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In this paper, we provide a review on the kernel method, which is one of theoptions for characterizing so-called exact tail asymptotic properties instationary probabilities of two-dimensional random walks, discrete orcontinuous (or mixed), in the quarter plane. Many two-dimensional queueingsystems can be modelled via these types of random walks. Stationaryprobabilities are one of the most sought statistical quantities in queueinganalysis. However, explicit expressions are available only for a very limitednumber of models. Therefore, tail asymptotic properties become more important,since they provide insightful information into the structure of the tailprobabilities, and often lead to approximations, performance bounds,algorithms, among possible others.  Characterizing tail asymptotics for random walks in the quarter plane is afundamental and also classical problem. Classical approaches are usually basedon a complete determination of the transformation for the unknown probabilitiesof interest, for example, a singular integral presentation for the unknownprobability generating function through boundary value problems\cite{FKM:82,Guillemin-Leeuwaarden:09}. In contrast to classical approaches(approaches based on the solution for the unknown probabilities or thetransform of the unknown probabilities), the kernel method, reviewed here, isvery efficient for two-dimensional problems, which only requires the localinformation about the location of the dominant singularity of the unknowntransformation function and the asymptotic property, through asymptoticanalysis in complex analysis, at the dominant singularity.  This kernel method reviewed in this paper is an extension of the classicalone.

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