Optimal Network Architectures for Minimizing Average Distance in k-ary n-dimensional Mesh Networks

A general expression for the average distance for meshes of any dimension and radix, including unequal radices in different dimensions, valid for any traffic pattern under zero-load condition is formulated rigorously to allow its calculation without network-level simulations. The average distance expression is solved analytically for uniform random traffic and for a set of local random traffic patterns. Hot spot traffic patterns are also considered and the formula is empirically validated by cycle true simulations for uniform random, local, and hot spot traffic. Moreover, a methodology to attain closed-form solutions for other traffic patterns is detailed. Furthermore, the model is applied to guide design decisions. Specifically, we show that the model can predict the optimal 3-D topology for uniform and local traffic patterns. It can also predict the optimal placement of hot spots in the network. The fidelity of the approach in suggesting the correct design choices even for loaded and congested networks is surprising. For those cases we studied empirically it is 100%.