Inequalities in the Theory of Queues

IT is a fair criticism of the theory of queues as it has developed through the years that, even in the simple cases for which explicit analytical solutions can be found, these solutions are often too complicated to be of practical use. It has been argued elsewhere (Kingman, 1966) that the criticism is to be met to some degree by the analysis of situations where robust approximations exist, such as that of "heavy traffic". It is, however, important to know how accurately such approximations represent the true solution, and the significance of inequalities for the various quantities of interest thus becomes apparent. To be useful, an inequality must have two properties which are to some extent incompatible with one another. If 0 is some quantity not perhaps easy to calculate, the inequality 0 < 6' will not be significant unless there is reason to hope that, in an appropriate sense, 6' is reasonably close to 0. This can of course best be determined if the inequality is one of a pair