| Within the past ten years, queueing theorists have shown considerable interest in the single-server multiple-queue model in which the server selects the next customer from whichever queue is longest. Researchers have focused on the property that this service discipline tends to equalize the queue lengths, making it a good discipline when buffer size requirements are important. It is particularly useful in applications where there is control over the service discipline but not the customer arrival rate, and applications of this model occur in telecommunications, finite-source inventory management, and, to a lesser degree, air traffic control.; Although researchers have focused on the performance of this model in the symmetric case where the arrival rates are approximately equal, there remains much to be learned about the performance of this model under asymmetric conditions. The primary finding of this thesis is that customers joining a queue characterized by a very small arrival rate relative to the other queue will incur a large delay penalty under this discipline. In fact, the delay is unbounded as the ratio of arrival rates at the two queues reach extreme (but finite) values. As a consequence, the longer queue discipline manifests a phenomena not normally observed in queueing systems: an infinitesimally small value of the traffic intensity leads to unboundedly large waiting times. These findings are proven analytically and verified through numerical simulation.; When the arrival rate ratios are unknown in advance, and when they might reach extreme Values, a better management strategy is to serve the longer queue while the arrival rate ratios are “reasonable” and then switch to an alternating discipline when they become “unreasonable.” The problem is quantifying the fuzzy term “reasonable.” Empirical results obtained via numerical simulation suggest that, for a traffic intensity of 3/4 or higher, a “reasonable” ratio occurs when arrivals at one queue lie within 30% and 70% of the total arrival rate. Outside these bounds, the longer queue discipline performs poorly, leading to extreme delays.; In demonstrating these results, the balance equations for a variant of the longer queue discipline are derived; these equations are verified by simulation; the probability that the server win switch from one queue to the other is analytically computed; a metric, called the virtual service time is defined and its lower and upper bounds are analytically computed and verified by simulation. The analytic results are limited to the M/M/1 model, but the simulation experiments suggest that M/G/1 models have qualitatively similar, but quantitatively different, performance characteristics. |
