# American Institute of Mathematical Sciences

• Previous Article
Transient analysis of N-policy queue with system disaster repair preventive maintenance re-service balking closedown and setup times
• JIMO Home
• This Issue
• Next Article
A robust reduced-order observers design approach for linear discrete periodic systems
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:

show all references

##### 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 $\alpha_1 = \alpha_2 = -2.5$)
 [1] Yoshiaki Kawase, Shoji Kasahara. Priority queueing analysis of transaction-confirmation time for Bitcoin. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1077-1098. doi: 10.3934/jimo.2018193 [2] Zhanyou Ma, Wuyi Yue, Xiaoli Su. Performance analysis of a Geom/Geom/1 queueing system with variable input probability. Journal of Industrial & Management Optimization, 2011, 7 (3) : 641-653. doi: 10.3934/jimo.2011.7.641 [3] Włodzimierz M. Tulczyjew, Paweł Urbański. Regularity of generating families of functions. Journal of Geometric Mechanics, 2010, 2 (2) : 199-221. doi: 10.3934/jgm.2010.2.199 [4] Simone Vazzoler. A note on the normalization of generating functions. Journal of Geometric Mechanics, 2018, 10 (2) : 209-215. doi: 10.3934/jgm.2018008 [5] Dequan Yue, Wuyi Yue, Zsolt Saffer, Xiaohong Chen. Analysis of an M/M/1 queueing system with impatient customers and a variant of multiple vacation policy. Journal of Industrial & Management Optimization, 2014, 10 (1) : 89-112. doi: 10.3934/jimo.2014.10.89 [6] Shaojun Lan, Yinghui Tang. Performance analysis of a discrete-time $Geo/G/1$ retrial queue with non-preemptive priority, working vacations and vacation interruption. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1421-1446. doi: 10.3934/jimo.2018102 [7] Huiyan Xue, Antonella Zanna. Generating functions and volume preserving mappings. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 1229-1249. doi: 10.3934/dcds.2014.34.1229 [8] Lijin Wang, Jialin Hong. Generating functions for stochastic symplectic methods. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 1211-1228. doi: 10.3934/dcds.2014.34.1211 [9] Zhanyou Ma, Pengcheng Wang, Wuyi Yue. Performance analysis and optimization of a pseudo-fault Geo/Geo/1 repairable queueing system with N-policy, setup time and multiple working vacations. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1467-1481. doi: 10.3934/jimo.2017002 [10] Dhanya Shajin, A. N. Dudin, Olga Dudina, A. Krishnamoorthy. A two-priority single server retrial queue with additional items. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2891-2912. doi: 10.3934/jimo.2019085 [11] Sofian De Clercq, Koen De Turck, Bart Steyaert, Herwig Bruneel. Frame-bound priority scheduling in discrete-time queueing systems. Journal of Industrial & Management Optimization, 2011, 7 (3) : 767-788. doi: 10.3934/jimo.2011.7.767 [12] Gang Chen, Zaiming Liu, Jinbiao Wu. Optimal threshold control of a retrial queueing system with finite buffer. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1537-1552. doi: 10.3934/jimo.2017006 [13] Bara Kim. Stability of a retrial queueing network with different classes of customers and restricted resource pooling. Journal of Industrial & Management Optimization, 2011, 7 (3) : 753-765. doi: 10.3934/jimo.2011.7.753 [14] Arthur Fleig, Lars Grüne. Strict dissipativity analysis for classes of optimal control problems involving probability density functions. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020053 [15] Domingo Gómez-Pérez, László Mérai. Algebraic dependence in generating functions and expansion complexity. Advances in Mathematics of Communications, 2020, 14 (2) : 307-318. doi: 10.3934/amc.2020022 [16] Rakesh Nandi, Sujit Kumar Samanta, Chesoong Kim. Analysis of $D$-$BMAP/G/1$ queueing system under $N$-policy and its cost optimization. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020135 [17] Yi Peng, Jinbiao Wu. On the $BMAP_1, BMAP_2/PH/g, c$ retrial queueing system. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020124 [18] Madhu Jain, Sudeep Singh Sanga. Admission control for finite capacity queueing model with general retrial times and state-dependent rates. Journal of Industrial & Management Optimization, 2020, 16 (6) : 2625-2649. doi: 10.3934/jimo.2019073 [19] Ana Paula S. Dias, Paul C. Matthews, Ana Rodrigues. Generating functions for Hopf bifurcation with $S_n$-symmetry. Discrete & Continuous Dynamical Systems, 2009, 25 (3) : 823-842. doi: 10.3934/dcds.2009.25.823 [20] Noah H. Rhee, PaweŁ Góra, Majid Bani-Yaghoub. Predicting and estimating probability density functions of chaotic systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 297-319. doi: 10.3934/dcdsb.2017144

2019 Impact Factor: 1.366

## Tools

Article outline

Figures and Tables