Symbolic Computation of Moments in Priority Queues

Analytical tools such as Laplace-Stieltjes transforms and z-transforms are commonly used to characterize queueing-theoretic quantities such as busy-period, waiting-time, and queue-size distributions. Many of these transforms, particularly in M/G/1 priority queueing applications, tend to be cumbersome expressions that involve both implicit and recursive functional relationships. Due to these complications, even the task of deriving moments becomes an algebraically intensive exercise. The focus of this paper is to describe a collection of efficient symbolic procedures useful for automating the tedious mathematical computations one encounters in working with these kinds of transforms. Central to this development is the introduction of a set-partition operator that enables moment expressions for delay cycles to be determined quickly and exactly, without having to derive any sort of higher-order derivative or Taylor-series expansion. Making use of this operator eliminates laborious derivations by hand and permits moments of various priority queueing-related quantities to be easily determined. In particular, the procedures are applied to the classical non-preemptive and preemptive resume queues, as well as two advanced variants of these models.