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We introduce three important factors for efficient video-on-demand (VoD) streaming strategy in broadband access networks such as a passive optical network (PON): an efficient streaming scheme for bandwidth savings, optimal use of deployed network bandwidth, and proactive use of user storage facilities. To achieve these goals, we propose efficient algorithms such as video adaptive streaming (VAST), video greedy adaptive streaming (VGAST), and video greedy adaptive streaming with proactive buffering (VGAST-PB). The VAST and VGAST algorithms use a patching scheme to save on network bandwidth for video streaming and increase the efficiency by adaptively changing the speed of patch streams and the patching window based on the time of the day and video popularity. VGAST-PB further improves the efficiency of VGAST by proactively buffering the popular videos. The proposed algorithms can be applied in fiber-to-the-x broadband access networks. To analyze the algorithms, we consider a typical 2.488 Gbps gigabit PON network providing VoD service to 512 households. We use a daily video request model and a video popularity model to determine the adaptive streaming speed and dynamic patching window. Numerical results show that, when the available network bandwidth is reduced below the required level due to background traffic, the efficient algorithms can con-siderably reduce the average user waiting time and the number of waiting requests.