Simple bounds for a transient queue

Bounds on performance of a queueing model can provide useful information to guarantee quality of service for communication networks. We study the bounds on the mean delay in a transient GI/GI/1 queue given the first two moments of the service time and the inter-arrival time, respectively. We establish a simple upper-bound, which then is used to show that the true transient mean-delay is at most four times larger than an asymptotic diffusion-approximation. We also prove that the tight lower-bound is zero as long as the service time and the inter-arrival time have finite variance and the load is below one. Tightness of the trivial lower-bound is in contrast to the stationary mean-delay, which has strictly positive lower-bound when the service time is sufficiently variable. We also show how our results can be applied to analyze the transient mean delay of packets in the real-world Internet.