Customers' optimal joining rules for the GI/M/s queue

A GI/M/s queue with a stationary balking sequence is considered. For the infinite horizon average reward criterion, it is shown that among all stationary joining policies the optimal ones are nonrandomized control limit rules of the form: join if and only if the queue size is smaller than some specific number. It is shown that, in general, exercising self-optimization by individual customers does not optimize public good. The M/M/s queue is then treated as an example, and a "direct" proof for the optimality of the control limit rule is given.