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We propose a new model for on-demand media streaming centered around the peer-to-peer (P2P) paradigm. The proposed P2P model can support a large number of clients with a low overall system cost. The P2P model allows for peers to share some of their resources with the system and in return, they get some incentives or rewards. We describe how to realize (or deploy) the proposed model. In addition, we present a new dispersion algorithm (for disseminating the media files into the system) and a searching algorithm (for locating peers with the required objects). We demonstrate the potential of the P2P model as an infrastructure for a large-scale on-demand media streaming service through an extensive simulation study on large, Internet-like, topologies. Starting with a limited streaming capacity (hence, low cost), the simulation shows that the capacity is rapidly increased and many clients can be served even if they come according to different arrival patterns such as constant rate arrivals, flash crowd arrivals, and Poisson arrivals.