Reliability and Birnbaum Importance for Sparsely Connected Circular Consecutive- $k$ Systems

A consecutive- k-out-of- n: F ( G ) system with sparse d consists of n components ordered in a line or a circle, while the system fails (works) iff, there exist at least k consecutive failed (working) components with sparse d for 0 ≤ d ≤ n - k. In this paper, a circular consecutive- k-out-of- n system with sparse d is considered. Some equations for system reliability and Birnbaum importance are derived by means of the finite Markov chain imbedding approach. Then the Birnbaum importance of components is compared in the situations where the system is under an IID model, and where one of the components is known to be failed, respectively. Finally, some numerical examples are followed to illustrate the results obtained in the paper.