A consecutive-k-n network is a generalization of the well-known consecutive-k-out-of-n system, and has many practical applications. This network consists of n + 2 nodes (node 0, the source, nodes 1, 2, …, n, and node n + 1, the target) and directed links from node i to node j (0 < i < j < n + 1, j - i < k). Because all nodes except the source and target, and all links are fallible, the network works if and only if there exists a working path from the source to the target. For the k = 2 case, based on identical node reliabilities and some assumptions on link reliabilities, Chen, Hwang and Li (1993) gave a recursive algorithm for the reliability of the consecutive-2-n network. In this paper we give a closed form equation for the reliability of the general consecutive-k-n network by means of a novel Markov chain method. Based on the equation, we propose an algorithm which is more efficient than other published ones for the reliability of the consecutive-k-n network.