Max Plus at Work: Modeling and Analysis of Synchronized Systems: A Course on Max-Plus Algebra and Its Applications

226 Pages·2005·12.56 MB·English

by  Bernd Heidergott

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Max Plus at Work
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PRINCETON SERIES IN APPLIED MATHEMATICS
EDITORS
Ingrid Daubechies, Princeton University
Wei nan E, Princeton University
Jan Karellenstra, Eindhoven University
Endre SOli, University of Oxford
TITLES IN THE SERIES
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Self-Regularity: A New Paradigm for Primal-Duo/Interior-Point Algorithms by Jiming Peng,
Conielis Roos, and Tamas Terlaky
Analytic Theory of Global Bifurcation: An Introduction by Boris Buffoni and John Toland
Entropy by Andreas Greven, Gerhard Keller, and Gerald Warnecke
Auxiliary Signal Design for Failure Detection by Stephen l. Campbell and Ramine
Nikoukhah
Thermodynamics: A Dynamical Systems Approach by Wassim M. Haddad, VijaySekhar
Chellaboina, and Sergey G. Nesesov
Optimization: Insights and Applications by Jan Brinkhuis and Vladimir Tikhomirov
Max Plus at Work: Modeling and Analysis of Synchronized Systems: A Course on Max-Plus
Algebra and Its Applications by Bernd Heidergott, Geert Jan Olsder, and Jacob van der
Woude
The Princeton Series in Applied Mathematics publishes high quality advanced texts and
monographs in all areas of applied mathematics. Books include those of a theoretical
and general nature as well as those dealing with the mathematics of specific applica
tions areas and real-world situations.
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Max Plus at Work
Modeling and Analysis of Synchronized
Systems: A Course on Max-Plus Algebra and
Its Applications
Bernd Heidergott
Geert Jan Olsder
Jacob van der Woude
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
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Copyright Q 2006 by Princeton University Press
Published by Princeton University Press, 41 William Street,
Princeton, New Jersey 08540
In the United Kingdom: Princeton University Press,
3 Market Place, Woodstock, Oxfords hire OX20 1S Y
All Rights Reserved
Library of Congress Cataloging-in-Publication Data
Heidergott, Bernd.
Max Plus at work: modeling and analysis of synchronized systems: a course on
Max-Plus algebra and its applications I Bernd Heidergott, Geert Jan Olsder,
Jacob van der Woude.
p.cm.-(Princeton series in applied mathematics)
Includes bibliographical references and index.
ISBN-13: 978-0-691-11763-8 (acid-free paper)
ISBN-10: Q-691-11763-2 (acid-free paper)
1. Matrices-Textbooks. 2. System theory-Textbooks. I. Olsder, Geert Jan. II.
Woude, J. W. van der. Ill. Title.
QA 188.H445 2006
512--dc22 2005048690
British Library Cataloging-in-Publication Data is available
The publisher would like to acknowledge the authors of this volume for providing
the camera-ready copy from which this book was printed.
Printed on acid-free paper.
pup.princeton.edu
Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
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Contents
Preface ix
Chapter 0. Prolegomenon 1
0.1 Introductory Example 1
0.2 On the Notation 3
0.3 On Eigenvalues and Eigenvectors 6
0.4 Some Modeling Issues 7
0.5 Counter and Dater Descriptions 8
0.6 Exercises 9
0.7 Notes 10
PART I. MAX-PLUS ALGEBRA 11
Chapter 1. Max-Plus Algebra 13
1.1 Basic Concepts and Definitions 13
1.2 Vectors and Matrices 17
1.3 A First Max-Plus Model 20
1.4 The Projective Space 24
1.5 Exercises 25
1.6 Notes 26
Chapter 2. Spectral Theory 28
2.1 Matrices and Graphs 28
2.2 Eigenvalues and Eigenvectors 36
2.3 Solving Linear Equations 42
2.4 Exercises 44
2.5 Notes 45
Chapter 3. Periodic Behavior and the Cycle-Time Vector 47
3.1 Cyclicity and Transient Time 48
3.2 The Cycle-Time Vector: Preliminary Results 56
3.3 The Cycle-Time Vector: General Results 62
3.4 A Sunflower Bouquet 67
3.5 Exercises 69
3.6 Notes 70
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vi CONTENTS
Chapter 4. Asymptotic Qualitative Behavior 72
4.1 Periodic Regimes 72
4.2 Characterization of the Eigenspace 74
4.3 Primitive Matrices 79
4.4 Limits in the Projective Space 80
4.5 Higher-Order Recurrence Relations 82
4.6 Exercises 83
4.7 Notes 84
Chapter 5. Numerical Procedures for Eigenvalues of Irreducible
Matrices 85
5.1 Karp's Algorithm 85
5.2 The Power Algorithm 91
5.3 Exercises 94
5.4 Notes 94
Chapter 6. A Numerical Procedure for Eigenvalues of Reducible
Matrices 95
6.1 Howard's Algorithm 96
6.2 Examples 102
6.3 Howard's Algorithm for Higher-Order Models 108
6.4 Exercises 110
6.5 Notes 111
PART II. TOOLS AND APPLICATIONS 113
Chapter 7. Petri Nets 115
7.1 Petri Nets and Event Graphs 115
7.2 The Autonomous Case 119
7.3 The Nonautonomous Case 122
· 7.4 Exercises 124
7.5 Notes 125
Chapter 8. The Dutch Railway System Captured in a Max-Plus Model 126
8.1 The Line System 126
8.2 Construction of the Timed Event Graph 130
8.3 State Space Description 132
8.4 Application of Howard's Algorithm 137
8.5 Exercises 138
8.6 Notes 139
Chapter 9. Delays, Stability Measures, and Results for the Whole
Network 140
9.1 Propagation of Delays 140
9.2 Results for the Whole Dutch Intercity Network 145
9.3 Other Modeling Issues 148
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CONTENTS vii
9.4 Exercises 151
9.5 Notes 152
Chapter 10. Capacity Assessment 153
I O.I Capacity Assessment with Different 'JYpes of Train_s 153
I 0.2 Capacity Assessment for a Series of TUnnels 154
10.3 Exercises 158
10.4 Notes 159
PART Ill. EXTENSIONS 161
Chapter 11. Stochastic Max-Plus Systems 163
II.I Basic Definitions and Examples 164
II.2 The Subadditive Ergodic Theorem 167
II.3 Matrices with Fixed Support 171
II.4 Beyond Fixed Support 174
11.5 Exercises 175
Il.6 Notes 176
Chapter 12. Min-Max-Plus Systems and Beyond 177
l2.I Min-Max-Plus Systems 177
I2.2 Links to Other Mathematical Areas 187
I2.3 Exercises 189
I2.4 Notes 190
Chapter 13. Continuous and Synchronized Flows on Networks 191
13.I Dater and Counter Descriptions 191
13.2 Continuous Aows without Capacity Constraints 192
I3.3 Continuous Flows with Capacity Constraints 197
13.4 Exercises 199
13.5 Notes 200
Bibliography 201
List of Symbols 206
Index 209
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Preface
This textbook is concerned with sequences of events, where events are viewed as
sudden changes in a process to be studied. Some examples of events are a message
arrives, a train leaves the station, and a door opens. This book deals with the model
ing, analysis, and timing of such events, all subject to synchronization constraints.
These constraints are relations that exist between the events, such as a message
must have been sent before it can arrive and a certain train should not depart before
another train has arrived (in order to allow the changeover of passengers).
Apart from the introductory Chapter 0, the book consists of three parts. Part I
(Chapters 1-6) deals with max-plus algebra, i.e., an algebra in which the evolution
of events can be described in a convenient way. Part II (Chapters 7-10) covers two
specific applications, both related to timetable design for railway networks. Part III
(Chapters 11-13) deals with various extensions. Later on in this Preface, we give a
brief description of each chapter.
The level of the book is last-year undergraduate student mathematics. The book
will be of interest for applied mathematicians, operations researchers, econome
tricians, and civil, electrical, and mechanical engineers with quantitative back
grounds. Most important is a basic knowledge of conventional algebra and matrix
calculus, with the addition of some knowledge of system theory (or recurrence re
lations). Some knowledge of stochastic processes is required for Chapter 11 only.
No prior knowledge of graph theory and the modeling tool of Petri nets is required.
Each chapter can be taught conveniently in two hours. The only exceptions to
this rule are Chapter 3 (which requires at least three hours) and Chapter 10 (for
which one hour is enough). With two hours per week, this would mean that fourteen
weeks would suffice to teach the whole book. Of course, depending on interest,
some chapters can be studied in more depth, requiring more than two hours, while
others can be skipped or can be treated superficially in one hour. A minimum course
would cover the material of Chapters 0, 1, 2, 7 and 8. From a mathematical point
of view, Chapter 3 is probably the most challenging in presenting key results. All
chapters contain an exercise section as well as a notes section with suggestions for
further reading and sometimes other remarks.
On the timescale of mathematical evolution, max-plus algebra and its applica
tions are a recent phenomenon. Originally, some results were published in jour
nals (e.g., [30], [43]), and even earlier traces exist. The first major leap forward in
this algebra was the appearance of the book [31] in 1979. The next book, written
from a system-theoretical perspective, was [5]. It can be viewed as a product of
the "French school." Neither book can be called a textbook. The current book is
believed to be the first textbook in the area of max-plus algebra and its applica-
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