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38
Synchronization and linearity: an algebra for discrete event systems
, 2001
"... The first edition of this book was published in 1992 by Wiley (ISBN 0 471 93609 X). Since this book is now out of print, and to answer the request of several colleagues, the authors have decided to make it available freely on the Web, while retaining the copyright, for the benefit of the scientific ..."
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Cited by 252 (10 self)
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The first edition of this book was published in 1992 by Wiley (ISBN 0 471 93609 X). Since this book is now out of print, and to answer the request of several colleagues, the authors have decided to make it available freely on the Web, while retaining the copyright, for the benefit of the scientific community. Copyright Statement This electronic document is in PDF format. One needs Acrobat Reader (available freely for most platforms from the Adobe web site) to benefit from the full interactive machinery: using the package hyperref by Sebastian Rahtz, the table of contents and all LATEX crossreferences are automatically converted into clickable hyperlinks, bookmarks are generated automatically, etc.. So, do not hesitate to click on references to equation or section numbers, on items of thetableofcontents and of the index, etc.. One may freely use and print this document for one’s own purpose or even distribute it freely, but not commercially, provided it is distributed in its entirety and without modifications, including this preface and copyright statement. Any use of thecontents should be acknowledged according to the standard scientific practice. The
Algebraic Tools for the Performance Evaluation of Discrete Event Systems
 IEEE Proceedings: Special issue on Discrete Event Systems
, 1989
"... In this paper, it is shown that a certain class of Petri nets called event graphs can be represented as linear "timeinvariant" finitedimensional systems using some particular algebras. This sets the ground on which a theory of these systems can be developped in a manner which is very ana ..."
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Cited by 69 (6 self)
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In this paper, it is shown that a certain class of Petri nets called event graphs can be represented as linear "timeinvariant" finitedimensional systems using some particular algebras. This sets the ground on which a theory of these systems can be developped in a manner which is very analogous to that of conventional linear system theory. Part 2 of the paper is devoted to showing some preliminary basic developments in that direction. Indeed, there are several ways in which one can consider event graphs as linear systems: these ways correspond to approaches in the time domain, in the event domain and in a twodimensional domain. In each of these approaches, a di#erent algebra has to be used for models to remain linear. However, the common feature of these algebras is that they all fall into the axiomatic definition of "dioids". Therefore, Part 1 of the paper is devoted to a unified presentation of basic algebraic results on dioids. 1 Introduction Definitions and examples of Discrete ...
Modeling and Analysis of Timed Petri Nets Using Heaps of Pieces
, 1997
"... We show that safe timed Petri nets can be represented by special automata over the (max,+) semiring, which compute the height of heaps of pieces. This extends to the timed case the classical representation a la Mazurkievicz of the behavior of safe Petri nets by trace monoids and trace languages. Fo ..."
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Cited by 42 (16 self)
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We show that safe timed Petri nets can be represented by special automata over the (max,+) semiring, which compute the height of heaps of pieces. This extends to the timed case the classical representation a la Mazurkievicz of the behavior of safe Petri nets by trace monoids and trace languages. For a subclass including all safe Free Choice Petri nets, we obtain reduced heap realizations using structural properties of the net (covering by safe state machine components). We illustrate the heapbased modeling by the typical case of safe jobshops. For a periodic schedule, we obtain a heapbased throughput formula, which is simpler to compute than its traditional timed event graph version, particularly if one is interested in the successive evaluation of a large number of possible schedules. Keywords Timed Petri nets, automata with multiplicities, heaps of pieces, (max,+) semiring, scheduling. I. Introduction The purpose of this paper 1 is to prove the following result: Timed safe Pe...
A spectral theorem for convex monotone homogeneous maps
 In Proceedings of the Satellite Workshop on MaxPlus Algebras, IFAC SSSC’01
, 2001
"... Abstract. We consider convex maps f: R n → R n that are monotone (i.e., that preserve the product ordering of R n), and nonexpansive for the supnorm. This includes convex monotone maps that are additively homogeneous (i.e., that commute with the addition of constants). We show that the fixed point ..."
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Cited by 17 (9 self)
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Abstract. We consider convex maps f: R n → R n that are monotone (i.e., that preserve the product ordering of R n), and nonexpansive for the supnorm. This includes convex monotone maps that are additively homogeneous (i.e., that commute with the addition of constants). We show that the fixed point set of f, when it is nonempty, is isomorphic to a convex infsubsemilattice of R n, whose dimension is at most equal to the number of strongly connected components of a critical graph defined from the tangent affine maps of f. This yields in particular an uniqueness result for the bias vector of ergodic control problems. This generalizes results obtained previously by Lanery, Romanovsky, and Schweitzer and Federgruen, for ergodic control problems with finite state and action spaces, which correspond to the special case of piecewise affine maps f. We also show that the length of periodic orbits of f is bounded by the cyclicity of its critical graph, which implies that the possible orbit lengths of f are exactly the orders of elements of the symmetric group
Algebraic System Analysis of Timed Petri Nets
, 1997
"... We show that Continuous Timed Petri Nets (CTPN) can be modeled by generalized polynomial recurrent equations in the (min,+) semiring. We establish a correspondence between CTPN and Markov decision processes. We survey the basic system theoretical results available: behavioral (inputoutput) properti ..."
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Cited by 15 (6 self)
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We show that Continuous Timed Petri Nets (CTPN) can be modeled by generalized polynomial recurrent equations in the (min,+) semiring. We establish a correspondence between CTPN and Markov decision processes. We survey the basic system theoretical results available: behavioral (inputoutput) properties, algebraic representations, asymptotic regime. A particular attention is paid to the subclass of stable systems (with asymptotic linear growth). 1 Introduction The fact that a subclass of Discrete Event Systems equations write linearly in the (min,+) or in the (max,+) semiring is now almost classical [9, 2]. The (min,+) linearity allows the presence of synchronization and saturation features but unfortunately prohibits the modeling of many interesting phenomena such as "birth" and "death" processes (multiplication of tokens) and concurrency. The purpose of this paper is to show that after some simplifications, these additional features can be represented by polynomial recurrences in the ...
Asymptotic Throughput of Continuous Timed Petri Nets
, 1995
"... We set up a connection between Continuous Timed Petri Nets (the fluid version of usual Timed Petri Nets) and Markov decision processes. We characterize the subclass of Continuous Timed Petri Nets corresponding to undiscounted average cost structure. This subclass satisfies consetration laws and show ..."
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Cited by 14 (5 self)
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We set up a connection between Continuous Timed Petri Nets (the fluid version of usual Timed Petri Nets) and Markov decision processes. We characterize the subclass of Continuous Timed Petri Nets corresponding to undiscounted average cost structure. This subclass satisfies consetration laws and shows a linear growth: one obtains as mere application of existing results for Dynamic Programming the existence of an asymptotic throughput. This rate can be computed using Howardtype 'algorithms, or by an extension of the well known cycle time formula for timed event graphs. We present an illustrating example and briefly sketch the relation with the discrete case.
Minplus methods in eigenvalue perturbation theory and generalised LidskiiVishikLjusternik theorem
, 2005
"... Abstract. We extend the perturbation theory of Viˇsik, Ljusternik and Lidskiĭ for eigenvalues of matrices, using methods of minplus algebra. We show that the asymptotics of the eigenvalues of a perturbed matrix is governed by certain discrete optimisation problems, from which we derive new perturba ..."
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Cited by 13 (1 self)
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Abstract. We extend the perturbation theory of Viˇsik, Ljusternik and Lidskiĭ for eigenvalues of matrices, using methods of minplus algebra. We show that the asymptotics of the eigenvalues of a perturbed matrix is governed by certain discrete optimisation problems, from which we derive new perturbation formulæ, extending the classical ones and solving cases which where singular in previous approaches. Our results include general weak majorisation inequalities, relating leading exponents of eigenvalues of perturbed matrices and minplus analogues of eigenvalues. 1.
Maxplus algebra
, 2006
"... Maxplus algebra has been discovered more or less independently by several schools, in relation with various mathematical fields. This chapter is limited to finite dimensional linear algebra. For more information, the reader may consult the books [CG79, Zim81, CKR84, BCOQ92, KM97, GM02]. The collect ..."
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Cited by 11 (4 self)
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Maxplus algebra has been discovered more or less independently by several schools, in relation with various mathematical fields. This chapter is limited to finite dimensional linear algebra. For more information, the reader may consult the books [CG79, Zim81, CKR84, BCOQ92, KM97, GM02]. The collections of articles [MS92, Gun98, LM05] give a good idea of current developments.
On the Burnside problem for Semigroups of Matrices in the (max,+) Algebra
, 1996
"... We show that the answer to the Burnside problem is positive for semigroups of matrices with entries in the (max,+)algebra (that is, the semiring (R[ f\Gamma1g; max; +)), and also for semigroups of (max,+)linear projective maps with rational entries. An application to the estimation of the Lyapuno ..."
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Cited by 11 (2 self)
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We show that the answer to the Burnside problem is positive for semigroups of matrices with entries in the (max,+)algebra (that is, the semiring (R[ f\Gamma1g; max; +)), and also for semigroups of (max,+)linear projective maps with rational entries. An application to the estimation of the Lyapunov exponent of certain products of random matrices is also discussed. 1. Introduction The "(max,+)algebra" is a traditional name for the semiring (R[f\Gamma1g; max; +), denoted Rmax in the sequel. This is a particular example of idempotent semiring (that is a semiring whose additive law satisfies a \Phi a = a), also known as dioid [17, 18, 2]. This algebraic structure has been popularized by its applications to Graph Theory and Operations Research [17, 8]. Linear operators in this algebra are central in HamiltonJacobi theory and in the study of exponential asymptotics [33]. The study of automata and semigroups of matrices over the analogous "tropical" semiring (N [ f+1g;min;+) has been ...
Resource Optimization and (min,+) Spectral Theory
 IEEE Trans. on Automat. Contr
, 1995
"... We show that certain resource optimization problems relative to Timed Event Graphs reduce to linear programs. The auxiliary variables which allow this reduction can be interpreted in terms of eigenvectors in the (min,+) algebra. KeywordsResource Optimization, Timed Event Graphs, (max,+) algebra, ..."
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Cited by 10 (2 self)
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We show that certain resource optimization problems relative to Timed Event Graphs reduce to linear programs. The auxiliary variables which allow this reduction can be interpreted in terms of eigenvectors in the (min,+) algebra. KeywordsResource Optimization, Timed Event Graphs, (max,+) algebra, spectral theory. I. INTRODUCTION Timed Event Graphs (TEGs) are a subclass of timed Petri nets which can be used to model deterministic discrete event dynamic systems subject to saturation and synchronization phenomena: typically, flexible manufacturing systems, multiprocessor systems, transportation networks [5], [1], [3], [2], [16], [17]. The most remarkable result about TEGs [4], [3], [1] is certainly the following: a TEG functioning at maximal speed reaches after a finite time a periodic regime. More precisely, let x denote the counter function of a given transition of the graph. That is, x(t) represents the number of firings of the transition up to time t, usually the number of parts...