Ordering conditional lifetimes of coherent systems. (English) Zbl 1111.60012

Summary: Consider a system of \(n\) components that has the property that there exists a number \(r\) \((r<n)\), such that if it is known that at most \(r\) components have failed, the system is still functioning with probability 1. Suppose that such a system is equipped with a warning light that comes up at the time of the failure of the \(r\)th component. The system is still working then, and we are interested in its residual life. In this paper we obtain some results which stochastically compare the residual lives of such systems with the same type, or with different types, of components. Some applications are given. In particular, we derive upper and lower bounds on the expected residual lives of such systems given that the warning light has not come up yet, and given that the component hazard rate functions are bounded from below or from above by a known constant.


60E15 Inequalities; stochastic orderings
60K10 Applications of renewal theory (reliability, demand theory, etc.)
62G30 Order statistics; empirical distribution functions
Full Text: DOI


[1] Asadi, M., 2004. On the mean residual life and the mean past life of the k-out-of-n systems. In: Proceedings of the 7th Iranian Statistical Conference, Invited Papers Volume. Allameh Tabatabaie University, Tehran, pp. 55-77.
[2] Asadi, M.; Bayramoglu, I., A note on the Mean residual life function of a parallel system, Comm. statist. theory methods, 34, 475-484, (2005) · Zbl 1062.62228
[3] Bairamov, I.; Ahsanullah, M.; Akhundov, I., A residual life function of a system having parallel or series structures, J. statist. theory appl., 1, 119-132, (2002)
[4] Barlow, R.E.; Proschan, F., Statistical theory of reliability and life testing: reliability models, (1975), Holt Rinehart and Winston, New York · Zbl 0379.62080
[5] Esary, J.D.; Marshall, A.W., Coherent life functions, SIAM J. appl. math., 18, 810-814, (1970) · Zbl 0198.24804
[6] Kochar, S.C.; Mukerjee, H.; Samaniego, F.J., The ‘signature’ of a coherent system and its application to comparisons among systems, Naval res. logist., 46, 507-523, (1999) · Zbl 0948.90067
[7] Korwar, R., On stochastic orders for the lifetime of a k-out-of-n system, Probab. eng. inform. sci., 17, 137-142, (2003) · Zbl 1065.90026
[8] Li, X.; Chen, J., Aging properties of the residual life length of k-out-of-n systems with independent but non-identical components, Appl. stochastic models business industry, 20, 143-153, (2004) · Zbl 1060.62115
[9] Li, X.; Zuo, M.J., On the behaviour of some new ageing properties based upon the residual life of k-out-of-n systems, J. appl. probab., 39, 426-433, (2002) · Zbl 1003.62089
[10] Li, X.; Zuo, M.J., Stochastic comparison of residual life and inactivity time at a random time, Stochastic models, 20, 229-235, (2004) · Zbl 1059.60091
[11] Mi, J.; Shaked, M., Stochastic dominance of random variables implies the dominance of their order statistics, J. Indian statist. assoc., 40, 161-168, (2002)
[12] Müller, A.; Stoyan, D., Comparison methods for stochastic models and risks, (2002), Wiley New York · Zbl 0999.60002
[13] Nanda, A.K.; Shaked, M., The hazard rate and the reversed hazard rate orders, with applications to order statistics, Ann. inst. statist. math., 53, 853-864, (2001) · Zbl 1006.60015
[14] Navarro, J.; Ruiz, J.M.; Sandoval, C.J., A note on comparisons among coherent systems with dependent components using signatures, Statist. probab. lett., 72, 179-185, (2005) · Zbl 1068.60026
[15] Samaniego, F.J., On the closure of the IFR class under formation of coherent systems, IEEE trans. reliability, R-34, 69-72, (1985) · Zbl 0585.62169
[16] Shaked, M.; Shanthikumar, J.G., Stochastic orders and their applications, (1994), Academic Press Boston · Zbl 0806.62009
[17] Shaked, M.; Suarez-Llorens, A., On the comparison of reliability experiments based on the convolution order, J. amer. statist. assoc., 98, 693-702, (2003) · Zbl 1040.62093
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.