Hard Fairness Versus Proportional Fairness in Wireless Communications: The Single-Cell Case

Abstract

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 C (bit/s/Hz) versus E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b </sub> /N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> . 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 C as a function of E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> /N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> . 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 C versus E <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> /N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> 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, despite the fact that the imposed fairness constraint is laxer; c) for low to moderate spectral efficiency, the stricter hard-fairness constraint incurs in a large throughput penalty with respect to PFS