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

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 {Xu, 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 X_n0Xu=l$\underline{\bm X} _n^0 \left| {X_u = l} \right. $ X_n1Xu=ll=0,1$\left({\underline{\bm X} _n^1 \left| {X_u = l} \right.} \right)l = 0,1 $ , where X_ni=Xn,1i,Xn,kii$\underline{\bm X} _n^i = \left({X_{n,1}^i,X_{n,k_i }^i } \right) $ and Xn,kii$X_{n,k_i }^i $ is the number of non overlapping occurrences of i-runs of length ki(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.