# On the reliability modeling of weighted *k*-out-of-*n* systems with randomly chosen components

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## Abstract

The weighted *k*-out-of-*n* (briefly denoted as weighted *k* / *n*) systems are among the most important kind of redundancy structures. We consider a weighted *k* / *n* system with dependent components where the system is built up from two classes \({\mathfrak {C}}_X\) and \({\mathfrak {C}}_Y\) of components that are categorized according to their weights and reliability functions. It is assumed that a random number *M*, \({M=0,1,\dots ,m}\), of the components are chosen from set \({\mathfrak {C}}_X\) whose components are distributed as \(F_X\) and the remaining \(n-M\) components selected from the set \({\mathfrak {C}}_Y\) whose components have distribution function \(F_Y\). We further assume that the structure of dependency of the components can be modeled by a copula function. The reliability of the system, at any time *t*, is expressed as a mixture of reliability of weighted *k* / *n* systems with fixed number of the components of types \({\mathfrak {C}}_X\) and \({\mathfrak {C}}_Y\) in terms of the probability mass function *M*. Some stochastic orderings are made between two different weighted *k* / *n* systems. It is shown that when the random mechanism of the chosen components for two systems are ordered in usual stochastic (*st*) order then, under some conditions, the lifetimes of the two systems are also ordered in *st* order. We also compare the lifetimes of two different systems in the sense of stochastic precedence concept. The results are examined by several illustrative examples under different conditions.

## Keywords

Reliability Weighted*k*-out-of-

*n*system Copulas Stochastic order Stochastic precedence

## Mathematics Subject Classification

90B25 62H20## Notes

### Acknowledgements

The authors express their sincere thanks to the Editor, and associate editor, and two referees for providing constructive comments which led to improvement of the paper. Asadi’s research work was carried out in IPM Isfahan branch and was in part supported by a Grant from IPM (No. 96620411).

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