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U. Narayan Bhat
An Introduction to
Queueing Theory
Modeling and Analysis in Applications
Birkha¨user
Boston • Basel • Berlin
U.NarayanBhat
ProfessorEmeritus
StatisticalScience&OperationsResearch
SouthernMethodistUniversity
Dallas,TX75275-0332
USA
ISBN:978-0-8176-4724-7 e-ISBN:978-0-8176-4725-4
DOI: 10.1007/978-0-8176-4725-4
LibraryofCongressControlNumber:2007941114
MathematicsSubjectClassification(2000):60J27,60K25,60K30,68M20,90B22,90B36
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Inmemoryofmyparents,
VaidyaP.IshwarandParvatiBhat
Contents
Preface ......................................................... xi
1 Introduction................................................. 1
1.1 BasicSystemElements...................................... 1
1.2 ProblemsinaQueueingSystem .............................. 2
1.3 AHistoricalPerspective ..................................... 4
1.4 ModelingExercises......................................... 11
2 SystemElementModels ....................................... 13
2.1 ProbabilityDistributionsasModels ........................... 13
2.1.1 DeterministicDistribution(D) ......................... 14
2.1.2 Exponentialdistribution;Poissonprocess(M)............ 14
2.2 IdentificationofModels ..................................... 18
2.2.1 CollectionofData ................................... 18
2.2.2 TestsforStationarity ................................. 18
2.2.3 TestsforIndependence ............................... 19
2.2.4 DistributionSelection ................................ 19
2.3 ReviewExercises .......................................... 20
3 BasicConceptsinStochasticProcesses........................... 23
3.1 StochasticProcess.......................................... 23
3.2 Point,Regenerative,andRenewalProcesses .................... 23
3.3 MarkovProcess............................................ 24
4 SimpleMarkovianQueueingSystems ........................... 29
4.1 AGeneralBirth-and-DeathQueueingModel.................... 29
4.2 TheQueueM/M/1......................................... 34
4.2.1 DepartureProcess.................................... 40
4.3 TheQueueM/M/s......................................... 43
4.4 TheFiniteQueueM/M/s/K ................................ 51
4.5 TheInfinite-ServerQueueM/M/∞........................... 58
viii Contents
4.6 Finite-SourceQueues ....................................... 59
4.7 OtherModels.............................................. 62
4.7.1 TheM/M/1/1System ............................... 62
4.7.2 MarkovianQueueswithBalking ....................... 64
4.7.3 MarkovianQueueswithReneging ...................... 66
4.7.4 Phase-TypeMachineRepair ........................... 66
4.8 Remarks .................................................. 68
4.9 Exercises ................................................. 68
5 ImbeddedMarkovChainModels............................... 75
5.1 ImbeddedMarkovChains ................................... 75
5.2 TheQueueM/G/1......................................... 76
5.3 TheQueueG/M/1......................................... 98
5.4 Exercises ................................................. 112
6 ExtendedMarkovModels ..................................... 115
6.1 TheBulkQueueM(X)/M/1 ................................. 115
6.2 TheBulkQueueM/M(X)/1 ................................. 118
6.3 TheQueuesM/E /1andE /M/1............................ 120
k k
6.4 TheBulkQueuesM/GK/1andGK/M/1 ..................... 123
6.5 TheQueuesE /G/1andG/E /1 ............................ 126
k k
6.6 TheQueueM/D/s ......................................... 126
6.7 TheQueueM/M/1withPriorityDisciplines ................... 127
6.8 Exercises ................................................. 138
7 QueueingNetworks .......................................... 141
7.1 Introduction ............................................... 141
7.2 TheMarkovianNodeNetwork ............................... 142
7.3 QueuesinSeries ........................................... 144
7.4 QueueswithBlocking....................................... 147
7.5 OpenJacksonNetworks ..................................... 150
7.6 ClosedJacksonNetworks.................................... 152
7.7 CyclicQueues ............................................. 154
7.8 OperationalLawsforPerformanceAnalysis .................... 155
7.9 Remarks .................................................. 157
7.10 Exercises ................................................. 158
8 RenewalProcessModels ...................................... 161
8.1 RenewalProcess ........................................... 161
8.2 RenewalProcessModelsforQueueingSystems................. 166
9 TheGeneralQueueG/G/1andApproximations.................. 169
9.1 TheGeneralQueueG/G/1 .................................. 169
9.2 Little’sLawL=λW ....................................... 173
9.3 Approximations............................................ 175
9.4 DiffusionApproximation .................................... 178
Contents ix
9.5 FluidApproximation........................................ 180
9.6 Remarks .................................................. 183
10 StatisticalInferenceforQueueingModels ........................ 185
10.1 Introduction ............................................... 185
10.2 Birth-and-DeathProcessModels.............................. 187
10.3 ImbeddedMarkovChainModelsforM/G/1andG/M/1 ........ 191
10.4 TheQueueG/G/1 ......................................... 193
10.5 OtherMethodsofEstimation................................. 194
10.6 TestsofHypotheses......................................... 197
10.7 ControlofTrafficIntensityinM/G/1andG/M/1 .............. 197
10.8 Remarks .................................................. 199
11 DecisionProblemsinQueueingTheory .......................... 201
11.1 Introduction ............................................... 201
11.2 PerformanceMeasuresforDecisionMaking.................... 202
11.3 DesignProblemsinDecisionMaking.......................... 202
11.4 ControlProblemsinDecisionMaking ......................... 205
12 ModelingandAnalysisUsingComputationalTools ................ 207
12.1 MeanValueAnalysis........................................ 207
12.2 TheConvolutionAlgorithm.................................. 211
12.2.1 ComputingOtherPerformanceMeasures ................ 213
12.3 Simulation ................................................ 214
12.4 MATLAB................................................. 217
12.5 Exercises ................................................. 223
Appendices
A PoissonProcess: PropertiesandRelatedDistributions ............. 229
A.1 PropertiesofthePoissonProcess ............................. 229
A.2 VariantsofthePoissonProcess ............................... 231
A.3 Hyperexponential(HE)Distribution........................... 233
A.4 ErlangDistribution(E )..................................... 234
k
A.5 MixedErlangDistributions .................................. 234
A.6 CoxianDistributions;Phase-TypeDistribution .................. 235
A.7 AGeneralDistribution ...................................... 236
A.8 SomeDiscreteDistributions.................................. 236
B MarkovProcess.............................................. 239
B.1 KolmogorovEquations...................................... 239
B.2 ThePoissonProcess ........................................ 240
B.3 ClassificationofStates ...................................... 242
B.4 Phase-TypeDistributions .................................... 243
x Contents
C ResultsfromMathematics..................................... 247
C.1 Riemann–StieltjesIntegral................................... 247
C.2 LaplaceTransforms......................................... 248
C.3 GeneratingFunctions ....................................... 250
References ...................................................... 253
Index........................................................... 265
Description:This introductory textbook is designed for a one-semester course on queueing theory that does not require a course in stochastic processes as a prerequisite. By integrating the necessary background on stochastic processes with the analysis of models, the work provides a sound foundational introducti
