  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
\cite{FKM:82,Guillemin-Leeuwaarden:09}. In contrast to classical approaches
(approaches based on the solution for the unknown probabilities or the
transform of the unknown 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.
