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.
In @cite_9 , the probability density function (pdf) and cumulative distribution function (cdf) of distances between nodes are derived for networks with uniformly random and Gaussian distributed nodes over a rectangular area. @cite_4 studies mean internodal distance properties for several kinds of multihop systems such as ring networks, Manhattan street networks, hypercubes and shufflenets. @cite_1 provides closed-form expressions for the distributions in @math -dimensional homogeneous PPPs and describes several applications of the results for large networks. @cite_5 derives the joint distribution of distances of nodes from a common reference point for planar networks with a finite number of nodes randomly distributed on a square. @cite_7 considers square random networks and determines the pdf and cdf of nearest neighbor and internodal distances. @cite_13 investigates one-dimensional multihop networks with randomly located nodes and analyzes the distributions of single-hop and multiple-hop distances.
@cite_4 @cite_7 @cite_9 @cite_1 @cite_5 @cite_13
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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. The spatial structure of transmitters in wireless networks plays a key role in evaluating the mutual interference and hence the performance. Although the Poisson point process (PPP) has been widely used to model the spatial configuration of wireless networks, it is not suitable for networks with repulsion. The Ginibre point process (GPP) is one of the main examples of determinantal point processes that can be used to model random phenomena where repulsion is observed. Considering the accuracy, tractability and practicability tradeoffs, we introduce and promote the @math -GPP, an intermediate class between the PPP and the GPP, as a model for wireless networks when the nodes exhibit repulsion. To show that the model leads to analytically tractable results in several cases of interest, we derive the mean and variance of the interference using two different approaches: the Palm measure approach and the reduced second moment approach, and then provide approximations of the interference distribution by three known probability density functions. Besides, to show that the model is relevant for cellular systems, we derive the coverage probability of the typical user and also find that the fitted @math -GPP can closely model the deployment of actual base stations in terms of the coverage probability and other statistics. In wireless networks, knowledge of internode distances is essential for performance analysis and protocol design. When determining distance distributions in random networks, the underlying nodal arrangement is almost universally taken to be a stationary Poisson point process. While this may be a good approximation in some cases, there are also certain shortcomings to this model, such as the fact that, in practical networks, the number of nodes in disjoint areas is not independent. This paper considers a more-realistic network model where a known and fixed number of nodes are independently distributed in a given region and characterizes the distribution of the Euclidean internode distances. The key finding is that, when the nodes are uniformly randomly placed inside a ball of arbitrary dimensions, the probability density function (pdf) of the internode distances follows a generalized beta distribution. This result is applied to study wireless network characteristics such as energy consumption, interference, outage, and connectivity. 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