2021/01/27 by Yiqiang Q. Zhao, Zhao, Yiqiang Q.
Decision Sciences · Business, Management and Accounting · Mathematics · #Probability and Risk Models #Advanced Queuing Theory Analysis #Random Matrices and Applications
paper · pdf · doi:10.48550/arxiv.2101.11661
In this paper, we provide a review on the kernel method, which is one of the\noptions for characterizing so-called exact tail asymptotic properties in\nstationary probabilities of two-dimensional random walks, discrete or\ncontinuous (or mixed), in the quarter plane. Many two-dimensional queueing\nsystems can be modelled via these types of random walks. Stationary\nprobabilities are one of the most sought statistical quantities in queueing\nanalysis. However, explicit expressions are available only for a very limited\nnumber of models. Therefore, tail asymptotic properties become more important,\nsince they provide insightful information into the structure of the tail\nprobabilities, and often lead to approximations, performance bounds,\nalgorithms, among possible others.\n Characterizing tail asymptotics for random walks in the quarter plane is a\nfundamental and also classical problem. Classical approaches are usually based\non a complete determination of the transformation for the unknown probabilities\nof interest, for example, a singular integral presentation for the unknown\nprobability generating function through boundary value problems\n citeFKM:82,Guillemin-Leeuwaarden:09. In contrast to classical approaches\n(approaches based on the solution for the unknown probabilities or the\ntransform of the unknown probabilities), the kernel method, reviewed here, is\nvery efficient for two-dimensional problems, which only requires the local\ninformation about the location of the dominant singularity of the unknown\ntransformation function and the asymptotic property, through asymptotic\nanalysis in complex analysis, at the dominant singularity.\n This kernel method reviewed in this paper is an extension of the classical\none.\n