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The recent drastic increase in mobile data traffic has pushed the mobile edge computing systems to the limit of their capacity. A promising solution to this problem is the task migration provided by unmanned aerial vehicles (UAV). Key factors to be taken into account in the design of UAV offloading schemes must include the number of tasks waiting in the system as well as their corresponding deadlines. An appropriate system cost which is used as an objective function to be minimized comprises two parts. First, an offloading cost which can be interpreted as the cost of using computational resources at the UAV. Second, a penalty cost due to potential task expiration. In order to minimize the expected (time average) cost over a time horizon, we formulate a Dynamic Programming (DP) equation and analyze it to describe properties of a candidate optimal offloading policy. The DP equation suffers from the well-known "Curse of Dimensionality" that makes computations intractable, especially when the state space is infinite. In order to reduce the computational burden, we identify three important properties of the optimal policy. Based on these properties, we show that it suffices to evaluate the DP equation on a finite subset of the state space only. We then show that the optimal task offloading decision associated with a state can be inferred from the decision taken at its "adjacent" states, further reducing the computational load. Finally, we provide numerical results to evaluate the influence of different parameters on the system performance as well as verify the theoretical results.