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In a typical video application, such as video-on-demand, videos are continuously streamed from a video server to a distributed set of receivers. The constant-quality video compression technique commonly used, variable bit rate (VBR) encoding, produces flows with multiple time-scale rate variability, so smoothing the VBR video traffic within an entire distribution tree presents a challenging task. This paper proposes a novel wavelet-based traffic smoothing (WTS) algorithm. Unlike existing algorithms, the WTS algorithm considers traffic smoothing at multiple resolutions. It results in a pruned version of a full tree, which corresponds to the original VBR traffic. Theoretical analysis and numerical evaluation demonstrate that: 1) WTS performs well across several metrics in smoothing bursty traffic and 2) for a video bit stream with N frames, the computational complexity of WTS is O(NlogN).