Simulation

338 Pages·2022·6.521 MB·English

by  Sheldon M. Ross

Simulation
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Simulation
Sixth Edition
Sheldon M. Ross
Epstein Department of Industrial and
Systems Engineering
University of Southern California
Los Angeles, CA, United States
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Contents
Preface ix
1 Introduction 1
Exercises 3
2 Elementsofprobability 5
2.1 Samplespaceandevents 5
2.2 Axiomsofprobability 5
2.3 Conditionalprobabilityandindependence 6
2.4 Randomvariables 9
2.5 Expectation 11
2.6 Variance 13
2.7 Chebyshev’sinequalityandthelawsoflargenumbers 15
2.8 Somediscreterandomvariables 17
2.9 Continuousrandomvariables 23
2.10Conditionalexpectationandconditionalvariance 30
Exercises 32
References 37
3 Randomnumbers 39
Introduction 39
3.1 Pseudorandomnumbergeneration 39
3.2 Usingrandomnumberstoevaluateintegrals 40
Exercises 44
References 45
4 Generatingdiscreterandomvariables 47
4.1 Theinversetransformmethod 47
4.2 GeneratingaPoissonrandomvariable 53
4.3 Generatingbinomialrandomvariables 54
4.4 Theacceptance–rejectiontechnique 55
4.5 Thecompositionapproach 57
4.6 Thealiasmethodforgeneratingdiscreterandomvariables 59
4.7 Generatingrandomvectors 62
Exercises 63
vi Contents
5 Generatingcontinuousrandomvariables 69
Introduction 69
5.1 Theinversetransformalgorithm 69
5.2 Therejectionmethod 73
5.3 Thepolarmethodforgeneratingnormalrandomvariables 82
5.4 GeneratingaPoissonprocess 86
5.5 GeneratinganonhomogeneousPoissonprocess 87
5.6 Simulatingatwo-dimensionalPoissonprocess 90
Exercises 94
References 97
6 Themultivariatenormaldistributionandcopulas 99
Introduction 99
6.1 Themultivariatenormal 99
6.2 Generatingamultivariatenormalrandomvector 101
6.3 Copulas 104
6.4 Generatingvariablesfromcopulamodels 109
Exercises 109
7 Thediscreteeventsimulationapproach 111
Introduction 111
7.1 Simulationviadiscreteevents 111
7.2 Asingle-serverqueueingsystem 112
7.3 Aqueueingsystemwithtwoserversinseries 115
7.4 Aqueueingsystemwithtwoparallelservers 116
7.5 Aninventorymodel 119
7.6 Aninsuranceriskmodel 120
7.7 Arepairproblem 122
7.8 Exercisingastockoption 124
7.9 Verificationofthesimulationmodel 126
Exercises 127
References 130
8 Statisticalanalysisofsimulateddata 133
Introduction 133
8.1 Thesamplemeanandsamplevariance 133
8.2 Intervalestimatesofapopulationmean 138
8.3 Thebootstrappingtechniqueforestimatingmeansquareerrors 141
Exercises 147
References 149
9 Variancereductiontechniques 151
Introduction 151
9.1 Theuseofantitheticvariables 153
Contents vii
9.2 Theuseofcontrolvariates 160
9.3 Variancereductionbyconditioning 166
9.4 Stratifiedsampling 180
9.5 Applicationsofstratifiedsampling 190
9.6 Importancesampling 199
9.7 Usingcommonrandomnumbers 212
9.8 Evaluatinganexoticoption 213
9.9 Appendix:Verificationofantitheticvariableapproachwhen
estimatingtheexpectedvalueofmonotonefunctions 217
Exercises 219
References 227
10 Additionalvariancereductiontechniques 229
Introduction 229
10.1TheconditionalBernoullisamplingmethod 229
10.2AsimulationestimatorbasedonanidentityofChen–Stein 233
10.3Usingrandomhazards 241
10.4Normalizedimportancesampling 246
10.5Latinhypercubesampling 250
Exercises 252
11 Statisticalvalidationtechniques 255
Introduction 255
11.1Goodnessoffittests 255
11.2Goodnessoffittestswhensomeparametersareunspecified 262
11.3Thetwo-sampleproblem 265
11.4ValidatingtheassumptionofanonhomogeneousPoissonprocess 271
Exercises 275
References 277
12 MarkovchainMonteCarlomethods 279
Introduction 279
12.1Markovchains 279
12.2TheHastings–Metropolisalgorithm 282
12.3TheGibbssampler 284
12.4ContinuoustimeMarkovchainsandaqueueinglossmodel 294
12.5Simulatedannealing 298
12.6Thesamplingimportanceresamplingalgorithm 300
12.7Couplingfromthepast 304
Exercises 306
References 308
Index 311
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Preface
Overview
In formulating a stochastic model to describe a real phenomenon, it used to be that
one compromised between choosing a model that is a realistic replica of the actual
situationandchoosingonewhosemathematicalanalysisistractable.Thatis,theredid
notseemtobeanypayoffinchoosingamodelthatfaithfullyconformedtothephe-
nomenon under study if it were not possible to mathematically analyze that model.
Similarconsiderationshaveledtotheconcentrationonasymptoticorsteady-statere-
sultsasopposedtothemoreusefulonesontransienttime.However,theadventoffast
and inexpensive computational power has opened up another approach—namely, to
trytomodelthephenomenonasfaithfullyaspossibleandthentorelyonasimulation
studytoanalyzeit.
Inthistextweshowhowtoanalyzeamodelbyuseofasimulationstudy.Inparticu-
lar,wefirstshowhowacomputercanbeutilizedtogeneraterandom(moreprecisely,
pseudorandom) numbers, and then how these random numbers can be used to gen-
erate the values of random variables from arbitrary distributions. Using the concept
of discrete events we show how to use random variables to generate the behavior
of a stochastic model over time. By continually generating the behavior of the sys-
temweshowhowtoobtainestimatorsofdesiredquantitiesofinterest.Thestatistical
questionsofwhentostopasimulationandwhatconfidencetoplaceintheresulting
estimatorsareconsidered.Avarietyofwaysinwhichonecanimproveontheusual
simulation estimators are presented. In addition, we show how to use simulation to
determinewhetherthestochasticmodelchosenisconsistentwithasetofactualdata.
New to this edition
• Newexercisesinmostchapters.
• ThenewSection5.2.1showshowwecansimulateorderstatisticsbyfirstsimulat-
ingbetarandomvariables.
• There are many new examples in the text. Example 9p is concerned with using
simulationtoestimatetheprobabilitythatasumofindependentandidenticallydis-
tributedrandomvariablesexceedssomevalue.ThisexamplegivestheAsmussen–
Kroese estimator along with an improvement of it. Example 9q uses simulation
arguments to obtain computational bounds on P(X =max(X ,...,X )) when
1 1 n
theX areindependentrandomvariables.
i