Aggregation Methods for Markov Reward Chains with Fast and Silent Transitions

We develop and compare reduction- and lumping-based aggregation methods for continuous-time Markov reward chains with different types of instantaneous transitions. Such models arise from the intermediate performance models of many high-level modeling formalisms like, for example, stochastic process algebras or (generalized) stochastic Petri nets. In our setting the instantaneous transitions are modeled as transitions parameterized with a real variable. They are called fast transitions when governed by explicit probabilities, and silent transitions when the probabilities are left unspecified. When the parameter tends to infinity the transitions introduce stochastic discontinuity in the model. We study the proposed two aggregation methods in three different settings of Markovian models with stochastic discontinuity, fast transitions, and silent transitions, respectively, and show that the methods coincide when all instantaneous transitions are eliminated.