Key points are not available for this paper at this time.
F. P. Kelly, A. K. Maulloo, D. K. H. Tan, Rate Control for Communication Networks: Shadow Prices, Proportional Fairness and Stability, The Journal of the Operational Research Society, Vol. 49, No. 3 (Mar., 1998), pp. 237-252
Kelly et al. (Sun,) studied this question.