Petri nets with k simultaneously enabled generally distributed timed transitions

We consider the problem of verifying quantitative reachability properties in stochastic models of concurrent activities with generally distributed durations. Models are specified as stochastic time Petri nets and checked against Boolean combinations of interval until operators imposing bounds on the probability that the marking process will satisfy a goal condition at some time in the interval [alpha, beta] after an execution that never violates a safety property. The proposed solution is based on the analysis of regeneration points in model executions: a regeneration is encountered after a discrete event if the future evolution depends only on the current marking and not on its previous history, thus satisfying the Markov property. We analyze systems in which multiple generally distributed timers can be started or stopped independently, but regeneration points are always encountered with probability 1 after a bounded number of discrete events. Leveraging the properties of regeneration points in probability spaces of execution paths, we show that the problem can be reduced to a set of Volterra integral equations, and we provide algorithms to compute their parameters through the enumeration of finite sequences of stochastic state classes encoding the joint probability density function (PDF) of generally distributed timers after each discrete event. The computation of symbolic PDFs is limited to discrete events before the first regeneration, and the repetitive structure of the stochastic process is exploited also before the lower bound a, providing crucial benefits for large time bounds. A case study is presented through the probabilistic formulation of Fischer's mutual exclusion protocol, a well-known real-time verification benchmark.