Reliability analysis of s-out-of-k multicomponent stress-strength system with dependent strength elements based on copula function

Jing Cai; Jianfeng Yang; Yongjin Zhang; Jing Cai; Jianfeng Yang; Yongjin Zhang

doi:10.3934/mbe.2023416

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 5: 9470-9488. doi: 10.3934/mbe.2023416

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Research article

Reliability analysis of s-out-of-k multicomponent stress-strength system with dependent strength elements based on copula function

1.
School of Data Science and Engineering, Guizhou Minzu University, Guiyang, China
2.
School of Data Science, Guizhou Institute of Technology, Guiyang, China
3.
School of Mathematics and Physics, Anhui University of Technology, Maanshan, China

Academic Editor: Hao Wang

Received: 19 January 2023 Revised: 27 February 2023 Accepted: 08 March 2023 Published: 20 March 2023

This paper considers the reliability analysis of a multicomponent stress-strength system which has $k$ statistically independent and identically distributed strength components, and each component is constructed by a pair of statistically dependent elements. These elements are exposed to a common random stress, and the dependence among lifetimes of elements is generated by Clayton copula with unknown copula parameter. The system is regarded to be operating only if at least $s$($1 \leq s \leq k$) strength variables in the system exceed the random stress. The maximum likelihood estimates (MLE) of unknown parameters and system reliability is established and associated asymptotic confidence interval is constructed using the asymptotic normality property and delta method, and the bootstrap confidence intervals are obtained using the sampling theory. Finally, Monte Carlo simulation is conducted to support the proposed model and methods, and one real data set is analyzed to demonstrate the applicability of our study.
- reliability analysis,
- s-out-of-k system,
- multicomponent,
- stress-strength model,
- copula
Citation: Jing Cai, Jianfeng Yang, Yongjin Zhang. Reliability analysis of s-out-of-k multicomponent stress-strength system with dependent strength elements based on copula function[J]. Mathematical Biosciences and Engineering, 2023, 20(5): 9470-9488. doi: 10.3934/mbe.2023416

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Abstract

This paper considers the reliability analysis of a multicomponent stress-strength system which has $k$ statistically independent and identically distributed strength components, and each component is constructed by a pair of statistically dependent elements. These elements are exposed to a common random stress, and the dependence among lifetimes of elements is generated by Clayton copula with unknown copula parameter. The system is regarded to be operating only if at least $s$($1 \leq s \leq k$) strength variables in the system exceed the random stress. The maximum likelihood estimates (MLE) of unknown parameters and system reliability is established and associated asymptotic confidence interval is constructed using the asymptotic normality property and delta method, and the bootstrap confidence intervals are obtained using the sampling theory. Finally, Monte Carlo simulation is conducted to support the proposed model and methods, and one real data set is analyzed to demonstrate the applicability of our study.

References

[1]	Z. W. Birnbaum, On a use of Mann-Whitney statistics, in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1 (1956), 13–17.
[2]	R. G. Srinivasa, M. Aslam, O. H. Arif, Estimation of reliability in multicomponent stress- strength based on two parameter exponentiated Weibull Distribution, Commun. Stat. Theory Methods, 46 (2017), 7495–7502. https://doi.org/10.1080/03610926.2016.1154155 doi: 10.1080/03610926.2016.1154155
[3]	A. Kohansal, On estimation of reliability in a multicomponent stress-strength model for a Kumaraswamy distribution based on progressively censored sample, Stat. Papers, 60 (2019), 2185–2224. https://doi.org/10.1007/s00362-017-0916-6 doi: 10.1007/s00362-017-0916-6
[4]	X. Bai, Y. Shi, Y. Liu, B. Liu, Reliability inference of stress-strength model for the truncated proportional hazard rate distribution under progressively Type-Ⅱ censored samples, Appl. Math. Modell. , 65 (2019), 377–389. https://doi.org/10.1016/j.apm.2018.08.020 doi: 10.1016/j.apm.2018.08.020
[5]	X. Bai, Y. Shi, Y. Liu, B. Liu, Reliability estimation of stress-strength model using finite mixture distributions under progressively interval censoring, J. Comput. Appl. Math. , 348 (2019), 509–524. https://doi.org/10.1016/j.cam.2018.09.023 doi: 10.1016/j.cam.2018.09.023
[6]	V. K. Sharma, Bayesian analysis of head and neck cancer data using generalized inverse Lindley stress-strength reliability model, Commun. Stat. Theory Methods, 47 (2018), 1155–1180. https://doi.org/10.1080/03610926.2017.1316858 doi: 10.1080/03610926.2017.1316858
[7]	W. Kuo, M. J. Zuo, Optimal Reliability Modeling: Principles and Applications, John Wiley & Sons, Hoboken, 2003.
[8]	R. G. Srinivasa, M. Aslam, O. H. Arif, Estimation of reliability in multicomponent stress- strength based on two parameter exponentiated Weibull Distribution, Commun. Stat. Theory Methods, 46 (2017), 7495–7502. https://doi.org/10.1080/03610926.2016.1154155 doi: 10.1080/03610926.2016.1154155
[9]	Y. Liu, Y. Shi, X. Bai, P. Zhan, Reliability estimation of a N-M-cold-standby redundancy system in a multicomponent stress-strength model with generalized half-logistic distribution, Phys. A, 490 (2018), 231–249. https://doi.org/10.1016/j.physa.2017.08.028 doi: 10.1016/j.physa.2017.08.028
[10]	F. Kızılaslan, Classical and Bayesian estimation of reliability in a multicomponent stress-strength model based on the proportional reversed hazard rate model, Math. Comput. Simul. , 136 (2017), 36–62. https://doi.org/10.1016/j.matcom.2016.10.011 doi: 10.1016/j.matcom.2016.10.011
[11]	L. Zhang, A. Xu, L. An, M. Li, Bayesian inference of system reliability for multicomponent stress-strength model under Marshall-Olkin Weibull distribution. Systems, 10 (2022), 1–14. https://doi.org/10.3390/systems10060196 doi: 10.3390/systems10060196
[12]	L. Wang, S. Dey, Y. M. Tripathi, S. J. Wu, Reliability inference for a multicomponent stress-strength model based on Kumaraswamy distribution, J. Comput. Appl. Math. , 376 (2020), 112823. https://doi.org/10.1016/j.cam.2020.112823 doi: 10.1016/j.cam.2020.112823
[13]	A. Kohansal, On estimation of reliability in a multicomponent stress-strength model for a Kumaraswamy distribution based on progressively censored sample, Stat. Papers, 60 (2019), 2185–2224. https://doi.org/10.1007/s00362-017-0916-6 doi: 10.1007/s00362-017-0916-6
[14]	S. Dey, J. Mazucheli, M. Z. Anis, Estimation of reliability of multicomponent stress-strength for a Kumaraswamy distribution, Commun. Stat. Theory Methods, 46 (2017), 1560–1572. https://doi.org/10.1080/03610926.2015.1022457 doi: 10.1080/03610926.2015.1022457
[15]	N. Jana, S. Bera, Interval estimation of multicomponent stress-strength reliability based on inverse Weibull distribution, Math. Comput. Simul. , 191 (2022), 95–119. https://doi.org/10.1016/j.matcom.2021.07.026 doi: 10.1016/j.matcom.2021.07.026
[16]	T. Zhu, Reliability estimation of s-out-of-k system in a multicomponent stress-strength dependent model based on copula function, J. Comput. Appl. Math. , 404 (2022), 113920. https://doi.org/10.1016/j.cam.2021.113920 doi: 10.1016/j.cam.2021.113920
[17]	M. K. Jha, S. Dey, R. M. Alotaibi, Y. M. Tripathi, Reliability estimation of a multicomponent stress-strength model for unit Gompertz distribution under progressive Type Ⅱ censoring, Qual. Reliab. Eng. Int. , 36 (2020), 965–987. https://doi.org/10.1002/qre.2610 doi: 10.1002/qre.2610
[18]	M. Nadar, F. Kızılaslan, Estimation of reliability in a multicomponent stress-strength model based on a Marshall-Olkin bivariate Weibull distribution, IEEE Trans. Reliab. , 65 (2015), 370–380. https://doi.org/10.1109/TR.2015.2433258 doi: 10.1109/TR.2015.2433258
[19]	F. Kızılaslan, M. Nadar, Estimation of reliability in a multicomponent stress-strength model based on a bivariate Kumaraswamy distribution, Stat. Papers, 59 (2018), 307–340. https://doi.org/10.1007/s00362-016-0765-8 doi: 10.1007/s00362-016-0765-8
[20]	A. Sklar, Functions de repartition an dimensions et leurs marges, Publications de l'Institut de Statistique de l'Université de Paris, Paris, 1959.
[21]	R. B. Nelsen, An Introduction to Copulas, Springer, New York, 2006.
[22]	G. K. Bhattacharyya, R. A. Johnson, Estimation of reliability in a multicomponent stress-strength model, J. Am. Stat. Assoc. , 69 (1974), 966–970. https://doi.org/10.1080/01621459.1974.10480238 doi: 10.1080/01621459.1974.10480238
[23]	O. Nave, S. Ajadi, Y. Lehavi, Analysis of the dynamics of fuel spray using asymptotic methods: The method of integral invariant manifolds, Appl. Math. Comput. , 218 (2012), 5877–5890. https://doi.org/10.1016/j.amc.2011.11.030 doi: 10.1016/j.amc.2011.11.030
[24]	A. Xu, S. Zhou, Y. Tang, A unified model for system reliability evaluation under dynamic operating conditions, IEEE Trans. Reliab. , 70 (2021), 65–72. https://doi.org/10.1109/TR.2019.2948173 doi: 10.1109/TR.2019.2948173
[25]	C. Luo, L. Shen, A. Xu, Modelling and estimation of system reliability under dynamic operating environments and lifetime ordering constraints, Reliab. Eng. Syst. Saf. , 218(2022), 108136. https://doi.org/10.1016/j.ress.2021.108136 doi: 10.1016/j.ress.2021.108136
[26]	P. Hall, Theoretical comparison of bootstrap confidence intervals, Ann. Stat. , 16 (1988), 927–953.
[27]	X. Shi, P. Zhan, Y. Shi, Statistical inference for a hybrid system model with incomplete observed data under adaptive progressive hybrid censoring, Concurrency Comput. Pract. Exper., 34 (2020), 5708. https://doi.org/10.1002/cpe.5708 doi: 10.1002/cpe.5708
[28]	A. Kohansal, S. Shoaee, Bayesian and classical estimation of reliability in a multicomponent stress-strength model under adaptive hybrid progressive censored data, Stat. Papers, 62 (2021), 309–359. https://doi.org/10.1007/s00362-019-01094-y doi: 10.1007/s00362-019-01094-y

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