Note—Comments on a Queueing Inequality

In a G/G/c/N system (a queue with general distributions of inter-arrival and service time, c servers and N − c ≥ 0 queueing positions), let B be the steady-state probability that an arriving customer finds all queue positions filled and p be the time average probability that all queue positions are filled. By assuming p = B, Matthew Sobel proved

where p is the traffic intensity. He also showed, by numerical examples, that the lower bound is a good approximation when p ≥ 1.5 and c ≥ 2.

In this paper, we show that the lower bound does not require the assumption p = B to hold, and that it follows from the conservation of load. This derivation also explains why the bound does not depend on N and is a good approximation in heavy traffic. We also show that the upper bound depends critically on the assumption that p = B.