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<p>This study considers Markovian queues with different service speeds, where the server speed can only be changed at control instants assumed to follow a Poisson process. Núñez-Queija et al. (Queueing Syst 100:233–235, 2022, Indag Math 34:990–1013, 2023) formulated a conjecture on the asymptotics of the stationary distribution for the scaled process of queue length and server speed as the control rate approaches 0. We completely resolve this conjecture by rigorously analyzing an intuitive explanation of the conjectured result. Furthermore, we extend this result to a renewal control model.</p>

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Proof of the conjecture on Markovian queues with Poisson control

  • 期刊论文

  • 发表日期：

    2025年1月4日

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  • Bara Kim,
  • Jeongsim Kim

摘要

This study considers Markovian queues with different service speeds, where the server speed can only be changed at control instants assumed to follow a Poisson process. Núñez-Queija et al. (Queueing Syst 100:233–235, 2022, Indag Math 34:990–1013, 2023) formulated a conjecture on the asymptotics of the stationary distribution for the scaled process of queue length and server speed as the control rate approaches 0. We completely resolve this conjecture by rigorously analyzing an intuitive explanation of the conjectured result. Furthermore, we extend this result to a renewal control model.