Note—Comments on a Queueing Inequality
Management Science · 1980 · Volume 26 · Issue 9
Abstract
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 .
Keywords
queues, multiserver, inequalities
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