Heavy-traffic approximations for linear networks operating under $alpha$-fair bandwidth-sharing policies

We consider the flow-level performance of a linear network supporting elastic traffic, where the service capacity is shared among the various classes of users according to a weighted alpha-fair policy. Assuming Poisson arrivals and exponentially distributed service requirements for each class, the dynamics of the user population may be described by a Markov process. While valuable stability results have been established for the family of alpha-fair policies, the distribution of the number of active users has remained intractable in all but a few special cases. In order to gain further insight in the flow-level performance in more general scenarios, we develop approximations for the mean number of users based on the assumption that one or two of the nodes experience heavy-traffic conditions. In case of just a single `bottleneck' node, we exploit the fact that this node approximately behaves as a two-class Discriminatory Processor-Sharing model. In the case that there are two nodes critically loaded, we rely on the observation that the joint workload process at these nodes is asymptotically independent of the fairness coefficient alpha, provided all classes have equal weights. In particular, the distribution of the joint workload process is roughly equal to that for an unweighted Proportional Fair policy, which is exactly known. In both cases, the numbers of users at non-bottleneck nodes can be approximated by that in an M/M/1 queue with reduced service capacity. Extensive numerical experiments indicate that the resulting approximations tend to be reasonably accurate across a wide range of parameters, even at relatively moderate load values. The approximations for the mean number of users also provide useful estimates for the mean transfer delays and user throughputs