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Applications of generalized inverses to Markov chains

Published online by Cambridge University Press:  01 July 2016

William Rising
William Rising*
Affiliation:
University of Louisville
*
∗Postal address: Department of Mathematics, University of Louisville, Louisville, KY 40292, USA.
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Abstract

First it is shown that any generalized inverse of the infinitesimal generator of an irreducible Markov chain can be used to compute the exact stationary distribution and all the expected first-passage times of the chain. In the special case of a single-server queue this allows all computations to be done with upper-triangular matrices.

Next it is shown that the effect of a perturbation of the infinitesimal generator on the stationary distribution and expected first-passage times can also be computed using generalized inverses. These results extend and generalize Schweitzer's [9] original work using fundamental matrices. It is then shown that any perturbation can be broken up into a series of perturbations each involving a single state.

Keywords

BISTABLE QUEUESFIRST-PASSAGE TIMES
Type
Research Article
Information
Advances in Applied Probability , Volume 23 , Issue 2 , June 1991 , pp. 293 - 302
Copyright
Copyright © Applied Probability Trust 1991 

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References

1. Carleial, A. B. and Hellman, M. E. (1975) Bistable behaviour of aloha-type systems. IEEE Trans. Comm. 45, 401–409.Google Scholar
2. Chung, Kai Lai (1960) Markov Chains with Stationary Transition Probabilities. Springer-Verlag, Berlin.Google Scholar
3. Cottrell, M., Fort, J.-C. and Malgouyres, G. (1983) Large deviations and rare events in the study of stochastic systems. IEEE Trans. Autom. Control 28, 907–920.Google Scholar
4. Haviv, M. and Van Der Heyden, L. (1984) Perturbation bounds for the stationary probabilities of a finite Markov chain. Adv. Appl. Prob. 16, 804–818.CrossRefGoogle Scholar
5. Kemeny, J. G. and Snell, J. L. (1960) Finite Markov Chains , Van Nostrand, Princeton.Google Scholar
6. Kleinrock, L. and Lam, S. J. (1975) Packet switching in a multiaccess broadcast channel. IEEE Trans. Comm. 23, 410–423.CrossRefGoogle Scholar
7. Nelson, R. (1987) Stochastic catastrophe theory in computer performance. J. Assoc. Comput. Mach. 34, 661–685.CrossRefGoogle Scholar
8. Onozato, Y. and Noguchi, S. (1983) On the thrashing cusp in slotted aloha systems. IEEE Trans. Comm. 33, 1171–1182.Google Scholar
9. Schweitzer, P. J. (1968) Perturbation and finite Markov chains. J. Appl. Prob. 5, 401–413.Google Scholar