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Optimal replacement of damaged devices

Published online by Cambridge University Press:  14 July 2016

M. Abdel-Hameed  and
I. N. Shimi
M. Abdel-Hameed
Affiliation:
University of North Carolina at Charlotte
I. N. Shimi
Affiliation:
Florida State University
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Abstract

A device is subject to a sequence of shocks occurring randomly at times n = 1, 2, ⃛. At each point in time, shocks occur according to a Poisson distribution with parameter λ. Shocks cause damage and damage accumulates additively. They can cause the device to fail, and the probability of such a failure depends on the accumulated damage. Failure occurs because of shocks and can occur only at times n = 1, 2, ⃛. The device can be replaced before or at failure. If the device fails it is immediately replaced at a fixed cost. Replacement before failure can only occur at times n = 1, 2, ⃛, and is done at a lower cost depending on the amount of accumulated damage at replacement. In this paper we determine the optimal replacement policy that minimizes the expected cost per unit time.

Keywords

POISSON PROCESSOPTIMAL REPLACEMENT POLICYOPTIMAL STOPPING TIMESCONVEX FUNCTIONSMARTINGALE AND SUPER-MARTINGALETOTAL POSITIVITYVARIATION DIMINISHING PROPERTY
Type
Research Papers
Information
Journal of Applied Probability , Volume 15 , Issue 1 , March 1978 , pp. 153 - 161
Copyright
Copyright © Applied Probability Trust 1978 

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References

[1] Abdel-Hameed, M. S. and Proschan, F. (1973) Non-stationary shock models. Stoch. Proc. Appl. 1, 383–404.Google Scholar
[2] Abdel-Hameed, M. S. and Proschan, F. (1975) Shock models with underlying birth process. J. Appl. Prob. 12, 18–28.CrossRefGoogle Scholar
[3] Abdel-Hameed, M. S. (1977) Optimality of the one step look-ahead stopping times. J. Appl. Prob. 14, 162–169.CrossRefGoogle Scholar
[4] Chung, K. L. (1968) A Course in Probability Theory. Academic Press, New York.Google Scholar
[5] Feldman, R. M. (1976) Optimal replacement with semi-Markov shock models. J. Appl. Prob. 13, 108–117.Google Scholar
[6] Karlin, S. (1968) Total Positivity, Vol. 1. Stanford University Press, Stanford.Google Scholar
[7] Knopp, K. (1956) Infinite Sequences and Series. Dover, New York.Google Scholar
[8] Neveu, J. (1965) Mathematical Foundation of the Calculus of Probability. Holden-Day, San Francisco.Google Scholar
[9] Taylor, H. M. (1975) Optimal replacement under additive damage and other failure models. Naval Res. Logist. Quart. 22, 1–18.CrossRefGoogle Scholar