3,844 research outputs found

    Closed form solutions for symmetric water filling games

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    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 users 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 with an arbitrary number of users. Even though the game is symmetric, there is an intrinsic hierarchical structure induced by the quantity of the resources available to the users. We use this hierarchical structure to perform a successive reduction of the 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

    Closed form solutions for symmetric water filling games

    Get PDF
    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 users 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 with an arbitrary number of users. Even though the game is symmetric, there is an intrinsic hierarchical structure induced by the quantity of the resources available to the users. We use this hierarchical structure to perform a successive reduction of the 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

    Worst-Case Robust Distributed Power Allocation in Shared Unlicensed Spectrum

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    This paper considers non-cooperative and fully-distributed power-allocation for selfish transmitter-receiver pairs in shared unlicensed spectrum when normalized-interference to each receiver is uncertain. We model each uncertain parameter by the sum of its nominal (estimated) value and a bounded additive error in a convex set, and show that the allocated power always converges to its equilibrium, called robust Nash equilibrium (RNE). In the case of a bounded and symmetric uncertainty region, we show that the power allocation problem for each user is simplified, and can be solved in a distributed manner. We derive the conditions for RNE's uniqueness and for convergence of the distributed algorithm; and show that the total throughput (social utility) is less than that at NE when RNE is unique. We also show that for multiple RNEs, the social utility may be higher at a RNE as compared to that at the corresponding NE, and demonstrate that this is caused by users' orthogonal utilization of bandwidth at RNE. Simulations confirm our analysis

    Competitive Spectrum Management with Incomplete Information

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    This paper studies an interference interaction (game) between selfish and independent wireless communication systems in the same frequency band. Each system (player) has incomplete information about the other player's channel conditions. A trivial Nash equilibrium point in this game is where players mutually full spread (FS) their transmit spectrum and interfere with each other. This point may lead to poor spectrum utilization from a global network point of view and even for each user individually. In this paper, we provide a closed form expression for a non pure-FS epsilon-Nash equilibrium point; i.e., an equilibrium point where players choose FDM for some channel realizations and FS for the others. We show that operating in this non pure-FS epsilon-Nash equilibrium point increases each user's throughput and therefore improves the spectrum utilization, and demonstrate that this performance gain can be substantial. Finally, important insights are provided into the behaviour of selfish and rational wireless users as a function of the channel parameters such as fading probabilities, the interference-to-signal ratio

    Coalitional Games for Transmitter Cooperation in MIMO Multiple Access Channels

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    Cooperation between nodes sharing a wireless channel is becoming increasingly necessary to achieve performance goals in a wireless network. The problem of determining the feasibility and stability of cooperation between rational nodes in a wireless network is of great importance in understanding cooperative behavior. This paper addresses the stability of the grand coalition of transmitters signaling over a multiple access channel using the framework of cooperative game theory. The external interference experienced by each TX is represented accurately by modeling the cooperation game between the TXs in \emph{partition form}. Single user decoding and successive interference cancelling strategies are examined at the receiver. In the absence of coordination costs, the grand coalition is shown to be \emph{sum-rate optimal} for both strategies. Transmitter cooperation is \emph{stable}, if and only if the core of the game (the set of all divisions of grand coalition utility such that no coalition deviates) is nonempty. Determining the stability of cooperation is a co-NP-complete problem in general. For a single user decoding receiver, transmitter cooperation is shown to be \emph{stable} at both high and low SNRs, while for an interference cancelling receiver with a fixed decoding order, cooperation is stable only at low SNRs and unstable at high SNR. When time sharing is allowed between decoding orders, it is shown using an approximate lower bound to the utility function that TX cooperation is also stable at high SNRs. Thus, this paper demonstrates that ideal zero cost TX cooperation over a MAC is stable and improves achievable rates for each individual user.Comment: in review for publication in IEEE Transactions on Signal Processin

    Real and Complex Monotone Communication Games

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    Noncooperative game-theoretic tools have been increasingly used to study many important resource allocation problems in communications, networking, smart grids, and portfolio optimization. In this paper, we consider a general class of convex Nash Equilibrium Problems (NEPs), where each player aims to solve an arbitrary smooth convex optimization problem. Differently from most of current works, we do not assume any specific structure for the players' problems, and we allow the optimization variables of the players to be matrices in the complex domain. Our main contribution is the design of a novel class of distributed (asynchronous) best-response- algorithms suitable for solving the proposed NEPs, even in the presence of multiple solutions. The new methods, whose convergence analysis is based on Variational Inequality (VI) techniques, can select, among all the equilibria of a game, those that optimize a given performance criterion, at the cost of limited signaling among the players. This is a major departure from existing best-response algorithms, whose convergence conditions imply the uniqueness of the NE. Some of our results hinge on the use of VI problems directly in the complex domain; the study of these new kind of VIs also represents a noteworthy innovative contribution. We then apply the developed methods to solve some new generalizations of SISO and MIMO games in cognitive radios and femtocell systems, showing a considerable performance improvement over classical pure noncooperative schemes.Comment: to appear on IEEE Transactions in Information Theor
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