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Stability of a cascade system with two stations and its extension for multiple stations

Author

Listed:
  • Masakiyo Miyazawa

    (Tokyo University of Science
    Chinese University of Hong Kong)

  • Evsey Morozov

    (Russian Academy of Sciences
    Petrozavodsk State University
    Moscow State University)

Abstract

We consider a two-station cascade system in which waiting or externally arriving customers at station 1 move to the station 2 if the queue size of station 1 including an arriving customer itself and a customer being served is greater than a given threshold level $$c_{1} \ge 1$$ c 1 ≥ 1 and if station 2 is empty. Assuming that external arrivals are subject to independent renewal processes satisfying certain regularity conditions and service times are i.i.d. at each station, we derive necessary and sufficient conditions for a Markov process describing this system to be positive recurrent in the sense of Harris. This result is extended to the cascade system with a general number k of stations in series. This extension requires certain traffic intensities of stations $$2,3,\ldots , k-1$$ 2 , 3 , … , k - 1 for $$k \ge 3$$ k ≥ 3 to be defined. We finally note that the modeling assumptions on the renewal arrivals and i.i.d. service times are not essential if the notion of the stability is replaced by a certain sample path condition. This stability notion is identical with the standard stability if the whole system is described by the Markov process which is a Harris irreducible T-process.

Suggested Citation

  • Masakiyo Miyazawa & Evsey Morozov, 2023. "Stability of a cascade system with two stations and its extension for multiple stations," Queueing Systems: Theory and Applications, Springer, vol. 104(3), pages 155-174, August.
    • Handle: RePEc:spr:queues:v:104:y:2023:i:3:d:10.1007_s11134-023-09883-x
    DOI: 10.1007/s11134-023-09883-x
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    References listed on IDEAS

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    1. Masakiyo Miyazawa & Evsey Morozov, 2022. "Stability condition of a cascade system with a general number of stations," Queueing Systems: Theory and Applications, Springer, vol. 100(3), pages 225-227, April.
    2. Bara Kim & Jeongsim Kim, 2023. "Stability of a cascade system with multiple stations," Queueing Systems: Theory and Applications, Springer, vol. 104(1), pages 53-64, June.
    3. E. Morozov & B. Steyaert, 2013. "Stability analysis of a two-station cascade queueing network," Annals of Operations Research, Springer, vol. 202(1), pages 135-160, January.
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