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Advances in Queueing Theory and Network Applications PDF

308 Pages·2009·7.941 MB·English
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Advances in Queueing Theory and Network Applications Wuyi Yue (cid:127) Yutaka Takahashi (cid:127) Hideaki Takagi Editors Advances in Queueing Theory and Network Applications 123 Editors WuyiYue HideakiTakagi DepartmentofIntelligencea ndI nformatics GraduateSchoolofSystems KonanUniversity andInformationEngineering Kobe UniversityofTsukuba 658-8501Japan Ibaraki [email protected] 305-8573Japan [email protected] YutakaTakahashi GraduateSchoolofInformatics KyotoUniversity Kyoto 606-8501Japan [email protected] ISBN:978-0-387-09702-2 e-ISBN:978-0-387-09703-9 DOI:10.1007/978-0-387-09703-9 LibraryofCongressControlNumber:2008939851 AMSCodes(2000):60K25,60K30,60K15,68M10,68M12,68M20,90B15,90B18,90B20,90B22 (cid:1)c SpringerScience+BusinessMedia,LLC2 009 Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY10013, USA),except forbrief excerpts inconnection with reviews orscholarly analysis. Usein connectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware, orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject toproprietaryrights. Printedonacid-freepaper springer.com Thebookwillproveusefultoacademicsand industrial scientistsaswellasengineers engagedinresearch.Itwillalsobenefitthose involvedinpost-graduate coursesin communications, computerscienceor engineering,whohaveinterestsinthe developmentandapplication ofqueueing theory. Thisbookmaybeusedaspre-requisite readingforothermoreadvancedcourseslike networkdesignandmanagementwhichare basedonperformancemodelingwith exampleapplications formodern communication andcomputernetworks.It couldalsobeusedasacoursebookon stochasticmodelsinmathematicsand operationsresearchdepartments. Contents Preface............................................................ ix PartI QueueingProcesses 1 TwoSidedDQBDProcessandSolutionstotheTailDecayRate ProblemandTheirApplicationstotheGeneralizedJoinShortest Queue ..................................................... 3 MasakiyoMiyazawa 2 Analytical Model of On-Demand Streaming Services Based onRenewalRewardTheory................................... 35 HiroshiToyoizumi PartII Single-ServerQueues 3 A Pure Decrement Service Geom/G/1 Queue with Multiple AdaptiveVacations .......................................... 49 ZhanyouMa,WuyiYue,andNaishuoTian 4 Performance Analysis of an M/M/1 Working Vacation Queue withSetupTimes............................................ 65 XiuliXuandNaishuoTian 5 ModelingofProductionSystemwithNonrenewalBatchInput, EarlySetup,andExtraJobs .................................. 77 HoWooLee,NoIkPark,SeWonLee,andJungWooBaek 6 PerformanceAnalysisofanM/E /1QueuewithBalkingandTwo k ServiceRatesBasedonaSingleVacationPolicy.................. 103 ChunyanLi,WuyiYue,andDequanYue PartIII MultipleQueues 7 MarkovianPollingSystems:FunctionalComputationforMean WaitingTimesanditsComputationalComplexity ................ 119 TetsujiHirayama vii viii Contents 8 PerformanceAnalysisofaTwo-StationMTO/MTSProduction System .................................................... 147 Kuo-HwaChangandYang-ShuLu PartIV Finite-BufferQueues 9 AnalysisofanM/M/c/NQueueingSystemwithBalking,Reneging, andSynchronousVacations ................................... 165 DequanYueandWuyiYue 10 Analysis of Mixed Loss-Delay M/M/m/K Queueing Systems withState-DependentArrivalRates ............................ 181 YoshinoriOzakiandHideakiTakagi 11 AsymptoticBehaviorofLossRateforFeedbackFiniteFluidQueue withDownwardJumps....................................... 195 YutakaSakumaandMasakiyoMiyazawa 12 ExplicitProbabilityDensityFunctionfortheLengthofaBusy PeriodinanM/M/1/KQueue ................................. 213 HideakiTakagiandAhmedM.K.Tarabia PartV NetworkApplications 13 PerformanceAnalysisofARQSchemesinSelf-SimilarTraffic...... 229 ShunfuJin,WuyiYue,andNaishuoTian 14 ModelingofP2PFileSharingwithaLevel-DependentQBDProcess. 247 SophieHautphenne,KenjiLeibnitz,andMarie-AngeRemiche 15 PerformanceAnalysisofaDecentralizedContentDeliverySystem withFECRecovery.......................................... 265 Kenji Kirihara, Hiroyuki Masuyama, Shoji Kasahara, and Yutaka Takahashi 16 BlockingProbabilitiesofMultipleClassesinIPNetworkswithQoS Routing.................................................... 281 Chia-HungWangandHsingLuh AbouttheEditorsandAuthors .................................... 303 Index .............................................................313 Preface A considerable amount of efforts needs to be devoted to performance modeling and analysis of emerging technologies and their applications in order to develop proper design and operation management of future multi-service networks where application-dependent Quality of Service (QoS) is ensured. To this end, extensive interdisciplinaryresearchinperformanceanalysisandsystemmanagementofcom- municationnetworksisessential. Included in this book are 16 chapters of high quality. All the manuscripts were selected, after peer-review, from among those submitted by prominent researchers working on queueing theory and network applications. The reviewers’ reports not only helped the editors qualify the articles for inclusion in the book, but also im- provedthequalityofthechapters. The chapters cover a wide variety of recent topics on queueing theory and network applications. These include single-server queues, finite-buffer queues, re- trial/balkingqueues,multiplequeuesaswellasoptimizationinqueues.Theyfurther presentnewtheoreticalresultsontimelytopicsrelatedtoprotocols,applicationser- vicesandroutingalgorithmsintheInternetandwireless-relatedissues. Webelievethatallofthesechaptersnotonlyprovidenovelideas,newanalytical models,simulationandexperimentalresultsinthisfieldbutalsoenhancethefuture researchactivitiesintheareaofQueueingTheoryandNetworkApplications. This book consists of five parts, which cover topics in Queueing Processes, Single-Server Queues, Multiple Queues, Finite-Buffer Queues, and Network Ap- plications.Abriefsummaryofeachchapterislistedasfollows[Chapters1–16,this book]. PartI:QueueingProcesses PartIdiscussesQueueingProcessesintwochapters,Chapters1and2. Chapter 1considersatwo-sideddoublyquasi-birth-and-deathprocess.Undera discrete time setting, this is a two dimensional skip free random walk on the half spacewhosesecondcomponentisnon-negativeintegervaluedandwhosefirstcom- ponentmaytakepositiveornegativeintegers.Themajorinterestofthischapteris ix x Preface in the tail decay rate of stationary distribution as one of the components goes ei- thertoinfinityortominusinfinity,providedthestationarydistributionexists.Two kindsofdecayrates,called’weak’and’exact’forthedoublyQBD(orDQBD)and characterizedinthepreviouspublicationbytheauthorareextendedtothetwosided DQBD,andareappliedtothegeneralizedshortestqueue.Thischaptershowsthat a weak decay rate, that is, the decay rate in the logarithmic sense, is completely specifiedintermsoftheprimitivedataforthegeneralizedshortestqueue. Chapter2proposesananalyticalmodelbasedonrenewalrewardtheorytoinves- tigatethedynamicsofon-demandstreamingservice,derivingtheaveragedownload rate. This chapter uses a simple method, combining multicast method and unicast methodthatcanreducethedownloadratefromthestreamingservereffectively.By modeling requests as Poisson arrivals, the dynamics of this streaming service are studied and the optimal sharing of unicast and multicast methods is derived. This chapteralsoshowshowtoestimatethefluctuationofdownloadratesofastreaming service. PartII:Single-ServerQueues PartIIonSingle-ServerQueuesincludesfourchapters,Chapters3–6. In Chapter 3, a Geom/G/1 queue with a pure decrementing service policy and multiple adaptive vacations is analyzed. The Probability Generating Func- tion (P.G.F.) of the queue length is obtained by using an embedded Markov chain method.TheP.G.F.ofthewaitingtimeisthenderivedandtheprobabilitiesforthe systembeinginvariousstatessuchasabusystate,anidlestateoravacationstate, are also derived. Finally, some special cases for the queueing model are given to demonstratethegeneralpropertiesofthequeuemodels. Chapter4investigatesanM/M/1workingvacationqueuewithsetuptimes,using a quasi birth and death process and a matrix-geometric solution method to derive thedistributionsforthestationaryqueuelengthandthewaitingtimeofacustomer inthesystem.Alsopresentedinthischapterarestochasticdecompositionstructures ofstationaryIndices. InChapter5,asingle-machineproductionsystemwithearlysetupandextrajob operationsisconsidered.Itiscontrolledbytwothresholds.Thefirstisusedtocon- trolthesetupstartingtimeandthesecondisusedtocontroltheproductionstarting time.ThesystemismodeledbytheBMAP/G/1queueandthemanufacturinglead time is analyzed. The factorization principle is used to derive its distribution and meanvalue. Chapter 6 presents an analysis of a state-dependent M/E /1 queue with balking k andsinglevacations.Customersareservedattwodifferentratesdependingonthe numberofcustomersinthesystem.Ifcustomersonarrivalfindanyothercustomers in the system, they decide to either enter the queue or balk with a constant proba- bility.Theservertakesasinglevacation whenthesystembecomes empty.First,a quasi-birthanddeathprocessisformulated.Then,theequilibriumconditionofthe systemisobtained.Byusingthematrixgeometricsolutionmethod,thesteady-state probability vectors are obtained. The computation of the steady-state probability Preface xi vectorsisalsodiscussed.Then,someperformancemeasuresarederivedexplicitly. Based on these performance analyses, a mathematical model is developed to opti- mizethecostofthesystem.Finally,somediscussiononthesensitivityofthemodel isgiventhroughnumericalexperiments. PartIII:MultipleQueues PartIIIdiscussesMultipleQueuesintwochapters,Chapters7and8. Chapter 7 considers Markovian polling systems in which a single server serves J stationswithPoissonarrivalsandgeneralservicetimes.Afteraserviceperiodat stationi,theserverselectsstation j withprobability p andvisitsthestationafter ij spendingaswitchovertime.Thischapterusesthefunctionalmethodthathasbeen proposedinapreviousresearchonmulticlassM/G/1typesystems.Theadvantages of the functional method are (1) its wide applicability to the analysis of M/G/1 typemulti-classqueues,and(2)itsrathersmallcomputationalcomplexitycompared withthebufferoccupancymethod. Chapter 8 considers a two-station hybrid system which handles make-to-order (MTO) and make-to-stock products (MTS). The first station represents an MTS systemproducingstandardproductsforordinarydemands,whichalsocanbesemi- finishedproductsforspecificdemandsprocessedinthesecondstation.Thesecond stationperformssomeadditionalworksonthestandardproductsforthespecificde- mands.Inthesystemconsideredinthischapter,theMTSsystemiscontrolledunder thebase-stockpolicy.Toevaluatetheperformanceofthesystem,thefillrateofthe ordinary demands and the response time of the specific demands are considered. The objective is to study the relation between base-stock level and the fill rate of theordinarydemandsandtheresponsetimeofthespecificdemands.Thesystemis analyzedbymodelingitasaninventory-queuemodel.Basedontheseanalyses,one candeterminetheoptimalbase-stocklevelnumericallyundertheconstraintsonthe fillrateoftheordinarydemandsandtheresponsetimeofthespecificdemands. PartIV:Finite-BufferQueues PartIVonFinite-BufferQueuesincludesfourchapters,Chapters9-12. Chapter 9 presents an analysis of an M/M/c/N queueing system with balk- ing, reneging and synchronous vacations of servers. By using the blocked matrix method,thesteady-stateprobabilityvectorisobtainedintermsoftheinverseoftwo matrices, whose computation is discussed. Then, the steady-state probabilities are calculatedbyusingtheelementsoftheinverseofthetwomatrices.Theconditional stationarydistributionofthequeuelengthandwaitingtimeisalsoderived. AnM/M/mqueuewithmixedlossanddelaycallswasanalyzedbyJ.W.Cohen halfacenturyago(1956),wherethetwotypesofcallshadidenticalconstantarrival andservicerates.ItisstraightforwardtoextendhisanalysistoanM/M/m/Kqueue. InChapter10,themodelisfurthergeneralizedsuchthatthecallarrivalratesdepend onthenumberofcallspresentinthesystemuponarrival.Thismodelincludesthe balkingandthefinitepopulationsizemodelsasspecialcases.Amethodispresented xii Preface tocalculatetheblockingprobabilityforlostcallsaswellasthedistributionofthe waitingtimeforaccepteddelayedcalls. Chapter 11 considers a feedback finite fluid queue (FFFQ, for short) with downward jumps, where the fluid flow rate and the jump size are controlled by a background Markov chain with a finite state space. The feedback means that the backgroundprocessmaychangeaccordingtothelevelofthebuffer,whichisused formodelingTCP/IPflow.Thematrixanalytictechniquehasbeensuccessfullyap- plied to an FFFQ without jumps. This chapter incorporates downward jumps into thisFFFQ,andshowsthatitslossprobabilitydecaysexponentiallyasthebuffersize getslargeunderanegativedriftcondition. In Chapter 12, a closed-form explicit expression is derived for the probability density function of the length of a busy period starting with i customers in an M/M/1/K queue, where K is the capacity of the system. The density function is givenasaweightedsumofK exponentialdistributionswithcoefficientscalculated fromK distinctzerosofapolynomialthatinvolvesChebyshevpolynomialsofthe secondkind.Themeanandsecondmomentofthebusyperiodarealsoshownex- plicitly. PartV:NetworkApplications PartVincludes4chapters,Chapters13–16onnetworkapplications. Chapter13presentsamethodtoanalyzetheperformanceofAutomaticRepeat reQuest (ARQ) schemes in self-similar traffic. A batch arrival queueing model is builtbytakingintoaccounttheself-similarnatureofamassive-scalewirelessmul- timediaserviceandbysupposingthatthebatchsizeisarandomvariablefollowing aParetodistribution.Asetupstrategyinthemodelisbuiltbyconsideringthedelay inthesettingupprocedureofadatalink.ThusabatcharrivalGeomX/G/1queueing systemwithsetupisbuiltinthischapter.Adiscrete-timeimbedded Markovchain isusedtoanalyzethestationarydistributionofthequeueingsystemandderivethe PGFsofthequeuelengthandthewaitingtimeofthesystem.Performancemeasures aregivenintermsoftheresponsetimeofdataframes,setupratiosandofferedloads fordifferentARQschemes.Numericalresultsaregiventoevaluatetheperformance ofthesystemandtoshowtheinfluenceoftheself-similardegreeandthedelayof thesetupprocedureontheseperformancemeasures Chapter14analyzesapeer-to-peer(P2P)filesharingsystembymeansofaso- calledlevel-dependentQuasi-Birth-and-Death(QBD)process.Thedisseminationof asinglefileconsistingofdifferentsegmentsisconsidered,andamodelisproposed fortheuploadqueuemanagementmechanismwithpeerscompetingforbandwidth. Byapplyinganefficientmatrix-analyticalgorithm,theperformanceofP2Pfiledif- fusioncanbeevaluatedintermsofthecorrespondingextinctionprobability,i.e.,the probabilitythatthesharingprocessends. Chapter 15 considers the performance of a decentralized content delivery sys- tem where video data is simultaneously delivered without duplication by multiple streamingvideoservers,resultinginalowsendingratepervideoserver.Byfocusing onamultiple-servervideostreamingservicereinforcedbyforwarderrorcorrection

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