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To achieve higher memory bandwidth in network-based multiprocessor architectures, multiple dynamic random access memories can be accessed simultaneously. In such architectures, not only resource utilization and latency are the critical issues but also a reordering mechanism is required to deliver the response transactions of concurrent memory accesses in-order. In this paper, we present a memory-efficient on-chip network architecture to cope with these issues efficiently. Each node of the network is equipped with a novel network interface (NI) to deal with out-of-order delivery, and a priority-based router to decrease the network latency. The proposed NI exploits a streamlined reordering mechanism to handle the in-order delivery and utilizes the advance extensible interface transaction-based protocol to maintain compatibility with existing intellectual property cores. To improve the memory utilization and reduce the memory latency, an optimized memory controller is integrated in the presented NI. Experimental results with synthetic test cases demonstrate that the proposed on-chip network architecture provides significant improvements in average network latency (16%), average memory access latency (19%), and average memory utilization (22%).