Behavioural equivalences for fluid stochastic Petri nets

We propose fluid equivalences that allow one to compare and reduce behaviour of labeled fluid stochastic Petri nets (LFSPNs) while preserving their discrete and continuous properties. We define 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 show that fluid trace equivalence preserves average potential fluid change volume for the transition sequences of every certain length. We prove that fluid bisimulation equivalence preserves the following aggregated (by such a bisimulation) probability functions: stationary probability mass for the underlying CTMC, as well as stationary fluid buffer empty probability, fluid density and distribution for the associated SFM. Hence, the equivalence guarantees identity of a number of discrete and continuous performance measures. Fluid bisimulation equivalence is then used to simplify the qualitative and quantitative analysis of LFSPNs that is accomplished by means of quotienting (by the equivalence) the discrete reachability graph and underlying CTMC. To describe the quotient associated SFM, the quotients of the probability functions are defined. We also characterize logically fluid trace and bisimulation equivalences with two novel fluid modal logics HMLflt and HMLflb, constructed on the basis of the well-known Hennessy-Milner Logic HML. These results can be seen as operational characterizations of the corresponding logical equivalences. The application example of a document preparation system demonstrates the behavioural analysis via quotienting by fluid bisimulation equivalence.