On the number of working components in a $\nu$-out-of-n system with components having random weights

In this paper, we consider a system having n components each has a random weight (capacity) with integer values. The system is defined to be in state $\nu$ or above at time instant t if the total weight of all working components is more than or equal to $\nu$. We obtain a recursive formula for the reliability of such system and explore the mean number of working components under the condition that the system is working at time t. An optimization problem on the optimal replacement time of $\nu$-out-of-n system with components having random weights is also formulated with some examples.