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doi: 10.3934/jimo.2018157

Analysis of strategic customer behavior in fuzzy queueing systems

School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, China

* Corresponding author.Email address: math_zjc@csu.edu.cn (Jingchuan Zhang)

Received  June 2018 Published  September 2018

Fund Project: This work is partially supported by the National Natural Science Foundation of China (11671404), and the Fundamental Research Funds for the Central Universities of Central South University (2017zzts061, 2017zzts386)

This paper analyzes the optimal and equilibrium strategies in fuzzy Markovian queues where the system parameters are all fuzzy numbers. In this work, tools from both fuzzy logic and queuing theory have been used to investigate the membership functions of the optimal and equilibrium strategies in both observable and unobservable cases. By Zadeh's extension principle and $α$-cut approach, we formulate a pair of parametric nonlinear programs to describe the family of crisp strategy. Then the membership functions of the strategies in single and multi-server models are derived. Furthermore, the grated mean integration method is applied to find estimate of the equilibrium strategy in the fuzzy sense. Finally, numerical examples are solved successfully to illustrate the validity of the proposed approach and a sensitivity analysis is performed, which show the relationship of these strategies and social benefits. Our finding reveals that the value of equilibrium and optimal strategies have no deterministic relationship, which are different from the results in the corresponding crisp queues. Since the performance measures of such queues are expressed by fuzzy numbers rather than by crisp values, the system managers could get more precise information.

Citation: Gang Chen, Zaiming Liu, Jingchuan Zhang. Analysis of strategic customer behavior in fuzzy queueing systems. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2018157
References:
[1]

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.

[2]

S.-P. Chen, Time value of delays in unreliable production systems with mixed uncertainties of fuzziness and randomness, European Journal of Operational Research, 255 (2016), 834-844. doi: 10.1016/j.ejor.2016.06.021.

[3]

D. J. Dubois, Fuzzy Sets and Systems: Theory and Applications, vol. 144, Academic Press, 1980.

[4]

A. Economou and A. Manou, Strategic behavior in an observable fluid queue with an alternating service process, European Journal of Operational Research, 254 (2016), 148-160. doi: 10.1016/j.ejor.2016.03.046.

[5]

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.

[6]

R. Hassin and M. Haviv, To queue or not to queue: Equilibrium behavior in queueing systems, vol. 59, Springer Science & Business Media, 2003. doi: 10.1007/978-1-4615-0359-0.

[7]

F. JolaiS. M. AsadzadehR. Ghodsi and S. Bagheri-Marani, A multi-objective fuzzy queuing priority assignment model, Applied Mathematical Modelling, 40 (2016), 9500-9513. doi: 10.1016/j.apm.2016.06.024.

[8]

J.-C. KeH.-I. Huang and C.-H. Lin, Parametric programming approach for batch arrival queues with vacation policies and fuzzy parameters, Applied Mathematics and Computation, 180 (2006), 217-232. doi: 10.1016/j.amc.2005.11.144.

[9]

J.-C. KeH.-I. Huang and C.-H. Lin, On retrial queueing model with fuzzy parameters, Physica A: Statistical Mechanics and its Applications, 374 (2007), 272-280. doi: 10.1016/j.physa.2006.05.057.

[10]

J.-C. KeH.-I. Huang and C.-H. Lin, Analysis on a queue system with heterogeneous servers and uncertain patterns, Journal of Industrial & Management Optimization, 6 (2010), 57-71. doi: 10.3934/jimo.2010.6.57.

[11]

J.-C. Ke and C.-H. Lin, Fuzzy analysis of queueing systems with an unreliable server: A nonlinear programming approach, Applied Mathematics and Computation, 175 (2006), 330-346. doi: 10.1016/j.amc.2005.07.024.

[12]

G. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic, vol. 4, Prentice Hall New Jersey, 1995.

[13]

R.-J. Li and E. Lee, Analysis of fuzzy queues, Computers & Mathematics with Applications, 17 (1989), 1143-1147. doi: 10.1016/0898-1221(89)90044-8.

[14]

Y. MaZ. Liu and Z. G. Zhang, Equilibrium in vacation queueing system with complementary services, Quality Technology & Quantitative Management, 14 (2017), 114-127. doi: 10.1080/16843703.2016.1191172.

[15]

G. C. Mahata and A. Goswami, Fuzzy inventory models for items with imperfect quality and shortage backordering under crisp and fuzzy decision variables, Computers & Industrial Engineering, 64 (2013), 190-199. doi: 10.1016/j.cie.2012.09.003.

[16]

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

[17]

L. J. Ratliff, C. Dowling, E. Mazumdar and B. Zhang, To observe or not to observe: Queuing game framework for urban parking, in Decision and Control (CDC), 2016 IEEE 55th Conference on, IEEE, 2016, 5286-5291. doi: 10.1109/CDC.2016.7799079.

[18]

M. SaffariS. Asmussen and R. Haji, The M/M/1 queue with inventory, lost sale, and general lead times, Queueing Systems, 75 (2013), 65-77. doi: 10.1007/s11134-012-9337-3.

[19]

R. ShoneV. 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.

[20]

E. SimhonY. HayelD. 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.

[21]

S. Stidham Jr, Optimal Design of Queueing Systems, CRC Press, 2009. doi: 10.1201/9781420010008.

[22]

B. VahdaniR. Tavakkoli-Moghaddam and F. Jolai, Reliable design of a logistics network under uncertainty: A fuzzy possibilistic-queuing model, Applied Mathematical Modelling, 37 (2013), 3254-3268. doi: 10.1016/j.apm.2012.07.021.

[23]

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.

[24]

Y. C. WangJ. S. Wang and F. H. Tsai, Analysis of discrete-time space priority queue with fuzzy threshold, Journal of Industrial and Management Optimization, 5 (2009), 467-479. doi: 10.3934/jimo.2009.5.467.

[25]

F. ZhangJ. Wang and B. Liu, Equilibrium joining probabilities in observable queues with general service and setup times, Journal of Industrial and Management Optimization, 9 (2013), 901-917. doi: 10.3934/jimo.2013.9.901.

[26]

R. Zhang, Y. A. Phillis and V. S. Kouikoglou, Fuzzy Control of Queuing Systems, Springer Science & Business Media, 2005.

[27]

S. Zhu and J. Wang, Strategic behavior and optimal strategies in an M/G/1 queue with bernoulli vacations, Journal of Industrial & Management Optimization, (2008), 56-64. doi: 10.3934/jimo.2018008.

[28]

H.-J. Zimmermann, Fuzzy Set Theory and Its Applications, Second edition, Kluwer Academic Publishers, Boston, MA, 1992.

show all references

References:
[1]

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.

[2]

S.-P. Chen, Time value of delays in unreliable production systems with mixed uncertainties of fuzziness and randomness, European Journal of Operational Research, 255 (2016), 834-844. doi: 10.1016/j.ejor.2016.06.021.

[3]

D. J. Dubois, Fuzzy Sets and Systems: Theory and Applications, vol. 144, Academic Press, 1980.

[4]

A. Economou and A. Manou, Strategic behavior in an observable fluid queue with an alternating service process, European Journal of Operational Research, 254 (2016), 148-160. doi: 10.1016/j.ejor.2016.03.046.

[5]

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.

[6]

R. Hassin and M. Haviv, To queue or not to queue: Equilibrium behavior in queueing systems, vol. 59, Springer Science & Business Media, 2003. doi: 10.1007/978-1-4615-0359-0.

[7]

F. JolaiS. M. AsadzadehR. Ghodsi and S. Bagheri-Marani, A multi-objective fuzzy queuing priority assignment model, Applied Mathematical Modelling, 40 (2016), 9500-9513. doi: 10.1016/j.apm.2016.06.024.

[8]

J.-C. KeH.-I. Huang and C.-H. Lin, Parametric programming approach for batch arrival queues with vacation policies and fuzzy parameters, Applied Mathematics and Computation, 180 (2006), 217-232. doi: 10.1016/j.amc.2005.11.144.

[9]

J.-C. KeH.-I. Huang and C.-H. Lin, On retrial queueing model with fuzzy parameters, Physica A: Statistical Mechanics and its Applications, 374 (2007), 272-280. doi: 10.1016/j.physa.2006.05.057.

[10]

J.-C. KeH.-I. Huang and C.-H. Lin, Analysis on a queue system with heterogeneous servers and uncertain patterns, Journal of Industrial & Management Optimization, 6 (2010), 57-71. doi: 10.3934/jimo.2010.6.57.

[11]

J.-C. Ke and C.-H. Lin, Fuzzy analysis of queueing systems with an unreliable server: A nonlinear programming approach, Applied Mathematics and Computation, 175 (2006), 330-346. doi: 10.1016/j.amc.2005.07.024.

[12]

G. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic, vol. 4, Prentice Hall New Jersey, 1995.

[13]

R.-J. Li and E. Lee, Analysis of fuzzy queues, Computers & Mathematics with Applications, 17 (1989), 1143-1147. doi: 10.1016/0898-1221(89)90044-8.

[14]

Y. MaZ. Liu and Z. G. Zhang, Equilibrium in vacation queueing system with complementary services, Quality Technology & Quantitative Management, 14 (2017), 114-127. doi: 10.1080/16843703.2016.1191172.

[15]

G. C. Mahata and A. Goswami, Fuzzy inventory models for items with imperfect quality and shortage backordering under crisp and fuzzy decision variables, Computers & Industrial Engineering, 64 (2013), 190-199. doi: 10.1016/j.cie.2012.09.003.

[16]

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

[17]

L. J. Ratliff, C. Dowling, E. Mazumdar and B. Zhang, To observe or not to observe: Queuing game framework for urban parking, in Decision and Control (CDC), 2016 IEEE 55th Conference on, IEEE, 2016, 5286-5291. doi: 10.1109/CDC.2016.7799079.

[18]

M. SaffariS. Asmussen and R. Haji, The M/M/1 queue with inventory, lost sale, and general lead times, Queueing Systems, 75 (2013), 65-77. doi: 10.1007/s11134-012-9337-3.

[19]

R. ShoneV. 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.

[20]

E. SimhonY. HayelD. 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.

[21]

S. Stidham Jr, Optimal Design of Queueing Systems, CRC Press, 2009. doi: 10.1201/9781420010008.

[22]

B. VahdaniR. Tavakkoli-Moghaddam and F. Jolai, Reliable design of a logistics network under uncertainty: A fuzzy possibilistic-queuing model, Applied Mathematical Modelling, 37 (2013), 3254-3268. doi: 10.1016/j.apm.2012.07.021.

[23]

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.

[24]

Y. C. WangJ. S. Wang and F. H. Tsai, Analysis of discrete-time space priority queue with fuzzy threshold, Journal of Industrial and Management Optimization, 5 (2009), 467-479. doi: 10.3934/jimo.2009.5.467.

[25]

F. ZhangJ. Wang and B. Liu, Equilibrium joining probabilities in observable queues with general service and setup times, Journal of Industrial and Management Optimization, 9 (2013), 901-917. doi: 10.3934/jimo.2013.9.901.

[26]

R. Zhang, Y. A. Phillis and V. S. Kouikoglou, Fuzzy Control of Queuing Systems, Springer Science & Business Media, 2005.

[27]

S. Zhu and J. Wang, Strategic behavior and optimal strategies in an M/G/1 queue with bernoulli vacations, Journal of Industrial & Management Optimization, (2008), 56-64. doi: 10.3934/jimo.2018008.

[28]

H.-J. Zimmermann, Fuzzy Set Theory and Its Applications, Second edition, Kluwer Academic Publishers, Boston, MA, 1992.

Figure 1.  The approximate membership function of optimal and equilibrium threshold $n$.
Figure 2.  The approximate membership function of optimal and equilibrium arrival rate $\lambda$.
Figure 3.  The approximate membership function of optimal social benefit $S$.
Table 1.  Fuzzy trapezoidal value for the input parameters $\tilde{\Lambda}$
$\tilde{\Lambda}$$P(\tilde{\Lambda})$$n_{e}$$n_{*}$$\lambda_{e}$$\lambda_{*}$
(1.0, 2.0, 3.0, 4.0)$2.50$$22$$14$$2.50$$2.50$
(1.1, 2.2, 3.3, 4.4)$2.75$$22$$13$$2.75$$2.75$
(1.2, 2.4, 3.6, 4.8)$3.00$$22$$13$$3.00$$3.00$
(1.3, 2.6, 3.9, 5.2)$3.25$$22$$12$$3.25$$3.25$
(1.4, 2.8, 4.2, 5.6)$3.50$$22$$11$$3.50$$3.50$
(1.5, 3.0, 4.5, 6.0)$3.75$$22$$10$$3.75$$3.75$
$\tilde{\Lambda}$$P(\tilde{\Lambda})$$n_{e}$$n_{*}$$\lambda_{e}$$\lambda_{*}$
(1.0, 2.0, 3.0, 4.0)$2.50$$22$$14$$2.50$$2.50$
(1.1, 2.2, 3.3, 4.4)$2.75$$22$$13$$2.75$$2.75$
(1.2, 2.4, 3.6, 4.8)$3.00$$22$$13$$3.00$$3.00$
(1.3, 2.6, 3.9, 5.2)$3.25$$22$$12$$3.25$$3.25$
(1.4, 2.8, 4.2, 5.6)$3.50$$22$$11$$3.50$$3.50$
(1.5, 3.0, 4.5, 6.0)$3.75$$22$$10$$3.75$$3.75$
Table 2.  Fuzzy trapezoidal value for the input parameters $\tilde{\mu}$
$\tilde{\mu}$$P(\tilde{\mu})$$n_{e}$$n_{*}$$\lambda_{e}$$\lambda_{*}$
(5.0, 6.0, 7.0, 8.0)$6.50$$22.75$$14$$2.5$$2.5$
(5.5, 6.6, 7.7, 8.8)$7.15$$25.03$$16$$2.5$$2.5$
(6.0, 7.2, 8.4, 9.6)$7.80$$27.30$$19$$2.5$$2.5$
(6.5, 7.8, 9.1, 10.4)$8.45$$29.58$$21$$2.5$$2.5$
(7.0, 8.4, 9.8, 11.2)$9.10$$31.85$$23$$2.5$$2.5$
(7.5, 9.0, 10.5, 12.0)$9.75$$34.13$$25$$2.5$$2.5$
$\tilde{\mu}$$P(\tilde{\mu})$$n_{e}$$n_{*}$$\lambda_{e}$$\lambda_{*}$
(5.0, 6.0, 7.0, 8.0)$6.50$$22.75$$14$$2.5$$2.5$
(5.5, 6.6, 7.7, 8.8)$7.15$$25.03$$16$$2.5$$2.5$
(6.0, 7.2, 8.4, 9.6)$7.80$$27.30$$19$$2.5$$2.5$
(6.5, 7.8, 9.1, 10.4)$8.45$$29.58$$21$$2.5$$2.5$
(7.0, 8.4, 9.8, 11.2)$9.10$$31.85$$23$$2.5$$2.5$
(7.5, 9.0, 10.5, 12.0)$9.75$$34.13$$25$$2.5$$2.5$
Table 3.  Fuzzy trapezoidal value for the input parameters $\tilde{R}$
$\tilde{R}$$P(\tilde{R})$$n_{e}$$n_{*}$$\lambda_{e}$$\lambda_{*}$
(10.0, 15.0, 20.0, 25.0)$17.50$$22$$14$$2.5$$2.5$
(11.0, 16.5, 22.0, 27.5)$19.25$$25$$16$$2.5$$2.5$
(12.0, 18.0, 24.0, 30.0)$21.00$$27$$17$$2.5$$2.5$
(13.0, 19.5, 26.0, 32.5)$22.75$$29$$18$$2.5$$2.5$
(14.0, 21.0, 28.0, 35.0)$24.50$$31$$20$$2.5$$2.5$
(15.0, 22.5, 30.0, 37.5)$26.25$$34$$21$$2.5$$2.5$
$\tilde{R}$$P(\tilde{R})$$n_{e}$$n_{*}$$\lambda_{e}$$\lambda_{*}$
(10.0, 15.0, 20.0, 25.0)$17.50$$22$$14$$2.5$$2.5$
(11.0, 16.5, 22.0, 27.5)$19.25$$25$$16$$2.5$$2.5$
(12.0, 18.0, 24.0, 30.0)$21.00$$27$$17$$2.5$$2.5$
(13.0, 19.5, 26.0, 32.5)$22.75$$29$$18$$2.5$$2.5$
(14.0, 21.0, 28.0, 35.0)$24.50$$31$$20$$2.5$$2.5$
(15.0, 22.5, 30.0, 37.5)$26.25$$34$$21$$2.5$$2.5$
Table 4.  Fuzzy trapezoidal value for the input parameters $\tilde{C}$
$\tilde{C}$$P(\tilde{C})$$n_{e}$$n_{*}$$\lambda_{e}$$\lambda_{*}$
(2.0, 4.0, 6.0, 8.0)$5.00$$22$$14$$2.5$$2.5$
(2.2, 4.4, 6.6, 8.8)$5.50$$20$$13$$2.5$$2.5$
(2.4, 4.8, 7.2, 9.6)$6.00$$18$$12$$2.5$$2.5$
(2.6, 5.2, 7.8, 10.4)$6.50$$17$$11$$2.5$$2.5$
(2.8, 5.6, 8.4, 11.2)$7.00$$16$$10$$2.5$$2.5$
(3.0, 6.0, 7.0, 12.0)$6.83$$16$$10$$2.5$$2.5$
$\tilde{C}$$P(\tilde{C})$$n_{e}$$n_{*}$$\lambda_{e}$$\lambda_{*}$
(2.0, 4.0, 6.0, 8.0)$5.00$$22$$14$$2.5$$2.5$
(2.2, 4.4, 6.6, 8.8)$5.50$$20$$13$$2.5$$2.5$
(2.4, 4.8, 7.2, 9.6)$6.00$$18$$12$$2.5$$2.5$
(2.6, 5.2, 7.8, 10.4)$6.50$$17$$11$$2.5$$2.5$
(2.8, 5.6, 8.4, 11.2)$7.00$$16$$10$$2.5$$2.5$
(3.0, 6.0, 7.0, 12.0)$6.83$$16$$10$$2.5$$2.5$
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