Aggregated Stochastic State Classes in Quantitative Evaluation of non-Markovian Stochastic Petri Nets

Abstract

Abstract-The method of stochastic state classes provides a new approach for the analysis of non-Markovian stochastic Petri Nets, which relies on the stochastic expansion of the graph of nondeterministic state classes based on Difference Bounds Matrix (DBM) which is usually employed in qualitative verification. In so doing, the method is able to manage multiple concurrent nonexponential (GEN) transitions and largely extends the class of models that are amenable to quantitative evaluation. However, its application requires that every cycle in the graph of nondeterministic state classes visits at least a regeneration point where all GEN transitions are newly enabled. In particular, this rules out models whose non-deterministic class graph includes cycles within a Continuous Time Markov Chain (CTMC) subordinated to the activity period of one or more GEN transitions.In this paper, we propose an extension that overcomes this limitation by aggregating together classes that are reached through firings that do not change the enabling status of GEN transitions. This enlarges the class of models that can be analysed through the method of stochastic state classes and makes it become a proper extension of the class of models that satisfies the so called enabling restriction.Index Terms-non-Markovian stochastic Petri nets, stochastic time Petri nets, steady state analysis, stochastic state classes.

Transient analysis of non-Markovian models using stochastic state classes

“…We formulate the analysis problem with reference to a variant of (non-Markovian) Stochastic Petri Nets (SPNs), which we call stochastic Time Petri Nets (sTPNs) [15,16]. As suggested by the name, an sTPN is here regarded as a non-deterministic Time Petri Net (TPN) [21,18,17] extended with a stochastic characterisation of timers and choices: while the TPN identifies a set of feasible behaviours, the stochastic extension associates them with a measure of probability.…”

“…The stochastic class graph can thus be enumerated and managed as a finite Discrete Time Markov Chain, opening the way to discrete time evaluation, both in transient and in steady state regime [15]. In addition, the distribution of times to fire observed at the entrance in a class supports the derivation of the average sojourn time within each class, thus enabling the reconstruction of continuous time steady state probabilities [16].…”

“…Following an aggregation approach such as that proposed for steady state analysis in [16], this condition can be relaxed so as to encompass cycles of immediate or exponential transitions that do not change the enabling status of generally distributed transitions. However, when the state class graph accepts cyclic behaviours that always maintain some GEN transition persistent, then the model accepts behaviours that never reach a regeneration and thus results in an unbounded number of transient stochastic classes.…”

“…More recently, [14][15][16] proposed a new analytical approach that manages the case of multiple concurrent generally distributed timers with possibly overlapping activity cycles, provided that every cyclic behaviour that changes the enabling status of generally distributed transitions visits at least one regeneration point [15]. The approach relies on the stochastic expansion of non-deterministic state-classes described by Difference Bounds Matrix (DBM) zones [17][18][19] which are commonly employed in qualitative verification tools.…”

“…The stochastic state class graph abstracts the behaviour of a GSMP into an embedded Discrete Time Markov Chain (DTMC) that samples the process after each transition and thus allows the evaluation of discrete time transient and steady state measures referred to the number of fired transitions. In addition, the distribution associated with stochastic classes allows the derivation of average sojourn times between subsequent transitions and thus enables the evaluation of steady state probabilities of the overall continuous time process [16]. However, for what pertains to continuous time transient behaviour, the approach only supports the derivation of the cumulative distribution of the time elapsed along a selected firing sequence that starts from a regenerative class where the process looses memory of its past history [15,20].…”