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Mobile Edge Computing (MEC) has been regarded as a promising paradigm to reduce service latency for data processing in the Internet of Things, by provisioning computing resources at the network edge. In this work, we jointly optimize the task partitioning and computational power allocation for computation offloading in a dynamic environment with multiple IoT devices and multiple edge servers. We formulate the problem as a Markov decision process with constrained hybrid action space, which cannot be well handled by existing deep reinforcement learning (DRL) algorithms. Therefore, we develop a novel Deep Reinforcement Learning called Dirichlet Deep Deterministic Policy Gradient (D3PG), which is built on Deep Deterministic Policy Gradient (DDPG) to solve the problem. The developed model can learn to solve multi-objective optimization, including maximizing the number of tasks processed before expiration and minimizing the energy cost and service latency.} More importantly, D3PG can effectively deal with constrained distribution-continuous hybrid action space, where the distribution variables are for the task partitioning and offloading, while the continuous variables are for computational frequency control. Moreover, the D3PG can address many similar issues in MEC and general reinforcement learning problems. Extensive simulation results show that the proposed D3PG outperforms the state-of-art methods.