Rate-Based Stochastic Fusion Calculus and Continuous Time Markov Chains

This paper presents a stochastic fusion calculus suitable to describe systems involving general patterns of interactions. We start from fusion calculus [8] which is a symmetric generalisation of the π-calculus, and present a rate-based stochastic fusion calculus, providing a concise and compositional way to describe the behaviour of complex systems by using probability distributions. We provide the semantics of stochastic fusion calculus by using rate-based transition systems [4] in the elegant and general variant proposed by De Nicola et al. [2]. The stochastic nature of the new transition systems is given by the fact that transition labels represent actions, and the transition result is a function associating a positive real value to each possible target process, expressing the stochastic rate of an exponential distribution modelling the duration of the transition. For two processes running in parallel, we define the distribution of their synchronisation using their apparent rates. Associativity of parallel composition is a particularly desirable property either in the context of network and distributed systems, either in the context of biological systems, where parallel composition is often used to model molecular populations. Following the approach proposed in [2], associativity of the parallel composition operator is guaranteed in the rate-based stochastic semantics of the fusion calculus (differently to what happens in the stochastic π-calculus [9] and in a previous formalisation of a stochastic fusion calculus [1]). We extend the notion of hyperbisimulation to stochastic fusion calculus, and prove that the stochastic hyperequivalence is a congruence. The rate-based transition system resulting from a stochastic fusion process leads the expression of a continuous time Markov chain which preserves the notion of hyperequivalence. The modelling power of the stochastic fusion calculus is suggested by an example where we formalise some of the one-to-many interactions occurring between a plant root and a particular kind of fungi in the arbuscular mycorrhizal symbiosis. A quantitative simulation is performed using the PRISM model checker on the continuous time Markov chain extracted from the rate-based transition system describing such interactions by means of stochastic fusion processes.