Formal Characterisation of Immediate Actions in SPA with Nondeterministic Branching

Stochastic Process Algebras (SPA) are process algebras in which the duration of each activity is given by a random variable. If the stochastic aspect is restricted to Markovian, i.e. exponentially distributed durations, nice algebraic foundations are available. They include a formal semantics and an equational theory for Markovian bisimulation, a congruence that can be seen as a stochastic counterpart of strong bisimulation. This paper extends that theory with a stochastic notion of Milner's observational congruence. We enrich a basic SPA with immediate actions that happen instantaneously if enabled. For the enriched calculus we will derive a sound and complete characterisation of Markovian observational congruence, a conservative extension of both Markovian bisimulation and observational congruence. The usefulness of immediate actions together with their equational theory will be illustrated by means of an example. Additionally, we will discuss some implementation and modelling issues arising from our results.