Stochastic stability under network utility maximization: General file size distribution

We prove the stochastic stability of resource allocation under Network Utility Maximization (NUM) under general arrival process and file size distribution with bounded support, for α-fair utilities with α sufficiently small and possibly different for different sources’ utility functions. In addition, our results imply that the system operating under α-fair utility is 1/(1 + α)-approximate stable for any α ∈ (0,∞) for any file size distribution with bounded support. Our results are in contrast to the recent stability result of Bramson (2005) for max-min fair (i.e. α = ∞) under general arrival process and file size distribution, and that of Massoulie (2006) for proportional fair (i.e. α = 1) under Poisson arrival process and phase-type distributions. To obtain our results, we develop an appropriate Lyapunov function for the fluid model established by Gromoll and Williams (2006)1 .