Max-plus algebra in the history of discrete event systems

This paper is a survey of the history of max-plus algebra and its role in the field of discrete event systems during the last three decades. It is based on the perspective of the authors but it covers a large variety of topics, where max-plus algebra plays a key role

Limit theorems for iterated random topical operators

Let A(n) be a sequence of i.i.d. topical (i.e. isotone and additively homogeneous) operators. Let $x(n,x_0)$ be defined by $x(0,x_0)=x_0$ and $x(n,x_0)=A(n)x(n-1,x_0)$. This can modelize a wide range of systems including, task graphs, train networks, Job-Shop, timed digital circuits or parallel processing systems. When A(n) has the memory loss property, we use the spectral gap method to prove limit theorems for $x(n,x_0)$. Roughly speaking, we show that $x(n,x_0)$ behaves like a sum of i.i.d. real variables. Precisely, we show that with suitable additional conditions, it satisfies a central limit theorem with rate, a local limit theorem, a renewal theorem and a large deviations principle, and we give an algebraic condition to ensure the positivity of the variance in the CLT. When A(n) are defined by matrices in the \mp semi-ring, we give more effective statements and show that the additional conditions and the positivity of the variance in the CLT are generic

Reachability problems for products of matrices in semirings

We consider the following matrix reachability problem: given $r$ square matrices with entries in a semiring, is there a product of these matrices which attains a prescribed matrix? We define similarly the vector (resp. scalar) reachability problem, by requiring that the matrix product, acting by right multiplication on a prescribed row vector, gives another prescribed row vector (resp. when multiplied at left and right by prescribed row and column vectors, gives a prescribed scalar). We show that over any semiring, scalar reachability reduces to vector reachability which is equivalent to matrix reachability, and that for any of these problems, the specialization to any $r\geq 2$ is equivalent to the specialization to $r=2$. As an application of this result and of a theorem of Krob, we show that when $r=2$, the vector and matrix reachability problems are undecidable over the max-plus semiring $(Z\cup\{-\infty\},\max,+)$. We also show that the matrix, vector, and scalar reachability problems are decidable over semirings whose elements are ``positive'', like the tropical semiring $(N\cup\{+\infty\},\min,+)$.Comment: 21 page

Exploring resource/performance trade-offs for streaming applications on embedded multiprocessors

Embedded system design is challenged by the gap between the ever-increasing customer demands and the limited resource budgets. The tough competition demands ever-shortening time-to-market and product lifecycles. To solve or, at least to alleviate, the aforementioned issues, designers and manufacturers need model-based quantitative analysis techniques for early design-space exploration to study trade-offs of different implementation candidates. Moreover, modern embedded applications, especially the streaming applications addressed in this thesis, face more and more dynamic input contents, and the platforms that they are running on are more flexible and allow runtime configuration. Quantitative analysis techniques for embedded system design have to be able to handle such dynamic adaptable systems. This thesis has the following contributions: - A resource-aware extension to the Synchronous Dataflow (SDF) model of computation. - Trade-off analysis techniques, both in the time-domain and in the iterationdomain (i.e., on an SDF iteration basis), with support for resource sharing. - Bottleneck-driven design-space exploration techniques for resource-aware SDF. - A game-theoretic approach to controller synthesis, guaranteeing performance under dynamic input. As a first contribution, we propose a new model, as an extension of static synchronous dataflow graphs (SDF) that allows the explicit modeling of resources with consistency checking. The model is called resource-aware SDF (RASDF). The extension enables us to investigate resource sharing and to explore different scheduling options (ways to allocate the resources to the different tasks) using state-space exploration techniques. Consistent SDF and RASDF graphs have the property that an execution occurs in so-called iterations. An iteration typically corresponds to the processing of a meaningful piece of data, and it returns the graph to its initial state. On multiprocessor platforms, iterations may be executed in a pipelined fashion, which makes performance analysis challenging. As the second contribution, this thesis develops trade-off analysis techniques for RASDF, both in the time-domain and in the iteration-domain (i.e., on an SDF iteration basis), to dimension resources on platforms. The time-domain analysis allows interleaving of different iterations, but the size of the explored state space grows quickly. The iteration-based technique trades the potential of interleaving of iterations for a compact size of the iteration state space. An efficient bottleneck-driven designspace exploration technique for streaming applications, the third main contribution in this thesis, is derived from analysis of the critical cycle of the state space, to reveal bottleneck resources that are limiting the throughput. All techniques are based on state-based exploration. They enable system designers to tailor their platform to the required applications, based on their own specific performance requirements. Pruning techniques for efficient exploration of the state space have been developed. Pareto dominance in terms of performance and resource usage is used for exact pruning, and approximation techniques are used for heuristic pruning. Finally, the thesis investigates dynamic scheduling techniques to respond to dynamic changes in input streams. The fourth contribution in this thesis is a game-theoretic approach to tackle controller synthesis to select the appropriate schedules in response to dynamic inputs from the environment. The approach transforms the explored iteration state space of a scenario- and resource-aware SDF (SARA SDF) graph to a bipartite game graph, and maps the controller synthesis problem to the problem of finding a winning positional strategy in a classical mean payoff game. A winning strategy of the game can be used to synthesize the controller of schedules for the system that is guaranteed to satisfy the throughput requirement given by the designer

Methods and Applications of (max,+) Linear Algebra

Projet META2Exotic semirings such as the ``$(\max,+)$ semiring'' $(\mathbb{R}\cup\{-\infty\},\max,+)$, or the ``tropical semiring'' $(\mathbb{N}\cup\{+\infty\},\min,+)$, have been invented and reinvented many times since the late fifties, in relation with various fields: performance evaluation of manufacturing systems and discrete event system theory; graph theory (path algebra) and Markov decision processes, Hamilton-Jacobi theory; asymptotic analysis (low temperature asymptotics in statistical physics, large deviations, WKB method); language theory (automata with multiplicities). Despite this apparent profusion, there is a small set of common, non-naive, basic results and problems, in general not known outside the $(\max,+)$ community, which seem to be useful in most applications. The aim of this short survey paper is to present what we believe to be the minimal core of $(\max,+)$ results, and to illustrate these results by typical applications, at the frontier of language theory, control, and operations research (performance evaluation of discrete event systems, analysis of Markov decision processes with average cost). Basic techniques include: solving all kinds of systems of linear equations, sometimes with exotic symmetrization and determinant techniques; using the $(\max,+)$ Perron-Frobenius theory to study the dynamics of $(\max,+)$ linear maps. We point out some open problems and current developments

Commande sous contraintes temporelles des réseaux de graphes d'événements temporisés en conflit

International audienceDans ce papier, nous abordons le problème de modélisation et de commande des systèmesàévénements discrets avec des ressources partagées représentés par une classe particulière des réseaux de Petri temporisés. Précisément, nous considérons des Réseaux de Graphes d'Evénements Temporisés en Conflit (RGETC) soumisà des contraintes tem-porelles strictes. Premièrement, une formalisation algébrique en termes de systèmesà com-mutation Max-Plus est proposée pour décrire le comportement dynamique des RGETCs. Deuxièmement, des lois de commande en boucle fermée sont calculées pour garantir le respect de ces contraintes de temps imposéesà certaines places du réseau. Des conditions suffisantes pour l'existence de telles lois de commande ontété fournies. Finalement, nous appliquons les résultats théoriques développés précédemment pour contrôler un système ferroviaire de croisement de trainà temps critique