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Max-plus algebra and system theory: Where we are and where to go now
- Annu. Rev. Control
, 1999
"... Abstract: More than sixteen years after the beginning of a linear theory for certain discrete event systems in which max-plus algebra and similar algebraic tools play a central role, this paper attempts to summarize some of the main achievements in an informal style based on examples. By comparison ..."
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Cited by 71 (19 self)
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Abstract: More than sixteen years after the beginning of a linear theory for certain discrete event systems in which max-plus algebra and similar algebraic tools play a central role, this paper attempts to summarize some of the main achievements in an informal style based on examples. By comparison with classical linear system theory, there are areas which are practically untouched, mostly because the corresponding mathematical tools are yet to be fabricated. This is the case of the geometric approach of systems which is known, in the classical theory, to provide another important insight to system-theoretic and control-synthesis problems, beside the algebraic machinery. A preliminary discussion of geometric aspects in the max-plus algebra and their use for system theory is proposed in the last part of the paper. Résumé: Plus de seize ans après le début d’une théorie linéaire de certains systèmes à événements discrets dans laquelle l’algèbre max-plus et autres outils algébriques assimilés jouent un rôle central, ce papier cherche àdécrire quelques uns des principaux résultats obtenus de façon informelle, en s’appuyant sur des exemples. Par comparaison avec la théorie classique des systèmes linéaires, il existe des domaines pratiquement vierges, surtout en raison du fait que les outils mathématiques correspondants restent à forger. C’est en particulier le cas de l’approche géométrique des systèmes qui, dans la théorie classique, est connue pour apporter un autre regard important sur les questions de théorie des systèmes et de synthèse de lois de commandes àcôté de la machinerie purement algébrique. Une discussion préliminaire sur les aspects géométriques de l’algèbre max-plus et leur utilité pour la théorie des systèmes est proposée dans la dernière partie du papier.
Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture
- J. of the American Mathematical Society
, 2001
"... A topical map is a map from Rn into itself verifying some conditions (see §1.2) and which, roughly speaking, behaves like a translation along some line, the amount of which is measured by a real number, called the average height (or average displacement) of the map. Then we look at a topical Iterate ..."
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Cited by 71 (5 self)
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A topical map is a map from Rn into itself verifying some conditions (see §1.2) and which, roughly speaking, behaves like a translation along some line, the amount of which is measured by a real number, called the average height (or average displacement) of the map. Then we look at a topical Iterated Function System (IFS),
Numerical computation of spectral elements in max-plus algebra
, 1998
"... We describe the specialization to max-plus algebra of Howard’s policy improvement scheme, which yields an algorithm to compute the solutions of spectral problems in the max-plus semiring. Experimentally, the algorithm shows a remarkable (almost linear) average execution time. ..."
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Cited by 55 (7 self)
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We describe the specialization to max-plus algebra of Howard’s policy improvement scheme, which yields an algorithm to compute the solutions of spectral problems in the max-plus semiring. Experimentally, the algorithm shows a remarkable (almost linear) average execution time.
Approximating the spectral radius of sets of matrices in the max-algebra is NP-hard
- THE IEEE TRANS. ON AUTOMATIC CONTROL
, 1999
"... The lower and average spectral radii measure the minimal and average growth rates, respectively, of long products of matrices taken from a finite set. The logarithm of the average spectral radius is traditionally called the Lyapunov exponent. When one forms these products in the maxalgebra, we obtai ..."
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Cited by 17 (5 self)
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The lower and average spectral radii measure the minimal and average growth rates, respectively, of long products of matrices taken from a finite set. The logarithm of the average spectral radius is traditionally called the Lyapunov exponent. When one forms these products in the maxalgebra, we obtain quantities that measure the performance of Discrete Event Systems. We show that approximating the lower and average maxalgebraic spectral radii is NP-hard.
Worst-case performance analysis of synchronous dataflow scenarios
- in CODES/ISSS
, 2010
"... Synchronous Dataflow (SDF) is a powerful analysis tool for regular, cyclic, parallel task graphs. The behaviour of SDF graphs however is static and therefore not always able to accurately capture the behaviour of modern, dynamic dataflow applications, such as embedded multimedia codecs. An approach ..."
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Cited by 16 (8 self)
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Synchronous Dataflow (SDF) is a powerful analysis tool for regular, cyclic, parallel task graphs. The behaviour of SDF graphs however is static and therefore not always able to accurately capture the behaviour of modern, dynamic dataflow applications, such as embedded multimedia codecs. An approach to tackle this limitation is by means of scenarios. In this paper we introduce a technique and a tool to automatically analyse a scenario-aware dataflow model for its worstcase performance. A system is specified as a collection of SDF graphs representing individual scenarios of behaviour and a finite state machine that specifies the possible orders of scenario occurrences. This combination accurately captures more dynamic applications and this way provides tighter results than an existing analysis based on a conservative static dataflow model, which is too pessimistic, while looking only at the ‘worst-case ’ individual scenario, without considering scenario transitions, can be too optimistic. We introduce a formal semantics of the model, in terms of (max, +) linear system-theory and in particular (max, +) automata. Leveraging existing results and algorithms from this domain, we give throughput analysis and state space generation algorithms for worst-case performance analysis. The method is implemented in a tool and the effectiveness of the approach is experimentally evaluated.
Deciding unambiguity and sequentiality from a finitely ambiguous max-plus automaton
- THEORET. COMPUT. SCI
, 2004
"... Finite automata with weights in the max-plus semiring are considered. The main result is: it is decidable whether a series that is recognized by a finitely ambiguous max-plus automaton is unambiguous, or is sequential. Furthermore, the proof is constructive. A collection of examples is given to illu ..."
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Cited by 11 (3 self)
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Finite automata with weights in the max-plus semiring are considered. The main result is: it is decidable whether a series that is recognized by a finitely ambiguous max-plus automaton is unambiguous, or is sequential. Furthermore, the proof is constructive. A collection of examples is given to illustrate the hierarchy of maxplus series with respect to ambiguity.
Asymptotic Behavior in a Heap Model with Two Pieces
- COMPUT. SCI
, 2000
"... In a heap model, solid blocks, or pieces, pile up according to the Tetris game mechanism. An optimal schedule is an infinite sequence of pieces minimizing the asymptotic growth rate of the heap. In a heap model with two pieces, we prove that there always exists an optimal schedule which is balanced, ..."
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Cited by 11 (5 self)
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In a heap model, solid blocks, or pieces, pile up according to the Tetris game mechanism. An optimal schedule is an infinite sequence of pieces minimizing the asymptotic growth rate of the heap. In a heap model with two pieces, we prove that there always exists an optimal schedule which is balanced, either periodic or Sturmian. We also consider the model where the successive pieces are chosen at random, independently and with some given probabilities. We study the expected growth rate of the heap. For a model with two pieces, the rate is either computed explicitly or given as an infinite series. We show an application for a system of two processes sharing a resource, and we prove that a greedy schedule is not always optimal.
A max-plus model of ribosome dynamics during mRNA translation. arXiv:1105.3580v1 [q-bio.QM
, 2011
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Extremal throughputs in free-choice nets
- 26th International Conference On Application and Theory of Petri Nets, LNCS
, 2004
"... We give a method to compute the throughput in a timed live and bounded free-choice Petri net under a total allocation (i.e. a 0-1 routing). We also characterize and compute the conflict-solving policies that achieve the smallest throughput in the special case of a 1-bounded net. They do not correspo ..."
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Cited by 6 (3 self)
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We give a method to compute the throughput in a timed live and bounded free-choice Petri net under a total allocation (i.e. a 0-1 routing). We also characterize and compute the conflict-solving policies that achieve the smallest throughput in the special case of a 1-bounded net. They do not correspond to total allocations, but still have a small period. Résumé Nous donnons une méthode pour calculer le débit d’un réseau de Petri à choix libres sous une allocation totale (i.e. un routage 0-1). Nous caractérisons aussi les politiques de résolution de conflits qui atteignent le débit minimal dans le cas des réseaux 1-bornés et nous montrons comment les calculer. Ce ne sont pas des allocations totales, mais des routages avec de petites périodes. Keywords: Free-choice Petri nets, timed and routed Petri nets, throughput. Mots-clés: Réseaux de Petri à choix libres, réseaux de Petri routés et temporisés, débit.
Asymptotic Analysis of Heaps of Pieces and application to Timed Petri Nets
- In PNPM'99, Saragoza
, 1999
"... What is the density of an infinite heap of pieces, if we let pieces fall down randomly, or if we select pieces to maximize the density? How many transitions of a safe timed Petri net can we fire per time unit? We reduce these questions to the computation of the average and optimal case Lyapunov expo ..."
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Cited by 6 (3 self)
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What is the density of an infinite heap of pieces, if we let pieces fall down randomly, or if we select pieces to maximize the density? How many transitions of a safe timed Petri net can we fire per time unit? We reduce these questions to the computation of the average and optimal case Lyapunov exponents of max-plus automata, and we present several techniques to compute these exponents. First, we introduce a completed "non-linear automaton", which essentially fills incrementally all the gaps that can be filled in a heap without changing its asymptotic height. Using this construction, when the pieces have integer valued shapes, and when any two pieces overlap, the Lyapunov exponents can be explicitly computed. We present two other constructions (partly based on Cartier-Foata normal forms of traces) which allow us to compute the optimal case Lyapunov exponent, assuming only that the pieces have integer valued shapes. 1 Introduction Heap models, where solid blocks are piled up according...
