  Proportional fairness is a popular service allocation mechanism to describe
and analyze the performance of data networks at flow level. Recently, several
authors have shown that the invariant distribution of such networks admits a
product form distribution under critical loading. Assuming exponential job size
distributions, they leave the case of general job size distributions as an open
question. In this paper we show the conjecture holds for a dense class of
distributions. This yields a key example of a stochastic network in which the
heavy traffic limit has an invariant distribution that does not depend on
second moments. Our analysis relies on a uniform convergence result for a fluid
model which may be of independent interest.
