Stochastic Petri networks provide a general formalism for describing the dynamics of discrete event systems. The present paper focuses on a subclass of stochastic Petri networks called stochastic event graphs, under the assumption that the variables used for their "timing" form stationary and ergodic sequences of random variables. We show that such stochastic event graphs can be seen as a $(\max, +)$ linear system in a random, stationary and ergodic environment. We then analyze the associated Lyapounov exponents and construct the stationary and ergodic regime of the increments, by proving an Oseledec-type multiplicative ergodic theorem. Finally, we show how to construct the stationary marking process from these results.
"Ergodic Theory of Stochastic Petri Networks." Ann. Probab. 20 (1) 375 - 396, January, 1992. https://doi.org/10.1214/aop/1176989932