Some conditional reliability properties of \(k\)-out-of-\(n\) system composed of different types of components with discrete independent lifetimes (Q2051526)

A technical system is said to have a \(k\)-out-of-\(n\) structure if it works when at least \(k\) of the \(n\) components operate. The author of this paper considers reliability properties of a \(k\)-out-of-\(n\) system consisting of \(l\) (\(1 \leq l\leq n\)) different types of components with discrete, independent lifetimes. Some conditional survival functions of the lifetime of a used system are obtained, which are used to calculate two conditional failure probabilities for \(k\)-out-of-\(n\) systems. Under some conditions, it is shown that they are equal to the unconditional failure probability of a \(k\)-out-of-\((n-r)\) system for \(r < n-k+1\).