Maximizing Weighted Energy Efficiency Over Parallel Gaussian Broadcast Channels

A power assignment over parallel Gaussian broadcast channels splits a power budget at the access point among all channel-user pairs subject to per-channel upper-bounds on the sum-power, and yields a rate allocation to all channel-user pairs. Its weighted energy efficiency (WEE) is the ratio of its weighted sum-rate over its sum-power plus a fixed positive overhead. The problem Max-WEE seeks a power assignment maximizing the WEE. Special variants of Max-WEE with unit weights or two users per channel have been extensively studied in the literature. But none of the existing algorithms for those special variants have known bounds on running time, mainly because they follow the general-purposed methods for fractional programming. In this paper, we first derive fundamental properties and closed-form expressions of maximum WEE. Then we devise a simple water-filling algorithm for Max-WEE. Assuming all users are presorted by weight, the water-filling algorithm has <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">linear</i> complexity in the number of channel-user pairs. Under a mild presorting condition, we further develop a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">linear</i> -complexity algorithm for Max-WEE subject to rate demand.