Lumping and reversed processes in cooperating automata

Performance evaluation of computer software or hardware architectures may rely on the analysis of a complex stochastic model whose specification is usually given in terms of a high level formalism such as queueing networks, stochastic Petri nets, stochastic automata or Markovian process algebras. Compositionality is a key-feature of many of these formalisms and allows the modeller to combine several (simple) components to form a complex architecture. However, although these formalisms lead to relative compact specifications of possibly complex models, the derivation of the performance indices may be computationally very time and space consuming since the set of possible states of the model tends to grow exponentially (or even faster) with the number of components. In this paper we focus on models with underlying continuous time Markov chains (CTMCs) and we introduce a notion of typed lumpability, which gives sufficient conditions under which a lumping of the process can be derived, allowing the exact computation of marginal stationary probabilities of the cooperating components. The peculiarity of our method relies on the fact that lumping is applied at the component-level rather than to the CTMC underlying the joint process, thus reducing both the memory requirements and the computational cost of the subsequent solution of the model. Moreover, we investigate the properties of the lumping of reversed automata and we prove that, if these are reversible, a conditional product-form solution of their cooperation with other non-blocking automata can be derived. Although conditional product-forms have been previously investigated by other authors with the aim of approximating non-product-form models, the contribution of this paper consists in giving sufficient conditions for this approach to yield exact results and providing examples to support the modeller's intuition.