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November  2020, 16(6): 2813-2842. doi: 10.3934/jimo.2019082

## Analysis of the queue lengths in a priority retrial queue with constant retrial policy

 1 Department of Telecommunications and Information Processing, Ghent University, Sint-Pietersnieuwstraat 41, 9000 Ghent, Belgium 2 Division of Policy and Planning Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8573, Japan

* Corresponding author: Arnaud Devos

Received  October 2018 Revised  March 2019 Published  July 2019

In this paper, we analyze a priority queueing system with a regular queue and an orbit. Customers in the regular queue have priority over the customers in the orbit. Only the first customer in the orbit (if any) retries to get access to the server, if the queue and server are empty (constant retrial policy). In contrast with existing literature, we assume different service time distributions for the high- and low-priority customers. Closed-form expressions are obtained for the probability generating functions of the number of customers in the queue and orbit, in steady-state. Another contribution is the extensive singularity analysis of these probability generating functions to obtain the stationary (asymptotic) probability mass functions of the queue and orbit lengths. Influences of the service times and the retrial policy are illustrated by means of some numerical examples.

Citation: Arnaud Devos, Joris Walraevens, Tuan Phung-Duc, Herwig Bruneel. Analysis of the queue lengths in a priority retrial queue with constant retrial policy. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2813-2842. doi: 10.3934/jimo.2019082
##### References:

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##### References:
Mean values of the queue and orbit lengths versus the mean class-2 service times ($\rho = 0.7$, $\beta_1 = 1$ and $\nu = 3$)
Mean values of the queue and orbit lengths versus the mean class-1 service times ($\rho = 0.7$, $\beta_2 = 1.5$ and $\nu = 3$)
Mean values of the queue and orbit lengths versus the mean class-1 service times ($\rho = 0.7$, $\beta_2 = 1.5$ and $R^*(s) = 1$)
Variances of the queue and orbit length versus the load ($\beta_1 = \beta_2 = 1.1$ and $\nu = 4$)
Regions for the tail behaviour as a function of the mean service times of both classes ($\lambda_1 = \lambda_2 = 0.3$ and $\nu = 3$)
Regions for the tail behaviour as a function of the mean service times of both classes ($\lambda_1 = \lambda_2 = 0.3$ and $R^*(s) = 1$)
Tail behaviour of the orbit length for different service time distributions ($\lambda_1 = \lambda_2 = 0.3$, $\beta_1 = 1.2$, $\beta_2 = 1$, $\nu = 3$ and $\alpha_1 = \alpha_2 = -2.5$)
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