Asymptotically Maximal Throughput in Tandem Systems with Flexible and Dedicated Servers

Abstract

For a system of two tandem queues with a finite intermediate buffer, we study the asymptotically maximal throughput as the number of servers in each station grows to infinity. First, we study the system with only dedicated servers, and then we examine the system with both dedicated and flexible servers. We assume that travel times between the two stations are negligible and that each server can only work on one customer at a time. We model the system as a birth–death Markov process, derive a closed form solution for the stationary distribution, and quantify the maximal asymptotic normalized throughput as the number of servers grows to infinity. We show that flexibility is more favorable for small systems, and as the number of servers grows, the benefits of flexibility decrease. Furthermore, we prove that when the number of servers goes to infinity, there is no need of flexibility at all, as the maximum value of the throughput is obtained. However, flexibility still has a secondary beneficial effect — a little flexibility (on the order of the square root of the number of dedicated servers at each station) guarantees that all dedicated servers are busy and results in faster convergence to the maximum throughput.