r-CONSECUTIVE-k-OUT-OF-n: F SYSTEMS WITH DEPENDENT COMPONENTS

Abstract

An "r-consecutive-k-out-of-n: F system" consists of n linearly ordered components. The system fails if and only if at least r non-overlapping sequences of k consecutive components fail. In this paper we examine this system in the case where the failure probability of a given component depends upon the state (good or failed) of the preceding one i.e. the states of the components form a Markov chain. First we give a recursive formula of the failure probability of such a system when the transition probabilities qi,0,qi,1 are not identical where qi,0 (respectively qi,1) is the probability that component i fails given that the preceding one fails (respectively works), for i = 1, 2, …, n. Secondly we treat a special case of the same system where qj,0 = qi,0 and qj,1 = qi,1 for j = mk + i (1 ≤ i ≤ k), and we call such a system an r-consecutive-k-out-of-n: F system with cycle (or period) k with Markov-dependent components, and in this case also we give a formula of the failure probability of the system.