Maximum Values in Queueing Processes

Motivated by extreme-value engineering in service systems, we develop and evaluate simple approximations for the distributions of maximum values of queueing processes over large time intervals. We provide approximations for several different processes, such as the waiting times of successive customers, the remaining workload at an arbitrary time, and the queue length at an arbitrary time, in a variety of models. All our approximations are based on extreme-value limit theorems. Our first approach is to approximate the queueing process by one-dimensional reflected Brownian motion (RBM). We then apply the extreme-value limit for RBM, which we derive here. Our second approach starts from exponential asymptotics for the tail of the steadystate distribution. We obtain an approximation by relating the given process to an associated sequence of i.i.d. random variables with the same asymptotic exponential tail. We use estimates of the asymptotic variance of the queueing process to determine an approximate number of variables in this associated i.i.d. sequence. Our third approach is to simplify GI/G/1 extremevalue limiting formulas in Iglehart (1972) by approximating the distribution of an idle period by the stationary-excess distribution of an interarrival time. We use simulation to evaluate the quality of these approximations for the maximum workload. From the simulations, we obtain a rough estimate of the time when the extreme value limit theorems begin to yield good approximations.