Asymptotic estimates for queueing systems with time-varying periodic transition rates

We consider the M t / M t / 1 queue, the multi-server queue (M t / M t / c t ), and queues with jumps of size one and two. Results are extensible to more general ergodic quasi-birth-death processes (QBDs) with time-varying periodic transition rates of period one. The estimates are asymptotic in the level of the process (the length of the queue). These asymptotic estimates highlight the connections between the asymptotic periodic distribution of a stable queue with time-varying rates and the same type of queue with constant rates. The estimates can also be used to approximate other performance measures such as the waiting time distribution. We illustrate the method with several examples.