The Annals of Applied Probability

Proportional fairness and its relationship with multi-class queueing networks

N. S. Walton

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We consider multi-class single-server queueing networks that have a product form stationary distribution. A new limit result proves a sequence of such networks converges weakly to a stochastic flow level model. The stochastic flow level model found is insensitive. A large deviation principle for the stationary distribution of these multi-class queueing networks is also found. Its rate function has a dual form that coincides with proportional fairness. We then give the first rigorous proof that the stationary throughput of a multi-class single-server queueing network converges to a proportionally fair allocation.

This work combines classical queueing networks with more recent work on stochastic flow level models and proportional fairness. One could view these seemingly different models as the same system described at different levels of granularity: a microscopic, queueing level description; a macroscopic, flow level description and a teleological, optimization description.

Article information

Ann. Appl. Probab., Volume 19, Number 6 (2009), 2301-2333.

First available in Project Euclid: 25 November 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K25: Queueing theory [See also 68M20, 90B22] 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx]
Secondary: 90K15 68K20

Multi-class queueing network proportional fairness bandwidth sharing stochastic flow level model insensitivity product form stationary distribution proportionally fair state space collapse


Walton, N. S. Proportional fairness and its relationship with multi-class queueing networks. Ann. Appl. Probab. 19 (2009), no. 6, 2301--2333. doi:10.1214/09-AAP612.

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  • [1] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • [2] Bonald, T. and Massoulie, L. (2001). Impact of fairness on internet performance. In Proceedings of ACM Sigmetrics 29 82–91. ACM, New York.
  • [3] Bonald, T. and Proutière, A. (2004). On performance bounds for balanced fairness. Performance Evaluation 55 25–50.
  • [4] Bramson, M. (1998). State space collapse with application to heavy traffic limits for multiclass queueing networks. Queueing Syst. 30 89–148.
  • [5] Chen, H. and Yao, D. D. (2001). Fundamentals of Queueing Networks: Performance, Asymptotics, and Optimization. Applications of Mathematics 46. Springer, New York.
  • [6] Dembo, A. and Zeitouni, O. (1998). Large Deviations Techniques and Applications, 2nd ed. Applications of Mathematics 38. Springer, New York.
  • [7] Doyle, P. G. and Snell, J. L. (1984). Random Walks and Electric Networks. Carus Mathematical Monographs 22. Mathematical Association of America, Washington, DC.
  • [8] De Veciana, G., Lee, T. J. and Konstantopoulos, T. (1999). Stability and performance analysis of networks supporting services with rate control—could the Internet be unstable? IEEE/ACM Transactions on Networking 9 2–14.
  • [9] Kang, W. N., Kelly, F. P., Lee, N. H. and Williams, R. J. (2007). State space collapse and diffusion approximation for a network operating under a fair bandwidth sharing policy. Ann. Appl. Probab. To appear.
  • [10] Kang, W. N., Kelly, F. P., Lee, N. H. and Williams, R. J. (2007). Product form stationary distributions for diffusion approximations to a flow level model operating under a proportional fair sharing policy. Performance Evaluation Review 35 36–38.
  • [11] Kelly, F. P. (1979). Reversibility and Stochastic Networks. Wiley, Chichester.
  • [12] Kelly, F. P. (1986). Blocking probabilities in large circuit-switched networks. Adv. in Appl. Probab. 18 473–505.
  • [13] Kelly, F. P. (1989). On a class of approximations for closed queueing networks. Queueing Syst. 4 69–76.
  • [14] Kelly, F. P. (1991). Network routing. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 337 343–367.
  • [15] Kelly, F. P. (1997). Charging and rate control for elastic traffic. European Transactions on Telecommunications 8 33–37.
  • [16] 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.
  • [17] Massoulié, L. (2007). Structural properties of proportional fairness: Stability and insensitivity. Ann. Appl. Probab. 17 809–839.
  • [18] Massoulie, L. and Roberts, J. (1998). Bandwidth sharing and admission control for elastic traffic. Telecommunication Systems 15 185–201.
  • [19] Massoulie, L. and Roberts, J. (1999). Bandwidth sharing: Objectives and algorithms. INFOCOM '99. Eighteenth Annual Joint Conference of the IEEE Computer and Communications Societies. Proceedings. IEEE 3 1395–1403.
  • [20] Pittel, B. (1979). Closed exponential networks of queues with saturation: The Jackson-type stationary distribution and its asymptotic analysis. Math. Oper. Res. 4 357–378.
  • [21] Proutière, A. (2003). Insensitivity and stochastic bounds in queueing networks—application to flow level traffic modelling in telecommunication networks. Ph.D. thesis, Ecole Doctorale de l’Ecole Polytechnique.
  • [22] Schassberger, R. (1978). Insensitivity of steady-state distributions of generalized semi-Markov processes with speeds. Adv. in Appl. Probab. 10 836–851.
  • [23] Schweitzer, P. J. (1979). Approximate analysis of multiclass closed networks of queues. In Proceedings of the International Conference on Stochastic Control and Optimization. Free Univ., Amsterdam.
  • [24] Shah, D. and Wischik, D. J. (2008). Heavy traffic analysis of optimal scheduling algorithms for switched networks. Ann. Appl. Probab. To appear.
  • [25] Srikant, R. (2004). The Mathematics of Internet Congestion Control. Birkhäuser, Boston, MA.
  • [26] Whittle, P. (1985). Partial balance and insensitivity. J. Appl. Probab. 22 168–176.