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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.

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.

Reviewer: Moshe Shaked (Tucson)

### MSC:

62N05 | Reliability and life testing |

60K10 | Applications of renewal theory (reliability, demand theory, etc.) |

60E15 | Inequalities; stochastic orderings |

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\textit{J. Navarro} and \textit{S. Eryilmaz}, J. Appl. Probab. 44, No. 1, 82--98 (2007; Zbl 1135.62084)

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### References:

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