About the stochastic and continuous Petri nets equivalence in the long run

Reliability analysis is often based on stochastic discrete event models like stochastic Petri nets (SPNs). For large dynamical systems with numerous components, the analytical expression of the SPNs steady state is full of complexities because of the combinatory explosion with discrete models. Moreover, the estimation of mean markings thanks to simulations is time consuming in case of rare events. For these reasons, Petri net fluidification may be an interesting alternative to provide a reasonable estimate of the asymptotic behavior of stochastic processes. Unfortunately, the steady states of SPNs and timed continuous Petri nets (contPNs) with the same structure, same initial marking and same firing rates are mainly often different. The region of SPN steady states (when firing rates are defined in a polyhedral area) contrasts with that of contPN ones. The purpose of this paper is to illuminate this issue in taking advantage of the piecewise-affine hybrid structure of contPNs. Regions and critical regions are defined in the marking space in order to characterize this structure. Based on this characterization, the main contribution is to propose a transformation of the considered SPN into a contPN with the same structure, modified firing rates and homothetic initial marking so that the corrected contPN converges partially to the same mean marking than the SPN. Consequently, a global understanding of an SPN steady state can be obtained according to the corrected contPN.