In this contribution, the performance of a multiuser system is analyzed in the context of frequency selective fading channels. Using game theoretic tools, a useful framework is provided in order to determine the optimal power allocation when users know only their own channel (while perfect channel state information is assumed at the base station). This scenario illustrates the case of decentralized schemes, where limited information on the network is available at the terminal. Various receivers are considered, namely the matched filter, the MMSE filter and the optimum filter. The goal of this paper is to extend previous work, and to derive simple expressions for the non-cooperative Nash equilibrium as the number of mobiles becomes large and the spreading length increases. To that end two asymptotic methodologies are combined. The first is asymptotic random matrix theory which allows us to obtain explicit expressions of the impact of all other mobiles on any given tagged mobile. The second is the theory of non-atomic games which computes good approximations of the Nash equilibrium as the number of mobiles grows.
In particular, the utility generally considered in those articles is justified in @cite_4 where the author describes a widely applicable model from first principles''. Conditions under which the utility will allow to obtain non-trivial Nash equilibria (i.e., users actually transmit at the equilibrium) are derived. The utility consisting of throughput-to-power ratio (detailed in Sec. ) is shown to satisfy these conditions. In addition, it possesses a propriety of reliability in the sense that the transmission occurs at non-negligible rates at the equilibrium. This kind of utility function had been introduced in previous works, with an economic leaning @cite_13 @cite_5 .
@cite_13 @cite_5 @cite_4
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A major challenge in the operation of wireless communications systems is the efficient use of radio resources. One important component of radio resource management is power control, which has been studied extensively in the context of voice communications. With the increasing demand for wireless data services, it is necessary to establish power control algorithms for information sources other than voice. We present a power control solution for wireless data in the analytical setting of a game theoretic framework. In this context, the quality of service (QoS) a wireless terminal receives is referred to as the utility and distributed power control is a noncooperative power control game where users maximize their utility. The outcome of the game results in a Nash (1951) equilibrium that is inefficient. We introduce pricing of transmit powers in order to obtain Pareto improvement of the noncooperative power control game, i.e., to obtain improvements in user utilities relative to the case with no pricing. Specifically, we consider a pricing function that is a linear function of the transmit power. The simplicity of the pricing function allows a distributed implementation where the price can be broadcast by the base station to all the terminals. We see that pricing is especially helpful in a heavily loaded system. Recent publications recognize that decentralized algorithms useful in wireless data applications can be obtained via microeconomics and game theory. In these studies, each agent maximizes, under appropriate rules and constraints, a quality-of-service (QoS) index. A key solution is a "Nash equilibrium"; i.e., an allocation from which no agent is better off by unilaterally "deviating". The actual maximization may be made by software which may not be directly "controllable" by a human user. The model and, especially, the chosen QoS index should be as general as possible, so that the derived results be applicable to a wide variety of channel conditions, modulation schemes, and other physical-layer characteristics. Likewise, the chosen index should exhibit predictable and reliable technical behavior, without exacting a high complexity cost. This note describes a model, and particularly, a QoS index which can accommodate a wide variety of physical layer situations. The proposed index is shown to exhibit solid technical behavior, be physically significant, intuitively appealing, and applicable to a wide variety of physical layer situations. A game in which terminals carrying multi-rate traffic seek to maximize this index is analyzed, and closed-form equilibrium conditions and power levels are derived "from first principles". All terminals want the same signal-to-interference ratio (SIR), but some cannot reach the necessary power level. At equilibrium, a number of terminals transmit full power, and others achieve the same optimal SIR. A basic rationale to search for these equilibria is provided.