Several proposals exist for the introduction of synchronization constraints into Queueing Networks (QN). We show that many monoclass QN with synchronizations can naturally be modelled with a subclass of Petri Nets (PN) called Free Choice nets (FC), for which a wide gamut of qualitative behavioural and structural results have been derived. We use some of these net theoretic results to characterize the ergodicity, boundedness and liveness of closed Free Choice Synchronized Queueing Networks (FCSQN). Moreover we define upper and lower throughput bounds based on the mean value of the service times, without any assumption on the probability distributions (thus including both the deterministic and the stochastic cases). We show that monotonicity properties exist between the throughput bounds and the parameters of the model in terms of population and service times. We propose (theoretically polynomial and practically linear complexity) algorithms for the computation of these bounds, based on ...

This paper addresses ergodicity and throughput bounds characterizations for a subclass of timed and stochastic Petri nets, interleaving qualitative and quantitative theories. The considered nets represent an extension of the well known subclass of marked graphs, defined as having a unique consistent firing count vector, independently of the stochastic interpretation of the net model. In particular, persistent and mono-T-semiflow nets subclasses are considered. Upper and lower throughput bounds are computed using linear programming problems defined on the incidence matrix of the underlying net. The bounds proposed here depend on the initial marking and the mean values of the delays but not on the probability distributions (thus including both the deterministic and the stochastic cases). From a different perspective, the considered subclasses of stochastic nets can be viewed as special classes of synchronized queueing networks, thus the proposed bounds can be applied to these networks. ...

A class of synchronized queueing networks with deterministic routing is identified to be equivalent to a subclass of timed Petri nets called marked graphs. First some structural and behavioral properties of marked graphs are recalled and used to show interesting properties of this class of performance models. In particular, ergodicity is derived from boundedness and liveness of the underlying Petri net representation, which can be efficiently computed in polynomial time on the net structure. In case of unbounded (i.e., non-strongly-connected) marked graphs, ergodicity is computed as a function of the average transition firing delays. Then the problem of computing both upper and lower bounds for the steady-state performance of timed and stochastic marked graphs is studied. In particular, linear programming problems defined on the incidence matrix of the underlying Petri nets are used to compute tight (i.e., attainable) bounds for the throughput of transitions for marked graphs with dete...

In this paper we overview some recent results obtained by the authors and collaborators on the performance bounds analysis of some stochastic Petri net systems. The mathematical model can be seen either as a result of the addition of a particular random timing interpretation to an "autonomous" Petri net or as a generalization of classical queueing networks with the addendum of a general synchronization primitive. It constitutes an adequate tool for both the validation of logical properties and the evaluation of performance measures of concurrent and distributed systems. Qualitative and quantitative understandings of Petri net models are stressed here making special emphasis on structural techniques for the analysis of logical and performance properties. Important aspects from the performance point of view, such as relative throughput of stations (transitions), and number of servers present at them, are related to Petri net concepts like P- or T-semiflows or liveness bounds...

This paper addresses the improvement of throughput upper bounds for live and bounded stochastic Petri nets, presented by the authors in previous works. The introduction of a greater amount of structural information, traps and implicit places, allows to improve the bounds using linear programming problems defined on the net structure, on the routing probabilities, and on the mean service time of transitions. The obtained bounds can be applied for the analysis of manufacturing systems modelled by means of stochastic Petri nets. An example is presented and evaluated using the introduced techniques. 1 INTRODUCTION Timed and stochastic Petri nets constitute an adequate model for the evaluation of performance measures of concurrent and distributed systems (see, e.g., [1, 2, 3, 4]). Nevertheless, one of the main problems in the actual use of these models for the evaluation of large systems is the explosion of the computational complexity of the analysis algorithms. The computation of upper ...

The problem of computing both upper and lower bounds for the steady-state performance of timed and stochastic Petri nets is studied. In particular, Linear Programming problems defined on the incidence matrix of underlying Petri net are used to compute bounds for the throughput of transitions for live and bounded nets with a unique possibility of steady-state behaviour. These classes of nets are defined and their characteristics are studied. The bounds proposed here depend on the initial marking and the mean values of the delays but not on the probability distributions (thus including both the deterministic and the stochastic cases); moreover they can be computed also for non-ergodic models. Connections between results and techniques typical of qualitative and quantitative analysis of Petri models are stressed. 1 Introduction In this paper, which is a continuation of the companion paper [CCCS89], we study the possibility of obtaining (upper and lower) bounds on the steady-state perform...

This paper addresses the computation of upper bounds for the throughput of transitions of live and safe deterministically or stochastically timed free choice nets. The obtained results are extensions of the marked graph case, presented by the authors in previous works. Polynomial complexity algorithms are derived using linear programming techniques. The obtained values are tight in the sense that, with the only knowledge of the net topology, the mean service times of transitions, and the routing rates at conflicts, is not possible to improve the bounds. Topics: Timed and stochastic nets. Analysis and synthesis, structure, and behaviour of nets. 1 Introduction One of the main problems in the actual use of timed and stochastic Petri net models for the performance evaluation of large systems is the explosion of the computational complexity of the analysis algorithms. In general, exact performance results are obtained from the numerical solution of a continuous time Markov chain ...

This paper addresses the computation of upper bounds for the steady-state throughput of stochastic Petri nets with immediate and exponentially distributed service times of transitions. We try to deeply bridge stochastic Petri net theory to untimed Petri net and queueing network theories. Previous results for general service time distributions are improved for the case of Markovian nets by considering the slowest embedded subnet (generated by the support of left annullers of the incidence matrix of the net). The obtained results for the case of live and bounded free choice nets are of special interest. For such nets, the subnets generated by the left annullers of the incidence matrix can be seen as embedded product-form closed monoclass queueing networks, and efficient algorithms exist for their analysis. 1 Introduction The computation of upper bounds for the throughput of transitions, defined as the average number of firings per unit time, of stochastic Petri nets with exponentially ...

: Pull control systems are a good way to control material flow in manufacturing systems. In this paper we are interested in four control mechanisms: the Base Stock Control System (BSCS), the Kanban Control System (KCS), the Generalized Kanban Control System (GKCS) and the Extended Kanban Control System (EKCS). The BSCS and the KCS are well known simple pull control mechanisms. The better performing GKCS and EKCS were recently defined as combinations of both the BSCS and the KCS. Most of the work on the modeling and analysis of these pull control systems is illustrated on systems having stages in series. The purpose of this paper is to provide a common modeling approach based on Petri nets for these four control systems in the case of assembly manufacturing systems. Keywords: Pull controlled production systems, Kanban systems, Petri nets. * Universite Française du Pacifique, CUNC BP4477, 98847 Noumea - NEW CALEDONIA, chaouiya@ufp.nc ** Laboratoirre MASI (URA 818 CNRS), IBP, Universi...

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