Stochastic Hybrid Approximations of Markovian Petri Nets

A Markovian Petri net (MPN) is a stochastic discrete event system (DES) frequently used for qualitative analysis, performance evaluation, and control purposes. In the past, the fluidization has been proposed in different DES formalisms (Markov chains, queuing networks, stochastic Petri nets (PNs), stochastic process algebra, etc.) as a relaxation technique for avoiding the state-explosion problem. Following that kind of approach, in this paper, a new hybrid PN model is defined as a partial relaxation of an original MPN. Nevertheless, it is shown through a simple example that such a partial relaxation can be worse than a full relaxation (given by a fully continuous PN), if certain conditions are not meet. Therefore, the rest of this paper is devoted to analyze the approximation of MPNs by means of hybrid PNs, where both the discrete and the continuous parts are stochastic. It is demonstrated that under certain conditions, the average and probability distribution function of the marking of a MPN can be approximated by those of the marking of the corresponding hybrid PN. The key point is to recognize the variability in behavior due to the time stochasticity. In some sense, a kind of functional central limit approach is obtained in a hybrid context.