Table Of ContentProbabilistic models for computer networks:
Tools and solved problems
José Alberto Hernández and Pablo Serrano
UniversidadCarlosIIIdeMadrid
2015
Copyright ©
978 1 291 54687 3
ISBN: - - - -
iii
This book is dedicated to all women that
make men better
Contents
1
1 Probability review
25
2 The exponential random variable
43
3 The Poisson process
61
4 Discrete-Time Markov Chains
89
5 Continuous-Time Markov Chains
117
6 Classical queueing theory
165
7 Open queueing networks
187
Index
Preface
About the authors
José Alberto Hernández and Pablo Serrano are full-time
lecturers and researchers at the Department of Telematic
Engineering of Universidad Carlos III de Madrid, Spain.
2009
In , they designed a course on performance evaluation
of networks for the second year of the Bachelor’s Degree
in Telematic Engineering at the university. This book com-
prises about four years of experience teaching this course.
The feedback collected from students has been very useful
in understanding which concepts require further explana-
tion and examples for a better understanding.
José Alberto Hernández completed the Bachelor’s de-
gree in Telecommunication Engineering at Universidad
2002
Carlos III de Madrid (Spain) in , and the Ph.D. degree
in Computer Science at Loughborough University (Leics,
2005 2005 2009
United Kingdom) in . From to , he was a
postdoctoral researcher and teaching assistant at Univer-
sidad Autónoma de Madrid, where he participated in a
number of both national and european research projects
concerning the modeling and performance evaluation of
2009
communication networks. In , he moved to Universi-
dad Carlos III de Madrid, where he became an Associate
Professor. His research interests include the areas at which
mathematical modeling and computer networks overlap.
80
He has published more than scientific articles in both
viii
peer-reviewed journals and conference in-proceedings, in
the areas of network performance and reliability, broad-
band networks, high-speed switched networks, optical
WDM, converged optical-wireless access networks and
energy efficiency.
Pablo Serrano obtained his Telecommunication Engi-
neering degree and his PhD from Universidad Carlos III
3 2002 2006
de Madrid (UC M) in and , respectively. He has
3 2002
been with the Telematics Department of UC M since ,
where he currently holds the position of Associate Pro-
2007
fessor. In he was a Visiting Researcher at the Com-
puter Network Research Group at Univ. of Massachusetts
Amherst partially supported by the Spanish Ministry of
Education under a José Castillejo grant. He has published
50
over scientific papers in peer-reviewed international
journal and conferences. He serves on the Editorial Board
of IEEE Communications Letters, and has served on the
TPC of a number of conferences and workshops including
IEEE Globecom and IEEE INFOCOM. His current work fo-
cuses on the performance evaluation of wireless networks.
Book structure
This book is split into seven chapters, not isolated but de-
pendent, as shown in Fig. . We suggest the reader to follow
the order presented in the figure.
The first chapter provides a basic review of some im-
portant probabilistic concepts and theorems, including
expectation, joint and conditional probability, the Bayes’
theorem, etc. Some well-known Probability Density Func-
tions are also introduced, most of them will be thoroughly
used across the book.
The second and third chapter focus on the exponen-
tial distribution and the Poisson process respectively. The
memoryless property of the exponential distribution comprises
ix
Figure1: Bookcontents
andrelationshipsbe-
1. Probability
tweenchapters. Review
2. Exponential 3. Poisson
4. DTMCs
Distribution Process
5. CTMCs
6. Classical 7. Open
Queueing Queueing
Theory networks
the first important milestone of the book, since it will play
a key role in the Poisson process, and continuous-time
Markov chains. We often say in class that the exponential
distribution and the Poisson prosess are like husband and
wife, since they go everywhere together. Milestone1: Memoryless
Then, the book introduces discrete-time Markov chains propertyoftheexponen-
tialdistribution
which are very useful in the analysis of a large number of
communication systems, protocols and network-related
situations. The Chapman-Kolmogorov equations and the con-
cept of limiting-state probabilities comprise the second big Milestone2: Chapman-
milestone of this book. Kolmogorovequations
andlimiting-stateprob-
The next step is to present continuous-time Markov chains
abilitiesofdiscrete-time
which combine discrete-time Markov chains with expo-
Markovchains
nential distributions. Before addressing this chapter, the
properties of the exponential distribution and the analy-
sis of discrete-time Markov chains must be very clear to
x
Milestone3: state- the reader. The third milestone in the book comprises the
transition-ratediagram, ability to correctly use a continuous-time Markov chain to
infinitesimalgenerator
model a given problem, including the state-transition-rate
andbalanceequationsof
diagram, the infinitesimal generator and the balance equations
continuous-timeMarkov
chains to obtain the steady-state probability vector of a Markov
chain.
Once continuous-time Markov chains are well under-
stood, then queueing theory comprises just an application of
them. Indeed, queueing theory relies on both continuous-
time Markov chains and the Little’s theorem, which is the
Milestone4: Little’s fourth important milestone in the book.
theorem Finally, open queueing networks builds upon queueing
theory and provide tools to analyse big complex networks
of queues in an easy manner. The fifth and final milestone
Milestone5: Burke’sand comprises both the Burke’s and Jackson’s theorem which allow
Jackson’stheorem for easining the analysis of open queueing networks.
Focus and readership
The book aims at providing a step-by-step learning, start-
ing from intuitive examples, introducing the theory and
strengthening the learning process with new exercises.
The goal is to involve the reader through the learning pro-
cess with small frequent challenges in every chapter, on
attempts to make him/her feel confident with the theory.
The length of the book has been deliverately intended
to remain short, giving priority to examples and problems
over long theorem proofs. The reader is recommended to
visit other classical books (those pointed out in the bibli-
ography) for further details concerning theorems and their
proofs, since we have more focused on intuition rather than
100
rigour. The book includes over solved examples and
exercises that cover the main concepts introduced in theory,
with an application to computer networks and communica-
tion protocols.
