Introduction to Probability Models

801 Pages·2006·3.69 MB·English

by  Sheldon M. Ross

Introduction to
Probability Models
Ninth Edition
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Introduction to
Probability Models
Ninth Edition
Sheldon M. Ross
UniversityofCalifornia
Berkeley,California
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Contents
Preface xiii
1. Introduction to Probability Theory 1
1.1. Introduction 1
1.2. SampleSpaceandEvents 1
1.3. ProbabilitiesDefinedonEvents 4
1.4. ConditionalProbabilities 7
1.5. IndependentEvents 10
1.6. Bayes’Formula 12
Exercises 15
References 21
2. Random Variables 23
2.1. RandomVariables 23
2.2. DiscreteRandomVariables 27
2.2.1. TheBernoulliRandomVariable 28
2.2.2. TheBinomialRandomVariable 29
2.2.3. TheGeometricRandomVariable 31
2.2.4. ThePoissonRandomVariable 32
2.3. ContinuousRandomVariables 34
2.3.1. TheUniformRandomVariable 35
2.3.2. ExponentialRandomVariables 36
2.3.3. GammaRandomVariables 37
2.3.4. NormalRandomVariables 37
v
vi Contents
2.4. ExpectationofaRandomVariable 38
2.4.1. TheDiscreteCase 38
2.4.2. TheContinuousCase 41
2.4.3. ExpectationofaFunctionofaRandomVariable 43
2.5. JointlyDistributedRandomVariables 47
2.5.1. JointDistributionFunctions 47
2.5.2. IndependentRandomVariables 51
2.5.3. CovarianceandVarianceofSumsofRandomVariables 53
2.5.4. JointProbabilityDistributionofFunctionsofRandom
Variables 61
2.6. MomentGeneratingFunctions 64
2.6.1. TheJointDistributionoftheSampleMeanandSample
VariancefromaNormalPopulation 74
2.7. LimitTheorems 77
2.8. StochasticProcesses 83
Exercises 85
References 96
3. Conditional Probability and Conditional
Expectation 97
3.1. Introduction 97
3.2. TheDiscreteCase 97
3.3. TheContinuousCase 102
3.4. ComputingExpectationsbyConditioning 105
3.4.1. ComputingVariancesbyConditioning 117
3.5. ComputingProbabilitiesbyConditioning 120
3.6. SomeApplications 137
3.6.1. AListModel 137
3.6.2. ARandomGraph 139
3.6.3. UniformPriors,Polya’sUrnModel,and
Bose–EinsteinStatistics 147
3.6.4. MeanTimeforPatterns 151
3.6.5. Thek-RecordValuesofDiscreteRandomVariables 155
3.7. AnIdentityforCompoundRandomVariables 158
3.7.1. PoissonCompoundingDistribution 161
3.7.2. BinomialCompoundingDistribution 163
3.7.3. ACompoundingDistributionRelatedtotheNegative
Binomial 164
Exercises 165
Contents vii
4. Markov Chains 185
4.1. Introduction 185
4.2. Chapman–KolmogorovEquations 189
4.3. ClassificationofStates 193
4.4. LimitingProbabilities 204
4.5. SomeApplications 217
4.5.1. TheGambler’sRuinProblem 217
4.5.2. AModelforAlgorithmicEfficiency 221
4.5.3. UsingaRandomWalktoAnalyzeaProbabilisticAlgorithm
fortheSatisfiabilityProblem 224
4.6. MeanTimeSpentinTransientStates 230
4.7. BranchingProcesses 233
4.8. TimeReversibleMarkovChains 236
4.9. MarkovChainMonteCarloMethods 247
4.10. MarkovDecisionProcesses 252
4.11. HiddenMarkovChains 256
4.11.1. PredictingtheStates 261
Exercises 263
References 280
5. The Exponential Distribution and the Poisson
Process 281
5.1. Introduction 281
5.2. TheExponentialDistribution 282
5.2.1. Definition 282
5.2.2. PropertiesoftheExponentialDistribution 284
5.2.3. FurtherPropertiesoftheExponentialDistribution 291
5.2.4. ConvolutionsofExponentialRandomVariables 298
5.3. ThePoissonProcess 302
5.3.1. CountingProcesses 302
5.3.2. DefinitionofthePoissonProcess 304
5.3.3. InterarrivalandWaitingTimeDistributions 307
5.3.4. FurtherPropertiesofPoissonProcesses 310
5.3.5. ConditionalDistributionoftheArrivalTimes 316
5.3.6. EstimatingSoftwareReliability 328
5.4. GeneralizationsofthePoissonProcess 330
5.4.1. NonhomogeneousPoissonProcess 330
5.4.2. CompoundPoissonProcess 337
5.4.3. ConditionalorMixedPoissonProcesses 343
viii Contents
Exercises 346
References 364
6. Continuous-Time Markov Chains 365
6.1. Introduction 365
6.2. Continuous-TimeMarkovChains 366
6.3. BirthandDeathProcesses 368
6.4. TheTransitionProbabilityFunctionP (t) 375
ij
6.5. LimitingProbabilities 384
6.6. TimeReversibility 392
6.7. Uniformization 401
6.8. ComputingtheTransitionProbabilities 404
Exercises 407
References 415
7. Renewal Theory and Its Applications 417
7.1. Introduction 417
7.2. DistributionofN(t) 419
7.3. LimitTheoremsandTheirApplications 423
7.4. RenewalRewardProcesses 433
7.5. RegenerativeProcesses 442
7.5.1. AlternatingRenewalProcesses 445
7.6. Semi-MarkovProcesses 452
7.7. TheInspectionParadox 455
7.8. ComputingtheRenewalFunction 458
7.9. ApplicationstoPatterns 461
7.9.1. PatternsofDiscreteRandomVariables 462
7.9.2. TheExpectedTimetoaMaximalRunofDistinctValues 469
7.9.3. IncreasingRunsofContinuousRandomVariables 471
7.10. TheInsuranceRuinProblem 473
Exercises 479
References 492
8. Queueing Theory 493
8.1. Introduction 493
8.2. Preliminaries 494
8.2.1. CostEquations 495
8.2.2. Steady-StateProbabilities 496
Contents ix
8.3. ExponentialModels 499
8.3.1. ASingle-ServerExponentialQueueingSystem 499
8.3.2. ASingle-ServerExponentialQueueingSystem
HavingFiniteCapacity 508
8.3.3. AShoeshineShop 511
8.3.4. AQueueingSystemwithBulkService 514
8.4. NetworkofQueues 517
8.4.1. OpenSystems 517
8.4.2. ClosedSystems 522
8.5. TheSystemM/G/1 528
8.5.1. Preliminaries:WorkandAnotherCostIdentity 528
8.5.2. ApplicationofWorktoM/G/1 529
8.5.3. BusyPeriods 530
8.6. VariationsontheM/G/1 531
8.6.1. TheM/G/1withRandom-SizedBatchArrivals 531
8.6.2. PriorityQueues 533
8.6.3. AnM/G/1OptimizationExample 536
8.6.4. TheM/G/1QueuewithServerBreakdown 540
8.7. TheModelG/M/1 543
8.7.1. TheG/M/1BusyandIdlePeriods 548
8.8. AFiniteSourceModel 549
8.9. MultiserverQueues 552
8.9.1. Erlang’sLossSystem 553
8.9.2. TheM/M/kQueue 554
8.9.3. TheG/M/kQueue 554
8.9.4. TheM/G/kQueue 556
Exercises 558
References 570
9. Reliability Theory 571
9.1. Introduction 571
9.2. StructureFunctions 571
9.2.1. MinimalPathandMinimalCutSets 574
9.3. ReliabilityofSystemsofIndependentComponents 578
9.4. BoundsontheReliabilityFunction 583
9.4.1. MethodofInclusionandExclusion 584
9.4.2. SecondMethodforObtainingBoundsonr(p) 593
9.5. SystemLifeasaFunctionofComponentLives 595
9.6. ExpectedSystemLifetime 604
9.6.1. AnUpperBoundontheExpectedLifeofa
ParallelSystem 608