Diffusion approximation of the stationary distribution of a two-level single server queue

We consider a single server queue that has a threshold to change its arrival process and service speed by its queue length, which is referred to as a two-level GI/G/1 queue. This model is motivated by an energy saving problem for a single server queue whose arrival process and service speed are controlled. To obtain its performance in tractable form, we study the limit of the stationary distribution of the queue length in this two-level queue under scaling in heavy traffic. Except for a special case, this limit corresponds to its diffusion approximation. It is shown that this limiting distribution is truncated exponential (or uniform if the drift is null) below the threshold level and exponential above it under suitably chosen system parameters and generally distributed interarrival times and workloads brought by customers. This result is proved under a mild limitation on arrival parameters using the so-called basic adjoint relationship (BAR) approach studied in Braverman, Dai, and Miyazawa (2017, 2024) and Miyazawa (2017, 2024). We also intuitively discuss about a diffusion process corresponding to the limit of the stationary distribution under scaling.