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On the residual lifelengths of the remaining components in a coherent system

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Abstract

In this note, we consider a coherent system with the property that, upon failure of the system, some of its components remain unfailed in the system. Under this condition, we study the residual lifetime of the live components of the system. Signature based mixture representation of the joint and marginal reliability functions of the live components are obtained. Various stochastic and aging properties of the residual lifetime of such components are investigated. Some characterization results on exponential distributions are also provided.

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References

  • Asadi M, Bairamov I (2006) The mean residual life function of a \(k\)-out-of-\(n\) structure at the system level. IEEE Trans Reliab 55:314–318

    Article  Google Scholar 

  • Asadi M, Goliforushani S (2008) On the mean residual life function of coherent systems. IEEE Trans Reliab 57:574–580

    Google Scholar 

  • Bairamov I, Arnold BC (2008) On the residual lifelengths of the remaining components in a \((n-k+1)\)-out-of-\(n\) system. Stat Probab Lett 78:945–952

    Article  MathSciNet  MATH  Google Scholar 

  • Barlow RE, Proschan F (1975) Statistical theory of reliability and life testing. Holt, Rinehart, and Winston, New York

    MATH  Google Scholar 

  • Eryilmaz S (2009) Reliability properties of consecutive \(k\)-out-of-\(n\) systems of arbitrarily dependent components. Reliab Eng Syst Saf 94:350–356

    Article  Google Scholar 

  • Eryilmaz S (2011) Estimation in coherent reliability systems through copulas. Reliab Eng Syst Saf  96:564–568

    Article  Google Scholar 

  • Eryilmaz S (2012) The number of failed components in a coherent system with exchangeable components. IEEE Trans Reliab 61:203–207

    Article  Google Scholar 

  • Eryilmaz S, Bairamov I (2012) On extreme residual lives after the failure of the system. Mathematical problems in engineering ID 342940, p 11. doi:10.1155/2012/342940

  • Khaledi BE, Shaked M (2007) Ordering conditional lifetimes of coherent systems. J Stat Plan Infer 137:1173–1184

    Article  MathSciNet  MATH  Google Scholar 

  • Kochar S, Mukerjee H, Samaniego FJ (1999) The “signature” of a coherent system and its application to comparison among systems. Nav Res Logist 46:507–523

    Article  MathSciNet  MATH  Google Scholar 

  • Li X, Zhang Z (2008) Some stochastic comparisons of conditional coherent systems. Appl Stoch Models Bus Ind 24:541–549

    Article  MathSciNet  MATH  Google Scholar 

  • Li X, Zhao P (2006) Some aging properties of the residual life of \(k\)-out-of-\(n\) systems. IEEE Trans Reliab 55:535–541

    Article  Google Scholar 

  • Mahmoudi M, Asadi M (2011) The dynamic signature of coherent systems. IEEE Trans Reliab 60:817–822

    Google Scholar 

  • Navarro J, Balakrishnan N, Samaniego FJ (2008) Mixture representations of residual lifetimes of used systems. J Appl Probab 45:1097–1112

    Google Scholar 

  • Navarro J, Hernandez PJ (2008) Mean residual life functions of finite mixtures and systems. Metrika 67:277–298

    Google Scholar 

  • Navarro J, Ruiz JM, Sandoval CJ (2005) A note on comparisons among coherent systems with dependent components using signatures. Stat Probab Lett 72:179–185

    Article  MathSciNet  MATH  Google Scholar 

  • Navarro J, Ruiz JM, Sandoval CJ (2007) Properties of coherent systems with dependent components. Commun Stat Theor Meth 36:175–191

    Article  MathSciNet  MATH  Google Scholar 

  • Rao CR, Shanbhag DN (1994) Choquet-Deny type functional equations with applications to stochastic models. Wiley, Chichester

    MATH  Google Scholar 

  • Samaniego FJ (1985) On closure of the IFR class under formation of coherent systems. IEEE Trans Reliab 34:69–72

    Article  MATH  Google Scholar 

  • Samaniego FJ, Balakrishnan N, Navarro J (2009) Dynamic signatures and their use in comparing the reliability of new and used systems. Nav Res Logist 56:577–591

    Article  MathSciNet  MATH  Google Scholar 

  • Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, New York

    Book  MATH  Google Scholar 

  • Zhang Z (2010a) Ordering conditional general coherent systems with exchangeable components. J Stat Plan Infer 140:454–460

    Article  MATH  Google Scholar 

  • Zhang Z (2010b) Mixture representations of inactivity times of conditional coherent systems and their applications. J Appl Probab 47:876–885

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang Z, Yang Y (2010) Ordered properties of the residual lifetime and inactivity time of \((n-k+1)\)-out-of-\(n\) systems under double monitoring. Stat Probab Lett 80:711–717

    Article  MATH  Google Scholar 

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Acknowledgments

We would like to thank the editor and the referee for constructive comments which improved the presentation of the paper. M. Asadi’s research was in part supported by a grant from IPM (No. 91620411). Z. Zhang’s research was supported by National Natural Science Foundation of China (11161028).

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Kelkin Nama, M., Asadi, M. & Zhang, Z. On the residual lifelengths of the remaining components in a coherent system. Metrika 76, 979–996 (2013). https://doi.org/10.1007/s00184-012-0427-3

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  • DOI: https://doi.org/10.1007/s00184-012-0427-3

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