Lifetime of Combined k-out-of-n, and Consecutive kc-out-of-n Systems

A combined k-out-of-n:F(G) & consecutive kc-out-of-n :F(G) system fails (functions) iff at least k components fail (function), or at least fcc consecutive components fail (function). Explicit formulas are given for the lifetime distribution of these combined systems whenever the lifetimes of components are exchangeable, and have an absolutely continuous joint distribution. The lifetime distributions of the aforementioned systems are represented as a linear combination of distributions of order statistics by using the concept of Samaniego's signature. Formulas for the mean lifetimes are given. Some numerical results are also presented.