Performance Limits of Solutions to Network Utility Maximization Problems

We study performance limits of solutions to utility maximization problems (e.g., max-min problems) in wireless networks as a function of the power budget $\bar{p}$ available to transmitters. Special focus is devoted to the utility and the transmit energy efficiency (i.e., utility over transmit power) of the solution. Briefly, we show tight bounds for the general class of network utility optimization problems that can be solved by computing conditional eigenvalues of standard interference mappings. The proposed bounds, which are based on the concept of asymptotic functions, are simple to compute, provide us with good estimates of the performance of networks for any value of $\bar{p}$ in many real-world applications, and enable us to determine points in which networks move from a noise limited regime to an interference limited regime. Furthermore, they also show that the utility and the transmit energy efficiency scales as $\Theta(1)$ and $\Theta(1/\bar{p})$, respectively, as $\bar{p}\to\infty$.