The kernel method tail asymptotics analytic approach for stationary probabilities of two-dimensional queueing systems

• Published in: Queueing systems Vol. 100; no. 1-2; pp. 95 - 131
• Main Author: Zhao, Yiqiang Q.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.02.2022; Springer Nature B.V
• Subjects: Algorithms; Asymptotic methods; Asymptotic properties; Business and Management; Combinatorial analysis; Combinatorics; Computer Communication Networks; Control; Kernels; Operations Research/Decision Theory; Probability; Probability Theory and Stochastic Processes; Queuing theory; Random walk; Supply Chain Management; Systems Theory; Two dimensional models; 60J10; Fundamental form; Analytic continuation; Generating functions; 30E15; Random walks in the quarter plane; Laplace-transform; Exact tail asymptotics; Kernel method; Stationary probabilities; Dominant singularity; 60K25; Asymptotic analysis; 05A15; Two-dimensional Markov chains
• ISSN: 0257-0130; 1572-9443
• DOI: 10.1007/s11134-021-09727-6

Many queueing systems can be modelled as two-dimensional random walks with reflective boundaries, discrete, continuous or mixed. Stationary probabilities are one of the most sought after statistical quantities in queueing analysis. However, explicit expressions are only available for a very limited number of models. Therefore, tail asymptotic properties become more important, since they provide insightful information on the structure of the tail probabilities, and often lead to approximations, performance bounds, algorithms, among possible other applications. In this survey, we provide key ideas of a kernel method, developed from the classical kernel method in analytic combinatorics, for studying so-called exact tail asymptotic properties in stationary probabilities for this type of random walk.