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.
Similar content being viewed by others
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
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
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
Dembińska A, Goroncy A (2020) Moments of order statistics from DNID discrete random variables with application in reliability. J Comput Appl Math 371:112703
Dembińska A, Jasiński K (2021) Maximum likelihood estimators based on discrete component lifetimes of a k-out-of-n system. TEST 30:407–428
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
Eryilmaz S, Koutras MV, Triantatyllou JS (2016) Mixed three-state \(k\)-out-of-\(n\) systems under double monitoring. IEEE Trans Rel 61:792–797
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
Jasiński K (2021) The number of failed components in a coherent working system when the lifetimes are discretely distributed. Metrika 84:1081–1094
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
Lai CD, Xie M (2007) Stochastic Ageing and Dependence for Reliability. Springer, New York
Miziuła P, Rychlik T (2014) Sharp bounds for lifetime variances of reliability systems with exchangeable components. IEEE Trans Rel 63:850–857
Navarro J, Rychlik T (2007) Reliability and expectation bounds for coherent systems with exchangeable components. J Multivar Anal 98:102–113
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
Parvardeh A, Balakrishnan N (2013) Conditional residual lifetimes of coherent systems. Statist Probab Lett 83:2664–2672
Samaniego FJ (1985) On closure of the IFR class under formation of coherent systems. IEEE Trans Rel R- 34:69–72
Shaked M, Shanthikumar JG (2007) Stochastic Orders. Springer, New York
Tank F, Eryilmaz S (2015) The distributions of sum, minima and maxima of generalized geometric random variables. Statist Papers 56:1191–1203
Weiss G (1962) On certain redundant systems which operate at discrete times. Technometrics 4:169–174
Young D (1970) The order statistics of the negative binomial distribution. Biometrika 57:181–186
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
Corresponding author
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
About this article
Cite this article
Jasiński, K. On conditional residual lifetimes of coherent systems consisting of components with discrete lifetimes. Metrika 86, 205–218 (2023). https://doi.org/10.1007/s00184-022-00871-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-022-00871-4