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 cross-references 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

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.

We present a survey of the main ergodic theory techniques which are used in the study of iterates of monotone and homogeneous stochastic operators. It is shown that ergodic theorems on discrete event networks (queueing networks and/or Petri nets) are a generalization of these stochastic operator theorems. Kingman's subadditive ergodic Theorem is the key tool for deriving what we call rst order ergodic results. We also show how to use backward constructions (also called Loynes schemes in network theory) in order to obtain second order ergodic results. We will propose a review of systems entering the framework insisting on two models, precedence constraints networks and Jackson type networks.

In a live and bounded Free Choice Petri net, pick a non-conflicting transition. Then there exists a unique reachable marking in which no transition is enabled except the selected one. For a routed live and bounded Free Choice net, this property is true for any transition of the net. Consider now a live and bounded stochastic routed Free Choice net, and assume that the routings and the firing times are independent and identically distributed. Using the above results, we prove the existence of asymptotic firing throughputs for all transitions in the net. Furthermore, the vector of the throughputs at the different transitions is explicitly computable up to a multiplicative constant. 1.

Stochastic timed Petri nets are a useful tool in performance analysis of concurrent systems such as parallel computers, communication networks and flexible manufacturing systems. In general, performance measures of stochastic timed Petri nets are difficult to obtain for problems of practical sizes. In this paper, we provide a method to compute efficiently upper and lower bounds for the throughputs and mean token numbers in general Markovian timed Petri nets. Our approach is based on uniformization technique and linear programming.

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 (input-output) 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 ...

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 Howard-type '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.

This paper addresses the computation of upper bounds for the steady-state throughput of stochastic Petri net systems with immediate and generally distributed timed transitions. It is achieved through the use of a kind of decomposition of the whole net system. Results are obtained deeply bridging stochastic Petri net theory to untimed Petri net and queueing network theories. Previous results are improved by considering some embedded product-form queueing networks (generated by the support of some left annullers of the incidence matrix of the net). The obtained results for the case of live and bounded free choice systems are of special interest. In this case, the subnets generated by the minimal left annullers of the incidence matrix always have a topology of product-form closed monoclass queueing networks. Keywords: Stochastic Petri net systems, throughput bounds, embedded product-form queueing networks. 1 Introduction

Approximate throughput computation of a class of discrete event systems (DES) modelled with stochastic weighted T-systems is considered. Stochastic weighted T-systems are the weighted extension of well known stochastic marked graphs Petri net subclass and are usually presented as a useful model to deal with bulk consumptions or productions of resources in manufacturing systems working on a cyclic basis. The iterative response time approximation technique that we present is deeply based on a previous structural decomposition and aggregation of the net model. Experimental results on several examples generally have and error of less than 5%. The state space is usually reduced by more than one order of magnitude; therefore, the analysis of otherwise intractable systems is possible.

The complexity of tracking perturbations in discrete event dynamic systems (DEDS) depends on the systems' perturbation propagation mechanism and on the length of the event trace. Existing perturbation propagation algorithms assume that all unperturbed event times are observed and that all perturbed times are required. This paper concerns a complementary approach, termed perturbation tracking (PT), that accurately tracks perturbations in systems for which only a subset of event times are known. We apply PT to a class of partiallyobserved, timed Petri nets and show that for accurate tracking it is necessary and sufficient to know the token holding times between observations. We conclude with an example, motivated by a practical software monitoring problem, that illustrates how this information can be derived from structural and event trace analysis. Not surprisingly, the perturbation propagation rules of our PT algorithm are closely related to the existing algorithms when all event timin...