The mean general residual lifetime of (n − k + 1)-out-of-n systems with exchangeable components

ABSTRACT

The study of the reliability properties of (n − k + 1)-out-of-n systems has gained a great deal of attention, from both theoretical and practical perspectives. In this article, we consider (n − k + 1)-out-of-n systems with exchangeable components and study the stochastic properties of two forms of residual lifetimes of such systems under the following conditions: n − r + 1 (r ⩽ k) components of the system are operating at time t > 0, and/or the rth (r < k) component has failed, but the system is working at time t. In addition, some results relating to the functions of the mean general residual lifetimes (MGRL) are derived for these systems. Finally, in accordance with the generalized Farlie–Gumbel–Morgenstern model, we present the reliability properties of the general residual lifetime of (n − k + 1)-out-of-n systems and investigate the asymptotic behavior of the proposed MGRL functions with exponential marginals.

KEYWORDS:

• Dependence structure
• Generalized FGM copula
• MTP₂
• Reliability
• Residual lifetime
• Stochastic order.

MATHEMATICS SUBJECT CLASSFICATION:

• 62N05
• 60E15
• 60E05

Acknowledgments

The authors would like to thank the editor and two anonymous referees, for their valuable and constructive comments, which improved the presentation of the article. We are also grateful to the office of Graduate Studies of the Birjand University of Technology for their support.