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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:
 [1] J. Abate and W. Whitt, Asymptotics for M/G/1 low-priority waiting-time tail probabilities, Queueing Systems, 25 (1997), 173-233.  doi: 10.1023/A:1019104402024.  Google Scholar [2] J. Abate, G. L. Choudhury and W. Whitt, Waiting-time tail probabilities in queues with long-tail service-time distributions, Queueing Systems, 16 (1994), 311-338.  doi: 10.1007/BF01158960.  Google Scholar [3] J. R. Artalejo, Accessible bibliography on retrial queues: Progress in 2000–2009, Mathematical and Computer Modelling, 51 (2010), 1071-1081.  doi: 10.1016/j.mcm.2009.12.011.  Google Scholar [4] J. R. Artalejo and A. Gómez-Corral, Retrial Queueing Systems, Springer-Verlag Berlin Heidelberg, 2008. doi: 10.1007/978-3-540-78725-9.  Google Scholar [5] I. Atencia and P. Moreno, A single-server retrial queue with general retrial times and Bernoulli schedule, Applied Mathematics and Computation, 162 (2005), 855-880.  doi: 10.1016/j.amc.2003.12.128.  Google Scholar [6] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, 1986. doi: 10.1017/CBO9780511721434.  Google Scholar [7] B. D. Choi, K. K. Park and C. E. M. Pearce, An M/M/1 retrial queue with control policy and general retrial times, Queueing Systems, 14 (1993), 275-292.  doi: 10.1007/BF01158869.  Google Scholar [8] B. D. Choi and Y. Chang, Single server retrial queues with priority calls, Mathematical and Computer Modelling, 30 (1999), 7-32.  doi: 10.1016/S0895-7177(99)00129-6.  Google Scholar [9] A. Devos, J. Walraevens and H. Bruneel, A priority retrial queue with constant retrial policy, International Conference on Queueing Theory and Network Applications, (2018), 3–21. doi: 10.1007/978-3-319-93736-6_1.  Google Scholar [10] G. Falin and J. G. Templeton, Retrial Queues, CRC Press, 1997. doi: 10.1007/978-1-4899-2977-8.  Google Scholar [11] K. Farahmand, Single line queue with repeated demands, Queueing Systems, 6 (1990), 223-228.  doi: 10.1007/BF02411475.  Google Scholar [12] G. Fayolle, A simple telephone exchange with delayed feedbacks, Proc. of the International Seminar on Teletraffic Analysis and Computer Performance Evaluation, (1986), 245–253. Google Scholar [13] P. Flajolet and A. Odlyzko, Singularity analysis of generating functions, SIAM Journal on Discrete Mathematics, 3 (1990), 216-240.  doi: 10.1137/0403019.  Google Scholar [14] P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009. doi: 10.1017/CBO9780511801655.  Google Scholar [15] S. Gao, A preemptive priority retrial queue with two classes of customers and general retrial times, Operational Research, 15 (2015), 233-251.  doi: 10.1007/s12351-015-0175-z.  Google Scholar [16] A. Gómez-Corral, Stochastic analysis of a single server retrial queue with general retrial times, Naval Research Logistics (NRL), 46 (1999), 561-581.  doi: 10.1002/(SICI)1520-6750(199908)46:5<561::AID-NAV7>3.0.CO;2-G.  Google Scholar [17] J. Kim, B. Kim and S. S. Ko, Tail asymptotics for the queue size distribution in an M/G/1 retrial queue, Journal of Applied Probability, 44 (2007), 1111-1118.  doi: 10.1239/jap/1197908829.  Google Scholar [18] H. Li and Y. Q. Zhao, Exact tail asymptotics in a priority queue characterizations of the non-preemptive model, Queueing Systems, 68 (2011), 165-192.  doi: 10.1007/s11134-011-9252-z.  Google Scholar [19] T. Maertens, J. Walraevens and H. Bruneel, Priority queueing systems: from probability generating functions to tail probabilities, Queueing Systems, 55 (2007), 27-39.  doi: 10.1007/s11134-006-9003-8.  Google Scholar [20] T. Phung-Duc, W. Rogiest, Y. Takahashi and H. Bruneel, Retrial queues with balanced call blending: Analysis of single-server and multiserver model, Annals of Operations Research, 239 (2016), 429-449.  doi: 10.1007/s10479-014-1598-2.  Google Scholar [21] T. Phung-Duc and W. Rogiest, Two way communication retrial queues with balanced call blending, International Conference on Analytical and Stochastic Modeling Techniques and Applications, (2012), 16–31. doi: 10.1007/978-3-642-30782-9_2.  Google Scholar [22] W. Shang, L. Liu and Q. Li, Tail asymptotics for the queue length in an M/G/1 retrial queue, Queueing Systems, 52 (2006), 193-198.  doi: 10.1007/s11134-006-5223-1.  Google Scholar [23] J. Walraevens, Discrete-Time Queueing Models With Priorities, Ph.D thesis, Ghent University, 2004. Google Scholar [24] J. Walraevens, D. Claeys and T. Phung-Duc, Asymptotics of queue length distributions in priority retrial queues, arXiv: 1801.06993, (2018). doi: 10.1016/j.peva.2018.10.004.  Google Scholar [25] J. Walraevens and B. Steyaert and H. Bruneel, Performance analysis of a single-server ATM queue with a priority scheduling, Computers & Operations Research, 30 (2003), 2003. doi: 10.1007/s10479-006-0053-4.  Google Scholar

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##### References:
 [1] J. Abate and W. Whitt, Asymptotics for M/G/1 low-priority waiting-time tail probabilities, Queueing Systems, 25 (1997), 173-233.  doi: 10.1023/A:1019104402024.  Google Scholar [2] J. Abate, G. L. Choudhury and W. Whitt, Waiting-time tail probabilities in queues with long-tail service-time distributions, Queueing Systems, 16 (1994), 311-338.  doi: 10.1007/BF01158960.  Google Scholar [3] J. R. Artalejo, Accessible bibliography on retrial queues: Progress in 2000–2009, Mathematical and Computer Modelling, 51 (2010), 1071-1081.  doi: 10.1016/j.mcm.2009.12.011.  Google Scholar [4] J. R. Artalejo and A. Gómez-Corral, Retrial Queueing Systems, Springer-Verlag Berlin Heidelberg, 2008. doi: 10.1007/978-3-540-78725-9.  Google Scholar [5] I. Atencia and P. Moreno, A single-server retrial queue with general retrial times and Bernoulli schedule, Applied Mathematics and Computation, 162 (2005), 855-880.  doi: 10.1016/j.amc.2003.12.128.  Google Scholar [6] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, Cambridge University Press, 1986. doi: 10.1017/CBO9780511721434.  Google Scholar [7] B. D. Choi, K. K. Park and C. E. M. Pearce, An M/M/1 retrial queue with control policy and general retrial times, Queueing Systems, 14 (1993), 275-292.  doi: 10.1007/BF01158869.  Google Scholar [8] B. D. Choi and Y. Chang, Single server retrial queues with priority calls, Mathematical and Computer Modelling, 30 (1999), 7-32.  doi: 10.1016/S0895-7177(99)00129-6.  Google Scholar [9] A. Devos, J. Walraevens and H. Bruneel, A priority retrial queue with constant retrial policy, International Conference on Queueing Theory and Network Applications, (2018), 3–21. doi: 10.1007/978-3-319-93736-6_1.  Google Scholar [10] G. Falin and J. G. Templeton, Retrial Queues, CRC Press, 1997. doi: 10.1007/978-1-4899-2977-8.  Google Scholar [11] K. Farahmand, Single line queue with repeated demands, Queueing Systems, 6 (1990), 223-228.  doi: 10.1007/BF02411475.  Google Scholar [12] G. Fayolle, A simple telephone exchange with delayed feedbacks, Proc. of the International Seminar on Teletraffic Analysis and Computer Performance Evaluation, (1986), 245–253. Google Scholar [13] P. Flajolet and A. Odlyzko, Singularity analysis of generating functions, SIAM Journal on Discrete Mathematics, 3 (1990), 216-240.  doi: 10.1137/0403019.  Google Scholar [14] P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009. doi: 10.1017/CBO9780511801655.  Google Scholar [15] S. Gao, A preemptive priority retrial queue with two classes of customers and general retrial times, Operational Research, 15 (2015), 233-251.  doi: 10.1007/s12351-015-0175-z.  Google Scholar [16] A. Gómez-Corral, Stochastic analysis of a single server retrial queue with general retrial times, Naval Research Logistics (NRL), 46 (1999), 561-581.  doi: 10.1002/(SICI)1520-6750(199908)46:5<561::AID-NAV7>3.0.CO;2-G.  Google Scholar [17] J. Kim, B. Kim and S. S. Ko, Tail asymptotics for the queue size distribution in an M/G/1 retrial queue, Journal of Applied Probability, 44 (2007), 1111-1118.  doi: 10.1239/jap/1197908829.  Google Scholar [18] H. Li and Y. Q. Zhao, Exact tail asymptotics in a priority queue characterizations of the non-preemptive model, Queueing Systems, 68 (2011), 165-192.  doi: 10.1007/s11134-011-9252-z.  Google Scholar [19] T. Maertens, J. Walraevens and H. Bruneel, Priority queueing systems: from probability generating functions to tail probabilities, Queueing Systems, 55 (2007), 27-39.  doi: 10.1007/s11134-006-9003-8.  Google Scholar [20] T. Phung-Duc, W. Rogiest, Y. Takahashi and H. Bruneel, Retrial queues with balanced call blending: Analysis of single-server and multiserver model, Annals of Operations Research, 239 (2016), 429-449.  doi: 10.1007/s10479-014-1598-2.  Google Scholar [21] T. Phung-Duc and W. Rogiest, Two way communication retrial queues with balanced call blending, International Conference on Analytical and Stochastic Modeling Techniques and Applications, (2012), 16–31. doi: 10.1007/978-3-642-30782-9_2.  Google Scholar [22] W. Shang, L. Liu and Q. Li, Tail asymptotics for the queue length in an M/G/1 retrial queue, Queueing Systems, 52 (2006), 193-198.  doi: 10.1007/s11134-006-5223-1.  Google Scholar [23] J. Walraevens, Discrete-Time Queueing Models With Priorities, Ph.D thesis, Ghent University, 2004. Google Scholar [24] J. Walraevens, D. Claeys and T. Phung-Duc, Asymptotics of queue length distributions in priority retrial queues, arXiv: 1801.06993, (2018). doi: 10.1016/j.peva.2018.10.004.  Google Scholar [25] J. Walraevens and B. Steyaert and H. Bruneel, Performance analysis of a single-server ATM queue with a priority scheduling, Computers & Operations Research, 30 (2003), 2003. doi: 10.1007/s10479-006-0053-4.  Google Scholar
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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