Simplified Reliabilities for Consecutive- k -out-of- n Systems

Reliabilities for consecutive-k-out-of-n systems are typically given in the form of recursive equations. Some attempts have been made to use combinatorics to obtain closed-form solutions, but the solutions contain $k - 1$ summations. In this paper we obtain closed form solutions with one summation over $n/ k$ terms. For $k = 2$ we are able to eliminate all summations. We apply our result to compute the reliability of a k-loop computer network.