Realizability of Schedules by Stochastic Time Petri Nets with Blocking Semantics

Abstract

This paper considers realizability of schedules by stochastic concurrent timed systems. Schedules are high level views of desired executions represented as partial orders decorated with timing constraints, while systems are represented as elementary stochastic time Petri nets. We first consider logical realizability: a schedule is realizable by a net $\mathcal {N}$ if it embeds in a time process of $\mathcal {N}$ that satisfies all its constraints. However, with continuous time domains, the probability of a time process that realizes a schedule is null. We hence consider probabilistic realizability up to $\alpha$ time units, that holds if the probability that $\mathcal {N}$ logically realizes S with constraints enlarged by $\alpha$ is strictly positive. Upon a sensible restriction guaranteeing time progress, logical 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.