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Mobile edge computing (MEC) has recently emerged as an effective paradigm to enhance the computing capability of capability-limited mobile devices (MDs). In this article, we consider an MEC system consisting of a number of MEC servers and one MD which generates a series of tasks characterized by their dependency relationships. We study computation scheduling and offloading problem of the tasks. To improve the task processing performance, we first propose a parallel transmission and execution (PTE) scheme, based on which we design a computation scheduling and offloading algorithm. Considering the fairness among tasks in terms of task transmission and execution time, we formulate the computation scheduling and offloading problem as a constrained worst-case latency optimization problem which minimizes the maximum completion time of all the tasks. As the original optimization problem cannot be solved conveniently, we first categorize the tasks into high priority tasks (HPTs), medium priority tasks (MPTs) and low priority tasks (LPTs) based on their task execution status and causal relationship. Then, a dynamic priority-based computation scheduling and offloading algorithm is proposed, which designs computation scheduling and offloading strategy for dynamically-changed HPTs and MPTs, respectively. In particular, for HPTs, a multiple knapsack-based heuristic algorithm is proposed, and a task weight and data size-based computation scheduling and offloading algorithm is further proposed for MPTs. Numerical results demonstrate the effectiveness of the proposed scheme. [ABSTRACT FROM AUTHOR]