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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

Max Plus at Work

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

Hardcover Published: 27th November 2005
ISBN: 9780691117638
Number Of Pages: 224
For Ages: 22+ years old

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Trains pull into a railroad station and must wait for each other before leaving again in order to let passengers change trains. How do mathematicians then calculate a railroad timetable that accurately reflects their comings and goings? One approach is to use max-plus algebra, a framework used to model Discrete Event Systems, which are well suited to describe the ordering and timing of events. This is the first textbook on max-plus algebra, providing a concise and self-contained introduction to the topic.

Applications of max-plus algebra abound in the world around us. Traffic systems, computer communication systems, production lines, and flows in networks are all based on discrete even systems, and thus can be conveniently described and analyzed by means of max-plus algebra.

The book consists of an introduction and thirteen chapters in three parts. Part One explores the introduction of max-plus algebra and of system descriptions based upon it. Part Two deals with a real application, namely the design of timetables for railway networks. Part Three examines various extensions, such as stochastic systems and min-max-plus systems. The text is suitable for last-year undergraduates in mathematics, and each chapter provides exercises, notes, and a reference section.

"Max Plus at Work is the best English textbook for learning eigenvector eigenvalues and the asymptotic regime of max-plus systems."--J. P. Quadrat, Director of Research, International Research Institute
"This book is very accessible, providing many examples and a clear road map for learning about max-plus algebra."--Bart De Schutter, Delft University of Technology

Prefacep. ix
Prolegomenonp. 1
Introductory Examplep. 1
On the Notationp. 3
On Eigenvalues and Eigenvectorsp. 6
Some Modeling Issuesp. 7
Counter and Dater Descriptionsp. 8
Exercisesp. 9
Notesp. 10
Max-Plus Algebrap. 11
Max-Plus Algebrap. 13
Basic Concepts and Definitionsp. 13
Vectors and Matricesp. 17
A First Max-Plus Modelp. 20
The Projective Spacep. 24
Exercisesp. 25
Notesp. 26
Spectral Theoryp. 28
Matrices and Graphsp. 28
Eigenvalues and Eigenvectorsp. 36
Solving Linear Equationsp. 42
Exercisesp. 44
Notesp. 45
Periodic Behavior and the Cycle-Time Vectorp. 47
Cyclicity and Transient Timep. 48
The Cycle-Time Vector: Preliminary Resultsp. 56
The Cycle-Time Vector: General Resultsp. 62
A Sunflower Bouquetp. 67
Exercisesp. 69
Notesp. 70
Asymptotic Qualitative Behaviorp. 72
Periodic Regimesp. 72
Characterization of the Eigenspacep. 74
Primitive Matricesp. 79
Limits in the Projective Spacep. 80
Higher-Order Recurrence Relationsp. 82
Exercisesp. 83
Notesp. 84
Numerical Procedures for Eigenvalues of Irreducible Matricesp. 85
Karp's Algorithmp. 85
The Power Algorithmp. 91
Exercisesp. 94
Notesp. 94
A Numerical Procedure for Eigenvalues of Reducible Matricesp. 95
Howard's Algorithmp. 96
Examplesp. 102
Howard's Algorithm for Higher-Order Modelsp. 108
Exercisesp. 110
Notesp. 111
Tools and Applicationsp. 113
Petri Netsp. 115
Petri Nets and Event Graphsp. 115
The Autonomous Casep. 119
The Nonautonomous Casep. 122
Exercisesp. 124
Notesp. 125
The Dutch Railway System Captured in a Max-Plus Modelp. 126
The Line Systemp. 126
Construction of the Timed Event Graphp. 130
State Space Descriptionp. 132
Application of Howard's Algorithmp. 137
Exercisesp. 138
Notesp. 139
Delays, Stability Measures, and Results for the Whole Networkp. 140
Propagation of Delaysp. 140
Results for the Whole Dutch Intercity Networkp. 145
Other Modeling Issuesp. 148
Exercisesp. 151
Notesp. 152
Capacity Assessmentp. 153
Capacity Assessment with Different Types of Trainsp. 153
Capacity Assessment for a Series of Tunnelsp. 154
Exercisesp. 158
Notesp. 159
Extensionsp. 161
Stochastic Max-Plus Systemsp. 163
Basic Definitions and Examplesp. 164
The Subadditive Ergodic Theoremp. 167
Matrices with Fixed Supportp. 171
Beyond Fixed Supportp. 174
Exercisesp. 175
Notesp. 176
Min-Max-Plus Systems and Beyondp. 177
Min-Max-Plus Systemsp. 177
Links to Other Mathematical Areasp. 187
Exercisesp. 189
Notesp. 190
Continuous and Synchronized Flows on Networksp. 191
Dater and Counter Descriptionsp. 191
Continuous Flows without Capacity Constraintsp. 192
Continuous Flows with Capacity Constraintsp. 197
Exercisesp. 199
Notesp. 200
Bibliographyp. 201
List of Symbolsp. 206
Indexp. 209
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780691117638
ISBN-10: 0691117632
Series: Princeton Series in Applied Mathematics
Audience: Tertiary; University or College
For Ages: 22+ years old
Format: Hardcover
Language: English
Number Of Pages: 224
Published: 27th November 2005
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 23.5 x 15.2  x 2.03
Weight (kg): 0.46

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