Approximations for multiserver systems with multiple priority classes

We first introduce approximations for the mean delay in an M/GI/$k$ queue with $m\geq 2$ priority classes (class $i$ jobs have preemptive priority over jobs of classes $i+1$ to $m$ for $1\leq i\leq m-1$). The abovementioned BB approximation was originally derived for exponential job size distributions [33] and was later applied to general job size distributions [23]. The BB approximation is the only (reasonable) approximation, for general job size distributions and for an arbitrary number of priority classes, that exists in the literature. Mitrani and King have proposed an approximation that works for exponential job size distributions and for an arbitrary number of priority classes [129], and this approach is also used by Nishida [140]. We refer to this approximation as MK-N. The MK-N approximation is defined and used only for exponential job size distributions, since MK-N requires an analysis of a multiserver queue with two priority classes, and an exact or nearly exact analysis has been known only for exponential job size distributions. Our analysis of an M/PH/$k$ queue with multiple priority classes via DR allows us to extend MK-N to PH job size distributions. We refer to the new approximation as DR-A. Below, we provide more details of the three approximations.