$m$-Consecutive-$k$, $l$-Out-of-$n$ Systems

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.