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Finite birth-and-death models in randomly changing environments

Published online by Cambridge University Press:  01 July 2016

D. P. Gaver ,
P. A. Jacobs  and
G. Latouche
D. P. Gaver*
Affiliation:
Naval Postgraduate School, Monterey
P. A. Jacobs*
Affiliation:
Naval Postgraduate School, Monterey
G. Latouche*
Affiliation:
Université Libre de Bruxelles
*
∗Postal address: Department of the Navy, Naval Postgraduate School, Monterey, CA 93943, USA.
∗Postal address: Department of the Navy, Naval Postgraduate School, Monterey, CA 93943, USA.
∗∗ Postal address: Université Libre de Bruxelles, C.P. 212, Boulevard du Triomphe, Bruxelles, Belgium.
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Abstract

An efficient computational approach to the analysis of finite birth-and-death models in a Markovian environment is given. The emphasis is upon obtaining numerical methods for evaluating stationary distributions and moments of first-passage times.

Keywords

BIRTH–DEATH PROCESSESMARKOVIAN ENVIRONMENTNUMERICAL METHODSSTATIONARY DISTRIBUTIONSFIRST-PASSAGE TIMES
Type
Research Article
Information
Advances in Applied Probability , Volume 16 , Issue 4 , December 1984 , pp. 715 - 731
Copyright
Copyright © Applied Probability Trust 1984 

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References

[1] Chu, B. B. and Gaver, D. P. (1977) Stochastic models for repairable redundant systems susceptible to common mode failure. SIAM, Proc. Internat. Conf. Nuclear Systems, Reliability Engineering and Risk Assessment, Gatlinburg, Tennessee, 342–367.Google Scholar
[2] Eisen, M. and Tainiter, M. (1963) Stochastic variations in queueing processes. Operat. Res. 11, 922–927.Google Scholar
[3] Feller, W. (1966) An Introduction to Probability Theory and its Applications, Vol. 2. Wiley, New York.Google Scholar
[4] Hajek, B. (1982) Birth-and-death processes on the integers with phases and general boundaries. J. Appl. Prob. 19, 488–499.CrossRefGoogle Scholar
[5] Karlin, S. and Taylor, H. M. (1975) A First Course in Stochastic Processes. Academic Press, London.Google Scholar
[6] Keilson, J., Sumita, U. and Zachman, M. (1981) Row-continuous finite Markov chains, structure and algorithms. Technical Report No. 8115, Graduate School of Management, University of Rochester.Google Scholar
[7] Kemeny, J. G. and Snell, J. L. (1960) Finite Markov Chains. Van Nostrand, Princeton, NJ.Google Scholar
[8] Neuts, M. F. (1980) The probabilistic significance of the rate matrix in matrix-geometric invariant vectors. J. Appl. Prob. 17, 291–296.Google Scholar
[9] Neuts, M. F. (1981) Matrix-Geometric Solutions in Stochastic Models - An Algorithmic Approach. The Johns Hopkins University Press, Baltimore, MD.Google Scholar
[10] Neuts, M. F. and Meier, K. S. (1981) On the use of phase type distributions in reliability modelling of systems with a small number of components. OR Spektrum 2, 227–234.Google Scholar
[11] Purdue, P. (1974) The M/M/l queue in a Markovian environment. Operat. Res. 22, 562–569.Google Scholar
[12] Torrez, W. C. (1979) Calculating extinction probabilities for the birth-and death chain in a random environment. J. Appl. Prob. 16, 709–720.Google Scholar
[13] Wilkinson, J. H. and Reinsch, C. (1971) Handbook for Automatic Computation, Vol. 2, Linear Algebra. Springer-Verlag, Berlin.Google Scholar
[14] Yechiali, U. (1973) A queueing-type birth-and-death process defined on a continuous-time Markov chain. Operat. Res. 21, 604–609.CrossRefGoogle Scholar