Fluid-flow solutions to the state space explosion problem

Achieving the appropiate performance requirements for computer–communication systems is as important as the correctness of the end-result. This is particularly difficult in the case of massively parallel computer systems such as the clusters of PCs behind the likes of Google and peer-to-peer filesharing networks such as Bittorrent. Measuring the performance of such systems using a mathematical model is invariably computationally intensive. Formal modelling techniques make possible the derivation of such performance measures but currently suffer from the state-space explosion problem, that is, models become intractably large even for systems of apparently modest complexity. This work develops a novel class of techniques aimed at addressing this problem by approximating a representation of massive state spaces as more computationally-tractable real variables (‘fluid-flow analysis’). Short introduction to performance modelling Accurate performance modelling at the system design stage has never been more important than in a technological age dominated by large and complex computer and communication networks. Measurements such as request throughput or server utilisation can be used, for example, to predict the location of bottlenecks in the passage of requests through a computer network, and suggest steps to improve the situation. A very useful mathematical tool for modelling many classes of systems is the continuous-time Markov chain (CTMC). CTMCs model the behaviour of a system by describing the set of possible states a system may be in and how the system moves between states over time. Models can be formalised directly into CTMCs, however there are many advantages to modelling a system using a higher-level abstraction, such as a stochastic process algebra (SPA) (e.g. PEPA [1], MTIPP [2], SPADES [3] and EMPA [4]), stochastic Petri net (SPN) [5, 6, 7] or queueing network [8, 9, 10, 11, 12]. Commonly, the model may be ‘solved’ through the computation of the steady-state analysis of an underlying CTMC1 (‘Markovian’ formalisms). This project focuses on the well-known and very popular SPA, PEPA (Peformance Evaluation Process Algebra), however the contributions are readily extensible to other Markovian formalisms. Despite the relative tractability of CTMCs, models of realistic complexity can easily result in underlying state spaces of computationally intractable size. This phenomenon is known as ‘state-space explosion’ and is the current bottle-neck in the field of performance analysis, limiting the size of models and thus the complexity of systems that can be efficiently analysed. Naturally, the demand for more accurate and finer-grained models of larger systems increases constantly, so there has been much research aimed at suppressing this explosion in some sense [13, 14, 15]. However, none of these approaches have succeeded in general at delivering the exponential speedup that is required to suppress this problem; the situation clearly demands a novel direction. 1In general, this requires the diagonalisation of a matrix with dimension equal to the number of states in the CTMC.