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In this paper, we propose a new congestion management strategy for lossless multistage interconnection networks that scales as network size and/or link bandwidth increase. Instead of eliminating congestion, our strategy avoids performance degradation beyond the saturation point by eliminating the HOL blocking produced by congestion trees. This is achieved in a scalable manner by using separate queues for congested flows. These are dynamically allocated only when congestion arises, and deallocated when congestion subsides. Performance evaluation results show that our strategy responds to congestion immediately and completely eliminates the performance degradation produced by HOL blocking while using only a small number of additional queues.