On approximations for queues, I: Extremal distributions

Many approximations for queueing characteristics such as the mean equilibrium queue length are based on two moments of the interarrivai and service times. To evaluate these approximations, we suggest looking at the set of all possible values of the queueing characteristics given the specified moment parameters. This set-valued function is useful for evaluating the accuracy of approximations. For several models, such as the GI/M/1 queue, the set of possible values for the mean queue length given limited-moment information can be conveniently described by simple extremal distributions. Here we calculate the set of possible values for the mean queue length in a GI/M/1 queue and show how it depends on the traffic intensity and the second moment. We also use extremal distributions to compare alternative parameters for approximations. The results provide useful insights about approximations for non-Markov networks of queues and other complex queueing systems. The general procedure is widely applicable to investigate the accuracy of approximations.

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