We investigate Markovian queues that are examined by a controller at random times determined by a Poisson process. Upon examination, the controller sets the service speed to be equal to the minimum of the current number of customers in the queue and a certain maximum service speed; this service speed prevails until the next examination time. We study the resulting two-dimensional Markov process of queue length and server speed, in particular two regimes with time scale separation, specifically for infinitely frequent and infinitely long examination times. In the intermediate regime the analysis proves to be extremely challenging. To gain further insight into the model dynamics we then analyse two variants of the model in which the controller is just an observer and does not change the speed of the server.

翻译：我们研究由泊松过程决定的随机时刻受控制器审查的马尔可夫排队系统。在审查时，控制器将服务速度设定为当前队列长度与某一最大服务速度的最小值，该服务速度将持续至下一次审查时刻。我们研究由此产生的队列长度与服务速度的二维马尔可夫过程，特别关注两种时间尺度分离的机制，即审查频率无限大与审查间隔无限长的情形。在中间机制中，分析证明极具挑战性。为深入理解模型动态，我们进一步分析该模型的两种变体，其中控制器仅作为观察者而不改变服务速度。