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:我们调查由控制器在由 Poisson 进程决定的随机时间随机检查的 Markovian 队列。 检查后, 控制器设定服务速度, 等于目前队列中客户人数的最小值和某种最大服务速度; 服务速度一直持续到下一次考试时间。 我们研究由此得出的队列长度和服务器速度的二维 Markov 进程, 特别是两个有时间尺度分离的系统, 特别是无限频繁和无限长的考试时间。 在中间系统中, 分析证明是极具挑战性的。 为了进一步了解模型动态, 我们然后分析两个模型的变体, 即控制器只是观察者, 不改变服务器的速度 。