Structure importance of consecutive-k-out-of-n systems

A consecutive-k-out-of-n:F system is an n-component system that fails when k consecutive components fail. The structure importance is a measure, which indicates the importance of a component relative to its positioning in the system. Through the relationship to the Fibonacci sequence with order k, a closed-form solution of structure importance for each component is obtained. We obtain a complete ordering of the components with respect to their structure importance for some consecutive-k-out-of-n:F systems, and a partial ordering for other systems.