Tighter upper bounds for the average message delay in a Geom[X]/Geom/N queueing system

In a previous paper by Rubin and Zhang (see ibid., vol.39, no.6, p.905-14, June 1991) upper and lower bounds for the average message system-size and delay in a Geom^[X]/Geom/N queueing system are presented. However, the upper bound for the average message delay is not tight in the low utilization range. In this note, the author presents much tighter upper bounds for the average message delay and system-size, using the convex property of the mean delay with respect to the mean arrival rate