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In network-on-chip (NoC), the packet arbitration policy in most switch designs is based either on the packet priority or on a round-robin scheme. Since the two arbitration policies cannot guarantee the quality of service (QoS) and the fairness simultaneously, this work proposes a novel round-based priority arbitration scheme, which not only allows the high-priority packets to be served first within a specific context called round, but also allows fairness for low-priority packets. The experiments show that this proposed arbitration scheme outperforms the packet priority scheme by as much as 15.3% in terms of fairness and is superior to the round-robin scheme by as much as 13.4% in terms of quality of service. Further, an estimation model for communication latency is also proposed for this arbitration scheme and the prediction accuracy is around 80%.