We consider a connection-level model of Internet congestion control, introduced by Massoulié and Roberts [Telecommunication Systems 15 (2000) 185–201], that represents the randomly varying number of flows present in a network. Here, bandwidth is shared fairly among elastic document transfers according to a weighted α-fair bandwidth sharing policy introduced by Mo and Walrand [IEEE/ACM Transactions on Networking 8 (2000) 556–567] [α∈(0, ∞)]. Assuming Poisson arrivals and exponentially distributed document sizes, we focus on the heavy traffic regime in which the average load placed on each resource is approximately equal to its capacity. A fluid model (or functional law of large numbers approximation) for this stochastic model was derived and analyzed in a prior work [Ann. Appl. Probab. 14 (2004) 1055–1083] by two of the authors. Here, we use the long-time behavior of the solutions of the fluid model established in that paper to derive a property called multiplicative state space collapse, which, loosely speaking, shows that in diffusion scale, the flow count process for the stochastic model can be approximately recovered as a continuous lifting of the workload process.
Under weighted proportional fair sharing of bandwidth (α=1) and a mild local traffic condition, we show how multiplicative state space collapse can be combined with uniqueness in law and an invariance principle for the diffusion [Theory Probab. Appl. 40 (1995) 1–40], [Ann. Appl. Probab. 17 (2007) 741–779] to establish a diffusion approximation for the workload process and hence to yield an approximation for the flow count process. In this case, the workload diffusion behaves like Brownian motion in the interior of a polyhedral cone and is confined to the cone by reflection at the boundary, where the direction of reflection is constant on any given boundary face. When all of the weights are equal (proportional fair sharing), this diffusion has a product form invariant measure. If the latter is integrable, it yields the unique stationary distribution for the diffusion which has a strikingly simple interpretation in terms of independent dual random variables, one for each of the resources of the network. We are able to extend this product form result to the case where document sizes are distributed as finite mixtures of exponentials and to models that include multi-path routing. We indicate some difficulties related to extending the diffusion approximation result to values of α≠1.
We illustrate our approximation results for a few simple networks. In particular, for a two-resource linear network, the diffusion lives in a wedge that is a strict subset of the positive quadrant. This geometrically illustrates the entrainment of resources, whereby congestion at one resource may prevent another resource from working at full capacity. For a four-resource network with multi-path routing, the product form result under proportional fair sharing is expressed in terms of independent dual random variables, one for each of a set of generalized cut constraints.
References
[1] Ben Fredj, S., Bonald, T., Proutière, A., Regnie, G. and Roberts, J. (2001). Statistical bandwidth sharing: A study of congestion at flow level. Computer Communication Review 31 111–121.
[2] Billingsley, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
[3] Bonald, T. and Massoulié, L. (2001). Impact of fairness on Internet performance. ACM Sigmetrics Performance Evaluation Review 29 82–91.
[4] Bonald, T. and Proutière, A. (2003). Insensitive bandwidth sharing in data networks. Queueing Syst. 44 69–100.
[5] Bramson, M. (1998). State space collapse with application to heavy traffic limits for multiclass queueing networks. Queueing Syst. 30 89–148.
[6] Bramson, M. (2009). Network stability under max–min fair bandwidth sharing. Preprint.
[7] Bu, T. and Towsley, D. (2001). Fixed point approximation for TCP behavior in an AQM network. ACM Sigmetrics Performance Ecaluation Review 29 216–225.
[8] Chiang, M., Shah, D. and Tang, A. K. (2006). Stochastic stability under network utility maximization: General file size distribution. In Proceedings of 44th Allerton Conference on Communication, Control and Computing, September 2006, Monticello, IL.
[9] Cottle, R. W., Pang, J.-S. and Stone, R. E. (1992). The Linear Complementarity Problem. Academic Press, Boston, MA.
[10] Dai, J. G. and Williams, R. J. (1995). Existence and uniqueness of semimartingale reflecting Brownian motions in convex polyhedrons. Theory Probab. Appl. 40 1–40. Correction: 50 (2006), 346–347.
[11] 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.
[12] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley, New York.
Mathematical Reviews (MathSciNet):
MR838085
[13] Gibbens, R. J., Sargood, S. K., Van Eijl, C., Kelly, F. P., Azmoodeh, H., Macfadyen, R. N. and Macfadyen, N. W. (2000). Fixed-point models for the end-to-end performance analysis of IP networks. In 13th ITC Specialist Seminar: IP Traffic Measurement, Modeling and Management, Sept 2000, Monterey, California.
[14] 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.
[15] Gromoll, H. C. and Williams, R. J. (2009). 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 for Thomas G. Kurtz. IMS Collections 4 253–265. Institute of Mathematical Statistics, Beachwood, OH.
[16] Harrison, J. M. (2003). Brownian models of open processing networks: Canonical representation of workload. Ann. Appl. Probab. 13 390–393.
[17] Harrison, J. M. and Williams, R. J. (1987). Multidimensional reflected Brownian motions having exponential stationary distributions. Ann. Probab. 15 115–137.
Mathematical Reviews (MathSciNet):
MR877593
[18] Harrison, J. M. and Williams, R. J. (1987). Brownian models of open queueing networks with homogeneous customer populations. Stochastics 22 77–115.
Mathematical Reviews (MathSciNet):
MR912049
[19] Han, H., Shakkottai, S., Hollot, C. V., Srikant, R. and Towsley, D. (2006). Multi-path TCP: A joint congestion control and routing scheme to exploit path diversity on the Internet. IEEE/ACM Transactions on Networking 14 1260–1271.
[20] Kang, W. and Williams, R. J. (2007). An invariance principle for semimartingale reflecting Brownian motions in domains with piecewise smooth boundaries. Ann. Appl. Probab. 17 741–779.
[21] Kang, W. N. and Williams, R. J. (2009). Diffusion approximation for an input-queued packet switch operating under a maximum weight algorithm. To appear.
[22] Kang, W., Kelly, F. P., Lee, N. H. and Williams, R. J. (2004). On fluid and Brownian approximations for an Internet congestion control model. In Proceedings of the 43rd IEEE Conference on Decision and Control. December 2004, 3938–3943.
[23] 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. ACM SIGMETRICS Performance Evaluation Review 35 36–38.
[24] Kelly, F. P. (1991). Loss networks. Ann. Appl. Probab. 1 319–378.
[25] Kelly, F. P. and Voice, T. (2005). Stability of end-to-end algorithms for joint routing and rate control. Computer Communication Review 35 5–12.
[26] 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.
[27] Key, P. and Massoulié, L. (2006). Fluid models of integrated traffic and multipath routing. Queueing Syst. 53 85–98.
[28] 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.
[29] Laws, C. N. (1992). Resource pooling in queueing networks with dynamic routing. Adv. in Appl. Probab. 24 699–726.
[30] Lin, X. and Shroff, N. (2004). On the stability region of congestion control. In Proceedings of the Allerton Conference on Communications, Control and Computing, September 2004, Monticello, IL.
[31] Lin, X., Shroff, N. B. and Srikant, R. (2008). On the connection-level stability of congestion-controlled communication networks. IEEE Trans. Inform. Theory 54 2317–2338.
[32] Massoulié, L. (2007). Structural properties of proportional fairness: Stability and insensitivity. Ann. Appl. Probab. 17 809–839.
[33] Massoulié, L. and Roberts, J. (2000). Bandwidth sharing and admission control for elastic traffic. Telecommunication Systems 15 185–201.
[34] Massoulié, L. and Roberts, J. (2002). Bandwidth sharing: Objectives and algorithms. IEEE/ACM Transactions on Networking 10 320–328.
[35] Mo, J. and Walrand, J. (2000). Fair end-to-end window-based congestion control. IEEE/ACM Transactions on Networking 8 556–567.
[36] Roughan, M., Erramilli, A. and Veitch, D. (2001). Network performance for TCP networks, Part I: Persistent sources. In Proceeding ITC’17, Brasil, September 2001.
[37] Schassberger, R. (1986). Two remarks on insensitive stochastic models. Adv. in Appl. Probab. 18 791–814.
Mathematical Reviews (MathSciNet):
MR857330
[38] Shah, D. and Wischik, D. (2007). Heavy traffic analysis of optimal scheduling algorithms for switched networks. Preprint.
[39] Srikant, R. (2004). On the positive recurrence of a Markov chain describing file arrivals and departures in a congestion-controlled network. In Presented at the IEEE Computer Communications Workshop.
[40] Tassiulas, L. and Ephremides, A. (1992). Stability properties of constrained queueing systems and scheduling policies for maximum throughput in multihop radio networks. IEEE Trans. Automat. Control 37 1936–1948.
[41] Williams, R. J. (1987). Reflected Brownian motion with skew symmetric data in a polyhedral domain. Probab. Theory Related Fields 75 459–485.
Mathematical Reviews (MathSciNet):
MR894900
[42] Williams, R. J. (1998). Diffusion approximations for open multiclass queueing networks: Sufficient conditions involving state space collapse. Queueing Systems Theory Appl. 30 27–88.
[43] Williams, R. J. (1998). An invariance principle for semimartingale reflecting Brownian motions in an orthant. Queueing Systems Theory Appl. 30 5–25.
[44] Ye, H. and Yao, D. D. (2008). Heavy-traffic optimality of a stochastic network under utility-maximizing resource allocation. Operations Research. 56 453–470.
