On approximations for queues, II: Shape constraints

This paper continues the investigation begun in Part I of approximations for queues that are based on a few parameters partially characterizing the arrival process and the service-time distribution. Part I provides insight into approximations for intractable systems by considering the set of all possible values of the mean queue length in the GI/M/1 queue given the service rate and the first two moments of the interarrival-time distribution. The distributions yielding the maximum and minimum values of the mean queue length turn out to be quite unusual, e.g., two-point distributions. This paper shows that the range of possible values can be reduced dramatically by imposing realistic shape constraints on the interarrival-time distribution with given first two moments. We found extremal distributions in the presence of shape constraints by restricting our attention to discrete distributions with all mass on a fixed finite set of points and solving nonlinear programs. The results strongly support the use of two-moment approximations in general queueing systems when the interarrival-time and service-time distributions are not too irregular.