Closed form solutions for water-filling problems in optimization and game frameworks

Abstract

We study power control in optimization and game frameworks. In the optimization framework there is a single decision maker who assigns network resources and in the game framework players share the network resources according to Nash equilibrium. The solution of these problems is based on so-called water-filling technique, which in turn uses bisection method for solution of non-linear equations for Lagrange multiplies. Here we provide a closed form solution to the water-filling problem, which allows us to solve it in a finite number of operations. Also, we produce a closed form solution for the Nash equilibrium in symmetric Gaussian interference game. In addition, to its mathematical beauty, the explicit solution allows one to study limiting cases when the crosstalk coefficient is either small or large. We provide an alternative simple proof of the convergence of the Iterative Water Filling Algorithm. Furthermore, it turns out that the convergence of Iterative Water Filling Algorithm slows down when the crosstalk coefficient is large. Using the closed form solution, we can avoid this problem. Finally, we compare the non-cooperative approach with the cooperative approach and show that the non-cooperative approach results in a more fair resource distribution.

Robust Power Control for Heterogeneous Users in Shared Unlicensed Bands

“…When the sum of transmit power levels of any user in all sub-channels is upper bounded to P max n , i.e., the users' strategies are nonlinear functions of their observations [46]. Hence, we cannot use Propositions 1-3 to analyze the RSG.…”

Equilibrium Traffic Flow Assignment in Case of Two Navigation Providers

“…In this way, the results from traffic flow assignment can be used in telecommunication systems and vice versa. That is why in this paper we will follow the results of studies by Altman et al [14,15]. We interpret transmitters from wireless networks as providers of navigation (navigators) in traffic flow networks and consider the traffic flow assignment problem as a non-zero-sum game.…”

“…We interpret transmitters from wireless networks as providers of navigation (navigators) in traffic flow networks and consider the traffic flow assignment problem as a non-zero-sum game. Pay-off functions are used in the proposed model close to those of Altman et al [14,15].…”

Fast Computation of Generalized Waterfilling Problems

“…In any (general) waterfilling problem (WFP), powers are allocated to the resources of the transmitting user in order to maximize the transmitting user's capacity (or mutual information) while satisfying the total power budget constraint. The user's resources can be the sub-carriers in Orthogonal Frequency Division Multiplexing (OFDM) or the normal frequency bands or the usage of the same sub-carriers in different time slots [1]. This implies that the resource's allocated power is inversely proportional to the noise level of the resource in WFP so as to maximize capacity [2], [3].…”