Geometric decay in level-expanding QBD models

Li, Liming; Miyazawa, Masakiyo; Zhao, Yiqiang Q.

doi:10.1007/s10479-007-0298-6

Cited by 4 publications

(4 citation statements)

References 7 publications

(11 reference statements)

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“…Once again, consider the RGfactorization in (15). Since all states in the process corresponding to the substochastic matrix are transient (for example, in Zhao [33] ), it follows from Proposition 5.3 in Kemeny et al [9] that k k < ∞, or I − is invertible.…”

Section: Proofsmentioning

confidence: 99%

“…Without this assumption, Kroese et al [11] provided a sufficient condition for exact geometric decay, which has been improved in Motyer and Taylor [22] , and Li et al [14] . Based on the above studies, Liu et al [15] obtained two sufficient conditions for the level-expanding QBD model. Applying these conditions, researchers are able to characterize exact geometric tail asymptotics in several two-dimensional queueing systems, such as join-the-shortest-queue models (Takahashi et al [28] , Haque [6] , Sakuma et al [26] , and Li et al [14] ), priority systems (Haque [6] , Miyazawa and Zhao [19] , and Xue and Alfa [32] ), the parallel queues with two types of demands (Haque [6] ), tandem queues (Haque [6] , Kroese et al [11] , and Tang and Zhao [29] ), a retrial queue (Li and Zhao [13] ), among possible others.…”

Section: Introductionmentioning

confidence: 99%

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Light-Tailed Behavior in QBD Processes with Countably Many Phases

Zhao

2009

Stochastic Models

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Generally speaking, analysis of tail asymptotics in two-dimensional queueing systems is very challenging. Earlier work based on complex analysis led to determinations of exact forms of tail asymptotics. Ideas of large deviations, a powerful tool for characterizing light-tailed decay rates or analysis of rough tail asymptotics, have been utilized recently to develop probabilistic methods to do exact tail asymptotic analysis. Another promising approach to do tail asymptotics analysis, both exact and rough, is the matrix-analytic method. In this article, we combine the matrixanalytic method with techniques from probability and analysis to characterize tail asymptotics in a QBD process with infinitely many phases. The main results include conditions on: (1) exact geometric decay; (2) light-tailed behavior without an exact geometric decay, which in general is not the focus of the large deviations method; and (3) upper and lower bounds for stationary probabilities. We apply the main results to two two-dimensional queueing systems, including a polling system and a gated random-order server queue to characterize their light-tailed behavior of the queue length processes.

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Section: Proofsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Light-Tailed Behavior in QBD Processes with Countably Many Phases

Zhao

2009

Stochastic Models

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“…For example, the compensation method was proposed by Adan [2], which provides a complete solution of the joint probability distribution in terms of an infinite series of product-form terms. Studies based on properties of Markov additive processes (including matrix-analytic methods) are vast, including McDonald [60], Takahashi, Fujimoto and Makimoto [75],, Foley and McDonald [25,26,27] (the exact tail asymptotics are based on detailed counting of the green function), Haque [31], Kroese, Scheinhardt and Taylor [43], Miyazawa [61], Miyazawa and Zhao [67], Haque, Liu and Zhao [32], Li and Zhao [50], Motyer and Taylor [68], Li, Miyazawa and Zhao [47], He, Li and Zhao [34], Liu, Miyazawa and Zhao [55], Tang and Zhao [76], Adan, Foley and McDonald [1], Khanchi [36,37], Kobayashi, Miyazawa and Zhao [42], Kobayashi and Miyazawa [41]. Not just for the rough decay, this type of method can also lead to conditions for exact geometric decay.…”

Section: Introductionmentioning

confidence: 99%

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

Zhao¹

2021

Preprint

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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 [18,29]. In contrast to classical approaches (approaches based on the solution for the unknown probabilities or the transform of the unknow 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, first introduced by Knuth [39] and further developed, more than 30 years after, by Banderier et al. [4], targeting at one-dimensional problems. The method combines analytic continuation with asymptotic analysis (for example, see [17,24]).

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“…Other methods for studying two-dimensional problems, including exact tail asymptotics, also exist, for example, based on large deviations, on properties of the Markov additive process (including matrixanalytic methods), or on asymptotic properties of the Green functions. References include Borovkov and Mogul'skii [5], McDonald [41], Foley and McDonald [17,18,19], Khanchi [24,25], Adan, Foley and McDonald [2], Raschel [52], Miyazawa [44,45,46], Kobayashi and Miyazawa [26], Takahashi, Fujimoto and Makimoto [53], Haque [21], Miyazawa [43], Miyazawa and Zhao [49], Kroese, Scheinhardt and Taylor [29], Haque, Liu and Zhao [22], Li and Zhao [33], Motyer and Taylor [51], Li, Miyazawa and Zhao [31], He, Li and Zhao [23], Liu, Miyazawa and Zhao [38], Tang and Zhao [54], Kobayashi, Miyazawa and Zhao [27], among others. For more references, people may refer to a recent survey on tail asymptotics of multi-dimensional reflecting processes for queueing networks by Miyazawa [47].…”

Section: Introductionmentioning

confidence: 99%

A Kernel Method for Exact Tail Asymptotics --- Random Walks in the Quarter Plane

Li¹,

Zhao²

2015

Preprint

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In this paper, we propose a kernel method for exact tail asymptotics of a random walk to neighborhoods in the quarter plane. This is a two-dimensional method, which does not require a determination of the unknown generating function(s). Instead, in terms of the asymptotic analysis and a Tauberian-like theorem, we show that the information about the location of the dominant singularity or singularities and the detailed asymptotic property at a dominant singularity is sufficient for the exact tail asymptotic behaviour for the marginal distributions and also for joint probabilities along a coordinate direction. We provide all details, not only for a "typical" case, the case with a single dominant singularity for an unknown generating function, but also for all non-typical cases which have not been studied before. A total of four types of exact tail asymptotics are found for the typical case, which have been reported in the literature. We also show that on the circle of convergence, an unknown generating function could have two dominant singularities instead of one, which can lead to a new periodic phenomena. Examples are illustrated by using this kernel method. This paper can be considered as a systematic summary and extension of existing ideas, which also contains new and interesting research results.

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Geometric decay in level-expanding QBD models

Cited by 4 publications

References 7 publications

Light-Tailed Behavior in QBD Processes with Countably Many Phases

Light-Tailed Behavior in QBD Processes with Countably Many Phases

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

A Kernel Method for Exact Tail Asymptotics --- Random Walks in the Quarter Plane

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