Real Time and Stochastic Time

We present a theory for the design and analysis of concurrent/distributed systems with real-time and stochastic time aspects. We start by presenting the model of Interactive Generalized Semi-Markov Processes (IGSMP): a compositional model for representing the class of stochastic processes known as Generalised Semi-Markov Processes (GSMPs), i.e. probabilistic timed systems where durations of delays are expressed by random variables with a general probability distribution. Technically, IGSMPs extend GSMPs with action transitions representing the ability of a process to interact with another process. Then, we introduce the calculus of Interactive Generalized Semi-Markov Processes, a stochastic process algebra which produces IGSMPs as semantic models of its terms. This is obtained by expressing the concurrent execution of delays through a simple probabilistic extension of Van Glabbeek and Vaandrageer's ST semantics based on dynamic names. We also present observational equivalence over IGSMPs, we observe that it is a congruence for all the operators of the calculus and we produce an axiomatization for this equivalence which is complete over finite-state strongly guarded processes. Finally, we present a case study on queuing systems G/G/1/q.