An interpolation approximation for the mean workload in a GI/G/1 queue

This paper develops a closed form approximation for the mean steady-state workload or virtual waiting time in a GI/G/1 queue, using the first two moments of the service-time distribution and the first three moments plus the density at the origin of the interarrival-time distribution, with default values provided in case information is unavailable. The approximation is based on light and heavy traffic limiting behavior. The essential ideas are exposed by using S.L. Brumelle’s formula to relate the mean workload to the mean waiting time and K.T. Marshall’s formula to relate the mean waiting time to the first two moments of the ideal period. Both formulas extend to general single server models without independence conditions, so this approach provides a basis for extensions, but a convenient exact expression for the second order heavy traffic term is evidently not possible even for GI/G/1. For the GI/G/1 second order heavy traffic term, an approximation is proposed, based on the relatively nice expression established for the GI/M/1 case by S. Halfin. The interpolation between light and heavy traffic limits, which can be applied to other performance measures and models whenever the limits can be determined or approximated, is chosen to satisfy differentiability and monotonicity regularity conditions.