Analysis of a batch service multi-server polling system with dynamic service control

Tao Jiang; Liwei Liu

doi:10.3934/jimo.2017073

Article Contents

2018, Volume 14, Issue 2: 743-757. Doi: 10.3934/jimo.2017073

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Analysis of a batch service multi-server polling system with dynamic service control

Tao Jiang^1, , and
Liwei Liu^2,

1.
College of Economics and Management, Shandong University of Science and Technology, Qingdao, Shandong, 266590, China
2.
Department of Mathematics, School of Science, Nanjing University of Science and Technology, Nanjing, Jiangsu, 210094, China

* Corresponding author: Tao Jiang
* Corresponding author: Tao Jiang

Received: March 31, 2016

Revised: August 31, 2016

Early access: August 2017

Published: March 2018

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Abstract

This paper considers a multi-server polling system with batch service of an unlimited size, i.e., the so called "Israeli queue" with multi-server, where the service rate of each server switches between a low and a high value depending on the number of groups standing in front of the servers upon its service completion. By means of matrix geometric method and LU-type RG factorization of the infinitesimal generator in irreducible QBD process, the explicit closed-form of rate matrix $R$ and the steady state distribution of the queue length are respectively derived. In terms of the results, some stationary performance measures are obtained. In addition, some numerical examples are presented.

Keywords:

Mathematics Subject Classification: Primary: 60K25; Secondary: 90B22.

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Figure 1. $L_q$ versus $\lambda$ ($p = 0.6, \theta= 0.2, \mu_1=3$)

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Figure 2. $L_q$ versus $\mu_1$ ($\lambda=3, p = 0.6, \theta= 0.2, $)

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Figure 3. $L_q$ versus $p$ ($\lambda=3, \mu_1=3, \theta= 0.2, $)

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Figure 4. $L_q$ versus $p$ ($\lambda=3, \mu_1=3, p= 0.6, $)

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