IDEAS home Printed from https://ideas.repec.org/a/inm/oropre/v48y2000i5p721-744.html
   My bibliography  Save this article

The Stability of Two-Station Multitype Fluid Networks

Author

Listed:
  • J. G. Dai

    (School of Industrial and Systems Engineering and School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0205)

  • J. H. Vande Vate

    (School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0205)

Abstract

This paper studies the fluid models of two-station multiclass queueing networks with deterministic routing. A fluid model is globally stable if the fluid network eventually empties under each nonidling dispatch policy. We explicitly characterize the global stability region in terms of the arrival and service rates. We show that the global stability region is defined by the nominal workload conditions and the “virtual workload conditions,” and we introduce two intuitively appealing phenomena---virtual stations and push starts---that explain the virtual workload conditions. When any of the workload conditions is violated, we construct a fluid solution that cycles to infinity, showing that the fluid network is unstable. When all the workload conditions are satisfied, we solve a network flow problem to find the coefficients of a piecewise linear Lyapunov function. The Lyapunov function decreases to zero, proving that the fluid level eventually reaches zero under any nonidling dispatch policy. Under certain assumptions on the inter arrival and service time distributions, a queueing network is stable or positive Harris recurrent if the corresponding fluid network is stable. Thus, the workload conditions are sufficient to ensure the global stability of two-station multiclass queueing networks with deterministic routing.

Suggested Citation

  • J. G. Dai & J. H. Vande Vate, 2000. "The Stability of Two-Station Multitype Fluid Networks," Operations Research, INFORMS, vol. 48(5), pages 721-744, October.
  • Handle: RePEc:inm:oropre:v:48:y:2000:i:5:p:721-744
    DOI: 10.1287/opre.48.5.721.12408
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/opre.48.5.721.12408
    Download Restriction: no

    File URL: https://libkey.io/10.1287/opre.48.5.721.12408?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
    ---><---

    References listed on IDEAS

    as
    1. J. G. Dai & G. Weiss, 1996. "Stability and Instability of Fluid Models for Reentrant Lines," Mathematics of Operations Research, INFORMS, vol. 21(1), pages 115-134, February.
    2. Hong Chen & Hanqin Zhang, 1997. "Stability of Multiclass Queueing Networks Under FIFO Service Discipline," Mathematics of Operations Research, INFORMS, vol. 22(3), pages 691-725, August.
    3. Muhammad El-Taha & Shaler Stidham, 1994. "Sample-path stability conditions for multiserver input-output processes," International Journal of Stochastic Analysis, Hindawi, vol. 7, pages 1-20, January.
    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. Jan A. Van Mieghem, 2003. "Commissioned Paper: Capacity Management, Investment, and Hedging: Review and Recent Developments," Manufacturing & Service Operations Management, INFORMS, vol. 5(4), pages 269-302, July.
    2. Schönlein, Michael & Makuschewitz, Thomas & Wirth, Fabian & Scholz-Reiter, Bernd, 2013. "Measurement and optimization of robust stability of multiclass queueing networks: Applications in dynamic supply chains," European Journal of Operational Research, Elsevier, vol. 229(1), pages 179-189.
    3. Dieter Armbruster & Daniel E. Marthaler & Christian Ringhofer & Karl Kempf & Tae-Chang Jo, 2006. "A Continuum Model for a Re-entrant Factory," Operations Research, INFORMS, vol. 54(5), pages 933-950, October.
    4. J. G. Dai & O. B. Jennings, 2004. "Stabilizing Queueing Networks with Setups," Mathematics of Operations Research, INFORMS, vol. 29(4), pages 891-922, November.
    5. David Gamarnik, 2002. "On Deciding Stability of Constrained Homogeneous Random Walks and Queueing Systems," Mathematics of Operations Research, INFORMS, vol. 27(2), pages 272-293, May.
    6. J. G. Dai & Caiwei Li, 2003. "Stabilizing Batch-Processing Networks," Operations Research, INFORMS, vol. 51(1), pages 123-136, February.

    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. Hong Chen & Hanqin Zhang, 2000. "Stability of Multiclass Queueing Networks Under Priority Service Disciplines," Operations Research, INFORMS, vol. 48(1), pages 26-37, February.
    2. J. G. Dai & Caiwei Li, 2003. "Stabilizing Batch-Processing Networks," Operations Research, INFORMS, vol. 51(1), pages 123-136, February.
    3. Legros, Benjamin & Jouini, Oualid & Akşin, O. Zeynep & Koole, Ger, 2020. "Front-office multitasking between service encounters and back-office tasks," European Journal of Operational Research, Elsevier, vol. 287(3), pages 946-963.
    4. Zhao, Yaping & Xu, Xiaoyun & Li, Haidong & Liu, Yanni, 2016. "Prioritized customer order scheduling to maximize throughput," European Journal of Operational Research, Elsevier, vol. 255(2), pages 345-356.
    5. Kevin D. Glazebrook & José Niño-Mora, 1997. "Assessing an intuitive condition for stability under a range of traffic conditions via a generalised Lu-Kumar network," Economics Working Papers 429, Department of Economics and Business, Universitat Pompeu Fabra, revised Jan 2000.
    6. Heng-Qing Ye & Jihong Ou & Xue-Ming Yuan, 2005. "Stability of Data Networks: Stationary and Bursty Models," Operations Research, INFORMS, vol. 53(1), pages 107-125, February.
    7. Koole, Ger & Pot, Auke, 2006. "Workload minimization in re-entrant lines," European Journal of Operational Research, Elsevier, vol. 174(1), pages 216-233, October.
    8. A. B. Dieker & J. Shin, 2013. "From Local to Global Stability in Stochastic Processing Networks Through Quadratic Lyapunov Functions," Mathematics of Operations Research, INFORMS, vol. 38(4), pages 638-664, November.
    9. Yongjiang Guo & Xiyang Hou & Yunan Liu, 2021. "A functional law of the iterated logarithm for multi-class queues with batch arrivals," Annals of Operations Research, Springer, vol. 300(1), pages 51-77, May.
    10. Pihnastyi, Oleh & Bondarenko, Kristina, 2016. "About the problem of selecting models for production line," MPRA Paper 91235, University Library of Munich, Germany, revised 07 Jan 2018.
    11. Dieter Armbruster & Daniel E. Marthaler & Christian Ringhofer & Karl Kempf & Tae-Chang Jo, 2006. "A Continuum Model for a Re-entrant Factory," Operations Research, INFORMS, vol. 54(5), pages 933-950, October.
    12. Dong, Ming & He, Fenglan, 2012. "A new continuous model for multiple re-entrant manufacturing systems," European Journal of Operational Research, Elsevier, vol. 223(3), pages 659-668.
    13. Łukasz Kruk & Tymoteusz Chojecki, 2022. "Instability of SRPT, SERPT and SJF multiclass queueing networks," Queueing Systems: Theory and Applications, Springer, vol. 101(1), pages 57-92, 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:inm:oropre:v:48:y:2000:i:5:p:721-744. 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: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    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.