On the effects of using the Grassmann-Taksar-Heyman method in iterative aggregation-disaggregation

Iterative aggregation-disaggregation (IAD) is an effective method for solving finite nearly completely decomposable (NCD) Markov chains. Small perturbations in the transition probabilities of these chains may lead to considerable changes in the stationary probabilities; NCD Markov chains are known to be ill-conditioned. During an IAD step, this undesirable condition is inherited by the coupling matrix and one confronts the problem of finding the stationary probabilities of a stochastic matrix whose diagonal elements are close to 1. In this paper, the effects of using the Grassmann-Taksar-Heyman (GTH) method to solve the coupling matrix formed in the aggregation step are investigated. Then the idea is extended in such a way that the same direct method can be incorporated into the disaggregation step. Finally, the effects of using the GTH method in the IAD algorithm on various examples are demonstrated, and the conditions under which it should be employed are explained.