Discrete Deterministic and Stochastic Petri Nets

1995

In this paper we present and compare two di erent approaches for the transient solution of Markov regenerative stochastic Petri Nets: the method based on Markov regenerative theory and the method of supplementary variables. In both cases the equations that govern the marking process of the non-Markovian stochastic Petri net are presented and then solved either in time-domain or using a Laplace-Stieltjes transformation. We develop expressions for asymptotic computational costs and storage requirements. We also perform experimental studies to compare accuracy, time, and space complexity of the methods.

Performance Evaluation, 1998

Stochastic Petri nets have been used to analyze the performance and reliability of complex systems comprising concurrency and synchronization. Various extensions have been proposed in literature in order to broaden their field of application to an increasingly larger range of real situations.

1983

Generalized Stochastic Petri Nets (GSPNs) are presented and are applied to the performance evaluation of multiprocessor systems. GSPNs are derived from standard Petri nets by partitioning the set of transitions into two subsets comprising timed and immediate transitions. An exponentially distributed random firing time is associated with each timed transition, whereas immediate transitions fire in zero time. It is shown that GSPNs are equivalent to continuous-time stochastic processes, and solution methods for the derivation of the steady state probability distribution are presented. Examples of application of GSPN models to the performance evaluation of multiprocessor systems show the usefulness and the effectiveness of this modeling tool. 1. INTRODUCTION Graph models have been proposed by many authors as a useful tool for the analysis of peculiar features of computer systems such as concurrency, synchronization, communication, and cooperation among subsystems. Much of the work in this field is related to original ideas developed by C. A. Petri in his Ph.D. dissertation [12]. These graph models are today generally known as Petri Nets (PNs). A PN comprises a set of places P, a set of transitions T, and a set of directed arcs A. In the graphical representation of PNs places are drawn as circles and transitions as bars. Arcs connect transitions to places and places to transitions. Places may contain tokens, which are drawn as black dots. The state of a PN is defined by the number of tokens contained in each place and is denoted by a

2005

Time Petri Nets describe the state of a timed system through a marking and a set of clocks. If clocks take values in a dense domain, state space analysis must rely on equivalence classes. These support verification of logical sequencing and quantitative timing of events, but they are hard to be enriched with a stochastic characterization of nondeterminism necessary for performance and dependability evaluation. Casting clocks into a discrete domain overcomes the limitation, but raises a number of problems deriving from the intertwined effects of concurrency and timing. We present a discrete-time variant of Time Petri Nets, called stochastic preemptive Time Petri Nets, which provides a unified solution for the above problems through the adoption of a maximal step semantics in which the logical location evolves through the concurrent firing of transition sets. We propose an analysis technique, which integrates the enumeration of a succession relation among sets of timed states with the calculus of their probability distribution. This enables a joint approach to the evaluation of performance and dependability indexes as well as to the verification of sequencing and timeliness correctness. Expressive and analysis capabilities of the model are demonstrated with reference to a real-time digital control system.

Lecture Notes in Computer Science, 1993

The conservative and the optimistic approaches of distributed discrete event simulation (DDES) are used as the starting point to develop an optimized simulation framework for studying the behaviour of large and complex timed transition Petri net (TTPN) models. This work systematically investigates the interdependencies among the DDES strategy (conservative, Time Warp) and the spatial decomposition of TTPNs into logical processes to be run concurrently on individual processing nodes in a message passing and shared memory multiprocessor environment. Partitioning heuristics are developed taking into account the structural properties of the TTPN model, and the simulation strategy is tuned accordingly in order to attain the maximum computational speedup. Implementations of the simulation framework have been undertaken for the Intel iPSC/860 hypercube, the Sequent Balance and a Transputer based multiprocessor. The simulation results show that the use of the Petri net formalism allows an automatic extraction of the parallelism and causality relations inherent to the model.

1992

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 of transitions. For the particularly interesting case of Markovian Petri net systems, some improvements of the bounds can be achieved. Marked graphs and free choice are net subclasses for which the obtained results have special quality, therefore an additional attention is focussed on them.

Journal of Circuits, Systems and Computers, 1998

Analytical modeling plays a crucial role in the analysis and design of computer systems. Stochastic Petri Nets represent a powerful paradigm, widely used for such modeling in the context of dependability, performance and performability. Many structural and stochastic extensions have been proposed in recent years to increase their modeling power, or their capability to handle large systems. This paper reviews recent developments by providing the theoretical background and the possible areas of application. Markovian Petri nets are rst considered together with very well established extensions known as Generalized Stochastic Petri nets and Stochastic Reward Nets. Key ideas for coping with large state spaces are then discussed. The challenging area of non-Markovian Petri nets is considered, and the related analysis techniques are surveyed together with the detailed elaboration of an example. Finally new models based on Continuous or Fluid Stochastic Petri Nets are brie y discussed.

Proceedings Title Proceedings of the 2012 Winter Simulation Conference, 2012

Long-run stochastic stability is a precondition for applying steady-state simulation output analysis methods to a stochastic Petri Net (SPN), and is of interest in its own right. A fundamental stability requirement for an irreducible SPN is that the markings of the net be recurrent, in that the marking process visits each marking infinitely often with probability 1. We study recurrence properties of irreducible non-Markovian SPNs with finite marking set. Our focus is on the "clocks" that govern the transition firings, and we consider SPNs in which zero, one, or at least two simultaneously-enabled transitions can have very heavy-tailed clock-setting distributions. We establish positive recurrence, null recurrence, and, perhaps surprisingly, possible transience of markings for these respective regimes. The transience result stands in strong contrast to Markovian or semi-Markovian SPNs, where irreducibility and finiteness of the marking set guarantee positive recurrence.

1993

Operational analysis techniques are used to partially characterize the behavior of timed Petri nets under very weak assumptions on their timing semantics. New operational inequalities are derived that are typical of the presence of synchronization and that were therefore not considered in queuing network models. An interesting application of the operational laws to the statement and the efficient solution of problems related to the estimation of performance bounds insensitive to the timing probability distributions is shown. The results obtained generalize and improve in a clear setting results that were derived in the last few years for several different subclasses of timed Petri nets. In particular, the extension to well-formed colored nets appears straightforward and allows an efficient exploitation of model symmetries

1999

This paper presents an algorithm of polynomial complexity to derive the throughput of a discrete event system via stochastic Petri net (SPN) models. The concept of flow nets, a subclass of SPN, is introduced to model a class of discrete event systems. The mathematical model for the throughput of flow nets is given. For a structurally non-competitive and acyclic flow net, the solution algorithm proceeds in four steps. First, divide the places and transitions into groups according to some rules. Next, list the flow equilibrium equation for each place, which shows the relation among the average flows of its input and output transitions. Then, deduce the relation of the average flow of any nonsource transition to that of all source transitions. Finally, determine the throughput of the model. By so doing, we significantly reduce the size of the linear programming problems. For a structurally competitive and cyclic flow net, a procedure is proposed to convert it to a structurally non-competitive and acyclic one in the sense of equivalent throughput. Besides, the paper also shows how to transform an SPN with shared resource to a flow net. An assembly system is used to illustrate the application of the technique for the analysis of throughput.

Theory and Practice of Logic Programming, 2006

The theory of Petri Nets provides a general framework to specify the behaviors of real-time reactive systems and Time Petri Nets were introduced to take also temporal specifications into account. We present in this paper a forward zone-based algorithm to compute the state space of a bounded Time Petri Net: the method is different and more efficient than the classical State Class Graph. We prove the algorithm to be exact with respect to the reachability problem. Furthermore, we propose a translation of the computed state space into a Timed Automaton, proved to be timed bisimilar to the original Time Petri Net. As the method produce a single Timed Automaton, syntactical clocks reduction methods (Daws and Yovine for instance) may be applied to produce an automaton with fewer clocks. Then, our method allows to model-check T-TPN by the use of efficient Timed Automata tools.

Proceedings of 1995 IEEE International Computer Performance and Dependability Symposium, 1995

Structure performance analysis theory and techniques is an essay to avoid the computational complexity problem associated to Markovian and discrete event simulation techniques. Even if a nished conceptual and technical framework is not yet available, important bene ts have been obtained not only from performance but also from correctness analysis point of view. In this survey we overview some of the achievements developed by the authors and collaborators towards a structure theory for performance evaluation of net based models. Concepts and techniques for the computation of performance bounds and approximate and exact evaluation are described in a semiformal/illustrative way through a selected collection of examples.

IFAC Proceedings Volumes, 2009

Reliability analysis is often based on stochastic discrete event models like Markov models or stochastic Petri nets. For complex dynamical systems with numerous components, analytical expressions of the steady state are tedious to work out because of the combinatory explosion with discrete models. The contribution of this paper is to approximate the steady state of mono Tsemiflow stochastic nets by mean of continuous Petri nets according to a modification of the maximal firing speed vector definition. This result is then used to accelerate convergence of stochastic simulations.

Fourth IEEE Region 10 International Conference TENCON

Stochastic Petri Nets (SPNs) have recently emerged as a principal performance modelling tool for distributed systems such as multiprocessors, local area networks, and automated manufacturing systems. Since the use of SPNs as an analytical tool is based on the generation of the entire state space, the technique becomes intractable for large systems. In such cases, discrete event simulation is the preferred tool for perfxmance evaluation. In this paper, we show how SPNs can be used as a simulation model. We present several efficient algorithms based on SPNs, for conducting discrete event simulations of distributed systems.

1989

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 de ned 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 de ned 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.