Today, we witness a transformation of radio access network topologies from strictly tree-structured towards meshed architectures. Yet, these edge networks follow mostly circuit-switched paradigms to support quality of service applications such as voice or video communication. In this work, we develop a novel quality of service aware routing framework to optimize the performance of edge networks in the packet-switched domain. Starting from the Internet?s best-effort routing paradigm and building on related work, we formulate a near-optimal multiclass minimum-delay routing algorithm. Our algorithm optimizes network-wide end-to-end delay and allows for differentiation of service. We exploit two orthogonal dimensions namely multipath routing and class-based service prioritization. We provide a precise description of our algorithm and a detailed experimental analysis against state of the art routing algorithms. Our algorithm is able to achieve excellent performance while maintaining the simplicity of a decentralized and distributed routing algorithm, thus matching the requirements for future radio access networks.