Stochastically Timed Process Algebra

Abstract

Timing aspects of concurrent and distributed systems can be expressed not only deterministically, but also probabilistically, which is particularly appropriate for shared-resource systems. When these aspects are modeled by using only exponentially distributed random variables, the stochastic process governing the system evolution over time turns out to be a Markov chain. From a process algebraic perspective, this limitation results in a simpler mathematical treatment both on the semantic side and on the stochastic side without sacrificing expressiveness. In this chapter, we introduce a Markovian process calculus with durational actions, then we discuss congruence properties, sound and complete axiomatizations, modal logic characterizations, and verification algorithms for Markovian versions of bisimulation equivalence, testing equivalence, and trace equivalence. We also examine a further property called exactness, which is related to Markov-chain-level aggregations induced by Markovian behavioral equivalences. Finally, we show how the linear-time/branching-time spectrum collapses in the Markovian case.