Relating the waiting time in a heavy-traffic queueing system to the queue length

Abstract

This study concerns the waiting time w k of the kth arrival to a single-server queueing system and the queue length l k just before the kth arrival. The first issue is whether the standard heavy-traffic limit distribution of these variables is the only possible limit. The second issue is the validity of the approximation w k ≅ υ ̄ l k , for large k, where ῡ is the average service time. The main results show that there are three types of heavy-traffic limiting distributions of the waiting times and queue lengths depending on whether the queueing systems are stable, marginally stable or unstable. Furthermore, these limit theorems justify the approximation w k ≅ υ ̄ l k for the three heavy-traffic regimes and they characterize the asymptotic distribution of the difference w k − υ ̄ l k . The results apply, in particular, to the GI⧸G⧸1 system and systems in which the service and interarrival times are stationary, regenerative, semi-stationary, asymptotically stationary and their sums satisfy certain functional limit laws. They also apply to queues that may not satisfy standard assumptions.