# American Institute of Mathematical Sciences

October  2018, 14(4): 1297-1322. doi: 10.3934/jimo.2018008

## Strategic behavior and optimal strategies in an M/G/1 queue with Bernoulli vacations

 1 Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, China 2 School of Mathematics & Information Science, Henan Polytechnic University, Jiaozuo, 454003, China

* Corresponding author:Jinting Wang

Received  August 2016 Revised  August 2017 Published  January 2018

Fund Project: This work is supported in part by the National Natural Science Foundation of China (Grant nos. 71571014,71390334.)

This paper considers an unobservable M/G/1 queue with Bernoulli vacations in which the server begins a vacation when the system is empty or upon completing a service. In the latter case, the server takes a vacation with p or serves the next customer, if any, with 1-p. We first give the steady-state equations and some performance measures, and then study the customer strategic behavior and obtain customers' Nash equilibrium strategies. From the viewpoint of the social planner, we derive the socially optimal joining probability, the socially optimal vacation probability and the socially optimal vacation rate. The socially optimal joining probability is found not greater than the equilibrium probability. In addition, if the vacation scheme does not incur any cost, the socially optimal decision is that the server does not take either a Bernoulli vacation or the normal vacation. On the other hand, if the server incurs the costs due to the underlying loss and the technology upgrade, proper vacations are beneficial to the social welfare maximization. Finally, sensitivity analysis is also performed to explore the effect of different parameters, and some managerial insights are provided for the social planner.

Citation: Sheng Zhu, Jinting Wang. Strategic behavior and optimal strategies in an M/G/1 queue with Bernoulli vacations. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1297-1322. doi: 10.3934/jimo.2018008
##### References:
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Hassin, Rational Queueing CRC Press, Raton, 2016.  Google Scholar [12] M. Haviv and Y. Kerner, On balking from an empty queue, Queueing Systems, 55 (2007), 239-249.  doi: 10.1007/s11134-007-9020-2.  Google Scholar [13] J. Keilson and L. D. Servi, Oscillating random walk models for GI/G/1 vacation systems with Bernoulli schedules, Journal of Applied Probability, 23 (1986), 790-802.  doi: 10.1017/S0021900200111933.  Google Scholar [14] Y. Kerner, Equilibrium joining probabilities for an M/G/1 queue, Games and Economic Behavior, 71 (2011), 521-526.  doi: 10.1016/j.geb.2010.06.002.  Google Scholar [15] B. K. Kumar and S. P. Madheswari, Analysis of an $M/M/N$ queue with Bernoulli service schedule, International Journal of Operational Research, 5 (2009), 48-72.  doi: 10.1504/IJOR.2009.024529.  Google Scholar [16] B. K. Kumar, R. Rukmani and S. R. A. Lakshmi, Performance analysis of an $M/G/1$ queueing system under Bernoulli vacation schedules with server setup and close down periods, Computers & Industrial Engineering, 66 (2013), 1-9.   Google Scholar [17] C. D. Liou, Optimization analysis of the machine repair problem with multiple vacations and working breakdowns, Journal of Industrial & Management Optimization, 11 (2014), 83-104.   Google Scholar [18] J. Liu and J. Wang, Strategic joining rules in a single server Markovian queue with Bernoulli vacation, Operational Research, 17 (2017), 413-434.  doi: 10.1007/s12351-016-0231-3.  Google Scholar [19] 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.  Google Scholar [20] P. Naor, The regulation of queue size by levying tolls, Econometrica, 37 (1969), 15-24.  doi: 10.2307/1909200.  Google Scholar [21] L. Servi, Average delay approximation of M/G/1 cyclic service queues with Bernoulli schedules, IEEE Journal on Selected Areas in Communications, 4 (1986), 813-822.   Google Scholar [22] Y. Shi and Z. Lian, Optimization and strategic behavior in a passenger-taxi service system, European Journal of Operational Research, 249 (2016), 1024-1032.  doi: 10.1016/j.ejor.2015.07.031.  Google Scholar [23] J. Wang, J. Cao and Q. Li, Reliability analysis of the retrial queue with server breakdowns and repairs, Queueing Systems, 38 (2001), 363-380.  doi: 10.1023/A:1010918926884.  Google Scholar [24] J. 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 [25] J. 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 [26] D. Yue, J. Yu and W. Yue, A Markovian queue with two heterogeneous servers and multiple vacations, Journal of Industrial & Management Optimization, 5 (2009), 453-465.  doi: 10.3934/jimo.2009.5.453.  Google Scholar [27] F. Zhang, J. Wang and B. Liu, On the optimal and equilibrium retrial rates in an unreliable retrial queue with vacations, Journal of Industrial & Management Optimization, 8 (2013), 861-875.  doi: 10.3934/jimo.2012.8.861.  Google Scholar

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##### References:
 [1] E. Altman and R. Hassin, Non-threshold equilibrium for customers joining an M/G/1 queue, Proceedings of 10th International Symposium on Dynamic Game and Applications, (2002), 56-64.   Google Scholar [2] A. Burnetas and A. Economou, Equilibrium customer strategies in a single server Markovian queue with setup times, Queueing Systems, 56 (2007), 213-228.  doi: 10.1007/s11134-007-9036-7.  Google Scholar [3] A. Burnetas and N. Apostolos, Customer equilibrium and optimal strategies in Markovian queues in series, Annals of Operations Research, 208 (2013), 515-529.  doi: 10.1007/s10479-011-1010-4.  Google Scholar [4] E. Cinlar, Introduction to Stochastic Processes Prentice-Hall, Englewood cliffs, 1975.  Google Scholar [5] 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), 967-982.  doi: 10.1016/j.peva.2011.07.001.  Google Scholar [6] H. M. Ghafir and C. B. Silio, Performance analysis of a multiple-access ring network, IEEE Transactions on Communications, 41 (1993), 1494-1506.  doi: 10.1109/26.237884.  Google Scholar [7] W. J. Gray, P. P. Wang and M. K. Scott, A vacation queueing model with service breakdowns, Applied Mathematical Modelling, 24 (2000), 391-400.  doi: 10.1016/S0307-904X(99)00048-7.  Google Scholar [8] P. Guo and R. Hassin, Strategic behavior and social optimization in Markovian vacation queues, Operations Research, 59 (2011), 986-997.  doi: 10.1287/opre.1100.0907.  Google Scholar [9] 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), 278-286.  doi: 10.1016/j.ejor.2012.05.026.  Google Scholar [10] R. Hassin and M. Haviv, To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems Kluwer Academic Publishers, Boston, 2003.  Google Scholar [11] R. Hassin, Rational Queueing CRC Press, Raton, 2016.  Google Scholar [12] M. Haviv and Y. Kerner, On balking from an empty queue, Queueing Systems, 55 (2007), 239-249.  doi: 10.1007/s11134-007-9020-2.  Google Scholar [13] J. Keilson and L. D. Servi, Oscillating random walk models for GI/G/1 vacation systems with Bernoulli schedules, Journal of Applied Probability, 23 (1986), 790-802.  doi: 10.1017/S0021900200111933.  Google Scholar [14] Y. Kerner, Equilibrium joining probabilities for an M/G/1 queue, Games and Economic Behavior, 71 (2011), 521-526.  doi: 10.1016/j.geb.2010.06.002.  Google Scholar [15] B. K. Kumar and S. P. Madheswari, Analysis of an $M/M/N$ queue with Bernoulli service schedule, International Journal of Operational Research, 5 (2009), 48-72.  doi: 10.1504/IJOR.2009.024529.  Google Scholar [16] B. K. Kumar, R. Rukmani and S. R. A. Lakshmi, Performance analysis of an $M/G/1$ queueing system under Bernoulli vacation schedules with server setup and close down periods, Computers & Industrial Engineering, 66 (2013), 1-9.   Google Scholar [17] C. D. Liou, Optimization analysis of the machine repair problem with multiple vacations and working breakdowns, Journal of Industrial & Management Optimization, 11 (2014), 83-104.   Google Scholar [18] J. Liu and J. Wang, Strategic joining rules in a single server Markovian queue with Bernoulli vacation, Operational Research, 17 (2017), 413-434.  doi: 10.1007/s12351-016-0231-3.  Google Scholar [19] 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.  Google Scholar [20] P. Naor, The regulation of queue size by levying tolls, Econometrica, 37 (1969), 15-24.  doi: 10.2307/1909200.  Google Scholar [21] L. Servi, Average delay approximation of M/G/1 cyclic service queues with Bernoulli schedules, IEEE Journal on Selected Areas in Communications, 4 (1986), 813-822.   Google Scholar [22] Y. Shi and Z. Lian, Optimization and strategic behavior in a passenger-taxi service system, European Journal of Operational Research, 249 (2016), 1024-1032.  doi: 10.1016/j.ejor.2015.07.031.  Google Scholar [23] J. Wang, J. Cao and Q. Li, Reliability analysis of the retrial queue with server breakdowns and repairs, Queueing Systems, 38 (2001), 363-380.  doi: 10.1023/A:1010918926884.  Google Scholar [24] J. 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 [25] J. 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 [26] D. Yue, J. Yu and W. Yue, A Markovian queue with two heterogeneous servers and multiple vacations, Journal of Industrial & Management Optimization, 5 (2009), 453-465.  doi: 10.3934/jimo.2009.5.453.  Google Scholar [27] F. Zhang, J. Wang and B. Liu, On the optimal and equilibrium retrial rates in an unreliable retrial queue with vacations, Journal of Industrial & Management Optimization, 8 (2013), 861-875.  doi: 10.3934/jimo.2012.8.861.  Google Scholar
Social welfare $SW$ vs. joining probability $q$ for $R = 4, C = 1, \lambda = 0.5, \beta_{1} = 1.2, \beta_{2} = 1, p = 0.1$
Equilibrium joining probability $q^{e}$ and socially optimal joining probability $q^{*}$ vs. potential arrival rate $\lambda$ for $R = 2, C = 1, \beta_{1} = 0.56, \beta_{2} = 0.25, \theta = 0.83, p = 0.1$
Equilibrium joining probability $q^{e}$ and socially optimal joining probability $q^{*}$ vs. the first moment of server time $\beta_{1}$ for $R = 2, C = 1, \lambda = 0.5, \beta_{2} = 0.25, \theta = 0.83, p = 0.1$
Equilibrium joining probability $q^{e}$ and socially optimal joining probability $q^{*}$ vs. $p$ for $R = 2, C = 1, \lambda = 0.5, \beta_{1} = 0.56, \beta_{2} = 0.25, \theta = 0.83$
Equilibrium joining probability $q^{e}$ and socially optimal joining probability $q^{*}$ vs. the second moment of server time $\beta_{2}$ for $R = 2, C = 1, \lambda = 0.5, \beta_{1} = 0.56, \theta = 0.83, p = 0.1$
Equilibrium joining probability $q^{e}$ and socially optimal joining probability $q^{*}$ vs. vacation rate $\theta$ for $R = 2, C = 1, \beta_{1} = 0.56, \beta_{2} = 0.25, \lambda = 0.5, p = 0.1$
Equilibrium joining probability $q^{e}$ and socially optimal joining probability $q^{*}$ vs. waiting cost per time $C$ for $R = 2, \theta = 0.83, \beta_{1} = 0.56, \beta_{2} = 0.25, \lambda = 0.5, p = 0.1$
Equilibrium joining probability $q^{e}$ and socially optimal joining probability $q^{*}$ vs. $R$ for $\theta = 0.83, \beta_{1} = 0.56, \beta_{2} = 0.25, \lambda = 0.5, p = 0.1$
Social welfare $SW$ vs. vacation probability $p$ for $R = 10, C = 1, \lambda = 0.8, \beta_{1} = 0.6, \beta_{2} = 1, q = 0.85, \theta = 1, C_{s} = 0.1$
Social welfare $SW$ vs. vacation rate $\theta$ for $R = 10, C = 1, \lambda = 0.8, \beta_{1} = 0.6, \beta_{2} = 1, q = 0.85, p = 0.3, C_{s} = 0.5$
Socially optimal vacation probability $p^{*}$ vs. $C_{b}$ for $R = 10, C = 1, \theta = 1, \lambda = 0.8, \beta_{1} = 0.6, \beta_{2} = 1$
Socially optimal vacation probability $p^{*}$ vs. joining probability $q$ for $R = 10, C = 1, \theta = 1, \lambda = 0.8, \beta_{1} = 0.6, \beta_{2} = 1$
Socially optimal vacation probability $p^{*}$ vs. vacation rate $\theta$ for $R = 10, C = 1, \theta = 1, \lambda = 0.8, \beta_{1} = 0.6, \beta_{2} = 1, C_{b} = 1$
Socially optimal vacation rate $\theta^{*}$ vs. $C_{s}$ for $R = 10, C = 1, \lambda = 0.8, \beta_{1} = 0.6, \beta_{2} = 1, p = 0.32$
Socially optimal vacation rate $\theta^{*}$ vs. joining probability $q$ for $R = 10, C = 1, \lambda = 0.8, \beta_{1} = 0.6, \beta_{2} = 1, p = 0.32$
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