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This paper identifies performance degradation in wormhole routed k-ary n-cube networks due to limited number of router-to-processor consumption channels at each node. Many recent research in wormhole routing have advocated the advantages of adaptive routing and virtual channel flow control schemes to deliver better network performance. This paper indicates that the advantages associated with these schemes cannot be realized with limited consumption capacity. To alleviate such performance bottlenecks, a new network interface design using multiple consumption channels is proposed. To match virtual multiplexing on network channels, we also propose each consumption channel to support multiple virtual consumption channels. The impact of message arrival rate at a node on the required number of consumption channels is studied analytically. It is shown that wormhole networks with higher routing adaptivity, dimensionality, degree of hot-spot traffic, and number of virtual lanes have to take advantage of multiple consumption channels to deliver better performance. The interplay between system topology, routing algorithm, number of virtual lanes, messaging overheads, and communication traffic is studied through simulation to derive the effective number of consumption channels required in a system. Using the ongoing technological trend, it is shown that wormhole-routed systems can use up to two-four consumption channels per node to deliver better system performance.