State space collapse and diffusion approximation for a network operating under a fair bandwidth sharing policy

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 .-fair bandwidth sharing policy introduced by Mo and Walrand [IEEE/ACM Transactions on Networking 8 (2000) 556.567] [..(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 (.=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 ..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.