Dominant poles and tail asymptotics in the critical Gaussian many-sources regime

TL;DR

This paper develops a novel dominant pole approximation method to derive tail asymptotics for a discrete queue in a critical Gaussian many-sources regime, overcoming limitations of classical DPA caused by pole clustering.

Contribution

A new DPA technique is introduced to handle pole clustering in Gaussian regimes, extending the applicability of asymptotic analysis methods.

Findings

Successfully derived tail asymptotics for the queue length.

Identified limitations of classical DPA in clustered pole scenarios.

Proposed a new DPA approach applicable to Gaussian scalings.

Abstract

The dominant pole approximation (DPA) is a classical analytic method to obtain from a generating function asymptotic estimates for its underlying coefficients. We apply DPA to a discrete queue in a critical many-sources regime, in order to obtain tail asymptotics for the stationary queue length. As it turns out, this regime leads to a clustering of the poles of the generating function, which renders the classical DPA useless, since the dominant pole is not sufficiently dominant. To resolve this, we design a new DPA method, which might also find application in other areas of mathematics, like combinatorics, particularly when Gaussian scalings related to the central limit theorem are involved.