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EQUILIBRIUM BALKING STRATEGIES IN THE REPAIRABLE M/M/1 G-RETRIAL QUEUE WITH COMPLETE REMOVALS

Published online by Cambridge University Press:  29 April 2019

Shan Gao ,
Deran Zhang ,
Hua Dong  and
Xianchao Wang
Shan Gao
Affiliation:
School of Mathematics and Statistics, Fuyang Normal College, Fuyang236037, P.R. China E-mail: sgao_09@Yeah.net
Deran Zhang
Affiliation:
Department of Electronics and Information Engineering, Bozhou University, Bozhou 236800, P.R. China E-mail: zhderan@163.com
Hua Dong
Affiliation:
School of Statistics, Qufu Normal University, Shandong 273165, P.R. China E-mail: sddh1978@126.com
Xianchao Wang
Affiliation:
School of Computer and Information Engineering, Fuyang Normal College, Fuyang236037, P.R. China E-mail: wxcdx@126.com
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Abstract

We consider an M/M/1 retrial queue subject to negative customers (called as G-retrial queue). The arrival of a negative customer forces all positive customers to leave the system and causes the server to fail. At a failure instant, the server is sent to be repaired immediately. Based on a natural reward-cost structure, all arriving positive customers decide whether to join the orbit or balk when they find the server is busy. All positive customers are selfish and want to maximize their own net benefit. Therefore, this system can be modeled as a symmetric noncooperative game among positive customers and the fundamental problem is to identify the Nash equilibrium balking strategy, which is a stable strategy in the sense that if all positive customers agree to follow it no one can benefit by deviating from it, that is, it is a strategy that is the best response against itself. In this paper, by using queueing theory and game theory, the Nash equilibrium mixed strategy in unobservable case and the Nash equilibrium pure strategy in observable case are considered. We also present some numerical examples to demonstrate the effect of the information together with some parameters on the equilibrium behaviors.

Keywords

balkingcomplete removalsNash equilibrium strategiesnegative customersretrial queues
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 35 , Special Issue 1: Learning, Optimization, and Theory of G-Networks , January 2021 , pp. 138 - 157
Copyright
Copyright © Cambridge University Press 2019

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References

1.Artalejo, J.R. (2000). G-networks: a versatile approach for work removal in queueing networks. European Journal of Operational Research, 126(2): 233–249.CrossRefGoogle Scholar
2.Artalejo, J.R. & Economou, A. (2004). Optimal control and performance analysis of an M X/M/1 queue with batches of negative customers. RAIRO Operations Research, 38(2): 121–151.CrossRefGoogle Scholar
3.Artalejo, J.R. & Gómez-Corral, A. (1999). Computation of the limiting distribution in queueing systems with repeated attempts and disasters. RAIRO Operations Research, 33(2): 371–382.CrossRefGoogle Scholar
4.Atencia, I & Moreno, P. (2005). A single-server G-queue in discrete-time with geometrical arrival and service process. Performance Evaluation, 59: 85–97.CrossRefGoogle Scholar
5.Boudali, O. & Economou, A. (2012). Optimal and equilibrium balking strategies in the single server Markovian queue with catastrophes. European Journal of Operational Research, 218: 708–715.CrossRefGoogle Scholar
6.Boudali, O. & Economou, A. (2013). The effect of catastrophes on the strategic customer behavior in queueing systems. Naval Research Logistics, 60: 571–587.CrossRefGoogle Scholar
7.Bountali, O. & Economou, A. (2017). Equilibrium joining strategies in batch service queueing systems. European Journal of Operational Research, 260(3): 1142–1151.CrossRefGoogle Scholar
8.Chao, X. (1995). A queueing network model with catastrophes and product form solution. Operations Research Letters, 18: 75–79.CrossRefGoogle Scholar
9.Do, N., Do, T., & Melikov, A. (2018). Equilibrium customer behavior in the M/M/1 retrial queue with working vacations and a constant retrial rate. Operational Research, https://doi.org/10.1007/s12351-017-0369-7.Google Scholar
10.Economou, A. & Kanta, S. (2008). Equilibrium balking strategies in the observable single-server queue with breakdowns and repairs. Operations Research Letters, 36: 696–699.CrossRefGoogle Scholar
11.Economou, A. & Kanta, S. (2011). Equilibrium customer strategies and social-profit maximization in the single-server constant retrial queue. Naval Research Logistics, 58: 107–122.CrossRefGoogle Scholar
12.Edelson, N.M. & Hildebrand, D.K. (1975). Congestion tolls for Poisson queuing processes. Econometrica, 43: 81–92.CrossRefGoogle Scholar
13.Fourneau, J.M. & Gelenbe, E. (2017). G-networks with adders. Future Internet, 9(3): 34. http://dx.doi.org/10.3390/fi9030034CrossRefGoogle Scholar
14.Gao, S., Dong, H., & Wang, X.C. (2018). Equilibrium and pricing analysis for an unreliable retrial queue with limited idle period and single vacation. Operational Research, https://doi.org/10.1007/s12351-018-0437-7.CrossRefGoogle Scholar
15.Gao, S., Wang, J., & Do, T.V. (2018). Analysis of a discrete-time repairable queue with disasters and working breakdowns. RAIRO-Operations Research, https://doi.org/10.1051/ro/2018057.Google Scholar
16.Gelenbe, E. (1989). Random neural networks with negative and positive signals and product form solution. Neural Computation, 1(4): 502–510.CrossRefGoogle Scholar
17.Gelenbe, E. (1991). Product-form queueing networks with negative and positive customers. Journal of Applied Probability, 28: 656–663.CrossRefGoogle Scholar
18.Gelenbe, E. (1993). G-Networks with signals and batch removal. Probability in the Engineering and Informatonal Sciences, 7: 335–342.CrossRefGoogle Scholar
19.Gelenbe, E. (1994). G-Networks: a unifying model for neural and queueing networks. Annals of Operations Research, 48: 433–461.CrossRefGoogle Scholar
20.Gelenbe, E. (2000). The first decade of G-networks. European Journal of Operational Research, 126: 231–232.CrossRefGoogle Scholar
21.Gelenbe, E. & Fourneau, J.M. (2002). G-Networks with resets. Performance Evaluation, 49: 179–192.CrossRefGoogle Scholar
22.Harrison, P.G. & Pitel, E. (1996). The M/G/1 queue with negative customers. Advances in Applied Probability, 28(2): 540–566.CrossRefGoogle Scholar
23.Hassin, R. (2016). Rational queueing. Boca Rotan, FL: Chapman and Hall/CRC.CrossRefGoogle Scholar
24.Hassin, R. & Haviv, M. (2003). To queue or not to queue: equilibrium behavior in queueing systems. Boston: Kluwer Academic Publishers.CrossRefGoogle Scholar
25.Kim, D.K. & Lee, D.H. (2014). The M/G/1 queue with disasters and working breakdowns. Applied Mathematical Modelling, 38: 1788–1798.CrossRefGoogle Scholar
26.Li, X., Wang, J., & Zhang, F. (2014). New results on equilibrium balking strategies in the single-server queue with breakdowns and repairs. Applied Mathematics and Computation, 241: 380–388.CrossRefGoogle Scholar
27.Lee, D.H., Yang, W.S., & Park, H.M. (2011). Geo/G/1 queues with disasters and general repair times. Applied Mathematical Modelling, 35(4): 1561–1570.CrossRefGoogle Scholar
28.Naor, P. (1969). The regulation of queue size by levying tolls. Econometrica, 37: 15–24.CrossRefGoogle Scholar
29.Wang, J. & Li, W.W. (2016). Noncooperative and cooperative joining strategies in cognitive radio networks With random access. IEEE Transactions on Vehicular Technology, 65(7): 5624–5636.CrossRefGoogle Scholar
30.Wang, J. & Zhang, F. (2011). Equilibrium analysis of the observable queues with balking and delayed repairs. Applied Mathematics and Computation, 218(6): 2716–2729.CrossRefGoogle Scholar
31.Wang, J. & Zhang, F. (2013). Strategic joining in M/M/1 retrial queues. European Journal of Operational Research, 230(1): 76–87.CrossRefGoogle Scholar
32.Wang, J. & Zhang, F. (2016). Monopoly pricing in a retrial queue with delayed vacations for local area network applications. Ima Journal of Management Mathematics, 27(2): 315–334.CrossRefGoogle Scholar
33.Wang, J., Huang, Y., & Dai, Z. (2011). A discrete-time on-off source queueing system with negative customers. Computers & Industrial Engineering, 61(4): 1226–1232.CrossRefGoogle Scholar
34.Wang, J., Zhang, X., & Huang, P. (2017). Strategic behavior and social optimization in a constant retrial queue with the N-policy. European Journal of Operational Research, 256: 841–849.CrossRefGoogle Scholar
35.Yang, W.S., Kim, J.D., & Chae, K.C. (2002). Analysis of M/G/1 stochastic clearing systems. Stochastic Analysis and Applications, 20: 1083–1100.CrossRefGoogle Scholar
36.Yu, S., Liu, Z., & Wu, J. (2016). Equilibrium strategies of the unobservable M/M/1 queue with balking and delayed repairs. Applied Mathematics and Computation, 290: 56–65.CrossRefGoogle Scholar