The Annals of Applied Probability

Network stability under max–min fair bandwidth sharing

Maury Bramson

Full-text: Open access


There has recently been considerable interest in the stability of different fair bandwidth sharing policies for models that arise in the context of Internet congestion control. Here, we consider a connection level model, introduced by Massoulié and Roberts [Telecommunication Systems 15 (2000) 185–201], that represents the randomly varying number of flows present in a network. The weighted α-fair and weighted max–min fair bandwidth sharing policies are among important policies that have been studied for this model. Stability results are known in both cases when the interarrival times and service times are exponentially distributed. Partial results for general service times are known for weighted α-fair policies; no such results are known for weighted max–min fair policies. Here, we show that weighted max–min fair policies are stable for subcritical networks with general interarrival and service distributions, provided the latter have 2+δ1 moments for some δ1>0. Our argument employs an appropriate Lyapunov function for the weighted max–min fair policy.

Article information

Ann. Appl. Probab., Volume 20, Number 3 (2010), 1126-1176.

First available in Project Euclid: 18 June 2010

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K25: Queueing theory [See also 68M20, 90B22] 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx] 90B15: Network models, stochastic

Bandwidth sharing max–min fair stability


Bramson, Maury. Network stability under max–min fair bandwidth sharing. Ann. Appl. Probab. 20 (2010), no. 3, 1126--1176. doi:10.1214/09-AAP649.

Export citation


  • [1] Bramson, M. (2008). Stability of Queueing Networks. Lecture Notes in Math. 1950. Springer, Berlin.
  • [2] Bonald, T. and Massoulié, L. (2001). Impact of fairness on Internet performance. In Proceedings of ACM Sigmetrics 82–91. ACM, New York.
  • [3] Chung, K. L. (1985). A Course in Probability Theory, 2nd ed. Academic Press, New York.
  • [4] Davis, M. H. A. (1993). Markov Models and Optimization. Monographs on Statistics and Applied Probability 49. Chapman & Hall, London.
  • [5] De Veciana, G., Lee, T. J. and Konstantopoulos, T. (2001). Stability and performance analysis of networks supporting elastic services. IEEE/ACM Transactions on Networking 9 2–14.
  • [6] Gromoll, H. C. and Williams, R. J. (2009). Fluid limits for networks with bandwidth sharing and general document size distributions. Ann. Appl. Probab. 19 243–280.
  • [7] Kang, W. N., Kelly, F. P., Lee, N. H. and Williams, R. J. (2009). State space collapse and diffusion approximation for a network operating under a fair bandwidth sharing policy. Ann. Appl. Probab. 19 1719–1780.
  • [8] Massoulié, L. (2007). Structural properties of proportional fairness: Stability and insensitivity. Ann. Appl. Probab. 17 809–839.
  • [9] Massoulié, L. and Roberts, J. (2000). Bandwidth sharing and admission control for elastic traffic. Telecommunication Systems 15 185–201.
  • [10] Meyn, S. P. and Tweedie, R. L. (1993). Generalized resolvents and Harris recurrence of Markov processes. In Doeblin and Modern Probability (Blaubeuren, 1991). Contemp. Math. 149 227–250. Amer. Math. Soc., Providence, RI.
  • [11] Nummelin, E. (1984). General Irreducible Markov Chains and Nonnegative Operators. Cambridge Tracts in Mathematics 83. Cambridge Univ. Press, Cambridge.
  • [12] Orey, S. (1971). Lecture Notes on Limit Theorems for Markov Chain Transition Probabilities. Van Nostrand-Reinhold, London.