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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  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 & Management Optimization, doi: 10.3934/jimo.2020120
References:
[1]

M. A. AbidiniO. BoxmaB. KimJ. 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.  Google Scholar

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M. A. AbidiniJ.-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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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A. Brugno, MAP/PH/1 systems with group service: Performance analysis under different admission strategies, (2017). Google Scholar

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A. BrugnoC. D'ApiceA. 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.  Google Scholar

[12]

A. BrugnoA. 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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

A. N. DudinR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

N. M. Edelson and D. K. Hilderbrand, Congestion tolls for Poisson queuing processes, Econometrica, 43 (1975), 81-92.  doi: 10.2307/1913415.  Google Scholar

[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.  Google Scholar

[21]

G. Falin and J. Templeton, Retrial Queues, Chapman & Hall, 1997. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

R. Hassin, Rational Queueing, CRC Press, Boca Raton, FL, 2016. doi: 10.1201/b20014.  Google Scholar

[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.  Google Scholar

[26]

L. Kleinrock, Queueing Systems. Volume 2: Computer Applications (Vol.66), New York: Wiley, 1976. Google Scholar

[27]

X. LiP. 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.  Google Scholar

[28]

X. LiQ. LiP. 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.   Google Scholar

[29]

A. ManouA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[33]

P. Naor, The regulation of queue size by levying tolls, Econometrica, 37 (1969), 15-24.  doi: 10.2307/1909200.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

Y. XuA. 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.  Google Scholar

[37]

X. Y. XuZ. T. LianX. 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.  Google Scholar

show all references

References:
[1]

M. A. AbidiniO. BoxmaB. KimJ. 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.  Google Scholar

[2]

M. A. AbidiniJ.-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.  Google Scholar

[3]

K. Anand and M. Paç, Quality-speed conundrum: Trade-offs in customer-intensive services, Management Science, 57 (2011), 40-56.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

A. Brugno, MAP/PH/1 systems with group service: Performance analysis under different admission strategies, (2017). Google Scholar

[11]

A. BrugnoC. D'ApiceA. 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.  Google Scholar

[12]

A. BrugnoA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

A. N. DudinR. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

N. M. Edelson and D. K. Hilderbrand, Congestion tolls for Poisson queuing processes, Econometrica, 43 (1975), 81-92.  doi: 10.2307/1913415.  Google Scholar

[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.  Google Scholar

[21]

G. Falin and J. Templeton, Retrial Queues, Chapman & Hall, 1997. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

R. Hassin, Rational Queueing, CRC Press, Boca Raton, FL, 2016. doi: 10.1201/b20014.  Google Scholar

[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.  Google Scholar

[26]

L. Kleinrock, Queueing Systems. Volume 2: Computer Applications (Vol.66), New York: Wiley, 1976. Google Scholar

[27]

X. LiP. 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.  Google Scholar

[28]

X. LiQ. LiP. 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.   Google Scholar

[29]

A. ManouA. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[33]

P. Naor, The regulation of queue size by levying tolls, Econometrica, 37 (1969), 15-24.  doi: 10.2307/1909200.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

Y. XuA. 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.  Google Scholar

[37]

X. Y. XuZ. T. LianX. 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.  Google Scholar

Figure 1.  The alternate renewal process in the retrial queue with batch service
Figure 2.  Transition rate diagram of the observable retrial queue with batch service
Figure 3.  Transition rate diagram of the unobservable retrial queue with batch service
Figure 4.  The three joining probabilities in observable case with respect to $ \lambda $, $ \mu = 1, \gamma = 2, k = 1.5, \theta = 5 $
Figure 5.  The three joining probabilities in observable case with respect to $ \mu $, $ \lambda = 0.7, \gamma = 2, k = 1.5, \theta = 5 $
Figure 6.  The three joining probabilities in observable case with respect to $ \gamma $, $ \lambda = 0.8, \gamma = 2, k = 2, \theta = 5 $
Figure 7.  The comparison of social benefits with respect to $ \lambda $, $ \mu = 1, \gamma = 2, k = 2, \theta = 5 $
Figure 8.  The comparison of social benefits with respect to $ \mu $, $ \lambda = 0.7, \gamma = 2, k = 1.5, \theta = 5 $
Figure 9.  The comparison of social benefits with respect to $ \gamma $, $ \lambda = 0.8, \mu = 1, k = 1.5, \theta = 5 $
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