开放封闭网络

NetworkDesignPerformanceNetworkDesignPerformanceQueuingTheory1©GraduateUniversity,ChineseacademyofQueueingSystemQueuingTheoryQueueingSystemQueuingTheory2©GraduateUniversity,ChineseacademyofOpenNetworksofClosedOpenNetworksofClosedNetworksofQueuingTheory3©GraduateUniversity,Chineseacademyof1.•Sofar,weanalyzed1.•Sofar,weanalyzedsingleoccupancy,delay,Movetowardinterconnectedqueuingperformancenetworkoptimizationandvariousnetwork•QueuingTheory4©GraduateUniversity,Chineseacademyof•Interconnectedqueueswithjobs•InterconnectedqueueswithjobsflowonequeuetoHere,wewillstudynetworksofqueueswithexponentialserversandPoissonexternal•QueuingTheory5©GraduateUniversity,ChineseacademyofQueuingTheory6QueuingTheory6©GraduateUniversity,ChineseacademyofAnOpenNetworkisonewherejobsarrivefromoutsidetooneormorequeuesandeventuallyleaveAnOpenNetworkisonewherejobsarrivefromoutsidetooneormorequeuesandeventuallyleavethenetworkfromsomeofthequeues.Ifanopennetworkhasmultiplejobclassesthenitmustbeopenforeachclassofjobs.QueuingTheory7©GraduateUniversity,ChineseacademyofAnClosedNetworkisonewherethereareaconstantnumberofjobsthatcontinuallycirculateinthenetworkwithnootherarrivalstothesystemordeparturesfromthesystem.AnClosedNetworkisonewherethereareaconstantnumberofjobsthatcontinuallycirculateinthenetworkwithnootherarrivalstothesystemordeparturesfromthesystem.IfaclosednetworkhasmultiplejobclassesthenitmustbeclosedforeachclassofjobsQueuingTheory8©GraduateUniversity,ChineseacademyofQueuingTheory9QueuingTheory9©GraduateUniversity,Chineseacademyof2.OpenNetworksof2.OpenNetworksofQueuingTheory©GraduateUniversity,Chineseacademyofl:totalmeanarrivall:totalmeanarrivalratetothemi:meanservicerateofithrsj:probabilitythatacustomerarrivingfromthesourcewillberoutedtoqueuejrjd:probabilitythatacustomerdepartingfromqueuejwillberoutedtothedestinationrjk:probabilitythatacustomerdepartingfromqueuejwillberoutedtoqueuekQueuingTheory©GraduateUniversity,ChineseacademyofStateDescriptionoftheStateDescriptionoftheQueuingTheory©GraduateUniversity,ChineseacademyofAverage istheaveragethroughputthroughqueue–Average istheaveragethroughputthroughqueue–meanrateofentering/leavingtheTraffic•Mfromrjiqj,i=,,...=+jQueuingTheory©GraduateUniversity,Chineseacademyof1•Findtheavgthroughputthrough1•FindtheavgthroughputthrougheachQueuingTheory©GraduateUniversity,ChineseacademyofGlobalBalanceEquations••)betheGlobalBalanceEquations••)betheprobabilityofbeinginWhatistherateofleavingArrivaltoanyFromQueuingTheory©GraduateUniversity,ChineseacademyofTheithUnitTheithUnitQueuingTheory©GraduateUniversity,ChineseacademyofGlobalBalanceEquations•WhatisGlobalBalanceEquations•WhatistherateofenteringQueuingTheory©GraduateUniversity,ChineseacademyofGlobalBalanceEquationsQueuingGlobalBalanceEquationsQueuingTheory©GraduateUniversity,ChineseacademyofLocalBalance•Usethese,plusLocalBalance•Usethese,plusglobalbalance,plustrafficequationstogetthesolutionQueuingTheory©GraduateUniversity,ChineseacademyofProductFormMp(n)=piMp(n))1ProductFormMp(n)=piMp(n))1-n=iiiir=miiNetworkbehavesasifallqueueswerestatisticallyindependentQueuingTheory©GraduateUniversity,Chineseacademyof2Inthenetworkofqueuesfromthepreviousexercise,assumePoissonarrivalsatanaveragerateof1,000packets/secandaverageservicetimes2Inthenetworkofqueuesfromthepreviousexercise,assumePoissonarrivalsatanaveragerateof1,000packets/secandaverageservicetimesateachqueueof0.2msec.WhatistheprobabilityofnopacketsbeingintheQueuingTheory©GraduateUniversity,ChineseacademyofNumberofcustomersanddelayNumberofcustomersanddelaytheQueuingTheory©GraduateUniversity,Chineseacademyof3Forthenetworkinthe3ForthenetworkinthetwopreviouswhattheaveragenumberofpacketsinthetheaveragedelaythroughtheQueuingTheory©GraduateUniversity,Chineseacademyof3.ClosedNetworksof3.ClosedNetworksofQueuingTheory©GraduateUniversity,Chineseacademyof•Closednetworksofqueuescan•Closednetworksofqueuescanbeusedtomodeldifferenttypesofsystems–inreality,systemisNumberofcustomersinthesystematanytimeisafixedvalueN•QueuingTheory©GraduateUniversity,Chineseacademyof•FinitepopulationmodelsEachof•FinitepopulationmodelsEachofKuserscanhaveatmost1callactiveatatime(max#ofcallsisK)SystemsunderheavyMulti-stagepacketswitchesw/finite#ofpacketsallowedin,andanewpacketalwaysreadytoenterwhenoneleavesWindow-basedflow–Maximum#ofpacketsintransitatany••QueuingTheory©GraduateUniversity,ChineseacademyofTrafficM,i=,TrafficM,i=,,...=jfromQueuingTheory©GraduateUniversity,ChineseacademyofGlobalBalanceQueuingTheoryGlobalBalanceQueuingTheory©GraduateUniversity,ChineseacademyofProductFormSolutionfor•ProductFormSolutionfor•G(M,N)isanormalizationQueuingTheory©GraduateUniversity,ChineseacademyofFindingQueuingTheoryFindingQueuingTheory©GraduateUniversity,ChineseacademyofExample•SeefigureinslideExample•Seefigureinslide“ClosedNetworksofComputersystemallows2activejobsatanygivenEachjobrequiresCPU&I/OWhenjobleavesCPU,thereare2Jobfinishedandinstantlyreplacedbyanother(probabilityp)JobrequiresI/O,thenmoreCPU(prob.1-QueuingTheory©GraduateUniversity,ChineseacademyofExample4.1–Getting••Example4.1–Getting••WritedownthetrafficWhatstatesareQueuingTheory©GraduateUniversity,ChineseacademyofExample4.1–StateExample4.1–StateQueuingTheory©GraduateUniversity,Chineseacademyof•Utilizationristheproportionoftime•UtilizationristheproportionoftimethattheisbusyistheActualarrivalratetoqueue•Inexample4.1,r1=QueuingTheory©GraduateUniversity,ChineseacademyofExample4.2–A••UsetheExample4.2–A••Usethesameset-upasinExampleLet4jobs/sec,1job/sec,p=CalculatethestateCalculatetheutilizationateachCalculatetheaveragenumberofcustomersineachqueuingsystemCalculatetheaveragedelaythrougheachqueuingQueuingTheory©GraduateUniversity,Chineseacademyof均值分析方法均值分析方法QueuingTheory©GraduateUniversity,Chineseacademyof均值分析方法令表示具有I个客户的均值分析方法令表示具有I个客户的网络中，第j个队列中的平均•表示客户在队列j中的平均花费时间列j的平均客户到达速率客户在队列j中的平均花费时间为•QueuingTheory©GraduateUniversity,Chineseacademyof均值分析方法由Little公均值分析方法由Little公式QueuingTheory©GraduateUniversity,Chineseacademyof均值分析方法均值分析方法1.首先由Traffic2.对3.QueuingTheory©GraduateUniversity,Chineseacademyof•NewprogramsarriveataCPUaccordingtoaPoissonprocessofrateaasshowninFig.1•NewprogramsarriveataCPUaccordingtoaPoissonprocessofrateaasshowninFig.1.AprogramspendsanexponentiallydistributedexecutiontimeofinCPU.Attheendofthisservicetime,theprogramexecutioniscompletewithprobabilityporitrequiresretrievingadditionalinformationfromsecondarystoragewithprobability1-p.Supposethattheretrievalofinformationfromsecondarystoragerequiresanexponentiallydistributedamountoftimewith.FindthetimethateachprogramspendsintheQueuingTheory©GraduateUniversity,ChineseacademyofQueuingTheoryQueuingTheor