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Optimal replacement in a shock model: discrete time

Published online by Cambridge University Press:  14 July 2016

Terje Aven  and
Simen Gaarder
Terje Aven*
Affiliation:
Rogaland College
Simen Gaarder
Affiliation:
University of Oslo
*
∗Postal address: Rogaland College, Box 2540, Ullandhaug, 4001 Stavanger, Norway.
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Abstract

A system is subject to a sequence of shocks occurring randomly at times n = 1, 2, ···; each shock causes a random amount of damage. The system might fail at any point in time n, and the probability of a failure depends on the history of the system. Upon failure the system is replaced by a new and identical system and a cost is incurred. If the system is replaced before failure a smaller cost is incurred. We study the problem of specifying a replacement rule which minimizes the long-run (expected) average cost per unit time. A special case, in which the system fails when the total damage first exceeds a fixed threshold, is analysed in detail.

Keywords

REPLACEMENTSTOPPING TIMEAVERAGE COST
Type
Short Communications
Information
Journal of Applied Probability , Volume 24 , Issue 1 , March 1987 , pp. 281 - 287
Copyright
Copyright © Applied Probability Trust 1987 

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Footnotes

∗∗

Present address: Gjensidige Insurance Co., Box 6738, St. Olavs pl., 0130 Oslo 1, Norway.

References

Abdel-Hameed, M. and Shimi, I. N. (1978) Optimal replacement of damaged devices. J. Appl. Prob. 15, 153–161.Google Scholar
Aven, T. (1983) Contribution to Failure Time Data Analysis and Optimal Maintenance Planning. , Univ of Oslo.Google Scholar
Aven, T. and Bergman, B. (1986) Optimal replacement times — a general set-up. J. Appl. Prob. 23, 432–442.Google Scholar
Bergman, B. (1978) Optimal replacement under a general failure model. Adv. Appl. Prob. 10, 431–451.Google Scholar
Bremaud, P. (1981) Point Processes and Queues. Springer-Verlag, New York.Google Scholar
Feldman, R. M. (1976) Optimal replacement with semi-Markov shock models. J. Appl. Prob. 13, 108–117.Google Scholar
Meyer, P. A. (1966) Probability and Potentials. Blaisdell, Waltham, Mass.Google Scholar
Nummelin, E. (1980) A general failure model: Optimal replacement with state dependent replacement and failure costs. Math. Operat. Res. 3, 381–387.10.1287/moor.5.3.381Google Scholar
Ross, S. M. (1970) Applied Probability Models with Optimization Applications. Holden-Day, San Fransisco.Google Scholar
Taylor, H. M. (1975) Optimal replacement under additive damage and other failure models. Naval Res. Logist Quart. 22, 1–18.Google Scholar
Yamada, K. (1980) Explicit formula of optimal replacement under additive shock processes. Stoch. Proc. Appl. 9, 193–208.Google Scholar
Zuckerman, D. (1978a) Optimal replacement policy for the case where damage process is a one-sided Lévy process. Stoch. Proc. Appl. 7, 141–151.Google Scholar
Zuckerman, D. (1978b) Optimal stopping in a semi-Markov shock model. J. Appl. Prob. 15, 629–634.Google Scholar
Zuckerman, D. (1979) Optimal replacement rule-discounted cost criterion. RAIRO 1, 67–74.Google Scholar