Qualitative Properties and Bounds for the Serial Covariances of Waiting Times in Single-Server Queues

For the stationary waiting time process of a GI/G/1 queue we prove the empirically obvious fact that if the service times are increased and the interarrival times decreased, then the correlation of waiting times of successive customers is increased. If the service and interarrival times are respectively larger or smaller than for given exponential service or interarrival times, then the serial covariances of waiting times are bounded by the known covariances for the given exponential case. Also, some general bounds for the covariances are given, and the heavy traffic case is considered. The results are very useful in simulation problems where the mean stationary waiting time is estimated by a sample and it is sought to determine the mean square error of the sample mean.