Queueing Systems

, Volume 47, Issue 1–2, pp 81–106 | Cite as

On Stochastic Bounds for Monotonic Processor Sharing Networks

  • T. Bonald
  • A. Proutière
Article

Abstract

We consider a network of processor sharing nodes with independent Poisson arrival processes. Nodes are coupled through their service capacity in that the speed of each node depends on the number of customers present at this and any other node. We assume the network is monotonic in the sense that removing a customer from any node increases the service rate of all customers. Under this assumption, we give stochastic bounds on the number of customers present at any node. We also identify limiting regimes that allow to test the tightness of these bounds. The bounds and the limiting regimes are insensitive to the service time distribution. We apply these results to a number of practically interesting systems, including the discriminatory processor sharing queue, the generalized processor sharing queue, and data networks whose resources are shared according to max–min fairness.

processor sharing networks stochastic bounds balance monotonicity insensitivity 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    F. Baskett, K.M. Chandy, R.R. Muntz and F.G. Palacios, Open, closed and mixed networks of queues with different classes of customers, J. Assoc. Comput. Mach. 22 (1975) 248–260.Google Scholar
  2. [2]
    D. Bertsekas and R. Gallager, Data Networks (Prentice Hall, New York, 1987).Google Scholar
  3. [3]
    D. Bertsimas, I.C. Paschalidis and J.N. Tsitsiklis, Large deviation analysis of the generalized processor sharing policy, Queueing Systems 32 (1999) 319–349.Google Scholar
  4. [4]
    P. Billingsley, Convergence of Probability Measures (Wiley, New York, 1968).Google Scholar
  5. [5]
    T. Bonald and L. Massoulié, Impact of fairness on Internet performance, in: Proc. of ACM SIGMETRICS, 2001.Google Scholar
  6. [6]
    T. Bonald and A. Proutière, Insensitivity in processor-sharing networks, Perform. Eval. 49 (2002) 193–209.Google Scholar
  7. [7]
    T. Bonald and A. Proutière, Insensitive bandwidth sharing in data networks, Queueing Systems 44(1) (2003) 69–100.Google Scholar
  8. [8]
    S. Borst, O.J. Boxma and P. Jelenkovic, Reduced-load equivalence and induced burstiness in GPS queues with long-tailed traffic flows, Queueing Systems 43(4) (2003) 273–306.Google Scholar
  9. [9]
    G.L. Choudhury, A. Mandelbaum, M.I. Reiman and W. Whitt, Fluid and diffusion limits for queues in slowly changing environments, Stochastic Models 13 (1997) 121–146.Google Scholar
  10. [10]
    J.W. Cohen, The multiple phase service network with generalized processor sharing, Acta Inform. 12 (1979) 245–284.Google Scholar
  11. [11]
    A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications (Jones & Barlett, Boston, MA, 1993).Google Scholar
  12. [12]
    G. Fayolle, I. Mitrani and R. lasnogorodski, Sharing a processor among many classes, J. Assoc. Comput. Mach. 27 (1980) 519–532.Google Scholar
  13. [13]
    P.R. Jelenkovic, The effect of multiple time scales and subexponentiality on the behavior of a broadband network multiplexer, Ph.D. Thesis, Columbia University, New York (October 1996).Google Scholar
  14. [14]
    F.P. Kelly, Reversibility and Stochastic Networks (Wiley, New York, 1979).Google Scholar
  15. [15]
    F.P. Kelly, A. Maulloo and D. Tan, Rate control for communication networks: Shadow prices, proportional fairness and stability, J. Oper. Res. Soc. 49 (1998) 237–252.Google Scholar
  16. [16]
    F.P. Kelly and R.J. Williams, Fluid models for a network operating under a fair bandwidth-sharing policy, to appear in Ann. Appl. Probab.Google Scholar
  17. [17]
    L. Kleinrock, Queueing Systems, Vol. 2 (Wiley, New York, 1975).Google Scholar
  18. [18]
    H.J. Kushner, Heavy-Traffic of Controlled Queueing and Commmunication Networks (Springer, Berlin, 2001).Google Scholar
  19. [19]
    L. Massoulié, Large deviation estimates for polling and weighted fair queueing systems, Adv. Perform. Anal. (1999).Google Scholar
  20. [20]
    L. Massoulié and J.W. Roberts, Bandwidth sharing and admission control for elastic traffic, Telecom. Systems 15 (2000) 185–201.Google Scholar
  21. [21]
    L. Massoulié and J.W. Roberts, Bandwidth sharing: objectives and algorithms, IEEE/ACM Trans. Networking 10(3) (2002) 320–328.Google Scholar
  22. [22]
    R. Núñez Queija, Processor-sharing models for integrated-services networks, Ph.D. Thesis, Eindhoven University of Technology (1999).Google Scholar
  23. [23]
    R.F. Serfozo, Introduction to Stochastic Networks (Springer, Berlin, 1999).Google Scholar
  24. [24]
    W. Whitt, Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Applications to Queues (Springer, Berlin, 2002).Google Scholar
  25. [25]
    G.G. Yin and Q. Zhang, Continuous-Time Markov Chains and Applications: A Singular Perturbation Approach (Springer, Berlin, 1998).Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • T. Bonald
    • 1
  • A. Proutière
    • 1
  1. 1.France Telecom R&DFrance

Personalised recommendations