Explosively increasing multimedia services and applications, e.g., automatic speech recognition (ASR), have aggravated the burden on the cloud server in mobile networks. To address the challenge, mobile edge computing has emerged for partially alleviating the workload of the cloud server and enhancing the quality of service of mobile users. In this paper, we aim to employ the technique of edge-cloud computing to accelerate the processing of ASR tasks generated by users in mobile networks. Particularly, we deploy a convolutional neural network based encoder in each edge server to extract features of the audio data. Based on some certain network constraints (i.e., user association and edge servers' storage/computing capacity), we propose a low-complexity and distributed iterative greedy method to address the formulated nonlinear mixed-integer nonconvex optimization problem. Simulation results demonstrate the effectiveness of the proposed scheme on reducing the total delay in the network.