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**Mixture representations of residual lifetimes of used systems.**
*(English)*
Zbl 1155.60305

J. Appl. Probab. 45, No. 4, 1097-1112 (2008); erratum ibid. 52, No. 4, 1183-1186 (2015).

Summary: 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.

### MSC:

60E15 | Inequalities; stochastic orderings |

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

90B25 | Reliability, availability, maintenance, inspection in operations research |

### Keywords:

coherent system; \(k\)-out-of-\(n\) system; order statistics; signature; stochastic ordering
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\textit{J. Navarro} et al., J. Appl. Probab. 45, No. 4, 1097--1112 (2008; Zbl 1155.60305)

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### References:

[1] | Barlow, R. E. and Proschan, F. (1975). Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York. · Zbl 0379.62080 |

[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 R. Soyer, T. Mazzuchi and N. D. Singpurwalla, Kluwer, Boston, pp. 1–29. |

[3] | David, H. A. and Nagaraja, H. N. (2003). Order Statistics , 3rd edn. Wiley, Hoboken, New Jersey. · Zbl 1053.62060 |

[4] | Khaledi, B. E. and Shaked, M. (2007). Ordering conditional lifetimes of coherent systems. J. Statist. Planning Infer. 137 , 1173–1184. · Zbl 1111.60012 |

[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. · Zbl 0948.90067 |

[6] | Navarro, J. (2008). Likelihood ratio ordering of order statistics, mixtures and systems. J. Statist. Planning Infer. 138 , 1242–1257. · Zbl 1144.60305 |

[7] | Navarro, J. and Eryilmaz, S. (2007). Mean residual lifetimes of consecutive-\(k\)-out-of-\(n\) systems. J. Appl. Prob. 44 , 82–98. · Zbl 1135.62084 |

[8] | Navarro, J. and Hernandez, P. J. (2008). Mean residual life functions of finite mixtures, order statistics and coherent systems. Metrika 67 , 277–298. · Zbl 1357.62304 |

[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. · Zbl 1121.60015 |

[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. · Zbl 1153.90386 |

[11] | Navarro, J. and Shaked, M. (2006). Hazard rate ordering of order statistics and systems. J. Appl. Prob. 43 , 391–408. · Zbl 1111.62098 |

[12] | Samaniego, F. J. (1985). On closure of the IFR class under formation of coherent systems. IEEE Trans. Reliab. 34, 69–72. · Zbl 0585.62169 |

[13] | Samaniego, F. (2007). System Signatures and Their Applications in Engineering Reliability (Internat. Ser. Operat. Res. Manag. Sci. 110 ). Springer, New York. · Zbl 1154.62075 |

[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. · Zbl 0409.90039 |

[15] | Shaked, M. and Shanthikumar, J. G. (2007). Stochastic Orders . Springer, New York. · Zbl 0806.62009 |

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