Extreme and high-level sojourns of the single server queue in heavy traffic

This study characterizes the behavior of large queue lengths in heavy traffic. We show that the distribution of the maximum queue length in a random time interval for a queueing system in heavy traffic converges to a novel extreme value distribution. We also study the processes that record the times that the queue length exceeds a high level and the cumulative time the queue is above the level. We show that these processes converge in distribution to compound Poisson processes.