Distance distribution between nodes in a 3D wireless network

In assessing the performance characteristics of a mobile, ad-hoc wireless network, the distance between nodes is of critical importance. Previous analysis of distance statistics for such networks have focused on the two-dimensional case, where nodes are distributed in the plane. Here we extend this analysis to three dimensions, deriving the probability distribution for distance between nearest neighboring nodes in a cubic volume. This distribution is further used to derive several measures that have proven useful in quantifying network reliability. The results are applicable to mobile networks of airborne or undersea vehicles, or even in multi-story buildings where nodes are distributed in all three spatial dimensions.