Distance distributions are a key building block in stochastic geometry modelling of wireless networks and in many other fields in mathematics and science. In this paper, we propose a novel framework for analytically computing the closed form probability density function (PDF) of the distance between two random nodes each uniformly randomly distributed in respective arbitrary (convex or concave) polygon regions (which may be disjoint or overlap or coincide). The proposed framework is based on measure theory and uses polar decomposition for simplifying and calculating the integrals to obtain closed form results. We validate our proposed framework by comparison with simulations and published closed form results in the literature for simple cases. We illustrate the versatility and advantage of the proposed framework by deriving closed form results for a case not yet reported in the literature. Finally, we also develop a Mathematica implementation of the proposed framework which allows a user to define any two arbitrary polygons and conveniently determine the distance distribution numerically.
In general, there are two types of distance distributions that are needed in the stochastic geometry modelling @cite_26 @cite_15 : (i) the distribution of the distance between a given reference node (located inside or outside the cell) and a random node located inside a cell, and (ii) the distribution of the distance between two random nodes (located in the same or different cells). An example of the former is the nearest neighbour distance distribution when the reference node (e.g., a base station) is located at the center of the cell. An example of the latter is the distribution of the distance between two randomly located device or machine type nodes in the same or different cells.
@cite_15 @cite_26
1989451648 2752397115
This paper derives the exact cumulative density function (cdf) of the distance between a randomly located node and any arbitrary reference point inside a regular L-sided polygon. Using this result, we obtain the closed-form probability density function of the Euclidean distance between any arbitrary reference point and its nth neighbor node when N nodes are uniformly and independently distributed inside a regular L-sided polygon. First, we exploit the rotational symmetry of the regular polygons and quantify the effect of polygon sides and vertices on the distance distributions. Then, we propose an algorithm to determine the distance distributions, given any arbitrary location of the reference point inside the polygon. For the special case when the arbitrary reference point is located at the center of the polygon, our framework reproduces the existing result in the literature. 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.