Queue length analysis of a Markov-modulated vacation queue with dependent arrival and service processes and exhaustive service policy

Gábor Horváth; Zsolt Saffer; Miklós Telek

doi:10.3934/jimo.2016077

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2017, Volume 13, Issue 3: 1365-1381. Doi: 10.3934/jimo.2016077

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Queue length analysis of a Markov-modulated vacation queue with dependent arrival and service processes and exhaustive service policy

Budapest University of Technology and Economics, Department of Networked Systems and Services, MTA-BME Information systems research group, Magyar Tudósok Körútja 2,1117 Budapest, Hungary

* Corresponding author: Gábor Horváth
* Corresponding author: Gábor Horváth

Received: August 31, 2015

Revised: May 31, 2016

Published: September 2017

The reviewing process of the paper was handled by Wuyi Yue and Yutaka Takahashi as Guest Editors.

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Abstract

The paper introduces a class of vacation queues where the arrival and service processes are modulated by the same Markov process, hence they can be dependent. The main result of the paper is the probability generating function for the number of jobs in the system. The analysis follows a matrix-analytic approach. A step of the analysis requires the evaluation of the busy period of a quasi birth death process with arbitrary initial level. This element can be useful in the analysis of other queueing models as well. We also discuss several special cases of the general model. We show that these special settings lead to simplification of the solution.

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Mathematics Subject Classification: Primary: 90B22; Secondary: 68M20, 60K25.

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Figure 1. Subset relations of the considered special vacation queue models

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Figure 2. Cycles in the evolution of the queue

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Figure 3. The mean number of jobs in the system

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Table 1. Vector β as a function of the vacation distribution

	Uniform	Exponential	Weibull
The general model	(0.546, 0.109, 0.345)	(0.543, 0.116, 0.341)	(0.539, 0.142, 0.319)
The MAP/MAP/1 vacation queue	(0.214, 0.097, 0.091, … 0.04, 0.382, 0.176)	(0.214, 0.097, 0.091, … 0.04, 0.382, 0.176)	(0.214, 0.097, 0.091, … 0.04, 0.382, 0.176)
QBD vac. queue	(0.546, 0.109, 0.345)	(0.543, 0.115, 0.342)	(0.53, 0.14, 0.33)
The indep. QBD vacation queue	(0.21, 0.101, 0.09, … 0.041, 0.373, 0.185)	(0.207, 0.104, 0.089, … 0.042, 0.367, 0.191)	(0.197, 0.113, 0.086, … 0.046, 0.348, 0.21)

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