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Traffic loads generated by many parallel applications can exhibit strong burstiness, which may significantly affect the performance of interconnection networks supporting communication among individual processors. As a step towards obtaining a deeper understanding of network performance under various traffic conditions, this paper develops an analytical model for pipelined circuit-switched k-ary n-cubes augmented with virtual channels support in the presence of bursty traffic by extending the application of the well-known Markov-modulated Poisson process (MMPP) to queuing networks. The model is then used to investigate the performance implications of using virtual channels when the network is subjected to bursty traffic. The results reveal the importance of virtual channel flow control for reducing the degrading effects of bursty traffic on network performance. Our analysis also shows that when the number of virtual channels reaches a certain threshold, adding more virtual channels does not lead to considerable performance gains.