Quantitative delay analysis of GI/G/1 queues with heavy-tailed traffic by means of Alpha Theory

Abstract

The GI/G/1 queue represents a versatile and crucial model for various telecommunication and computer systems applications, but its mathematical complexity makes it difficult to analyse its steady-state behaviour. As an alternative, bounds and approximations for the stationary mean delay have been proposed. However, when dealing with heavy-tailed traffic, even the use of these latter methods becomes problematic. Due to divergent second-order moments, they provide uninformative intervals or indeterminate/divergent estimates, and are therefore unfeasible or useless. This paper presents a new analytical methodology based on Alpha Theory, a development of NonStandard Analysis, to overcome these limitations. Our approach extends classical delay bounds and approximations tools, enabling direct numerical applicability even in challenging scenarios with infinite second-order moments. Additionally, utilizing recently introduced Bounded Algorithmic Numbers (a fixed-length representation format for numbers containing infinite and infinitesimal values, other than finite ones), a discrete-event simulation of the queue is presented. Various simulative tests, under several heavy-tailed traffic conditions and different scheduling policies, are carried out to investigate the behaviour of the queue in terms of mean delay. The obtained results agree with theoretical predictions, even in the case of infinite values. The implementation turns out to be accurate, ensuring satisfying convergence speed and numerical stability.