Refinements to Heavy Traffic Limit Theorems in Queueing Theory

We consider single server M/G/1 and GI/G/1 queues in the limit of heavy traffic. We develop a procedure for obtaining the full asymptotic series of the stationary distribution of unfinished work in powers of one minus the traffic intensity. The leading term in this series is the (exponential) density obtained from the heavy traffic limit theorem. We show that the correction terms have different forms in different regions of the state space. These corrections are constructed using the method of matched asymptotic expansions. We assume that the method of matched asymptotic expansions is valid.