An Upper Bound for the Equilibrium Mean Wait in a Stationary GI/G/1 Queue

Various bounds have been proposed for the mean value of the equilibrium waiting time random variable, E{W}, in a stationary GI/G/1 queueing system. In this note we establish a new upper bound which is derived using a convexity argument similar to that of Marshall (Marshall, K. T. 1968. Some inequalities in queueing. Oper. Res.16 651–665.) but applied to an algebraic expression for E{W} previously used by Kingman (Kingman, J. F. C. 1962. Some inequalities for the queue GI/G/1. Biometrika49 315–324.), This bound and others have been compared numerically in a variety of systems; two graphs are presented to illustrate the results of such computations.