We consider an M/M/1 queue with a removable server that dynamically chooses its service rate from a set of finitely many rates. If the server is off, the system must warm up for a random, exponentially distributed amount of time, before it can begin processing jobs. We show under the average cost criterion, that work conserving policies are optimal. We then demonstrate the optimal policy can be characterized by a threshold for turning on the server and the optimal service rate increases monotonically with the number in system. Finally, we present some numerical experiments to provide insights into the practicality of having both a removable server and service rate control.

中文翻译：

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具有可移动服务器和动态服务费率的 M/M/1 队列的最佳控制策略

我们考虑一个米/米/1 带有可移动服务器的队列，该服务器从一组有限多个速率中动态选择其服务速率。如果服务器关闭，则系统必须预热一段随机的、呈指数分布的时间，然后才能开始处理作业。我们在平均成本标准下表明，工作保存策略是最优的。然后我们证明了最优策略可以通过打开服务器的阈值来表征，并且最优服务率随着系统中的数量单调增加。最后，我们提出了一些数值实验，以深入了解同时拥有可移动服务器和服务速率控制的实用性。