Kernel Method -- An Analytic Approach for Tail Asymptotics in Stationary Probabilities of 2-Dimensional Queueing Systems

In this paper, we provide a review on the kernel method, which is one of the options for characterizing so-called exact tail asymptotic properties in stationary probabilities of two-dimensional random walks, discrete or continuous (or mixed), in the quarter plane. Many two-dimensional queueing systems can be modelled via these types of random walks. Stationary probabilities are one of the most sought statistical quantities in queueing analysis. However, explicit expressions are available only for a very limited number of models. Therefore, tail asymptotic properties become more important, since they provide insightful information into the structure of the tail probabilities, and often lead to approximations, performance bounds, algorithms, among possible others. Characterizing tail asymptotics for random walks in the quarter plane is a fundamental and also classical problem. Classical approaches are usually based on a complete determination of the transformation for the unknown probabilities of interest, for example, a singular integral presentation for the unknown probability generating function through boundary value problems [18, 29]. In contrast to classical approaches (approaches based on the solution for the unknown probabilities or the transform of the unknow probabilities), the kernel method, reviewed here, is very efficient for two-dimensional problems, which only requires the local information about the location of the dominant singularity of the unknown transformation function and the asymptotic property, through asymptotic analysis in complex analysis, at the dominant singularity. This kernel method reviewed in this paper is an extension of the classical one, first introduced by Knuth [39] and further developed, more than 30 years after, by Banderier et al. [4], targeting at one-dimensional problems. The method combines analytic continuation with asymptotic analysis (for example, see [17, 24]).