Linear stochastic Petri networks

Summary form only given, as follows. The author reviews results on stochastic Petri nets based on the so called (max,+) approach. In this approach, the state variables of the network are the epochs at which transitions fire, to be opposed to the marking state variables of the conventional approach. It is recalled that, within this framework, stochastic event graphs can be seen as (max,+)-linear systems in a random medium, and it is indicated how to translate the graphical description of the network into a standard (max,+)-linear recurrence of order 1. The author then reviews basic stability results and shows their relations with the spectral theory of random (max,+)-matrices. The author shows in particular that cycle times can be seen as (max,+)-Lyapunov exponents and stationary regimes as (max,+)-stochastic eigenvectors. The author then shows how this approach provides structural results on the throughout and the stationary marking processes. For instance, the author gives sufficient conditions on the distribution function of the firing times for the throughput to be a concave function of the initial marking. Finally, the author reviews a method of fast SIMD simulation based on this representation and an implementation currently under development on the Connection Machine