Comparison properties of stochastic decision free Petri nets

Stochastic comparison problems arising in the analysis of stochastic decision free Petri nets are treated. The analysis is based on the evolution equations satisfied by firing times. Various structural properties are obtained, including the association of the firing times and their stochastic and convex monotonicity with respect to the holding time sequences, the initial marking, and the topology. The association and the stochastic monotonicity properties are extended to the counters using an inversion relation. It is also proven that the counters and the throughput are stochastically concave in the initial marking, provided that the holding times are independent, identically distributed, and belong to a subclass of the log-concave distribution functions that are introduced. Various bounds for the asymptotic cycle time and the throughput are then derived from these stochastic ordering results. Comparative results are presented for the marking and response times