Reduction techniques for discrete-time Markov chains on totally ordered state space using stochastic comparisons

We propose in this paper two methods to compute Markovian bounds for monotone functions of a discrete time homogeneous Markov chain evolving in a totally ordered state space. The main interest of such methods is to propose algorithms to simplify analysis of transient characteristics such as the output process of a queue, or sojourn time in a subset of states. Construction of bounds are based on two kinds of results: well-known results on stochastic comparison between Markov chains with the same state space; and the fact that in some cases a function of Markov chain is again a homogeneous Markov chain but with smaller state space. Indeed, computation of bounds uses knowledge on the whole initial model. However, only part of this data is necessary at each step of the algorithms.