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

TL;DR

This paper analyzes the stability conditions of a multi-station cascade queueing system with threshold-based customer movement, extending the results from two stations to multiple stations and providing necessary and sufficient conditions for positive recurrence.

Contribution

It derives comprehensive stability criteria for a cascade queueing system with multiple stations, extending previous two-station results and considering general traffic intensities.

Findings

Necessary and sufficient conditions for system stability are established.

Extension of stability analysis from two stations to multiple stations in series.

Stability criteria are valid under general renewal arrivals and i.i.d. service times.

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 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.