On the Throughput Allocation for Proportional Fairness in Multirate IEEE   802.11 DCF
  This paper presents a modified proportional fairness (PF) criterion suitable for mitigating the \textit{rate anomaly} problem of multirate IEEE 802.11 Wireless LANs employing the mandatory Distributed Coordination Function (DCF) option. Compared to the widely adopted assumption of saturated network, the proposed criterion can be applied to general networks whereby the contending stations are characterized by specific packet arrival rates, $\lambda_s$, and transmission rates $R_d^{s}$.   The throughput allocation resulting from the proposed algorithm is able to greatly increase the aggregate throughput of the DCF while ensuring fairness levels among the stations of the same order of the ones available with the classical PF criterion. Put simply, each station is allocated a throughput that depends on a suitable normalization of its packet rate, which, to some extent, measures the frequency by which the station tries to gain access to the channel. Simulation results are presented for some sample scenarios, confirming the effectiveness of the proposed criterion. 
F. Daneshgaran, M. Laddomada, F. Mesiti, and M. Mondin