Universiteit Leiden

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Steady-State Analysis of Large Scale Systems

Promotor: W.Th.F. den Hollander Co-Promotor: F.M. Spieksma

L.C. Smit
25 mei 2016
Thesis in Leiden Repository

The general area of research of this dissertation concerns large systems with random aspects to their behavior that can be modeled and studied in terms of the stationary distribution of Markov chains. As the state spaces of such systems become large, their behavior gets hard to analyze, either via mathematical theory or via computer simulation. In this dissertation a class of Markov chains that we call successively lumpable is specified for which we show that the stationary probabilities can be obtained by successively computing the stationary probabilities of a propitiously constructed sequence of Markov chains. Each of the latter chains has a smaller state space and this yields significant computational improvements. These types of Markov chains have applications in many areas of applied probability comprising computer science, queueing theory, inventory theory, reliability and the theory of branching processes. To elaborate the applicability of the method we present explicit solutions for well-known queueing models. We compare the method both in speed and applicability with other methods and derive some additional properties and a numerical analysis to compute the associated product form, if it exists. Also, we handle some possibilities to extend the applicibility, for example by removing transitions from the network.

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