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Proportional fairness is a popular service allocation mechanism to describeand analyze the performance of data networks at flow level. Recently, severalauthors have shown that the invariant distribution of such networks admits aproduct form distribution under critical loading. Assuming exponential job sizedistributions, they leave the case of general job size distributions as an openquestion. In this paper we show the conjecture holds for a dense class ofdistributions. This yields a key example of a stochastic network in which theheavy traffic limit has an invariant distribution that does not depend onsecond moments. Our analysis relies on a uniform convergence result for a fluidmodel which may be of independent interest.

Insensitivity of Proportional Fairness in Critically Loaded Bandwidth Sharing Networks