Stochastic comparison of Markov queueing networks using coupling.

Abstract

Stochastic comparison is a method to prove bounds on performance metrics of stochastic models. Here, coupling can be used to define two processes on a common probability space, which makes it possible to compare the steady-state distributions of the processes. Two processes are stochastically related if their steady-state distributions satisfy a certain comparison relation. Such a stochastic relation can be more general than a stochastic order. In this thesis, a thorough description of stochastic comparison using coupling for the probability kernels of Markov processes is presented. Necessary and sufficient conditions for the stochastic comparison of stochastically related Markov queueing networks are given, in particular for the coordinate-wise and the summation relation. Also, an example of a Jackson network with breakdowns is studied, and an explicit coupling which preserves a subrelation of the coordinate-wise order relation is constructed. This allows to conclude that the steady-state distributions of the breakdown models are coordinate-wise comparable. Keywords: coupling, stochastic comparison, stochastic order, stochastic relation, Strassen’s theorem, Markov queueing network, Jackson network, probability kernel