Stability of a cascade system with two stations and its extension for multiple stations
| Autor: | Miyazawa, Masakiyo, Morozov, Evsey |
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| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 a customer being served is greater than a given threshold level $C_{1} \ge 1$ 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 the actual traffic intensities of stations $2,3,\ldots, k-1$ for $k \ge 3$. 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. Comment: Accepted for publication in Queueing Systems |
| Databáze: | arXiv |
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