For an n -state, homogeneous, ergodic Markov chain with a transition matrix T , its stationary distribution vector and mean first passage matrix are two important characteristics that provide the long-term and short-term perspectives, respectively, on the behavior of the chain. It has been shown by C. D. Meyer that any Perron complement of T can be regarded as the transition matrix of some homogeneous ergodic Markov chain with fewer states and that such reduced chains obtained from the Perron complements of T can be exploited to compute in parallel the stationary distribution vector for the entire chain.
Motivated by Meyer's work, we study the relationship between the mean first passage matrix for the entire chain and the mean first passage matrices for the reduced chains obtained from the Perron complements of T. We then formulate an algorithm for computing in parallel the mean first passage matrix for the entire chain via those reduced chains. We estimate the asymptotic number of multiplication operations that is necessary to implement our algorithm, which suggests that significant savings could be achieved. We also investigate the stability issues related to both Meyer's algorithm and ours. Analysis in this regard shows that in general the computation involving the reduced chains is at least as stable as that on the entire chain.