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

Masakiyo Miyazawa () and Evsey Morozov
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Masakiyo Miyazawa: Tokyo University of Science
Evsey Morozov: Russian Academy of Sciences

Queueing Systems: Theory and Applications, 2023, vol. 104, issue 3, No 2, 155-174

Abstract: 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.

Keywords: Cascade system; Renewal arrivals; Multiple stations; Stability; Markov process; Harris positive recurrent (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1007/s11134-023-09883-x

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