In recent years there have been several large-scale deployments of P2P live video systems. Existing and future P2P live video systems will offer a large number of channels, with users switching frequently among the channels. In this paper, we develop infinite-server queueing network models to analytically study the performance of multi-channel P2P streaming systems. Our models capture essential aspects of multi-channel video systems, including peer channel switching, peer churn, peer bandwidth heterogeneity, and Zipf-like channel popularity. We apply the queueing network models to two P2P streaming designs: the isolated channel design (ISO) and the View-Upload Decoupling (VUD) design. For both of these designs, we develop efficient algorithms to calculate critical performance measures, develop an asymptotic theory to provide closed-form results when the number of peers approaches infinity, and derive near-optimal provisioning rules for assigning peers to groups in VUD. We use the analytical results to compare VUD with ISO. We show that VUD design generally performs significantly better, particularly for systems with heterogeneous channel popularities and streaming rates.