhicharedistributedandeasytoimplement,bufferedcrossbarsprovidethroughput,rateanddelayguarantees.Althoughthecrossbarismorecomplexthanbefore,thebandwidthandpincountisthesameasbefore,theCIOQarchitectureismaintained,andnomemoryneedstorunfasterthantwicetheline-rate.Thisprovidesasimplepathtoscalecrossbarbasedrouters.[1]N.McKeown,A.Mekkittikul,V.Anantharam,andJ.Walrand,ÒAchieving100%ThroughputinanInput-QueuedSwitch,ÓIEEETransactionson,Vol.47,No.8,Aug.1999.[2]S-T.Chuang,A.Goel,N.McKeown,andB.Prabhakar,ÒMatchingOutputQueueingwithaCombinedInputOutputQueuedSwitchÓ,IEEEJ.Select.AreasCommun.,17,No.6,pp.1030Ð1039.AshortversionappearsinTheProceedingsofInfocomÕ99[3]S.Iyer,R.Zhang,andN.McKeown,ÒRouterswithaSingleStageofBufferingÓ,ACMSIGCOMMÕ02,Pittsburgh,USA,Sep.2002.[4]N.McKeown,ÒiSLIP:ASchedulingAlgorithmforInput-QueuedIEEETransactionsonNetworking,Vol.7,No.2,Apr.1999.[5]Y.Tamir,andH.C.Chi,ÒSymmetriccrossbararbitersforVLSIcommuni-cationswitches,ÓIEEETransactionsonParallelandDistributedSystemsVol.4,No.1,pp.13Ð27,1993.[6]J.Dai,andB.Prabhakar,ÒThethroughputofdataswitcheswithandwithoutspeedupÓ,IEEEINFOCOM2000,Vol.2,p.556Ð564,TelAviv,Mar.2000.[7]J.E.Hopcroft,R.M.Karp,ÓAnAlgorithmforMaximumMatchinginBipartiteGraphs,ÓSocietyforIndustrialandAppliedMath-ematicsJ.Computers,Vol.2,pp.225Ð231.[8]T.E.Anderson,S.S.Owicki,J.B.Saxe,andC.P.Thacker,ÒHighspeedswitchschedulingforlocalareanetworks,ÓACMTransactionsonCom-puterSystems,Vol.11,No.4,pp.319Ð352,Nov.1993.[9]K.YoshigoeandK.J.Christensen,ÒDesignandEvaluationofaParallel-PolledVirtualOutputQueuedSwitch,ÓProceedingsoftheIEEE2001InternationalConferenceonCommunications,pp.112Ð116,June2001.[10]Rojas-Cessa,E.Oki,Z.Jing,andH.J.Chao,ÒCIXB-1:CombinedInput-One-cell-CrosspointBufferedSwitch,ÓIEEEWorkshoponHighPerformanceSwitchingandRouting,Dallas,TX,July2001.[11]Rojas-Cessa,E.Oki,andH.J.Chao,ÒCIXOB-k:CombinedInput-Crosspoint-OutputBufferedPacketSwitch,ÓinIEEEGlobecom,SanAntonio,Texas,Nov.2001.[12]LotÞMhamdi,MounirHamdi,ÒMCBF:AHigh-PerformanceSchedul-ingAlgorithmforBufferedCrossbarSwitchesÓ,IEEECommunicationsLetters,2003.[13]D.Stephens,H.Zhang,ÒImplementingDistributedPacketFairQueueinginaScalableSwitchArchitectureÓ,inProc.INFOCOM[14]N.Chrysos,M.Katevenis,ÒWeightedFairnessinBufferedCrossbarScheduling,ÓProceedingsoftheIEEEWorkshoponHighPerformanceSwitchingandRouting,Torino,Italy,pp.17Ð22,June2003.[15]T.Javidi,RMagill,andT.Hrabik,ÒAHighThroughputSchedulingAlgorithmforaBufferedCrossbarSwitchFabricÓ,inProc.IEEEInternationalConferenceonCommunications,Vol.5,pp.1586Ð1591.[16]B.Magill,C.Rohrs,R.Stevenson,ÒOutput-QueuedSwitchEmulationbyFabricsWithLimitedMemoryÓ,inIEEEJournalonSelectedAreasinCommunications,pp.606Ð615,May2003.[17]S.Iyer,N.McKeown,ÒAnalysisoftheParallelPacketSwitchArchitec-tureÓ,inIEEE/ACMTransactionsonNetworking,Vol.11-2,pp.314Ð324,Apr.2003.[18]N.McKeown,C.Calamvokis,S.Chuang,ÒA2.5Tb/sLCSSwitchCoreÓ,HotChips,StanfordUniversity,Aug.2001.[19]A.Demers,S.Keshav,andS.Shenker,ÒAnalysisandsimulationofafairqueuingalgorithmÓ,ACMComputerCommunicationReview,pp.3Ð12,1989.[20]A.K.Parekh,andR.G.Gallager,ÒAgeneralizedprocessorsharingapproachtoßowcontrolinintegratedservicesnetworks:ThesinglenodeIEEE/ACMTransactiononNetworking,Vol.1,No.3,pp.344Ð357,June1993.[21]D.Gale,andL.S.Shapley,ÒCollegeAdmissionsandtheStabilityofMarriage,ÓAmericanMathematicalMonthly,Vol.69,pp.9Ð15,1962.[22]L.Tassiulas,andA.Ephremides,ÒStabilityPropertiesofConstrainedQueueingSystemsandSchedulingPoliciesforMaximumThroughputinMultihopRadioNetworks,ÓIEEETrans.onTutomaticControl,37,pp.1936Ð1949,Dec.1992.[23]H.J.Kushner,StochasticStabilityandControl,AcademicPress.1967.[24]G.Fayolle,ÒOnrandomwalksarisinginqueuingsystems:ergodicityandtransienceviaquadraticformsaslyapunovfunctions-PartIÓ,QueueingSystems,Vol.5,pp.167Ð184,1989.[25]E.Leonardi,M.Mellia,F.Neri,andM.A.Marsan,ÒOnthestabilityofinput-queuedswitcheswithspeed-up,ÓIEEE/ACMTransactionson,Vol.9,No.1,pp.104Ð118,February2001. Inputs R 2 3 Outputs R R2 R3 Modified Buffered Crossbar Output Crosspoints 3 3 3 Fig.4.ThearchitectureofamodiÞedbufferedcrossbarwiththreeports.THROUGHPUTFORANARBITRARYSCHEDULINGALGORITHMLemma2:Considerasystemofqueueswhoseevo-lutionisdescribedbyadiscretetimemarkovchain(DTMC)whichisaperiodicandirreduciblewithstatevectorSupposethatalowerbounded,non-negativefunctioncalledLyapunovfunction,existssuchthatthatF(Yn+1)|Yn].Supposealsothatthereexist,suchthat&#xTj /;ò 1;&#x Tf ;.45;r 0;&#x TD ;C,,F(Yn+1)F(Yn)|Yn],(2)thenallstatesoftheDTMCarepositiverecurrentandforevery,thereexistssuchthat.Proof:ThisisastraightforwardextensionofFosterÕscriteriaandfollowsfrom[22][23][24][25]. WewillusetheabovelemmainprovingTheorem1.Theorem8:Underanarbitraryschedulingalgorithm,thebufferedcrossbargives100%throughputwithspeedupoftwo.Proof:Intherestoftheproofwewillassumethatallindicesi,j,kvaryfrom,..N.DenotetheoccupancyVOQattime.Also,letdenotethecombinedoccupancyoftheVOQandthecrosspointattime.BydeÞnition,Observethatfrom(3)=1ifacelldepartsfromVOQattimeandzerootherwise.Also,letifacellarrivesVOQandzerootherwise.Then,+1)=.Henceforth,wewilldropthetimethesymbolfor,andrefertothemasrespectively,sinceintherestoftheproof,wewillonlybeconcernedwiththearrivalsanddeparturesofcellsatatf1(n+1))Xij(n+1)+1)Thenwegetgetf1(n+1)andsimilarly,we f1(n+1))AikDik]Xij(n)(6)DenoteEij(n)=1ifacelldepartsfromthecombinedqueueofVOQandthecrosspoint,andzerootherwise.Notethatonlywhenacelldepartsfromthetotheoutputattime,sincealldeparturestotheoutputmustoccurfromthecrosspoint.Alsorecallthatthearrivalratetothecombinedqueue,VOQisthesameasthearrivalratetoVOQ.Sowecanwrite+1)=.Againwewilldropthetimefromthesymbolfor,andrefertothemasrespectively.Then,similartothederivationin(6),wecanderiveusingsingf2(n+1))AkjDkj]Zij(n)(7)Sofrom(6)and(7),(7),F(n+1))AikDik]Xij(n)+E[AkjEkj]Zij(n)=2N3+2i,jXij(n)kE[AikDik]+Zij(n)kE[AkjEkj]Re-substitutingZij=Xij+Bij,wegetgetf(n+1))AikDik]+Xij(n)+Bij(n)kE[AkjEkj]=2N3+2i,jXij(n)kE[AikDik+AkjEkj]+Bij(n)kE[AkjEkj]WecansubstitutesubstituteAikDik+AkjEkj]andSj=kE[AkjEkj]andre-writethisas,as,F(n+1)But,wealsohavefromequation1,,Cij(n+1))kZkj(n+1)Thisisinfacttheconditionalexpectationgivenknowledgeofthestateofallqueuesandcrosspointsattime.Forsimplicityintherestoftheproof(sinceweonlyusetheconditionalexpectation),wewilldroptheconditionalexpectationsignandsimplyusethesymbolforexpectationasitsmeaningisclear.InSectionII,itwasshownthatforabufferedcrossbarwithspeedupoftwo,isstrictlynegativewhenthetrafÞcisadmissible.SotheÞrstproductterminsidethesummationsigninequation(8)Similarly,ifthetrafÞcisadmissible,thenthenAkj]1.Also,when,thenfrom(1)andcase1oftheorem1insectionII,weknowthattheoutputwillreceiveatleastonecellandsoatleastonecellmusthavedepartedoneofthecrosspointsdestinedtooutputattime.AndsowhenthetrafÞcisadmissibleand,then.Thisimpliesthatthesecondproductterminsidethesummationsigninequation(8),Inbothcases,areequaltozeroonlyifrespectively.NowwewanttouseLemma2andshowthatthewholerighthandsideofequation(8)isstrictlynegative.AllthatneedstobedoneistoensurethatoneoftheVOQsXinthesummationinequation(8)islargeenoughsothatcannegatethepositiveInordertoshowthis,let,,i,j,..N.Chooseany,andletijk(1+ (1+ correspondstotheconstantinLemma2.Recall.ThentheaboveinequalitycanonlybesatisÞedifthereexistssuchthat:(1+ AsshowninsectionII,whenTherefore,wehave(1+Ifwesubstitutethisinequation8,thenforallsuchthatthatF(n+1)correspondtothevariableinLemma2andset.Alsoitiseasytoseethat,that,F(n+1)FromLemma2,forevery,thereexistssuchthat.FromdeÞnition2,theschedulingalgorithmgives100%throughput. ROOFFORBeforeweprovethetheorem,wewillneedthefollowingLemma3:Theslacknessofacellwaitingontheinputsideisnon-decreasingfromtimeslottotimeslot.Proof:betheslacknessofcellwhichbelongsVOQatthebeginningofatimeslot.Duringthearrivalcanincreasebyatmostonebecauseanarrivingcellmightbeinsertedaheadofinitsinputprioritylist.Duringthedeparturephase,willdecreasebyatmostone.So,candecreasebyatmosttwoinasingletimeFromLemma1,increasesbyatleastoneperschedul-ingphase.Withtwoschedulingphasespertimeslot,increasesbyatleasttwo.Takingintoaccountarrivals,depar-turesandbothschedulingphases,cannotdecreasefromtimeslottotimeslot. Lemma4:Theslacknessofanewlyarrivingcellnon-negative.Proof:Consideranycellthatisinsertedwithaslacknessof.Followingthearrivalphase,byatleastoneineachofthetwoschedulingphases.Andinthedeparturephase,willdecreasebyone.Therefore,attheendofthetimeslot,increasesbyatleastone.Forexample,ifarrivingcell,isinsertedwithaslacknessofzero,thenattheendofthetimeslot,theslacknessofcellbeatleastone.FromLemma3andthefactthattheslacknessofanarrivingcellwillincreasebyoneattheendofthetimeslotrelativetotheslacknessofthecellwhenitarrived,weknowthatiftheslacknessofacellislessthanone,thenitsslacknessmusthavebeennegativewhenthecellwasinserted.LetbetheÞrsttimethatanarrivingcellisinsertedwithnegativeslackness.Considertwocases:Case1:Ifcellwasinsertedattheheadofthepriorityiszero.SincetheoutputcushionisdeÞnedasanon-negativevalue,theslacknessofthecellisnon-negativewheninserted,whichcontradictsourassump-Case2:Ifcellwasnotinsertedattheheadoftheprioritylist,cellmustbeinsertedimmediatelybehindanothercell,,destinedtothesameoutputascellwasinsertedbeforetime,itmusthavebeeninsertedwithnon-negativeslackness.Attheendofthetimeslotcellwasinserted,itsslacknessincreasedbyone.FromLemma3,theslacknessofcellisstillatleastoneattime.Butsince,and,then.Sotheslacknessofthecellmustalsobenon-negativewheninserted,whichagaincontradictsourassumption. Theorem2:(SufÞciency)AbufferedcrossbarwithaspeedupoftwocanmimictherestrictedPIFO-OQswitch,regardlessoftheincomingtrafÞcpattern.Proof:SupposethattheCIOQswitchhassuccessfullymimickedtheOQswitchupuntiltimeslot.Considerthebeginningoftimeslot.Wemustshowthatanycellreachingitsdeparturetimeiseither:(1)alreadyattheoutputsideoftheswitch,or(2)willbetransferredtotheoutputduringtimeFromLemma3andLemma4,weknowthatacellalwayshasanon-negativeslackness.Therefore,whenacellreachesitsdeparturetime(i.e.itsoutputcushionhasreachedzero),itsinputthreadmustalsoequalzero.Thismeanseither:(1)thatthecellisalreadyatitsoutput,andmaydepartontime,(2)thatthecellisinthecrosspointbufferor(3)thatthecellissimultaneouslyattheheadofitsinputprioritylist(becauseitsinputthreadiszero),andhastheearliestdeparturetime(becauseithasreacheditsdeparturetime).Incase(3),theinputschedulingphaseisguaranteedtotransferthecelltothecrosspoint.Sincethecellisinthecrosspointaftertheinputschedulingphaseinbothcases(2)and(3),andhastheearliestdeparturetime,itwillbeselectedintheoutputschedulingphase.Thecellwillthenreachtheoutputduringthetimeslot,andthereforethecelldepartsontime. ROOFFORInwhatfollows,considerthefollowingvirtualÞnishtimeassignmentpolicywhenacellarrivestothecrosspoint.AssumeacellwhicharrivestothecrosspointattimeWithoutlossofgeneralityletthisbethecellfrominputtooutputCase1:Ifthecellisstillpresentinoutputthebufferedcrossbar,thentheoutputofthebufferedcrossbarwillassignthevirtualÞnishtime, Case2:Ifthecellisnotpresentinoutputandthecellisnottransferredtothecrosspointintheschedulingphaseimmediatelyafteritsarrival,thentheoutputofthebufferedcrossbarwillassignavirtualÞnishtimeof Case3:Ifthecellisnotpresentintheoutputofthebufferedcrossbarandthecellistransferredtothecrosspointimmediatelyafteritsarrival,thentheoutputofthebufferedcrossbarwillassignthevirtualÞnishtime, WefurtherassumethatthebufferedcrossbarhasspeeduptwoandtheoutputpickscellswiththesmallestvirtualÞnishtimefromthenon-emptycrosspoints.Lemma5:ThevirtualÞnishtimeofeverycellisthesameintheWRR-bufferedcrossbarswitchandtheWRR-OQProof:AssumethatthebufferedcrossbarhascorrectlycalculatedthevirtualÞnishtimeofallcellswhichhavearrivedtothecrosspointsupuntiltimeandtheoutputshavechosenthecellsfromtheircrosspointswhichhavethesmallestÞnishtimeineveryschedulingphase.FromtheresultsinsectionIV,thismeansthatthebufferedcrossbarwithaspeeduptwo,hasmimickedtheWRR-OQswitchupuntil.LetbetheÞrsttimethatthevirtualÞnishingtimeofacellcalculatedisdifferentfromthevirtualÞnishingtimecalculatedbytheWRR-OQswitch.Considerthatcell whicharrivestothecrosspointattimewasincorrectlycalculated.Withoutlossofgeneralityletthisbethefrominputtooutput.Weconsiderthreecases.Case1:Ifthecellisstillpresentinoutputthebufferedcrossbar,thenthismeansthatitwasalsopresentintheWRR-OQswitchwhenthecellarrived.SoboththeWRR-OQswitchandtheoutputofthebufferedcrossbarwillassignthesamevirtualÞnishtime, contradictsourassumption.Case2:Ifthecellisnotpresentinoutputandthecellisnottransferredtothecrosspointintheschedulingphaseimmediatelyafteritsarrival,thenitmusthavebeeninsertedbehindthecellintheinputprioritylistoritwasinsertedtotheheadoftheinputprioritylist,butthecrosspointcontainedthecell.SincethebufferedcrossbarswitchhasmimickedtheWRR-OQswitchupuntil,thismeansthatthecell,wasalsopresentintheWRR-OQswitchattime.TheoutputofthebufferedcrossbarassignsavirtualÞnishtimeof matchesthevirtualÞnishtimeassignedbytheWRR-OQswitch.Theassignmentisthesame,whichcontradictsourCase3:Ifthecellisnotpresentintheoutputofthebufferedcrossbarandthecellistransferredtothecrosspointimmediatelyafteritsarrival,thensincethebufferedcrossbarswitchhasmimickedtheWRR-OQswitchupuntil,neitherswitchhascellsinthesystemfrominputdestinedtooutput.SoboththeWRR-OQswitchandtheoutputofthebufferedcrossbarwillassignthesamevirtualÞnishtime, whichagaincontradictsourassumption.SothevirtualÞnishtimeofacellattimecanalsobecorrectlycalculated. Theabovelemma,andthefactthattheWRR-OQswitchisaspecialcaseoftherestrictedPIFO-OQpolicyimplythefollowingtheorem.Theorem3:(SufÞciency)AbufferedcrossbarcanmimicanOQswitchusingaweightedroundrobinpolicywithspeeduptwo,regardlessoftheincomingtrafÞcpattern.ROOFFORBeforeweprovethetheorem,wewillneedthefollowingLemma6:AfterthemodiÞedarrivalphase,allcellsinthewillhaveearlierdepartureorderthananycellqueuedatinputdestinedforoutputProof:Assumethattheabovepropertyholdsupuntil.LetbetheÞrsttimethatanycellinthecrosspointdoesnothavetheearliestdepartureorderascomparedtoanycellqueuedatinputdestinedforoutput.Attimetherecanbeatmostonenewlyarrivingcelltoaninput.Ifthearrivingcellhasaearlierdepartureorderthanthecellinthecorrespondingcrosspoint,thenthemodiÞedarrivalphaseallowscelltoswapwiththecellinthecorrespondingcrosspoint,whichcontradictsourassumption. Lemma7:Theslacknessofacelldecreasesbyatleastoneineachschedulingphase.Proof:SinceLemma6guaranteesthatthecellinthecrosspointhastheearliestdepartureordercomparedwithanycellqueuedinthecorrespondinginputqueue,Lemma1still Lemma8:Theslacknessofacellwaitingontheinputsideisnon-decreasingfromtimeslottotimeslot.Proof:GivenLemma7theonlyotherdifferenceascomparedtoLemma3isinthemodiÞedarrivalphase.Irrespectiveofwhetheraswapoccurredornot,thereisonlyonenewlyarrivingcelltodealwithi.e.ifaswapdoesnotoccur,thenitisacellwhichjustarrivedattheinput,elseifaswapoccurs,thenthenewlyarrivingcellisthecellfromtheswappedcrosspoint.TherestoftheproofissimilartoLemma3. Lemma9:Theslacknessofanewlyarrivingcellnon-negative.Proof:AsdescribedinLemma8,weonlyneedtobeconcernedaboutinsertingonenewlyarrivingcelltotheprioritylistattheinputirrespectiveofwhetheraswapoccurredornot.TherestoftheproofissimilartoLemma4. Theorem4:AbufferedcrossbarcanmimicaPIFO-OQswitch(andhencegivedelayguarantees)withspeedupthree,regardlessoftheincomingtrafÞcpattern.Proof:GivenLemma8andLemma9,theproofisexactlythesameastheproofforTheorem2. ROOFFORProof:Aninputcanreceiveatmostgrantsoveranyconsecutiveschedulingphases.IftheinputaddsnewgrantstothetailofagrantFIFO,andreadsonegrantfromtheheadofthegrantFIFOineachschedulingphase,thenthegrantFIFOwillnevercontainmorethangrants.EachtimetheinputtakesagrantfromthegrantFIFO,itsendsthecorrespondingcelltothesetofcrosspointsforitsoutput.BecausethegrantFIFOisservedonceperphase,acellthatisgrantedatschedulingphasewillreachtheoutputcrosspointbyphaseWeneedtoverifythattheper-outputbuffersinthecrossbarneveroverßow.Ifthecrosspointschedulerissuesagrantat,thenthecorrespondingcellwillreachtheoutputcrosspointbetweenphases.Therefore,duringschedulingphase,theonlycellswhichcanbeintheoutputcrosspointarecellswhichweregrantedbetweenphases.Withbuffersperoutput,thebufferswillneveroverßow,andeachcellfacesadelayofatmostphases,ortimeslots(because PracticalAlgorithmsforPerformanceGuaranteesinBufferedCrossbarsShang-TseChuang,SundarIyer,NickMcKeownComputerSystemsLaboratory,StanfordUniversity,Stanford,CA94305-9030.stchuang,sundaes,nickmThispaperisabouthighcapacityswitchesandroutersthatgiveguaranteedthroughput,rateanddelayguaran-tees.Manyroutersarebuiltusinginputqueueingorcombinedinputandoutputqueueing(CIOQ),usingcrossbarswitchingfabrics.Butsuchroutersrequireimpracticallycomplexschedul-ingalgorithmstoprovidethedesiredguarantees.Weexplore