Optimal repair of a series system with fixed repair times

Consider a system consisting of n components in series, subject to failures, with one repairman. No preemption is permitted during a repair. The repairman has a list of components and when he becomes available he repairs the failed component which is closest to the top of the list. The goal is to minimize the mean time until all the components operate - the mean down time. For systems consisting of two components, we assume that the operation and repair times are generally distributed and that the operation times are stochastically ordered. For systems consisting of n components we assume exponentially distributed operation times with rate λ_i and fixed repair times. We prove that lists in which the components are ordered in an increasing order of their expected operation times are optimal.