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We propose a novel notion of fluid bisimulation equivalence that allows one to compare and reduce the behavior of labeled fluid stochastic Petri nets (LF-SPNs) while preserving their discrete and continuous properties. The underlying stochastic model for the discrete part of the LFSPNs is a continuous time Markov chain (CTMC). The performance analysis of the continuous part of the LFSPNs is accomplished via the associated stochastic fluid models. For the fluid bisimulation on the discrete markings of two LFSPNs, we require it to be a (strong) Markovian bisimulation. On the continuous markings, for every pair of Markovian bisimilar discrete markings, the fluid flow rates of the continuous places in the first LFSPN should coincide with those of the corresponding continuous places in the second LFSPN. We prove that the resulting fluid bisimulation equivalence preserves fluid density and distribution, as well as discrete and continuous performance measures.

We define a number of stochastic equivalences in the dtsPBC framework, which is a discrete time stochastic extension of finite Petri box calculus (PBC) enriched with iteration. These equivalences allow the identification of stochastic processes that have similar behaviour but are differentiated by the semantics of the calculus. We explain how the equivalences we propose can be used to reduce transition systems of expressions, and demonstrate how to apply the equivalences to compare the stationary behaviour. The equivalences guarantee a coincidence of performance indices for stochastic systems, and can be used for performance analysis simplification. We use a case study to outline a method of modelling, performance evaluation and behaviour preserving reduction of concurrent computing systems, and apply it to the dining philosophers system.

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