Interpolation approximations for M |G|∞ arrival processes

We present an approximate analysis of a discrete–time queue with correlated arrival processes of the so–called M |G|∞ type. The proposed heuristic approximations are developed around asymptotic results in the heavy and light traffic regimes. Investigation of the system behavior in light traffic quantifies the differences between the gradual M |G|∞ inputs and the point arrivals of a classical GI|GI|1 queue. In heavy traffic, salient features are effectively captured by the exponential distribution and the Mittag–Leffler special function, under short– and long–range dependence respectively. By interpolating between the heavy and light traffic extremes we derive approximations to the queue size distribution, applicable to all traffic intensities. We examine the accuracy of these expressions and discuss possible extensions of our results in several numerical examples.