Tail asymptotics for a generalized two-demand queueing model—a kernel method

Abstract

In this paper, we consider a generalized two-demand queueing model, the same model studied in Wright (Adv. Appl. Prob., 24, 986-1007, 1992). Using this model, we show how the kernel method can be applied to a two-dimensional queueing system for exact tail asymptotics in the stationary joint distribution and also in the two marginal distributions. We demonstrate in detail how to locate the dominant singularity and how to determine the detailed behavior of the unknown generating function around the dominant singularity for a bivariate kernel, which is much more challenging than the analysis for a one-dimensional kernel. This information is the key for characterizing exact tail asymptotics in terms of asymptotic analysis theory. This approach does not require a determination or presentation of the unknown generating function(s).

Light tail asymptotics in multidimensional reflecting processes for queueing networks

“…This method can also be used to get exact asymptotics, but it has been limited to relatively simple models such as tandem or parallel queues (e.g., see Flatto and McKean 1977). Its recent studies can be found in Guillemin and van Leeuwaarden (2011) and Li and Zhao (2010), where the approach is called a kernel method.…”

Tail Asymptotics of the Stationary Distribution of a Two-Dimensional Reflecting Random Walk with Unbounded Upward Jumps

“…This analytic approach is called the kernel method (see, e.g. [18], [23], and [26]). In the kernel method, the generating function is more convenient to use than the moment generating function.…”

“…In recent years this classical approach has been renewed in combination with other methods (see, e.g. [18] and [23]). However, its application is limited to skip-free processes because of technical reasons (see Remark 3).…”

Exact tail asymptotics for fluid models driven by an M/M/c queue

“…In this paper, we applied the kernel method to investigate exact tail asymptotic properties of the joint stationary probabilities and the marginal distribution of the fluid queue driven by an M/M/c queue. Different from the model studied in [11] and [10], for which tail asymptotic properties are symmetric between the level and the phase, since both level and phase processes are discrete, the tail asymptotics for φ c−1 (α) and ψ(z) are asymmetric in this paper, since the phase process is discrete and level process is continuous.…”