In wireless networks, the knowledge of nodal distances is essential for several areas such as system configuration, performance analysis and protocol design. In order to evaluate distance distributions in random networks, the underlying nodal arrangement is almost universally taken to be an infinite Poisson point process. While this assumption is valid in some cases, there are also certain impracticalities to this model. For example, practical networks are non-stationary, and the number of nodes in disjoint areas are not independent. This paper considers a more realistic network model where a finite number of nodes are uniformly randomly distributed in a general d-dimensional ball of radius R and characterizes the distribution of Euclidean distances in the system. The key result is that the probability density function of the distance from the center of the network to its nth nearest neighbor follows a generalized beta distribution. This finding is applied to study network characteristics such as energy consumption, interference, outage and connectivity.
The minimum necessary aggregate link capacity in a telecommunication network is directly proportional to the mean distance between nodes. The mean internodal distance is therefore an important network characteristic. It is shown that most network topologies, including those constructed at random, display mean internodal distances comparable to those of many carefully designed networks. Thus, careful selection of network topology to minimize the mean internodal distance may be important in only the most sensitive applications. Furthermore, even in such sensitive applications, an almost randomly chosen network topology may be the best choice. > Separation distance between nodes is an important index in characterizing the optimum transmission range, the most probable Euclidean distance between two random selected nodes, the node degree and the network connectivity of wireless ad hoc networks. However, because nodes are randomly deployed, the separation distances between nodes in the wireless ad hoc networks are also random. Thus, in this paper, we present methodologies to analyze three distance-related probability distributions: the distribution of the distance to the k-th nearest neighbor, the distribution of the distance between two random selected nodes and the joint distribution of the distances between nodes and a common reference node.  The distribution of Euclidean distances in Poisson point processes is determined. The main result is the density function of the distance to the n-nearest neighbor of a homogeneous process in Ropfm, which is shown to be governed by a generalized Gamma distribution. The result has many implications for large wireless networks of randomly distributed nodes The calculation of two-hop connectivity be- tween two terminals for randomly deployed wireless networks requires the joint probability distribution of the distances between these terminals and the terminal that is acting as a relay. In general the distances are not independent since a common terminal is involved. The marginal distributions for link distances are known for various random deployment models. However, the joint distribution of two or more link distances is not known. In this paper, the derivation of the joint distribution is given in general form and in a new form suitable for computation for a network of terminals randomly deployed in a square area. 