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Some renewal-theoretic investigations in the theory of sojourn times in finite semi-Markov processes

Published online by Cambridge University Press:  14 July 2016

Attila Csenki
Attila Csenki*
Affiliation:
Aston University
*
∗ Postal address: Department of Computer Science and Applied Mathematics, Aston University, Aston Triangle, Birmingham B4 7ET, UK.
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Abstract

In this note, an irreducible semi-Markov process is considered whose finite state space is partitioned into two non-empty sets A and B. Let MB(t) stand for the number of visits of Y to B during the time interval [0, t], t > 0. A renewal argument is used to derive closed-form expressions for the Laplace transform (with respect to t) of a certain family of functions in terms of which the moments of MB(t) are easily expressible. The theory is applied to a small reliability model in conjunction with a Tauberian argument to evaluate the behaviour of the first two moments of MB(t) as t →∞.

Keywords

RENEWAL THEORYLAPLACE TRANSFORMRELIABILITYTAUBERIAN THEOREM
Type
Research Papers
Information
Journal of Applied Probability , Volume 28 , Issue 4 , December 1991 , pp. 822 - 832
Copyright
Copyright © Applied Probability Trust 1991 

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References

[1] Bhat, U. N. (1984) Elements of Applied Stochastic Processes. Wiley, New York.Google Scholar
[2] Csenki, A. (1992) The joint distribution of sojourn times in finite Markov processes. Adv. Appl. Prob. 24 (2).Google Scholar
[3] Dubner, H. and Abate, J. (1968) Numerical inversion of Laplace transforms by relating them to the finite Fourier cosine transform. J. ACM 15, 115–123.Google Scholar
[4] Karlin, S. and Taylor, H. M. (1975) A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
[5] Rubino, G. and Sericola, B. (1989) Distribution of operational times in fault-tolerant systems modeled by semi-Markov reward processes. Technical Report 996, I.N.R.I.A., Campus de Beaulieu, 35042 Rennes Cedex, France, March 1989.Google Scholar
[6] Rubino, G. and Sericola, B. (1989) Accumulated reward over the N first operational periods in fault-tolerant computing systems. Technical Report 1028, I.N.R.I.A., Campus de Beaulieu, 35042 Rennes Cedex, France, May 1989.Google Scholar
[7] Rubino, G. and Sericola, B. (1991) Successive operational periods as measures of dependability. Dependable Computing and Fault-Tolerant Systems 4, 239–254.Google Scholar
[8] Rubino, G. and Sericola, B. (1989) Sojourn times in finite Markov processes. J. Appl. Prob. 26, 744–756.Google Scholar
[9] Singh, C. and Billinton, R. (1977) System Reliability Modelling and Evaluation. Hutchinson, London.Google Scholar