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In this paper, we address the problem of data-burst grooming in optical burst-switched (OBS) networks. In OBS networks, IP packets with the same edge-node destination are assembled into larger packets called data bursts. Depending on the core node's switching technology, data bursts are required to have a minimum length. On the other hand, each IP packet in a burst has a time delay constraint, called maximum end-to-end delay, which determines the upper time limit before which the packet must reach its destination. Thus, a data burst cannot wait indefinitely until a sufficient number of IP packets are assembled and the minimum burst length requirement is met. In order to satisfy the packet maximum end-to-end delay requirement, many bursts will be timed out and released before they reach the minimum-length requirement. Under such circumstances, padding overhead must be added to these short bursts, called sub-bursts. Excessive padding results in high overhead and high data-burst blocking probability. One approach to minimize the amount of padding overhead, while maintaining the end-to-end delay requirement of IP packets, is to groom multiple sub-bursts together. That is, sub-bursts with different destinations are aggregated together at the edge node and transmitted as a single burst until they are separated at some downstream node. In this paper, we present an edge-node architecture enabling burst-grooming capability. We also develop two basic grooming approaches, namely No routing overhead (NoRO) and minimum total overhead (MinTO). Through a comprehensive simulation study, we show that, in general, our proposed grooming algorithms can significantly improve the performance compared to the case of no grooming. However, careful considerations must be given to network-loading conditions and the number of sub-bursts allowed to be groomed together. We show that although simple greedy algorithms can reduce network overhead, they may alter the traffic characteristics and increase its burstiness, resulting in high packet blocking probability.