# On the saturation rule for the stability of queues

Abstract : This paper focuses on the stability of open queueing systems under stationary ergodic assumptions. It defines a set of conditions, the monotone separable framework, ensuring that the stability region is given by the following saturation rule : saturate the queues which are fed by the external arrival stream, look ate the intensity [??] of the departure stream in this saturated system, then stability holds whenever the intensity of the arrival process, say l satisfies the condition [??], whereas the network is unstable if [??]. Whenever the stability condition is satisfied, it is also shown that certain state variables associated with the network admit a finite stationary regime which is constructed pathwise using a Loynes type bacward argument. This framework involves two main pathwise properties, external monotonicity and separability, which are satisfied by several classical queueing networks. The main tool for the proof of this rule is sub-additive ergodic theory.
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Reports
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https://hal.inria.fr/inria-00074656
Contributor : Rapport de Recherche Inria <>
Submitted on : Wednesday, May 24, 2006 - 4:00:25 PM
Last modification on : Tuesday, July 14, 2020 - 11:04:05 AM
Long-term archiving on: : Tuesday, April 12, 2011 - 6:06:12 PM

### Identifiers

• HAL Id : inria-00074656, version 1

### Citation

François Baccelli, Serguei Foss. On the saturation rule for the stability of queues. [Research Report] RR-2015, INRIA. 1993. ⟨inria-00074656⟩

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