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

Richard F. Serfozo, Wladyslaw Szczotka and Krzysztof Topolski

Stochastic Processes and their Applications, 1994, vol. 52, issue 1, 119-134

Abstract: This study concerns the waiting time wk of the kth arrival to a single-server queueing system and the queue length lk 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 3, for large k, where [upsilon] 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 3 for the three heavy-traffic regimes and they characterize the asymptotic distribution of the difference 1. The results apply, in particular, to the GI[+45 degree rule]G[+45 degree rule]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.

Keywords: Single-server; queueing; system; Waiting; times; Heavy; traffic; Weak; convergence; Brownian; motion; Little's; law (search for similar items in EconPapers)
Date: 1994
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