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Max Plus at Work

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
Copyright Date: 2006
Pages: 224
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  • Book Info
    Max Plus at Work
    Book Description:

    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.

    eISBN: 978-1-4008-6523-9
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-iv)
  2. Table of Contents
    (pp. v-viii)
  3. Preface
    (pp. ix-xii)
    Bernd Heidergott, Geert Jan Olsder and Jacob van der Woude
  4. Chapter Zero Prolegomenon
    (pp. 1-10)

    In this book we will model, analyze, and optimize phenomena in which the order of events is crucial. The timing of such events, subject to synchronization constraints, forms the core. This zeroth chapter can be viewed as an appetizer for the other chapters to come.

    Consider a simple railway network between two cities, each with a station, as indicated in Figure 0.1. These stations are calledS1andS2, respectively, and are connected by two tracks. One track runs fromS1toS2, and the travel time for a train along this track is assumed to be 3...


    • Chapter One Max-Plus Algebra
      (pp. 13-27)

      In the previous chapter we described max-plus algebra in an informal way. The present chapter contains a more rigorous treatment of max-plus algebra. In Section 1.1 basic concepts are introduced, and algebraic properties of max-plus algebra are studied. Matrices and vectors over max-plus algebra are introduced in Section 1.2, and an important model, calledheap of piecesorheap model, which can be described by means of max-plus algebra, is presented in Section 1.3. Finally, the projective space, a mathematical framework most convenient for studying limits, is introduced in Section 1.4.

      Define$\varepsilon \overset{def}{\mathop{=}}\,-\infty $and$e\overset{def}{\mathop{=}}\,0$, and denoted by ℝmax...

    • Chapter Two Spectral Theory
      (pp. 28-46)

      This chapter is devoted to spectral theory of matrices over the max-plus semiring. In Section 2.1 we will study the relation between graphs and matrices over the max-plus semiring. The basic observation is that any square matrix can be translated into a weighted graph (to be defined shortly) and that products and powers of matrices over the max-plus semiring have entries with a nice graph-theoretical interpretation. This interpretation will be further studied in Section 2.2. The key result will be that, under mild conditions, a square matrix over the max-plus semiring possesses a unique eigenvalue that equals the maximal average...

    • Chapter Three Periodic Behavior and the Cycle-Time Vector
      (pp. 47-71)

      This chapter deals with sequences {x(k) :k∈ ℕ} generated by

      x(k+ 1) =Ax(k),

      fork≥ 0, whereA$\mathbb{R}_{\max }^{n\times n}$andx(0) =x0$\mathbb{R}_{\max }^{n}$is the initial condition. The sequences are then equivalently described by

      x(k) =Akx0, (3.1)

      for allk≥ 0.

      Definition 3.1Let{x(k) :k∈ ℕ}be a sequence in$\mathbb{R}_{\max }^{n}$,and assume that for all jṉ the quantity ƞj, defined by\[\underset{k\to \infty }{\mathop{\lim }}\,\frac{{{x}_{j}}(k)}{k}\],exists. The vector ƞ = (ƞ12,…,ƞn)is called the cycle-time vectorof the sequence x(k).If all ƞj’s...

    • Chapter Four Asymptotic Qualitative Behavior
      (pp. 72-84)

      As in the previous chapter, we will study in this chapter sequences {x(k) :k∈ ℕ} given through

      x(k+ 1) =Ax(k),k∈ ℕ, (4.1)

      with initial vectorx(0) =x0$\mathbb{R}_{\max }^{n}$andA$\mathbb{R}_{\max }^{n\times n}$. Provided thatAis irreducible with unique eigenvalue λ and associated eigenvectorv, it follows forx(O) = v andk≥ 0 thatx(k) =A⊗kx(0) = λ⊗kv. In words, the vectorsx(k) are proportional tov, and we may therefore say that thequalitativeasymptotic behavior ofx(k) is completely characterized...

    • Chapter Five Numerical Procedures for Eigenvalues of Irreducible Matrices
      (pp. 85-94)

      In this chapter we discuss two numerical procedures for irreducible matrices over max-plus algebra. The first one, calledKarp’s algorithm, will be presented in Section 5.1 and yields the eigenvalue of an irreducible matrix. The second one, called apower algorithm, to be presented in Section 5.2, yields the eigenvalue and a corresponding eigenvector. Notice that we have already encountered an algorithm for computing the eigenvalue in Chapter 2. Indeed, by Theorem 2.9 the eigenvalue of an irreducible matrixAis equal to the maximal average circuit weight of the communication graph ofA.

      We start in this chapter from...

    • Chapter Six A Numerical Procedure for Eigenvalues of Reducible Matrices
      (pp. 95-112)

      The generalized eigenmode of a square matrix has been introduced and studied in Chapter 3. More specifically, in Sections 3.2 and 3.3 the existence of a generalized eigenmode of a square regular matrix has been proved by making use of its normal form. As the proofs in Sections 3.2 and 3.3 are constructive, a conceptual algorithm has been obtained by which a generalized eigenmode in principle can be computed. See in particular the proof of Corollary 3.16. However, the obtained algorithm heavily relies on a normal form of the matrix involved.

      In this chapter an alternative algorithm is presented.Howard’s...


    • Chapter Seven Petri Nets
      (pp. 115-125)

      In this chapter we will give a brief introduction to Petri nets as a modeling tool. We will show that a subclass of Petri nets, the so-called event graphs, is a suitable modeling aid for the construction of max-plus linear systems (i.e., for the construction of equations like (0.9) or (4.7)). In Section 7.1, the definitions of a Petri net and a timed event graph will be given. The construction of max-plus linear systems, starting from an event graph description of a model, will be treated in Section 7.2 for the autonomous case (i.e., when no external inputs are considered),...

    • Chapter Eight The Dutch Railway System Captured in a Max-Plus Model
      (pp. 126-139)

      This chapter and the next deal with the application of max-plus algebra in a study of the timetable of the Dutch railway system. The starting point is the railway track layout, consisting of a number of lines along which trains run up and down, and the requested synchronization data (i.e., which trains should wait for which other trains in order to allow passengers to transfer from one to the other). It will be assumed that this data is provided. In addition, it is assumed that a timetable is given with a period of one hour (or, a frequency of one...

    • Chapter Nine Delays, Stability Measures, and Results for the Whole Network
      (pp. 140-152)

      This chapter is a follow-up to the previous one. Once a timetable is given, we are interested in its sensitivity with respect to disturbances in the system. A question that came up during one of the discussions at the Dutch railway headquarters was, how many minutes can all changeover times be increased such that a timetable with a period of sixty minutes still can be maintained? The underlying reason for this question was that an increase in age of the average passenger is expected during the coming years due to the baby boom after World War II. Older passengers walk...

    • Chapter Ten Capacity Assessment
      (pp. 153-160)

      This chapter illustrates the application of max-plus algebra to models with sharing of and competition for resources. Section 10.1 will describe the occupation of a railway track (being the resource) by two types of trains. Slow and fast trains alternately use the track. The heaps of pieces approach, as introduced in Section 1.3, provides useful insights. Section 10.2 will deal with a real-life study in which the competition for resources stems from the situation in which a double-track railway line passes through three tunnels, each of them essentially functioning as a single-track section. Section 10.2.1 will deal with a stylized...


    • Chapter Eleven Stochastic Max-Plus Systems
      (pp. 163-176)

      This chapter is devoted to the study of sequences {x(k) :k∈ ℕ} satisfying the recurrence relation

      x(k+ 1) =A(k) ⊗x(k),k≥0, (11.1)

      wherex(0) =x0$\mathbb{R}_{\max }^{n}$is the initial value and {A(k) :k∈ ℕ} is a sequence ofnxnmatrices over$\mathbb{R}_{\max }$· In order to develop a meaningful mathematical theory, we need some additional assumptions on {A(k) :k∈ ℕ}. The approach presented in this chapter assumes that {A(k) :k∈ ℕ} is a sequence of random matrices$\mathbb{R}_{\max }^{n\times n}$in defined on a common probability space....

    • Chapter Twelve Min-Max-Plus Systems and Beyond
      (pp. 177-190)

      In this chapter min-max-plus systems will be studied. Such systems can be viewed as an extension of max-plus systems in the sense that in addition to the max and plus operators, the min(imization) operator is now also allowed. This gives more flexibility with respect to modeling issues. At the end of this chapter, we will briefly discuss the imbedding of min-max-plus systems in the even more general class of nonexpansive systems.

      Min-max-plus systems are described by expressions in which the three operations minimization, maximization, and addition appear. They can be viewed as an extension of max-plus expressions in the sense...

    • Chapter Thirteen Continuous and Synchronized Flows on Networks
      (pp. 191-200)

      So far, we have formulated timed events as discrete flows on networks. In this section, we consider a continuous version of such flows.

      One possible way to define, describe, and analyze such continuous flows is by limit arguments in timed event graphs (Chapter 7). In such an approach tokens are split up into mini-tokens (say, one original token consists ofNidentical minitokens); the original corresponding place is replaced byNplaces in series, with one mini-token in each of them and with transitions in between. The original holding times are divided byN(firing times remain zero). A transition...

  8. Bibliography
    (pp. 201-205)
  9. List of Symbols
    (pp. 206-208)
  10. Index
    (pp. 209-213)