On the transport capacity of a multiple access Gaussian channel

We address the problem of maximizing the transport capacity of a wireless network, defined as the sum, over all transmitters, of the products of the transmission rate with a reward r(x), which is a function of the distance x separating the transmitter and its receiver. When r(x) = x, this product is measured in bps × meters, and is the natural measure of the usefulness of a transmission in a multihop wireless ad hoc network. We first consider a single transmitter-receiver pair, and determine the optimal distance between the two that maximizes the rate-reward product, for reward functions of the form r(x) = x^ρ and when the signal power decays with distance according to a power law. We then calculate the scheme that maximizes the transport capacity in a multiple access network consisting of a single receiver and a number of transmitters, each placed at a fixed distance from the receiver, and each with a fixed power constraint. We conclude by showing that when the per-transmitter power constraints are substituted with a single constraint on the sum of the powers, the maximum transport capacity and the power allocation scheme that achieves it can be found by solving a convex optimization problem.