On a Repairman Assignment Problem

Consider a model with m parallel queues, each with its own server, and the same finite capacity B. All service times are exponentially distributed with the same rate. Arriving customers have to be assigned to one of the non-empty queues. It is well known that, if the decision is based on the current queue lengths, assigning to the shortest queue is optimal, for a large class of reward functions. Also the arrivals can be quite general, even allowing the arrival stream to depend in some sense on the queue lengths.