Stochastic Monotonicity in Queueing Networks

Stochastic monotonicity is one of the sufficient conditions for stochastic comparisons of Markov chains. On a partially ordered state space, several stochastic orderings can be defined by means of increasing sets. The most known is the strong stochastic (sample-path) ordering, but weaker orderings (weak and weak*) could be defined by restricting the considered increasing sets. When the strong ordering could not be defined, weaker orderings represent an alternative as they generate less constraints. Also, they may provide more accurate bounds. The main goal of this paper is to provide an intuitive event formalism added to stochastic comparisons methods in order to prove the stochastic monotonicity for multidimensional Continuous Time Markov Chains (CTMC). We use the coupling by events for the strong monotonicity. For weaker monotonicity, we give a theorem based on generator inequalities using increasing sets. We prove this theorem, and we present the event formalism for the definition of the increasing sets. We apply our formalism on queueing networks, in order to establish monotonicity properties.

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