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October  2019, 15(4): 1599-1615. doi: 10.3934/jimo.2018113

Equilibrium and optimal balking strategies for low-priority customers in the M/G/1 queue with two classes of customers and preemptive priority

Biao Xu 1, , Xiuli Xu 2,, and  Zhong Yao 3,4,

1. 

School of Economics and Management, Beihang University, Beijing 100191, China
2. 

College of Science, Yanshan University, Hebei, Qinhuangdao 066004, China
3. 

School of Economics and Management, Beihang University, Beijing 100191, China
4. 

Institute of Economics and Business, Beihang University, Beijing 100191, China

* Corresponding author: Xiuli Xu

Received  October 2016 Revised  June 2018 Published  October 2019 Early access  August 2018

This paper investigates the low-priority customers' strategic behavior in the single-server queueing system with general service time and two customer types. The priority system is preemptive resume, which means that if a high-priority customer enters the system that are serving a low-priority customer, the arriving customer preempts the service facility and the preempted customer returns to the head of the queue for his own class. The customer who is preempted resumes service at the point of interruption upon reentering the system. The low-priority customer's dilemma is whether to join or balk based on a linear reward-cost structure. Two cases are distinguished based on the different levels of information that the low-priority customers acquire before joining the system. The equilibrium threshold strategy in the observable case and the equilibrium balking strategy as well as the socially optimal balking strategy in the unobservable case for the low-priority customers are derived finally.

Keywords: M/G/1 queue, two classes, equilibrium, optimization, priority, preemption.
Mathematics Subject Classification: Primary: 60K25; Secondary: 90B22.
Citation: Biao Xu, Xiuli Xu, Zhong Yao. Equilibrium and optimal balking strategies for low-priority customers in the M/G/1 queue with two classes of customers and preemptive priority. Journal of Industrial & Management Optimization, 2019, 15 (4) : 1599-1615. doi: 10.3934/jimo.2018113
References:
[1]

E. Altaman and R. Hassin, Non-threshold equilibrium for customers joining an M/G/1 queue, International Symposium on Dynamic Games & Applications, 2002 (2002), 56-64.   Google Scholar

[2]

J. Altmann, H. Daanen, H. Oliver and A. S.-B. Suarez, How to market-manage a QoS network, Proceedings - IEEE INFOCOM 1, 2002 (2002), 56-64.  doi: 10.1109/INFCOM.2002.1019270.  Google Scholar

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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

[5]

G. Brouns and J. Wal, Optimal threshold policies in a two-class preemptive priority queue with admission and termination control, Queueing System, 54 (2006), 21-33.  doi: 10.1007/s11134-006-8307-z.  Google Scholar

[6]

F. Chen and V. Kulkarni, Individual, class-based, and social optimal admission policies in two-priority queues, Stochastic Models, 23 (2007), 97-127.  doi: 10.1080/15326340601142180.  Google Scholar

[7]

P. Chen and Y. Zhou, Equilibrium balking strategies in the single server queue with setup times and breakdowns, Operational Research: An International Journal, 15 (2015), 213-231.  doi: 10.1007/s12351-015-0174-0.  Google Scholar

[8]

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

[9]

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[10]

J. Erlichman and R. Hassin, Equilibrium solutions in the observable M/M/1 queue with overtaking, International ICST Conference on Performance Evaluation Methodologies & Tools, (1975). doi: 10.4108/ICST.VALUETOOLS2009.8039.  Google Scholar

[11]

S. Gavirneni and V. Kulkarni, Self-selecting priority queues with burr distributed waiting costs, Production & Operations Management, 25 (2016), 979-992.   Google Scholar

[12]

W. Gilland and D. Warsing, The impact of revenue-maximizing priority pricing on customer delay costs, Decision Sciences, 40 (2009), 89-120.   Google Scholar

[13]

R. Hassin, Rational Queueing, CRC press, Boca Raton, 2016. doi: 10.1201/b20014.  Google Scholar

[14]

M. Haviv and Y. Kerner, On balking from an empty queue, Queueing Systems Theory & Applications, 55 (2007), 239-249.  doi: 10.1007/s11134-007-9020-2.  Google Scholar

[15]

V. Hsu, S. Xu and B. Jukic, Optimal scheduling and incentive compatible pricing for a service system with quality of service guarantees, Manufacturing & Service Operations Management, 11 (2008), 375-396.  doi: 10.1287/msom.1080.0226.  Google Scholar

[16]

Y. Kerner, Equilibrium joining probabilities for an M/G/1 queue, Games & Economic Behavior, 71 (2011), 521-526.  doi: 10.1016/j.geb.2010.06.002.  Google Scholar

[17]

Y. Kim and M. Mannino, Optimal incentive-compatible pricing for M/G/1 queues, Operations Research Letters, 31 (2003), 459-461.  doi: 10.1016/S0167-6377(03)00060-9.  Google Scholar

[18]

D. Lee and S. Park, Performance analysis of queueing strategies for multiple priority calls inmultiservice personal communications services, Computer Communications, 23 (2000), 1069-1083.   Google Scholar

[19]

L. Li, J. Wang and F. Zhang, Equilibrium customer strategies in Markovian queues with partial breakdowns, Computers & Industrial Engineering, 66 (2013), 751-757.  doi: 10.1016/j.cie.2013.09.023.  Google Scholar

[20]

C. Liu and R. Berry, A priority queue model for competition with shared spectrum, Communication, Control & Computing, 2014 (2014), 629-636.  doi: 10.1109/ALLERTON.2014.7028514.  Google Scholar

[21]

M. Mandjes, Pricing strategies under heterogeneous service requirements, Computer Networks, 42 (2003), 231-249.   Google Scholar

[22]

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

[23]

A. Printezis and A. Burnetas, Priority option pricing in an M/M/m queue, Operations Research Letters, 36 (2008), 700-704.  doi: 10.1016/j.orl.2008.07.001.  Google Scholar

[24]

S. Ross, Introduction to Probability Models, Academic Press, Boston, 1989.  Google Scholar

[25]

W. Sun, P. Guo and N. Tian, Relative priority policies for minimizing the cost of queueing systems with service discrimination, Applied Mathematical Modelling, 33 (2009), 4241-4258.  doi: 10.1016/j.apm.2009.03.012.  Google Scholar

[26]

W. Sun, Sy. Li and C.-G. E., Equilibrium and optimal balking strategies of customers in Markovian queues with multiple vacations and N-policy, Applied Mathematical Modelling, 40 (2016), 284–301. doi: 10.1016/j.apm.2015.04.045.  Google Scholar

[27]

H. Takagi, Unified and refined analysis of the response time and waiting time in the M/M/m FCFS preemptive-resume priority queue, Journal of Industrial and Management Optimization, 13 (2017), 1945-1973.  doi: 10.3934/jimo.2017026.  Google Scholar

[28]

R. Tian, D. Yue and W. Yue, Optimal balking strategies in an M/G/1 queueing system with a removable server under N-policy, Journal of Industrial and Management Optimization, 11 (2015), 715-731.  doi: 10.3934/jimo.2015.11.715.  Google Scholar

[29]

B. Xu and X. Xu, Equilibrium strategic behavior of customers in the M/M/1 queue with partial failures and repairs, Operational Research: An International Journal, 18 (2018), 273-292.  doi: 10.1007/s12351-016-0264-7.  Google Scholar

[30]

B. Xu, X. Xu and X. Wang, Optimal balking strategies for high-priority customers in M/G/1 queues with 2 classes of customers, Journal of Applied Mathematics and Computing, 51 (2016), 623-642.  doi: 10.1007/s12190-015-0923-5.  Google Scholar

[31]

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

[32]

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), 901-917.  doi: 10.3934/jimo.2013.9.901.  Google Scholar

show all references

References:
[1]

E. Altaman and R. Hassin, Non-threshold equilibrium for customers joining an M/G/1 queue, International Symposium on Dynamic Games & Applications, 2002 (2002), 56-64.   Google Scholar

[2]

J. Altmann, H. Daanen, H. Oliver and A. S.-B. Suarez, How to market-manage a QoS network, Proceedings - IEEE INFOCOM 1, 2002 (2002), 56-64.  doi: 10.1109/INFCOM.2002.1019270.  Google Scholar

[3]

H. Berg, M. Mandjes and R. Nunez-Queija, Pricing and distributed QoS control for elastic network traffic, Operations Research Letters, 35 (2007), 297-307.  doi: 10.1016/j.orl.2006.03.018.  Google Scholar

[4]

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

[5]

G. Brouns and J. Wal, Optimal threshold policies in a two-class preemptive priority queue with admission and termination control, Queueing System, 54 (2006), 21-33.  doi: 10.1007/s11134-006-8307-z.  Google Scholar

[6]

F. Chen and V. Kulkarni, Individual, class-based, and social optimal admission policies in two-priority queues, Stochastic Models, 23 (2007), 97-127.  doi: 10.1080/15326340601142180.  Google Scholar

[7]

P. Chen and Y. Zhou, Equilibrium balking strategies in the single server queue with setup times and breakdowns, Operational Research: An International Journal, 15 (2015), 213-231.  doi: 10.1007/s12351-015-0174-0.  Google Scholar

[8]

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

[9]

N. Edelson and D. Hildebrand, Congestion tolls for Poisson queueing processes, Econometrica, 43 (1975), 81-92.  doi: 10.2307/1913415.  Google Scholar

[10]

J. Erlichman and R. Hassin, Equilibrium solutions in the observable M/M/1 queue with overtaking, International ICST Conference on Performance Evaluation Methodologies & Tools, (1975). doi: 10.4108/ICST.VALUETOOLS2009.8039.  Google Scholar

[11]

S. Gavirneni and V. Kulkarni, Self-selecting priority queues with burr distributed waiting costs, Production & Operations Management, 25 (2016), 979-992.   Google Scholar

[12]

W. Gilland and D. Warsing, The impact of revenue-maximizing priority pricing on customer delay costs, Decision Sciences, 40 (2009), 89-120.   Google Scholar

[13]

R. Hassin, Rational Queueing, CRC press, Boca Raton, 2016. doi: 10.1201/b20014.  Google Scholar

[14]

M. Haviv and Y. Kerner, On balking from an empty queue, Queueing Systems Theory & Applications, 55 (2007), 239-249.  doi: 10.1007/s11134-007-9020-2.  Google Scholar

[15]

V. Hsu, S. Xu and B. Jukic, Optimal scheduling and incentive compatible pricing for a service system with quality of service guarantees, Manufacturing & Service Operations Management, 11 (2008), 375-396.  doi: 10.1287/msom.1080.0226.  Google Scholar

[16]

Y. Kerner, Equilibrium joining probabilities for an M/G/1 queue, Games & Economic Behavior, 71 (2011), 521-526.  doi: 10.1016/j.geb.2010.06.002.  Google Scholar

[17]

Y. Kim and M. Mannino, Optimal incentive-compatible pricing for M/G/1 queues, Operations Research Letters, 31 (2003), 459-461.  doi: 10.1016/S0167-6377(03)00060-9.  Google Scholar

[18]

D. Lee and S. Park, Performance analysis of queueing strategies for multiple priority calls inmultiservice personal communications services, Computer Communications, 23 (2000), 1069-1083.   Google Scholar

[19]

L. Li, J. Wang and F. Zhang, Equilibrium customer strategies in Markovian queues with partial breakdowns, Computers & Industrial Engineering, 66 (2013), 751-757.  doi: 10.1016/j.cie.2013.09.023.  Google Scholar

[20]

C. Liu and R. Berry, A priority queue model for competition with shared spectrum, Communication, Control & Computing, 2014 (2014), 629-636.  doi: 10.1109/ALLERTON.2014.7028514.  Google Scholar

[21]

M. Mandjes, Pricing strategies under heterogeneous service requirements, Computer Networks, 42 (2003), 231-249.   Google Scholar

[22]

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

[23]

A. Printezis and A. Burnetas, Priority option pricing in an M/M/m queue, Operations Research Letters, 36 (2008), 700-704.  doi: 10.1016/j.orl.2008.07.001.  Google Scholar

[24]

S. Ross, Introduction to Probability Models, Academic Press, Boston, 1989.  Google Scholar

[25]

W. Sun, P. Guo and N. Tian, Relative priority policies for minimizing the cost of queueing systems with service discrimination, Applied Mathematical Modelling, 33 (2009), 4241-4258.  doi: 10.1016/j.apm.2009.03.012.  Google Scholar

[26]

W. Sun, Sy. Li and C.-G. E., Equilibrium and optimal balking strategies of customers in Markovian queues with multiple vacations and N-policy, Applied Mathematical Modelling, 40 (2016), 284–301. doi: 10.1016/j.apm.2015.04.045.  Google Scholar

[27]

H. Takagi, Unified and refined analysis of the response time and waiting time in the M/M/m FCFS preemptive-resume priority queue, Journal of Industrial and Management Optimization, 13 (2017), 1945-1973.  doi: 10.3934/jimo.2017026.  Google Scholar

[28]

R. Tian, D. Yue and W. Yue, Optimal balking strategies in an M/G/1 queueing system with a removable server under N-policy, Journal of Industrial and Management Optimization, 11 (2015), 715-731.  doi: 10.3934/jimo.2015.11.715.  Google Scholar

[29]

B. Xu and X. Xu, Equilibrium strategic behavior of customers in the M/M/1 queue with partial failures and repairs, Operational Research: An International Journal, 18 (2018), 273-292.  doi: 10.1007/s12351-016-0264-7.  Google Scholar

[30]

B. Xu, X. Xu and X. Wang, Optimal balking strategies for high-priority customers in M/G/1 queues with 2 classes of customers, Journal of Applied Mathematics and Computing, 51 (2016), 623-642.  doi: 10.1007/s12190-015-0923-5.  Google Scholar

[31]

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

[32]

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), 901-917.  doi: 10.3934/jimo.2013.9.901.  Google Scholar

Figure 1.  The service process of an arbitrary arriving class-1 customer
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Figure 2.  The service process of an arbitrary arriving class-2 customer
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Figure 3.  The equilibrium thresholds of the class-1 customers vs.$K_1$ for ${\lambda _2} = 0.5, E\left[ {{G_1}} \right] = E\left[ {{G_2}} \right] = 1, {C_1} = 8$
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Figure 4.  The equilibrium thresholds of the class-1 customers vs.$C_1$ for ${\lambda _2} = 0.5, E\left[ {{G_1}} \right] = E\left[ {{G_2}} \right] = 1, {K_1} = 100$
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Figure 5.  The equilibrium thresholds of the class-1 customers vs.$E\left[ {{G_1}} \right]$ for ${\lambda _2} = 0.5, E\left[ {{G_2}} \right] = 1, {K_1} = 100, {C_1} = 5$
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Figure 6.  The expected net social benefit vs.$q$ for ${\lambda _1} = {\lambda _2} = 0.5, E\left[ {{G_1}} \right] = E\left[ {{G_2}} \right] = 1, E\left[ {{G_1^2}} \right] = E\left[ {{G_2^2}} \right] = 1.2, {C_1} = 10, {K_2} = 100, {C_2} = 50$
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Figure 7.  The expected net social benefit vs.$q$ for ${\lambda _1} = {\lambda _2} = 0.5, E\left[ {{G_1}} \right] = E\left[ {{G_2}} \right] = 1, E\left[ {{G_1^2}} \right] = E\left[ {{G_2^2}} \right] = 1.2, {K_1} = 10, {K_2} = 100, {C_2} = 50$
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Figure 8.  Equilibrium and socially optimal joining probabilities of the class-1 customers vs. ${K_1}$ for ${\lambda _1} = {\lambda _2} = 0.5, E\left[ {{G_1}} \right] = E\left[ {{G_2}} \right] = 1, E\left[ {{G_1^2}} \right] = E\left[ {{G_2^2}} \right] = 1.2, {C_1} = 10$
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Figure 9.  Equilibrium and socially optimal joining probabilities of the class-1 customers vs. ${C_1}$ for ${\lambda _1} = {\lambda _2} = 0.5, E\left[ {{G_1}} \right] = E\left[ {{G_2}} \right] = 1, E\left[ {{G_1^2}} \right] = E\left[ {{G_2^2}} \right] = 1.2, {K_1} = 50$
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Figure 10.  Equilibrium and socially optimal joining probabilities of the class-1 customers vs. ${\lambda _1}$ for ${\lambda _2} = 0.2, E\left[ {{G_1}} \right] = E\left[ {{G_2}} \right] = 1, E\left[ {{G_1^2}} \right] = E\left[ {{G_2^2}} \right] = 1.2, {K_1} = 15, {C_1} = 10$
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