On approximating the stochastic behaviour of Markovian process algebra models
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Markov chains offer a rigorous mathematical framework to describe systems that exhibit stochastic behaviour, as they are supported by a plethora of methodologies to analyse their properties. Stochastic process algebras are high-level formalisms, where systems are represented as collections of interacting components. This compositional approach to modelling allows us to describe complex Markov chains using a compact high-level specification. There is an increasing need to investigate the properties of complex systems, not only in the field of computer science, but also in computational biology. To explore the stochastic properties of large Markov chains is a demanding task in terms of computational resources. Approximating the stochastic properties can be an effective way to deal with the complexity of large models. In this thesis, we investigate methodologies to approximate the stochastic behaviour of Markovian process algebra models. The discussion revolves around two main topics: approximate state-space aggregation and stochastic simulation. Although these topics are different in nature, they are both motivated by the need to efficiently handle complex systems. Approximate Markov chain aggregation constitutes the formulation of a smaller Markov chain that approximates the behaviour of the original model. The principal hypothesis is that states that can be characterised as equivalent can be adequately represented as a single state. We discuss different notions of approximate state equivalence, and how each of these can be used as a criterion to partition the state-space accordingly. Nevertheless, approximate aggregation methods typically require an explicit representation of the transition matrix, a fact that renders them impractical for large models. We propose a compositional approach to aggregation, as a means to efficiently approximate complex Markov models that are defined in a process algebra specification, PEPA in particular. Regarding our contributions to Markov chain simulation, we propose an accelerated method that can be characterised as almost exact, in the sense that it can be arbitrarily precise. We discuss how it is possible to sample from the trajectory space rather than the transition space. This approach requires fewer random samples than a typical simulation algorithm. Most importantly, our approach does not rely on particular assumptions with respect to the model properties, in contrast to otherwise more efficient approaches.