Exact tail asymptotics for the Israeli queue with retrials and non-persistent customers

Song, Yang; Lu, Huijun

doi:10.1016/j.orl.2021.08.006

Cited by 4 publications

(1 citation statement)

References 12 publications

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“…The Tauberian-like theorems are developed from the work in analytic combinatorics, for which references can be found in Flajolet and Sedgewick [24]. The extended version of the kernel method has been successfully applied to various models such as: Li and Zhao [51] for the generalized two-demand queueing model; Li, Tavakoli and Zhao [49] for genus 0 random walks in the quarter plane; Ye [80] and Ye, Li and Zhao [81] for a longer-queue-serve-first system; Zafari [82] for the generalized join-the-shortest-queue model considered earlier by several other researchers including [25], [35], [46] and [64]; Dai and Zhao [12] for a revisit of the wireless 3-hop networks with stealing considered in [28]; Song, Liu and Dai [72] for a discrete-time preemptive priority queueing model; Song, Liu and Zhao [73] for a revisit of the retrial queue with two input streams and two orbits considered by Avrachenkov, Nain and Yechiali [3]; Dai, Dawson and Zhao [9] for extending the kernel method to continuous random walks; Dai, Kong and Song [11] for a two-stage queue; Dai, Dawson and Zhao [10] for extending the kernel method to a 3-dimensional tandem queueing system; Li and Zhao [52] for a systematic treatment of the kernel method for random walks in the quarter plane; Li, Liu and Zhao [53] for a 2-demand model modulated by a two-state Markov chain, which is a special case of the random walk in the quarter plane modulate by a finite-state Markov chain; Song and Lu [74] for the Israeli queue with retrials and non-persistent customers.…”

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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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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The Israeli queue with a capacitated server: modeling and approximations

Perel,

Kaspi

2024

Ann Oper Res

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The Israeli Queue is a batch service polling system where a single server attends to multiple queues based on seniority. Each arriving customer belongs to one of several classes. Upon arrival, a customer either joins an existing queue for their class or initiates a new queue if they are the first of their class to arrive. Customers from the class with the most senior member are served together as a batch, with the service time remaining constant regardless of the batch size. This service model is found in applications like advanced elevator systems and on-demand shared mobility, where passengers heading to the same destination can share a ride. However, in many real-world scenarios, the vehicle capacities are small and constraining, which calls for a deeper exploration of the Israeli queue with a capacitated server (IQCS). In this paper, we formally define the IQCS and address the challenges of creating a mathematically tractable model to represent it. To approximate the IQCS, we develop a quasi-birth-death process and derive approximations for key performance measures. To validate our approach, we implement a simulation model and use it to compare the IQCS, the approximate model, and the original Israeli Queue. Our results across various scenarios demonstrate the accuracy of the approximate model. Nonetheless, the presence of a remaining gap underscores the ongoing challenge of precisely and efficiently modeling the IQCS, posing an open question for the research community.

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Discrete-Time Informal Queue with an Infinite Number of Groups for Resource Management of a Data Center

Goswami,

Mund

2024

Methodol Comput Appl Probab

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Exact tail asymptotics for the Israeli queue with retrials and non-persistent customers

Cited by 4 publications

References 12 publications

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

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

The Israeli queue with a capacitated server: modeling and approximations

Discrete-Time Informal Queue with an Infinite Number of Groups for Resource Management of a Data Center

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