Birnbaum Reliability Importance for (n,f,k) and ⟨n,k,f⟩ System

Abstract

The ( n, f, k): F/⟨ n, k, f ⟩: F system is the combination of most popular consecutive-k-out-of-n and f-out-of-n system and its failure is caused by two different failure criteria. The ( n, f, k): F (⟨ n, k, f ⟩: F) system consists of n components ordered in a line or circle, while the system fails if and only if there exist at least f failed components or (and) at least k consecutive failed components. In this paper, we consider the sequence {X_u, u ⩾ 1} of {0, 1}-valued Markov Bernoulli trials (MBT) and study the Birnbaum reliability importance for components of ( n, f, k): F(G), and ⟨ n, k, f ⟩: F(G) system through the conditional joint distribution of , where and is the number of non overlapping occurrences of i-runs of length k_i(i = 0, 1) in n MBT. The formula for evaluation of exact Birnbaum reliability importance is developed and the results are demonstrated numerically. We also bring out the important inter-relations between the reliability and reliability importance of four systems as f-out-of-n: F, consecutive-k-out-of-n: F, ( n, f, k): F and ⟨ n, k, f ⟩: F systems.

Keywords:

• Birnbaum reliability importance
• (n,f,k) and ⟨n,k,f⟩ system
• Runs

Mathematics Subject Classification:

• 62N05
• 37A50