Optimizing buffer size for the retrial queue: two state space collapse results in heavy traffic

We study a single server queueing model with admission control and retrials. In the heavy traffic limit, the main queue and retrial queue lengths jointly converge to a degenerate two-dimensional diffusion process. When this model is considered with holding and rejection costs, formal limits lead to a free boundary curve that determines a threshold on the main queue length as a function of the retrial queue length, above which arrivals must be rejected. However, it is known to be a notoriously difficult problem to characterize this curve. We aim instead at optimizing the threshold on the main queue length independently of the retrial queue length. Our main result shows that in the small and large retrial rate limits, this problem is governed by the Harrison–Taksar free boundary problem, which is a Bellman equation in which the free boundary consists of a single point. We derive the asymptotically optimal buffer size in these two extreme cases, as the scaling parameter and the retrial rate approach their limits.