The Reliability of Weighted k-out-of-n Systems with Randomized Components of Several Batches

One of the most important types of redundancy structures is weighted k-out-of-n system (which briefly denoted as weighted k/n system)‎. ‎In this paper‎, ‎a weighted k/n system‎ ‎with dependent components is considered and it is assumed that the components of the system are selected from p, p≥3, ‎batches‎. ‎The components in the batches are different in terms of their weights and reliability‎ ‎functions‎. ‎It is supposed that the selection of components from p-1 batches is based on a random‎ ‎vector with dimension p-1‎, ‎and the remaining components are chosen from the last batch‎. ‎The structure of dependency of components is also determined using a copula function‎. ‎Under these circumstances‎, ‎a mixture representation is given for the reliability function of the system based on the reliability functions of weighted k/n systems with a fixed number of the components in the batches‎. ‎Some stochastic orderings are presented‎, ‎under some conditions‎, ‎for comparing two different weighted k/n systems with randomized components‎. ‎The results are illustrated using some examples and graphs‎.