We establish heavy-traffic limits for the arrival and workload processes in a single-server queue with a time-varying arrival-rate function. We establish limits at and before a critical point, the onset of critical loading, where the arrival-rate function approaches its critical value from below. We extend results by Newell (1968) and Mandelbaum and Massey (1995) and present alternative views of the interesting scaling constants that arise in these limits.
Whitt, W. (2016). Heavy-traffic limits for a single-server queue leading up to a critical point. Operations Research Letters, 44(6), 796–800. https://doi.org/10.1016/j.orl.2016.10.005