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Heavy-traffic limits for stationary network flows

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  • 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
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Martin I. Reiman, 1984. "Open Queueing Networks in Heavy Traffic," Mathematics of Operations Research, INFORMS, vol. 9(3), pages 441-458, August.
    4. James R. Jackson, 1957. "Networks of Waiting Lines," Operations Research, INFORMS, vol. 5(4), pages 518-521, August.
    5. 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.
    6. Sigman, Karl, 1990. "The stability of open queueing networks," Stochastic Processes and their Applications, Elsevier, vol. 35(1), pages 11-25, June.
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    Cited by:

    1. Ward Whitt & Wei You, 2022. "New decomposition approximations for queueing networks," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 365-367, April.

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