<formula formulatype="inline"><tex Notation="TeX">$m$</tex></formula>-Consecutive-<formula formulatype="inline"><tex Notation="TeX">$k$</tex></formula>, <formula formulatype="inline"><tex Notation="TeX">$l$</tex></formula>-Out-of-<formula formulatype="inline"><tex Notation="TeX">$n$</tex></formula> Systems

Abstract

Here we present the structure functions for the linear and circular <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$m$</tex></formula> -consecutive- <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$k$</tex></formula> , <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$l$</tex></formula> -out-of- <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n$</tex></formula> :F (G) systems, which consists of <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n$</tex></formula> linearly or circularly ordered components such that the system fails (works) iff there are at least <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$m$</tex></formula> times <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$l$</tex></formula> -overlapping runs of consecutive <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$k$</tex> </formula> 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.