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Mixture Representations of Residual Lifetimes of Used Systems

Part of: Special processes Distribution theory - Probability

Published online by Cambridge University Press:  14 July 2016

Jorge Navarro ,
N. Balakrishnan  and
Francisco J. Samaniego
Jorge Navarro*
Affiliation:
Universidad de Murcia
N. Balakrishnan*
Affiliation:
McMaster University
Francisco J. Samaniego*
Affiliation:
University of California, Davis
*
∗Postal address: Facultad de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain. Email address: jorgenav@um.es
∗∗Postal address: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario L8S 4K1, Canada. Email address: bala@mcmaster.ca
∗∗∗Postal address: University of California, Davis, 1 Shields Avenue, Davis, CA 95616, USA. Email address: fjsamaniego@ucdavis.edu
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Abstract

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The representation of the reliability function of the lifetime of a coherent system as a mixture of the reliability function of order statistics associated with the lifetimes of its components is a very useful tool to study the ordering and the limiting behaviour of coherent systems. In this paper, we obtain several representations of the reliability functions of residual lifetimes of used coherent systems under two particular conditions on the status of the components or the system in terms of the reliability functions of residual lifetimes of order statistics.

Keywords

Coherent systemk-out-of-n systemorder statisticssignaturestochastic ordering

MSC classification

Secondary: 60E15: Inequalities; stochastic orderings 60K10: Applications (reliability, demand theory, etc.)
Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 4 , December 2008 , pp. 1097 - 1112
Copyright
Copyright © Applied Probability Trust 2008 

References

[1] Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
[2] Boland, P. J. and Samaniego, F. (2004). The signature of a coherent system and its applications in reliability. In Mathematical Reliability: An Expository Perspective, eds Soyer, R., Mazzuchi, T. and Singpurwalla, N. D., Kluwer, Boston, pp. 1–29.Google Scholar
[3] David, H. A. and Nagaraja, H. N. (2003). Order Statistics, 3rd edn. Wiley, Hoboken, New Jersey.CrossRefGoogle Scholar
[4] Khaledi, B. E. and Shaked, M. (2007). Ordering conditional lifetimes of coherent systems. J. Statist. Planning Infer. 137, 1173–1184.Google Scholar
[5] Kochar, S., Mukerjee, H., and Samaniego, F. J. (1999). The “signature” of a coherent system and its application to comparison among systems. Naval Res. Logistics 46, 507–523.Google Scholar
[6] Navarro, J. (2008). Likelihood ratio ordering of order statistics, mixtures and systems. J. Statist. Planning Infer. 138, 1242–1257.Google Scholar
[7] Navarro, J. and Eryilmaz, S. (2007). Mean residual lifetimes of consecutive-k-out-of-n systems. J. Appl. Prob. 44, 82–98.CrossRefGoogle Scholar
[8] Navarro, J. and Hernandez, P. J. (2008). Mean residual life functions of finite mixtures, order statistics and coherent systems. Metrika 67, 277–298.CrossRefGoogle Scholar
[9] Navarro, J., Ruiz, J. M. and Sandoval, C.J. (2007). Properties of coherent systems with dependent components. Commun. Statist. Theory Meth. 36, 175–191.CrossRefGoogle Scholar
[10] Navarro, J., Samaniego, F. J., Balakrishnan, N. and Bhattacharya, D. (2008). On the application and extension of system signatures to problems in engineering reliability. Naval Res. Logistics 55, 313–327.CrossRefGoogle Scholar
[11] Navarro, J. and Shaked, M. (2006). Hazard rate ordering of order statistics and systems. J. Appl. Prob. 43, 391–408.CrossRefGoogle Scholar
[12] Samaniego, F. J. (1985). On closure of the IFR class under formation of coherent systems. IEEE Trans. Reliab. 34, 69–72.Google Scholar
[13] Samaniego, F. (2007). System Signatures and Their Applications in Engineering Reliability (Internat. Ser. Operat. Res. Manag. Sci. 110). Springer, New York.CrossRefGoogle Scholar
[14] Satyarananaya, A. and Prabhakar, A. (1978). A new topological formula and rapid algorithm for reliability analysis of complex networks. IEEE Trans. Reliab. 30, 82–100.Google Scholar
[15] Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders. Springer, New York.Google Scholar