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We investigate the use of fixed-point methods for predicting the performance of multiple TCP flows sharing geostationary satellite links. The problem formulation is general in that it can address both error-free and error-prone links, proxy mechanisms such as split-TCP connections and account for asymmetry and different satellite network configurations. We apply the method in the specific context of bandwidth on demand (BoD) satellite links. The analytical approximations show good agreement with simulation results, although they tend to be optimistic when the link is not saturated. The main constraint upon the method applicability is the limited availability of analytical models for the MAC-induced packet delay under non-homogeneous load and prioritization mechanisms.