We address deterministic resource allocation in point-to-point multi-terminal AWGN channels without inter-terminal interference, with particular focus on optimizing quantile transmission rates for cell-edge terminal service. Classical utility-based approaches -- such as minimum rate, sumrate, and proportional fairness -- are either overconservative, or inappropriate, or do not provide a rigorous and/or interpretable foundation for fair rate optimization at the edge. To overcome these challenges, we employ Conditional Value-at-Risk (CVaR), a popular coherent risk measure, and establish its equivalence with the sum-least-$α$th-quantile (SL$α$Q) utility. This connection enables an exact convex reformulation of the SL$α$Q maximization problem, facilitating analytical tractability and precise and interpretable control over cell-edge terminal performance. Utilizing Lagrangian duality, we provide (for the first time) parameterized closed-form solutions for the optimal resource policy -- which is of waterfilling-type -- as well as the associated (auxiliary) Value-at-Risk variable. We further develop a novel inexact dual subgradient descent algorithm of minimal complexity to determine globally optimal resource policies, and we rigorously establish its convergence. The resulting edge waterfilling algorithm iteratively and efficiently allocates resources while explicitly ensuring transmission rate fairness across (cell-edge) terminals. Several (even large-scale) numerical experiments validate the effectiveness of the proposed method for enabling robust quantile rate optimization at the edge.