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Due to peer instability and time-varying peer upload bandwidth availability in live peer-to-peer (P2P) streaming channels, it is preferable to provision adequate levels of stable upload capacities at dedicated streaming servers, in order to guarantee the streaming quality in all channels. Most commercial P2P streaming systems have resorted to the practice of over-provisioning upload capacities on streaming servers. In this paper, we have performed a detailed analysis on 400 GB and 7 months of run-time traces from UUSee, a commercial P2P streaming system, and observed that available capacities on streaming servers are not able to keep up with the increasing demand imposed by hundreds of channels. We propose a novel online server capacity provisioning algorithm that proactively adjusts the server capacities available to each of the concurrent channels, such that the supply of server bandwidth in each channel dynamically adapts to the forecasted demand, taking into account the number of peers, the streaming quality, and the priorities of channels. The algorithm is able to learn over time, and has full ISP awareness to maximally constrain P2P traffic within ISP boundaries. To evaluate the effectiveness of our solution, our experimental studies are based on an implementation of the algorithm with actual channels of P2P streaming traffic, with real-world traces replayed within a server cluster.