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On simulation of non-Markovian stochastic Petri Nets with heavy-tailed firing times

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.