Copula-Based Reliability for Weighted-$k$-Out-of-$n$ Systems Having Randomly Chosen Components of $m$ Different Types

In this article, we consider a weighted-
$k$
-out-of-
$n$
system having
$m \geq 2$
type of components each with its own positive integer-valued weight, in which the random lifetimes of components are dependent. This system is supposed to work with performance level
$k$
if and only if the total weight of functioning components of all types is at least
$k$
. The structure of dependence of the system component lifetimes is modeled by the copula function. Also, it is assumed that the random numbers
$N_i$
of components are chosen from class
$D_i$
for type
$i$
. The reliability of the system is obtained as a mixture of the reliability of weighted-
$k$
-out-of-
$n$
systems consisting of
$m$
types of components with fixed number of them in terms of the probability mass function of the random vector
$(N_1, \ldots,N_{m-1})$
. Then, a copula-based expression for component importance in each class is obtained, and illustrative examples are presented.