Average connectivity properties of wireless ad hoc networks

We consider the average radio coverage area size of a connected cluster Ω(α, r) in a uniformly randomly deployed wireless network over a D-dimensional infinite field (D ≥ 1), where r is the radio distance, and α the nodal deployment density. We show that Ω_N(y)=▵αΩ(α, r) is a function of y=▵αΦ(r) only, where Φ(r) denotes the volume of a sphere with radius r. We provide an explicit form of Ω_N(y) for arbitrary D as the sum of three terms, dominated by one that exhibits exponential behavior. For D = 1, we show that Ω_N(y) = exp (y/2) + (y/2) - 1. Our simulations validate our 1-d solution, and show that the exponent for 2-d deployment is smaller than y.