Koppelman Louisiana State University Baton Rouge LA 70803 Abstract An analysis of 731niteinputbuffered banyan networks offered traf731c having geo metrically distributed message lengths is presented This is one of the few multistagenetwork analyses ID: 22476
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AnAnalysisofBanyanNetworksOfferedTrafficWithGeometricallyDistributedMessageLengthsI-PyenLyawandDavidM.Koppelman*LouisianaStateUniversity,BatonRouge,LA70803Abstract:Ananalysisof®nite-input-bufferedbanyannetworksofferedtraf®chavinggeo-metricallydistributedmessagelengthsispresented.Thisisoneofthefewmultistage-networkanalysesfornetworksofferednon-unit-lengthmessagesandistheonlyonethattheauthorsareawareoffor®nite-input-bufferedbanyannetworks.Intheanalysis,networkswitchingelementsaremodeledusingtwostate-machines,oneforqueueheads(HOL's),theotherforentirequeues.Anetworkismodeledusingoneswitching-elementmodeltorepresenteachstage.Togetherthesemodeltheeffectthatnon-unit-lengthmessageshaveonbanyans.Solu-tionsareobtainediteratively.Networkperformance®gureswereobtainedwiththisanalysisandcomparedtosimulationresults.The®guresshowthattheanalysiscanpredicttheeffectofmessagelengthonthroughputanddelay,includingtheperformancedegradationcausedbylongermessages. ThisworkissupportedinpartbytheLouisianaBoardofRegentsthroughtheLouisianaEducationQualitySupportFund,contractnumberLEQSF(1993-95)-RD-A-07andbytheNationalScienceFoundationunderGrantNo.MIP-9410435. NTRODUCTIONBanyannetworks,unique-pathmultistageinterconnectionnetworks[1],arewidelycon-sideredforuseincommunicationandparallel-computingsystems.Performanceanalysesofsuchnetworksareneededbothforevaluationofsystemdesignsandforunderstandingthenetworksthemselves.Manybanyan-networkanalysismethodshavebeenreported;thebulkoftheworkwasfornetworksofferedunit-length(single-packet)messages.Forexample,Patelanalyzedunbufferedbanyans[8],Jenq,single-bufferedbanyans[3],Yoon,Lee,andLiu,®nite-bufferedbanyans[9],andMunandYoundescribeda®nite-bufferedbanyananalysiswhichworkswellatheavytraf-®cloads[7].(Alloftheseanalyseswereforinput-bufferednetworks.)Asisreadilyobservedfromsimulation,messagelengthhasastrongeffectonperformance.Sincenetworksusedforcommunicationswitchesandparallelcomputersmustcarrymessageshavingvaryinglengths,anon-uniform-message-lengthanalysisisneeded.SuchananalysishadbeenperformedbyKruskal,Snir,andWeiss[4]forin®nite-bufferednetworks.Thenetworkstheyanalyzedhaveoutput-bufferedswitchingelements(SE's)inwhichqueuescanbesimultaneouslyfedbyanynumberofSEinputs.Exact®rst-stageswitching-elementqueuestatedistributionswerefound.Anempiricallyderivedformulawasthenusedto®ndthewaitingtimeinsubsequentstages.Becausemessagesenteringaqueuearenotblocked,awidevarietyofoffered-traf®cmodelscanbeanalyzed,includingthosewithgeometricallydistributedmessagelengths.Theiranalysisofsuchtraf®cshowsthatdelayincreasesasaveragemessagelengthisincreased(whileholdingtraf®cintensityconstant)[5].Theiranalysis,however,isnotapplicabletomanyofthenetworksthatmightactuallybeused.Actualnetworksuse®nitequeuesandmayalsousecrossbarsthatblock.Thesecreateabackpressureeffect[6]whichresultsinadistributionofmessageswithinthenetworkverydifferentthanthenetworksKruskaletalanalyze.Themodelingofmessagedistributionisanimportantpartofanalysis,andsoadifferentmodelmustbeusedfor®nite-buffered,blocking-crossbarnetworks. Themodelusedherecapturesthefollowingbehaviorofnon-unit-lengthmessagesinthesenetworks.Amessage,whilepassingfromaqueueinoneSEtoanother,willbetheonlymessageusingitsSEoutput.Messagepacketsfollowingthe®rstpacketenterthenext-stageSEat,whatamountsto,anarrivalrateofone,tendingto®llthequeuethere.Thus,theprobabilitythatthe®rstpacketofamessage(theheadpacket)will®ndthenext-stagequeuefullislowerthanthecorrespondingprobabilityforotherpacketsinthemessage.Tocapturethisbehavior,eachstageismodeledbytwostatemachines.Oneforthequeueheadsinaswitchingelement,thehead-of-line(HOL)model,theotherforasingle(entire)queue,thequeuemodel.(CombinedHOL/queueSEmodelshavebeenusedearlier,forexamplebyHui[2]toanalyzenetworkswithunit-lengthmessages.TheHOLandqueuemodelsherearedifferent.)TheHOLmodelencodesthestateofthequeues'headslots(whethertheslotisemptyorhasaheadornon-headpacket,aswellasitsdestination).Itisusedto®ndthedistributionofthesestates,fromwhicharrivalandserviceratesforthequeuemodelsarecomputed.Thequeuemodelhastwosetsofstates:onesetisforqueuesintowhichamessageisentering(thatis,theheadpacketofthemessagehasenteredbutthelastpacket,the[tailpacket],hasnotyetentered),andonesetofstatesisforqueuesintowhichnomessageisentering.TransitionsfortheHOLmodelarebased,amongotherthings,ontheprobabilitythatamessageusingaswitching-elementoutputwillend.Transitionsarealsobasedontheprobabilitythattherewillbespaceinthenext-stagequeuegiventhetypeofpacket,headornon-head.Thesespaceprobabilitiesaredeterminedfromthequeuemodel.Thesemodelthebehaviordescribedabove.Theremainderofthispaperisorganizedasfollows.Inthenextsectionnetwork-structureandotherpreliminariesappear.TheanalysisisdescribedinSection3[p.4],theanalysisiscomparedwithsimulationinSection4[p.13].ConclusionsfollowinSection5[p.15].RELIMINARIES2.1NETWORKTRUCTUREAnalyzednetworkswillbespeci®edbya3-tuple,n;a;m.Suchanetworkconsistsof stages,numbered1,theinputstage,,theoutputstage.Eachstagecontainsswitchingelements(SE's).EachSEconsistsof-slotqueues,eachconnectedtothecrossbarinputs.LinksconnectSE'sinadjacentstagesand®rst-andlast-stageSE'stonetworkinputsandoutputs,respectively.Thelinkscanbeconnectedinanypatternforwhichthereisexactlyonepathbetweenallnetworkinput/outputpairs.See[1,2,6]fordetails.2.2MESSAGETRUCTUREANDLOWONTROLDataarrivesatnetworkinputsintheformofEachmessageconsistsofanumberof®xed-lengthpacketswhichpassthroughthenetworkasaunit.The®rstpacketiscalledheadpacketandthelastpacketiscalledthetailpacket;areusedtorefertoheadpacketsandtailpackets,respectively.Symbols refertopacketswhichfollowtheheadpacketandprecedethetailpacket,respectively.Switching-elementqueuesusea®rst-in,®rst-out,servicediscipline.Eachqueueslotholdsexactlyonepacket.Theslotthatholds,orwouldhold,thenextpackettoleaveiscalledthehead-of-line(HOL)slot.ApacketoccupyingtheHOLslotiscalledtheHOLpacket.Eachmessagehasadestination,thenetworkoutputtowhichitisbound.AHOLpacket'sneededlinkisthelink(connectedtotheSEoutput)onthepathtothepacket'sdestination.AHOLpacket'snext-stagequeueisthenext-stagequeuethatisonthepathtothepacket'sdestination.Timeisdividedintocycles.AHOLpacketwillmovetothenext-stagequeue(ornetworkoutput)duringacycleifthereisspaceanditwinscontentionfortheneededcrossbaroutput.Thereisalwaysspaceatanetworkoutput.Thereisspaceinaqueueifthereisatleastoneslotfreeduringthecycle.Anyqueuespacevacatedwillbeavailableatthenextcycle.Anon-headpacketwillalwayswincontention.Otherwise,foreachSEoutputlink,onewinnerwillberandomlychosenfromthoseHOLpacketsneedingthelink.2.3TRAFFICThestatisticsofmessagesarrivingatthenetworkinputsareindependentandidentically distributed.Duringacycleanetworkinputcaneitherbeidle,haveaheadpacketarriving,orhaveanon-headpacketarriving.Messagearrivaltimesandlengthsaredescribedbyathree-statediscrete-timeMarkovchain,withstateslabeled,(idle);,(start);and,(active).Letdenoteatransitionfromstateandletdenotethecorrespondingtransitionprobability.AninputnotreceivingapacketduringacycleismodeledbytransitionsAheadpacketarrivalismodeledbytransitions;amessagecontinuingismodeledbytransitions.Transitionprobabilitiesare,and.Thistraf®cmodelgeneratesmessageswithanexpectedlengthofpacketsanda¯owrateof+1)packetspercycle,where;Messagedestinationsarerandomlychosen(onceforeachmessage)anduniformlydistributedoveralloutputs.Asaconsequenceofthisdestinationdistributionandthebanyannetwork'sstructure,thelinkneededbyaheadpacketenteringaHOLslotwillbeuniformlydistributedovertheSEoutputs.NALYSIS3.1OVERVIEWAnetworkismodeledbyMarkov-chainpairs,eachofwhichcharacterizesastage.Onepairmember,calledthequeuemodel,characterizesaswitching-elementqueue.Theother,calledtheHOLmodel,characterizesaswitching-elementHOLsystem.AllqueueandHOLmodelsmakingupanetworkarestatisticallyindependentofeachother.Thestatedistributionsaresolvedbyiterativelycomputingstatedistributionsandtransitionprobabilities.TheHOLmodelisimportantbecausemessagesaretransmittedinseveralconsecutivepacketswhichcannotbeinterrupted.Ifthecorrelationamongpacketsbypriorcontention,capturedintheHOLmodel,isinsteadneglected,predictionswillbeinaccurate,especiallyforlargemessage HH);();H); H OutputsDestination of Non-Head PacketFigure1.HOLStateExample.3.2HOLMHOL-modelstatesarelabeled;:::;,where;:::;a; .PairindicatesthestateoftheHOLslotinSEqueue.Slotcontentsisindicatedbydenotesaheadpacket, denotesanon-headpacket,anddenotesanemptyqueue.Theswitching-elementoutputneededbytheHOLslotisindicatedby;:::;a; ,whereanintegerreferstoaswitching-elementoutputanddenotesanemptyqueue.AHOL-modelstateismadeuponlyofpairsairs(d;l;:::;a .AnexampleofaHOL-modelstateisillustratedinFigure1.ThenumberofHOLstatesislarge,evenforsystemswithsmallswitchingelements.Forpurposesofanalysis,thesetofHOLstatescanbepartitionedintoamuchsmallersetofequivalenceclassessuchthatonlyonememberofeachclassneedbeconsidered.Seetheappendixfordetails.Theprobabilitythatastage-HOLmodelisinstate;:::;isdenoted;:::;.Letbetwoequivalentstates.Statetransitionswillbede®nedsothat,aswillbeexplainedbelow.HOL-modeltransitionprobabilitieswillbespeci®edasaproductofHOLfactors.ForeachHOL-modeltransitionprobabilitythereareHOLfactors,oneforeachofthequeuesinaSE.LettheHOLfactorassociatedwithqueueinstageforatransitionfrombedenotedandlettheHOL-modeltransitionprobabilitybedenoted.Then,betheprobabilityofstage-HOL-modelstateattime.Thenthestateprobability isde®nedtobeforall,whereisthesetofallHOL-modelstates.ThevalueforaHOLfactorisdeterminedbythequeue'sroleinthetransition.Aqueueissaidtobeactiveinatransitionfromstateifitcontainsatleastonepacketincouldhavewonthecontentionorretainscontroloftheportinthetransitionto.(NotethatitisnotalwayspossibletodetermineifaqueuewinscontentioninatransitionbetweenHOL-modelstates.)De®netobetrueifqueueisactiveandfalseotherwise.Thenisgivenbythelogicalexpression;:::;;:::;)=(2);:::;i;:::;a;a(lx=Lx=H)^(dx=Dx)]where^istheconjunctionoperator,isthelogicalimplicationoperator,andistheuniversalquanti®er.(E.g.,expressionisequivalentto^^HOLfactorsforactivequeuesaredeterminedbytheprobabilityofspaceinthenextstage,theprobabilitiesthataHOLslotcontainsaheadpacket,non-headpacket,orisempty,andthenumberofqueuesneedingthesameport.Aqueuewhichisnotactiveiseitheremptyordidnotwincontention.TheHOLfactorfortheformercaseisbasedontheprobabilityofanarrivaltothequeue,theHOLfactorforthelatercaseis1.HOL-factorexpressionswillbegivenafterarrivalandspaceprobabilitiesareintroduced.tobetheprobabilitythataheadpacketintheHOLslotofanactivestage-queuewill®ndspaceinitsnext-stagequeue.Similarly,de®ne tobetheprobabilitythatanon-headpacketintheHOLslotofanactivestage-queuewill®ndspaceinitsnext-stagequeue.tobetheprobabilityofhead-packetarrivaltoanemptystage-Similarly,de®netobetheprobabilityofhead-packetarrivaltoastage-queueHOLslotgiventhattheslotheldatailpacketinthepreviouscycle. Thequeue-HOLfactorforatransitionfromstate;:::;tostate;:::;isgivenby;:::;;:::; C(HtvT;i=E,li=H,A(Ht; Hdi)vT;i=E,li= H,A(Ht;vE ;i=H,li=E;1;i=li=H,Di=di, ))+ 1 C(Ht;i=li=H,Di=di,A(Ht;Hdi) C(HtvT ;i=li=H,Di6di,A(Ht; Hdi)vT ;i=H,li= H,A(Ht;Hdi) C(Ht);i= H,li=H,Di=di,A(Ht;1� Hdi)i=li= isthenumberofqueueswithhead-slotpacketsboundforoutputport Hdi ,andspace-conditionalHOLfactorwillbeusedinthecomputationofarrivalrates.Thestage-space-conditionalHOLfactor,isgivenby(3)when Hdi Hdi6 (1)=1,and(1)=1.Thecorrespondingspace-conditionaltransitionprobabilityisgivenby3.3QQueue-modelstatesarelabeledx;y,and .Thesymboldenotesthenumberofpacketsinthequeue.Thesymbolifthelastoccupiedslotholdsthe tailpacketofamessage, ifthelastoccupiedslotholdsanon-tailpacketofamessage,and.Theprobabilitythatstage-queuewillbeinstatex;yisdenotedx;yQueue-modeltransitionprobabilitiesareafunctionofarrivalrate,servicerate,andexpectedmessagelength.De®netobetheservicerate,theprobabilitythatastage-packetisabletomoveforward.Leti;j,denotethearrivalrate,theprobabilityanewmessagewillbereadytomoveintoastage-queuegiventhatthequeueisinstate.Fourdistinctvaluesofi;jwillbecomputedperqueue:foranemptyqueue,forafullqueue,foraqueuewithoneslotfree,andforaqueuewith1topackets.Fornotationalsimplicity,i;j,willbeusedfor;thiswillbecalledthequeuearrivalThestationaryprobabilitiesmustsatisfythefollowingequations: T)sj+pj(I2;T)sjr2pj(I1; Tj(I0r0j(I1;T)sjr1j(I1; T)sj)pj(Ij(Ii�1;Tsj)ri�1+pj(Ii�1; (�sj)(1�ri;ji;j )sj+pj(IiT)sjripj(I j(Ii�1;Tsj)ri�1j(Ii�1; i;j )sj)pj(Im�1;Tj(Im�2;Tsj)rm�2+pj(Im�2; Tsj)+pj(Im�1;Tsjrm�1jrm�1j(Im�1; T)sj+pj(I)sjpj(Im�1; Tj(Im�2;Tsj)rm�2j(Im�2; Tsjj(Im�1;T)sjrm�1j(Im�1; T)sjj(I )sj pj(Ij(Im�1;Tsj)rm�1+pj(Im�1; Tsj)+pj(Isj)pj(I j(Im�1;Tsj)rm�1j(Im�1; Tsjj(I 3.4COMPUTATIONOFATESThequeue-andHOL-arrivalrates,servicerate,andspaceprobabilitiesareafunctionofHOL-andqueue-modelstateprobabilities.Theempty-queuearrivalrate,,iscomputedfromthestage-HOL-modelstatedistribution.Let bethesetofHOLstatesinwhichnoHOLpacketsaredestinedforaparticularport,withoutlossofgenerality,1.Thesetisgivenby ;:::;.Similarly,letbethesetofstatesinwhichatleastoneHOLpacketisdestinedfortheport,;:::;.Anemptystage-queueattimecoincideswithstage-HOLstate attime.Theempty-queuearrivalrateisthen PH2pj�1(H1)Tj�1(H1) PH .Forthe®rststage,thearrivalratefornewmessages.Thequantityiscomputedsothatthe¯owrateintoaHOLslotisthenetwork¯owrate,.Thistraf®cisdividedintothreecomponents.Flowentering:anemptyHOLslot,aHOLslotcontainingaheadpacket,andaHOLslotcontaininganon-headpacket.LetbethesetofstatesinwhichaparticularqueueHOLslot,say1,isempty,;:::;2Hg.LetbethesetofstatesinwhichtheHOLslothasaheadpacketthatisnotblockedbyamessageinprogress,;:::;.Let bethesetofstatesinwhichtheHOLslothasanon-head packet, H=fHjH=H1; ;:::;2Hg.Thenischosensothat ((H pj(H) holds.Solvingyields ()�PH pj(H) H)�PHpj(H)vE H0pj(H)H (+PH pj(H) Thestage-queue-modelservicerateisequivalenttothestage-HOL-modelservicerate.TheHOL-modelservicerateistheprobabilitythataHOLpacketwilladvance.Anon-headpacketwilladvanceifthereisspace;aheadpacketwilladvanceifthereisspaceanditwinscontention.Theservicerateisgivenby (PH pj(H) H Theempty-queuearrivalprobabilityisequivalenttothecorrespondingprobabilityusedintheHOLmodel,thatis,.Thenormalqueuearrivalprobability,,isfoundbyconsideringtheprevious-stageHOLsystem.Ifanon-emptyqueuehaslessthanitemsandthelastoccupiedslotholdsatailpacketthenanypacketthathadenteredthequeueinthepreviouscyclewasnotblockedandhadendedamessage.Thisfactisusedtoobtainaprevious-stageHOL-systemdistributionwhichisinturnusedto®ndthearrivalprobability.LetbethesetofstatesinwhichatleastoneHOLslothasapacketboundforaparticularqueue,say1,andnoHOLslothasanon-headpacketboundforthequeue.;:::;=1).Let bethesetofstatesinwhichnoHOLslothasanon-headpacketboundforthequeue, ;:::;=1).Usingthespace-conditionaltransitionprobabilities, PHtHt !1pj�1(Ht)T0j�1(Ht .Thearrivalprobabilityiscomputedsothatthe¯owrateintothequeueisx;j ) .Thefull-queuearrivalprobabilityisfoundsothatthefractionoftimetheprevious-stageHOLmodelhasaheadpacketboundforaparticularqueuematchesthecorrespondingquantityinthequeuemodel:x;j .Forthe®rststage,Stage-spaceprobabilities arecomputedsothe¯owrateleavingastage-HOLslotisequaltothe¯owrateenteringastage-+1)queue.Theprobabilitythattherewillbeaheadpacketreadytoenterastage-+1)queueisx;jtheprobabilitythatitissuccessfulisx;jThisyieldsthehead-packetspaceprobability, x;j.Similarreasoningisusedforthenon-headspaceprobability, HpjI ) PmxjI .Forthelaststage, The¯owrateisdeterminedbythearrivingtraf®candthefractionoftimethattraf®cisnotblocked: ): 3.5COMPUTATIONOFELAYofamessageisde®nedtobethenumberofcyclesthattheheadpacketisinthenetwork.Thenormalizeddelayisde®nedasthedelaydividedbythenumberofstages.Thewaitingtimeofamessageinaqueueisde®nedtobethenumberofcyclesthattheheadpacketspendsinthequeue.(Thisincludeswhatotherscallservicetime.)Thedelaythenisthesumofthewaitingtimes.Thetotaltimeamessagespendsinthenetworkisthedelayplusthemessagelengthminusone.Theexpectedwaitingtimeisfoundby®rstcomputing,for,thewaitingtime,i;j,foraheadpacketarrivingatastage-queuethathadpacketsinthepreviouscycle.Theexpectedwaitingtimeistheweightedsumover.Let PH2( betheserviceprobabilityofaHOLpacketthatisaheadpacket.Theexpectedwaitingtimeofaheadpacketatastage-queueHOLslotisthen.TheexpectedwaitingtimeofaHOLpacketofunknowntypeis.Ifthequeuehadpacketsinthecyclebeforeamessagearrivedtheni;j.Theexpectedwaitingtime,,isthengivenby i;j i;jThenormalizeddelayisthen3.6ANALYSISROCEDUREStationarydistributionsfortheHOLandqueuemodelsareobtainedthroughiteration,usingthefollowingprocedure.Stateprobabilitiesareinitializeduniformly.Thatis,ifastatemodelstatesthentheprobabilityofeachstateisinitializedto.Space,arrival,andserviceprobabilitiesareinitializedto.Otherinitializationsarepossible;allthosetestedyieldedthesameresults.Aftertheseconditerationnewvaluesforthestateprobabilities,serviceandarrival 2 4 6 8 10 12 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Average Message LengthNormalized Throughput : analysis (2 x 2 switching element) : simulation (2 x 2 switching element) : analysis (4 x 4 switching element) 1 3 5 7 9 11 2 4 6 8 10 12 0 2 4 6 8 10 12 14 16 18 Average Message LengthNormalized Delay : analysis (2 x 2 switching element) : simulation (2 x 2 switching element) : analysis (4 x 4 switching element) 1 3 5 7 9 11 Figure2.Throughputanddelayv.messagelengthfornetworks,rates,andspaceprobabilitiesarecomputedusingtheaverageofvaluescomputedintheprevioustwoiterations.(Atthe®rsttwoiterationsinitialvaluesareused.)Iterationproceedsuntilthedifferencebetweenvaluescomputedforquantitiesinconsecutiveiterationsissuf®cientlysmall.ESULTSTheanalysiswastestedbycomparingitspredictionsagainstthoseofasimulator.Comparisonsofpredictedthroughputanddelayweremadeforavarietyofnetworkandtraf®cmodels.Networksize,arrivalrate,queuesize,switching-elementsize,andmessagesizewerevaried.Thesimulatorusesthesamenetworkandtraf®cmodelastheanalysis.Simulationswereperformedfor100,000cycles;simulatoroutputincludesdelayandthroughput.Con®denceintervalswerecomputedassumingthatsimulatorthroughputanddelaycomputedforaseriesofidenticalrunsarenormallydistributed.Thecon®denceintervalsareextremelysmall.Theanalysiswasperformedforlessthan1000iterationsinmostcases.Thenumberofiterationswaschosensothatcorrespondingprobabilitiesdifferedbylessthaninthelasttwoiterationsof 2 4 6 8 10 12 0 2 4 6 8 10 12 14 16 18 Size of QueueNormalized Delay : analysis 3 5 7 9 11 0.4 0.6 0.8 1 0 2 4 6 8 10 12 14 16 18 Packet Arrival RateNormalized Delay : analysis 0.10.30.50.70.9 Figure3.(a)Delayv.arrivalrateinnetworks.(b)Delayv.queuesizeinnetworksfor.Bothformessagelength8.eachanalysis.Thepredictionofmessage-lengtheffectsonnetworksusingelementscanbeseeninFigure2.The®gureshowsthenormalizedthroughputanddelayfornetworksofferedsaturatingtraf®c,thatis,.Theeffectofmessagesizeisclearlymodeled.Aswithotheranalysesofthistype,thethroughputisoverestimated.Thedelayinsimulatedandanalyzedsystemscloselymatch.Thepredictionofdelayatvaryingarrivalratesforanaveragemessagelengthof8areplottedinFigure3(.Thedelaycomputedcloselymatchessimulations.Athigherarrivalratesthethroughputcomputed(notshown)istoohigh.AscanbeseeninFigure3(theeffectofqueuesizeonnetworkperformanceispredicted.Inthat®gure,delayisplottedagainstqueuesizefornetworks.Notethatthequeuesizerangesfromsmaller-thantolarger-thantheaveragemessagelength.Delayisaccuratelypredictedforsmallerqueuesizes,butdivergesforlargerqueues.Theeffectofnetworksizewasalsotested.Aswithunit-message-lengthanalyses,thethroughputpredictionisincreasinglyoverestimatedasthenumberofstagesincreases.Thedelayprediction,incontrast,remainsclosetothedelaysobtainedfromsimulation. Ageometricallydistributed-message-lengthbanyan-networkanalysishasbeenpre-sented.Thisisoneofthefewbanyannetworkanalysesthatconsideranythingotherthan®xed-lengthmessages.Thisisofvaluebecauseinrealparallelcomputersandcommunicationnetworksmessagesizesvary.Intheanalysis,thebanyan-networkswitchingelementsarecapturedbytwostatemodels:onemodelingasinglequeue,theothermodelingallofaswitching-element'squeueheads.Banyannetworkswith®nitequeuesizesandarbitraryswitching-elementsizecanbeanalyzed.Theanalysiswastestedagainstsimulations.Theresultsshowthatmessage-lengtheffectsareeffectivelymodeled.Inparticular,thenegativeimpactthatlongmessageshaveonnetworkperformanceispredicted.bepermutationsofswitching-elementinputandoutputlabels,respectively.Thenstates;:::;aresaidtobeequivalentifthereexistsaninput-labelpermutationandanoutput-labelpermutationsuchthat;:::;;:::;isthesymboltowhichismappedunderpermutationForalltransitionprobabilityisanyoftheswitching-elementmappingsdescribedabove.Inspectionofequations(1-3)willrevealthatpermutingswitching-elementlabelswillhavenoeffect.Forexample,considerpredicatein(3).Clearly,where.Otherreferencestoswitching-elementinputsandoutputsarealsoindependentofabsoluteorrelativepositionandsoareunaffectedbythepermutations.Adetailedproofisomitted. 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ListofFiguresFigure1.HOLStateExample.Figure2.Throughputanddelayv.messagelengthfornetworks,Figure3.(a)Delayv.arrivalrateinnetworks.(b)Delayv.queuesizeinnetworksfor.Bothformessagelength8. LooseEnds ABLEOF1Introduction......................2Preliminaries.....................2.1NetworkStructure..................2.2MessageStructureandFlowControl............2.3Traf®cModel....................3Analysis.......................3.1Overview.....................3.2HOLModel....................3.3QueueModel....................3.4ComputationofRates.................3.5ComputationofDelay.................3.6AnalysisProcedure..................4Results.......................5Conclusions......................6Appendix.......................7References......................