Optimal consecutive-k-out-of-n : G cycle for n ≤ 2k + 1

A cyclic consecutive-k-out-of-n. G system consists of n components lying on a cycle. Those components are exchangeable but may have different working probabilities. The system works if and only if there are k consecutive components at work. What is the optimal assignment of components for maximizing the reliability of the system? Does the optimal assignment depend on the working probability values of components? For k ≤ n ≤ 2k + 1, Zuo and Kuo in 1990 proposed a solution independent from the working probability values of components, called the invariant optimal assignment. However, their proof is incomplete, pointed out recently by Jalali et al. [The Optimal Consecutive-k-out-of-n: G Line, for n ≤ 2k, manuscript, 1999]. We present a complete proof in this paper.