RELATING THE WAITING TIME IN A HEAVY-TRAFFIC QUEUING SYSTEM TO THE QUEUE LENGTH

[摘要] 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) congruent to ($) over bar vl(k), for large k, where ($) over bar v is the average service time. The main results show that there art 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) congruent to ($) over bar vl(k) for the three heavy-traffic regimes and they characterize the asymptotic distribution of the difference W-k - ($) over bar vl(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.

[发布日期] 1994-08-01

[关键词] SINGLE-SERVER QUEUING SYSTEM;WAITING TIMES;HEAVY TRAFFIC;WEAK CONVERGENCE;BROWNIAN MOTION;LITTLES LAW