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We address the problem of allocating transmission rates to a set of network sessions with end-to-end bandwidth and delay requirements. We give a unified convex programming formulation that captures both average and probabilistic delay requirements. Moreover, we present a distributed algorithm and establish its convergence to the global optimum of the overall rate allocation problem. In our algorithm, session sources selfishly update their rates as to maximize their individual benefit (utility minus bandwidth cost), the network partitions end-to-end delay requirements into local per-link delays, and the links adjust their prices to coordinate the sources' and network's decisions, respectively. This algorithm relies on a network utility maximization (NUM) approach, and can be viewed as a generalization of TCP and active queue management (AQM) algorithms to handle end-to-end QoS. We extend our results to deterministic delay requirements when nodes employ Packet-level Generalized Processor Sharing (PGPS) schedulers.