IDEAS home Printed from https://ideas.repec.org/a/spr/queues/v106y2024i1d10.1007_s11134-023-09900-z.html
   My bibliography  Save this article

Adaptive service rate control of an M/M/1 queue with server breakdowns

Author

Listed:
  • Yi Zheng

    (Pennsylvania State University)

  • Juxihong Julaiti

    (Pennsylvania State University)

  • Guodong Pang

    (George R. Brown School of Engineering, Rice University)

Abstract

We study service rate control problems for an M/M/1 queue with server breakdowns in which the breakdown rate is assumed to be a function of the service rate. Assuming that the queue has infinite capacity, we first establish the optimality equations for the discounted cost problem and characterize the optimal rate control policies. Then, we characterize the ergodicity of the controlled queue and establish the optimality conditions for the average-cost (ergodic) control problem using the vanishing discounted method. We next study the ergodic control problem when the queue has a finite capacity and establish a verification theorem by directly involving the stationary distribution of the controlled Markov process. For practical applications, we consider the adaptive service rate control problem for the model with finite capacity. Studying this problem is useful because the relationship between the server breakdown rate and the service rate is costly to observe in practice. We propose an adaptive (self-tuning) control algorithm, assuming that the relationship between the server breakdown rate and the service rate is linear with unknown parameters. We prove that the regret vanishes under the algorithm and the proposed policies are asymptotically optimal. In addition, numerical experiments are conducted to validate the algorithm.

Suggested Citation

  • Yi Zheng & Juxihong Julaiti & Guodong Pang, 2024. "Adaptive service rate control of an M/M/1 queue with server breakdowns," Queueing Systems: Theory and Applications, Springer, vol. 106(1), pages 159-191, February.
    • Handle: RePEc:spr:queues:v:106:y:2024:i:1:d:10.1007_s11134-023-09900-z
    DOI: 10.1007/s11134-023-09900-z
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11134-023-09900-z
    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/s11134-023-09900-z?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. Ananda Weerasinghe, 2014. "Diffusion Approximations for G / M / n + GI Queues with State-Dependent Service Rates," Mathematics of Operations Research, INFORMS, vol. 39(1), pages 207-228, February.
    2. Arnoud V. den Boer & Bert Zwart, 2014. "Simultaneously Learning and Optimizing Using Controlled Variance Pricing," Management Science, INFORMS, vol. 60(3), pages 770-783, March.
    3. N. Bora Keskin & Assaf Zeevi, 2014. "Dynamic Pricing with an Unknown Demand Model: Asymptotically Optimal Semi-Myopic Policies," Operations Research, INFORMS, vol. 62(5), pages 1142-1167, October.
    4. Eugene A. Feinberg & Mark E. Lewis, 2007. "Optimality Inequalities for Average Cost Markov Decision Processes and the Stochastic Cash Balance Problem," Mathematics of Operations Research, INFORMS, vol. 32(4), pages 769-783, November.
    5. Omar Besbes & Assaf Zeevi, 2015. "On the (Surprising) Sufficiency of Linear Models for Dynamic Pricing with Demand Learning," Management Science, INFORMS, vol. 61(4), pages 723-739, April.
    6. Boxiao Chen & Xiuli Chao & Cong Shi, 2021. "Nonparametric Learning Algorithms for Joint Pricing and Inventory Control with Lost Sales and Censored Demand," Mathematics of Operations Research, INFORMS, vol. 46(2), pages 726-756, May.
    7. Baric{s} Ata & Shiri Shneorson, 2006. "Dynamic Control of an M/M/1 Service System with Adjustable Arrival and Service Rates," Management Science, INFORMS, vol. 52(11), pages 1778-1791, November.
    8. Barιş Ata, 2005. "Dynamic Power Control in a Wireless Static Channel Subject to a Quality-of-Service Constraint," Operations Research, INFORMS, vol. 53(5), pages 842-851, October.
    9. Guodong Pang & Ward Whitt, 2009. "Service Interruptions in Large-Scale Service Systems," Management Science, INFORMS, vol. 55(9), pages 1499-1512, September.
    10. Jennifer M. George & J. Michael Harrison, 2001. "Dynamic Control of a Queue with Adjustable Service Rate," Operations Research, INFORMS, vol. 49(5), pages 720-731, October.
    11. Ravi Kumar & Mark E. Lewis & Huseyin Topaloglu, 2013. "Dynamic service rate control for a single‐server queue with Markov‐modulated arrivals," Naval Research Logistics (NRL), John Wiley & Sons, vol. 60(8), pages 661-677, December.
    12. Qi (George) Chen & Stefanus Jasin & Izak Duenyas, 2019. "Nonparametric Self-Adjusting Control for Joint Learning and Optimization of Multiproduct Pricing with Finite Resource Capacity," Mathematics of Operations Research, INFORMS, vol. 44(2), pages 601-631, May.
    13. Cong Shi & Weidong Chen & Izak Duenyas, 2016. "Technical Note—Nonparametric Data-Driven Algorithms for Multiproduct Inventory Systems with Censored Demand," Operations Research, INFORMS, vol. 64(2), pages 362-370, April.
    14. B. Avi-Itzhak & P. Naor, 1963. "Some Queuing Problems with the Service Station Subject to Breakdown," Operations Research, INFORMS, vol. 11(3), pages 303-320, June.
    15. Omar Besbes & Assaf Zeevi, 2009. "Dynamic Pricing Without Knowing the Demand Function: Risk Bounds and Near-Optimal Algorithms," Operations Research, INFORMS, vol. 57(6), pages 1407-1420, December.
    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. Thomas Loots & Arnoud V. den Boer, 2023. "Data‐driven collusion and competition in a pricing duopoly with multinomial logit demand," Production and Operations Management, Production and Operations Management Society, vol. 32(4), pages 1169-1186, April.
    2. Yang, Xiangyu & Zhang, Jianghua & Hu, Jian-Qiang & Hu, Jiaqiao, 2024. "Nonparametric multi-product dynamic pricing with demand learning via simultaneous price perturbation," European Journal of Operational Research, Elsevier, vol. 319(1), pages 191-205.
    3. Jianqing Fan & Yongyi Guo & Mengxin Yu, 2024. "Policy Optimization Using Semiparametric Models for Dynamic Pricing," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 119(545), pages 552-564, January.
    4. Boxiao Chen & Xiuli Chao & Cong Shi, 2021. "Nonparametric Learning Algorithms for Joint Pricing and Inventory Control with Lost Sales and Censored Demand," Mathematics of Operations Research, INFORMS, vol. 46(2), pages 726-756, May.
    5. Yiwei Chen & Cong Shi, 2023. "Network revenue management with online inverse batch gradient descent method," Production and Operations Management, Production and Operations Management Society, vol. 32(7), pages 2123-2137, July.
    6. Boxiao Chen & David Simchi-Levi & Yining Wang & Yuan Zhou, 2022. "Dynamic Pricing and Inventory Control with Fixed Ordering Cost and Incomplete Demand Information," Management Science, INFORMS, vol. 68(8), pages 5684-5703, August.
    7. Joon Suk Huh & Ellen Vitercik & Kirthevasan Kandasamy, 2024. "Bandit Profit-maximization for Targeted Marketing," Papers 2403.01361, arXiv.org, revised Jul 2024.
    8. Qi Feng & J. George Shanthikumar, 2022. "Developing operations management data analytics," Production and Operations Management, Production and Operations Management Society, vol. 31(12), pages 4544-4557, December.
    9. Athanassios N. Avramidis & Arnoud V. Boer, 2021. "Dynamic pricing with finite price sets: a non-parametric approach," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 94(1), pages 1-34, August.
    10. Huashuai Qu & Ilya O. Ryzhov & Michael C. Fu & Eric Bergerson & Megan Kurka & Ludek Kopacek, 2020. "Learning Demand Curves in B2B Pricing: A New Framework and Case Study," Production and Operations Management, Production and Operations Management Society, vol. 29(5), pages 1287-1306, May.
    11. Athanassios N. Avramidis, 2020. "A pricing problem with unknown arrival rate and price sensitivity," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(1), pages 77-106, August.
    12. Gur, Yonatan & Macnamara, Gregory & Saban, Daniela, 2020. "On the Disclosure of Promotion Value in Platforms with Learning Sellers," Research Papers 3865, Stanford University, Graduate School of Business.
    13. den Boer, Arnoud V., 2015. "Tracking the market: Dynamic pricing and learning in a changing environment," European Journal of Operational Research, Elsevier, vol. 247(3), pages 914-927.
    14. Ruben Geer & Arnoud V. Boer & Christopher Bayliss & Christine S. M. Currie & Andria Ellina & Malte Esders & Alwin Haensel & Xiao Lei & Kyle D. S. Maclean & Antonio Martinez-Sykora & Asbjørn Nilsen Ris, 2019. "Dynamic pricing and learning with competition: insights from the dynamic pricing challenge at the 2017 INFORMS RM & pricing conference," Journal of Revenue and Pricing Management, Palgrave Macmillan, vol. 18(3), pages 185-203, June.
    15. Benjamin Legros, 2022. "The principal-agent problem for service rate event-dependency," Post-Print hal-03605421, HAL.
    16. Wang Chi Cheung & David Simchi-Levi & He Wang, 2017. "Technical Note—Dynamic Pricing and Demand Learning with Limited Price Experimentation," Operations Research, INFORMS, vol. 65(6), pages 1722-1731, December.
    17. Yining Wang & Xi Chen & Xiangyu Chang & Dongdong Ge, 2021. "Uncertainty Quantification for Demand Prediction in Contextual Dynamic Pricing," Production and Operations Management, Production and Operations Management Society, vol. 30(6), pages 1703-1717, June.
    18. Barιş Ata & Deishin Lee & Erkut Sönmez, 2019. "Dynamic Volunteer Staffing in Multicrop Gleaning Operations," Operations Research, INFORMS, vol. 67(2), pages 295-314, March.
    19. Sentao Miao & Xi Chen & Xiuli Chao & Jiaxi Liu & Yidong Zhang, 2022. "Context‐based dynamic pricing with online clustering," Production and Operations Management, Production and Operations Management Society, vol. 31(9), pages 3559-3575, September.
    20. Doan, Xuan Vinh & Lei, Xiao & Shen, Siqian, 2020. "Pricing of reusable resources under ambiguous distributions of demand and service time with emerging applications," European Journal of Operational Research, Elsevier, vol. 282(1), pages 235-251.

    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:106:y:2024:i:1:d:10.1007_s11134-023-09900-z. 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.