The MacLaurin series for the GI/G/1 queue

Abstract

We derive the MacLaurin series for the moments of the system time and the delay with respect to the parameters in the service time or interarrival time distributions in the GI/G/1 queue. The coe cients in these series are expressed in terms of the derivatives of the interarrival time density function evaluated at zero and the moments of the service time distribution, which can be easily calculated through a simple recursive procedure. The light tra c derivatives can be obtained from these series. For the M/G/1 queue, we are able to recover the formulas for the moments of the system time and the delay, including the Pollaczek-Khinchin mean-value formula.

The mean waiting time of a GI/G/1 queue in light traffic via random thinning

“…They also extend their results to GI/G/k queues in [6]. In [8], the authors obtain a set of recursive formulas for the MacLaurin series of the moments of the waiting time ofa GI/G/I queue in light traffic via scaling; the issue of convergence of the series is discussed in [10]. In [9], the MacLaurin series of the mean waiting time of a GI /G /1 queue in light traffic via scaling is obtained in a closed form using a different method, and the authors also suggest the idea of applying Pade approximation to the series to extrapolate the mean waiting time beyond light traffic.…”

On the transient delays of M/G/1 queues

“…where X + = max{X, 0}. Recently, Gong and Hu (1992) derived the Maclaurin series for the moments of the transient delay with respect to the parameters in the service-time or interarrival-time distributions in the GI/G/1 queue. To be more specific, they show…”

“…In this paper, we too consider the M/G/1 queue and assume that it is initially empty, that is, D 0 = 0. We first take the approach in Gong and Hu (1992) to derive an identity on the moments of D n .…”

Tail probability of transient and stationary waiting times in (max,+)-linear systems

“…Ayhan and Seo [3] generalize the results of [6] There are several other approaches in the literature where series expansions are used as the key mathematical tool in the analysis of single-server queues. Gong and Hu [9] develop a MacLaurin series for the stationary waiting time in a GI=G=1 queue. This technique combined with the idea of computing a rational approximant from such MacLaurin expansions is applied to investigate various performance functions and system characteristics, see [13], [14], and [20].…”