A linear weighted (n, f, k) system for non-homogeneous Markov-dependent components

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 w_u for u = 1, 2, …, n and w = (w₁, w₂, …, w_n). 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.

KEYWORDS:

• Marginal reliability importance
• joint reliability importance
• weighted (n, f, k) system
• non-homogeneous Markov-dependent components
• probability generating function

Acknowledgements

The authors thank the associate editor, Dr. David W. Coit, and three anonymous reviewers for their constructive comments that improved the article.

Funding

The work of the first author was supported in part by the National Science Foundation of China under grant NSFC# 71571178 and Chinese Thousand Youth Talents Program.

Additional information

Notes on contributors

Xiaoyan Zhu

Xiaoyan Zhu received a B.S. degree from Tsinghua University, Beijing, China in 2000. She received M.S. and Ph.D. degrees in Industrial Engineering from Texas A&M University, College Station, in 2002 and 2005, respectively. Her research interests are in system reliability optimization, importance measure--based decision making, and network and integer programming with applications to large-scale problems in transportation and supply chain management.

Mahmoud Boushaba

Mahmoud Boushaba is a Professor in the Department of Mathematics, Ecole Normale Supérieure de Constantine, Algeria. He received his Ph.D. degree from the University of Constantine in 2003. He is currently the head of Ecole Normale Supérieure de Constantine. He has more than 20 years of teaching experience. He has published over 20 peer-reviewed technical publications. His area of research is reliability theory and applied probability. He is a reviewer for IEEE Transactions on Reliability and International Journal of System Sciences and Communications in Statistics: Theory and Methods.