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We consider power and server allocation in a multibeam satellite downlink which transmits data to N different ground locations over N time-varying channels. Packets destined for each ground location are stored in separate queues and the server rate for each queue i depends on the power pi (t) allocated to that server and the channel state ci (t) according to a concave rate-power curve μi(pi, ci). We establish the capacity region of all arrival rate vectors (λ1,...,λN) which admit a stabilizable system. We then develop a power-allocation policy which stabilizes the system whenever the rate vector lies within the capacity region. Such stability is guaranteed even if the channel model and the specific arrival rates are unknown. Furthermore, the algorithm is shown to be robust to arbitrary variations in the input rates and a bound on average delay is established. As a special case, this analysis verifies stability and provides a performance bound for the Choose-the-K-Largest-Connected-Queues policy when channels can be in one of two states (ON or OFF) and K servers are allocated at every timestep (K N). These results are extended to treat a joint problem of routing and power allocation in a system with multiple users and satellites and a throughput maximizing algorithm for this joint problem is constructed. Finally, we address the issue of interchannel interference and develop a modified policy when power vectors are constrained to feasible activation sets. Our analysis and problem formulation is also applicable to power control for wireless systems.