Optimal consecutive-k-out-of-n systems under a fixed budget.

A consecutive-k-out-of-n system is a graph with n vertices where the system fails if and only if some path of k consecutive vertices all fail. Assume that q sub i is the failing probability of the i sup (th) vertex and the states of the vertices are statistically independent. The problem is to find the best system, i.e., a set {q sub i,...,q sub n}sub h and its assignment to the vertices of the system graph, which maximizes the reliability of the system, under the constraint that the product of q sub i 's is a constant. We solve the problem for the line, the cycle (except when n and k are relatively prime), and any tree.