Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-27T22:11:03.488Z Has data issue: false hasContentIssue false
  • English
  • Français

The identifiability problem for repairable systems subject to competing risks

Part of: Markov processes Operations research and management science Special processes Stochastic processes Ergodic theory

Published online by Cambridge University Press:  01 July 2016

Tim Bedford  and
Bo H. Lindqvist
Tim Bedford*
Affiliation:
Strathclyde University
Bo H. Lindqvist*
Affiliation:
Norwegian University of Science and Technology
*
∗ Postal address: Department of Management Science, Strathclyde University, Graham Hills Building, 40 George Street, Glasgow G1 1QE, UK. Email address: [email protected]
∗∗ Postal address: Department of Mathematical Sciences, Norwegian University of Science and Technology, N-7491 Trondheim, Norway. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Within reliability theory, identifiability problems arise through competing risks. If we have a series system of several components, and if that system is replaced or repaired to as good as new on failure, then the different component failures represent competing risks for the system. It is well known that the underlying component failure distributions cannot be estimated from the observable data (failure time and identity of failed component) without nontestable assumptions such as independence. In practice many systems are not subject to the ‘as good as new’ repair regime. Hence, the objective of this paper is to contrast the identifiability issues arising for different repair regimes. We consider the problem of identifying a model within a given class of probabilistic models for the system. Different models corresponding to different repair strategies are considered: a partial-repair model, where only the failing component is repaired; perfect repair, where all components are as good as new after a failure; and minimal repair, where components are only minimally repaired at failures. We show that on the basis of observing a single socket, the partial-repair model is identifiable, while the perfect- and minimal-repair models are not.

Keywords

Competing risksidentifiabilityreliabilitymarked point processergodicityMarkov chainjoint survival distribution

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces 90B25: Reliability, availability, maintenance, inspection
Secondary: 60G35: Signal detection and filtering 37A30: Ergodic theorems, spectral theory, Markov operators 60K10: Applications (reliability, demand theory, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 36 , Issue 3 , September 2004 , pp. 774 - 790
Copyright
Copyright © Applied Probability Trust 2004 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bedford, T. and Meilijson, I. (1997). A characterisation of marginal distributions of (possibly dependent) stochastic variables which right-censor each other. Ann. Statist. 25, 1622–1645.Google Scholar
[2] Crowder, M. J. (2001). Classical Competing Risks. Chapman and Hall/CRC, Boca Raton, FL.CrossRefGoogle Scholar
[3] Doob, J. L. (1953). Stochastic Processes. John Wiley, New York.Google Scholar
[4] Dorado, C., Hollander, M. and Sethuraman, J. (1997). Nonparametric estimation for a general repair model. Ann. Statist. 25, 1140–1160.Google Scholar
[5] Gumbel, E. J. (1960). Bivariate exponential distributions. J. Amer. Statist. Assoc. 55, 698–707.Google Scholar
[6] Sigman, K. (1995). Stationary Marked Point Processes. An Intuitive Approach. Chapman and Hall, New York.Google Scholar