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This paper introduces a robust buffer occupancy-based connection admission control (CAC) mechanism to alleviate both uplink and downlink congestions in converged IP and broadcasting networks. The scheme also ensures a fair share of downlink bandwidth among competing satellite terminals (subnetworks) in the event of congestion. The proposed scheme is dubbed Weighted Fair CAC (W-FCAC). It accepts or rejects connections based on an adaptive fuzzy-based approach. The use of the fuzzy-based mechanism is for the purpose of overcoming issues related to instantaneous link capacity assessment, flow characterization and the associated high computational complexity, and use of traffic descriptors for new flows. Additionally, the adaptive fuzzy logic makes the proposed CAC approach robust to traffic dynamics. These features make the scheme highly suitable for DVB-S2 environments where the link capacity frequently fluctuates due to the adaptive coding/modulation of the physical layer during noisy periods. Simulation results elucidate that the proposed W-FCAC scheme prevents downlink congestion and fairly allocates network resources among satellite terminals. It also minimizes the frequency of congestion events while maintaining efficient utilization of network resources.