# American Institute of Mathematical Sciences

November  2021, 17(6): 3309-3332. doi: 10.3934/jimo.2020120

## Strategic joining in a single-server retrial queue with batch service

 1 Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, China 2 Department of Management Science, School of Management Science and Engineering, Central, University of Finance and Economics, Beijing, 100081, China 3 Department of Decision Sciences, Western Washington University, Bellingham, WA 98225, USA 4 Beedie School of Business, Simon Fraser University Burnaby, BC V5A 1S6, Canada

* Corresponding author: Jinting Wang

Received  November 2019 Revised  April 2020 Published  November 2021 Early access  June 2020

Fund Project: This work was supported in part by the National Natural Science Foundation of China (Grant nos. 71871008 and 71571014) and the Fundamental Research Funds for the Central Universities under grant 2019YJS196

We consider a single-server retrial queue with batch service where potential customers arrive according to a Poisson process. The service process has two stages: busy period and admission period, which are corresponding to whether the server is in service or not, respectively. In such an alternate renewal process, if arrivals find busy period, they make join-or-balk decisions. Those joining ones will stay in the orbit and attempt to get into server at a constant rate. While, if arrivals find the server is in an admission period, they get into the server directly. At the end of each admission period, all customers in the server will be served together regardless of the size of the batch. The reward of each arrival after completion of service depends on the service size. Furthermore, customers in the orbit fail to get into the server before the end of each service cycle will be forced to leave the system. We study an observable scenario that customers are informed about the service period upon arrivals and an unobservable case without this information. The Nash equilibrium joining strategies are identified and the social- and profit- maximization problems are obtained, respectively. Finally, the optimal joining strategies in the observable queue and the comparison of social welfare between the two queues are illustrated by numerical examples.

Citation: Ke Sun, Jinting Wang, Zhe George Zhang. Strategic joining in a single-server retrial queue with batch service. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3309-3332. doi: 10.3934/jimo.2020120
##### References:
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Gómez-Corral, Retrial Queueing Systems: A Computational Approach, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-78725-9. [7] N. T. J. Bailey, On queueing processes with bulk service, Journal of the Royal Statistical Society: Series B, 16 (1954), 80-87.  doi: 10.1111/j.2517-6161.1954.tb00149.x. [8] O. Bountali and A. Economou, Equilibrium joining strategies in batch service queueing systems, European Journal of Operational Research, 260 (2017), 1142-1151.  doi: 10.1016/j.ejor.2017.01.024. [9] O. Bountali and A. Economou, Equilibrium threshold joining strategies in partially observable batch service queueing systems, European Journal of Operational Research Annals of Operations Research, 260 (2017), 1142-1151.  doi: 10.1007/s10479-017-2630-0. [10] A. Brugno, MAP/PH/1 systems with group service: Performance analysis under different admission strategies, (2017). [11] A. Brugno, C. D'Apice, A. Dudin and R. Manzo, Analysis of an MAP/PH/1 queue with flexible group service, International Journal of Applied Mathematics and Computer Science, 27 (2017), 119-131.  doi: 10.1515/amcs-2017-0009. [12] A. Brugno, A. N. Dudin and R. Manzo, Analysis of a strategy of adaptive group admission of customers to single server retrial system, Journal of Ambient Intelligence and Humanized Computing, 9 (2018), 123-135.  doi: 10.1007/s12652-016-0419-7. [13] M. L. Chaudhry and J. G. C. Templeton, A First Course in Bulk Queues, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983. [14] R. K. Deb and R. F. Serfozo, Optimal control of batch service queues, Advances in Applied Probability, 5 (1973), 340-361.  doi: 10.2307/1426040. [15] F. Downton, Waiting time in bulk service queues, Journal of the Royal Statistical Society: Series B, 17 (1955), 256-261.  doi: 10.1111/j.2517-6161.1955.tb00199.x. [16] A. N. Dudin, R. Manzo and R. Piscopo, Single server retrial queue with group admission of customers, Comput. Oper. Res., 61 (2015), 89-99.  doi: 10.1016/j.cor.2015.03.008. [17] A. Economou and S. Kanta, Equilibrium customer strategies and social-profit maximization in the single-server constant retrial queue, Naval Research Logistics, 58 (2011), 107-122.  doi: 10.1002/nav.20444. [18] A. Economou and A. Manou, Equilibrium balking strategies for a clearing queueing system in alternating environment, Annals of Operations Research, 208 (2013), 489-514.  doi: 10.1007/s10479-011-1025-x. [19] N. M. Edelson and D. K. Hilderbrand, Congestion tolls for Poisson queuing processes, Econometrica, 43 (1975), 81-92.  doi: 10.2307/1913415. [20] G. I. Falin, The M/M/1 retrial queue with retrials due to server failures, Queueing Systems, 58 (2008), 155-160.  doi: 10.1007/s11134-008-9065-x. [21] G. Falin and J. Templeton, Retrial Queues, Chapman & Hall, 1997. [22] P. F. Guo and R. Hassin, Strategic behavior and social optimization in Markovian vacation queues: The case of heterogeneous customers, European Journal of Operational Research, 222 (2012), 278-286.  doi: 10.1016/j.ejor.2012.05.026. [23] R. Hassin and M. Haviv, To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems, International Series in Operations Research & Management Science, 59. Kluwer Academic Publishers, Boston, MA, 2003. doi: 10.1007/978-1-4615-0359-0. [24] R. Hassin, Rational Queueing, CRC Press, Boca Raton, FL, 2016. doi: 10.1201/b20014. [25] B. Kim and J. Kim, Analysis of the waiting time distribution for polling systems with retrials and glue periods, Annals of Operations Research, 277 (2019), 197-212.  doi: 10.1007/s10479-018-2800-8. [26] L. Kleinrock, Queueing Systems. Volume 2: Computer Applications (Vol.66), New York: Wiley, 1976. [27] X. Li, P. F. Guo and Z. T. Lian, Quality-speed competition in customer-intensive services with boundedly rational customers, Production and Operations Management, 25 (2016), 1885-1901.  doi: 10.1111/poms.12583. [28] X. Li, Q. Li, P. Guo and Z. Lian, On the uniqueness and stability of equilibrium in quality-speed competition with boundedly-rational customers: The case with general reward function and multiple servers, Production and Operations Management, 193 (2017), 726-736. [29] A. Manou, A. Economou and F. Karaesmen, Strategic customers in a transpotation station: When is it optimal to wait?, Operations Research, 62 (2014), 910-925.  doi: 10.1287/opre.2014.1280. [30] J. Medhi, Waiting time distribution in a Poisson queue with a general bulk service rule, Management Science, 21 (1975), 727-847.  doi: 10.1287/mnsc.21.7.777. [31] E. Morozov and T. Phung-Duc, Stability analysis of a multiclass retrial system with classical retrial policy, Performance Evaluation, 112 (2017), 15-26.  doi: 10.1016/j.peva.2017.03.003. [32] E. Morozov, A. Rumyantsev, S. Dey and T. Deepak, Performance analysis and stability of multiclass orbit queue with constant retrial rates and balking, Performance Evaluation, 134 (2019), 102005. [33] P. Naor, The regulation of queue size by levying tolls, Econometrica, 37 (1969), 15-24.  doi: 10.2307/1909200. [34] J. T. Wang and F. Zhang, Equilibrium analysis of the observable queues with balking and delayed repairs, Applied Mathematics and Computation, 218 (2011), 2716-2729.  doi: 10.1016/j.amc.2011.08.012. [35] J. T. Wang and F. Zhang, Strategic joining in M/M/1 retrial queues, European Journal of Operational Research, 230 (2013), 76-87.  doi: 10.1016/j.ejor.2013.03.030. [36] Y. Xu, A. Scheller-Wolf and K. Sycara, The benefit of introducing variability in single-server queues with application to quality-based service domains, Operations Research, 63 (2015), 233-246.  doi: 10.1287/opre.2014.1330. [37] X. Y. Xu, Z. T. Lian, X. Li and P. F. Guo, A Hotelling queue model with probabilistic service, Operations Research Letters, 44 (2016), 592-597.  doi: 10.1016/j.orl.2016.07.003.

show all references

##### References:
 [1] M. A. Abidini, O. Boxma, B. Kim, J. Kim and J. Resing, Performance analysis of polling systems with retrials and glue periods, Queueing Systems, 87 (2017), 293-324.  doi: 10.1007/s11134-017-9545-y. [2] M. A. Abidini, J.-P. Dorsman and J. Resing, Heavy traffic analysis of a polling model with retrials and glue periods, Stochastic Models, 34 (2018), 464-503.  doi: 10.1080/15326349.2018.1530601. [3] K. Anand and M. Paç, Quality-speed conundrum: Trade-offs in customer-intensive services, Management Science, 57 (2011), 40-56. [4] J. R. Artalejo, A queueing system with returning customers and waiting line, Operations Research Letters, 17 (1995), 191-199.  doi: 10.1016/0167-6377(95)00017-E. [5] J. R. Artalejo and A. Gómez-Corral, Steady state solution of a single-server queue with linear repeated requests, Journal of Applied Probability, 34 (1997), 223-233.  doi: 10.2307/3215189. [6] J. R. Artalejo and A. Gómez-Corral, Retrial Queueing Systems: A Computational Approach, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-78725-9. [7] N. T. J. Bailey, On queueing processes with bulk service, Journal of the Royal Statistical Society: Series B, 16 (1954), 80-87.  doi: 10.1111/j.2517-6161.1954.tb00149.x. [8] O. Bountali and A. Economou, Equilibrium joining strategies in batch service queueing systems, European Journal of Operational Research, 260 (2017), 1142-1151.  doi: 10.1016/j.ejor.2017.01.024. [9] O. Bountali and A. Economou, Equilibrium threshold joining strategies in partially observable batch service queueing systems, European Journal of Operational Research Annals of Operations Research, 260 (2017), 1142-1151.  doi: 10.1007/s10479-017-2630-0. [10] A. Brugno, MAP/PH/1 systems with group service: Performance analysis under different admission strategies, (2017). [11] A. Brugno, C. D'Apice, A. Dudin and R. Manzo, Analysis of an MAP/PH/1 queue with flexible group service, International Journal of Applied Mathematics and Computer Science, 27 (2017), 119-131.  doi: 10.1515/amcs-2017-0009. [12] A. Brugno, A. N. Dudin and R. Manzo, Analysis of a strategy of adaptive group admission of customers to single server retrial system, Journal of Ambient Intelligence and Humanized Computing, 9 (2018), 123-135.  doi: 10.1007/s12652-016-0419-7. [13] M. L. Chaudhry and J. G. C. Templeton, A First Course in Bulk Queues, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983. [14] R. K. Deb and R. F. Serfozo, Optimal control of batch service queues, Advances in Applied Probability, 5 (1973), 340-361.  doi: 10.2307/1426040. [15] F. Downton, Waiting time in bulk service queues, Journal of the Royal Statistical Society: Series B, 17 (1955), 256-261.  doi: 10.1111/j.2517-6161.1955.tb00199.x. [16] A. N. Dudin, R. Manzo and R. Piscopo, Single server retrial queue with group admission of customers, Comput. Oper. Res., 61 (2015), 89-99.  doi: 10.1016/j.cor.2015.03.008. [17] A. Economou and S. Kanta, Equilibrium customer strategies and social-profit maximization in the single-server constant retrial queue, Naval Research Logistics, 58 (2011), 107-122.  doi: 10.1002/nav.20444. [18] A. Economou and A. Manou, Equilibrium balking strategies for a clearing queueing system in alternating environment, Annals of Operations Research, 208 (2013), 489-514.  doi: 10.1007/s10479-011-1025-x. [19] N. M. Edelson and D. K. Hilderbrand, Congestion tolls for Poisson queuing processes, Econometrica, 43 (1975), 81-92.  doi: 10.2307/1913415. [20] G. I. Falin, The M/M/1 retrial queue with retrials due to server failures, Queueing Systems, 58 (2008), 155-160.  doi: 10.1007/s11134-008-9065-x. [21] G. Falin and J. Templeton, Retrial Queues, Chapman & Hall, 1997. [22] P. F. Guo and R. Hassin, Strategic behavior and social optimization in Markovian vacation queues: The case of heterogeneous customers, European Journal of Operational Research, 222 (2012), 278-286.  doi: 10.1016/j.ejor.2012.05.026. [23] R. Hassin and M. Haviv, To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems, International Series in Operations Research & Management Science, 59. Kluwer Academic Publishers, Boston, MA, 2003. doi: 10.1007/978-1-4615-0359-0. [24] R. Hassin, Rational Queueing, CRC Press, Boca Raton, FL, 2016. doi: 10.1201/b20014. [25] B. Kim and J. Kim, Analysis of the waiting time distribution for polling systems with retrials and glue periods, Annals of Operations Research, 277 (2019), 197-212.  doi: 10.1007/s10479-018-2800-8. [26] L. Kleinrock, Queueing Systems. Volume 2: Computer Applications (Vol.66), New York: Wiley, 1976. [27] X. Li, P. F. Guo and Z. T. Lian, Quality-speed competition in customer-intensive services with boundedly rational customers, Production and Operations Management, 25 (2016), 1885-1901.  doi: 10.1111/poms.12583. [28] X. Li, Q. Li, P. Guo and Z. Lian, On the uniqueness and stability of equilibrium in quality-speed competition with boundedly-rational customers: The case with general reward function and multiple servers, Production and Operations Management, 193 (2017), 726-736. [29] A. Manou, A. Economou and F. Karaesmen, Strategic customers in a transpotation station: When is it optimal to wait?, Operations Research, 62 (2014), 910-925.  doi: 10.1287/opre.2014.1280. [30] J. Medhi, Waiting time distribution in a Poisson queue with a general bulk service rule, Management Science, 21 (1975), 727-847.  doi: 10.1287/mnsc.21.7.777. [31] E. Morozov and T. Phung-Duc, Stability analysis of a multiclass retrial system with classical retrial policy, Performance Evaluation, 112 (2017), 15-26.  doi: 10.1016/j.peva.2017.03.003. [32] E. Morozov, A. Rumyantsev, S. Dey and T. Deepak, Performance analysis and stability of multiclass orbit queue with constant retrial rates and balking, Performance Evaluation, 134 (2019), 102005. [33] P. Naor, The regulation of queue size by levying tolls, Econometrica, 37 (1969), 15-24.  doi: 10.2307/1909200. [34] J. T. Wang and F. Zhang, Equilibrium analysis of the observable queues with balking and delayed repairs, Applied Mathematics and Computation, 218 (2011), 2716-2729.  doi: 10.1016/j.amc.2011.08.012. [35] J. T. Wang and F. Zhang, Strategic joining in M/M/1 retrial queues, European Journal of Operational Research, 230 (2013), 76-87.  doi: 10.1016/j.ejor.2013.03.030. [36] Y. Xu, A. Scheller-Wolf and K. Sycara, The benefit of introducing variability in single-server queues with application to quality-based service domains, Operations Research, 63 (2015), 233-246.  doi: 10.1287/opre.2014.1330. [37] X. Y. Xu, Z. T. Lian, X. Li and P. F. Guo, A Hotelling queue model with probabilistic service, Operations Research Letters, 44 (2016), 592-597.  doi: 10.1016/j.orl.2016.07.003.
The alternate renewal process in the retrial queue with batch service
Transition rate diagram of the observable retrial queue with batch service
Transition rate diagram of the unobservable retrial queue with batch service
The three joining probabilities in observable case with respect to $\lambda$, $\mu = 1, \gamma = 2, k = 1.5, \theta = 5$
The three joining probabilities in observable case with respect to $\mu$, $\lambda = 0.7, \gamma = 2, k = 1.5, \theta = 5$
The three joining probabilities in observable case with respect to $\gamma$, $\lambda = 0.8, \gamma = 2, k = 2, \theta = 5$
The comparison of social benefits with respect to $\lambda$, $\mu = 1, \gamma = 2, k = 2, \theta = 5$
The comparison of social benefits with respect to $\mu$, $\lambda = 0.7, \gamma = 2, k = 1.5, \theta = 5$
The comparison of social benefits with respect to $\gamma$, $\lambda = 0.8, \mu = 1, k = 1.5, \theta = 5$
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