Product form stationary distributions for diffusion approximations to a flow-level model operating under a proportional fair sharing policy

Abstract

We consider a flow-level model of Internet congestion control introduced by Massoulié and Roberts [2]. We assume that bandwidth is shared amongst elastic documents according to a weighted proportional fair bandwidth sharing policy. With 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. In [1], under a mild local traffic condition, we establish a diffusion approximation for the workload process (and hence for the flow count process) in this model. We first recall that result in this paper. We then state results showing that when all of the weights are equal (proportional fair sharing) the diffusion has a product form invariant distribution with a strikingly simple interpretation in terms of dual random variables, one for each of the resources of the network. This result can be extended to the case where document sizes are distributed as finite mixtures of exponentials, and to models that include multi-path routing (these extensions are not described here, but can be found in [1]).

Short-term fairness and long-term QoS

“…This result complements the work of Kang et al [4] who have conjectured the distribution on the number of files in the network in a heavy-traffic regime using Brownian motion models. Assuming there is a single-hop flow on each link (local traffic assumption), it has been shown that…”

“…Using the Foster-Lyapunov criterion, we derive an upper-bound on the expected number of files in the network under the proportionally-fair policy, and the upper bound increases linearly with the number of flows in the network. This result complements the work of Kang et al [4] who have obtained a conjecture on the distribution on the number of files in the network in a heavy-traffic regime using Brownian motion models. Our upper bound matches the result in [4] in certain symmetric networks.…”

“…This result complements the work of Kang et al [4] who have obtained a conjecture on the distribution on the number of files in the network in a heavy-traffic regime using Brownian motion models. Our upper bound matches the result in [4] in certain symmetric networks. It should be noted that the result in [4] is a generalization of earlier results on product-form solutions obtained by [9] and [2].…”

“…Our upper bound matches the result in [4] in certain symmetric networks. It should be noted that the result in [4] is a generalization of earlier results on product-form solutions obtained by [9] and [2].…”

“…Despite the above limitations, the expressions obtained in Theorem 2 and in [4] can be used to study the delay performance of proportionally-fair resource allocation. In [13], optimal policies for a line network with two links were proposed.…”