An asymptotic convergence result for the aggregation of closed fork-join generalized stochastic Petri nets

This paper is concerned with the aggregation of a closed generalized stochastic Petri net (GSPN) featuring a synchronized parallel structure. In closed networks the analysis of synchronized parallel structures is complicated due to the absence of product-form solutions for the underlying continuous-time Markov chain (CTMC). The number of states in the underlying CTMC grows rapidly with the number of tokens in the GSPN rendering the analysis prohibitive. Aggregation techniques are commonly employed to reduce the number of states to a manageable level. In this paper we apply stochastic complementation to the aggregation of a closed fork-join GSPN with no closed-form solution. It is shown that the marking-dependent firing rates converge to constant values as the number of tokens tends to infinity. For closed fork-join GSPNs with a large number of tokens this result provides a reduced complexity aggregation technique.