Numerical Methods for Solving Discrete Event Systems: With Applications to Queueing Systems

370 Pages·2022·3.755 MB·English

by  Winfried Grassmann, Javad Tavakoli

CMS/CAIMS Books in Mathematics
Canadian
Mathematical Society
Winfried Grassmann
Société mathématique
Javad Tavakoli du Canada
Numerical Methods
for Solving Discrete
Event Systems
With Applications to Queueing Systems
CMS/CAIMS Books in Mathematics
Volume 5
Series Editors
Karl Dilcher
Department of Mathematics and Statistics, Dalhousie University, Halifax,
NS, Canada
Frithjof Lutscher
Department of Mathematics, University of Ottawa, Ottawa, ON, Canada
Nilima Nigam
Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada
Keith Taylor
Department of Mathematics and Statistics, Dalhousie University, Halifax, NS,
Canada
Associate Editors
Ben Adcock
Department of Mathematics, Simon Fraser University, Burnaby, BC, Canada
Martin Barlow
University of British Columbia, Vancouver, BC, Canada
Heinz H. Bauschke
University of British Columbia, Kelowna, BC, Canada
Matt Davison
Department of Statistical and Actuarial Sciences, Western University, London,
ON, Canada
Leah Keshet
Department of Mathematics, University of British Columbia, Vancouver,
BC, Canada
Niky Kamran
Department of Mathematics and Statistics, McGill University, Montreal,
QC, Canada
Mikhail Kotchetov
Memorial University of Newfoundland, St. John's, Canada
Raymond J. Spiteri
Department of Computer Science, University of Saskatchewan, Saskatoon,
SK, Canada
CMS/CAIMS Books in Mathematics is a collection of monographs and graduate-
level textbooks published in cooperation jointly with the Canadian Mathematical
Society-SocietémathématiqueduCanadaandtheCanadianAppliedandIndustrial
Mathematics Society-Societé Canadienne de Mathématiques Appliquées et Indus-
trielles.Thisseries offers authorsthejoint advantageofpublishingwithtwomajor
mathematical societies and with a leading academic publishing company. The se-
ries is edited by Karl Dilcher, Frithjof Lutscher, Nilima Nigam, and Keith Taylor.
The series publishes high-impact works across the breadth of mathematics and its
applications. Books in this series will appeal to all mathematicians, students and
established researchers. The series replaces the CMS Books in Mathematics series
that successfully published over 45 volumes in 20 years.
Winfried Grassmann Javad Tavakoli
(cid:129)
Numerical Methods
for Solving Discrete Event
Systems
With Applications to Queueing Systems
123
Winfried Grassmann JavadTavakoli
Department ofComputer Science Department ofMathematics
University of Saskatchewan University of British Columbia
Saskatoon, SK,Canada Kelowna, BC,Canada
ISSN 2730-650X ISSN 2730-6518 (electronic)
CMS/CAIMS Booksin Mathematics
ISBN978-3-031-10081-9 ISBN978-3-031-10082-6 (eBook)
https://doi.org/10.1007/978-3-031-10082-6
MathematicsSubjectClassification: 60J27,60J10,60J22,60K25,65C40
©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature
SwitzerlandAG2022
Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether
thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseof
illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and
transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar
ordissimilarmethodologynowknownorhereafterdeveloped.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom
therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse.
The publisher, the authors, and the editorsare safeto assume that the adviceand informationin this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authors or the editors give a warranty, expressed or implied, with respect to the material contained
hereinorforanyerrorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregard
tojurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
Preface
This book shows how to formulate and solve discrete event systems by numerical
methods without using discrete event simulation. Discrete event systems are pre-
dominantly stochastic systems that are changed by discrete events. Examples of
discreteeventsystemsincludequeueingsystems,wherearrivalsanddeparturesare
discrete events, and inventory models, where usages and replenishments are dis-
creteevents.Mostsystemsweconsiderhaveseveralstatevariables,suchasqueues
and inventories. We provide numerical methods to find transient and steady-state
solutions for such systems.
The book is method rather than model oriented. In other words, the book helps
researcherswhoalreadyhaveamodeltheyneedtosolve,andtheyarenowlooking
for applicable methods to solve this model. The stress is on numerical tools as
opposedtoformulas.Weprovidealgorithmsthathopefullycanbeapplied,possibly
in a modified version, to solve the problems in question. Of course, for many
researchers,thefirstapproachtosolvetheirproblemisMonteCarlosimulation,but
since there is an ample literature in this area, we do not cover it here. Instead, our
algorithms are deterministic in the sense that no random numbers are used. We
formulate our systems as Markov chains and solve them as such. This has certain
advantages,especiallywhendealingwithsmallersystems,andincaseswherehigh
precision is required.
The use of Markov chains and their properties naturally leads to a better
understanding of important concepts. It helps to gain insights into the concepts of
recurrence and transience, and the concept of a statistical equilibrium. For this
reason, this book may also serve people that merely want to have a better under-
standing of Markovian systems, and their use to solve discrete event systems.
Aprimeexampleofadiscreteeventsystemisaqueueingsystem,andindeedwe
use many results from queueing theory in this book. However, we go beyond the
standard models of queueing theory, such as one-server and multi-server queues.
Wearelookingatsituationswherethereareseveralqueues,andwherearrivalsare
not necessarily Poisson and service times are not necessarily exponential. We
realize that dealing with such systems forces us to use numerical methods, which
also allows us to avoid complex mathematics.
v
vi Preface
In traditional queueing theory, the emphasis is on mathematical derivations
rather than on algorithms. This is understandable when considering the history of
queueingtheoryandcomputing.Whentheclassicalresultsofqueueingtheorywere
derived,computersweresloworevennon-existent,andcomputations werecostly.
Today, even a simple laptop can do millions of multiplications per second. This
completely changes the situation. If you have no computers, you cannot use
solution methods that require millions of additions and multiplications, but the
derivation of formulas is not hampered by the lack of computing power. Today,
even a simple laptop provides enough computing power to find transient and
equilibrium solutions of non-trivial discrete event systems.
Thoughdeterministicnumericalmethodsgainedenormouslyfromtheincreased
speed of the computers, simulation gained much more. The reason is the curse of
dimensionality. Essentially, the numerical effort to find numerical solutions
increases exponentially with the number of state variables needed to formulate the
model. This severely limits the types of problems that can be solved by deter-
ministic numerical methods. In simulation, the computational effort only increases
linearly with the numberof state variables, meaning that for large models, discrete
eventsimulationisthemethodofchoice.However,wefoundthatforsmallmodels,
deterministic methods have many advantages over simulations, including shorter
execution times.
Concentrating on smaller models is not necessarily a disadvantage. There are
manyinstanceswheremodelswithveryfewstatevariablesareappropriate,andfor
these models, solution methods based on probability theory are a good fit. Indeed,
modelswithmanystatevariablesareoftendifficulttointerpretbecauseoftheirhigh
dimensionality, which may hide essential features. For this reason, simple models
are often more instructive.
Instead of using the discrete event paradigm, many people use stochastic Petri
net[65]orstochasticactivitynets[79].Personally,wefoundtheseapproachesboth
lessgeneralandmoredifficulttounderstandthandiscreteeventsystems,astheyare
described by Casandras and Lafortune [9]. Of course, discrete event systems are
familiar to anyone using simulating queues or similar stochastic systems.
Hereistheoutlineofthisbook.InChapter1,weintroducebasicconceptswhich
are needed to understand the remainder of this book. Chapters 2–5 show how to
formulate discrete event systems, and how to convert them into Markov chains by
usingsupplementaryvariables.Chapter6introducescomputationalcomplexityand
discussesthedifferenttypesoferrorsthemodelerhastoconsider.Inparticular,the
chapterdealswithroundingerrors,anditshowsthatroundingerrorsaremagnified
bysubtraction.Fortunately,whendealingwithprobabilities,subtractionscanoften
beavoided.Chapter7discussestransientsolutions,withparticularemphasisonthe
randomization method. Chapter 8 introduces the classifications of states, and it
showshowtouseeigenvaluestofindtransientsolutionsofdiscreteevent systems.
Bothtopicsarerequiredinordertoestimatehowfastasystemconvergestowardits
equilibriumsolution.Chapter9describeshowtofindequilibriumsolutions,bothby
directanditerativemethods.Thedirectmethodwepreferisaneliminationmethod
we call state reduction method or state elimination method, which is numerically
Preface vii
very stable, an important property because the discrete event systems lead to
Markov chains with many states, and rounding errors increase as the number of
states increases. State elimination also has a probabilistic interpretation in terms of
censoring,amethodcloselyrelatedtoembedding.ThisleadstoChapter10,which
shows how to use censoring and embedding to reduce the state space. Chapter 11
covers systems that are almost or completely independent. It also covers
product-formsolutionswherethestatevariablescanbeconsideredasindependent,
though only in systems in a statistical equilibrium. Chapter 12 deals with systems
withaninfinitestatespace,withemphasisonthematrixanalyticmethodandrelated
methods.
Since this book does not use advanced mathematical methods, it should be
accessibletonon-specialists wholook foreffectivemethodstosolve thestochastic
systems they encounter in their research. It should also be useful to graduate stu-
dents for their projects and theses. We believe that this book is also useful as a
text/reference for any typical stochastic process and related topic courses.
Aarau, Switzerland Winfried Grassmann
Kelowna, BC, Canada Javad Tavakoli
Contents
1 Basic Concepts and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 The Definition of a Discrete Event System. . . . . . . . . . . . . . . . 1
1.1.1 State Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Markov Chains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Discrete-time Markov Chains (DTMCs). . . . . . . . . . . . 5
1.2.2 Continuous-time Markov Chains (CTMCs) . . . . . . . . . 8
1.3 Random Variables and Their Distributions . . . . . . . . . . . . . . . . 10
1.3.1 Expectation and Variance . . . . . . . . . . . . . . . . . . . . . . 10
1.3.2 Sums of Random Variables. . . . . . . . . . . . . . . . . . . . . 12
1.3.3 Some Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.4 Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 The Kendall Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5 Little’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.6 Sets and Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 Systems with Events Generated by Poisson or by Binomial
Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.1 The Binomial and the Poisson Process. . . . . . . . . . . . . . . . . . . 28
2.2 Specification of Poisson Event Systems . . . . . . . . . . . . . . . . . . 29
2.3 Basic Principles for Generating Transition Matrices . . . . . . . . . 31
2.4 One-dimensional Discrete Event Systems. . . . . . . . . . . . . . . . . 32
2.4.1 Types of One-dimensional Discrete Event
Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4.2 The M=M=1=N Queue . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4.3 Birth–Death Processes, with Extensions. . . . . . . . . . . . 35
2.4.4 A Simple Inventory Problem. . . . . . . . . . . . . . . . . . . . 36
ix
x Contents
2.5 Multidimensional Poisson Event Systems. . . . . . . . . . . . . . . . . 37
2.5.1 Types of Multidimensional Systems . . . . . . . . . . . . . . 38
2.5.2 First Example: A Repair Problem . . . . . . . . . . . . . . . . 39
2.5.3 Second Example: Modification of the Repair
Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.6 Immediate Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.6.1 An Example Requiring Immediate Events . . . . . . . . . . 44
2.6.2 A Second Example with Immediate Events:
A Three-way Stop . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.7 Event-based Formulations of the Equilibrium Equations . . . . . . 47
2.8 Binomial Event Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.8.1 The Geom=Geom=1=N Queue. . . . . . . . . . . . . . . . . . . 50
2.8.2 Compound Events and Their Probabilities . . . . . . . . . . 52
2.8.3 The Geometric Tandem Queue . . . . . . . . . . . . . . . . . . 53
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3 Generating the Transition Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1 The Lexicographic Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2 The Transition Matrix for Systems with Cartesian
State Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3 The Lexicographic Code Used for Non-Cartesian
State Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.4 Dividing the State Space into Subspaces . . . . . . . . . . . . . . . . . 68
3.5 Alternative Enumeration Methods . . . . . . . . . . . . . . . . . . . . . . 70
3.6 The Reachability Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4 Systems with Events Created by Renewal Processes. . . . . . . . . . . . 75
4.1 The Renewal Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.1.1 Remaining Lifetimes . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.1.2 The Age Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.1.3 The Number of Renewals . . . . . . . . . . . . . . . . . . . . . . 80
4.2 Renewal Event Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2.1 Description of Renewal Event Systems . . . . . . . . . . . . 81
4.2.2 The Dynamics of Renewal Event Systems. . . . . . . . . . 84
4.3 Generating the Transition Matrix . . . . . . . . . . . . . . . . . . . . . . . 87
4.3.1 The Enumeration of States in Renewal Event
Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.3.2 Ages Used as Supplementary State Variables. . . . . . . . 89
4.3.3 Remaining Lifetimes used as Supplementary
State Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.3.4 Using both Age and Remaining Life
as Supplementary State Variables . . . . . . . . . . . . . . . . 93
Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93