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Queues with inspection cost: To see or not to see?

Author

Listed:
  • Jake Clarkson

    (Centre Inria d’Université Côte d’Azur)

  • Konstantin Avrachenkov

    (Centre Inria d’Université Côte d’Azur)

  • Eitan Altman

    (Centre Inria d’Université Côte d’Azur)

Abstract

Consider an M/M/1-type queue where joining attains a known reward, but a known waiting cost is paid per time unit spent queueing. In the 1960 s, Naor showed that any arrival optimally joins the queue if its length is less than a known threshold. Yet acquiring knowledge of the queue length often brings an additional cost, e.g., website loading time or data roaming charge. Therefore, our model presents any arrival with three options: join blindly, balk blindly, or pay a known inspection cost to make the optimal joining decision by comparing the queue length to Naor’s threshold. In a recent paper, Hassin and Roet-Green prove that a unique Nash equilibrium always exists and classify regions where the equilibrium probabilities are nonzero. We complement these findings with new closed-form expressions for the equilibrium probabilities in the majority of cases. Further, Hassin and Roet-Green show that minimising inspection cost maximises social welfare. Envisaging a queue operator choosing where to invest, we compare the effects of lowering inspection cost and increasing the queue-joining reward on social welfare. We prove that the former dominates and that the latter can even have a detrimental effect on social welfare.

Suggested Citation

  • Jake Clarkson & Konstantin Avrachenkov & Eitan Altman, 2025. "Queues with inspection cost: To see or not to see?," Queueing Systems: Theory and Applications, Springer, vol. 109(2), pages 1-34, June.
  • Handle: RePEc:spr:queues:v:109:y:2025:i:2:d:10.1007_s11134-025-09940-7
    DOI: 10.1007/s11134-025-09940-7
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    References listed on IDEAS

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    1. Young, H Peyton, 1993. "The Evolution of Conventions," Econometrica, Econometric Society, vol. 61(1), pages 57-84, January.
    2. Refael Hassin & Ricky Roet-Green, 2017. "The Impact of Inspection Cost on Equilibrium, Revenue, and Social Welfare in a Single-Server Queue," Operations Research, INFORMS, vol. 65(3), pages 804-820, June.
    3. Hassin, Refael, 1986. "Consumer Information in Markets with Random Product Quality: The Case of Queues and Balking," Econometrica, Econometric Society, vol. 54(5), pages 1185-1195, September.
    4. Ming Hu & Yang Li & Jianfu Wang, 2018. "Efficient Ignorance: Information Heterogeneity in a Queue," Management Science, INFORMS, vol. 64(6), pages 2650-2671, June.
    5. Edelson, Noel M & Hildebrand, David K, 1975. "Congestion Tolls for Poisson Queuing Processes," Econometrica, Econometric Society, vol. 43(1), pages 81-92, January.
    6. Kerner, Yoav, 2011. "Equilibrium joining probabilities for an M/G/1 queue," Games and Economic Behavior, Elsevier, vol. 71(2), pages 521-526, March.
    7. David Lingenbrink & Krishnamurthy Iyer, 2019. "Optimal Signaling Mechanisms in Unobservable Queues," Operations Research, INFORMS, vol. 67(5), pages 1397-1416, September.
    8. Antonis Economou, 2022. "How much information should be given to the strategic customers of a queueing system?," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 421-423, April.
    9. Naor, P, 1969. "The Regulation of Queue Size by Levying Tolls," Econometrica, Econometric Society, vol. 37(1), pages 15-24, January.
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    11. Rouba Ibrahim, 2018. "Sharing delay information in service systems: a literature survey," Queueing Systems: Theory and Applications, Springer, vol. 89(1), pages 49-79, June.
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