We examine the problem of how best to route jobs among a number of queues whose servers are subject to random periods of unavailability. The optimal routing policy is computed by modelling the system as a discrete-time, finite-state Markov decision process and solving the resulting dynamic programming equations. In a series of numerical experiments, the performance of various heuristic policies is compared with that of the optimal policy. A particular heuristic, using an 'index policy', is shown to be close to optimal.