In this correspondence, we determine the optimal power allocation, that achieves any specified point on the boundary of the capacity region, for sets of parallel Gaussian multiple-access channels (MACs) and sets of parallel Gaussian broadcast channels (BCs). The power allocation is across the parallel channels and, within each channel, across the users. In both cases, there is a single constraint on the total power used. First, the allocation for the parallel MACs is determined, in a simple form, using the Karush-Kuhn-Tucker (KKT) conditions and a simple Lagrangian argument. Using this result, the allocation for the parallel BCs is derived using recent findings on the duality of the MAC and BC.

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

We study a setting in which a spatially distributed set of sources is creating data that must be delivered to a spatially distributed set of sinks. A network of wireless nodes is responsible for sensing the data, transporting them over a wireless channel to the sink locations, and delivering them to the sinks. The problem is to find the optimal placement of these nodes, so that the minimum number of nodes is needed. Our approach is to assume a massively dense network, in which there are so many sources, sinks, and wireless nodes, that it does not make sense to model the network in terms of microscopic parameters, such as their individual placements, but rather in terms of macroscopic parameters, such as their spatial densities. We use calculus of variations to describe the optimal spatial density of nodes, that minimizes their total number but can still support the whole traffic, in terms of a scalar, nonlinear, partial differential equation. The derivation is under a general model for the physical layer, that encompasses many cases of interest. Our work can help in the design of networks that are deployed in the most efficient manner, not only avoiding the formation of bottlenecks, but also striking the optimal balance between avoiding congestion and having the data packets follow short routes.

We study the transport capacity of a Gaussian multiple access channel, which consists of a set of transmitters and a single receiver. The transport capacity is defined as the sum, over all transmitters, of the product of the transmission rate with a reward r(x), which is a function of the distance x between the transmitter and the receiver, and quantifies the usefulness of the transmitting information over a distance x. Assuming that the sum of the transmitter powers is upper bounded, we present in closed form the optimal power allocation among the transmitters, that maximizes the transport capacity. We then present simple expressions for the optimal power allocation and induced transport capacity, as the number of transmitters approaches infinity. We also study the transport capacity of a Gaussian broadcast channel, which consists of a single transmitter and multiple receivers. Here, the transport capacity is defined as the sum, over all receivers, of the product of the transmission rate with a reward r(x). We determine in closed form the maximum possible transport capacity and the distribution of the available transmitter power among the receivers that achieve it. Although this result has already been reported in the literature, our derivation is shorter, and leads to simpler expressions. Our results can be used to gain intuition and develop good design principles in a variety of settings. For example, they apply to the uplink and downlink channel of cellular networks, and also to sensor networks which consist of multiple sensors that communicate with a single central station.