Browsing by Subject "Stationary Probability"

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    On the effects of using the Grassmann-Taksar-Heyman method in iterative aggregation-disaggregation
    (SIAM, 1996) Dayar T.; Stewart, W. J.
    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.
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    Quasi lumpability, lower-bounding coupling matrices, and nearly completely decomposable Markov chains
    (SIAM, 1997) Dayar T.; Stewart, W. J.
    In this paper, it is shown that nearly completely decomposable (NCD) Markov chains are quasi-lumpable. The state space partition is the natural one, and the technique may be used to compute lower and upper bounds on the stationary probability of each NCD block. In doing so, a lower-bounding nonnegative coupling matrix is employed. The nature of the stationary probability bounds is closely related to the structure of this lower-bounding matrix. Irreducible lower-bounding matrices give tighter bounds compared with bounds obtained using reducible lower-bounding matrices. It is also noticed that the quasi-lumped chain of an NCD Markov chain is an ill-conditioned matrix and the bounds obtained generally will not be tight. However, under some circumstances, it is possible to compute the stationary probabilities of some NCD blocks exactly.