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Imperfect repair

Published online by Cambridge University Press:  14 July 2016

Mark Brown  and
Frank Proschan
Mark Brown*
Affiliation:
City College, CUNY
Frank Proschan*
Affiliation:
Florida State University
*
∗ Postal address: Department of Mathematics, City College, CUNY, New York, NY 10031, U.S.A.
∗∗ Postal address: Department of Statistics and Statistical Consulting Center, The Florida State University, Tallahassee, FL 32306, U.S.A.
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Abstract

A device is repaired at failure. With probability p, it is returned to the ‘good-as-new' state (perfect repair), with probability 1 – p, it is returned to the functioning state, but it is only as good as a device of age equal to its age at failure (imperfect repair). Repair takes negligible time. We obtain the distribution Fp of the interval between successive good-as-new states in terms of the underlying life distribution F. We show that if F is in any of the life distribution classes IFR, DFR, IFRA, DFRA, NBU, NWU, DMRL, or IMRL, then Fp is in the same class. Finally, we obtain a number of monotonicity properties for various parameters and random variables of the stochastic process. The results obtained are of interest in the context of stochastic processes in general, as well as being useful in the particular imperfect repair model studied.

Keywords

LIFE DISTRIBUTIONSSTOCHASTIC PROCESSMONOTONICITYPRESERVATION PROPERTYPROPORTIONAL HAZARDSMAINTENANCE POLICYRENEWAL PROCESSES
Type
Research Papers
Information
Journal of Applied Probability , Volume 20 , Issue 4 , December 1983 , pp. 851 - 859
Copyright
Copyright © Applied Probability Trust 1983 

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Footnotes

Research performed under the support of the Air Force Office of Scientific Research, AFSC, USAF, under Grant AFOSR F49620-79-C-0157 while this author was Professor of Statistics at Florida State University.

Research performed under the support of the Air Force Office of Scientific Research, USAF, AFSC, under Grant AFOSR 81-0038.

References

Barlow, R. E. and Proschan, F. (1965) Mathematical Theory of Reliability. Wiley, New York.Google Scholar
Block, H. W., Borges, W. S. and Savits, T. H. (1983) Age dependent minimal repair. Submitted to J. Appl. Prob. Google Scholar
Blumenthal, S., Greenwood, J. A. and Herbach, L. H. (1976) A comparison of the bad as old and superimposed renewal models. Management Sci. 23, 280–285.CrossRefGoogle Scholar
Brown, M. (1980) Bounds, inequalities, and monotonicity properties for some specialized renewal processes. Ann. Prob. 8, 227–240.Google Scholar
Neuts, M. F. (1967) Waiting times between record observations. J. Appl. Prob. 4, 206–208.Google Scholar
Resnick, S. I. (1973) Extremal processes and record value times. J. Appl. Prob. 10, 864–868.Google Scholar
Shorrock, R. W. (1972) A limit theorem for inter-record times. J. Appl. Prob. 9, 219–223.CrossRefGoogle Scholar
Strawderman, W. and Holmes, P. T. (1970) On the law of the iterated logarithm for inter-record times. J. Appl. Prob. 7, 432–439.Google Scholar