Congestion in large balanced multirate networks

In this paper, we obtain analytical approximations for various performance measures for a large fluid stochastic network that operates under a balanced fair bandwidth allocation policy. Balanced fairness results in the insensitivity of the stationary distribution of the number in the system to the precise distribution of file sizes. Balanced fairness has been shown to coincide with proportional fairness in large systems. The model we consider is that of servers operating under balanced fair rate allocations that are accessed by a large number of independent heterogeneous flows characterized by their arrival rate and general distributions of the file sizes; and a maximum service rate associated with each type of flow. The largeness of the system is parameterized by a scaling parameter that scales the arrival rates and capacity in such a way that the ratio is fixed. By exploiting a connection of the congestion probabilities with multirate Erlang loss systems, we use local limit large deviation methods to obtain accurate approximations as the scaling increases. The paper first discusses the single link case which is then extended to the case of a parking lot model that is a special case of tree networks.