Insensitivity of proportional fairness in critically loaded bandwidth sharing networks

Proportional fairness is a popular service allocation mechanism to describe and analyze the performance of data networks at flow level. Several authors have shown that the invariant distribution of networks operating according to proportional fairness admits a product form distribution under critical loading. They focus however on exponential job size distributions, leaving the case of general job size distributions as an open question. Motivated by this, we consider a network operating under proportional fairness where the job size distributions are of phase-type. We establish a heavy-traffic process limit theorem and show that the invariant distribution of the limit process is determined by the first moments of the job sizes only. Our analysis relies on a uniform convergence result for a fluid model.

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