IDEAS home Printed from https://ideas.repec.org/a/spr/queues/v109y2025i1d10.1007_s11134-024-09931-0.html
   My bibliography  Save this article

Second-order bounds for the M/M/s queue with random arrival rate

Author

Listed:
  • Wouter J. E. C. Eekelen

    (University of Chicago)

  • Grani A. Hanasusanto

    (University of Illinois Urbana-Champaign)

  • John J. Hasenbein

    (The University of Texas at Austin)

  • Johan S. H. Leeuwaarden

    (Tilburg University)

Abstract

Consider an M/M/s queue with the additional feature that the arrival rate is a random variable of which only the mean, variance, and range are known. Using semi-infinite linear programming and duality theory for moment problems, we establish for this setting tight bounds for the expected waiting time. These bounds correspond to an arrival rate that takes only two values. The proofs crucially depend on the fact that the expected waiting time, as function of the arrival rate, has a convex derivative. We apply the novel tight bounds to a rational queueing model, where arriving individuals decide to join or balk based on expected utility and only have partial knowledge about the market size.

Suggested Citation

  • Wouter J. E. C. Eekelen & Grani A. Hanasusanto & John J. Hasenbein & Johan S. H. Leeuwaarden, 2025. "Second-order bounds for the M/M/s queue with random arrival rate," Queueing Systems: Theory and Applications, Springer, vol. 109(1), pages 1-31, March.
  • Handle: RePEc:spr:queues:v:109:y:2025:i:1:d:10.1007_s11134-024-09931-0
    DOI: 10.1007/s11134-024-09931-0
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11134-024-09931-0
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s11134-024-09931-0?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. James E. Smith, 1995. "Generalized Chebychev Inequalities: Theory and Applications in Decision Analysis," Operations Research, INFORMS, vol. 43(5), pages 807-825, October.
    2. Athanassios N. Avramidis & Alexandre Deslauriers & Pierre L'Ecuyer, 2004. "Modeling Daily Arrivals to a Telephone Call Center," Management Science, INFORMS, vol. 50(7), pages 896-908, July.
    3. Arie Harel, 1990. "Convexity Properties of the Erlang Loss Formula," Operations Research, INFORMS, vol. 38(3), pages 499-505, June.
    4. Ramandeep S. Randhawa, 2016. "Optimality gap of asymptotically derived prescriptions in queueing systems," Queueing Systems: Theory and Applications, Springer, vol. 83(1), pages 131-155, June.
    5. Richard R. Weber, 1980. "Note---On the Marginal Benefit of Adding Servers to G/GI/m Queues," Management Science, INFORMS, vol. 26(9), pages 946-951, September.
    6. Edelson, Noel M & Hildebrand, David K, 1975. "Congestion Tolls for Poisson Queuing Processes," Econometrica, Econometric Society, vol. 43(1), pages 81-92, January.
    7. Ying Chen & John J. Hasenbein, 2020. "Knowledge, congestion, and economics: Parameter uncertainty in Naor’s model," Queueing Systems: Theory and Applications, Springer, vol. 96(1), pages 83-99, October.
    8. Naor, P, 1969. "The Regulation of Queue Size by Levying Tolls," Econometrica, Econometric Society, vol. 37(1), pages 15-24, January.
    9. Yan Chen & Ward Whitt, 2021. "Extremal GI/GI/1 queues given two moments: exploiting Tchebycheff systems," Queueing Systems: Theory and Applications, Springer, vol. 97(1), pages 101-124, February.
    10. Oldrich A. Vasicek, 1977. "Technical Note—An Inequality for the Variance of Waiting Time under a General Queuing Discipline," Operations Research, INFORMS, vol. 25(5), pages 879-884, October.
    11. Yan Chen & Ward Whitt, 2022. "Correction to: Extremal GI/GI/1 queues given two moments: exploiting Tchebycheff systems," Queueing Systems: Theory and Applications, Springer, vol. 102(3), pages 553-556, December.
    12. Erick Delage & Yinyu Ye, 2010. "Distributionally Robust Optimization Under Moment Uncertainty with Application to Data-Driven Problems," Operations Research, INFORMS, vol. 58(3), pages 595-612, June.
    13. Geurt Jongbloed & Ger Koole, 2001. "Managing uncertainty in call centres using Poisson mixtures," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 17(4), pages 307-318, October.
    14. J. Michael Harrison & Assaf Zeevi, 2005. "A Method for Staffing Large Call Centers Based on Stochastic Fluid Models," Manufacturing & Service Operations Management, INFORMS, vol. 7(1), pages 20-36, September.
    15. Richard R. Weber, 1983. "Technical Note—A Note on Waiting Times in Single Server Queues," Operations Research, INFORMS, vol. 31(5), pages 950-951, October.
    16. A. E. Eckberg, 1977. "Sharp Bounds on Laplace-Stieltjes Transforms, with Applications to Various Queueing Problems," Mathematics of Operations Research, INFORMS, vol. 2(2), pages 135-142, May.
    17. Ying Chen & John J. Hasenbein, 2017. "Staffing large-scale service systems with distributional uncertainty," Queueing Systems: Theory and Applications, Springer, vol. 87(1), pages 55-79, October.
    18. Hassin, Refael & Haviv, Moshe & Oz, Binyamin, 2023. "Strategic behavior in queues with arrival rate uncertainty," European Journal of Operational Research, Elsevier, vol. 309(1), pages 217-224.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. van Eekelen, Wouter, 2023. "Distributionally robust views on queues and related stochastic models," Other publications TiSEM 9b99fc05-9d68-48eb-ae8c-9, Tilburg University, School of Economics and Management.
    2. Hassin, Refael & Haviv, Moshe & Oz, Binyamin, 2023. "Strategic behavior in queues with arrival rate uncertainty," European Journal of Operational Research, Elsevier, vol. 309(1), pages 217-224.
    3. Ward Whitt, 2006. "Staffing a Call Center with Uncertain Arrival Rate and Absenteeism," Production and Operations Management, Production and Operations Management Society, vol. 15(1), pages 88-102, March.
    4. Ying Chen & John J. Hasenbein, 2020. "Knowledge, congestion, and economics: Parameter uncertainty in Naor’s model," Queueing Systems: Theory and Applications, Springer, vol. 96(1), pages 83-99, October.
    5. Legros, Benjamin & van Leeuwaarden, J.S.H. & Fransoo, Jan C., 2025. "Managing reusable resources with usage time limits," Other publications TiSEM 10a58c0a-9a6c-49a4-8a41-d, Tilburg University, School of Economics and Management.
    6. Boris N. Oreshkin & Nazim Réegnard & Pierre L’Ecuyer, 2016. "Rate-Based Daily Arrival Process Models with Application to Call Centers," Operations Research, INFORMS, vol. 64(2), pages 510-527, April.
    7. Tevfik Aktekin & Refik Soyer, 2012. "Bayesian analysis of queues with impatient customers: Applications to call centers," Naval Research Logistics (NRL), John Wiley & Sons, vol. 59(6), pages 441-456, September.
    8. Noah Gans & Haipeng Shen & Yong-Pin Zhou & Nikolay Korolev & Alan McCord & Herbert Ristock, 2015. "Parametric Forecasting and Stochastic Programming Models for Call-Center Workforce Scheduling," Manufacturing & Service Operations Management, INFORMS, vol. 17(4), pages 571-588, October.
    9. Achal Bassamboo & Ramandeep S. Randhawa & Assaf Zeevi, 2010. "Capacity Sizing Under Parameter Uncertainty: Safety Staffing Principles Revisited," Management Science, INFORMS, vol. 56(10), pages 1668-1686, October.
    10. Refael Hassin, 2022. "Profit maximization and cost balancing in queueing systems," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 429-431, April.
    11. Barrow, Devon & Kourentzes, Nikolaos, 2018. "The impact of special days in call arrivals forecasting: A neural network approach to modelling special days," European Journal of Operational Research, Elsevier, vol. 264(3), pages 967-977.
    12. Rouba Ibrahim & Pierre L'Ecuyer, 2013. "Forecasting Call Center Arrivals: Fixed-Effects, Mixed-Effects, and Bivariate Models," Manufacturing & Service Operations Management, INFORMS, vol. 15(1), pages 72-85, May.
    13. Soonhui Lee & Tito Homem-de-Mello & Anton Kleywegt, 2012. "Newsvendor-type models with decision-dependent uncertainty," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 76(2), pages 189-221, October.
    14. Alex Roubos & Ger Koole & Raik Stolletz, 2012. "Service-Level Variability of Inbound Call Centers," Manufacturing & Service Operations Management, INFORMS, vol. 14(3), pages 402-413, July.
    15. Viet Anh Nguyen & Fan Zhang & Shanshan Wang & Jose Blanchet & Erick Delage & Yinyu Ye, 2021. "Robustifying Conditional Portfolio Decisions via Optimal Transport," Papers 2103.16451, arXiv.org, revised Apr 2024.
    16. Manlu Chen & Ming Hu & Jianfu Wang, 2022. "Food Delivery Service and Restaurant: Friend or Foe?," Management Science, INFORMS, vol. 68(9), pages 6539-6551, September.
    17. 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.
    18. 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.
    19. Legros, Benjamin & Fransoo, Jan C., 2023. "Admission and pricing optimization of on-street parking with delivery bays," Other publications TiSEM 6d41ee5c-27dc-4d34-aff1-4, Tilburg University, School of Economics and Management.
    20. Achal Bassamboo & Assaf Zeevi, 2009. "On a Data-Driven Method for Staffing Large Call Centers," Operations Research, INFORMS, vol. 57(3), pages 714-726, June.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:queues:v:109:y:2025:i:1:d:10.1007_s11134-024-09931-0. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.