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Wireless data traffic, especially video traffic, continues to increase at a rapid rate. Innovative network architectures and protocols are needed to improve the efficiency of data delivery and the quality of experience (QoE) of mobile users. Mobile edge computing (MEC) is a new paradigm that integrates computing capabilities at the edge of the wireless network. This paper presents a computation-capable and programmable wireless access network architecture to enable more efficient and robust video content delivery based on the MEC concept. It incorporates in-network data processing and communications under a unified software-defined networking platform. To address the multiple resource management challenges that arise in exploiting such integration, we propose a framework to optimize the QoE for multiple video streams, subject to wireless transmission capacity and in-network computation constraints. We then propose two simplified algorithms for resource allocation. The evaluation results demonstrate the benefits of the proposed algorithms for the optimization of video content delivery.