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July  2015, 11(3): 715-731. doi: 10.3934/jimo.2015.11.715

## Optimal balking strategies in an M/G/1 queueing system with a removable server under N-policy

 1 College of Science, Yanshan University, Qinhuangdao, 066004, China, China 2 Department of Intelligence and Informatics, Konan University, Kobe 658-8501

Received  February 2013 Revised  June 2014 Published  October 2014

In this paper, we consider the balking behavior of customers in an M/G/1 queueing system with a removable server under N-policy, where the server may be turned off when no customers are present, and be turned on when the queue length reaches size $N$. Arriving customers decide whether to join the system or balk, based on a linear reward-cost structure that incorporates their desire for service, as well as their unwillingness for waiting. For the unobservable and partially observable queues, we first analyze the stationary behavior of the system; then derive the equilibrium mixed strategies and compare them to the socially optimal strategies. We take the number $N$ as a decision variable and discuss the optimal operations policy in equilibrium states. Finally, we present two examples to demonstrate some of the phenomena in the considered models.
Citation: Ruiling Tian, Dequan Yue, Wuyi Yue. Optimal balking strategies in an M/G/1 queueing system with a removable server under N-policy. Journal of Industrial & Management Optimization, 2015, 11 (3) : 715-731. doi: 10.3934/jimo.2015.11.715
##### References:
 [1] A. Burnetas and A. Economou, Equilibrium customer strategies in a single server Markovian queue with setup times,, Queueing Systems, 56 (2007), 213. doi: 10.1007/s11134-007-9036-7. [2] A. Economou, A. Gómez-Corral and S. Kanta, Optimal balking strategies in single-server queues with general service and vacation times,, Performance Evaluation, 68 (2011), 967. doi: 10.1016/j.peva.2011.07.001. [3] N. M. Edelson and D. K. Hildebrand, Congestion tolls for Poisson queueing processes,, Econometrica, 43 (1975), 81. doi: 10.2307/1913415. [4] A. Economou and S. Kanta, Equillibrium balking strategies in the observable single-server queue with breakdowns and repairs,, Operations Research Letters, 36 (2008), 696. doi: 10.1016/j.orl.2008.06.006. [5] A. Economou and S. Kanta, Optimal balking strategies and pricing for the single server Markovian queue with compartmented waiting space,, Queueing Systems, 59 (2008), 237. doi: 10.1007/s11134-008-9083-8. [6] P. Guo and R. Hassin, Strategic behavior and social optimization in Markovian vacation queues,, Operations Research, 59 (2011), 986. doi: 10.1287/opre.1100.0907. [7] P. 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. doi: 10.1016/j.ejor.2012.05.026. [8] P. Guo and Q. Li, Strategic behavior and social optimization in partially-observable Markovian vacation queues,, Operations Research Letters, 41 (2013), 277. doi: 10.1016/j.orl.2013.02.005. [9] P. Guo and P. Zipkin, Analysis and comparison of queues with different levels of delay information,, Management Science, 53 (2007), 962. doi: 10.1287/mnsc.1060.0686. [10] R. Hassin and M. Haviv, Equilibrium Behavior in Queueing Systems: To Queue or Not to Queue,, Kluwer Academic Publishers, (2003). doi: 10.1007/978-1-4615-0359-0. [11] M. Haviv and Y. Kerner, On balking from an empty queue,, Queueing Systems, 55 (2007), 239. doi: 10.1007/s11134-007-9020-2. [12] Y. Kerner, Equilibrium joining probabilities for an M/G/1 queue,, Games and Economic Behavior, 71 (2011), 521. doi: 10.1016/j.geb.2010.06.002. [13] L. Li, J. Wang and F. Zhang, Equilibrium customer strategies in Markovian queues with partial breakdowns,, Computers & Industrial Engineering, 66 (2013), 751. doi: 10.1016/j.cie.2013.09.023. [14] P. Naor, The regulation of queue size by levying tolls,, Econometrica, 37 (1969), 15. doi: 10.2307/1909200. [15] W. Stein, A. Rapoport, D. Seale, H. Zhang and R. Zwick, Batch queues with choice of arrivals: Equilibrium analysis and experimental study,, Games and Economic Behavior, 59 (2007), 345. doi: 10.1016/j.geb.2006.08.008. [16] W. Sun and N. Tian, Contrast of the equilibrium and socially optimal strategies in a queue with vacations,, Journal of Computational Information Systems, 4 (2008), 2167. [17] H. Takagi, Queueing Analysis-A Foundation of Performance Evaluation, Vol. 1: Vacation and Prioriry Systems, Part I,, North-Holland, (1991). [18] N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications,, Springer, (2006). [19] R. Tian and D. Yue, Optimal balking strategies in an Markvian queue with a single vacation,, Journal of Information and Computational Science, 9 (2012), 2827. [20] F. Zhang, J. Wang and B. Liu, Equilibrium joining probabilities in observable queues with general service and setup times,, Journal of Industrial and Management Optimization, 9 (2013), 901. doi: 10.3934/jimo.2013.9.901. [21] F. Zhang, J. Wang and B. Liu, Equilibrium balking strategies in Markovian queues with working vacations,, Applied Mathematical Modelling, 37 (2013), 8264. doi: 10.1016/j.apm.2013.03.049. [22] F. Zhang, J. Wang and B. Liu, On the optimal and equilibrium retrial rates in an unreliable retrial queue with vacations,, Journal of Industrial and Management Optimization, 8 (2012), 861. doi: 10.3934/jimo.2012.8.861.

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##### References:
 [1] A. Burnetas and A. Economou, Equilibrium customer strategies in a single server Markovian queue with setup times,, Queueing Systems, 56 (2007), 213. doi: 10.1007/s11134-007-9036-7. [2] A. Economou, A. Gómez-Corral and S. Kanta, Optimal balking strategies in single-server queues with general service and vacation times,, Performance Evaluation, 68 (2011), 967. doi: 10.1016/j.peva.2011.07.001. [3] N. M. Edelson and D. K. Hildebrand, Congestion tolls for Poisson queueing processes,, Econometrica, 43 (1975), 81. doi: 10.2307/1913415. [4] A. Economou and S. Kanta, Equillibrium balking strategies in the observable single-server queue with breakdowns and repairs,, Operations Research Letters, 36 (2008), 696. doi: 10.1016/j.orl.2008.06.006. [5] A. Economou and S. Kanta, Optimal balking strategies and pricing for the single server Markovian queue with compartmented waiting space,, Queueing Systems, 59 (2008), 237. doi: 10.1007/s11134-008-9083-8. [6] P. Guo and R. Hassin, Strategic behavior and social optimization in Markovian vacation queues,, Operations Research, 59 (2011), 986. doi: 10.1287/opre.1100.0907. [7] P. 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. doi: 10.1016/j.ejor.2012.05.026. [8] P. Guo and Q. Li, Strategic behavior and social optimization in partially-observable Markovian vacation queues,, Operations Research Letters, 41 (2013), 277. doi: 10.1016/j.orl.2013.02.005. [9] P. Guo and P. Zipkin, Analysis and comparison of queues with different levels of delay information,, Management Science, 53 (2007), 962. doi: 10.1287/mnsc.1060.0686. [10] R. Hassin and M. Haviv, Equilibrium Behavior in Queueing Systems: To Queue or Not to Queue,, Kluwer Academic Publishers, (2003). doi: 10.1007/978-1-4615-0359-0. [11] M. Haviv and Y. Kerner, On balking from an empty queue,, Queueing Systems, 55 (2007), 239. doi: 10.1007/s11134-007-9020-2. [12] Y. Kerner, Equilibrium joining probabilities for an M/G/1 queue,, Games and Economic Behavior, 71 (2011), 521. doi: 10.1016/j.geb.2010.06.002. [13] L. Li, J. Wang and F. Zhang, Equilibrium customer strategies in Markovian queues with partial breakdowns,, Computers & Industrial Engineering, 66 (2013), 751. doi: 10.1016/j.cie.2013.09.023. [14] P. Naor, The regulation of queue size by levying tolls,, Econometrica, 37 (1969), 15. doi: 10.2307/1909200. [15] W. Stein, A. Rapoport, D. Seale, H. Zhang and R. Zwick, Batch queues with choice of arrivals: Equilibrium analysis and experimental study,, Games and Economic Behavior, 59 (2007), 345. doi: 10.1016/j.geb.2006.08.008. [16] W. Sun and N. Tian, Contrast of the equilibrium and socially optimal strategies in a queue with vacations,, Journal of Computational Information Systems, 4 (2008), 2167. [17] H. Takagi, Queueing Analysis-A Foundation of Performance Evaluation, Vol. 1: Vacation and Prioriry Systems, Part I,, North-Holland, (1991). [18] N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications,, Springer, (2006). [19] R. Tian and D. Yue, Optimal balking strategies in an Markvian queue with a single vacation,, Journal of Information and Computational Science, 9 (2012), 2827. [20] F. Zhang, J. Wang and B. Liu, Equilibrium joining probabilities in observable queues with general service and setup times,, Journal of Industrial and Management Optimization, 9 (2013), 901. doi: 10.3934/jimo.2013.9.901. [21] F. Zhang, J. Wang and B. Liu, Equilibrium balking strategies in Markovian queues with working vacations,, Applied Mathematical Modelling, 37 (2013), 8264. doi: 10.1016/j.apm.2013.03.049. [22] F. Zhang, J. Wang and B. Liu, On the optimal and equilibrium retrial rates in an unreliable retrial queue with vacations,, Journal of Industrial and Management Optimization, 8 (2012), 861. doi: 10.3934/jimo.2012.8.861.
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