Asymptotic critical transmission ranges for connectivity in wireless ad hoc networks with Bernoulli nodes

Wireless ad hoc networks with Bernoulli nodes provide a unified model of various important problems including fault-tolerance, randomized construction of virtual backbone, randomized broadcast routing, and randomized wake/sleep management. We assume that the wireless ad hoc network consists of n nodes which are distributed independently and uniformly in a unit-area disk and are active (or available) independently with some constant probability ρ. Let ρ_n denote the random variable which is the smallest transmission range at which the active nodes form a connected network, and p_n´ denote the random variable which is the smallest transmission range at which the active nodes form a connected network and each inactive node is adjacent to at least one active node, ρ_n is referred to as the critical transmission range for connectivity of active modes, and ρ_n´ is referred to as the critical transmission range for connectivity of all nodes. In this paper, we derive the precise asymptotic distributions of ρ_n and ρ_n´.