Transport capacity of Gaussian multiple access and broadcast channels with a large number of nodes

We determine the maximum transport capacity of a Gaussian broadcast channel and of a Gaussian multiple access channel. Transport capacity is defined as the sum, over all simultaneous transmissions, of the product of the transmission rate with a reward r(x), where x is the distance between the receiver and the transmitter. In the case of the broadcast channel, the optimization is performed over all allocations of the transmitter power to the signals intended for the different receivers. In the case of the multiple access channel, the optimization is performed over all allocations of power at the transmitters that satisfy a sum constraint. Our approach is to assume a large number of receivers (or transmitters) and formulate the problem within the framework of the calculus of variations