Contextual lumpability

Quantitative analysis of computer systems is often based on Markovian models. Among the formalisms that are used in practice, Markovian process algebras have found many applications, also thanks to their compositional nature that allows one to specify systems as interacting individual automata that carry out actions. Nevertheless, as with all state-based modelling techniques, Markovian process algebras suffer from the well-known state space explosion problem. State aggregation, specifically lumping, is one of the possible methods for tackling this problem. In this paper we revisit the notion of Markovian bisimulation which has previously been shown to induce a lumpable relation in the underlying Markov process. Here we consider the coarser relation of contextual lumpability, and taking the specific example of strong equivalence in PEPA, we propose a slightly relaxed definition of Markovian bisimulation, named lumpable bisimilarity, and prove that this is a characterisation of the notion of contextual lumpability for PEPA components. Moreover, we show that lumpable bisimilarity induces the largest contextual lumping over the Markov process underlying any PEPA component. We provide an algorithm for lumpable bisimilarity and study both its time and space complexity.