The Brownian approximation for rate-control throttles and the G/G/1/C queue

This paper studies approximations to describe the performance of a rate-control throttle based on a token bank, which is closely related to the standard G/G/1/C queue and the two-node cyclic network of ·/G/1/∞ queues. Several different approximations for the throttle are considered, but most attention is given to a Brownian or diffusion approximation. The Brownian approximation is supported by a heavy-traffic limit theorem (as the traffic intensity approaches the upper limit for stability) for which an upper bound on the rate of convergence is established. Means and squared coefficients of variation associated with renewal-process approximations for the overflow processes are also obtained from the Brownian approximation. The accuracy of the Brownian approximation is investigated by making numerical comparisons with exact values. The relatively simple Brownian approximation for the job overflow rate is not very accurate for small overflow rates, but it nevertheless provides important insights into the way the throttle design parameters should depend on the arrival-process characteristics in order to achieve a specified overflow rate. This simple approximation also provides estimates of the sensitivity of the overflow rates to the model parameters.