GERT analysis of m-consecutive-k-out-of-n:F system with overlapping runs and (k 1)-step Markov dependence

An m-consecutive-k-out-of-n:F system is a system of n linearly ordered components which fails if and only if at least m non-overlapping sequences of k consecutive components fail. When m = 1, we have the classic consecutive-k-out-of-n:F system about which there is an extensive literature. In this paper, we study the situation in which a system consisting of n linearly ordered sequence of components fails if and only if there are at least m overlapping runs of k consecutive failed components. Graphical Evaluation and Review Technique (GERT) analysis is used for reliability evaluation of the system for both, i.i.d. components and (k?1)-step Markov dependent components, in a unified manner. Software Mathematica is used for systematic computation. Illustrative numerical examples are presented to substantiate the theory.