Figure 1.
The service process of an arbitrary arriving class-1 customer
-
Previous Article
Online ordering strategy for the discrete newsvendor problem with order value-based free-shipping
- JIMO Home
- This Issue
-
Next Article
Optimal pricing of perishable products with replenishment policy in the presence of strategic consumers
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
| 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 |
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.
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 and 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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [9] |
N. Edelson and D. Hildebrand,
Congestion tolls for Poisson queueing processes, Econometrica, 43 (1975), 81-92.
doi: 10.2307/1913415. |
| [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. |
| [11] |
S. Gavirneni and V. Kulkarni,
Self-selecting priority queues with burr distributed waiting costs, Production & Operations Management, 25 (2016), 979-992.
|
| [12] |
W. Gilland and D. Warsing,
The impact of revenue-maximizing priority pricing on customer delay costs, Decision Sciences, 40 (2009), 89-120.
|
| [13] |
R. Hassin, Rational Queueing, CRC press, Boca Raton, 2016.
doi: 10.1201/b20014. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [21] |
M. Mandjes,
Pricing strategies under heterogeneous service requirements, Computer Networks, 42 (2003), 231-249.
|
| [22] |
P. Naor,
The regulation of queue size by levying tolls, Econometrica, 37 (1969), 15-24.
doi: 10.2307/1909200. |
| [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. |
| [24] |
S. Ross, Introduction to Probability Models, Academic Press, Boston, 1989. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [9] |
N. Edelson and D. Hildebrand,
Congestion tolls for Poisson queueing processes, Econometrica, 43 (1975), 81-92.
doi: 10.2307/1913415. |
| [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. |
| [11] |
S. Gavirneni and V. Kulkarni,
Self-selecting priority queues with burr distributed waiting costs, Production & Operations Management, 25 (2016), 979-992.
|
| [12] |
W. Gilland and D. Warsing,
The impact of revenue-maximizing priority pricing on customer delay costs, Decision Sciences, 40 (2009), 89-120.
|
| [13] |
R. Hassin, Rational Queueing, CRC press, Boca Raton, 2016.
doi: 10.1201/b20014. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [21] |
M. Mandjes,
Pricing strategies under heterogeneous service requirements, Computer Networks, 42 (2003), 231-249.
|
| [22] |
P. Naor,
The regulation of queue size by levying tolls, Econometrica, 37 (1969), 15-24.
doi: 10.2307/1909200. |
| [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. |
| [24] |
S. Ross, Introduction to Probability Models, Academic Press, Boston, 1989. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
Figure 2.
The service process of an arbitrary arriving class-2 customer
Figure 3.
The equilibrium thresholds of the class-1 customers vs.$K_1$ for ${\lambda _2} = 0.5, E\left[ {{G_1}} \right] = E\left[ {{G_2}} \right] = 1, {C_1} = 8$
Figure 4.
The equilibrium thresholds of the class-1 customers vs.$C_1$ for ${\lambda _2} = 0.5, E\left[ {{G_1}} \right] = E\left[ {{G_2}} \right] = 1, {K_1} = 100$
Figure 5.
The equilibrium thresholds of the class-1 customers vs.$E\left[ {{G_1}} \right]$ for ${\lambda _2} = 0.5, E\left[ {{G_2}} \right] = 1, {K_1} = 100, {C_1} = 5$
Figure 6.
The expected net social benefit vs.$q$ for ${\lambda _1} = {\lambda _2} = 0.5, E\left[ {{G_1}} \right] = E\left[ {{G_2}} \right] = 1, E\left[ {{G_1^2}} \right] = E\left[ {{G_2^2}} \right] = 1.2, {C_1} = 10, {K_2} = 100, {C_2} = 50$
Figure 7.
The expected net social benefit vs.$q$ for ${\lambda _1} = {\lambda _2} = 0.5, E\left[ {{G_1}} \right] = E\left[ {{G_2}} \right] = 1, E\left[ {{G_1^2}} \right] = E\left[ {{G_2^2}} \right] = 1.2, {K_1} = 10, {K_2} = 100, {C_2} = 50$
Figure 8.
Equilibrium and socially optimal joining probabilities of the class-1 customers vs. ${K_1}$ for ${\lambda _1} = {\lambda _2} = 0.5, E\left[ {{G_1}} \right] = E\left[ {{G_2}} \right] = 1, E\left[ {{G_1^2}} \right] = E\left[ {{G_2^2}} \right] = 1.2, {C_1} = 10$
Figure 9.
Equilibrium and socially optimal joining probabilities of the class-1 customers vs. ${C_1}$ for ${\lambda _1} = {\lambda _2} = 0.5, E\left[ {{G_1}} \right] = E\left[ {{G_2}} \right] = 1, E\left[ {{G_1^2}} \right] = E\left[ {{G_2^2}} \right] = 1.2, {K_1} = 50$
Figure 10.
Equilibrium and socially optimal joining probabilities of the class-1 customers vs. ${\lambda _1}$ for ${\lambda _2} = 0.2, E\left[ {{G_1}} \right] = E\left[ {{G_2}} \right] = 1, E\left[ {{G_1^2}} \right] = E\left[ {{G_2^2}} \right] = 1.2, {K_1} = 15, {C_1} = 10$
| [1] |
Sheng Zhu, Jinting Wang. Strategic behavior and optimal strategies in an M/G/1 queue with Bernoulli vacations. Journal of Industrial and Management Optimization, 2018, 14 (4) : 1297-1322. doi: 10.3934/jimo.2018008 |
| [2] |
Shaojun Lan, Yinghui Tang. Performance analysis of a discrete-time $ Geo/G/1$ retrial queue with non-preemptive priority, working vacations and vacation interruption. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1421-1446. doi: 10.3934/jimo.2018102 |
| [3] |
Hideaki 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, 2017, 13 (4) : 1945-1973. doi: 10.3934/jimo.2017026 |
| [4] |
Dhanya Shajin, A. N. Dudin, Olga Dudina, A. Krishnamoorthy. A two-priority single server retrial queue with additional items. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2891-2912. doi: 10.3934/jimo.2019085 |
| [5] |
Chia-Huang Wu, Kuo-Hsiung Wang, Jau-Chuan Ke, Jyh-Bin Ke. A heuristic algorithm for the optimization of M/M/$s$ queue with multiple working vacations. Journal of Industrial and Management Optimization, 2012, 8 (1) : 1-17. doi: 10.3934/jimo.2012.8.1 |
| [6] |
Dequan Yue, Wuyi Yue, Gang Xu. Analysis of customers' impatience in an M/M/1 queue with working vacations. Journal of Industrial and Management Optimization, 2012, 8 (4) : 895-908. doi: 10.3934/jimo.2012.8.895 |
| [7] |
Shaojun Lan, Yinghui Tang, Miaomiao Yu. System capacity optimization design and optimal threshold $N^{*}$ for a $GEO/G/1$ discrete-time queue with single server vacation and under the control of Min($N, V$)-policy. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1435-1464. doi: 10.3934/jimo.2016.12.1435 |
| [8] |
Ahmed M. K. Tarabia. Transient and steady state analysis of an M/M/1 queue with balking, catastrophes, server failures and repairs. Journal of Industrial and Management Optimization, 2011, 7 (4) : 811-823. doi: 10.3934/jimo.2011.7.811 |
| [9] |
Dequan Yue, Wuyi Yue, Guoxi Zhao. Analysis of an M/M/1 queue with vacations and impatience timers which depend on the server's states. Journal of Industrial and Management Optimization, 2016, 12 (2) : 653-666. doi: 10.3934/jimo.2016.12.653 |
| [10] |
Arnaud Devos, Joris Walraevens, Tuan Phung-Duc, Herwig Bruneel. Analysis of the queue lengths in a priority retrial queue with constant retrial policy. Journal of Industrial and Management Optimization, 2020, 16 (6) : 2813-2842. doi: 10.3934/jimo.2019082 |
| [11] |
Veena Goswami, M. L. Chaudhry. Explicit results for the distribution of the number of customers served during a busy period for $M^X/PH/1$ queue. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021168 |
| [12] |
Jerim Kim, Bara Kim, Hwa-Sung Kim. G/M/1 type structure of a risk model with general claim sizes in a Markovian environment. Journal of Industrial and Management Optimization, 2012, 8 (4) : 909-924. doi: 10.3934/jimo.2012.8.909 |
| [13] |
Chandra Shekhar, Amit Kumar, Shreekant Varshney, Sherif Ibrahim Ammar. $ \bf{M/G/1} $ fault-tolerant machining system with imperfection. Journal of Industrial and Management Optimization, 2021, 17 (1) : 1-28. doi: 10.3934/jimo.2019096 |
| [14] |
Ruiling Tian, Dequan Yue, Wuyi Yue. Optimal balking strategies in an M/G/1 queueing system with a removable server under N-policy. Journal of Industrial and Management Optimization, 2015, 11 (3) : 715-731. doi: 10.3934/jimo.2015.11.715 |
| [15] |
Sung-Seok Ko, Jangha Kang, E-Yeon Kwon. An $(s,S)$ inventory model with level-dependent $G/M/1$-Type structure. Journal of Industrial and Management Optimization, 2016, 12 (2) : 609-624. doi: 10.3934/jimo.2016.12.609 |
| [16] |
Yuan Zhao, Wuyi Yue. Cognitive radio networks with multiple secondary users under two kinds of priority schemes: Performance comparison and optimization. Journal of Industrial and Management Optimization, 2017, 13 (3) : 1449-1466. doi: 10.3934/jimo.2017001 |
| [17] |
Shan Gao, Jinting Wang. On a discrete-time GI$^X$/Geo/1/N-G queue with randomized working vacations and at most $J$ vacations. Journal of Industrial and Management Optimization, 2015, 11 (3) : 779-806. doi: 10.3934/jimo.2015.11.779 |
| [18] |
Yung Chung Wang, Jenn Shing Wang, Fu Hsiang Tsai. Analysis of discrete-time space priority queue with fuzzy threshold. Journal of Industrial and Management Optimization, 2009, 5 (3) : 467-479. doi: 10.3934/jimo.2009.5.467 |
| [19] |
Qingqing Ye. Algorithmic computation of MAP/PH/1 queue with finite system capacity and two-stage vacations. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2459-2477. doi: 10.3934/jimo.2019063 |
| [20] |
Zhanyou Ma, Wenbo Wang, Linmin Hu. Performance evaluation and analysis of a discrete queue system with multiple working vacations and non-preemptive priority. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1135-1148. doi: 10.3934/jimo.2018196 |
2020 Impact Factor: 1.801
Tools
Metrics
Other articles
by authors
[Back to Top]