Stochastic equivalence for performance analysis of concurrent systems in dtsiPBC

Abstract

We propose an extension with immediate multiactions of discrete time stochastic Petri box calculus (dtsPBC), pre- sented by I.V. Tarasyuk. The resulting algebra dtsiPBC is a discrete time analogue of stochastic Petri box calculus (sPBC) with immediate multiactions, proposed by H. MaciV. Valero and others within a continuous time domain. The step operational semantics is constructed via labeled probabilistic transition systems. The denotational seman- tics is defined on the basis of a subclass of labeled discrete time stochastic Petri nets with immediate transitions. A consistency of the both semantics is demonstrated. In order to evaluate performance, the corresponding semi-Markov chains and (reduced) discrete time Markov chains are analyzed. We define step stochastic bisimulation equivalence of expressions and prove that it can be applied to reduce their transition systems and underlying semi-Markov chains while preserving the functionality and performance characteristics. We explain how this equivalence may help to sim- plify performance analysis of the algebraic processes. In a case study, a method of modeling, performance evaluation and behaviour preserving reduction of concurrent systems is outlined and applied to the shared memory system.