This book is, in essence, the dissertation I submitted to the University of Edinburgh in early January 1994. My examiners, Peter Harrison of the Imperial College, and Stuart Anderson of the University of Edinburgh, suggested some corrections and revisions. Apart from those changes, most chapters remain unaltered except for minor corrections and reformatting. The exceptions are the first and final chapter. Since the final chapter discusses several possible directions for future work, it is now supplemented with a section which reviews the progress which has been made in each of these directions since January 1994. There are now many more people interested in stochastic process algebras and their application to performance modelling. Moreover, since these researchers have backgrounds and motivations different from my own some of the most interesting new developments are outside the areas identified in the original conclusions of the thesis. Therefore the book concludes with a brief overview of the current status of the field which includes many recent references. This change to the structure of the book is reflected in the summary given in Chapter 1. No other chapters of the thesis have been updated to reflect more recent developments. A modified version of Chapter 8 appeared in the proceedings of the 2nd International Workshop on Numerical Solution of Markov Chains, January 1995. I would like to thank my supervisor, Rob Pooley, for introducing me to performance modelling and giving me the job which brought me to Edinburgh initially. Many colleagues on the IMSE project provided stimulating discussions which influenced this work. My second supervisor, Julian Bradfield, provided support and advice in large quantities for which I am very grateful. Many other people also influenced this work through helpful comments, discussions and encouragement; they include

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In this paper we present an overview of the field of deterministic approximation of Markov processes, both in discrete and continuous times. We will discuss mean field approximation of discrete time Markov chains and fluid approximation of continuous time Markov chains, considering the cases in which the deterministic limit process lives in continuous time or discrete time. We also consider some more advanced results, especially those relating to the limit stationary behaviour. We assume a knowledge of modelling with Markov chains, but not of more advanced topics in stochastic processes

In this paper we present a new technique for performance modelling and a tool supporting this approach. Performance Evaluation Process Algebra (PEPA) [1] is an algebraic language which can be used to build models of computer systems which capture information about the performance of the system. The PEPA language serves two purposes as a formal description language for computer system models. The performance-related information in the model may be used to predict the performance of the system whereas the behavioural information in the model may be exploited when reasoning about the functional behaviour of the system (e.g. when finding deadlocks or when exhibiting equivalences between sub-components). In this paper we concentrate on the performance aspects of the language. A method of reasoning about PEPA models proceeds by considering the derivation graph obtained from the model using the underlying operational semantics of the PEPA language. The derivation graph is systematically reduced to a form where it can be treated as the state transition diagram of the underlying stochastic (in fact, Markovian) process. From this can be obtained the infinitesimal generator matrix of the Markov process. A steady state probability distribution for the system can then be obtained, if it exists. We have implemented a prototype tool which supports this methodology from the initial checking of the well-formedness of the PEPA model through the creation of the state transition diagrams to the calculation of performance measures based on the infinitesimal generator matrix. The tool is implemented in Standard ML [2] and provides an interface to the Maple Symbolic Algebra package [3] for the solution of matrix equations.