The Annals of Applied Probability

Fluid limits for networks with bandwidth sharing and general document size distributions

H. Christian Gromoll and Ruth J. Williams

Full-text: Open access

Abstract

We consider a stochastic model of Internet congestion control, introduced by Massoulié and Roberts [Telecommunication Systems 15 (2000) 185–201], that represents the randomly varying number of flows in a network where bandwidth is shared among document transfers. In contrast to an earlier work by Kelly and Williams [Ann. Appl. Probab. 14 (2004) 1055–1083], the present paper allows interarrival times and document sizes to be generally distributed, rather than exponentially distributed. Furthermore, we allow a fairly general class of bandwidth sharing policies that includes the weighted α-fair policies of Mo and Walrand [IEEE/ACM Transactions on Networking 8 (2000) 556–567], as well as certain other utility based scheduling policies. To describe the evolution of the system, measure valued processes are used to keep track of the residual document sizes of all flows through the network. We propose a fluid model (or formal functional law of large numbers approximation) associated with the stochastic flow level model. Under mild conditions, we show that the appropriately rescaled measure valued processes corresponding to a sequence of such models (with fixed network structure) are tight, and that any weak limit point of the sequence is almost surely a fluid model solution. For the special case of weighted α-fair policies, we also characterize the invariant states of the fluid model.

Article information

Source
Ann. Appl. Probab., Volume 19, Number 1 (2009), 243-280.

Dates
First available in Project Euclid: 20 February 2009

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1235140339

Digital Object Identifier
doi:10.1214/08-AAP541

Mathematical Reviews number (MathSciNet)
MR2498678

Zentralblatt MATH identifier
1169.60025

Subjects
Primary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx]
Secondary: 60F17: Functional limit theorems; invariance principles 90B15: Network models, stochastic

Keywords
Bandwidth sharing α-fair flow level Internet model congestion control simultaneous resource possession fluid model workload measure valued process invariant manifold

Citation

Gromoll, H. Christian; Williams, Ruth J. Fluid limits for networks with bandwidth sharing and general document size distributions. Ann. Appl. Probab. 19 (2009), no. 1, 243--280. doi:10.1214/08-AAP541. https://projecteuclid.org/euclid.aoap/1235140339


Export citation

References

  • [1] Bonald, T. and Massoulié, L. (2001). Impact of fairness on Internet performance. In Proceedings of ACM Sigmetrics 2001 82–91.
  • [2] Bramson, M. (2005). Stability of networks for max-min fair routing. Presentation at the 13th INFORMS Applied Probability Conference, Ottawa.
  • [3] Chiang, M., Shah, D. and Tang, A. (2006). Stochastic stability under fair bandwidth allocation: General file size distribution. In Proceedings of the 44th Allerton Conference 899–908.
  • [4] Dai, J. G. (1995). On positive Harris recurrence of multiclass queueing networks: A unified approach via fluid limit models. Ann. Appl. Probab. 5 49–77.
  • [5] de Veciana, G., Konstantopoulos, T. and Lee, T.-J. (2001). Stability and performance analysis of networks supporting elastic services. IEEE/ACM Transactions on Networking 9 2–14.
  • [6] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Wiley, New York.
  • [7] Gromoll, H. C., Puha, A. L. and Williams, R. J. (2002). The fluid limit of a heavily loaded processor sharing queue. Ann. Appl. Probab. 12 797–859.
  • [8] Gromoll, H. C. and Williams, R. J. (2008). Fluid model for a data network with α-fair bandwidth sharing and general document size distributions: Two examples of stability. In Proceedings of Markov Processes and Related Topics— A Festschrift (T. G. Kurtz, S. N. Ethier, J. Feng and R. H. Stockbridge, eds.). Institute of Mathematical Statistics. Forthcoming.
  • [9] Hansen, J., Reynolds, C. and Zachary, S. (2007). Stability of processor sharing networks with simultaneous resource requirements. J. Appl. Probab. 44 636–651.
  • [10] Harrison, J. M. (2000). Brownian models of open processing networks: Canonical representation of workload. Ann. Appl. Probab. 10 75–103. Corr. 13 (2003) 390–393.
  • [11] Kallenberg, O. (1986). Random Measures, 4th ed. Academic Press, London.
  • [12] Kelly, F. P. and Williams, R. J. (2004). Fluid model for a network operating under a fair bandwidth-sharing policy. Ann. Appl. Probab. 14 1055–1083.
  • [13] Key, P. and Massoulié, L. (2006). Fluid models of integrated traffic and multipath routing. Queueing Syst. 53 85–98.
  • [14] Lakshmikantha, A., Beck, C. L. and Srikant, R. (2004). Connection level stability analysis of the internet using the sum of squares (sos) techniques. In Conference on Information Sciences and Systems (Princeton, New Jersey).
  • [15] Lin, X. and Shroff, N. (2004). On the stability region of congestion control. In Proceedings of the 42nd Allerton Conference on Communications, Control and Computing 1266–1275.
  • [16] Lin, X., Shroff, N. and Srikant, R. (2008). On the connection-level stability of congestion-controlled communication networks. IEEE Trans. Inform. Theory 54 2317–2338.
  • [17] Massoulié, L. (2007). Structural properties of proportional fairness: Stability and insensitivity. Ann. Appl. Probab. 17 809–839.
  • [18] Massoulié, L. and Roberts, J. (2000). Bandwidth sharing and admission control for elastic traffic. Telecommunication Systems 15 185–201.
  • [19] Mo, J. and Walrand, J. (2000). Fair end-to-end window-based congestion control. IEEE/ACM Transactions on Networking 8 556–567.
  • [20] Puha, A. L. and Williams, R. J. (2004). Invariant states and rates of convergence for a critical fluid model of a processor sharing queue. Ann. Appl. Probab. 14 517–554.
  • [21] Srikant, R. (2004). On the positive recurrence of a Markov chain describing file arrivals and departures in a congestion-controlled network. Presentation the IEEE Computer Communications Workshop.
  • [22] Ye, H.-Q. (2003). Stability of data networks under an optimization-based bandwidth allocation. IEEE Trans. Automat. Control 48 1238–1242.
  • [23] Ye, H.-Q., Ou, J. and Yuan, X.-M. (2005). Stability of data networks: Stationary and bursty models. Oper. Res. 53 107–125.