Hard fairness versus proportional fairness in wireless communications: the single-cell case

We consider a wireless communication system formed by a single cell with one base station and $K$ user terminals. User channels are characterized by frequency-selective fading due to small-scale effects, modeled as a set of $M$ parallel block-fading channels, and a frequency-flat distance-dependent path loss. We compare delay-limited systems with variable-rate systems under fairness constraints, in terms of the achieved system spectral efficiency ${ssr C}$ (bit/s/Hz) versus $E_b/N_0$. The considered delay-limited systems impose hard-fairness: every user transmits at its desired rate on all blocks, independently of its fading conditions. The variable-rate system imposes proportional fairness via the popular Proportional Fair Scheduling (PFS) algorithm, currently implemented in 3G wireless for data (delay-tolerant) applications. We find simple iterative resource allocation algorithms that converge to the optimal delay-limited throughput for orthogonal (frequency-division multiple access (FDMA)/time-division multiple access (TDMA)) and optimal (superposition/interference cancellation) signaling. In the limit of large $K$ and finite $M$ we find closed-form expressions for ${ssr C}$ as a function of $E_b/N_0$. We show that in this limit, the optimal allocation policy consists of letting each user transmit on its best subchannel only. Also, we find a simple closed-form expression for the throughput of PFS in a cellular environment, that holds for any $K$ and $M$. Finally, we obtain closed-form expressions for ${ssr C}$ versus $E_b/N_0$ in the low and high spectral efficiency regimes. The conclusions of our analysis in terms of system design guidelines are as follows: a) if hard fairness is a requirement, orthogonal access incurs a large throughput penalty with respect to the optimal (superposition coding) strategy, especially in the regime of high spectral efficiency; b) for high spectral efficiency, PFS does not provide any significant gain and may even perform- worse than the optimal delay-limited system, de