Sub-stochastic matrix analysis and performance bounds

On the one hand, the state space of complex Markovian models can often be partitioned between a small subset with a high steady-state probability and a large subset with a low steady-state probability. On the other hand, performance evaluation and reliability analysis require the computation of performance indices, often defined as functions of instantaneous rewards on the states of the model. Thus the time and space complexity of this computation would be greatly decreased by avoiding the explicit representation of a large part of the subset associated to the low probability. In this report, we present a method to derive bounds on such rewards directly from bounds on the parameters of the model transition rates or probabilities. The method is based on the analysis of an aggregated Markov chain and on the properties of strong stochastic comparison for discrete as well as continuous Markov (sub-)chains. We also propose a specific method when the reward is the output rate towards a subset of states of a continuous Markov chain. Finally we illustrate our approach on some examples in order to show its interest.