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Statistical Papers

, Volume 60, Issue 3, pp 805–821 | Cite as

Stochastic and ageing properties of coherent systems with dependent identically distributed components

  • M. KelkinnamaEmail author
  • M. Asadi
Regular Article

Abstract

In the study of reliability and stochastic properties of technical systems a realistic assumption is to consider the dependency between the components of the system. We investigate the reliability and stochastic properties of a coherent system where the component lifetimes of the system are identically distributed and the structural dependency of the components is expressed using a copula. We use the notion of distortion function to explore several ageing and stochastic properties of the residual and inactivity time of coherent systems and order statistics. Some illustrative examples are also provided.

Keywords

Residual lifetime Inactivity time Survival copula Dependence Signature vector Reliability 

Notes

References

  1. Barlow RE, Proschan F (1981) Statistical theory of reliability and life testing. Silver SpringGoogle Scholar
  2. David H, Nagaraja HN (2003) Order statistics. Wiley, HobokenCrossRefzbMATHGoogle Scholar
  3. Eryilmaz S (2010) Number of working components in consecutive k-out-of-n system while it is working. Commun Stat Simul Comput 39:683–692MathSciNetCrossRefzbMATHGoogle Scholar
  4. Eryilmaz S (2011) Estimation in coherent reliability systems through copulas. Reliab Eng Syst Saf 96:564–568CrossRefGoogle Scholar
  5. Eryilmaz S (2012) The number of failed components in a coherent system with exchangeable components. IEEE Trans Reliab 61:203–207CrossRefGoogle Scholar
  6. Eryilmaz S (2013) On residual lifetime of coherent systems aftyer the rth failure. Stat Pap 5:243–250CrossRefzbMATHGoogle Scholar
  7. Eryilmaz S, Zuo MJ (2010) Computing and applying the signature of a system with two common failure criteria. IEEE Trans Reliab 59:576–580CrossRefGoogle Scholar
  8. Gupta N, Kumar S (2014) Stochastic comparisons of component and system redundancies with dependent components. Oper Res Lett 42:284289MathSciNetCrossRefzbMATHGoogle Scholar
  9. Gupta N, Misra N, Kumar S (2014) Stochastic comparisons of residual lifetimes and inactivity times of coherent systems with dependent identically distributed components. Eur J Oper Res 240:425–430MathSciNetCrossRefzbMATHGoogle Scholar
  10. Gurler S (2012) On residual lifetimes in sequential (n-k+1)-out-of-n systems. Stat Pap 53:23–31MathSciNetCrossRefzbMATHGoogle Scholar
  11. Karlin S (1968) Total positivity. Stanford University Press, StanfordzbMATHGoogle Scholar
  12. Kelkinnama M, Asadi M, Tavangar M (2015) New developments on stochastic properties of coherent systems. IEEE Trans Reliab 64:1276–1286CrossRefGoogle Scholar
  13. Lai CD, Xie M (2006) Stochastic ageing and dependence for reliability. Springer, New YorkzbMATHGoogle Scholar
  14. Navarro J (2008) Likelihood ratio ordering of order statistics, mixtures and systems. J Stat Plan Inference 138:1242–1257MathSciNetCrossRefzbMATHGoogle Scholar
  15. Navarro J (2016) Distribution-free comparisons of residual lifetimes of coherent systems based on copula properties. Stat Pap.  https://doi.org/10.1007/s00362-016-0789-0
  16. Navarro J, Lai CD (2007) Ordering properties of systems with two dependent components. Commun Stat Theory Methods 36:645–655MathSciNetCrossRefzbMATHGoogle Scholar
  17. Navarro J, Rychlik T (2007) Reliability and expectation bounds for coherent systems with exchangeable components. J Multivar Anal 98:102–113MathSciNetCrossRefzbMATHGoogle Scholar
  18. Navarro J, Rychlik T (2010) Comparisons and bounds for expected lifetimes of reliability systems. Eur J Oper Res 207:309317MathSciNetCrossRefzbMATHGoogle Scholar
  19. Navarro J, Spizzichino F (2010) Comparisons of series and parallel systems with components sharing the same copula. Appl Stoch Models Bus Ind 26:775791MathSciNetCrossRefzbMATHGoogle Scholar
  20. Navarro J, Rubio R (2011) A note on necessary and sufficient conditions for ordering properties of coherent systems with exchangeable components. Naval Res Logist 58:478–488MathSciNetCrossRefzbMATHGoogle Scholar
  21. Navarro J, Gomis MC (2015) Comparisons in the mean residual life order of coherent systems with identically distributed components. Appl Stoch Models Bus Ind 32:33–47.  10.1002/asmb.2121 MathSciNetCrossRefzbMATHGoogle Scholar
  22. Navarro J, Ruiz JM, Sandoval CJ (2005) A note on comparisons among coherent systems with dependent components using signatures. Stat Prob Lett 72:179–185MathSciNetCrossRefzbMATHGoogle Scholar
  23. Navarro J, Ruiz JM, Sandoval CJ (2007) Properties of coherent systems with dependent components. Commun Stat Theory Methods 36:175–191MathSciNetCrossRefzbMATHGoogle Scholar
  24. Navarro J, Balakrishnan N, Samaniego FJ (2008) Mixture representations of residual lifetimes of used systems. J Appl Prob 45:1097–1112MathSciNetCrossRefzbMATHGoogle Scholar
  25. Navarro J, Ruiz JM, Sandoval CJ (2008) Properties of systems with two exchangeable pareto components. Stat Pap 49:177–190MathSciNetCrossRefzbMATHGoogle Scholar
  26. Navarro J, Samaniego FJ, Balakrishnan N, Bhattacharya D (2008) On the applications and extension of system signatures in engineering reliability. Naval Res Logist 55:313–327MathSciNetCrossRefzbMATHGoogle Scholar
  27. Navarro J, Aguila Y, Sordo MA, Suarez-Llorens A (2013) Stochastic ordering properties for systems with dependent identically distributed components. Appl Stoch Models Bus Ind 29:264–278MathSciNetCrossRefzbMATHGoogle Scholar
  28. Navarro J, Aguila Y, Sordo MA, Suarez-Llorens A (2014) Preservation of reliability classes under the formation of coherent systems. Appl Stoch Models Bus Ind 30:444–454MathSciNetCrossRefGoogle Scholar
  29. Navarro J, Aguila Y, Sordo MA, Suarez-Llorens A (2016) Preservation of stochastic orders under the formation of generalized distorted distributions. Applications to coherent systems. Methodol Comput Appl Prob 18:529–545MathSciNetCrossRefzbMATHGoogle Scholar
  30. Nelsen RB (2006) An introduction to copulas. Springer, New YorkzbMATHGoogle Scholar
  31. Samaniego FJ (2007) System signatures and their applications in engineering reliability. Springer, New YorkCrossRefzbMATHGoogle Scholar
  32. Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, New YorkCrossRefzbMATHGoogle Scholar
  33. Tavangar M, Bairamov I (2012) On the residual lifetimes of coherent systems with exchangeable components, Pakistan. J Stat 28:303–313Google Scholar
  34. Zhang Z (2010) Mixture representations of inactivity times of conditional coherent systems and their applications. J Appl Prob 47:876–885MathSciNetCrossRefzbMATHGoogle Scholar
  35. Zhang Z (2010) Ordering conditional general coherent systems with exchangeable components. J Stat Plan Inference 140:454–460MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Department of Mathematical SciencesIsfahan University of TechnologyEsfahānIran
  2. 2.Department of StatisticsUniversity of IsfahanEsfahānIran
  3. 3.School of MathematicsInstitute of Research in Fundamental Sciences (IPM)TehranIran

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