We consider the problem of controlling the service and/or arrival rate in queues, with the objective of minimizing the total expected cost to reach state zero. We present a unified, simple method for proving that an optimal policy is monotonic in the number of customers in the system. Applications to average-cost minimization over an infinite horizon are given. Both exponential and non-exponential models are considered; the essential characteristic is a left-skip-free transition structure and a nondecreasing (not necessarily convex) holding-cost function. Some of our results are insensitive to service-time distributions.
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