Arranging Queues in Series

For given external arrival process and given service-time distributions, the object is to determine the order of infinite-capacity single-server queues in series that minimizes the long-run average sojourn time per customer. We gain additional insight into this queueing design problem, and congestion in open queueing networks more generally, primarily by performing simulation experiments. We develop a new parametric-decomposition approximation for departure processes, which can be used in general queueing network algorithms as well as in this design problem. For this design problem, we conclude that the key issue is variability: The order tends to matter when the service-time distributions have significantly different variability, and not otherwise. The order also matters less in light traffic, even with a relative difference criterion. Arranging the queues in order of increasing service-time variability, using the squared coefficient of variation as a partial characterization of variability, seems to be an effective simple heuristic. In all cases of two queues in series, the simulation results indicate that the same order is optimal for all combinations of the traffic intensities, suggesting that the light-traffic asymptotics in Greenberg and Wolff (1988) should usually be effective for identifying the best order. Comparisons with simulations for two queues also indicate that the parametric-decomposition approximations provide quite accurate quantitative estimates of the expected sojourn time with each order. However, for more than two queues, the approximations need not properly describe the congestion at a bottleneck queue.