IDEAS home Printed from https://ideas.repec.org/a/spr/queues/v95y2020i1d10.1007_s11134-019-09645-8.html
   My bibliography  Save this article

Heavy-traffic limits for stationary network flows

Author

Listed:
  • Ward Whitt

    (Columbia University)

  • Wei You

    (HKUST)

Abstract

This paper studies stationary customer flows in an open queueing network. The flows are the processes counting customers flowing from one queue to another or out of the network. We establish the existence of unique stationary flows in generalized Jackson networks and convergence to the stationary flows as time increases. We establish heavy-traffic limits for the stationary flows, allowing an arbitrary subset of the queues to be critically loaded. The heavy-traffic limit with a single bottleneck queue is especially tractable because it yields limit processes involving one-dimensional reflected Brownian motion. That limit plays an important role in our new nonparametric decomposition approximation of the steady-state performance using indices of dispersion and robust optimization.

Suggested Citation

  • Ward Whitt & Wei You, 2020. "Heavy-traffic limits for stationary network flows," Queueing Systems: Theory and Applications, Springer, vol. 95(1), pages 53-68, June.
    • Handle: RePEc:spr:queues:v:95:y:2020:i:1:d:10.1007_s11134-019-09645-8
    DOI: 10.1007/s11134-019-09645-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s11134-019-09645-8
    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/s11134-019-09645-8?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. J. G. Dai & Viên Nguyen & Martin I. Reiman, 1994. "Sequential Bottleneck Decomposition: An Approximation Method for Generalized Jackson Networks," Operations Research, INFORMS, vol. 42(1), pages 119-136, February.
    2. James R. Jackson, 1957. "Networks of Waiting Lines," Operations Research, INFORMS, vol. 5(4), pages 518-521, August.
    3. Amarjit Budhiraja & Chihoon Lee, 2009. "Stationary Distribution Convergence for Generalized Jackson Networks in Heavy Traffic," Mathematics of Operations Research, INFORMS, vol. 34(1), pages 45-56, February.
    4. Hong Chen & Avi Mandelbaum, 1991. "Discrete Flow Networks: Bottleneck Analysis and Fluid Approximations," Mathematics of Operations Research, INFORMS, vol. 16(2), pages 408-446, May.
    5. Martin I. Reiman, 1984. "Open Queueing Networks in Heavy Traffic," Mathematics of Operations Research, INFORMS, vol. 9(3), pages 441-458, August.
    6. Sigman, Karl, 1990. "The stability of open queueing networks," Stochastic Processes and their Applications, Elsevier, vol. 35(1), pages 11-25, June.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Glynn, Peter W. & Wang, Rob J., 2023. "A heavy-traffic perspective on departure process variability," Stochastic Processes and their Applications, Elsevier, vol. 166(C).
    2. Ward Whitt & Wei You, 2022. "New decomposition approximations for queueing networks," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 365-367, April.

    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. Amarjit Budhiraja & Chihoon Lee, 2009. "Stationary Distribution Convergence for Generalized Jackson Networks in Heavy Traffic," Mathematics of Operations Research, INFORMS, vol. 34(1), pages 45-56, February.
    2. Shaler Stidham, 2002. "Analysis, Design, and Control of Queueing Systems," Operations Research, INFORMS, vol. 50(1), pages 197-216, February.
    3. Ward Whitt & Wei You, 2022. "New decomposition approximations for queueing networks," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 365-367, April.
    4. Itai Gurvich, 2014. "Validity of Heavy-Traffic Steady-State Approximations in Multiclass Queueing Networks: The Case of Queue-Ratio Disciplines," Mathematics of Operations Research, INFORMS, vol. 39(1), pages 121-162, February.
    5. Dimitris Bertsimas & David Gamarnik & Alexander Anatoliy Rikun, 2011. "Performance Analysis of Queueing Networks via Robust Optimization," Operations Research, INFORMS, vol. 59(2), pages 455-466, April.
    6. Ward Whitt, 2001. "The Reflection Map with Discontinuities," Mathematics of Operations Research, INFORMS, vol. 26(3), pages 447-484, August.
    7. Amarjit Budhiraja & Jiang Chen & Sylvain Rubenthaler, 2014. "A Numerical Scheme for Invariant Distributions of Constrained Diffusions," Mathematics of Operations Research, INFORMS, vol. 39(2), pages 262-289, May.
    8. Bitran, Gabriel R. & Morabito, Reinaldo., 1994. "Open queueing networks : optimization and performance evaluation models for discrete manufacturing systems," Working papers 3743-94., Massachusetts Institute of Technology (MIT), Sloan School of Management.
    9. Jennifer Sommer & Joost Berkhout & Hans Daduna & Bernd Heidergott, 2017. "Analysis of Jackson networks with infinite supply and unreliable nodes," Queueing Systems: Theory and Applications, Springer, vol. 87(1), pages 181-207, October.
    10. Chang Cao & J. G. Dai & Xiangyu Zhang, 2022. "State space collapse for multi-class queueing networks under SBP service policies," Queueing Systems: Theory and Applications, Springer, vol. 102(1), pages 87-122, October.
    11. Josh Reed & Yair Shaki, 2015. "A Fair Policy for the G / GI / N Queue with Multiple Server Pools," Mathematics of Operations Research, INFORMS, vol. 40(3), pages 558-595, March.
    12. Saulius Minkevičius & Igor Katin & Joana Katina & Irina Vinogradova-Zinkevič, 2021. "On Little’s Formula in Multiphase Queues," Mathematics, MDPI, vol. 9(18), pages 1-15, September.
    13. Rosario Delgado, 2010. "State space collapse and stability of queueing networks," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 72(3), pages 477-499, December.
    14. Budhiraja, Amarjit & Lee, Chihoon, 2007. "Long time asymptotics for constrained diffusions in polyhedral domains," Stochastic Processes and their Applications, Elsevier, vol. 117(8), pages 1014-1036, August.
    15. Natalia Chernova & Sergey Foss & Bara Kim, 2012. "On the stability of a polling system with an adaptive service mechanism," Annals of Operations Research, Springer, vol. 198(1), pages 125-144, September.
    16. Francisco Castro & Hamid Nazerzadeh & Chiwei Yan, 2020. "Matching queues with reneging: a product form solution," Queueing Systems: Theory and Applications, Springer, vol. 96(3), pages 359-385, December.
    17. Ari Arapostathis & Hassan Hmedi & Guodong Pang, 2021. "On Uniform Exponential Ergodicity of Markovian Multiclass Many-Server Queues in the Halfin–Whitt Regime," Mathematics of Operations Research, INFORMS, vol. 46(2), pages 772-796, May.
    18. Zhen Xu & Jiheng Zhang & Rachel Q. Zhang, 2019. "Instantaneous Control of Brownian Motion with a Positive Lead Time," Mathematics of Operations Research, INFORMS, vol. 44(3), pages 943-965, August.
    19. Amarjit Budhiraja & Xin Liu, 2012. "Stability of Constrained Markov-Modulated Diffusions," Mathematics of Operations Research, INFORMS, vol. 37(4), pages 626-653, November.
    20. Xin Liu, 2019. "Diffusion approximations for double-ended queues with reneging in heavy traffic," Queueing Systems: Theory and Applications, Springer, vol. 91(1), pages 49-87, February.

    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:95:y:2020:i:1:d:10.1007_s11134-019-09645-8. 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.