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Generalised Markovian Analysis of Timed Transition Systems

William J. Knottenbelt

Master's Thesis
Department of Computer Science, University of Cape Town
May, 1996

This dissertation concerns analytical methods for assessing the performance of concurrent systems. More specifically, it focuses on the efficient generation and solution of large Markov chains which are derived from models of unrestricted timed transition systems. Timed transition systems may be described using several high-level formalisms, including Generalised Stochastic Petri nets, queueing networks and Queueing Petri nets. A system modelled with one of these formalisms may be mapped onto a Markov chain through a process known as state space generation. The Markov chain thus generated can then be solved for its steady-state distribution by numerically determining the solution to a large set of sparse linear equations known as the steady-state equations.

Existing techniques of state space generation are surveyed and a new space-saving probabilistic dynamic state management technique is proposed and analysed in terms of its reliability and space complexity. State space reduction techniques involving on-the-fly elimination of vanishing states are also considered. Linear equation solvers suitable for solving large sparse sets of linear equations are surveyed, including direct methods, classical iterative methods, Krylov Subspace techniques and decomposition-based techniques. Emphasis is placed on Krylov subspace techniques and the Aggregation-Isolation technique, which is a recent decomposition-based algorithm applicable to solving general Markov chains.

Since Markov chains derived from real life models may have very large state spaces, it is desirable to automate the performance analysis sequence. Consequently, the new state management technique and several linear equation solvers have been implemented in the Markov chain analyser DNAmaca. DNAmaca accepts a high-level model description of a timed transition system, generates the state space, derives and solves the steady-state equations and produces performance statistics. DNAmaca is described in detail and examples of timed transition systems which have been analysed with DNAmaca are presented.

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