In this contribution, the performance of a multi-user 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). We consider the realistic case of frequency selective channels for uplink CDMA. 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 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.
Resource allocation is of major interest in the context of multi-user systems. In the uplink multiuser systems, it is important for users to transmit with enough power to achieve their requested quality of service, but also to minimize the amount of interference caused to other users. Thus, an efficient power allocation mechanism allows to prevent an excessive consumption of the limited ressources of the users.
The most straightforward way to design a power allocation (PA) mechanism is as a centralized procedure, with the base station receiving training sequences from the users and signaling back the optimal power allocation for each user. Power control schemes in cellular systems were first introduced for TDMA/FDMA [1], [2]; more recently an optimal scheme was derived for Code Division Multiple Access (CDMA) [3]. In order to achieve the optimal capacity, the users may also be sorted according to some rule of precedence [4]. However, this involves a non negligible overhead and numerous non informational transmissions. In addition, the complexity of centralized schemes increases drastically with the number of users. As discussed in [5], centralized algorithms generally do not have a practical use for real systems, but provide useful bounds on the performance that can be attained by distributed algorithms.
A way to avoid the constraints of a centralized procedure is to implement a decentralized one where each user calculates its estimation of the optimal transmission power according to its local knowledge of the system. This is, for example, the case in ad-hoc networks applications. Most of the time, a distributed algorithm means an iterative version of a centralized one. Mobiles update their power allocation according to some rule based on the limited information they retrieve from the system. Supposing that an optimal power allocation exists, a distributed iterative algorithm is derived from a differential equation in [6] and its convergence is proven analytically. A distributed version of the algorithm of [2] is presented in [7]. Building on these results, a general framework for power control in cellular systems is given in [8]. A review of different methods of centralized and distributed power control in CDMA systems is given in [5].
In this context, a natural framework is game theory, which studies competition (as well as cooperation) between independent actors. Tools of game theory have already been frequently used as a central framework for modeling competition and cooperation in networking, see for example [9] and references therein. Building on the framework of [8], a game theoretic approach was introduced in [10], [11]. Numerous works on power allocation games have followed since, a selection of which we present in Sec. II.
Game theory can be used to treat the case of any number of players. However, as the size of the system increases, the number of parameters increases drastically and it is difficult to gain insight on the expressions obtained.
In order to obtain expressions depending only on few parameters, we consider the system in an asymptotic setting, letting both the number of users and the spreading factor tend to infinity with a fixed ratio. We use tools of random matrix theory [12] to analyze the system in this limit. Random matrix theory is a field of mathematical physics that has been recently applied to wireless communications to analyze various measures of interest such as capacity or Signal to Interference plus Noise Ratio (SINR). Interestingly, it enables to single out the main parameters of interest that determine the performance in numerous models of communication systems with more or less involved models of attenuation [13], [14], [15], [16]. In addition, these asymptotic results provide good approximations for the practical finite size case, as shown by simulations.
In the asymptotic regime, the non-cooperative game becomes a non-atomic one, in which the impact (through interference) of any single mobile on the performance of other mobiles is negligible. In the networking game context, the related solution concept is often called Wardrop equilibrium [17]; it is often much easier to compute than the original Nash equilibrium [9], and yet, the former equilibrium is a good approximation for the latter, see details in [18]. In this paper, we derive the nonatomic equilibrium, which generally corresponds to a non-uniform PA for the users.
The non-atomic Nash equilibrium is studied in this paper for several linear receivers, namely the matched filter and the MMSE filter, as well as nonlinear filters, such as the successive interference cancellation (SIC) [19] version of those filters. However, in order to perform SIC, the users need to know their decoding order, in order to adjust their rates. In this paper, we introduce ways of obtaining an ordering of the users in a distributed manner. The ordering can be determined simply in a distributed manner under weak hypotheses. This gi
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