In this paper we obtain approximations for various performance measures in a multirate link sharing bandwidth under an insensitive sharing mechanism called balanced fairness. Balanced fairness can be viewed as the large system limit of proportional fairness. For a large system, we obtain closed form expressions for the calculation of long run fraction of time that the system is congested, the probability that an arriving flow will not obtain its maximum bit rate and the average fraction of time that an arriving flow is not allocated its maximum bit rate while in the system. The techniques are based on local limit theorems for convolution measures.