Logical characterization of fluid equivalences

Abstract

We investigate fluid equivalences that allow one to compare and reduce behaviour of labeled fluid stochastic Petri nets (LFSPNs) with a single continuous place while preserving their discrete and continuous properties. We propose a linear-time relation of fluid trace equivalence and its branching-time counterpart, fluid bisimulation equivalence. Both fluid relations take into account the essential features of the LFSPNs behaviour, such as functional activity, stochastic timing and fluid flow. We consider the LFSPNs whose continuous markings have no influence to the discrete ones, i.e. every discrete marking determines completely both the set of enabled transitions, their firing rates and the fluid flow rates of the incoming and outgoing arcs for each continuous place. Moreover, we require that the discrete part of the LFSPNs should be continuous time stochastic Petri nets. The underlying stochastic model for the discrete part of the LFSPNs is continuous time Markov chains (CTMCs). The performance analysis of the continuous part of LFSPNs is accomplished via the associated stochastic fluid models (SFMs). We characterize logically fluid trace and bisimulation equivalences with two novel fluid modal logics HM L f lt and HM L f lb , constructed on the basis of the well-known Hennessy-Milner Logic (HML). These characterizations guarantee that two LFSPNs are fluid (trace or bisimulation) equivalent iff they satisfy the same formulas of the respective logic, i.e. they are logically equivalent. The results imply operational characterizations of the logical equivalences.