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Combined input-crosspoint buffered switches provide high-performance switching and relax arbitration timing for packet switches with high-speed ports. It has been shown that a switch with one-cell crosspoint buffers and round-robin selection at input and output ports provides 100% throughput under uniform traffic. However, under admissible traffic patterns with nonuniform distributions, only weight-based arbitration schemes are reported to provide high throughput by these switches. This paper introduces two arbitration schemes based on round-robin arbitration for combined input-crosspoint buffered packet switches. The proposed schemes provide nearly 100% throughput for several admissible traffic patterns, including uniform and unbalanced traffic, using one-cell crosspoint buffers.