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Buffered crossbar switches are special crossbar switches with a small exclusive buffer at each crosspoint of the crossbar. They demonstrate unique advantages, such as variable length packet handling and distributed scheduling, over traditional unbuffered crossbar switches. The current main approach for buffered crossbar switches to provide performance guarantees is to emulate push-in-first-out output queued switches. However, such an approach has several drawbacks, and in particular it has difficulty in providing tight constant performance guarantees. To address the issue, we propose in this paper the guaranteed-performance asynchronous packet scheduling (GAPS) algorithm for buffered crossbar switches. GAPS intends to provide tight performance guarantees, and requires no speedup. It directly handles variable length packets without segmentation and reassembly, and makes scheduling decisions in a distributed manner. We show by theoretical analysis that GAPS achieves constant performance guarantees. We also prove that GAPS has a bounded crosspoint buffer size of 3L, where L is the maximum packet length. Finally, we present simulation data to verify the analytical results and show the effectiveness of GAPS.