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Two major goals of queue management are flow fairness and queue-length stability However, most prior works dealt with these goals independently. In this paper, we show that both goals can be effectively achieved at the same time. We propose a novel scheme that realizes flow fairness and queue-length stability. In the proposed scheme, high-bandwidth flows are identified via a multilevel caching technique. Then, we calculate the base drop probability for resolving congestion with a stable queue, and apply it to individual flows differently depending on their sending rates. Via extensive simulations, we show that the proposed scheme effectively realizes flow fairness between unresponsive and TCP flows, and among heterogeneous TCP flows, while maintaining a stable queue.