Table Of ContentQi-Ming He
Fundamentals of
Matrix-Analytic
Methods
Fundamentals of Matrix-Analytic Methods
Qi-Ming He
Fundamentals
of Matrix-Analytic Methods
Qi-MingHe
DepartmentofManagementSciences
UniversityofWaterloo
Waterloo,ON,Canada
ISBN978-1-4614-7329-9 ISBN978-1-4614-7330-5(eBook)
DOI10.1007/978-1-4614-7330-5
SpringerNewYorkHeidelbergDordrechtLondon
LibraryofCongressControlNumber:2013936739
©SpringerScience+BusinessMediaNewYork2014
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart
of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionor
informationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar
methodologynowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerpts
inconnectionwithreviewsorscholarlyanalysisormaterialsuppliedspecificallyforthepurposeofbeing
enteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthework.Duplication
ofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheCopyrightLawofthe
Publisher’s location, in its current version, and permission for use must always be obtained from
Springer.PermissionsforusemaybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.
ViolationsareliabletoprosecutionundertherespectiveCopyrightLaw.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexempt
fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
While the advice and information in this book are believed to be true and accurate at the date of
publication,neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityfor
anyerrorsoromissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,with
respecttothematerialcontainedherein.
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
To my parents
To Rongjing Feng, Zechuan He,
and Patrick Zebing He
Preface
This is a textbook focusing on matrix-analytic methods. In this book, the basic
theory is accompanied by a large number of examples and exercises. The book is
suitable for senior undergraduates and graduate students in science, engineering,
management,business,andfinance.Inaddition,anumberofrecentdevelopments
arecollected.Thus,thebookcanalsobeofvaluetoresearchersandpractitioners.
Marcel F. Neuts pioneered matrix-analytic methods in the study of queueing
models in the 1970s. Since then, matrix-analytic methods have become an indis-
pensabletoolinstochasticmodelingandhavefoundapplicationsintheanalysisand
design of manufacturing systems, telecommunications networks, risk/insurance
models, reliability models, and inventory and supply chain systems. The power
andpopularityofmatrix-analyticmethodscomefromtheirflexibilityinstochastic
modeling, capacity for analytic exploration, natural algorithmic thinking, and
tractabilityinnumericalcomputation.
SomeExistingWorkonMatrix-AnalyticMethods
A number of monographs on matrix-analytic methods have been published. For a
summary of earlier work on matrix-analytic methods, we refer to Neuts (1981,
1989)andreferencestherein.WerefertoLatoucheandRamaswami(1999)forthe
developmentsinthe1980sand1990s.Forspecializedsubjects,wereferto(i)Ost
(2001) for matrix-analytic methods and telecommunications networks; (ii)
Grassmann (2000) and Bini, Latouche, and Meini (2005) for matrix-analytic
methods and numerical computation; (iii) Breuer and Baum (2005) for matrix-
analytic methods and queueing theory; (iv) Tian and Zhang (2006) for matrix-
analytic methods and vacation queues; (v) Artalejo and Gomez-Corral (2008) for
matrix-analytic methods and retrial queues; (vi) Alfa (2010) for matrix-analytic
methodsanddiscretetimequeues;and(vii)Li(2010)formatrix-analyticmethods,
Wiener-Hopffactorization,andstructuredMarkovchains.
Since 1995, seven international conferences on matrix-analytic methods in
stochastic models have been held. Seven conference proceedings have been
vii
viii Preface
published:ChakravarthyandAlfa(1996);AlfaandChakravarthy(1998);Latouche
and Taylor (2000, 2002); Bini, Latouche, and Meini (2005); He, Takine, Taylor,
and Zhang (2008); and Latouche, Ramaswami, Sethuraman, Sigman, Squillante,
and Yao (2013). The proceedings represent the developments of matrix-analytic
methodsandtheirapplicationsindifferenttimeperiods.
FeaturesoftheBook
Thistextbookcoversthreefundamentalaspectsofmatrix-analyticmethodsandtwo
of their important applications areas. Each chapter in the book begins with basic
conceptsandwell-knownmodelsinstochasticprocesses,followedbyconceptsand
models in matrix-analytic methods. The book takes a constructive approach to
introduce stochastic models, which is intuitive, easy to identify and interpret
parametersandvariables,andmathematicallyrigorous.Thefactthattheapproach
ismathematicallyrigorousisespeciallyusefulforengineeringstudentstodevelop
advancedsolutionskills.
Thebookfocusesonthebasictheory,probabilisticandintuitiveinterpretations
of results, and computational methods. For key results whose proof requires
advanced mathematics, the book presents them without a proof, and reference(s)
wheretheoriginalproofcanbefoundareprovided.Furthermore,probabilisticand
intuitive interpretations to all the basic results are given, in order to help readers
gain in-depth understanding of the subjects. A large collection of computational
methodsinmatrix-analyticmethodsareincluded.
Therearealargenumberofexercisesinthebook.Someexercisesareforreview.
They are given for readers to review basic calculus, linear algebra, probability
theory, and matrix theory. Some exercises are very simple and are intended for
understandingfundamentalconceptsandideas.Someexercisesaretypicalandare
intendedforpracticingtheuseofthetheorydeveloped.Finally,someexercisesare
forextensionsofthetheoryandarepresentedforinformation.Exercisesofthistype
aredifficultandaremarkedbyanasterisk.
TopicsandOrganizationoftheBook
The selection of the topics reflects my preference, and is driven by my desire to
includesometopicsthathavenotbeeninpreviousmonographs.Thebookconsists
ofatheoreticalpartandanapplicationpart.ThetheoreticalpartincludesChaps.1,
2,and3.Chapter1definestheexponentialdistributionand,fromthere,introduces
phase-type distributions. Chapter 2 defines the Poisson process and, from there,
introduces Markovian arrival processes. Chapter 3 defines the birth-and-death
process and, from there, introduces Markov chains of the QBD, GI/M/1, and
M/G/1 type. The application part includes Chaps. 4 and 5. Chapter 4 defines
theM/M/1queueand,fromthere,introducesmorecomplexqueueingmodelsthat
canbeanalyzedusingmatrix-analyticmethods.Chapter5focusesoninventoryand
supplychainmodels.
Preface ix
ReadingtheBook
Readers are expected to have a solid background in basic probability theory,
Markov chains, Poisson processes, and matrix theory. In each chapter, a list of
textbooks is provided for background reading. Readers should read the first parts
ofChaps.1,2,and3tolearnthebasictheoryofmatrix-analyticmethods.Sections
1.1,1.2,and1.3covertheessentialsofphase-typedistributions.Sections2.1,2.2,
2.3,and2.4covertheessentialsofMarkovianarrivalprocesses.Sections3.1,3.2,
3.3, 3.4, and 3.5 cover the essentials of structured Markov chains. Other sections
in these chapters cover topics that are not normally discussed in books about
matrix-analytic methods. In particular, Sect. 1.4 introduces multivariate phase-
typedistributions.Section1.5presentstheEM-algorithmforparameterestimation
ofphase-typedistributions.Section2.5introduces marked Markovianarrivalpro-
cesses. Sections 3.6, 3.7 and 3.8 define and analyze Markov chains with a tree
structure. Section3.9 deals with tail asymptotics of Markov chains with infinitely
manybackgroundphases.
ThefirstpartinChaps.4and5dealwithsimpleandtypicalstochasticmodelsin
queueing theory and inventory management, respectively. The remaining parts of
thechaptersinvestigateadvancedmodels.Inparticular,Sects.4.6and4.9introduce
and investigate queues with multiple types of customers. Sections 5.3 and 5.4
introduce inventory management systems and develop algorithms for computing
the optimal policy, which is not normally covered in books on matrix-analytic
methods.
References
AlfaAS(2010)Queueingtheoryfortelecommunications:discretetimemodellingofasinglenode
system.Springer,NewYork
Alfa AS, Chakravarthy SR (1998) Advances in matrix analytic methods for stochastic models.
NotablePublications,NewJersey
Artalejo JR, Gomez-Corral A (2008) Retrial queueing systems: a computational approach.
Springer,Berlin
Bini D, Meini B, Latouche G (2005) Stochastic models: fifth international conference on
matrixanalyticmethods,21(No2–3)
BiniDA,LatoucheG,MeiniB(2005)NumericalmethodsforstructuredMarkovchains.Oxford
UniversityPress,Oxford
Breuer L, Baum D (2005) An introduction to queueing theory and matrix-analytic methods.
Springer,Dordrecht
ChakravarthySR,AlfaAS(1996)Matrix-analyticmethodsinstochasticmodels.MarcelDekker,
NewYork
GrassmannWK(2000)Computationalprobability.Kluwer,Massachusetts
He QM, Takine T, Taylor P, Zhang HQ (2008) Annals of operations research: matrix-analytic
methodsinstochasticmodels160
Latouche G, Taylor (2000) Advances in algorithmic methods for stochastic models. Notable
Publications,NewJersey
LatoucheG,Taylor(2002)Matrix-analyticmethods:theoryandapplications. WorldScientific,
NewJersey
