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Queueing Systems

, Volume 55, Issue 1, pp 1–8 | Cite as

Equilibria of a class of transport equations arising in congestion control

  • Francois Baccelli
  • Ki Baek Kim
  • David R. McDonald
Article

Abstract

This paper studies a class of transport equations arising from stochastic models in congestion control. This class contains two cases of loss models as particular cases: the rate-independent case where the packet loss rate is independent of the throughput of the flow and the rate-dependent case where it depends on it. This class of equations covers both the case of persistent and of non-persistent flows. For the first time, we give a direct proof of the fact that there is a unique density solving the associated differential equation. This density and its mean value are provided as closed form expressions.

Keywords

Density Stationary solutions ODE Uniqueness PDE Congestion control 

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References

  1. 1.
    O. Diekmann, The cell size distribution and semigroups of linear operators. In Lecture Notes in Biomathematics: The Dynamics of Physiologically Structured Populations, J.A.J. Metz (ed.) (Springer, 1986).Google Scholar
  2. 2.
    R. Stevens and G. Wright, TCP Illustrated (Addison Wesley, 2001).Google Scholar
  3. 3.
    F. Baccelli, D.R. McDonald, and J. Reynier, A mean-field model for multiple TCP connections through a buffer implementing RED. Performance Evaluation 49 (2002) 77–97.CrossRefGoogle Scholar
  4. 4.
    F. Baccelli and D. McDonald, A square Root Formula for the Rate of Non Persistent HTTP Flows. INRIA Report Number 5301, (2004).Google Scholar
  5. 5.
    F. Baccelli, K.B. Kim, and D.D. Vleeschauwer, Analysis of the competition between wired, DSL and wireless users in an access network (IEEE Infocom 05, Miami, 2005).Google Scholar
  6. 6.
    B. Perthame and L. Ryzhik, Exponential decay for the fragmentation or cell-division equation. J. Differential Equations 210(1) (2005) 155–177.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2006

Authors and Affiliations

  • Francois Baccelli
    • 1
  • Ki Baek Kim
    • 2
  • David R. McDonald
    • 3
  1. 1.INRIA-ENS, Département d’InformatiqueEcole Normale SupérieureParis cedex 05France
  2. 2.INRIA-ENS, Département d’InformatiqueEcole Normale SupérieureParis cedex 05France
  3. 3.Department of Mathematics and StatisticsUniversity of Ottawa, 585 King Edward AvenueOttawaCanada

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