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The Stability of Two-Station Multitype Fluid Networks

Author

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  • 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
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    References listed on IDEAS

    as
    1. 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.
    2. 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.
    3. 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.
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    Cited by:

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    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.
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    6. J. G. Dai & Caiwei Li, 2003. "Stabilizing Batch-Processing Networks," Operations Research, INFORMS, vol. 51(1), pages 123-136, February.

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