  We consider a single server queue which has a threshold to change its arrival
process and service rate by its queue length, which is referred to as a
two-level single server queue. This model is motivated by an energy saving
problem for a single server queue whose arrival process and service rate may be
changed. To get its performance in tractable form, we study in the limit of the
stationary distribution of this two-level queue under a certain scaling in
heavy traffic, which corresponds to a diffusion approximation. It is shown that
this limiting distribution is truncated exponential below the threshold level
and exponential above it under suitably chosen system parameters. This result
is proved using the so called BAR approach studied in Braverman, Dai and
Miyazawa (2017, 2023) and Miyazawa (2017, 2023). We also intuitively discuss
about a diffusion process corresponding to the limit of the stationary
distributions.
