Composition and Equivalence of Markovian and Non-Markovian Models

Compositional modeling and the aggregation of components according to equivalence relations based on stochastic bisimulation are often used to handle the problem of state space explosion in Markov models. The paper presents a general class of equivalence relations between Markov models that include stochastic bisimulation or %basically lump ability as specific cases and proves the congruence property of the new equivalence with respect to the composition of components. It is shown that the equivalence relates Markovian and non-Markovian representations but requires some restrictions for the composition which are automatically observed if stochastic bisimulation is used as equivalence relation. Nevertheless, the approach offers the possibility of state space reduction beyond stochastic bisimulation without loosing the possibility of analyzing the resulting stochastic process by means of numerical methods.