Analysis of a batch arrival retrial queue with impatient customers subject to the server disasters

Yi Peng; Jinbiao Wu

doi:10.3934/jimo.2020067

Article Contents

2021, Volume 17, Issue 4: 2243-2264. Doi: 10.3934/jimo.2020067

This issue Previous Article A complementarity model and algorithm for direct multi-commodity flow supply chain network equilibrium problem Next Article Convergence properties of inexact Levenberg-Marquardt method under Hölderian local error bound

Analysis of a batch arrival retrial queue with impatient customers subject to the server disasters

Yi Peng^1, and
Jinbiao Wu^2, ,

1.
School of Mathematical Science, Changsha Normal University, Changsha 410100, Hunan, China
2.
School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, China

^* Corresponding author: Jinbiao Wu
^* Corresponding author: Jinbiao Wu

Received: August 2019

Revised: January 2020

Early access: March 2020

Published: July 2021

The first author is supported by Provincial Natural Science Foundation of Hunan under Grant 2019JJ50677 and the Program of Hehua Excellent Young Talents of Changsha Normal University

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Abstract

We consider an $ M^X/G/1 $ retrial queue with impatient customers subject to disastrous failures at which times all customers in the system are lost. When the server finishes serving a customer and finds the orbit empty, the server becomes dormant until $ N $ or more customers accumulate. If a coming batch of customers finds the server idle, one of the arriving customers begins his service immediately and the rest join the orbit to repeat their request later. Otherwise, if the server is dormant or busy or down, all customers of the coming batch enter the orbit. When the server is under repair, customers in the orbit can become impatient after waiting a random amount of time and leave the system. By using the characteristic method for partial differential equations, the steady-state distributions of the server state and the number of customers in the orbit are obtained along with various performance measures. In addition, the reliability of the system is analyzed detailed. Finally, an application to telecommunication networks is provided and the effects of various parameters on the system performance are demonstrated numerically.

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

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Figure 1. The mean system size versus $ p $ with $ (\alpha, \delta, \theta) = (0.1, 3, 0.7) $

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Figure 2. The mean system size versus $ \delta $ with $ (\alpha, p, \theta) = (0.1, 0.1, 0.7) $

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Figure 3. The mean system size versus $ \theta $ with $ (\alpha, p, \delta) = (0.1, 0.1, 3) $

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Figure 4. The mean system size versus $ \alpha $ with $ (\theta, p, \delta) = (0.7, 0.1, 3) $

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