Asymptotic Analysis of Maximum Queue Length in Fluid Systems with Self-Similar Gaussian Inputs

In this paper, we study the asymptotic behavior of the maximum queue length in a fluid queue driven by a self-similar Gaussian process with non-stationary increments. We derive almost sure and in-probability convergence of the normalized maximum queue length, described by a function [Formula: see text], where [Formula: see text] is the Hurst parameter. We obtain sharp tail bounds for the distribution of the queue length, using Malliavin calculus, which explicitly depend on [Formula: see text] and the drift parameter [Formula: see text]. Our results generalize classical results on fractional Brownian motion (fBm) to broader classes such as sub-fractional Brownian motion (sub-fBm) and bi-fractional Brownian motion (bi-fBm). The key techniques introduced in this framework include Gaussian supremum localization, path-penalty large deviations, and metric entropy. Systems that exhibit long-range dependence and self-similarity are developed based on these results.