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Peer-to-peer networks are an increasingly popular solution for the distribution of media content to a large number of users, with limited investments for network infrastructures. In this paper we focus on mesh-based, also referred to as generic, unstructured, or data-driven, peer-to-peer video streaming systems. We carry out a performance analysis of these systems, analyzing their sensitivity to the most critical system parameters. The analysis focuses on the number of neighbors of each peer, the average number of peers in the system, the permanence time of peers, and the amount of buffered video. Differently from existing studies we found that, depending on peers' permanence time, an optimal number of neighbors can be found. Moreover, moving towards a higher number of neighbors, the corresponding increase of signaling traffic causes a decrease of overall performance.