Proof of the conjecture on Markovian queues with Poisson control

Bara Kim; Jeongsim Kim

doi:10.1007/s11134-024-09933-y

Proof of the conjecture on Markovian queues with Poisson control

Bara Kim
, Jeongsim Kim^*

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

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.

Original language	English
Article number	5
Journal	Queueing Systems
Volume	109
Issue number	1
DOIs	https://doi.org/10.1007/s11134-024-09933-y
Publication status	Published - 2025 Mar

Bibliographical note

Publisher Copyright:
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.

Keywords

Asymptotics
Markovian queues
Regenerative processes

ASJC Scopus subject areas

Statistics and Probability
Computer Science Applications
Management Science and Operations Research
Computational Theory and Mathematics

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10.1007/s11134-024-09933-y

Cite this

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title = "Proof of the conjecture on Markovian queues with Poisson control",

abstract = "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. N{\'u}{\~n}ez-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.",

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N1 - Publisher Copyright: © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.

PY - 2025/3

Y1 - 2025/3

N2 - 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.

AB - 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.

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KW - Regenerative processes

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M1 - 5

ER -

Proof of the conjecture on Markovian queues with Poisson control

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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