On approximations for queues, III: Mixtures of exponential distributions

To evaluate queueing approximations based on a few parameters (e.g., the first two moments) of the interarrival-time and service-time distributions, we examine the set of all possible values of the mean queue length given this partial information. In general, the range of possible values given such partial information can be large, but if in addition shape constraints are imposed on the distributions, then the range can be significantly reduced. The effect of shape constraints on the interarrival-time distribution in a GI/M/1 queue was investigated in Part II (see "On Approximations for Queues, II: Shape Constraints," this issue) by restricting attention to discrete probability distributions with probability on a fixed finite set of points and then solving nonlinear programs. In this paper we show how one kind of shape constraint — assuming that the distribution is a mixture of exponential distributions — can be examined analytically. By considering GI/G/1 queues in which both the interarrival-time and service-time distributions are mixtures of exponential distributions with specified first two moments, we show that additional information about the distributions is more important for the interarrivai time than for the service time.