Stabilizing Performance in a Single-Server Queue with Time-Varying Arrival Rate

We consider a general Gt/Gt/1 single-server queue with unlimited waiting space and a time-varying arrival rate, where the the service rate at each time is subject to control. We study the rate-matching control, where the the service rate is made proportional to the arrival rate. We show that the model with the rate-matching control can be regarded as a deterministic time transformation of a stationary G/G/1 model, so that the queue length distribution is stabilized as time evolves. However, the time-varying virtual waiting time is not stabilized. We show that the time-varying expected virtual waiting time with the rate-matching service-rate control becomes inversely proportional to the arrival rate in a heavy-traffic limit. We also show that no control that stabilizes the queue length asymptotically in heavy-traffic can also stabilize the virtual waiting time. Then we consider two square-root service-rate controls. We show that these alternative square-root service-rate controls stabilize the waiting time when the arrival rate changes very slowly relative to the average service time, so that a pointwise stationary approximation is appropriate