Reductibility considerations for an aggregated Markov chain

Aggregation of synchronized structures such as closed fork-join generalized stochastic Petri nets is often a challenging problem due to the absence of closed-form solutions. This paper provides a full characterization of the aggregation properties of a closed fork-join subnet with multiple branches by resorting to the asymptotic properties of the underlying continuous-time Markov chain (CTMC) as the number of tokens tends to infinity. The asymptotic analysis provides valuable insight into the aggregation of closed fork-join structures. The relationship between the firing rates of the original closed fork-join subnet and the approximate reducibility of the aggregated CTMC is investigated and illustrated with numerical examples.