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A linear weighted system for non-homogeneous Markov-dependent components

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  • Xiaoyan Zhu
  • Mahmoud Boushaba

Abstract

We study a linear weighted (n, f, k) system, denoted by L(n, f, k, w) system and consider the situation where components are non-homogeneous Markov-dependent. An L(n, f, k, w) system consists of n components ordered in a line, and each component u has a positive integer weight wu for u = 1, 2, …, n and w = (w1, w2, …, wn). The L(n, f, k, w):F (G) system fails (works) if the total weight of failed (working) components is at least f or the total weight of consecutive failed (working) components is at least k. For the L(n, f, k, w):F system with non-homogeneous Markov-dependent components, we derive closed-form formulas for the system reliability, the marginal reliability importance measure of a single component, and the joint reliability importance measure of multiple components using a conditional probability generating function method. We extend these results to the L(n, f, k, w):G systems, the weighted consecutive-k-out-of-n systems, and the weighted f-out-of-n systems. Our numerical examples and a case study on a bridge system demonstrate the use of derived formulas and provide the insights on the L(n, f, k, w) systems and the importance measures. In addition, the two failure modes associated with the L(n, f, k, w):F systems are analyzed by comparing to the single failure mode associated with the weighted consecutive-k-out-of-n:F systems and the single failure mode associated with the weighted f-out-of-n:F systems.

Suggested Citation

  • Xiaoyan Zhu & Mahmoud Boushaba, 2017. "A linear weighted system for non-homogeneous Markov-dependent components," IISE Transactions, Taylor & Francis Journals, vol. 49(7), pages 722-736, July.
    • Handle: RePEc:taf:uiiexx:v:49:y:2017:i:7:p:722-736
    DOI: 10.1080/24725854.2016.1269977
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    References listed on IDEAS

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    1. Xiaoyan Zhu & Qingzhu Yao & Way Kuo, 2012. "Patterns of the Birnbaum importance in linear consecutive--out-of- systems," IISE Transactions, Taylor & Francis Journals, vol. 44(4), pages 277-290.
    2. K. K. Kamalja & R. L. Shinde, 2014. "On the Reliability of (n, f, k) and 〈n, f, k〉 Systems," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 43(8), pages 1649-1665, April.
    3. Qingzhu Yao & Xiaoyan Zhu & Way Kuo, 2011. "Heuristics for component assignment problems based on the Birnbaum importance," IISE Transactions, Taylor & Francis Journals, vol. 43(9), pages 633-646.
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    Cited by:

    1. Zhu, Xiaoyan & Boushaba, Mahmoud & Coit, David W. & Benyahia, Azzeddine, 2017. "Reliability and importance measures for m-consecutive-k, l-out-of-n system with non-homogeneous Markov-dependent components," Reliability Engineering and System Safety, Elsevier, vol. 167(C), pages 1-9.
    2. Xiaoyan Zhu & Mahmoud Boushaba & Abdelmoumene Boulahia & Xian Zhao, 2019. "A linear m-consecutive-k-out-of-n system with sparse d of non-homogeneous Markov-dependent components," Journal of Risk and Reliability, , vol. 233(3), pages 328-337, June.

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