Stability of a cascade system with multiple stations

Bara Kim; Jeongsim Kim

doi:10.1007/s11134-023-09871-1

Stability of a cascade system with multiple stations

Bara Kim, Jeongsim Kim

Department of Mathematics

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

1 Citation (Scopus)

Abstract

This paper considers an m-station cascade system with renewal arrival processes and general service time distributions. Miyazawa and Morozov (Queueing Syst 100:225–227, 2022a; Stability of a cascade system with two stations and its extension for multiple stations, 2022b. arXiv:2203.14294v1) formulated a conjecture on the stability of this cascade system. In this paper, we show that this conjecture is not true for m≥ 3. We also show that when m≥ 3 , the stability condition is not determined solely by the moments of the interarrival and service times, but also depends on their distributions. In addition, we provide the necessary and sufficient condition for the stability of this cascade system by modifying a result of Miyazawa and Morozov (2022).

Original language	English
Pages (from-to)	53-64
Number of pages	12
Journal	Queueing Systems
Volume	104
Issue number	1-2
DOIs	https://doi.org/10.1007/s11134-023-09871-1
Publication status	Published - 2023 Jun

Bibliographical note

Funding Information:
We are grateful to the reviewers for their valuable comments and suggestions. B. Kim’s research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1A2B5B01001864). J. Kim’s research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1F1A1A01065568).

Funding Information:
We are grateful to the reviewers for their valuable comments and suggestions. B. Kim’s research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1A2B5B01001864). J. Kim’s research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1F1A1A01065568).

Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.

Keywords

Cascade system
Positive Harris recurrence
Stability

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/s11134-023-09871-1

Cite this

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title = "Stability of a cascade system with multiple stations",

abstract = "This paper considers an m-station cascade system with renewal arrival processes and general service time distributions. Miyazawa and Morozov (Queueing Syst 100:225–227, 2022a; Stability of a cascade system with two stations and its extension for multiple stations, 2022b. arXiv:2203.14294v1) formulated a conjecture on the stability of this cascade system. In this paper, we show that this conjecture is not true for m≥ 3. We also show that when m≥ 3 , the stability condition is not determined solely by the moments of the interarrival and service times, but also depends on their distributions. In addition, we provide the necessary and sufficient condition for the stability of this cascade system by modifying a result of Miyazawa and Morozov (2022).",

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note = "Funding Information: We are grateful to the reviewers for their valuable comments and suggestions. B. Kim{\textquoteright}s research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1A2B5B01001864). J. Kim{\textquoteright}s research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1F1A1A01065568). Funding Information: We are grateful to the reviewers for their valuable comments and suggestions. B. Kim{\textquoteright}s research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1A2B5B01001864). J. Kim{\textquoteright}s research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1F1A1A01065568). Publisher Copyright: {\textcopyright} 2023, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.",

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N2 - This paper considers an m-station cascade system with renewal arrival processes and general service time distributions. Miyazawa and Morozov (Queueing Syst 100:225–227, 2022a; Stability of a cascade system with two stations and its extension for multiple stations, 2022b. arXiv:2203.14294v1) formulated a conjecture on the stability of this cascade system. In this paper, we show that this conjecture is not true for m≥ 3. We also show that when m≥ 3 , the stability condition is not determined solely by the moments of the interarrival and service times, but also depends on their distributions. In addition, we provide the necessary and sufficient condition for the stability of this cascade system by modifying a result of Miyazawa and Morozov (2022).

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Stability of a cascade system with multiple stations

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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