This paper provides a mathematical model of data center performance based on the recently introduced Quantitative Theory of Bottleneck Structures (QTBS). Using the model, we prove that if the traffic pattern is \textit{interference-free}, there exists a unique optimal design that both minimizes maximum flow completion time and yields maximal system-wide throughput. We show that interference-free patterns correspond to the important set of patterns that display data locality properties and use these theoretical insights to study three widely used interconnects---fat-trees, folded-Clos and dragonfly topologies. We derive equations that describe the optimal design for each interconnect as a function of the traffic pattern. Our model predicts, for example, that a 3-level folded-Clos interconnect with radix 24 that routes 10\% of the traffic through the spine links can reduce the number of switches and cabling at the core layer by 25\% without any performance penalty. We present experiments using production TCP/IP code to empirically validate the results and provide tables for network designers to identify optimal designs as a function of the size of the interconnect and traffic pattern.