Hostname: page-component-77c89778f8-vpsfw Total loading time: 0 Render date: 2024-07-17T02:59:55.215Z Has data issue: false hasContentIssue false
  • English
  • Français

On reliability prediction and semi-renewal processes

Part of: Operations research and management science Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Michael Tortorella
Michael Tortorella*
Affiliation:
Rutgers University
*
∗Current address: RUTCOR 148, Rutgers University, 640 Bartholomew Road, Piscataway, NJ 08854, USA. Email address: mtortore@rci.rutgers.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We analyze the notion of ‘reliability prediction’ by studying in detail a key property that is tacitly assumed to make reliability prediction possible. The analysis leads in turn to a special type of point process for which the connection of future to past can be explicitly displayed. In this type of process, the semi-renewal process, all finite-dimensional distributions are completely determined by the distribution of the time to the first event in the process. The theory provides a heretofore unappreciated unification of the two most commonly used reliability prediction models for maintained systems, namely, the renewal and revival processes. We show that familiar results from renewal theory extend and generalize to semi-renewal processes.

Keywords

Renewalrelevationbad-as-oldrecord value processsemigroup

MSC classification

Secondary: 60G55: Point processes 90B25: Reliability, availability, maintenance, inspection
Type
Research Article
Information
Journal of Applied Probability , Volume 47 , Issue 2 , June 2010 , pp. 394 - 406
Copyright
Copyright © Applied Probability Trust 2010 

References

[1] Ascher, H. E. and Feingold, H. (1984). Repairable Systems Reliability. Marcel Dekker, New York.Google Scholar
[2] Baxter, L. A. and Chlouverakis, G. (1990). A model for a repairman who introduces faults. Working paper, Stern School of Business, New York University.Google Scholar
[3] Baxter, L. A. and Chlouverakis, G. (1990). Optimal maintenance with an imperfect repairman. Working paper, Stern School of Business, New York University.Google Scholar
[4] Baxter, L. A., Kijima, M. and Tortorella, M. (1996). A point process model for the reliability of a maintained system subject to general repair. Commun. Statist. Stoch. Models 12, 37–65.Google Scholar
[5] Brown, M. and Proschan, F. (1983). Imperfect repair. J. Appl. Prob. 20, 851–859.Google Scholar
[6] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.Google Scholar
[7] Karlin, S. and Taylor, H. M. (1975). A First Course in Stochastic Processes, 2nd edn. Academic Press, New York.Google Scholar
[8] Kijima, M. (1989). Some results for repairable systems with general repair. J. Appl. Prob. 26, 89–102.Google Scholar
[9] Kijima, M. and Sumita, U. (1986). A useful generalization of renewal theory: counting processes governed by nonnegative Markovian increments. J. Appl. Prob. 23, 71–88.Google Scholar
[10] Shanthikumar, J. G. and Baxter, L. A. (1985). Closure properties of the relevation transform. Naval Res. Logistics 32, 185–189.Google Scholar
[11] Shorrock, R. W. (1972). On record values and record times. J. Appl. Prob. 9, 316–326.Google Scholar
[12] Snyder, D. L. (1975). Random Point Processes. John Wiley, New York.Google Scholar
[13] Tortorella, M. (2010). Strong and weak minimal repair and the revival process in reliability theory. Submitted.Google Scholar
[14] Yosida, K. (1971). Functional Analysis, 3rd edn. Springer, New York.Google Scholar