An Asynchronous Calculus for Generative-Reactive Probabilistic Systems

• Authors: A. Aldini;M. Bravetti
• Venue: An Asynchronous Calculus for Generative-Reactive Probabilistic Systems
• Year: 2000

The experience in the formal specification and analysis of real concurrent systems via process algebras revealed the importance of using languages expressing time (the advancing speed of processes), probabilistic external/internal choices and multi-way synchronization. In this paper we address the open problem of developing a process algebra with all the mechanisms above in the context of discrete time, i.e. where time is not continuous but is represented by a sequence of discrete steps as in Discrete Time Markov Chains (DTMCs). Such a problem is solved in a natural way by resorting to a variant of CSP which employs a probabilistic asynchronous parallel operator and a synchronization mechanism based on a mixture of the classical generative and reactive models of probability. Differently from existing discrete time process algebras, where parallel processes are executed in synchronous locksteps, the parallel operator that we adopt allows processes with different probabilistic advancing speeds (mean number of actions executed per time unit) to be modeled. Moreover the generative-reactive synchronization mechanism that we adopt is based on an asymmetric master-slave cooperation discipline which leads to a clear notion of control over action synchronizations based on probabilistic external/internal choices. Since semantic models of processes turn out to be a mixture of the generative and reactive probabilistic models (including external non-determinism only), in the case of fully specified systems we obtain DTMCs which can be analyzed through standard techniques for deriving performance measures. Moreover we show that when evaluating steady state based performance measures of systems, our approach provides an exact solution even if advancing speeds are considered not to be probabilistic, without incurring in the state space explosion problem which arises with standard synchronous approaches. We then present a case study on multi-path routing showing the expressiveness of our calculus and, in particular, that it makes it particularly easy to produce scalable specifications. Finally, we show that the introduction of an auxiliary operator leads to a sound and complete axiomatization of probabilistic bisimulation over finite processes of our calculus, which is a smooth extension of the axiom system for a standard process algebra. In particular, with respect to the previous proposals in the literature, the new operator leads to a clean axiomatization of action restriction in the generative model.