Abstract
In certain multiqueue systems, a resourceful server may decide on which queue to attend to next based on criteria that may include the equilibrium queue lengths of some or all queues. The server's decision will be unaffected by queue length only when the distribution of queue length is uniform. In studying this problem, we present first the simple case of the M/GI/1 queue to determine conditions under which equilibrium queue lengths can either be uniformly distributed, or have components that are uniform, and also present a method for designing such a queueing process. It is shown that there is an inherent connection between probability distributions { k j } of the number of arriving customers during a service that are geometric and equilibrium queue length distributions { p j } that are geometric. In particular, we present a simple probabilistic proof via a recurrence equation that says that the M/M/1 queue is the only M/GI/1 queue with both { k j } and { p j } geometric.
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