Table Of ContentWILEY-INTERSCIENCE SERIES IN SYSTEMS
AND OPTIMIZATION
Advisory Editors
Sheldon Ross
Dcpanment of Industrial Enginccrlng and Opcntions Research, Univmity of California,
Berkeley, CA 94720, USA
Richard Weber
Swistical Laboratory, Cambridge Univenity, 16 Mill Lane, Cambridge CB2 I RX, UK
CHAO/MIYA ZA WA/PINEDO-Queueing Networks: Customers, Signals and Product
Fonn Solutions
GITTINS-Multi-armed Bandit Allocation Indices
KALUWA LLACE-Stochastic Programming
KAMP/HASLER-Recursive Neural Networks for Associative Memory
KIBZUN/KAN-Stochastic Programming Problems with Probability and Quantile
Functions
RU STEM-Algorithms for Nonlinear Prognmming and Multiple-objective Decisions
VAN DIJK-Queueing Networks and Product Forms: A Systems Approach
WHITn.,E-OptirnaJ Control: Buics and Beyond
WHITTLE-Risk-sensitive Optimal Control
WHITn.,E-Neural Nets and Q-•-,tic Cartiers
The concept of a system as an entity in its own right has emerged with increasing
force in the past few decades in. for example, the areas of electrical and control
engineering, economics, ecology, urban structures, automation theory, operational
research and industry. The more definite concept of a large-scale system is implicit
in these applications, but is particularly evident in such fields as the study of
communication networks, computer networks, and neural networks. The Wiley
lntcrsciencc Series in Systems and Optimiz.ation has been established to serve the
needs of researchers in these rapidly developing fields. It is intended for works
concerned with developments in quantitative systems theory, applications of such
theory in areas of interest, or associated methodology.
Queueing
Networks
Customers, Signals and
Product Form Solutions
XiuU Chao
New Jer.1ey Institute of Technology, USA
Masakiyo Miyazawa
Science University of To/cyo, Japan
Michael Pinedo
Yn
Nn, Ulelversity, USA
I ' ;J'tftW
/
; '
' '
I.T.R.C.
• I
JOHN WILEY & SONS, LID
Chichester• New Yodc • Weinheim •Brisbane• Singapore• Toronto
,,,.,
Copyright O 1999 John Wiley & Sons Ltd
Baffins Lane, Chichester,
West SIWCX, POl9 IUD, England
Nation.,/ 01243 779777
/"'•""'''"""' (+44) 124) 779777
Reprinted December I '199
e-mail (for orders and CUSIOmlf service ei1qu1ries): e1-boolu@will)'.co.u1t
Visit our Home PageonhapJlwww.wilcyco.uk or hap:/lwww.wilcy.ClOUl
All Ripb Reserved. No plll1 of !his publical.ion may be rcprodllllld, IIORCI in a Rlricval s)lltan. o,
lnlll.Slll.illcd, in any form or by any mcam, clmronic, mcdlaniGal. pboeowpying. -.lin1, .-m1 o,
Olhawilc, c,u;cpt wider die tcnnl oflhe Copyright, Daips and l'lllnll Act 1911 or under die tanm of
a licence issued by the Copyright Liunsing Ag,mcy, 90 Toelalham CGurt Road, Loodoa WIP 9HE, llK.
without the pcnnission in writin11 ofdle Publisher.
Ncllha the aulhon nor John Wiley & Sona ud IICccpl any rapomlbillty or liability for km o, dam-.
oc:usioncd lo any penon or propcny through 111in11 lhc maaial, ilulructionl. IIIClbodl or iclcu
conlaincd hcRin, or acting o, rchinin1 from acting • • n:sult of IUdl u.sc. The aulbar and Publisher
CllpRISSly diKlaim all implied wunnliCI, including merchaatability o, ft-for any particular
PWJIOIC-
Other WIiiy Editorial Offl«1
John Wiley & Sona, lw:, 605 Third Avenue,
New York, NY 10158-001:Z, USA
Weinhcim • Brisbane • Sinppore • Toronlo
Chm,Xiuli
Queueing nctwoou : c111tomer1, slanal1, and product ftJnn IOlll&ionl
/ Xiuli Chm, Muakiyo Miyazawa, Micha.:! Pmodo.
p, cm. --(Wilcy-lntencimce series in systems and optimiu&iun)
Includes biblios,aphical n:famc:a and index.
ISBN 0-471-98309-8 (hardcover: alk. paper)
I. Pruducuon 1chodulin1. 2. Qucucin1 lhcory. I. Mi,-1,
Mualtlyo II. Pinedo, Mlahlll. Ill. lllla. IV Serltl.
TKl57.5.C453 1999 99-26511
651."3--dcll CIP
A ca&aloguc rocord for lhi1 book 11 available from lhc Brltilh Liblwy
ISBN0471987'06
Produced PoatScripc fila 1upplied by the authon
Prinred and bound in Otc.1 Britain by Anwny Rowe, Cblppeallaa
Thi■ book 11 prlnred un acid-free paper raponsibly manufactured lhlm -tainablc fOR:IIJy in which Ill
lust two nca .,. plaored rm oacll one II.led for paper pn,du,clJun.
To our families
Contents
PREFACE xi
1 INTRODUCTION
I.I Introduction .
1.2 Queueing networks with signals ~
1.3 Applications .......... . 5
1.4 Definitions of the basic concepts 9
1.5 Outline of the book 12
1.6 Reference notes 14
2 FUNDAMENTALS 15
2.1 Markov chains . . . . . . . . . . . . . . . . . 15
2.2 Counting processes driven by Markov chains 22
2.3 Local balance equations . . . . . . . . . . 28
2.4 Time reversed processes and reversibility 35
2.5 Product forms and decomposability .......... . 46
2.6 The M/Mr/1 queue: Batch services .......... . 47
2. 7 The MX /MY/ I queue: Batch arrivals and batch services 49
2.8 Reference notes 54
Exercises ...... . 55
3 QUASI-REVERSIBILITY 58
3.1 Continuous time Markov chains revisited 58
3.2 Quasi-reversibility of queues . . . . . . . 61
3.3 Quasi-reversibility of queues with signals 6~
3.4 Reference notes 76
Exercises ................ . 76
4 NETWORKS OF QUASI-REVERSIBLE QUEUES 78
4.1 An intuitive approach to quasi-reversibility . 78
4.2 Connecting nodes into networks . . . . . . . . . 84
4.3 Networks of quasi-reversible nodes . . . . . . . . 88
4.4 Networks with signals and instantaneous movements 95
VII
VIII
4.5 Networks with positive and negative signals ...... . 105
4.5. I Single class of positive and negative signals .. . 105
4.5.2 Multiple classe'.'> of positive and negative signals . 111
4.6 Reference notes 116
Exercises ........................ . 116
5 NETWORKS WITH EXPONENTIAL SERVICE TIMES 119
5.1 Jackson and Kelly networks ........ . 119
5.2 Networks with negative signals ....... . 128
5.) Networks with multi-server nodes and signals 140
5.4 Reference notes ............... . 144
Exercises 144
6 MULTIPLE CUSTOMER CLASSES AND ARBITRARY
SERVICE TIMES 146
6.1 BCMP and networks with symmetric services .... . 146
6.2 Quasi-reversibility 111' symmetric queues ....... . 152
6.3 Networks with negative signals and single-server nodes 156
6.4 Quasi-reversibility of symmetric queues with signals 161
6.5 Networks with signals and multi-server nodes 166
6.6 Reference notes 170
Exercises .................. . 170
7 NETWORKS WITH BATCH SERVICES AND NEGATIVE
SIGNALS 173
7.1 Batch service and customer coalescence 173
7.2 Batch service and negative signals 178
7.3 Special cases and examples . 184
7 .4 Refe re nee notes 193
Exercises . . . . . . . . . . 193
8 BATCH ARRIVALS, BATCH SERVICES AND CONCURRENT
MOVEMENTS 195
1-1. I lntrodudion . . . . . . . . . . 195
1-1.2 Assembly-transfer networks . 198
8.3 Concurrent batch movements 203
8.4 Examples . . . 210
8.5 Reference notes 215
Exercises ... 215
9 STATE-DEPENDENT AND HISTORY-DEPENDENT
TRANSITIONS 216
9.1 State-dependent Jackson networks . . . . . . . . . . 216
IX
9.2 Linear networks with batch movements . . . . . . . . . . . . 223
9.3 Nonlinear networks with state-dependent transicions . . . . . . 232
9.4 Networks with negative signals and state-dependent transitions 239
9.5 Networks with positive and negative signals and state-dependent
transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 I
9.6 Networks with routing depending upon service requirements . . . 258
9.7 Networks with routing depending on the number of intc:rruptions
by signals . . . 264
9.8 Reference notes 266
Exercises . . . 267
10 LOCAL BALANCE REVISITED 269
10.1 Biased local balance ..... 269
10.2 Quasi-reversibility and biased local balance . . . . . . . . . . . 273
I 0.3 Refined local balances with respect to customer cla-;ses and sc:rvict:
positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
10.4 Cross local balances with respect to positive and negative dfects 281
10.5 Cross local balance equations in networks with signals . . 288
10.6 Construction of networks with state-dependent transitions . 295
I 0. 7 Time-reversed processes and the Arrival Theorem 306
10.8 Reference notes 308
Exercises . . . . . . . . . . . . . . . . . . . . . 308
11 CHARACTERIZATION OF PRODUCT FORM AND STABILITY
ISSUES 310
I I. I Introduction and notation . . . . . . . . . . . . . . . 310
11.2 Necessary and sufficient conditions for product form 313
11.3 Quasi-reversibility revisited .......... . 321
I 1.4 Networks with multiple classes of transitions . 127
11.5 Solving traffic equations: An iterative approach 330
I 1.6 Existence of solutiom,: A fixed point approach . 335
I I .7 Reference notes 341
Exercises .......... . 341
12 DISCRETE TIME NETWORKS 344
12.1 Discrete time modeling . . . . . . . . . . . . . . 344
12.2 Discrete time quasi-reversibility . . . . . . . . . 348
12.3 Networks of discrete time quasi-reversible queues 360
12.4 Discrete time linear networks with concurrent movements . 364
12.5 A discrete time nonlinear network with concurrent movements 374
12.5.1 Model description . . . . . . . . . . 374
12.5.2 Modification of the network . . . . . 377
12.5.3 Characterization of the product form . 378
12.6 Batch transfer networks . . . . . . . . . . . . . . . . . . . . . 382
12.7 Examples of batch transfer networks . . . . . . . . . . . . . . 391
12.8 State-dependent routing and structural reversibility properties . 395
12.9 Reference notes 403
404
APPENDICES 408
A BASIC CONCEPTS IN PROBABILITY THEORY 408
A. I Probability space . . 408
A.2 Stochastic processes .. . 410
A.3 Expectations ...... . 411
A.4 Conditional expectations 412
A.5 Reference notes ..... 414
B MARKOV AND POISSON PROCESSES 415
B. I Sojourn times in a Markov chain . 415
B.2 Poisson processes ........ . 421
B.3 Markov driven Poisson processes . 424
B.4 Reference notes ......... . 429
C BROUWER'S FIXED POINT THEOREM 430
C. I Euclidean. space 430
C.2 Fixed point . . 431
C.3 Reference notes 433
REFERENCES 434
INDEX 441
Preface
Over the last four decades many books have been written on queueing theory with
a number of these focusing on queueing networks. So the reader muy ask what
the motivation behind this book is. This book basil'ally rnnlains material that is
relatively new. The content is mostly based on results lhat have appeared in the
literature during the last ten years.
The focus of the book is mainly theoretical; we establish a broad framework
and try to present results that are as general as possible. The modeling frame
work adopted is, we believe, more general than the frameworks used in most other
queueing network books. We assume throughout the book that a queueing system
not only has customers moving around. but also signals instructing customers what
to do. With a customer one usually associates a "bo<ly"; with a signal one usually
does not.
Since the structure of the models considered is more general than that in any
existing text. we believe that the application domain is also wider than that of any
other text. We do refer regularly to examples that motivate the theoretical mod
els. These examples come mainly from areas such as manufacturing. distribution,
computers and telecommunications. They include production models with quality
control stations that scrap multiple items at the same time, computer virus models.
circuit switched telecommunication networks. and predator-prey models.
The book is meant for Masters and Ph.D. students in Engineering. Operations
Research, and Probability and Statistics. At the end of each chapter there are ex
ercises. where the student is asked either to prove results that are presented in the
text without proofs or to consider models thal require some extension of the theory.
The bcx,k can be used for a Masters/Ph.D. level course in. for example. electrical
engineering with students that are somewhat theoretically inclined. Students do not
need a course in elementary queueing as a prercqui~ilc. However. they do need a
course in stochastit· processes. Parts of the hook have been used in Masters/Ph.D.
level courses at Columbia University and at Science University of Tokyo.
In the references we include only those that are directly related to the material
covered in the text. We believe that references on topics that are similar but not
directly related can be found in other queueing network books; including all of
them would detract from the thrust of the book.
XI
XII
We have developed and will continue to maintain a website which contains
information that could be of interest lo readers of this book. The address is
http://www.i.uwda.Jut.ac jpr qnet
This website includes material that could be useful to an instructor who teaches a
course in queueing networks. It provides hints to the exercises in the book as well
as some additional exercise problems. It also has links to other related websites.
This book has benefited greatly from joint work and discussions with our re
search collaborators, including Richard Boucherie, Issei Kino, Hideo Osawa,
Richard F. Serfozo, Karl Sigman, Hiroyuki Takada, Peter G. Taylor, Nico M. van
Dijk, Ronald W. Wolff, Hideaki Yamashita, and Shaohui Zheng. We arc indebted
to our students al Columbia University and Science University of Tokyo for their
detailed comments and suggestions. We are also grateful for the helpful advice
from Richard Freedman and Kanji Yoneyama. Finally, we acknowledge Yasutaka
Haltori for drawing many of the figures in the book.
XIULI CHAO
MASAKIYO MIYAZAWA
MICHAEL PINEDO
