The Distribution of Path Losses for Uniformly Distributed Nodes in a Circle

Abstract

When simulating a wireless network, users/nodes are usually assumed to be distributed uniformly in space. Path losses between nodes in a simulated network are generally calculated by determining the distance between every pair of nodes and applying a suitable path loss model as a function of this distance (power of distance with an environment‐specific path loss exponent) and adding a random component to represent the log‐normal shadowing. A network with N nodes consists of N(N − 1)/2 path loss values. In order to generate statistically significant results for system‐level simulations, Monte Carlo simulations must be performed where the nodes are randomly distributed at the start of every run. This is a time‐consuming operation which need not be carried out if the distribution of path losses between the nodes is known. The probability density function (pdf) of the path loss between the centre of a circle and a node distributed uniformly within a the circle is derived in this work.