GERT Analysis of $m$-Consecutive-$k$ -Out-of-$n$ Systems

An m-consecutive-k-out-of-n:F system, introduced by W.S. Griffith, consists of an ordered linear sequence of n i.i.d. components that fails iff there are at least m non-overlapping runs of k consecutive failed components. However, a situation may occur in which an ordered linear sequence of n i.i.d. components fails iff there are at least m non-overlapping runs of at least k consecutive failed components. We call such a system an m-consecutive-at least-k-out-of-n:F system. This paper presents a graphical evaluation and review technique (GERT) analysis of both types of systems providing closed form explicit formulae for reliability evaluation in a unified manner. GERT, besides providing a visual picture of the system, helps to analyse the system in a less inductive manner. Numerical examples for each system are studied in detail by computing the reliability for various combinations of sets of values of the parameters involved. It is observed that m-consecutive-at least-k-out-of-n:F systems are more reliable than m-consecutive-k-out-of-n:F systems as the number of possible state combinations leading to system's failure are larger in the latter. Mathematica is used for systematic computations. Numerical investigations illustrate the efficiency of GERT in reliability analysis of such systems. In comparison with the existing formulae of m-consecutive-k-out-of-n:F systems for i.i.d. components, the formula obtained by GERT analysis, to be referred to as GERT-F, is much more efficient owing to its significantly low computational time, and easy implementation