The proportional fair sharing (PFS) algorithm has been used in multi-user systems as an attempt to balance fairness and performance of the system throughput. Motivated by the cellular downlink scheduling problem, it is shown that when the rates of each user are i.i.d., the performance of the PFS scheduling algorithms is asymptotically equivalent to a purely greedy scheduling algorithm. The mean asymptotic throughput of the PFS algorithm is characterized and the rate of convergence to this limit is derived under i.i.d. models. Additionally the asymptotic covariance matrix about the convergence point is stated.