article

On converse Lyapunov theorems for fluid network models

Authors:

Michael Schönlein,

Fabian WirthAuthors Info & Claims

Queueing Systems: Theory and Applications, Volume 70, Issue 4

Pages 339 - 367

https://doi.org/10.1007/s11134-012-9279-9

Published: 01 April 2012 Publication History

Abstract

We consider the class of closed generic fluid network (GFN) models, which provides an abstract framework containing a wide variety of fluid networks. Within this framework a Lyapunov method for stability of GFN models was proposed by Ye and Chen. They proved that stability of a GFN model is equivalent to the existence of a functional on the set of paths that is decaying along paths. This result falls short of a converse Lyapunov theorem in that no state-dependent Lyapunov function is constructed. In this paper we construct state-dependent Lyapunov functions in contrast to path-wise functionals. We first show by counterexamples that closed GFN models do not provide sufficient information that allow for a converse Lyapunov theorem. To resolve this problem we introduce the class of strict GFN models by forcing closed GFN models to satisfy a concatenation and a semicontinuity condition. For the class of strict GFN models we define a state-dependent Lyapunov function and show that a converse Lyapunov theorem holds. Finally, it is shown that common fluid network models, like general work-conserving and priority fluid network models as well as certain linear Skorokhod problems define strict GFN models.

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Cited By

Schönlein M(2019)A Lyapunov view on positive Harris recurrence of multiclass queueing networksOperations Research Letters10.1016/j.orl.2015.03.00443:3(299-303)Online publication date: 1-Jan-2019
https://dl.acm.org/doi/10.1016/j.orl.2015.03.004
Moka SJuneja SHill RKuhl M(2013)Regenerative simulation for multiclass open queueing networksProceedings of the 2013 Winter Simulation Conference: Simulation: Making Decisions in a Complex World10.5555/2675983.2676067(643-654)Online publication date: 8-Dec-2013
https://dl.acm.org/doi/10.5555/2675983.2676067

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cover image Queueing Systems: Theory and Applications

Queueing Systems: Theory and Applications Volume 70, Issue 4

April 2012

109 pages

ISSN:0257-0130

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Copyright © Copyright © 2012 Springer Science+Business Media, LLC.

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J. C. Baltzer AG, Science Publishers

United States

Publication History

Published: 01 April 2012

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Cited By

Schönlein M(2019)A Lyapunov view on positive Harris recurrence of multiclass queueing networksOperations Research Letters10.1016/j.orl.2015.03.00443:3(299-303)Online publication date: 1-Jan-2019
https://dl.acm.org/doi/10.1016/j.orl.2015.03.004
Moka SJuneja SHill RKuhl M(2013)Regenerative simulation for multiclass open queueing networksProceedings of the 2013 Winter Simulation Conference: Simulation: Making Decisions in a Complex World10.5555/2675983.2676067(643-654)Online publication date: 8-Dec-2013
https://dl.acm.org/doi/10.5555/2675983.2676067

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