On the Consecutive k-of-n System.

Abstract
We consider the consecutive k-of-n system in which there are n components linearly ordered. Each component either functions or fails and the system is said to be failed if any k consecutive components are failed. Let r(p) = r(p(1), ..., p(n)) denote the probability that the system does not fail given that the components are independent, component i functions with probability p(i), i = 1, ..., n. The function r(p) is called the reliability function. We study the above system both when the components are linearly ordered and also when they are arranged in a circular order. We consider the case where all p(i) are identical and present a recursion for obtaining the reliability of a consecutive k-of-n in terms of the reliability of a consecutive k - 1 of n system. This yields simple explicit formulas when k is small and differs from a recursion. We show how upper and lower bounds on r(p) can be simply obtained. We consider a dynamic version in which each component independently functions for random time having distribution F. We show that when F is increasing failure rate (IFR), then system lifetime is also IFR only in the circular case when k = 2. We consider a sequential optimization model in the linear k = 2 case. In this model, components are put in place one at a time with complete knowledge as to whether the previous component has worked or not.

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