Information Theoretic Approximations for M/G/1 and G/G/1 Queuing Systems.

The behavior of single server queuing systems is characterized by various performance distributions, including distributions of queue length, waiting time, residence time, busy period, number served in a busy period, etc. In principle, if the arrival and service time distributions are known exactly, then these performance distributions can be computed using standard techniques. We consider the problem of estimating the distributions when only the first few moments of the service time distribution are known. Our approach uses standard relations to compute moments of the performance distributions from the known moments of the service time distribution and the principles of maximum entropy and minimum cross-entropy to estimate the performance distributions themselves. For MG1 systems with known average arrival rates, we derive analytic results for cases when one or two moments of the service time distribution are known, and we show how one can compute results using as many moments of the service time distribution as are available. For GG1 systems, our results are limited to the case in which only the average arrival and service rates are known. Among the results obtained with this information theoretic approach is a new light-load approximation for the MM1 busy period probability density. Throughout the paper, we illustrate our approach using MM1, MH21, and MD1 examples. Author