Realizability of schedules by stochastic time Petri nets with blocking semantics

This paper considers realizability of expected schedules by production systems with concurrent tasks, bounded resources that have to be shared among tasks, and random behaviors and durations. Schedules are high level views of desired executions of systems represented as partial orders decorated with timing constraints. Production systems (production cells, train networks…) are modeled as stochastic time Petri nets STPNs with an elementary (1-bounded) semantics. We detail their interleaved operational semantics, and then propose a non-interleaved semantics through the notion of time processes. We then consider boolean realizability: a schedule S is realizable by a net N if S embeds in a time process of N that satisfies all its constraints. However, with continuous time domains, the probability of a time process with exact dates is null. We hence consider probabilistic realizability up to α time units, that holds if the probability that N realizes S with constraints enlarged by α is strictly positive. Upon a sensible restriction guaranteeing time progress, boolean and probabilistic realizability of a schedule can be checked on the finite set of symbolic prefixes extracted from a bounded unfolding of the net. We give a construction technique for these prefixes and show that they represent all time processes of a net occurring up to a given maximal date. We then show how to verify existence of an embedding and compute the probability of its realization.