Connectivity of three dimensional wireless sensor networks using geometrical probability

This paper investigates the connectivity properties of three dimensional wireless sensor network in which N nodes are independently and uniformly (i.u.d.) distributed either on the surface of a sphere of radius R or inside the volume of a ball of radius R. Our approach utilizes the geometrical probability results for the conditional probability that a random node falls inside a ball centered at an arbitrary sensor node. We obtain exact expressions for the mean node degree and node isolation probability for the two topologies, which can be easily evaluated analytically or numerically. We also illustrate an upper bound for the probability of connectivity. The analysis is validated by comparison with existing results and Monte Carlo simulations.