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Aggregation Methods for Markov Reward Chains with Fast and Silent Transitions

We analyze derivation of Markov reward chains from intermediate performance models that arise from formalisms for compositional performance analysis like stochastic process algebras, (generalized) stochastic Petri nets, etc. The intermediate models are typically extensions of continuous-time Markov reward chains with instantaneous labeled transitions. We give stochastic meaning to the intermediate models using stochastically discontinuous Markov reward chains, for which there are two prominent methods for aggregation: lumping and reduction to a pure Markov reward chain. As stochastically discontinuous Markov reward chains are not intuitive in nature, we consider Markov reward chains extended with transitions that are parameterized by a real variable. These transitions are called fast transitions, when they are governed by explicit probabilities, and silent transitions, when the probabilities are left unspecified. In the asymptotic case when the parameter tends to infinity, the models have a behavior of a stochastic discontinuous Markov reward chains. For all Markovian models, we develop two aggregation methods, one based on reduction and another one based on lumping and we give a comparative analysis between them.