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Mobile edge computing (MEC) is emerging to support delay-sensitive 5G applications at the edge of mobile networks. When a user moves erratically among multiple MEC nodes, the challenge of how to dynamically migrate its service to maintain service performance (i.e., user-perceived latency) arises. However, frequent service migration can significantly increase operational cost, incurring the conflict between improving performance and reducing cost. To address these mis-aligned objectives, this paper studies the performance optimization of mobile edge service placement under the constraint of long-term cost budget. It is challenging because the budget involves the future uncertain information (e.g., user mobility). To overcome this difficulty, we devote to leveraging the power of prediction and advocate predictive service placement with predicted near-future information. By using two-timescale Lyapunov optimization method, we propose a T-slot predictive service placement (PSP) algorithm to incorporate the prediction of user mobility based on a frame-based design. We characterize the performance bounds of PSP in terms of cost-delay trade-off theoretically. Furthermore, we propose a new weight adjustment scheme for the queue in each frame named PSP-WU to exploit the historical queue information, which greatly reduces the length of queue while improving the quality of user-perceived latency. Rigorous theoretical analysis and extensive evaluations using realistic data traces demonstrate the superior performance of the proposed predictive schemes.