Growth of wireless ad hoc networks

The utility of a node in an energy-limited wireless ad hoc network is defined as a positive linear function of the number of bits that the node sends as a source and the number of bits that it receives as a destination. First, we show that under the one-to-one traffic model in which every node wants to send traffic to a randomly chosen destination node, a utility that grows asymptotically at least as c_n(N/logN)^(n-1)2/ is achievable for every node when the nodes are distributed randomly on the surface of a fixed sphere. In this expression, N denotes the number of nodes, n denotes the transmit power fall-off exponent, and c_n is a constant that depends on n and is independent of N. Second, we introduce a "dollars-per-Joule pricing" system for wireless ad hoc networks, in which each node can charge any other node a price per Joule of energy that it expends on the other node\´s traffic. Under this pricing system, we extend the definition of the utility of a node to include the revenue that the node raises and the payments that it makes on the network. We show that the core capacity region of a wireless ad hoc network is non-empty under this dollars-per-Joule pricing system. Further, we show that there exists a sequence of utility vectors in the core capacity region such that the average of the utilities of the nodes grows asymptotically at least as c_n(N/logN)^(n-1)2/ under the one-to-one traffic model.