Computing marginal and joint Birnbaum, and Barlow–Proschan importances in weighted-k-out-of-n:G systems

• Reliability importance in weighted k-out-of-n:G system is studied.
• Marginal and joint importances are computed.
• Universal generating function method is used.

A weighted-k-out-of-n:G system is a system that consists of n binary components, each with its own positive weight, and operates only when the total weight of working components is at least k. Such a structure is useful when the components have different contributions to the performance of the entire system. This paper is concerned with both marginal and joint Birnbaum, and Barlow–Proschan (BP) importances of the components in weighted- k-out-of-n:G systems. The method of universal generating function is used for computing marginal and joint Birnbaum importances. The method for computing BP-importance is based on a direct probabilistic approach. Extensive numerical calculations are presented. By the help of these calculations and illustrations, it is possible to observe how the marginal and joint importances change with respect to the weights of components.