On the choice of the stochastic comparison method for multidimensional Markov chains analysis

The $\preceq{\Phi}$ stochastic comparison $(\preceq{\Phi} \in {\preceq{st}$, $\preceq{wk}$, $\preceq{wk^}})$ of multidimensional Continuous Time Markov Chains (CTMC)s is an efficient but a complex method for the performability evaluation of computer systems. Different techniques can be applied for the stochastic comparison of Markov chains. The coupling is an intuitive method, and may be applied by comparing the evolution of sample paths due to events to establish the $\preceq{st}$ ordering. The increasing set method is based on the comparison of transition rates for a family of increasing sets. It is a more general formalism as it can be applied for all of these orderings : $\preceq{st}$, $\preceq{wk}$, and $\preceq{wk^}$. The goal of this paper is to identify the relationships between these orderings, in order to determine the method to apply for establishing comparisons between models. Altough the $\preceq{st}$ ordering between random variables implies $\preceq{wk}$ and $\preceq{wk^}$ orderings, this result could not be generalized to the comparison of stochastic processes. However even the $\preceq_ {st}$ ordering does not exist between processes, the $\preceq{wk}$ and the $\preceq{wk^}$ constraints could be satisfied. In this paper, we aim to give the intuition to choose the most suitable method with respect to the underlying performability study.