successive lumping procedure for a class of Markov chains

A class of Markov chains we call successively lumpable is specified for which it is shown that the stationary probabilities can be obtained by successively computing the stationary probabilities of a propitiously constructed sequence of Markov chains. Each of the latter chains has (typically much) smaller state space and this yields signifi-cant computational improvements. We discuss how the results for discrete time Markov chains extend to semi-Markov processes and continuous time Markov processes. Finally we will study applications of successively lumpable Markov chains to classical reliability and queueing models