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With the increasing diversity of applications supported over optical networks, new service guarantees must be offered to network customers. Among the emerging data-intensive applications are those which require their data to be transferred before a predefined deadline. We call these deadline-driven requests (DDRs). In such applications, data-transfer finish time (which must be accomplished before the deadline) is the key service guarantee that the customer wants. In fact, the amount of bandwidth allocated to transfer a request is not a concern for the customer as long as its service deadline is met. Hence, the service provider can choose the bandwidth (transmission rate) to provision the request. In this case, even though DDRs impose a deadline constraint, they provide scheduling flexibility for the service provider since it can choose the transmission rate while achieving two objectives: 1) satisfying the guaranteed deadline; and 2) optimizing the network's resource utilization. We investigate the problem of provisioning DDRs with flexible transmission rates in wavelength-division multiplexing (WDM) mesh networks, although this approach is generalizable to other networks also. We investigate several (fixed and adaptive to network state) bandwidth-allocation policies and study the benefit of allowing dynamic bandwidth adjustment, which is found to generally improve network performance. We show that the performance of the bandwidth-allocation algorithms depends on the DDR traffic distribution and on the node architecture and its parameters. In addition, we develop a mathematical formulation for our problem as a mixed integer linear program (MILP), which allows choosing flexible transmission rates and provides a lower bound for our provisioning algorithms.