Stability of networks under general file size distribution with alpha fair rate allocation

Convex optimization has been widely used to model bandwidth allocation policies among TCP flows in the Internet. When the offered load is less than capacity, stochastic stability of networks using such policies has been established for exponentially distributed file sizes. The problem has remained open for general file size distributions, which is very relevant as it is well known that Internet file sizes follow heavy-tailed distributions. In this paper, building on existing results on the fluid model of the system, we use a partial differential equation to characterize the dynamics. The equation keeps track of residual file size and therefore is suitable to study networks under general file size distributions. For alpha fair bandwidth allocation, with any positive alpha parameter, a Lyapunov function is constructed to prove stability when the offered load does not exceed capacity. The result gives an affirmative answer to this open problem and sets a foundation for further analysis of such systems.