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Mean residual lifetimes of consecutive-\(k\)-out-of-\(n\) systems. (English) Zbl 1135.62084

The authors study some reliability properties of consecutive-\(k\)-out-of-\(n\) systems with exchangeable component lifetimes. Some monotonicity and asymptotic properties of the associated mean residual life function are derived, and some ordering properties among the lifetimes of such systems are obtained.
For example, the authors show that, under some assumptions, the mean residual life function of a consecutive-\(k\)-out-of-\(n:G\) system (that is, a system that functions if and only if at least \(k\) consecutive components function) is asymptotically equivalent to that of a series system of \(k\) components. When the component lifetimes are independent and identically distributed, the authors show, for \(2k>n\), that consecutive-\(k\)-out-of-\(n\) system lifetimes are ordered in the likelihood ratio order.

MSC:

62N05 Reliability and life testing
60K10 Applications of renewal theory (reliability, demand theory, etc.)
60E15 Inequalities; stochastic orderings
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[1] Aki, S. and Hirano, K. (1996). Lifetime distribution and estimation problems of consecutive \(k\)-out-of-\(n:F\) systems. Ann. Inst. Statist. Math. 48, 185–199. · Zbl 0857.62094
[2] Balakrishnan, N. and Koutras, M. V. (2002). Runs and Scans with Applications . John Wiley, New York. · Zbl 0991.62087
[3] Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing . Holt, Rinehart and Winston, New York. · Zbl 0379.62080
[4] Block, H. W., Li, Y. and Savits, T. H. (2003). Initial and final behaviour of failure rate functions for mixtures and systems. J. Appl. Prob. 40, 721–740. · Zbl 1046.62111
[5] Boland, P. J. and Papastavridis, S. (1999). Consecutive \(k\)-out-of-\(n:F\) systems with cycle \(k\). Statist. Prob. Lett. 44, 155–160. · Zbl 0939.62104
[6] Boland, P. J. and Samaniego, F. J. (2004). Stochastic ordering results for consecutive \(k\)-out-of-\(n:F\) systems. IEEE Trans. Reliab. 53, 7–10.
[7] Chen, R. W. and Hwang, F. K. (1985). Failure distributions of consecutive \(k\)-out-of-\(n:F\) systems. IEEE Trans. Reliab. 34, 338–341. · Zbl 0588.62181
[8] David, H. A. and Nagaraja, H. N. (2003). Order Statistics , 3rd edn. Wiley, New York. · Zbl 1053.62060
[9] Derman, C., Lieberman, G. J. and Ross, S. M. (1982). On the consecutive \(k\)-out-of-\(n:F\) system. IEEE Trans. Reliab. 31, 57–63. · Zbl 0478.90029
[10] Eryilmaz, S. (2005). On the distribution and expectation of success runs in nonhomogeneous Markov dependent trials. Statist. Papers 46, 117–128. · Zbl 1072.60056
[11] Fu, J. C. and Lou, W. Y. W. (2003). Distribution Theory of Runs and Patterns and Its Applications . World Scientific, River Edge, NJ. · Zbl 1030.60063
[12] Kochar, S., Mukerjee, H. and Samaniego, F. J. (1999). The ‘signature’ of a coherent system and its application to comparisons among systems. Naval Res. Logistics 46, 507–523. · Zbl 0948.90067
[13] Navarro, J. and Hernandez, P. J. (2005). Mean residual life functions of finite mixtures and systems. Submitted. · Zbl 1357.62304
[14] Navarro, J. and Rychlik, T. (2006). Reliability and expectation bounds for coherent systems with exchangeable components. J. Multivariate Anal. 98, 102–113. · Zbl 1102.62111
[15] Navarro, J. and Shaked, M. (2006). Hazard rate ordering of order statistics and systems. J. Appl. Prob. 43, 391–408. · Zbl 1111.62098
[16] Navarro, J., Belzunce, F. and Ruiz, J. M. (1997). New stochastic orders based on double truncation. Prob. Eng. Inf. Sci. 11, 395–402. · Zbl 1096.60513
[17] Navarro, J., Ruiz, J. M. and Sandoval, C. J. (2005). A note on comparisons among coherent systems with dependent components using signature. Statist. Prob. Lett. 72, 179–185. · Zbl 1068.60026
[18] Navarro, J., Ruiz, J. M. and Sandoval, C. J. (2007). Modelling coherent systems under dependence. To appear in Commun. Statist. Theory Meth. · Zbl 1121.60015
[19] Samaniego, F. J. (1985). On closure of the IFR class under formation of coherent systems. IEEE Trans. Reliab. 34, 69–72. · Zbl 0585.62169
[20] Shaked, M. and Shanthikumar, J. G. (1994). Stochastic Orders and Their Applications . Academic Press, San Diego, CA. · Zbl 0806.62009
[21] Shanthikumar, J. G. (1985). Lifetime distribution of consecutive \(k\)-out-of-\(n:F\) systems with exchangeable life times. IEEE Trans. Reliab. 34, 480–483. · Zbl 0588.62180
[22] Tong, Y. L. (1985). A rearrangement inequality for the longest run with an application to network reliability. J. Appl. Prob. 22, 386–393. JSTOR: · Zbl 0564.60085
[23] Wondmagegnehu, E., Navarro, J. and Hernández, P. J. (2005). Bathtub shaped failure rates from mixtures: a practical point of view. IEEE Trans. Reliab. 54, 270–275.
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