Transform Methods for Heavy-Traffic Analysis

The drift method was recently developed to study queuing systems in steady state. It was used successfully to obtain bounds on the moments of the scaled queue lengths that are asymptotically tight in heavy traffic and in a wide variety of systems, including generalized switches, input-queued switches, bandwidth-sharing networks, and so on. In this paper, we develop the use of transform techniques for heavy-traffic analysis, with a special focus on the use of moment-generating functions. This approach simplifies the proofs of the drift method and provides a new perspective on the drift method. We present a general framework and then use the moment-generating function method to obtain the stationary distribution of scaled queue lengths in heavy traffic in queuing systems that satisfy the complete resource pooling condition. In particular, we study load balancing systems and generalized switches under general settings.