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April  2019, 15(2): 689-703. doi: 10.3934/jimo.2018065

Optimal information policy in discrete-time queues with strategic customers

Veena Goswami 1, and  Gopinath Panda 2,

1. 

School of Computer Applications, Kalinga Institute of Industrial Technology, Bhubaneswar-751024, India
2. 

School of Mathematical Sciences, National Institute of Science Education and Research, Bhubaneswar-752050, India

Received  May 2016 Revised  March 2018 Published  June 2018

This paper studies optimal information revelation policies in discrete-time $Geo/Geo/1$ queue. Revealing the queue length information to arriving customers plays an important role in their decision making, that is, whether to join the system or balk. We consider policies where a service provider discloses information to some customers and conceals it from others, depending upon the number of waiting customers. This partial information disclosure policy helps the service provider minimize the idle period of the system and maximize the revenue.

Keywords: Equilibrium threshold, balking, strategic customers, Bernoullischeduled, social benefit.
Mathematics Subject Classification: Primary: 60K25, 68M20; Secondary: 90B22.
Citation: Veena Goswami, Gopinath Panda. Optimal information policy in discrete-time queues with strategic customers. Journal of Industrial & Management Optimization, 2019, 15 (2) : 689-703. doi: 10.3934/jimo.2018065
References:
[1]

Z. Aksin, M. Armony and V. Mehrotra, The modern call center: A multi-disciplinary perspective on operations management research, Production and Operations Management, 16 (2007), 665-688. Google Scholar

[2]

O. Boudali and A. Economou, Optimal and equilibrium balking strategies in the single server Markovian queue with catastrophes, European Journal of Operational Research, 218 (2012), 708-715. doi: 10.1016/j.ejor.2011.11.043. Google Scholar

[3]

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

[4]

A. Di Crescenzo, A probabilistic analogue of the mean value theorem and its applications to reliability theory, Journal of Applied Probability, 36 (1999), 706-719. doi: 10.1239/jap/1032374628. Google Scholar

[5]

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

[6]

S. Gao and J. Wang, Equilibrium balking strategies in the observable ${Geo/Geo/1}$ queue with delayed multiple vacations, RAIRO-Operations Research, 50 (2016), 119-129. doi: 10.1051/ro/2015019. Google Scholar

[7]

A. Glazer and R. Hassin, On the economics of subscriptions, European Economic Review, 19 (1982), 343-356. doi: 10.1016/S0014-2921(82)80059-7. Google Scholar

[8]

P. Guo, W. Sun and Y. Wang, Equilibrium and optimal strategies to join a queue with partial information on service times, European Journal of Operational Research, 214 (2011), 284-297. doi: 10.1016/j.ejor.2011.04.011. Google Scholar

[9]

R. Hassin and M. Haviv, To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems, Springer Science & Business Media, 2003. doi: 10.1007/978-1-4615-0359-0. Google Scholar

[10]

R. Hassin and R. Roet-Green, Equilibrium in a Two Dimensional Queueing Game: When Inspecting the Queue is Costly, Technical report, Tel Aviv University, Israel, 2011.Google Scholar

[11]

J. J. Hunter, Mathematical Techniques of Applied Probability. Vol. 2, Discrete Time Models: Techniques and Applications, Academic Press, 1983. Google Scholar

[12]

J. H. Large and T. W. Norman, Markov perfect Bayesian equilibrium via ergodicity, Working paper.Google Scholar

[13]

Z. Liu, Y. Ma and Z. G. Zhang, Equilibrium mixed strategies in a discrete-time markovian queue under multiple and single vacation policies, Quality Technology & Quantitative Management, 12 (2015), 369-382. doi: 10.1080/16843703.2015.11673387. Google Scholar

[14]

Y. Ma, W.-q. Liu and J.-h. Li, Equilibrium balking behavior in the ${Geo/Geo/1}$ queueing system with multiple vacations, Applied Mathematical Modelling, 37 (2013), 3861-3878. doi: 10.1016/j.apm.2012.08.017. Google Scholar

[15]

Y. Ma and Z. Liu, Pricing analysis in ${Geo/Geo/1}$ queueing system, Mathematical Problems in Engineering, 2015 (2015), Art. ID 181653, 5 pp. doi: 10.1155/2015/181653. Google Scholar

[16]

P. Naor, The Regulation of Queue Size by Levying Tolls, Econometrica, 37 (1969), 15-24. doi: 10.2307/1909200. Google Scholar

[17]

R. Shone, V. A. Knight and J. E. Williams, Comparisons between observable and unobservable ${M/M/1}$ queues with respect to optimal customer behavior, European Journal of Operational Research, 227 (2013), 133-141. doi: 10.1016/j.ejor.2012.12.016. Google Scholar

[18]

E. Simhon, Y. Hayel, D. Starobinski and Q. Zhu, Optimal information disclosure policies in strategic queueing games, Operations Research Letters, 44 (2016), 109-113. doi: 10.1016/j.orl.2015.12.005. Google Scholar

[19]

W. Sun, P. Guo and N. Tian, Equilibrium threshold strategies in observable queueing systems with setup/closedown times, Central European Journal of Operations Research, 18 (2010), 241-268. doi: 10.1007/s10100-009-0104-4. Google Scholar

[20]

F. Wang, J. Wang and F. Zhang, Equilibrium customer strategies in the ${Geo/Geo/1}$ queue with single working vacation, Discrete Dynamics in Nature and Society, 2014 (2014), Art. ID 309489, 9 pp. doi: 10.1155/2014/309489. Google Scholar

[21]

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

[22]

M. E. Woodward, Communication and Computer Networks: Modelling with Discrete-Time Queues, IEEE Computer Soc. Press, 1994.Google Scholar

[23]

T. Yang, J. Wang and F. Zhang, Equilibrium balking strategies in the ${Geo/Geo/1}$ queues with server breakdowns and repairs, Quality Technology & Quantitative Management, 11 (2014), 231-243. doi: 10.1080/16843703.2014.11673341. Google Scholar

[24]

F. Zhang, J. Wang and B. Liu, Equilibrium balking strategies in Markovian queues with working vacations, Applied Mathematical Modelling, 37 (2013), 8264-8282. doi: 10.1016/j.apm.2013.03.049. Google Scholar

show all references

References:
[1]

Z. Aksin, M. Armony and V. Mehrotra, The modern call center: A multi-disciplinary perspective on operations management research, Production and Operations Management, 16 (2007), 665-688. Google Scholar

[2]

O. Boudali and A. Economou, Optimal and equilibrium balking strategies in the single server Markovian queue with catastrophes, European Journal of Operational Research, 218 (2012), 708-715. doi: 10.1016/j.ejor.2011.11.043. Google Scholar

[3]

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

[4]

A. Di Crescenzo, A probabilistic analogue of the mean value theorem and its applications to reliability theory, Journal of Applied Probability, 36 (1999), 706-719. doi: 10.1239/jap/1032374628. Google Scholar

[5]

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

[6]

S. Gao and J. Wang, Equilibrium balking strategies in the observable ${Geo/Geo/1}$ queue with delayed multiple vacations, RAIRO-Operations Research, 50 (2016), 119-129. doi: 10.1051/ro/2015019. Google Scholar

[7]

A. Glazer and R. Hassin, On the economics of subscriptions, European Economic Review, 19 (1982), 343-356. doi: 10.1016/S0014-2921(82)80059-7. Google Scholar

[8]

P. Guo, W. Sun and Y. Wang, Equilibrium and optimal strategies to join a queue with partial information on service times, European Journal of Operational Research, 214 (2011), 284-297. doi: 10.1016/j.ejor.2011.04.011. Google Scholar

[9]

R. Hassin and M. Haviv, To Queue or Not to Queue: Equilibrium Behavior in Queueing Systems, Springer Science & Business Media, 2003. doi: 10.1007/978-1-4615-0359-0. Google Scholar

[10]

R. Hassin and R. Roet-Green, Equilibrium in a Two Dimensional Queueing Game: When Inspecting the Queue is Costly, Technical report, Tel Aviv University, Israel, 2011.Google Scholar

[11]

J. J. Hunter, Mathematical Techniques of Applied Probability. Vol. 2, Discrete Time Models: Techniques and Applications, Academic Press, 1983. Google Scholar

[12]

J. H. Large and T. W. Norman, Markov perfect Bayesian equilibrium via ergodicity, Working paper.Google Scholar

[13]

Z. Liu, Y. Ma and Z. G. Zhang, Equilibrium mixed strategies in a discrete-time markovian queue under multiple and single vacation policies, Quality Technology & Quantitative Management, 12 (2015), 369-382. doi: 10.1080/16843703.2015.11673387. Google Scholar

[14]

Y. Ma, W.-q. Liu and J.-h. Li, Equilibrium balking behavior in the ${Geo/Geo/1}$ queueing system with multiple vacations, Applied Mathematical Modelling, 37 (2013), 3861-3878. doi: 10.1016/j.apm.2012.08.017. Google Scholar

[15]

Y. Ma and Z. Liu, Pricing analysis in ${Geo/Geo/1}$ queueing system, Mathematical Problems in Engineering, 2015 (2015), Art. ID 181653, 5 pp. doi: 10.1155/2015/181653. Google Scholar

[16]

P. Naor, The Regulation of Queue Size by Levying Tolls, Econometrica, 37 (1969), 15-24. doi: 10.2307/1909200. Google Scholar

[17]

R. Shone, V. A. Knight and J. E. Williams, Comparisons between observable and unobservable ${M/M/1}$ queues with respect to optimal customer behavior, European Journal of Operational Research, 227 (2013), 133-141. doi: 10.1016/j.ejor.2012.12.016. Google Scholar

[18]

E. Simhon, Y. Hayel, D. Starobinski and Q. Zhu, Optimal information disclosure policies in strategic queueing games, Operations Research Letters, 44 (2016), 109-113. doi: 10.1016/j.orl.2015.12.005. Google Scholar

[19]

W. Sun, P. Guo and N. Tian, Equilibrium threshold strategies in observable queueing systems with setup/closedown times, Central European Journal of Operations Research, 18 (2010), 241-268. doi: 10.1007/s10100-009-0104-4. Google Scholar

[20]

F. Wang, J. Wang and F. Zhang, Equilibrium customer strategies in the ${Geo/Geo/1}$ queue with single working vacation, Discrete Dynamics in Nature and Society, 2014 (2014), Art. ID 309489, 9 pp. doi: 10.1155/2014/309489. Google Scholar

[21]

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

[22]

M. E. Woodward, Communication and Computer Networks: Modelling with Discrete-Time Queues, IEEE Computer Soc. Press, 1994.Google Scholar

[23]

T. Yang, J. Wang and F. Zhang, Equilibrium balking strategies in the ${Geo/Geo/1}$ queues with server breakdowns and repairs, Quality Technology & Quantitative Management, 11 (2014), 231-243. doi: 10.1080/16843703.2014.11673341. Google Scholar

[24]

F. Zhang, J. Wang and B. Liu, Equilibrium balking strategies in Markovian queues with working vacations, Applied Mathematical Modelling, 37 (2013), 8264-8282. doi: 10.1016/j.apm.2013.03.049. Google Scholar

Figure 1.  Various time epochs in late-arrival system with delayed access (LAS-DA)
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Figure 2.  State transition diagram of the $Geo/Geo/1$ model with selective threshold policy $\xi_D$.
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Figure 3.  Various time epochs in early arrival system (EAS)
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Figure 4.  Uniformed policy ($\xi_-$) is optimal for the $Geo/Geo/1/30$ queue with $\lambda = 0.5, \mu = 0.6, R = 50.$
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Figure 5.  Informed policy ($\xi_+$) is optimal for the $Geo/Geo/1/30$ queue with $\lambda = 0.65, \mu = 0.6, R = 50.$
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Figure 6.  Expected waiting time for different joining probabilities for $\lambda = 0.65, \mu = 0.6, D = 5.$
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Figure 7.  Expected waiting time for different joining probabilities for $\lambda = 0.5, \mu = 0.6, D = 5.$
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