A Discrete Queue, Fourier Sampling on Szegö Curves and Spitzer Formulas

Abstract

We consider a discrete-time multi-server queue for which the moments of the stationary queue length can be expressed in terms of series over the zeros in the closed unit disk of a queue-specific characteristic function. In many important cases these zeros can be considered to be located on a queue-specific curve, called generalized Szegö curve. By adopting a special parametrization of these Szegö curves, the relevant zeros occur as equidistant samples of a 2π-periodic function whose Fourier coefficients can be determined analytically. Thus the series occurring in the expressions for the moments can be written as Fourier aliasing series with terms given in analytic form. This gives rise to formulas for e.g. the mean and variance of the queue length that are reminiscent of Spitzer's identity for the moment generating function of the steady-state waiting time for a G/G/1 queue. Indeed, by considering the queue under investigation as a G/G/1 queue, the new formulas for the mean and variance also follow from Spitzer's identity. The approach in this paper can also be used to compute the probability distribution function of the queue length in analytic form.

Analytic Computation Schemes for the Discrete-Time Bulk Service Queue

“…In [22] the c l are shown to have exponential decay for β ≥ 1 (which covers in fact all practically relevant instances). It is further shown that for 0 ≤ β < 1 the c l have exponential decay if and only if…”

“…The roots lie inside J, and satisfy (22). Since |A(z)| 1/s < |z| for all z ∈ J, it follows from Rouché's theorem that for each w k , the function z − w k A 1/s (z) has as many zeros inside J as z.…”

“…Now assume that C z l−1 [A l/s (z)] decays exponentially. We shall sketch the proof that Condition 3.1 is valid; full details can be found in [22], proof of Lemma 4.1. We consider for w in a neighbourhood of 0 the equation…”

“…2 we plotted the S A,s for q = 0.82 < 2( √ 2 − 1), and the dots indicate the roots z k obtained by calculating the sum in (23) up to l = 50. When q is increased, such that q > 2( √ 2 − 1), S A,s turns from a smooth Jordan curve containing zero into two separate closed curves (see [22]), and (23) no longer holds.…”

“…We showed in [22] that (50) can be alternatively derived using Fourier sampling. We note here that He and Sohraby [21] also derived a root-free expression of a similar type for the stationary queue length distribution, using combinatorial arguments and Ballot theorems.…”