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Broadcast satellite networks are going to play an important role in the global information infrastructure. Satellites can provide direct-to-user Internet services and as well as traffic trunks in the middle of the network. About 98% of Internet traffic is TCP traffic. TCP works well in terrestrial fiber networks, but does not work well in satellite and in hybrid (satellite-terrestrial) networks. We analyze the problems that cause this degraded performance and review the solutions proposed in the literature. Because an end-to-end solution is difficult in the hybrid network, we propose a solution based on connection splitting. While much research has been done on improving TCP throughput for a single connection, we consider the case when multiple connections with different round trip times compete with each other. The crucial part of our scheme is the flow control algorithm at the hybrid gateways (i.e., the interfaces between the satellite and terrestrial networks). Because the end-to-end connection is split at the hybrid gateways, the data packets are buffered at the TCP layer rather than at the IP layer as in a normal router. Furthermore, the TCP layer not only buffers data waiting for transmission, but also buffers data transmitted but not yet acknowledged. The flow control algorithm should avoid stall caused by buffer exhaustion, guarantee fairness among all competing connections and maintain high utilization of the satellite link. A performance evaluation demonstrates that our scheme meets these requirements.