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Content delivery networks (CDNs) cache and deliver hundreds of trillions of user requests each day from hundreds of thousands of servers around the world. The traffic served by CDNs can be partitioned into hundreds of traffic classes, each with different user access patterns, popularity distributions, object sizes, and performance requirements. Midgress is the cache miss traffic between the CDN's servers and the content provider origins. A major goal of a CDN is to minimize its midgress, since higher midgress translates to higher bandwidth costs and increased user-perceived latency. We propose algorithms that provision traffic classes to servers such that midgress is minimized. Using extensive traces from Akamai's CDN, we show that our midgress-aware traffic provisioning schemes can reduce midgress by nearly 20% in comparison with the midgress-unaware schemes currently in use. We also propose an efficient heuristic for traffic provisioning that achieves near-optimal midgress and is suitable for use in production settings. Further, we show how our algorithms can be extended to other settings that require minimum caching performance per traffic class and minimum content duplication for fault tolerance. Finally, our paper provides a strong case for implementing midgress-aware traffic provisioning in production CDNs.