Product-form and stochastic Petri nets: a structural approach

Stochastic Petri nets (SPNs) with product-form solution are nets for which there is an analytic expression of the steady-state probabilities with respect to place markings, as it is the case for product-form queueing networks with respect to queue lengths. The most general kind of SPNs with product-form solution introduced by Coleman et al. (and denoted here by SΠ-nets) suffers a serious drawback: the existence of such a solution depends on the values of the transition rates. Thus since their introduction, it is an open question to characterize SΠ-nets with product-form solution for any values of the rates. A partial characterization has been obtained by Henderson et al. However, this characterization does not hold for every initial marking and it is expressed in terms of the reachability graph. In this paper, we obtain a purely structural characterization of SΠ-nets for which a product-form solution exists for any value of probabilistic parameters of the SPN and for any initial marking. This structural characterization leads to the definition of SΠ2-nets (Stochastic Parametric Product-form Petri nets). We also design a polynomial time (with respect to the size of the net structure) algorithm to check whether a SPN is a SΠ2-net. Then, we study qualitative properties of Π-nets and Π2-nets, the non-stochastic versions of SΠ-nets and SΠ2-nets: we establish two results on the complexity bounds for the liveness and the reachability problems, which are central problems in Petri nets theory. This set of results complements previous studies on these classes of nets and improves the applicability of product-form solutions for SPNs.