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We investigate the problem of smoothing multiplexed network traffic when either a streaming server transmits data to multiple clients or a storage server accesses data from multiple storage devices or other servers. We introduce efficient algorithms for lexicographically optimally smoothing the aggregate bandwidth requirements over a shared network link. Possible applications include improvement in the bandwidth utilization of network links and reduction in the energy consumption of server hosts. In the data transmission problem, we consider the case in which the clients have different buffer capacities and unlimited bandwidth constraints or unlimited buffer capacities and different bandwidth constraints. For the data access problem, we handle the general case of a shared buffer capacity and individual network bandwidth constraints. Previous approaches for the data access problem handled either the case of only a single stream or did not compute the lexicographically optimal schedule. By provably minimizing the variance of the required aggregate bandwidth, lexicographically optimal smoothing makes the maximum resource requirements within the network more predictable and increases the useful resource utilization. It also improves fairness in sharing a network link among multiple users and makes new requests from future clients more likely to be successfully admitted without the need for rescheduling previously accepted traffic. With appropriate hardware and system support, data traffic smoothing can also reduce the energy consumption of the host processor and the communication links. Overall, we expect that efficient resource management at the network edges will better meet quality of service requirements without restricting the scalability of the system.