An r consecutive k out of n: F system is a system of n linearly arranged components which fails if r non-overlapping sequences of k components fail. When r=1 we have the classic consecutive k out of n: F system about which there is an extensive literature. In this research we study the situation where there are k distinct components with failure probabilities qi for i=1,...,k and where the failure probability of the jth component (j=mk+i (1[less-than-or-equals, slant]i[less-than-or-equals, slant]k)) is qi. We call such a system an r consecutive k out of n: F system with cycle (or period) k. We obtain exact expressions for the failure probability of an r consecutive k out of n: F system and in particular show that it is independent of the order of the k components if n is a multiple of k. Interesting applications are given for the arrangement of sport competitions and inspection procedures in quality control.
