The (True) Concurrent Markov Property and Some Applications to Markov Nets

We study probabilistic safe Petri nets, a probabilistic extension of safe Petri nets interpreted under the true-concurrent semantics. In particular, the likelihood of processes is defined on partial orders, not on firing sequences. We focus on memoryless probabilistic nets: we give a definition for such systems, that we call Markov nets, and we study their properties. We show that several tools from Markov chains theory can be adapted to this true-concurrent framework. In particular, we introduce stopping operators that generalize stopping times, in a more convenient fashion than other extensions previously proposed. A Strong Markov Property holds in the concurrency framework. We show that the Concurrent Strong Markov property is the key ingredient for studying the dynamics of Markov nets. In particular we introduce some elements of a recurrence theory for nets, through the study of renewal operators. Due to the concurrency properties of Petri nets, Markov nets have global and local renewal operators, whereas both coincide for sequential systems.