Sankhya B

, Volume 77, Issue 1, pp 141–164 | Cite as

The Inactivity Time of Exchangeable Components of k-out-of-n Structures

Article
First Online:
Received:
Revised:

Abstract

Most of the research, on the study of the reliability properties of technical systems, assume that the components of the systems operate independently. However, in real life situation, it is more reasonable to assume that there is dependency among the components of the system. In this paper, we consider a (n−k+1)-out-of-n structure in which the component lifetimes are dependent random variables. We investigate stochastic properties of the inactivity time of the failed components of the system, extending some existing results in the literature where the components of the system are assumed to be independent and identically distributed. The results are then extended to the case where the system has an arbitrary coherent structure with exchangeable components.

Keywords

Multivariate life distributions coherent systems joint reliability function exchangeable random variables multivariate stochastic order signature vector. 

AMS (2000) subject classification.

Primary 60E15 Secondary 60K10. 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Asadi M. (2006). On the mean past lifetime of the components of a parallel system. J. Statist. Plann. Inference, 136, 1197–1206.CrossRefMATHMathSciNetGoogle Scholar
  2. Asadi M. and Bairamov I. (2005). A note on the mean residual life function of a parallel system. Commun. Stat. Theor. Methods, 34, 475–484.CrossRefMATHMathSciNetGoogle Scholar
  3. Asadi M. and Bayramoglu I. (2006). On the mean residual life function of the k-out-of-n systems at the system level. IEEE Trans. Reliab., 55, 314–318.CrossRefGoogle Scholar
  4. Asadi M. and Goliforushani S. (2008). On the mean residual life function of coherent systems. IEEE Trans. Reliab., 57, 574–580.CrossRefGoogle Scholar
  5. Bairamov I., Ahsanullah M., and Akhundov I. (2002). A residual life function of a system having parallel or series structures. J. Stat. Theory Appl., 1, 119–132.MathSciNetGoogle Scholar
  6. Belzunce F., Gurler S., and Ruiz J.M. (2011). Revisiting multivariate likelihood ratio ordering results for order statistics. Probab. Engrg. Inform. Sci., 25, 3, 355–368.CrossRefMATHMathSciNetGoogle Scholar
  7. Castillo E., Sarabia J.M., and Hadi A.S. (1997). Fitting continuous bivariate distributions to data. The Statistician, 46, 355–369.Google Scholar
  8. Goliforushani S. and Asadi M. (2011). Stochastic ordering among inactivity times of coherent systems. Sankhya, 73, 241–262.CrossRefMATHMathSciNetGoogle Scholar
  9. Gumbel E.G. (1960). Bivariate exponential distribution. J. Amer. Statist. Assoc., 55, 698–707.CrossRefMATHMathSciNetGoogle Scholar
  10. Joe H. (1997). Multivariate Models and Dependence Concepts. Chapman & Hall, London.CrossRefMATHGoogle Scholar
  11. Karlin S. and Rinott Y. (1980). Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions. J. Multivariate Anal., 10, 467–498.CrossRefMATHMathSciNetGoogle Scholar
  12. Khaledi B.-E. and Shaked M. (2007). Ordering conditional lifetimes of coherent systems. J. Statist. Plann. Inference, 37, 1173–1184.CrossRefMathSciNetGoogle Scholar
  13. Kochar S.C., Mukerjee H., and Samaniego F.J. (1999). The ‘signature’ of a coherent system and its application to comparisons among systems. Naval Res. Logist., 46, 507–523.CrossRefMATHMathSciNetGoogle Scholar
  14. Kotz S., Balakrishnan N., and Johnson N.L. (2000). Continuous Multivariate Distributions. Vol. 1, 2nd edn. Wiley, New York.CrossRefMATHGoogle Scholar
  15. Li X. and Zhang Z. (2008). Some stochastic comparisons of conditional coherent systems. J. Appl. Stoch. Model. Bus. Ind., 24, 541–549.CrossRefMATHGoogle Scholar
  16. Li X. and Zhao P. (2006). Some aging properties of the residual life of k-out-of-n systems. IEEE Trans. Reliab., 55, 535–541.CrossRefGoogle Scholar
  17. Li X. and Zhao P. (2008). Stochastic comparison on general inactivity time and general residual life of k-out-of-n systems. Commun. Stat. Simul. Compt., 37, 1005–1019.CrossRefMATHGoogle Scholar
  18. Müller A. and Stoyan D. (2002). Comparison Methods for Stochastic Models and Risks. Wiley, New York.MATHGoogle Scholar
  19. Navarro J., Balakrishnan N., and Samaniego F.J. (2008). Mixture representations of residual lifetimes of used systems. J. Appl. Probab., 45, 1097–1112.CrossRefMATHMathSciNetGoogle Scholar
  20. Navarro J. and Rubio R. (2010). Comparisons of coherent systems using stochastic precedence. Test, 19, 469–486.CrossRefMATHMathSciNetGoogle Scholar
  21. Navarro J. and Rubio R. (2011). A note on necessary and sufficient conditions for ordering properties of coherent systems with exchangeable components. Naval Res. Logist., 58, 478–489.CrossRefMATHMathSciNetGoogle Scholar
  22. Navarro J. and Rubio R. (2012). Comparisons of coherent systems with nonidentically distributed components. J. Statist. Plann. Inference, 142, 1310–1319.CrossRefMATHMathSciNetGoogle Scholar
  23. Navarro J., Ruiz J.M., and Sandoval C.J. (2005). A note on comparisons among coherent systems with dependent components using signature. Statist. Probab. Lett., 72, 179–185.CrossRefMATHMathSciNetGoogle Scholar
  24. Navarro J., Ruiz J.M., and Sandoval C.J. (2007). Properties of coherent systems with dependent components. Commun. Stat. Theor. Methods, 36, 175–191.CrossRefMATHMathSciNetGoogle Scholar
  25. Nelsen R.B. (2006). An Introduction to Copulas, 2nd edn. Springer, New York.MATHGoogle Scholar
  26. Rezapour M., Salehi E.T., and Alamatsaz M.H. (2013). Stochastic comparison of residual and past lifetimes of (n−k+1)-out-of-n systems with dependent components. Commun. Stat. Theor. Methods, 42, 2185–2199.CrossRefMATHMathSciNetGoogle Scholar
  27. Sadegh M.K. (2011). A note on the mean residual life function of a coherent system with exchangeable or nonidentical components. J. Statist. Plann. Inference, 141, 3267–3275.CrossRefMATHMathSciNetGoogle Scholar
  28. Samaniego F.J. (1985). On the closure of the IFR class under formation of coherent systems. IEEE Trans. Reliab., R-34, 69–72.CrossRefGoogle Scholar
  29. Shaked M. and Shanthikumar J.G. (2007). Stochastic Orders. Springer, New York.CrossRefMATHGoogle Scholar
  30. Spizzichino F. and Navarro J. (2012). Recent Advances in System Reliability. Springer-Verlag, London, Lisnianski A. and Frenkel I (eds.), p. 33–48.CrossRefGoogle Scholar
  31. Tavangar M. and Asadi M. (2010). A study on the mean past lifetime of the components of (n−k+1)-out-of-n system at the system level. Metrika, 72, 59–73.CrossRefMATHMathSciNetGoogle Scholar
  32. Tavangar M. and Bairamov I. (2012). On the residual lifetimes of coherent systems with exchangeable components. Pak. J. Stat., 28, 303–313.MathSciNetGoogle Scholar
  33. Zhang Z. (2010). Ordering conditional general coherent systems with exchangeable components. J. Statist. Plann. Inference, 140, 454–460.CrossRefMATHMathSciNetGoogle Scholar
  34. Zhang Z. (2011). Ordering new conditional residual lifetimes of k-out-of-n systems. Commun. Stat. Theor. Methods, 40, 1591–1600.CrossRefMATHGoogle Scholar
  35. Zhang Z. and Chang Y. (2011). MRL and MPL functions of parallel systems with INID components under double monitoring. Commun. Stat. Simul. Compt., 40, 1514–1523.CrossRefMATHMathSciNetGoogle Scholar
  36. Zhang Z. and Yang Y. (2010). Ordered properties on the residual life and inactivity time of (n−k+1)-out-of-n systems under double monitoring. Statist. Probab. Lett., 80, 711–717.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Indian Statistical Institute 2014

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of IsfahanIsfahanIran

Personalised recommendations