Generalised Markovian analysis of timed transition systems

This dissertation concerns analytical methods for assessing the performance of concurrent systems. More speci cally, it focuses on the e cient 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-they 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. ii Acknowledgements I would like to express my thanks and appreciation to: My supervisor, Prof Pieter S. Kritzinger, for his guidance and skilful management throughout the past 18 months. My family for their love, support and encouragement. My fellow DNA lab colleagues and my fellow Masters students in the Department of Computer Science, for their companionship and encouragement. Michael Sczittnick, the author of USENUM, for the use of his Markov Chain analyser and his assistance with USENUM model speci cations. DNA lab member Heinz Kabutz, for extensively beta-testing DNAmaca on several models produced during the course of his masters thesis. Peter Kemper, from the University of Dortmund, Germany, for helping me to verify the results produced by DNAmaca by running USENUM models with large state spaces using the computer resources at the University of Dortmund. Farooq Khan and Prof Djamal Zeghlache, both from RST department of L'Institut National des T el ecommunications, Evry, France, for beta-testing DNAmaca and for suggesting the teletra c switch application. Abderezak Touzene, from King Saud University, Saudi Arabia, for his advice and assistance while implementing his Aggregation-Isolation algorithm. DNA lab sta member Andrew Hutchison, for his willing assistance with modelling test cases using the MicroSnap queueing network analyser. iii Roya Ulrich, currently at the International Computer Science Institute, Berkeley, California, for her help with USENUM and performance analysis concepts. The Electrical Engineering department at the University of Cape Town for the use of their 4-processor SPARC 20 during the results-collection phase. Sandi Donno and Aleks Strez, our local system adminstrators, for keeping my computing environment running smoothly. The South African Foundation for Research and Development (FRD) for funding this research. iv