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AnIntroduction

ToStochastic

Modeling

HowardM.Taylor

SamuelKarlin

AnIntroductionto

StochasticModeling

ThirdEdition

AnIntroductionto

StochasticModeling

ThirdEdition

HowardM.Taylor

StatisticalConsultant

Onancock,Viginia

SamuelKarlin

DepartmentofMathematics

StanfordUniversity

Stanford,California

O

AcademicPress SanDiego London Boston New York Sydney Tokyo^ Toronto

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Taylor,HowardM. Anintroductiontostochasticmodeling/HowardM.Taylor,Samuel Karlin.-3rded. p. cm.

Includesbibliographicalreferences(p. - )andindex.

ISBN-13:978-0-12-684887-8 ISBN-10:0-12-684887- 1.Stochasticprocesses. I.Karlin,Samuel. II.Title. QA274.T35 1998 003'.76--dc2l

ISBN-13: 978-0-12-684887- ISBN-10:0-12-684887-

PRINTEDINTHEUNITEDSTATESOFAMERICA

05060708IP

• I Introduction Preface ix - 1.StochasticModeling
  • 2.ProbabilityReview
    • 3.TheMajorDiscreteDistributions
  • 4.ImportantContinuousDistributions
  • 5.SomeElementaryExercises
  • 6.UsefulFunctions,Integrals,andSums
    • Expectation II ConditionalProbabilityandConditional
    • 1.TheDiscreteCase
  • 2.TheDiceGameCraps
    • 3.RandomSums
  • 4.ConditioningonaContinuousRandomVariable
    • 5.Martingales*
• III MarkovChains:Introduction - 1.Definitions - 2.TransitionProbabilityMatricesofaMarkovChain - 3.SomeMarkovChainModels - 4.FirstStepAnalysis - 5.SomeSpecialMarkovChains - 6.FunctionalsofRandomWalksandSuccessRuns - 7.AnotherLookatFirstStepAnalysis* vi Contents - 8.BranchingProcesses* - 9.BranchingProcessesandGeneratingFunctions*
  • IVTheLongRunBehaviorofMarkovChains - 1.RegularTransitionProbabilityMatrices - 2.Examples - 3.TheClassificationofStates - 4.TheBasicLimitTheoremofMarkovChains - 5.ReducibleMarkovChains*
    • VPoissonProcesses - 1.ThePoissonDistributionandthePoissonProcess - 2.TheLawofRareEvents - 3.DistributionsAssociatedwiththePoissonProcess - 4.TheUniformDistributionandPoissonProcesses - 5.SpatialPoissonProcesses - 6.CompoundandMarkedPoissonProcesses
  • VI ContinuousTimeMarkovChains - 1.PureBirthProcesses - 2.PureDeathProcesses - 3.BirthandDeathProcesses - Processes 4.TheLimitingBehaviorofBirthandDeath - 5.BirthandDeathProcesseswithAbsorbingStates - 6.FiniteStateContinuousTimeMarkovChains - 7.APoissonProcesswithaMarkovIntensity*
• VII RenewalPhenomena - RelatedConcepts 1.DefinitionofaRenewalProcessand - 2.SomeExamplesofRenewalProcesses - Process 3.ThePoissonProcessViewedasaRenewal - 4.TheAsymptoticBehaviorofRenewalProcesses Contents vii - Processes 5.GeneralizationsandVariationsonRenewal - 6.DiscreteRenewalTheory*
  • VIII BrownianMotionandRelatedProcesses - 1.BrownianMotionandGaussianProcesses - 2.TheMaximumVariableandtheReflectionPrinciple - 3.VariationsandExtensions - 4.BrownianMotionwithDrift - 5.TheOrnstein-UhlenbeckProcess*
    • IX QueueingSystems - 1.QueueingProcesses
      • 2.PoissonArrivals,ExponentialServiceTimes
      • 3.GeneralServiceTimeDistributions
      • 4.VariationsandExtensions
      • 5.OpenAcyclicQueueingNetworks
      • 6.GeneralOpenNetworks
• FurtherReading
• AnswerstoExercises
• Index

PrefacetotheFirstEdition

Stochasticprocessesarewaysofquantifyingthedynamicrelationshipsof sequencesofrandomevents.Stochasticmodelsplayanimportantrolein elucidatingmanyareasofthenaturalandengineeringsciences.Theycan beusedtoanalyzethevariabilityinherentinbiologicalandmedical processes,todealwithuncertaintiesaffectingmanagerialdecisionsand withthecomplexitiesofpsychologicalandsocialinteractions,andtopro- videnewperspectives,methodology,models,andintuitiontoaidinother mathematicalandstatisticalstudies. Thisbookisintendedasabeginningtextinstochasticprocessesforstu- dentsfamiliarwithelementaryprobabilitycalculus.Itsaimistobridge thegapbetweenbasicprobabilityknow-howandanintermediate-level courseinstochasticprocesses-forexample,AFirstCourseinStochastic Processes,bythepresentauthors. Theobjectivesofthisbookarethree:(1)tointroducestudentstothe standardconceptsandmethodsofstochasticmodeling;(2)toillustratethe richdiversityofapplicationsofstochasticprocessesinthesciences;and (3)toprovideexercisesintheapplicationofsimplestochasticanalysisto appropriateproblems. Thechaptersareorganizedaroundseveralprototypeclassesofsto- chasticprocessesfeaturingMarkovchainsindiscreteandcontinuous time,Poissonprocessesandrenewaltheory,theevolutionofbranching events,andqueueingmodels.AftertheconcludingChapterIX,wepro- videalistofbooksthatincorporatemoreadvanceddiscussionsofseveral ofthemodelssetforthinthistext.

instructors,hopethatstudentswouldposeandanswerforthemselvesas theyreadatext.Answerstotheexercisesaregivenattheendofthebook sothatstudentsmaygaugetheirunderstandingastheygoalong. Problemsaremoredifficult.Someinvolveextensivealgebraicorcal- culusmanipulation.Manyare"wordproblems"whereinthestudentis asked,ineffect,tomodelsomedescribedscenario.Asinformulatinga model,thefirststepinthesolutionofawordproblemisoftenasentence oftheform"Letx=...."Amanualcontainingthesolutionstotheprob- lemsisavailablefromthepublisher. Areasonablestrategyonthepartoftheteachermightbetoholdstu- dentsresponsibleforalloftheexercises,buttorequiresubmittedsolu- tionsonlytoselectedproblems.Everystudentshouldattemptarepresen- tativeselectionoftheproblemsinordertodevelophisorherabilityto carryoutstochasticmodelinginhisorherareaofinterest. Asmallnumberofproblemsarelabeled"ComputerChallenges."These callformorethanpencilandpaperfortheiranalyses,andeithersimula- tion,numericalexploration,orsymbolmanipulationmayprovehelpful. ComputerChallengesaremeanttobeopen-ended,intendedtoexplore whatconstitutesananswerintoday'sworldofcomputingpower.They mightbeappropriateaspartofanhonorsrequirement. Becauseourfocusisonstochasticmodeling,insomeinstanceswehave omittedaproofandcontentedourselveswithaprecisestatementofa resultandexamplesofitsapplication.Allsuchomittedproofsmaybe foundinAFirstCourseinStochasticProcesses,bythepresentauthors. Inthismoreadvancedtext,theambitiousstudentwillalsofindadditional materialonmartingales,Brownianmotion,andrenewalprocesses,and presentationsofseveralotherclassesofstochasticprocesses.

TotheInstructor

Ifpossible,werecommendhavingstudentsskimthefirsttwochapters,re- ferringasnecessarytotheprobabilityreviewmaterial,andstartingthe coursewithChapterIII,onMarkovchains.Aonequartercourseadapted tothejunior-seniorlevelcouldconsistofacursory(one-week)reviewof ChaptersIandII,followedinorderbyChaptersIIIthroughVI.Forinter- estedstudents,ChaptersVII,VIII,andIXdiscussothercurrentlyactive areasofstochasticmodeling.Starredsectionscontainmaterialofamore advancedorspecializednature.

Acknowledgments

Manypeoplehelpedtobringthistextintobeing.Wegratefullyacknowl- edgethehelpofAnnaKarlin,ShelleyStevens,KarenLarsen,and LaurieannShoemaker.ChapterIXwasenrichedbyaseriesoflectureson queueingnetworksgivenbyRalphDisneyatTheJohnsHopkinsUniver- sityin1982.AlanKarr,IvanJohnstone,LukeTierney,BobVanderbei, andothersbesidesourselveshavetaughtfromthetext,andwehaveprof- itedfromtheircriticisms.Finally,wearegratefulforimprovementssug- gestedbytheseveralgenerationsofstudentswhohaveusedthebookover thepastfewyearsandhavegivenustheirreactionsandsuggestions.

ChapterI

Introduction

1. StochasticModeling

Aquantitativedescriptionofanaturalphenomenoniscalledamathe- maticalmodelofthatphenomenon.Examplesabound,fromthesimple equationS=Zgt2describingthedistanceStraveledintimetbyafalling objectstartingatresttoacomplexcomputerprogramthatsimulatesa biologicalpopulationoralargeindustrialsystem. Inthefinalanalysis,amodelisjudgedusingasingle,quitepragmatic, factor,themodel'susefulness.Somemodelsareusefulasdetailedquanti- tativeprescriptionsofbehavior,asforexample,aninventorymodelthat isusedtodeterminetheoptimalnumberofunitstostock.Anothermodel inadifferentcontextmayprovideonlygeneralqualitativeinformation abouttherelationshipsamongandrelativeimportanceofseveralfactors influencinganevent.Suchamodelisusefulinanequallyimportantbut quitedifferentway.Examplesofdiversetypesofstochasticmodelsare spreadthroughoutthisbook. Suchoftenmentionedattributesasrealism,elegance,validity,and reproducibilityareimportantinevaluatingamodelonlyinsofarasthey bearonthatmodel'sultimateusefulness.Forinstance,itisbothunrealis- ticandquiteineleganttoviewthesprawlingcityofLosAngelesasageo- metricalpoint,amathematicalobjectofnosizeordimension.Yetitis quiteusefultodoexactlythatwhenusingsphericalgeometrytoderivea minimum-distancegreatcircleairroutefromNewYorkCity,another "point."

2 I Introduction

Thereisnosuchthingasthebestmodelforagivenphenomenon.The pragmaticcriterionofusefulnessoftenallowstheexistenceoftwoor moremodelsforthesameevent,butservingdistinctpurposes.Consider light.Thewaveformmodel,inwhichlightisviewedasacontinuousflow, isentirelyadequatefordesigningeyeglassandtelescopelenses.Incon- trast,forunderstandingtheimpactoflightontheretinaoftheeye,the photonmodel,whichviewslightastinydiscretebundlesofenergy,is preferred.Neithermodelsupersedestheother;botharerelevantand useful. Theword"stochastic"derivesfromtheGreed toaim,to guess)andmeans"random"or"chance."Theantonymis"sure,""deter- ministic,"or"certain."Adeterministicmodelpredictsasingleoutcome fromagivensetofcircumstances.Astochasticmodelpredictsasetof possibleoutcomesweightedbytheirlikelihoods,orprobabilities.Acoin flippedintotheairwillsurelyreturntoearthsomewhere.Whetheritlands headsortailsisrandom.Fora"fair"coinweconsiderthesealternatives equallylikelyandassigntoeachtheprobability12. However,phenomenaarenotinandofthemselvesinherentlystochas- ticordeterministic.Rather,tomodelaphenomenonasstochasticorde- terministicisthechoiceoftheobserver.Thechoicedependsontheob- server'spurpose;thecriterionforjudgingthechoiceisusefulness.Most oftentheproperchoiceisquiteclear,butcontroversialsituationsdoarise. Ifthecoinoncefallenisquicklycoveredbyabooksothattheoutcome "heads"or"tails"remainsunknown,twoparticipantsmaystillusefully employprobabilityconceptstoevaluatewhatisafairbetbetweenthem; thatis,theymayusefullyviewthecoinasrandom,eventhoughmostpeo- plewouldconsidertheoutcomenowtobefixedordeterministic.Asaless mundaneexampleoftheconversesituation,changesinthelevelofalarge populationareoftenusefullymodeleddeterministically,inspiteofthe generalagreementamongobserversthatmanychanceeventscontribute totheirfluctuations. Scientificmodelinghasthreecomponents:(i)anaturalphenomenon understudy,(ii)alogicalsystemfordeducingimplicationsaboutthephe- nomenon,and(iii)aconnectionlinkingtheelementsofthenaturalsystem understudytothelogicalsystemusedtomodelit.Ifwethinkofthese threecomponentsintermsofthegreat-circleairrouteproblem,thenat- uralsystemistheearthwithairportsatLosAngelesandNewYork;the logicalsystemisthemathematicalsubjectofsphericalgeometry;andthe

4 I Introduction

Thenextprinciple,thelongrunrelativefrequencyinterpretationof probability,isabasicbuildingblockinmodernstochasticmodeling,made preciseandjustifiedwithintheaxiomaticstructurebythelawoflarge numbers.Thislawassertsthattherelativefractionoftimesinwhichan eventoccursinasequenceofindependentsimilarexperimentsap- proaches,inthelimit,theprobabilityoftheoccurrenceoftheeventonany singletrial. Theprincipleisnotrelevantinallsituations,however.Whenthesur- geontellsapatientthathehasan80-20chanceofsurvival,thesurgeon means,mostlikely,that80percentofsimilarpatientsfacingsimilar surgerywillsurviveit.Thepatientathandisnotconcernedwiththelong run,butinvividcontrast,isvitallyconcernedonlyintheoutcomeofhis, thenext,trial. Returningtothegroupexperiment,wewillsupposenextthatthecoinis flippedintotheairand,uponlanding,isquicklycoveredsothatnoonecan seetheoutcome.WhatisPr{H}now?Severalinthegrouparguethatthe outcomeofthecoinisnolongerrandom,thatPr{H}iseither0or1,and thatalthoughwedon'tknowwhichitis,probabilitytheorydoesnotapply. Othersarticulateadifferentview,thatthedistinctionbetween"ran- dom"and"lackofknowledge"isfuzzy,atbest,andthatapersonwitha sufficientlylargecomputerandsufficientinformationaboutsuchfactors astheenergy,velocity,anddirectionusedintossingthecoincouldhave predictedtheoutcome,headsortails,withcertaintybeforethetoss. Therefore,evenbeforethecoinwasflipped,theproblemwasalackof knowledgeandnotsomeinherentrandomnessintheexperiment. Inarelatedapproach,severalpeopleinthegrouparewillingtobetwith eachother,atevenodds,ontheoutcomeofthetoss.Thatis,theyarewill- ingtousethecalculusofprobabilitytodeterminewhatisafairbet,with- outconsideringwhethertheeventunderstudyisrandomornot.Theuse- fulnesscriterionforjudgingamodelhasappeared. Whiletherestofthemobweredebating"random"versus"lackof knowledge,"onemember,Karen,lookedatthecoin.Herprobabilityfor headsisnowdifferentfromthatofeveryoneelse.Keepingthecoincov- ered,sheannouncestheoutcome"Tails,"whereuponeveryonementally assignsthevaluePr{H}=0.Butthenhercompanion,Mary,speaksup andsaysthatKarenhasahistoryofprevarication. Thelastscenarioexplainswhytherearehorseraces;differentpeople assigndifferentprobabilitiestothesameevent.Forthisreason,probabil-

1. StochasticModeling 5

itiesusedinoddsmakingareoftencalledsubjectiveprobabilities.Then, oddsmakingformsthethirdprincipleforassigningprobabilityvaluesin modelsandforinterpretingthemintherealworld. Themodernapproachtostochasticmodelingistodivorcethedefinition ofprobabilityfromanyparticulartypeofapplication.Probabilitytheory isanaxiomaticstructure(seeSection2.8),apartofpuremathematics.Its useinmodelingstochasticphenomenaispartofthebroaderrealmofsci- enceandparallelstheuseofotherbranchesofmathematicsinmodeling deterministicphenomena. Tobeuseful,astochasticmodelmustreflectallthoseaspectsofthe phenomenonunderstudythatarerelevanttothequestionathand.Inad- dition,themodelmustbeamenabletocalculationandmustallowthede- ductionofimportantpredictionsorimplicationsaboutthephenomenon.

1.1. StochasticProcesses

AstochasticprocessisafamilyofrandomvariablesXwheretisapara- meterrunningoverasuitableindexsetT.(Whereconvenient,wewill writeX(t)insteadofX,.)Inacommonsituation,theindextcorresponds todiscreteunitsoftime,andtheindexsetisT={0,1,2,...}.Inthis case,X,mightrepresenttheoutcomesatsuccessivetossesofacoin,re- peatedresponsesofasubjectinalearningexperiment,orsuccessiveob- servationsofsomecharacteristicsofacertainpopulation.Stochastic processesforwhichT=[0,c)areparticularlyimportantinapplications. Heretoftenrepresentstime,butdifferentsituationsalsofrequentlyarise. Forexample,tmayrepresentdistancefromanarbitraryorigin,andX,may countthenumberofdefectsintheinterval(0,t]alongathread,orthe numberofcarsintheinterval(0,t]alongahighway. Stochasticprocessesaredistinguishedbytheirstatespace,ortherange ofpossiblevaluesfortherandomvariablesXbytheirindexsetT,andby thedependencerelationsamongtherandomvariablesX,.Themostwidely usedclassesofstochasticprocessesaresystematicallyandthoroughly presentedforstudyinthefollowingchapters,alongwiththemathemati- caltechniquesforcalculationandanalysisthataremostusefulwiththese processes.Theuseoftheseprocessesasmodelsistaughtbyexample. Sampleapplicationsfrommanyanddiverseareasofinterestareaninte- gralpartoftheexposition.