Lumping-Based Equivalences in Markovian Automata and Applications to Product-Form Analyses

The analysis of models specified with formalisms like Markovian process algebras or stochastic automata can be based on equivalence relations among the states. In this paper we introduce a relation called exact equivalence that, differently from most aggegation approaches, induces an exact lumping on the underlying Markov chain instead of a strong lumping. We prove that this relation is a congruence for Markovian process algebras and stochastic automata whose synchronisation semantics can be seen as the master/slave synchronisation of the Stochastic Automata Networks (SAN). We show the usefulness of this relation by proving that the class of quasi-reversible models is closed under exact equivalence. Quasi-reversibility is a pivotal property to study productform models, i.e., models whose equilibrium behaviour can be computed very efficiently without the problem of the state space explosion. Hence, exact equivalence turns out to be a theoretical tool to prove the productform of models by showing that they are exactly equivalent to models which are known to be quasi-reversible.