Population models from PEPA descriptions

Stochastic process algebras (e.g. PEPA [10], EMPA [1], TIPP [9]) emerged about 15 years ago as system description techniques for performance modelling. They have enjoyed some considerable success in this arena. For example, PEPA has been used to study the performance of a wide variety of systems [12, 2, 3, 18, 13]. This analysis has been based on the generation of a continuous time Markov chain (CTMC) underlying the labelled transition system of the process algebra model, derived via the interleaving structured operational semantics. The CTMC facilitates steady state and transient analysis numerically. From these distributions many performance metrics can be derived, such as utilization, throughput, and mean time to congestion. Unfortunately, as with all state-based modelling techniques, CTMCs, and consequently stochastic process algebras, suffer from problems of state space explosion. Such models can be regarded as being at the individual level as all details of all the components of the model are recorded in the state of the model and its subsequent analysis. Although developed for performance modelling of computer and communication systems, stochastic process algebras have proved to be useful for modelling other systems as well. In particular, in recent years there has been considerable interest in using stochastic process algebras and similar formalisms for expressing models for systems biology [16, 17, 4, 6]. These models also exhibit problems of state space explosion, as both the types of components and the number of components of each type are typically large. In recent work we have been exploring an alternative mapping from a system description in the PEPA stochastic process algebra. In this mapping we aim to capture the behaviour of PEPA components at a population level. Rather than capturing individual behaviours as happens in the CTMC semantics, we instead map to a set of non-linear ordinary differential equations (ODEs) [5, 11]. This incurs some loss of information with respect to, for example, the steady state probability distribution of the CTMC. Nevertheless the solution of the set of ODEs can still give us useful quantitative information about the system.