Research Papers

Dynamic Reliability Evaluation of Nonrepairable Multistate Weighted k-Out-of-n System With Dependent Components Based on Copula

[+] Author and Article Information
Xinchen Zhuang

School of Aeronautics,
Northwestern Polytechnical University,
Xi'an Shaanxi 710072, China
e-mail: zhuangxinchen@126.com

Tianxiang Yu

School of Aeronautics,
Northwestern Polytechnical University,
Xi'an Shaanxi 710072, China
e-mail: tianxiangyu@nwpu.edu.cn

Linjie Shen

School of Aeronautics,
Northwestern Polytechnical University,
Xi'an Shaanxi 710072, China
e-mail: s_linjie@126.com

Bozhi Guo

School of Aeronautics,
Northwestern Polytechnical University,
Xi'an Shaanxi 710072, China
e-mail: 1090519629@qq.com

1Corresponding author.

Manuscript received February 21, 2017; final manuscript received January 12, 2018; published online April 18, 2018. Assoc. Editor: Faisal Khan.

ASME J. Risk Uncertainty Part B 4(4), 041001 (Apr 18, 2018) (7 pages) Paper No: RISK-17-1032; doi: 10.1115/1.4039243 History: Received February 21, 2017; Revised January 12, 2018

As a common type system, multistate weighted k-out-of-n system is of great importance in reliability engineering. The components are usually treated as independent from each other. It is usually not that case in real life and the components are dependent. On the other hand, the performance of the components degrades over time, leading to the change of the components' weight at the same time. As a result, the present paper provides a method to evaluate the dynamic reliability of multistate weighted k-out-of-n: G system with s-dependent components. The degradation of the components follows a Markov process and the components are nonrepairable. Copula function is used to model the s-dependence of the components. The LZ-transform for a discrete-state continuous-time Markov process is combined, and the explicit expression for the survival function and the mean time to failure (MTTF) of the system is obtained. A small electricity generating system is studied based on our method in the illustration, and detailed comparison result is made for dependent case and independent case. Dynamic reliability with varied levels of electricity generation conforming to the actual situation for this generating system is also calculated.

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Eryilmaz, S. , 2014, “ Multivariate Copula Based Dynamic Reliability Modeling With Application to Weighted k-Out-of-n Systems of Dependent Components,” Struct. Saf., 51, pp. 23–28. [CrossRef]
Eryilmaz, S. , 2016, “ A Reliability Model for a Three-State Degraded System Having Random Degradation Rates,” Reliab. Eng. Syst. Saf., 156, pp. 59–63. [CrossRef]
Samaniego, F. J. , and Shaked, M. , 2008, “ Systems With Weighted Components,” Stat. Probab. Lett., 78(6), pp. 815–823. [CrossRef]
Wu, J. S. , and Chen, R. J. , 1994, “ An Algorithm for Computing the Reliability of a Weighted-k-Out-of-n System,” IEEE Trans. Reliab., 43(2), pp. 327–328. [CrossRef]
Li, W. , and Zuo, M. J. , 2008, “ Reliability Evaluation of Multi-State Weighted k-Out-of-n Systems,” Reliab. Eng. Syst. Saf., 93(1), pp. 160–167. [CrossRef]
Wang, Y. , Li, L. , Huang, S. , and Chang, Q. , 2012, “ Reliability and Covariance Estimation of Weighted k-Out-of-n Multi-State Systems,” Eur. J. Oper. Res., 221(1), pp. 138–147. [CrossRef]
Eryilmaz, S. , 2015, “ Mean Time to Failure of Weighted k-Out-of-n: G Systems,” Commun. Stat. Simul. Comput., 44, pp. 2705–2713.
Amrutkar, K. P. , and Kamalja, K. K. , 2014, “ Reliability and Importance Measures of Weighted k-Out-of-n:F System,” Int. J. Reliab. Qual. Saf. Eng., 21(3), p. 1450015. [CrossRef]
Rahmani, R. A. , Izadi, M. , and Khaledi, B. E. , 2016, “ Importance of Components in k-Out-of-n System With Components Having Random Weights,” J. Comput. Appl. Math., 296, pp. 1–9. [CrossRef]
Meshkat, R. S. , and Mahmoudi, E. , 2017, “ Joint Reliability and Weighted Importance Measures of a k-Out-of-n System With Random Weights for Components,” J. Comput. Appl. Math., 326, pp. 273–283. [CrossRef]
Rahmani, R.-A. , Izadiv, M. , and Khaledi, B.-E. , 2016, “ Stochastic Comparisons of Total Capacity of Weighted k-Out-of-n Systems,” Stat. Probab. Lett., 117, pp. 216–220. [CrossRef]
Ram, M. , and Manglik, M. , 2016, “ Performance Evaluation of a Multi-State System Covering Imperfect Fault Coverage,” Commun. Stat. Simul. Comput., 45(9), pp. 3259–3280. [CrossRef]
Li, W. , and Zuo, M. J. , 2008, “ Optimal Design of Multi-State Weighted k-Out-of-n Systems Based on Component Design,” Reliab. Eng. Syst. Saf., 93(11), pp. 1673–1681. [CrossRef]
Eryilmaz, S. , 2013, “ Mean Instantaneous Performance of a System With Weighted Components That Have Arbitrarily Distributed Lifetimes,” Reliab. Eng. Syst. Saf., 119, pp. 290–293. [CrossRef]
Eryilmaz, S. , 2013, “ On Reliability Analysis of a k-Out-of-n System With Components Having Random Weights,” Reliab. Eng. Syst. Saf., 109, pp. 41–44. [CrossRef]
Wang, L. , Wang, X. J. , Li, Y. L. , Lin, G. P. , and Qiu, Z. P. , 2017, “ Structural Time-Dependent Reliability Assessment of the Vibration Active Control System With Unknown‐but‐Bounded Uncertainties,” Struct. Control Health Monit., 24(10), p. e1965. [CrossRef]
Wang, L. , Wang, X. J. , Chen, X. , and Wang, R. X. , 2015, “ Time-Variant Reliability Model and its Measure Index of Structures Based on a Non-Probabilistic Interval Process,” Acta Mech., 216(10), pp. 3221–3241. [CrossRef]
Wang, L. , Wang, X. J. , Chen, X. , and Wang, R. X. , 2015, “ Time-Dependent Reliability Modeling and Analysis Method for Mechanics Based on Convex Process,” Math. Probl. Eng., 2015, pp. 1–16.
Liu, Y. W. , and Kailash, K. C. , 2006, “ Reliability Measures for Dynamic Multistate Non-Repairable Systems and Their Applications to System Performance Evaluation,” IIE Trans., 38(6), pp. 511–520. [CrossRef]
Eryilmaz, S. , and AliRıza, B. , 2014, “ An Algorithmic Approach for the Dynamic Reliability Analysis of Non-Repairable Multi-State Weighted k-Out-of-n: G System,” Reliab. Eng. Syst. Saf., 131, pp. 61–65. [CrossRef]
Eryilmaz, S. , 2015, “ Capacity Loss and Residual Capacity in Weighted k-Out-of-n: G Systems,” Reliab. Eng. Syst. Saf., 136, pp. 140–144. [CrossRef]
Faghih-Roohi, S. , Xie, M. , Ng, K. M. , and Richard, C. M. , 2014, “ Dynamic Availability Assessment and Optimal Component Design of Multi-State Weighted k-Out-of-n Systems,” Reliab. Eng. Syst. Saf., 123, pp. 57–62. [CrossRef]
Foiondella, L. , and Xing, L. D. , 2015, “ Discrete and Continuous Reliability Models for Systems With Identically Distributed Correlated Components,” Reliab. Eng. Syst. Saf., 133(1), pp. 1–10. [CrossRef]
Chen, Y. , and Yang, Q. , 2005, “ Reliability of Two-Stage Weighted k-Out-of-n System With Components in Common,” IEEE Trans. Reliab., 54(3), pp. 431–440. [CrossRef]
Li, X. H. , You, Y. P. , and Fang, R. , 2016, “ On Weighted k-Out-of-n Systems With Statistically Dependent Component Lifetimes,” Probab. Eng. Inf. Sci., 30(4), pp. 533–546. [CrossRef]
Zhu, X. , and Boushaba, M. A. , 2017, “ Linear Weighted (n, f, k) System for Non-Homogeneous Markov-Dependent Components,” IISE Trans., 49(7), pp. 722–736. [CrossRef]
Levitin, G. , 2004, “ A Universal Generating Function Approach for the Analysis of Multi-State Systems With Dependent Elements,” Reliab. Eng. Syst. Saf., 84(3), pp. 285–292. [CrossRef]
Eryilmaz, S. , 2014, “ Modeling Dependence Between Two Multi-State Components Via Copulas,” IEEE Trans. Reliab., 63(3), pp. 715–720. [CrossRef]
Nelsen, R. B. , 2006, An Introduction to Copulas: Dependence, Springer, London, Chap. 2.
Tang, X. S. , Li, D. Q. , Zhou, C. B. , Phoon, K. K. , and Zhang, L. M. , 2013, “ Impact of Copulas for Modeling Bivariate Distributions on System Reliability,” Struct. Saf., 44, pp. 80–90. [CrossRef]
Navarro, J. , and Spizzichino, F. , 2010, “ Comparisons of Series and Parallel Systems With Components Sharing the Same Copula,” Appl. Stochastic Models Bus. Ind., 26(6), pp. 775–791. [CrossRef]
Shih, J. H. , and Louis, T. A. , 1995, “ Inferences on the Association Parameter in Copula Models for Bivariate Survival Data,” Biometrics, 51(4), pp. 1384–1399. [CrossRef] [PubMed]
Akaike, H. , 1974, “ A New Look at the Statistical Model Identification,” IEEE Trans. Autom. Control, 19(6), pp. 716–723. [CrossRef]
Lisnianski, A. , and Frenkel, I. , 2012, Recent Advances in System Reliability, Springer, London, Chap. 6. [CrossRef]
Lisnianski, A. , 2004, “ Universal Generating Function Technique and Random Process Methods for Multi-State System Reliability Analysis,” Second International Workshop in Applied Probability (IWAP2004), Piraeus, Greece, Mar. 22–25, pp. 237–242.


Grahic Jump Location
Fig. 1

Reliability of the system under dependence and independence for k = 5

Grahic Jump Location
Fig. 2

Reliability of the system under dependence and independence for k = 7

Grahic Jump Location
Fig. 3

Reliability of the system under dependence and independence for k = 9

Grahic Jump Location
Fig. 4

Reliability for the system with varied level




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