Cost of proportional fairness in multiuser networks

Maintaining fairness in multiuser networks is viable at the expense of reducing the overall spectral efficiency in the network. This paper considers a general multiuser network and analyzes the disparity between the sum-rate capacity and the sum-rate achievable under the proportionally fair allocation of the resources among the users. The analysis demonstrates that the ratio of the fairness-constrained sum-rate to the sum-rate capacity in an n-user network scales with n at a rate larger than n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1/2</sup> . Moreover, a lower bound on this ratio is obtained that is valid for any arbitrary network topology and user connectivity.