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 l. 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.

[1]  Wei Wu,et al.  Numerical Experiments with Iteration and Aggregation for Markov Chains , 1992, INFORMS J. Comput..

[2]  William J. Stewart,et al.  Iterative aggregation/disaggregation techniques for nearly uncoupled markov chains , 1985, JACM.

[3]  W. Stewart,et al.  ITERATIVE METHODS FOR COMPUTING STATIONARY DISTRIBUTIONS OF NEARLY COMPLETELY DECOMPOSABLE MARKOV CHAINS , 1984 .

[4]  T. Y. WilliamJ,et al.  Numerical Methods in Markov Chain Modeling , 1992, Operational Research.

[5]  Herbert A. Simon,et al.  Aggregation of Variables in Dynamic Systems , 1961 .

[6]  William J. Stewart,et al.  A comparison of numerical techniques in Markov modeling , 1978, CACM.

[7]  C. Paige,et al.  Computation of the stationary distribution of a markov chain , 1975 .

[8]  G. Stewart,et al.  A Two-Stage Iteration for Solving Nearly Completely Decomposable Markov Chains , 1994 .

[9]  Winfried K. Grassmann,et al.  Regenerative Analysis and Steady State Distributions for Markov Chains , 1985, Oper. Res..

[10]  Sergio Pissanetzky,et al.  Sparse Matrix Technology , 1984 .

[11]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[12]  William J. Stewart,et al.  Introduction to the numerical solution of Markov Chains , 1994 .

[13]  R. Plemmons,et al.  Comparison of Some Direct Methods for Computing Stationary Distributions of Markov Chains , 1984 .

[14]  C. D. Meyer.  Sensitivity of the Stationary Distribution of a Markov Chain , 1994, SIAM J. Matrix Anal. Appl..

[15]  G. Stewart,et al.  On a direct method for the solution of nearly uncoupled Markov chains , 1991 .

[16]  Carl D. Meyer,et al.  Stochastic Complementation, Uncoupling Markov Chains, and the Theory of Nearly Reducible Systems , 1989, SIAM Rev..

[17]  C. O'Cinneide.  Entrywise perturbation theory and error analysis for Markov chains , 1993 .

[18]  Daniel P. Heyman,et al.  Further comparisons of direct methods for computing stationary distributions of Markov chains , 1987 .