Queueing Theory, Volume 2: Advanced Trends

318 Pages·2021·5.087 MB·English

by  Vladimir Anisimov, Nikolaos Limnios

Queueing Theory 2
SCIENCES
Mathematics, Field Director – Nikolaos Limnios
Queuing Theory and Applications, Subject Head – Vladimir Anisimov
Queueing Theory 2
Advanced Trends
Coordinated by
Vladimir Anisimov
Nikolaos Limnios
First published 2020 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
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ISBN 978-1-78945-004-0
ERC code:
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PE1_21 Application of mathematics in industry and society
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Chapter 1. Stability Analysis of Queueing Systems based on
SynchronizationoftheInputandMajorizingOutputFlows . . . . . . 1
LarisaAFANASEVA
1.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2.Modeldescription . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.Auxiliaryserviceprocess . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4.Instabilityresultforthecaseρ≥1 . . . . . . . . . . . . . . . . . . . . . 9
1.5.Stochasticboundednessforthecaseρ<1 . . . . . . . . . . . . . . . . 10
1.6.Queueingsystemwithunreliableserversandpreemptiveresume
servicediscipline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.7.Discrete-timequeueingsystemwithinterruptionsandpreemptive
repeatdifferentservicediscipline . . . . . . . . . . . . . . . . . . . . . . . . 15
1.8.Queueingsystemwithapreemptiveprioritydiscipline . . . . . . . . . . 18
1.9.Queueingsystemwithsimultaneousserviceofacustomerbyarandom
numberofservers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.10.Applicationstotransportsystemsanalysis . . . . . . . . . . . . . . . . 23
1.11.Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.12.Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
1.13.References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
vi QueueingTheory2
Chapter2.QueueingModelsinServices–Analyticaland
SimulationApproach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
SrinivasR.CHAKRAVARTHY
2.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.2.Phase-typedistributionsandthebatchMarkovianarrivalprocess . . . . 34
2.2.1.Phase-typedistributions . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2.2.SomeusefulresultsrelatedtocontinuousPH distributions . . . . 36
2.2.3.ThebatchMarkovianarrivalprocess . . . . . . . . . . . . . . . . . 40
2.3.GenerationofMAP processesfornumericalpurposes . . . . . . . . . 42
2.4.AnalysisofselectedqueueingmodelsofBMAP/G/ctype . . . . . . 44
2.4.1.MAP/PH/1queueingmodel . . . . . . . . . . . . . . . . . . . . 44
2.4.2.Thesystemperformancemeasures . . . . . . . . . . . . . . . . . . 48
2.4.3.IllustrativenumericalexamplesforMAP/PH/1 . . . . . . . . . 49
2.4.4.MAP/M/cqueueingmodel . . . . . . . . . . . . . . . . . . . . . 55
2.4.5.Thesystemperformancemeasures . . . . . . . . . . . . . . . . . . 57
2.4.6.IllustrativenumericalexamplesforMAP/M/c . . . . . . . . . . 57
2.5.SimulatedmodelsofBMAP/G/ctypequeues . . . . . . . . . . . . . 58
2.5.1.SimulatedmodelvalidationusingMAP/M/ctypequeues . . . . 59
2.5.2.SimulatedmodelvalidationusingMAP/PH/1type
queues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.5.3.SelectedsimulatedmodelsofBMAP/G/ctypequeues . . . . . 59
2.6.AnalysisofselectedqueueingmodelsofBMAP/G/ctypewith
avacation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.6.1.MAP/PH/1queueingmodelwithavacation . . . . . . . . . . . 66
2.6.2.Thesystemperformancemeasures . . . . . . . . . . . . . . . . . . 69
2.6.3.IllustrativenumericalexamplesforMAP/PH/1with
avacation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
2.6.4.Validationofthesimulatedmodelforvacationtypequeues . . . . 74
2.6.5.SelectedsimulatedmodelsofBMAP/G/ctypequeueswith
avacation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.7.Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
2.8.References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Chapter3.DistributionsandRandomProcessesRelatedto
QueueingandReliabilityModels . . . . . . . . . . . . . . . . . . . . . . . 81
BoyanDIMITROV
3.1.Someusefulnotations,relationshipsandinterpretations . . . . . . . . . 81
3.2.Unreliableservicemodelandreliabilitymaintenance . . . . . . . . . . 85
3.3.Characterizationsofexponentialandgeometricdistributionsvia
propertiesofservicetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.3.1.Instantrepairs: characterizationofgeometricdistribution . . . . . 89
Contents vii
3.3.2.Instantrepairs: characterizationsoftheexponential
distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.3.3.Varioussimplifyingconditions . . . . . . . . . . . . . . . . . . . . 101
3.3.4.Unreliableservice,repairtimesincluded . . . . . . . . . . . . . . 111
3.4.Probabilitydistributionsalmosthavinglackofmemoryproperty . . . . 115
3.4.1.Servicetimeonanunreliableserver: instantaneousrepairs . . . . 116
3.4.2.PropertiesofALMdistributions,andequivalentpresentations . . 119
3.4.3.Periodicityinnaturalphenomena . . . . . . . . . . . . . . . . . . . 126
3.5.Randomprocesseswithaperiodicnature . . . . . . . . . . . . . . . . . 126
3.5.1.Countingprocesses. . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.5.2.CharacterizationofanNPP . . . . . . . . . . . . . . . . . . . . . . 128
3.5.3.Applicationsinriskmodeling . . . . . . . . . . . . . . . . . . . . . 131
3.6.Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.7.References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
Chapter4.TheImpactofInformationStructureonStrategic
BehaviorinQueueingSystems . . . . . . . . . . . . . . . . . . . . . . . . 137
AntonisECONOMOU
4.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4.2.Game-theoreticalframeworkinqueueing . . . . . . . . . . . . . . . . . 139
4.3.Theunobservablemodel . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
4.4.Theobservablemodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
4.5.Comparisonoftheunobservableandtheobservablemodels. . . . . . . 151
4.6.Partiallyobservablemodels . . . . . . . . . . . . . . . . . . . . . . . . . 153
4.7.Heterogeneouslyobservablemodels . . . . . . . . . . . . . . . . . . . . 158
4.8.Observable-with-delaymodels . . . . . . . . . . . . . . . . . . . . . . . 162
4.9.Conclusionsandliteraturereviewforfurtherstudy . . . . . . . . . . . . 167
4.10.Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
4.11.References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
Chapter5.Non-extensiveMaximumEntropyFormalismsand
InductiveInferenceofaStableM/G/1QueuewithHeavyTails . . . . 171
DemetresD.KOUVATSOSandIsmailA.MAGEED
5.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.2.GeneralsystemsandinductiveMEformalisms . . . . . . . . . . . . . . 175
5.2.1.“Classical”Shannon’sEMEformalismwithshort-range
interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.2.2.Rényi’sandTsallis’sNMEformalismswithlong-range
interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
5.3.NMEformalismsandEMEconsistencyaxioms . . . . . . . . . . . . . 177
5.4.AstableM/G/1queuewithlong-rangeinteractions. . . . . . . . . . . . 179
5.4.1.Background: Shannon’sEMEstateprobabilityofastableM/G/1
queue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
viii QueueingTheory2
5.4.2.Tsallis’andRényi’sNMEstateprobabilitiesofastableM/G/1
queue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
5.4.3.ExactRényi’sandTsallis’NMEstateprobabilitieswithdistinct
GEq-typeservicetimedistributions . . . . . . . . . . . . . . . . . . . . . 183
5.5.Numericalexperimentsandinterpretations . . . . . . . . . . . . . . . . 188
5.6.Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
5.7.Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
5.8.Appendix: Rényi’sNMEformalismsversusEMEconsistency
axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
5.8.1.Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
5.8.2.Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
5.8.3.Systemindependence . . . . . . . . . . . . . . . . . . . . . . . . . 197
5.8.4.Subsetindependence . . . . . . . . . . . . . . . . . . . . . . . . . . 198
5.9.References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Chapter6.InventorywithPositiveServiceTime: aSurvey . . . . . . 201
AchyuthaKRISHNAMOORTHY,DhanyaSHAJINandViswanathC.NARAYANAN
6.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
6.2.Queueinginventorymodels . . . . . . . . . . . . . . . . . . . . . . . . . 203
6.2.1.Single-commodityqueueing-inventorysystems. . . . . . . . . . . 206
6.2.2.Productioninventorysystems . . . . . . . . . . . . . . . . . . . . . 215
6.2.3.Multicommodityqueueing-inventorysystem . . . . . . . . . . . . 217
6.2.4.Retrialqueueswithinventory . . . . . . . . . . . . . . . . . . . . . 219
6.2.5.Queuesrequiringadditionalitemsforservice . . . . . . . . . . . . 222
6.2.6.Queueing-inventory: someworkinprogressandsuggestionsfor
futurestudies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
6.3.Acknowledgment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
6.4.References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Chapter7.AStabilityAnalysisMethodofRegenerativeQueueing
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
EvseyMOROZOVandBartSTEYAERT
7.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
7.2.Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
7.3.Thesingle-serversystem. . . . . . . . . . . . . . . . . . . . . . . . . . . 244
7.4.Thezero-delayedmultiserversystem . . . . . . . . . . . . . . . . . . . . 248
7.5.Thedelayedmultiserversystem: finitenessofthefirstregeneration
period . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
7.6.Instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
7.6.1.Somecommentsonthemethod . . . . . . . . . . . . . . . . . . . . 260
7.7.Relatedresearch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
7.8.Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
7.9.References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266
Contents ix
Chapter8.TransientAnalysisofMarkovianQueueingSystems:
aSurveywithFocusonClosed-formsandUniformization . . . . . . 269
GerardoRUBINO
8.1.Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
8.2.BasicsonMarkovianqueues . . . . . . . . . . . . . . . . . . . . . . . . 272
8.2.1.Markovmodels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
8.2.2.Uniformization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
8.3.Firstexamples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
8.3.1.TheEhrenfestmodelincontinuous-time. . . . . . . . . . . . . . . 275
8.3.2.TheM/M/∞model . . . . . . . . . . . . . . . . . . . . . . . . . 276
8.3.3.Aqueuewithnoserverandcatastrophes . . . . . . . . . . . . . . 277
8.3.4.ThefundamentalM/M/1model . . . . . . . . . . . . . . . . . . . 278
8.3.5.M/M/1withboundedwaitingroom: theM/M/1/H model . . 282
8.3.6.Comments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
8.4.Anuniformization-basedpathfortheM/M/1withmatrixgenerating
functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
8.4.1.Generalcase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
8.4.2.MeannumberofcustomersattimetintheM/M/1 . . . . . . . . 287
8.5.Anuniformization-basedpathusingduality . . . . . . . . . . . . . . . . 290
8.5.1.Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
8.5.2.Thepathtowardthetransientstatedistributionsusingduality. . . 293
8.5.3.ApplicationtotheM/M/1queueingsystem . . . . . . . . . . . . 294
8.5.4.ApplicationtotheM/M/1/H queueingsystem . . . . . . . . . . 295
8.5.5.ApplicationtoanM/M/1/H modelwithcatastrophes . . . . . . 297
8.6.Othertransientresults . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
8.6.1.BusyperiodoftheM/M/1 . . . . . . . . . . . . . . . . . . . . . . 299
8.6.2.MaxbacklogoftheM/M/1overafinitetimeinterval . . . . . . 299
8.6.3.M/E/1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
8.7.Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
8.8.References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
ListofAuthors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
Preface
Vladimir ANISIMOV1 andNikolaos LIMNIOS2
1AmgenInc.,London,UnitedKingdom
2UniversityofTechnologyofCompiègne,France
Queueing theory is a huge and very rapidly developing branch of science
belonging to probability theory and stochastic modelling that originated a long time
ago from the pioneering works by Erlang (1909) on the analysis of the models for
telephone communication using Poisson processes. Later on, these results were
extended further in different directions in the works of such famous mathematicians
asPollaczek,Khinchin,Kendall,Kleinrockandmanyothers.
Nowadays, queueing theory is rapidly growing in various areas including a
theoreticalanalysisofqueueingmodelsandnetworksofrathercomplicatedstructure
using rather sophisticated mathematical models and various types of stochastic
processes. It also includes very wide areas of modern applications: computing and
telecommunicationnetworks,trafficengineering,mobiletelecommunications,etc.
The aim of this second volume, together with Volume 1, is to reflect the current
cutting-edge thinking and established practices in the analysis and applications of
queueingmodels.
Thisvolumeincludes8chapterswrittenbyexpertswell-knownintheirareas.
Twochapters,Chapters1and7,aredevotedtoinvestigatingastabilityanalysisof
sometypesofmultiserverregenerativequeueingsystemswithheterogeneousservers
andaregenerativeinputflowusingsynchronizationoftheinputandmajorizingoutput
flows; and a stability analysis of regenerative queueing systems based on a renewal
analysistechnique,whichisillustratedonclassicalGI/G/1andGI/G/mqueueing
systems.
QueueingTheory2,
coordinatedbyVladimirANISIMOV,NikolaosLIMNIOS.©ISTEEditions2020.