American Institute of Mathematical Sciences

Advanced
  • 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

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  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

[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

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
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The service process of an arbitrary arriving class-2 customer
Figure Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

Figure 10.  Equilibrium and socially optimal joining probabilities of the class-1 customers vs. ${\lambda _1}$ for ${\lambda _2} = 0.2, E\left[ {{G_1}} \right] = E\left[ {{G_2}} \right] = 1, E\left[ {{G_1^2}} \right] = E\left[ {{G_2^2}} \right] = 1.2, {K_1} = 15, {C_1} = 10$
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

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
[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 & 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 & 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 & Management Optimization, 2019  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 & 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 & 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 & 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 & 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 & Management Optimization, 2016, 12 (2) : 653-666. doi: 10.3934/jimo.2016.12.653
[10]

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 & Management Optimization, 2012, 8 (4) : 909-924. doi: 10.3934/jimo.2012.8.909
[11]

Chandra Shekhar, Amit Kumar, Shreekant Varshney, Sherif Ibrahim Ammar. $ \bf{M/G/1} $ fault-tolerant machining system with imperfection. Journal of Industrial & Management Optimization, 2019  doi: 10.3934/jimo.2019096
[12]

Sung-Seok Ko, Jangha Kang, E-Yeon Kwon. An $(s,S)$ inventory model with level-dependent $G/M/1$-Type structure. Journal of Industrial & Management Optimization, 2016, 12 (2) : 609-624. doi: 10.3934/jimo.2016.12.609
[13]

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 & Management Optimization, 2015, 11 (3) : 715-731. doi: 10.3934/jimo.2015.11.715
[14]

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 & Management Optimization, 2015, 11 (3) : 779-806. doi: 10.3934/jimo.2015.11.779
[15]

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 & Management Optimization, 2019  doi: 10.3934/jimo.2019082
[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 & Management Optimization, 2017, 13 (3) : 1449-1466. doi: 10.3934/jimo.2017001
[17]

Yung Chung Wang, Jenn Shing Wang, Fu Hsiang Tsai. Analysis of discrete-time space priority queue with fuzzy threshold. Journal of Industrial & Management Optimization, 2009, 5 (3) : 467-479. doi: 10.3934/jimo.2009.5.467
[18]

Qingqing Ye. Algorithmic computation of MAP/PH/1 queue with finite system capacity and two-stage vacations. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2459-2477. doi: 10.3934/jimo.2019063
[19]

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 & Management Optimization, 2020, 16 (3) : 1135-1148. doi: 10.3934/jimo.2018196
[20]

Kai Li, Yuqian Pan, Bohai Liu, Bayi Cheng. The setting and optimization of quick queue with customer loss. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1539-1553. doi: 10.3934/jimo.2019016

2019 Impact Factor: 1.366

Tools

Metrics

  • PDF downloads (194)
  • HTML views (1377)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences