Skip to main content

On conditional residual lifetimes of coherent systems consisting of components with discrete lifetimes

Metrika (2022)Cite this article

Abstract

In this paper, we consider a coherent system composed of components whose lifetimes are independent and identically discretely distributed random variables. We study several aging and stochastic properties of the conditional residual lifetime of the system under the condition that some of its components have failed by time t. Moreover, we compare the conditional residual lifetimes of two coherent systems by using various stochastic orders.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Availability of data and material

Not applicable.

References

  • Barlow RE, Proschan F (1975) Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston

    MATH  Google Scholar 

  • Davies K, Dembińska A (2019) On the number of failed components in a \(k\)-out-of-\(n\) system upon system failure when the lifetimes are discretely distributed. Reliab Eng Syst Saf 188:47–61

    Article  Google Scholar 

  • Dembińska A (2018) On reliability analysis of \(k\)-out-of-\(n\) systems consisting of heterogeneous components with discrete lifetimes. IEEE Trans Rel 67:1071–1083

    Article  Google Scholar 

  • Dembińska A, Goroncy A (2020) Moments of order statistics from DNID discrete random variables with application in reliability. J Comput Appl Math 371:112703

    MathSciNet  Article  Google Scholar 

  • Dembińska A, Jasiński K (2021) Maximum likelihood estimators based on discrete component lifetimes of a k-out-of-n system. TEST 30:407–428

    MathSciNet  Article  Google Scholar 

  • Dembińska A, Nikolov NI, Stoimenova E (2021) Reliability properties of \(k\)-out-of-\(n\) systems with one cold standby unit. J Comput Appl Math 388:113289

    MathSciNet  Article  Google Scholar 

  • Eryilmaz S, Koutras MV, Triantatyllou JS (2016) Mixed three-state \(k\)-out-of-\(n\) systems under double monitoring. IEEE Trans Rel 61:792–797

    Article  Google Scholar 

  • Goliforushani S, Asadi M, Balakrishnan N (2012) On the residual and inactivity times of the components of used coherent systems. J Appl Probab 49:385–404

    MathSciNet  Article  Google Scholar 

  • Jasiński K (2021) The number of failed components in a coherent working system when the lifetimes are discretely distributed. Metrika 84:1081–1094

    MathSciNet  Article  Google Scholar 

  • Kochar SC, Mukerjee H, Samaniego FJ (1999) The signature of a coherent system and its application to comparisons among systems. Naval Res Logist 46:507–523

    MathSciNet  Article  Google Scholar 

  • Lai CD, Xie M (2007) Stochastic Ageing and Dependence for Reliability. Springer, New York

    Google Scholar 

  • Miziuła P, Rychlik T (2014) Sharp bounds for lifetime variances of reliability systems with exchangeable components. IEEE Trans Rel 63:850–857

    Article  Google Scholar 

  • Navarro J, Rychlik T (2007) Reliability and expectation bounds for coherent systems with exchangeable components. J Multivar Anal 98:102–113

    MathSciNet  Article  Google Scholar 

  • Navarro J, Samaniego FJ, Balakrishnan N, Bhattacharya D (2008) On the application and extension of system signatures in engineering reliability. Naval Res Logist 55:313–327

    MathSciNet  Article  Google Scholar 

  • Parvardeh A, Balakrishnan N (2013) Conditional residual lifetimes of coherent systems. Statist Probab Lett 83:2664–2672

    MathSciNet  Article  Google Scholar 

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

    Article  Google Scholar 

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

    Book  Google Scholar 

  • Tank F, Eryilmaz S (2015) The distributions of sum, minima and maxima of generalized geometric random variables. Statist Papers 56:1191–1203

    MathSciNet  Article  Google Scholar 

  • Weiss G (1962) On certain redundant systems which operate at discrete times. Technometrics 4:169–174

    MathSciNet  MATH  Google Scholar 

  • Young D (1970) The order statistics of the negative binomial distribution. Biometrika 57:181–186

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

I would like to express my sincere thanks to two anonymous referees for their constructive comments and suggestions which improved the presentation of the paper.

Funding

Not applicable.

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100, Toruń, Poland

    Krzysztof Jasiński

Authors
  1. Krzysztof Jasiński
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Krzysztof Jasiński.

Ethics declarations

Conflicts of interest/Competing interests

The author states that there is no conflict of interest.

Code availability

Not applicable.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Jasiński, K. On conditional residual lifetimes of coherent systems consisting of components with discrete lifetimes. Metrika (2022). https://doi.org/10.1007/s00184-022-00871-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00184-022-00871-4

Keywords

  • Coherent systems
  • Residual lifetime
  • Discrete lifetime distribution
  • Order statistics
  • Stochastic orders

Mathematics Subject Classification

  • 90B25
  • 60K10
  • 60E15