Random-to-Random Nodal Distance Distributions in Finite Wireless Networks

Most performance metrics in wireless networks, such as outage probability, link capacity, etc., are functions of the distances between communicating/interfering nodes. A probabilistic distance-based model is definitely needed in quantifying these metrics, which eventually involves the nodal distance distribution (NDD) in a finite network intrinsically depending on the network coverage and nodal spatial distribution. Recently, the NDD from a reference node to a uniformly distributed node has been extended to the networks in the shape of arbitrary polygons. In contrast, the NDD between two uniformly distributed nodes (Ran2Ran) is still confined to the networks in certain specific shapes, including disks, triangles, rectangles, rhombuses, trapezoids, and regular polygons, which greatly limits its applicable network scenarios. By extending a tool in integral geometry, called Kinematic Measure, and using decomposition and recursion methods, this paper shows a systematic, algorithmic approach to Ran2Ran NDDs, which can handle arbitrarily-shaped networks, including convex, concave, disjoint, and tiered networks. Besides validating our approach through extensive simulations and comparisons with the known results if applicable, we also demonstrate its potentials in handling nonuniform nodal distributions, and in modeling two wireless networks of particular interest in the current literature, where the existing approaches are inapplicable.