Hard Fairness Versus Proportional Fairness in Wireless Communications:   The Multiple-Cell Case
  We consider the uplink of a cellular communication system with $K$ users per cell and infinite base stations equally spaced on a line. The system is conventional, i.e., it does not make use of joint cell-site processing. A hard fairness (HF) system serves all users with the same rate in any channel state. In contrast, a system based on proportional fairness serves the users with variable instantaneous rates depending on their channel state. We compare these two options in terms of the system spectral efficiency \textsf{C} (bit/s/Hz) versus $E_b/N_0$. Proportional fair scheduling (PFS) performs generally better than the more restrictive HF system in the regime of low to moderate SNR, but for high SNR an optimized HF system achieves throughput comparable to that of PFS system for finite $K$. The hard-fairness system is interference limited. We characterize this limit and validate a commonly used simplified model that treats outer cell interference power as proportional to the in-cell total power and we analytically characterize the proportionality constant. In contrast, the spectral efficiency of PFS can grow unbounded for $K \to \infty$ thanks to the multiuser diversity effect. We also show that partial frequency/time reuse can mitigate the throughput penalty of the HF system, especially at high SNR. 
Daeyoung Park and Giuseppe Caire