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This paper presents a queuing model to evaluate the latency associated with file transfers or replications in peer-to-peer (P2P) computer systems. The main contribution of this paper is a modeling framework for the peers that accounts for the file size distribution, the search time, load distribution at peers, and number of concurrent downloads allowed by a peer. We propose a queuing model that models the nodes or peers in such systems as M/G/1/K processor sharing queues. The model is extended to account for peers which alternate between online and offline states. The proposed queuing model for the peers is combined with a single class open queuing network for the routers interconnecting the peers to obtain the overall file transfer latency. We also show that in scenarios with multipart downloads from different peers, a rate proportional allocation strategy minimizes the download times.