I would start by viewing it as tree structure optimization problem. Draw a tree where each node is a physical server and the root node is the domain name server. Now try to maximize throughput of random lookups while minimizing height (complexity). For each level of the tree, come up with a list of everything you can think of that might affect the traversal (processing\lookup) time when a node (server) in that level is entered. Also create a list of everything you can think of that might affect the lines (connections) between nodes. This exercise should give you a good idea of what you need to learn and help generate more specific questions.