$m$ -Consecutive- $k$ , $l$

Here we present the structure functions for the linear and circular m-consecutive- k, l-out-of- n:F (G) systems, which consists of n linearly or circularly ordered components such that the system fails (works) iff there are at least m times l-overlapping runs of consecutive k failed (working) components. The duality of these F and G systems is obtained, and the proofs are given. The generalized reliability formulae for the systems are obtained by using the finite Markov chain imbedding approach for statistically independent and Markov-dependent cases. Some miscellaneous results on the reliability of these systems and numerical examples are provided to illustrate the results obtained in the paper.