Markovian queues with Poisson control

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