IDEAS home Printed from https://ideas.repec.org/a/spr/queues/v96y2020i3d10.1007_s11134-020-09677-5.html
   My bibliography  Save this article

Equilibrium points and equilibrium sets of some $$\textit{GI}/M/1$$ GI / M / 1 queues

Author

Listed:
  • N. Hemachandra

    (IIT Bombay)

  • Kishor Patil

    (INRIA Sophia Antipolis)

  • Sandhya Tripathi

    (Washington University School of Medicine at St. Louis)

Abstract

Queues can be seen as a service facility where quality of service (QoS) is an important measure for the performance of the system. In many cases, the queue implements the optimal admission control (either discounted or average) policy in the presence of holding/congestion cost and revenue collected from admitted customers. In this paper, users offer an arrival rate at stationarity that depends on the QoS they experience. We study the interaction between arriving customers and such a queue under two different QoS measures—the asymptotic rate of the customers lost and the fraction of customers lost in the long run. In particular, we investigate the behaviour of equilibrium points and equilibrium sets associated with this interaction and their interpretations in terms of business cycles. We provide sufficient conditions for existence of equilibrium sets for M/M/1 queue. These conditions further help us to identify the relationship among system parameters for which equilibrium sets exist. Next, we consider $$\textit{GI}/M/1$$ GI / M / 1 queues and provide a sufficient condition for existence of multiple optimal revenue policies. We then specialize these results to study the equilibrium sets of (i) a D/M/1 queue and (ii) a queue where the arrival rate is locally continuous. The equilibrium behaviour in the latter case is more interesting as there may be multiple equilibrium points or sets. Motivated by such queues, we introduce a weaker version of monotonicity and investigate the existence of generalized equilibrium sets.

Suggested Citation

  • N. Hemachandra & Kishor Patil & Sandhya Tripathi, 2020. "Equilibrium points and equilibrium sets of some $$\textit{GI}/M/1$$ GI / M / 1 queues," Queueing Systems: Theory and Applications, Springer, vol. 96(3), pages 245-284, December.
    • Handle: RePEc:spr:queues:v:96:y:2020:i:3:d:10.1007_s11134-020-09677-5
    DOI: 10.1007/s11134-020-09677-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11134-020-09677-5
    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/s11134-020-09677-5?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. Martin L. Puterman & Lyn C. Thomas, 1987. "Note---A Note on Computing Optimal Control Limits for GI/M/1 Queueing Systems," Management Science, INFORMS, vol. 33(7), pages 939-943, July.
    2. Nandyala Hemachandra & Kamma Sri Naga Rajesh & Mohd. Abdul Qavi, 2016. "A model for equilibrium in some service-provider user-set interactions," Annals of Operations Research, Springer, vol. 243(1), pages 95-115, August.
    3. Shaler Stidham, 2014. "The Price of Anarchy for a Network of Queues in Heavy Traffic," International Series in Operations Research & Management Science, in: P. Simin Pulat & Subhash C. Sarin & Reha Uzsoy (ed.), Essays in Production, Project Planning and Scheduling, edition 127, chapter 5, pages 91-121, Springer.
    4. Naor, P, 1969. "The Regulation of Queue Size by Levying Tolls," Econometrica, Econometric Society, vol. 37(1), pages 15-24, January.
    5. Martin J. Osborne & Ariel Rubinstein, 1994. "A Course in Game Theory," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262650401, April.
    6. Robert C. Rue & Matthew Rosenshine, 1981. "Some properties of optimal control policies for entry to an M/M/1 queue," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 28(4), pages 525-532, 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. Ghosh, Souvik & Hassin, Refael, 2021. "Inefficiency in stochastic queueing systems with strategic customers," European Journal of Operational Research, Elsevier, vol. 295(1), pages 1-11.
    2. L D Smith & D C Sweeney & J F Campbell, 2009. "Simulation of alternative approaches to relieving congestion at locks in a river transportion system," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 60(4), pages 519-533, April.
    3. Refael Hassin, 2022. "Profit maximization and cost balancing in queueing systems," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 429-431, April.
    4. Shi, Yi & Deng, Yawen & Wang, Guoan & Xu, Jiuping, 2020. "Stackelberg equilibrium-based eco-economic approach for sustainable development of kitchen waste disposal with subsidy policy: A case study from China," Energy, Elsevier, vol. 196(C).
    5. Cheng‐Kuang Wu & Yi‐Ming Chen & Dachrahn Wu & Ching‐Lin Chi, 2020. "A Game Theory Approach for Assessment of Risk and Deployment of Police Patrols in Response to Criminal Activity in San Francisco," Risk Analysis, John Wiley & Sons, vol. 40(3), pages 534-549, March.
    6. Kyle Y. Lin, 2003. "Decentralized admission control of a queueing system: A game‐theoretic model," Naval Research Logistics (NRL), John Wiley & Sons, vol. 50(7), pages 702-718, October.
    7. Müller, Christoph, 2020. "Robust implementation in weakly perfect Bayesian strategies," Journal of Economic Theory, Elsevier, vol. 189(C).
    8. Hitoshi Matsushima, 2019. "Implementation without expected utility: ex-post verifiability," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 53(4), pages 575-585, December.
    9. Kyle Y. Lin & Sheldon M. Ross, 2003. "Admission Control with Incomplete Information of a Queueing System," Operations Research, INFORMS, vol. 51(4), pages 645-654, August.
    10. Dasgupta Utteeyo, 2011. "Are Entry Threats Always Credible?," The B.E. Journal of Economic Analysis & Policy, De Gruyter, vol. 11(1), pages 1-41, December.
    11. Baran Han, 2018. "The role and welfare rationale of secondary sanctions: A theory and a case study of the US sanctions targeting Iran," Conflict Management and Peace Science, Peace Science Society (International), vol. 35(5), pages 474-502, September.
    12. Carlos Pimienta & Jianfei Shen, 2014. "On the equivalence between (quasi-)perfect and sequential equilibria," International Journal of Game Theory, Springer;Game Theory Society, vol. 43(2), pages 395-402, May.
    13. Asheim, Geir & Søvik, Ylva, 2003. "The semantics of preference-based belief operators," Memorandum 05/2003, Oslo University, Department of Economics.
    14. Wang, Yafeng & Graham, Brett, 2009. "Generalized Maximum Entropy estimation of discrete sequential move games of perfect information," MPRA Paper 21331, University Library of Munich, Germany.
    15. Karbowski, Adam, 2011. "O kilku modelach samolubnego karania w ekonomii behawioralnej [Evolution of altruism in the light of behavioral economics]," MPRA Paper 69604, University Library of Munich, Germany.
    16. Ricardo F. Reis & Phillip C. Stocken, 2007. "Strategic Consequences of Historical Cost and Fair Value Measurements," Contemporary Accounting Research, John Wiley & Sons, vol. 24(2), pages 557-584, June.
    17. Fajardo, Val Andrei & Drekic, Steve, 2015. "Controlling the workload of M/G/1 queues via the q-policy," European Journal of Operational Research, Elsevier, vol. 243(2), pages 607-617.
    18. Costello, Christopher & Molina, Renato, 2021. "Transboundary marine protected areas," Resource and Energy Economics, Elsevier, vol. 65(C).
    19. Martin Hellwig & Felix Bierbrauer, 2009. "Public Good Provision in a Large Economy," 2009 Meeting Papers 1062, Society for Economic Dynamics.
    20. Jonathan Hofer, 2024. "Red light game identifies ineffective criminal deterrence," American Journal of Economics and Sociology, Wiley Blackwell, vol. 83(3), pages 673-682, May.

    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:96:y:2020:i:3:d:10.1007_s11134-020-09677-5. 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.