Rescaled Markov Processes and Fluid Limits

  • Philippe Robert
Part of the Applications of Mathematics book series (SMAP, volume 52)


It is in general quite difficult to have a satisfactory description of an ergodic Markov process describing a stochastic network. When the dimension of the state space d is greater than 1, the geometry complicates a lot any investigation: Analytical tools of Chapter 2 for dimension 1 cannot be easily generalized to higher dimensions. Note that the natural order on the real line plays an important role for Wiener-Hopf methods. The example of queueing networks seen in Chapter 4 for which the stationary distribution has a product form should be seen as an interesting exception, but an exception. In the same way as in Chapter 3, it is possible nevertheless to get some insight on the behavior of these processes through some limit theorems. In this chapter, limit results consist in speeding up time and scaling appropriately the process itself with some parameter. The behavior of such rescaled stochastic processes is analyzed when the scaling parameter goes to infinity. In the limit one gets a sort of caricature of the initial stochastic process which is defined as a fluid limit (see the rigorous definition below). As it will be seen, a fluid limit keeps the main characteristics of the initial stochastic process while some stochastic fluctuations of second order vanish with this procedure. In “good cases”, a fluid limit is a deterministic function, solution of some ordinary differential equation. As it can be expected, the general situation is somewhat more complicated. These ideas of rescaling stochastic processes have emerged recently in the analysis of stochastic networks, to study their ergodicity properties in particular. See Rybko and Stolyar[RS92] for example. In statistical physics, these methods are quite classical, see Comets[Com91].


Markov Process Poisson Process Stochastic Differential Equation Fluid Limit Invariant Distribution 
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Bibliographical Notes

  1. [RS92]
    A.N. Rybko and A.L. Stolyar, On the ergodicity of random processes that describe the functioning of open queueing networks, Problems on Information Transmission 28 (1992), no. 3, 3–26.MathSciNetGoogle Scholar
  2. [Com91]
    F. Comets, Limites hydrodynamiques,Astérisque (1991), no. 201–203, Exp. No. 735, 167–192 (1992), Séminaire Bourbaki, Vol. 1990/91.Google Scholar
  3. [JMR94]
    A. Jean-Marie and Ph. Robert, On the transient behavior of some single server queues, Queueing Systems, Theory and Applications 17 (1994), 129–136.Google Scholar
  4. [HK94]
    P.J. Hunt and T.G. Kurtz, Large loss networks, Stochastic Processes and their Applications 53 (1994), .363–378.Google Scholar
  5. [Dai95]
    J.G. Dai, On positive Harris recurrence of multiclass queueing networks: a unified approach via fluid limit models, Annals of Applied Probability 5 (1995), 1, 49–77.Google Scholar
  6. [CM91]
    H. Chen and A. Mandelbaum, Discrete flow networks: bottleneck analysis and fluid approximations, Mathematics of Operation Research 16 (1991), no. 2, 408–446.Google Scholar
  7. [Wil95]
    R.J. Williams, Semimartingale reflecting Brownian motions in the orthant, Stochastic networks (New York) (F.P. Kelly and R.J. Williams, eds.), IMA Volumes in Mathematics and Its Applications, no, 71, Springer Verlag, 1995.Google Scholar
  8. [Bra94]
    M. Bramson, Instability of FIFO queueing networks, Annals of Applied Probability 4 (1994), no. 2, 414–431.Google Scholar
  9. [Kes73]
    H. Kesten, Randon difference equations and renewal theory for products of random matrices, Acta Mathematica 131 (1973), 207–248.Google Scholar
  10. [RS92]
    Op. cit. page 232.Google Scholar
  11. [Bra94]
    Op. cit. page 261.Google Scholar
  12. [Ma193]
    V.A. Malyshev, Networks and dynamical systems, Advances in Applied Probability 25 (1993), no. 1, 140–175.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [DW94]
    P. Dupuis and R.J. Williams, Lyapounov functions for semimartingale reflecting Brownian motions, Annals of Applied Probability 22 (1994), no. 2, 680–702.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [Com91]
    Op. cit. page 232.Google Scholar
  15. [IM93]
    I.A. Ignatyuk and V.A. Malyshev, Classification of random walks in 7G+, Selecta Mathematica 12 (1993), no. 2, 129–194.MathSciNetzbMATHGoogle Scholar
  16. [FIMM91]
    G. Fayolle, I.A. Ignatyuk, V.A. Malyshev, and M.V. Menshikov, Random walks in two-dimensional complexes, Queueing Systems, Theory and Applications 9 (1991), no. 3, 269–300.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Philippe Robert
    • 1
  1. 1.Domaine de VoluceauINRIALe ChesnayFrance

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