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

• Published in: Queueing systems Vol. 104; no. 3-4; pp. 155 - 174
• Main Authors: Miyazawa, Masakiyo, Morozov, Evsey
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.08.2023; Springer Nature B.V
• Subjects: Arrivals; Business and Management; Computer Communication Networks; Control; Customers; Markov analysis; Markov processes; Operations Research/Decision Theory; Probability Theory and Stochastic Processes; Queuing theory; Stability; Supply Chain Management; Systems Theory; Multiple stations; Markov process; Cascade system; Stability; Harris positive recurrent; Renewal arrivals
• ISSN: 0257-0130; 1572-9443
• DOI: 10.1007/s11134-023-09883-x

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 ≥ 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 , … , k - 1 for 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.