On the Reliability of (n, f, k) and 〈n, f, k〉 Systems

The (n, f, k)(⟨n, f, k⟩) 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 (n, f, k): F, (n, f, k): G, ⟨n, f, k⟩: F, and ⟨n, f, k⟩: G systems and give method of evaluating the reliability of these systems using joint distribution of X_ni=(Xn,ki,Xn,1i)$\underline{\rm X}_{\rm n}^{\rm i}=({\rm X}_{{\rm n},{\rm k}}^{\rm i},{\rm X}_{{\rm n},1}^i)$, (i = 0, 1) where Xin, k is the number of occurrences of non-overlapping i-runs of length k in the sequence of Bernoulli trials. We obtain the probability generating function of joint distribution of X_n1=(Xn,k1,Xn,11)$\underline{\rm X}_{\rm n}^1=({\rm X}_{{\rm n},{\rm k}}^1,{\rm X}_{{\rm n},1}^1)$ using the method of conditional probability generating function for the sequence of Markov Bernoulli trials and m-dependent Markov Bernoulli trials. An algorithm based on the simple theory of matrix polynomials is developed to get the exact probability distribution of X_n1$\underline{\rm X}_{\rm n}^1$ from the corresponding probability generating function derived. Further we discuss the use of exact distribution of X_n0$\underline{\rm X}_{\rm n}^0$ to evaluate the reliability of (n, f, k): F and ⟨n, f, k⟩: F system when the components are arranged in line and circle. To demonstrate the computational efficiency of an algorithm developed we evaluate the reliability of (n, f, k) and ⟨n, f, k⟩ system when the system has linear as well as circular arrangement of components.