We analyze a peer-assisted Video-on-Demand system in which users contribute their upload bandwidth to the redistribution of a video that they are downloading or that they have cached locally. Our target is to characterize the additional bandwidth that servers must supply to immediately satisfy all requests to watch a given video. We develop an approximate fluid model to compute the required server bandwidth in the sequential delivery case, as well as in controlled Bit-Torrent like swarms. Our approach is able to capture several stochastic effects related to peer churn, upload bandwidth heterogeneity, non-stationary traffic conditions, which have not been documented or analyzed before. At last, we provide important hints for the design of efficient peer-assisted VoD systems under server capacity constraints.