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The emergence of new applications on the Internet like voice-over-IP, peer-to-peer, and video-on-demand has created highly dynamic and changing traffic patterns. In order to route such traffic with quality-of-service (QoS) guarantees without requiring detection of traffic changes in real-time or reconfiguring the network in response to it, a routing and bandwidth allocation scheme has been recently proposed that allows preconfiguration of the network such that all traffic patterns permissible within the network's natural ingress-egress capacity constraints can be handled in a capacity efficient manner. The scheme routes traffic in two phases. In the first phase, incoming traffic is sent from the source to a set of intermediate nodes and then, in the second phase, from the intermediate nodes to the final destination. The traffic in the first phase is distributed to the intermediate nodes in predetermined proportions that depend on the intermediate nodes. In this paper, we develop linear programming formulations and a fast combinatorial algorithm for routing under the scheme so as to maximize throughput (or, minimize maximum link utilization). We compare the throughput performance of the scheme with that of the optimal scheme among the class of all schemes that are allowed to even make the routing dependent on the traffic matrix. For our evaluations, we use actual Internet Service Provider topologies collected for the Rocketfuel project. We also bring out the versatility of the scheme in not only handling widely fluctuating traffic but also accommodating applicability to several widely differing networking scenarios, including i) economical Virtual Private Networks (VPNs); ii) supporting indirection in specialized service overlay models like Internet Indirection Infrastructure (i3); iii) adding QoS guarantees to services that require routing through a netwo- - rk-based middlebox; and iv) reducing IP layer transit traffic and handling extreme traffic variability in IP-over-optical networks without dynamic reconfiguration of the optical layer. The two desirable properties of supporting indirection in specialized service overlay models and static optical layer provisioning in IP-over-optical networks are not present in other approaches for routing variable traffic, such as direct source-destination routing along fixed paths.