The presence of exponentiality in entropy maximized M/G/1 queues

It is known that entropy maximized M/GI/1 queues yield queue length distributions that are geometric when the constraints involve only the first two moments of the service time distribution. By proving that geometric queue length is equivalent to exponential service in M/GI/1 queues, we show that using the entropy maximization procedure with only two service time moments is equivalent to using exponentially distributed service times. Thus, by identifying the parameter of the equivalent service time distribution, we can compute such entropy maximized solutions via classical M/M/1 theory. In case an additional constraint is used to preserve the probability of an empty queue, the service time distribution becomes a mixture of two distributions, one of which is exponential, and the other an impulse function of unit mass at the origin. In this case, except for the probability of an empty queue, the remaining queue length probabilities follow a geometric sequence. In either case, our results demonstrate the presence of exponentiality in the service times of such entropy maximized M/GI/1 queues.