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13
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 56 (18 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...
Approximating the spectral radius of sets of matrices in the maxalgebra is NPhard
 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 NPhard.
Worstcase 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 scenarioaware 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 ‘worstcase ’ individual scenario, without considering scenario transitions, can be too optimistic. We introduce a formal semantics of the model, in terms of (max, +) linear systemtheory and in particular (max, +) automata. Leveraging existing results and algorithms from this domain, we give throughput analysis and state space generation algorithms for worstcase 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 maxplus automaton
 THEORET. COMPUT. SCI
, 2004
"... Finite automata with weights in the maxplus semiring are considered. The main result is: it is decidable whether a series that is recognized by a finitely ambiguous maxplus 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 maxplus semiring are considered. The main result is: it is decidable whether a series that is recognized by a finitely ambiguous maxplus 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.
Analytic Expansions of (max,+) Lyapunov Exponents
, 1998
"... We give an explicit analytic series expansion of the (max; +)Lyapunov exponent fl(p) of a sequence of independent and identically distributed random matrices in this algebra, generated via a Bernoulli scheme depending on a small parameter p. A key assumption is that one of the matrices has a unique ..."
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Cited by 10 (1 self)
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We give an explicit analytic series expansion of the (max; +)Lyapunov exponent fl(p) of a sequence of independent and identically distributed random matrices in this algebra, generated via a Bernoulli scheme depending on a small parameter p. A key assumption is that one of the matrices has a unique eigenvector. This allows us to use a representation of this exponent as the mean value of a certain random variable, and then a discrete analogue of the socalled lighttraffic perturbation formulas to derive the expansion. We show that it is analytic under a simple condition on p. This also provides a closed form expression for all derivatives of fl(p) at p = 0 and approximations of fl(p) of any order, together with an error estimate for nite order Taylor approximations. Several extensions of this are discussed, including expansions of multinomial schemes depending on small parameters (p 1, ..., p m ) and expansions for exponents associated with iterates of a class of random operators...
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 maxplus automata, and we present several techniques to compute these exponents. First, we introduce a completed "nonlinear 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 CartierFoata 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...
Series Expansions of Lyapunov Exponents and Forgetful Monoids
, 2000
"... We consider Lyapunov exponents of random iterates of monotone homogeneous maps. We assume that the images of some iterates are lines, with positive probability. Using this memoryloss property which holds generically for random products of matrices over the maxplus semiring, and in particular, for ..."
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Cited by 5 (0 self)
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We consider Lyapunov exponents of random iterates of monotone homogeneous maps. We assume that the images of some iterates are lines, with positive probability. Using this memoryloss property which holds generically for random products of matrices over the maxplus semiring, and in particular, for Tetrislike heaps of pieces models, we give a series expansion formula for the Lyapunov exponent, as a function of the probability law. In the case of rational probability laws, we show that the Lyapunov exponent is an analytic function of the parameters of the law, in a domain that contains the absolute convergence domain of a partition function associated to a special "forgetful" monoid, defined by generators and relations.
Optimal Sequences in a Heap Model with Two Pieces
 Paris
, 1998
"... . In a heap model, solid blocks, or pieces, pile up according to the Tetris game mechanism. An optimal sequence is an infinite sequence of pieces minimizing the asymptotic average height of the heap. In a heap model with two pieces, we prove that there always exists an optimal sequence which is eith ..."
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Cited by 3 (3 self)
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. In a heap model, solid blocks, or pieces, pile up according to the Tetris game mechanism. An optimal sequence is an infinite sequence of pieces minimizing the asymptotic average height of the heap. In a heap model with two pieces, we prove that there always exists an optimal sequence which is either periodic or Sturmian. We completely characterize the cases where the optimal is periodic and the ones where it is Sturmian. The proof is constructive, providing an explicit optimal sequence. We show an application for a system of two processes sharing a resource, and we prove that a greedy schedule is not always optimal. CONTENTS 1. Introduction 2 2. Heap Model 3 3. Optimal Behavior 6 4. Sturmian Word 7 5. (Max,+) Automaton 8 5.1. (Max,+) and (min,+) spectral theory 8 5.2. Deterministic automaton 9 5.3. Cayley automaton 9 5.4. Completed automaton 9 6. Heap Models with Two Pieces: Realizations of Dimensions 2 or 3 11 6.1. Bicompleted heap automaton 11 6.2. Minimal realization 12 7. Heap M...
Series Expansions of Lyapunov Exponents and
, 2000
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.