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Price Decisions in a Two-Server Queue Considering Customer Retrial Behavior: Profit-Driven Versus Social-Driven

Author

Listed:
  • Xiaoli Cai

    (College of Management Science, Chengdu University of Technology, Chengdu 610059, China)

  • Miaomiao Yu

    (School of Mathematics Sciences, Sichuan Normal University, Chengdu 610066, China)

  • Yunling Yang

    (Department of Mathematics, Faculty 3-Mathematics and Computer Science, University of Bremen, 28359 Bremen, Germany)

Abstract

This study investigates price decisions in a queue with two servers, where customers exhibit retrial behavior. There is no waiting space. Arrival customers have the option to either join the system or balk; when the two servers are occupied, those who decide to enter become repeat customers. Two scenarios where the waiting lines in orbit are unobservable and observable are considered. We analyze customers’ behavior and derive their Nash equilibrium strategies under both cases. Additionally, we examine optimal pricing decisions aimed at maximizing profit and social welfare, respectively. Moreover, we demonstrate that these objectives often lead to divergent outcomes. Compared to a single-server queue, the reduction in customers’ sojourn time is more obvious when the waiting line is unobservable. Under certain conditions—such as a large potential market size, high customer impatience, or a low retrial rate—increasing the number of service personnel can enhance both profit and social welfare. Notably, a profit-maximizing manager is more incentivized to increase servers than the social planner. These findings provide valuable insights for balancing operational efficiency, profitability, and customer satisfaction in queue management systems.

Suggested Citation

  • Xiaoli Cai & Miaomiao Yu & Yunling Yang, 2025. "Price Decisions in a Two-Server Queue Considering Customer Retrial Behavior: Profit-Driven Versus Social-Driven," Mathematics, MDPI, vol. 13(8), pages 1-21, April.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:8:p:1310-:d:1636196
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    References listed on IDEAS

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    1. Hanukov, Gabi, 2022. "Improving efficiency of service systems by performing a part of the service without the customer's presence," European Journal of Operational Research, Elsevier, vol. 302(2), pages 606-620.
    2. Antonis Economou & Spyridoula Kanta, 2011. "Equilibrium customer strategies and social–profit maximization in the single‐server constant retrial queue," Naval Research Logistics (NRL), John Wiley & Sons, vol. 58(2), pages 107-122, March.
    3. Refael Hassin, 1996. "On the Advantage of Being the First Server," Management Science, INFORMS, vol. 42(4), pages 618-623, April.
    4. Ioannis Dimitriou, 2023. "A single server retrial queue with event-dependent arrival rates," Annals of Operations Research, Springer, vol. 331(2), pages 1053-1088, December.
    5. Mor Armony & Erica Plambeck & Sridhar Seshadri, 2009. "Sensitivity of Optimal Capacity to Customer Impatience in an Unobservable M/M/S Queue (Why You Shouldn't Shout at the DMV)," Manufacturing & Service Operations Management, INFORMS, vol. 11(1), pages 19-32, June.
    6. Uri Yechiali, 1972. "Customers' Optimal Joining Rules for the GI/M/s Queue," Management Science, INFORMS, vol. 18(7), pages 434-443, March.
    7. Edelson, Noel M & Hildebrand, David K, 1975. "Congestion Tolls for Poisson Queuing Processes," Econometrica, Econometric Society, vol. 43(1), pages 81-92, January.
    8. Mor Armony & Constantinos Maglaras, 2004. "Contact Centers with a Call-Back Option and Real-Time Delay Information," Operations Research, INFORMS, vol. 52(4), pages 527-545, August.
    9. Naor, P, 1969. "The Regulation of Queue Size by Levying Tolls," Econometrica, Econometric Society, vol. 37(1), pages 15-24, January.
    10. Zhao, Chen & Wang, Zhongbin, 2023. "The impact of line-sitting on a two-server queueing system," European Journal of Operational Research, Elsevier, vol. 308(2), pages 782-800.
    11. Zhongbin Wang & Jinting Wang, 2019. "Information heterogeneity in a retrial queue: throughput and social welfare maximization," Queueing Systems: Theory and Applications, Springer, vol. 92(1), pages 131-172, June.
    12. Wenhui Zhou & Weixiang Huang & Vernon N. Hsu & Pengfei Guo, 2023. "On the Benefit of Privatization in a Mixed Duopoly Service System," Management Science, INFORMS, vol. 69(3), pages 1486-1499, March.
    13. Yu Zhang & Jinting Wang, 2023. "Managing retrial queueing systems with boundedly rational customers," Journal of the Operational Research Society, Taylor & Francis Journals, vol. 74(3), pages 748-761, March.
    14. Shiliang Cui & Xuanming Su & Senthil Veeraraghavan, 2019. "A Model of Rational Retrials in Queues," Operations Research, INFORMS, vol. 67(6), pages 1699-1718, November.
    15. Wang, Jinting & Zhang, Feng, 2013. "Strategic joining in M/M/1 retrial queues," European Journal of Operational Research, Elsevier, vol. 230(1), pages 76-87.
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