Hostname: page-component-848d4c4894-mwx4w Total loading time: 0 Render date: 2024-07-04T20:32:24.706Z Has data issue: false hasContentIssue false

Asymptotic failure rate of a Markov deteriorating system with preventive maintenance

Published online by Cambridge University Press:  14 July 2016

Sophie Mercier*
Affiliation:
Université de Marne-la-Vallée
Michel Roussignol*
Affiliation:
Université de Marne-la-Vallée
*
Postal address: Laboratoire d’Analyse et de Mathématiques Appliquées (CNRS-UMR 8050), Université de Marne-la-Vallée, 5, boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée cedex 2, France.
Postal address: Laboratoire d’Analyse et de Mathématiques Appliquées (CNRS-UMR 8050), Université de Marne-la-Vallée, 5, boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée cedex 2, France.

Abstract

We consider a system with a finite state space subject to continuous-time Markovian deterioration while running that leads to failure. Failures are instantaneously detected. This system is submitted to sequential checking and preventive maintenance: up states are divided into ‘good’ and ‘degraded’ ones and the system is sequentially checked through perfect and instantaneous inspections until it is found in a degraded up state and stopped to allow maintenance (or until it fails). Time between inspections is random and is chosen at each inspection according to the current degradation degree of the system. Markov renewal equations fulfilled by the reliability of the maintained system are given and an exponential equivalent is derived for the reliability. We prove the existence of an asymptotic failure rate for the maintained system, which we are able to compute. Sufficient conditions are given for the preventive maintenance policy to improve the reliability and the asymptotic failure rate of the system. A numerical example illustrates our study.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 2003 

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] Asmussen, S. (1987). Applied Probability and Queues. John Wiley, Chichester.Google Scholar
[2] Barlow, R. E., and Proschan, F. (1996). Mathematical Theory of Reliability (Classics Appl. Math. 17). SIAM, Philadelphia, PA.Google Scholar
[3] Barlow, R. E., Hunter, L. C., and Proschan, F. (1963). Optimal checking procedures. J. Soc. Indust. Appl. Math. 11, 10781095.CrossRefGoogle Scholar
[4] Bloch-Mercier, S. (2002). A preventive maintenance policy with sequential checking procedure for a Markov deteriorating system. Europ. J. Operat. Res. 142, 548576.CrossRefGoogle Scholar
[5] Çinlar, E. (1975). Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
[6] Cocozza-Thivent, C. (1997). Processus Stochastiques et Fiabilité des Systèmes (Math. Appl. 28). Springer, Berlin.Google Scholar
[7] Cocozza-Thivent, C. (2000). A model for a dynamic preventive maintenance policy. J. Appl. Math. Stoch. Anal. 13, 321346.CrossRefGoogle Scholar
[8] Cocozza-Thivent, C., and Kalashnikov, V. (1996). The failure rate in reliability: approximations and bounds. J. Appl. Math. Stoch. Anal. 9, 497530.CrossRefGoogle Scholar
[9] Cocozza-Thivent, C., and Kalashnikov, V. (1997). The failure rate in reliability: numerical treatment. J. Appl. Math. Stoch. Anal. 10, 2145.CrossRefGoogle Scholar
[10] McCall, J. J. (1965). Maintenance policies for stochastically failing equipment. A survey. Manag. Sci. 11, 493524.CrossRefGoogle Scholar
[11] Pierskalla, W. P., and Voelker, J. A. (1976). A survey of maintenance models: the control and surveillance of deteriorating systems. Naval Res. Logistics Quart. 23, 353388.CrossRefGoogle Scholar
[12] Scarf, P. A. (1997). On the application of mathematical models in maintenance. Europ. J. Operat. Res. 99, 493506.CrossRefGoogle Scholar
[13] Seneta, E. (1981). Non-negative Matrices and Markov Chains, 2nd edn. Springer, New York.CrossRefGoogle Scholar
[14] Shaked, M., and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications. Academic Press, Boston, MA.Google Scholar
[15] Sherif, Y. S., and Smith, M. L. (1981). Optimal maintenance models for systems subject to failure—a review. Naval Res. Logistics Quart. 28, 4774.CrossRefGoogle Scholar
[16] Valdez-Flores, C., and Feldman, R. M. (1989). A survey of preventive maintenance models for stochastically deteriorating single-unit systems. Naval Res. Logistics 36, 419446.3.0.CO;2-5>CrossRefGoogle Scholar