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Heavy Traffic Limits for Join-the-Shortest-Estimated-Queue Policy Using Delayed Information

Author

Listed:
  • Rami Atar

    (Viterbi Faculty of Electrical Engineering Technion–Israel Institute of Technology, Haifa 32000, Israel)

  • David Lipshutz

    (Viterbi Faculty of Electrical Engineering Technion–Israel Institute of Technology, Haifa 32000, Israel)

Abstract

We consider a load-balancing problem for a network of parallel queues in which information on the state of the queues is subject to a delay. In this setting, adopting a routing policy that performs well when applied to the current state of the queues can perform quite poorly when applied to the delayed state of the queues. Viewing this as a problem of control under partial observations, we propose using an estimate of the current queue lengths as the input to the join-the-shortest-queue policy. For a general class of estimation schemes, under heavy traffic conditions, we prove convergence of the diffusion-scaled process to a solution of a so-called diffusion model, in which an important step toward this goal establishes that the estimated queue lengths undergo state-space collapse. In some cases, our diffusion model is given by a novel stochastic delay equation with reflection, in which the Skorokhod boundary term appears with delay. We illustrate our results with examples of natural estimation schemes, discuss their implementability, and compare their relative performance using simulations.

Suggested Citation

  • Rami Atar & David Lipshutz, 2021. "Heavy Traffic Limits for Join-the-Shortest-Estimated-Queue Policy Using Delayed Information," Mathematics of Operations Research, INFORMS, vol. 46(1), pages 268-300, February.
    • Handle: RePEc:inm:ormoor:v:46:y:2021:i:1:p:268-300
    DOI: 10.1287/moor.2020.1056
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    References listed on IDEAS

    as
    1. Jamol Pender & Richard Rand & Elizabeth Wesson, 2020. "A Stochastic Analysis of Queues with Customer Choice and Delayed Information," Mathematics of Operations Research, INFORMS, vol. 45(3), pages 1104-1126, August.
    2. Patrick Eschenfeldt & David Gamarnik, 2018. "Join the Shortest Queue with Many Servers. The Heavy-Traffic Asymptotics," Mathematics of Operations Research, INFORMS, vol. 43(3), pages 867-886, August.
    3. Varun Gupta & Neil Walton, 2019. "Load Balancing in the Nondegenerate Slowdown Regime," Operations Research, INFORMS, vol. 67(1), pages 281-294, January.
    4. Anton Braverman, 2020. "Steady-State Analysis of the Join-the-Shortest-Queue Model in the Halfin–Whitt Regime," Mathematics of Operations Research, INFORMS, vol. 45(3), pages 1069-1103, August.
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