Subcarrier allocation based on correlated equilibrium in multicell OFDMA systems
 Jianchao Zheng^{1}Email author,
 Yueming Cai^{1} and
 Dan Wu^{1}
https://doi.org/10.1186/168714992012233
© Zheng et al.; licensee Springer. 2012
Received: 14 January 2012
Accepted: 11 July 2012
Published: 25 July 2012
Abstract
Abstract
In uplink orthogonal frequency division multiple access (OFDMA) systems, efficient resource allocation can greatly improve system performance. In this article, taking throughput, intercell interference and complexity into account, we present a gametheoretical approach to perform distributed subcarrier allocation for multicell OFDMA systems with limited base station (BS) coordination. Firstly, we construct a multicell resource allocation game. Creatively, the subcarrier is viewed as a game player to choose the most satisfying user and the BS acts as a referee or a coordinator. Then, we introduce the correlated equilibrium which helps the noncooperative players coordinate their strategies, hence, achieves better performance compared to Nash equilibrium. Particularly, we point out the condition under which the correlated equilibrium is Pareto efficient. Moreover, we propose a novel subcarrier allocation algorithm based on the noregret procedure that guarantees convergence to correlated equilibrium, in which the BS coordinates the players’ strategies and provides only partial information exchange. Extensive simulation results are provided to demonstrate the effectiveness of the proposed algorithm.
Keywords
Introduction
Orthogonal frequency division multiple access (OFDMA) has emerged as one of the most promising multiple access techniques for high data rate transmission over wireless channels due to its ability to mitigate multipath fading and its efficient implementation using IFFT and FFT blocks. The most recently proposed next generation wireless communication technologies, such as wireless wide area network (WWAN) standards, 3GPP2 ultra mobile broadband (UMB), IEEE 802.20 mobile broadband wireless access (MBWA), 3GPP LTE and worldwide interoperability for microwave access (WiMAX) are all OFDMA based[1].
In an OFDMA system, the spectrum is orthogonally divided into timefrequency resource blocks (RBs), which increases flexibility in resource allocation, thereby allowing high spectral efficiency. Exploiting all RBs simultaneously in every cell to achieve universal frequency reuse becomes a key objective toward the deployment of 4G networks[2]. Focusing on the universal frequency reuse scheme, intercell interference is a major impairment that limits the system throughput[3]. In multicell environment, one of the major issues to research is how to maximize the performance by controlling the cochannel interference among the neighboring cells[4]. Interference coordination can be fulfilled by allocating system resources with interference awareness in terms of frequency, time, transmit power and space, etc.[5, 6]. Due to limited and precious radio resources in cellular networks, interference aware resource allocation is a challenging problem and has received much attention from both research and standardization communities in recent years[5–9]. Moreover, because any change of resource allocation in a specific cell will affect the performance of the nearby cells, joint resource allocation considering both throughput and interference over a cluster of neighboring cells via BS coordination is a promising solution.
Recently, BS coordination, where neighboring BSs connected through highspeed wireline links only share channel state information (CSI) and can jointly compute their transmit power and user scheduling, has been proposed as a major technique to mitigate cochannel interference, since it shifts the signal processing burden to the BSs[10]. Many works have been done on coordinated resource allocation in cellular wireless networks, including both centralized and distributed procedures[11]. Centralized algorithms (e.g.,[12–14]) require global information to decide the user assignment and transmit power in each cell. The problem is often formulated as an optimization task subject to bit rate, power level, or other types of constraints[5, 8]. However, since most of the optimization problems in cellular networks can be proved to be NPhard[15, 16] (as a mixed integernonlinear problem), standard optimization techniques do not apply directly and even centralized algorithms cannot guarantee that the globally optimal solution is found. In addition, even if computational issues were to be resolved, the optimal solution still requires a central controller updated with instantaneous intercell channel gains which would create serious signaling overhead issues, thus hinder it from practice[17]. Consequently, distributed algorithms (e.g.,[11, 18–20]) are more attractive as they do not require a central controller and may demand less information exchange and computational complexity.
Game theory, which is naturally the dominant paradigm for analyzing the decentralized framework, is recently adopted by many researchers to seek for a satisfactory solution to the problem of resource allocation and/or interference coordination[4, 21–23]. Kwon and Le[4] design the utility function that represents the weighted sum of the data rates and the power consumption in a cell. The problem of maximizing the utility under the maximum power constraint is modeled as a noncooperative resource allocation game, in which the BS is viewed as the game player. Liang et al.[21] focus on the adaptive allocation of subcarrier, bit, and power among BSs of a downlink multicell OFDMA systems. The utility function takes both data rate and power consumption into account. However, the authors have not formulated the problem from the perspective of interference mitigation. In[22], a noncooperative game in which each user selfishly tries to minimize its own transmitted power subject to a transmission rate constraint is proposed. Nevertheless, the proposed game is not guaranteed to converge to a Nash equilibrium, and therefore a virtual referee is introduced to monitor the resource allocation and force it to a stable and efficient equilibrium point. AlZahrani et al.[23] consider a transmit power adaptation method using a noncooperative game theory approach to reduce the intercell interference in the OFDM networks. The throughput is enhanced by finding the optimum transmit power for each cochannel user using game theorybased scheme. However, no subcarrier allocation is discussed.
The existing works based on game theory mainly concentrate on power control, while subcarrier allocation is more or less simplified. Moreover, the power allocation is easier to be solved by continuous game method, while the discrete game applying to subcarrier allocation is much harder that few works consider. Thus in this work we make a gametheoretic study on the distributed subcarrier allocation algorithm in the uplink multicell OFDMA systems. Note that the pure noncooperative game may result in nonconvergence or some undesirable Nash equilibria with low system and individual performance. To enhance the performance, we introduce an important generalization of the Nash Equilibrium, known as the correlated equilibrium, which is more preferable than Nash equilibrium since it directly considers the ability of agents to coordinate their actions. This coordination can lead to better performance than if each agent was required to act in isolation[24].
The main contributions of this article are summarized as follows:

In this article, we formulate the subcarrier allocation problem in a novel point of view that each subcarrier performs as a game player to choose the most satisfying user, which guarantees the fairness from the perspective of the subcarriers. Therefore, it is different from the traditional subcarrier allocation, in which the subcarriers are allocated passively.

An efficient distributed learning algorithm is developed to perform subcarrier allocation for the multicell scenario, which achieves a good performance, jointly considering the throughput, interference and fairness. The proposed algorithm exhibits low complexity and converges to the set of correlated equilibria with probability one.

In general, the outcomes of individual optimization might not always be as good as those of system optimization. To solve this problem, the BS is introduced as a referee or coordinator, which is in charge of monitoring and improving the outcome of noncooperative competition among the distributed players. Thus, strictly speaking, the approach is limited cooperation among BSs by adopting distributed algorithms, which is recognized as a good tradeoff between the performance gain nd the relevant cost, considering that they demand less information exchange and computational complexity.
The rest of this article is organized as follows. In Section “System model and problem formulation”, we present the system model and a novel utility function considering both throughput and interference. In Section “Correlated equilibrium for joint strategy selection”, we study the correlated equilibrium. Then, we construct a distributed subcarrier allocation algorithm based on noregret procedure in Section “Distributed learning algorithm for joint strategy selection” and prove that the algorithm converges to a set of correlated equilibria. Simulation results are shown in Section “Simulation results and analysis” and finally conclusions are drawn in Section “Conclusion”.
System model and problem formulation
System model
 (1)
In each singlecell OFDMA network, K is always much larger than the number of users and no subcarrier can simultaneously support transmission for more than one user. Hence, all the users can simultaneously transmit data to the BS on one or more subcarriers without interfering others.
 (2)
Each user is served by only one BS which locates in his cell. Thus, ${\mathcal{N}}_{l}\cap {\mathcal{N}}_{{l}^{\prime}}=\varnothing $, for l ≠ l ^{ ′ }.
 (3)
The bandwidth of each subchannel is less than the coherence bandwidth of the channel so that each subcarrier experiences flat fading.
 (4)
The subcarriers are perfectly orthogonal such that no intersymbol interference between adjacent symbols occurs.
 (5)
Perfect synchronization is assumed so that there will not be intersubcarrier interference.
 (6)
The BS periodically estimates the uplink channel gains on all subcarriers for all the users through pilot signals. And all the CSI needed can be accurately tracked by the BS.
 (7)
The network is geographically static in the sense that the time scale of algorithm convergence is shorter than the channel’s coherence time. Thus, the channel gains on subcarriers remain unchanged during one implementation of the algorithm [26].
Problem formulation
In this section, we model the subcarrier allocation as a multiplayer discrete, finite strategy game, in which the subcarriers are considered as players. Therefore, there is a shift in perspective from a user’s view of allocation to a subcarrier’s view, and the subcarriers can choose the most satisfying users for themselves. It means that the fairness from the perspective of the subcarriers can be ensured.
Each BS is assumed to have access to all available subcarriers, i.e., the frequency reuse factor is 1. Consequently, each BS has K available subcarriers. Note that different cells have different user sets, which means that the same subcarrier in different cells has different strategy sets, thereby the same subcarriers in different cells should be treated as different players. Thus, we denote the subcarrier k in cell l by k_{ l } for distinguishing. And the strategy of player k_{ l } is denoted by${S}_{{k}_{l}}$, while the joint strategy by opponents of player k_{ l }is denoted by${S}_{{k}_{l}}$. Hence,$S=({S}_{{k}_{l}},{S}_{{k}_{l}})$ is the joint strategy of all players, also known as a strategy profile.${U}_{{k}_{l}}({S}_{{k}_{l}},{S}_{{k}_{l}})$ or${U}_{{k}_{l}}\left(S\right)$ denotes the utility function of player k_{ l }.
where$i={S}_{{k}_{l}}$ and$j={S}_{{k}_{{l}^{\prime}}}$ denote the users chosen by subcarrier k_{ l }and k l_{ ′ }, respectively. Obviously, user i and user j are in different cells. Subcarrier k_{ l } and k l_{ ′ }are the same frequency band which are allocated by different BSs. l and l^{ ′ } denote the cell index.
 (1)
$\mathcal{N}=\{1,2,\dots ,K\}$ is the index set of the players (we use player, subcarrier interchangeably). In fact, the same subcarrier in different cells acts as the different players making their own decisions alone. Therefore, the number of the total players is KL.
 (2)
${\Omega}_{{k}_{l}}$ is the strategy space of player k _{ l }. Obviously, ${\Omega}_{{k}_{l}}={\mathcal{N}}_{l}$. Therefore, the space for the joint strategy profile is defined by $\mathbb{S}={\Omega}_{1}\times {\Omega}_{2}\times \cdots \times {\Omega}_{K}\times {\Omega}_{K+1}\times {\Omega}_{K+2}\times \cdots \times {\Omega}_{2K}\times \cdots \times {\Omega}_{\mathit{\text{KL}}}$.
 (3)
${U}_{{k}_{l}}:\mathbb{S}\to \mathbb{R}$ is the individual utility mapping the joint strategy space to the set of the real number.
Remark 1
Although the subcarriers can decide which user to choose at their will, they have no capability of performing strategy selection. Therefore, the subcarriers are virtual game players actually. In essence, the game is managed by BSs who act as the coordinators or referees, and BSs allocate the subcarriers according to the equilibrium point of the game.
Correlated equilibrium for joint strategy selection
In order to analyze the outcome of the proposed game, we focus on an important generalization of the Nash equilibrium, known as the correlated equilibrium that a strategy profile is chosen randomly according to a certain distribution given to the players by some “coordinator” or “referee”. Each player is givenprivatelyinstructions for his own play only and the joint distribution is known to all of them. It is to the players’ best interests to conform to this recommended strategy, and the distribution is called the correlated equilibrium[28].
Correlated equilibrium
Definition 1
The inequality means that when the recommendation to player k_{ l } is to choose action${S}_{{k}_{l}}$, then choosing the any other action instead cannot obtain a higher expected utility.
Theorem 1
For the multicell subcarrier allocation game G, a correlated equilibrium always exists.
Proof
The result from[30] shows that every finite game has a correlated equilibrium. Hence, Theorem 1 is justified, and enables the application of the proposed game. □
Remark 2
The set of correlated equilibria is nonempty, closed and convex in G. In fact, every Nash equilibrium is a correlated equilibrium and Nash equilibrium corresponds to the special case where the action of the different players is independent, i.e.,$p({S}_{{k}_{l}},{S}_{{k}_{l}})=p\left({S}_{1}\right)\times \cdots p\left({S}_{k}\right)\times \cdots p\left({S}_{\mathit{\text{KL}}}\right)$. Moreover, the set of correlated equilibrium distributions of G is a convex polytope and the Nash equilibia all lie on the boundary of the polytope[31].
Optimal correlated equilibrium
where${E}_{p}\left(\right)$ is the expectation over p. The constraints guarantee that the solution is within the correlated equilibrium set.
Theorem 2
In the proposed game G, the correlated equilibrium which maximizes the expected sum of utilities of the subcarriers, P^{∗}, is Pareto efficient.
Proof
If the resulting correlated equilibrium P^{∗} is not Pareto efficient, there must exist a different probability distribution P^{ ′ }such that$\sum _{S\in \mathbb{S}}{P}^{\prime}\left(S\right){U}_{{k}_{l}}\left(S\right)\ge \sum _{S\in \mathbb{S}}{P}^{\ast}$$\left(S\right){U}_{{k}_{l}}\left(S\right)$,$\forall {k}_{l}\in {\mathcal{K}}_{l}$,$\forall l\in \mathcal{L}$ and$\sum _{S\in \mathbb{S}}{P}^{\prime}\left(S\right){U}_{{k}_{l}}\left(S\right)>$$\sum _{S\in \mathbb{S}}{P}^{\ast}\left(S\right){U}_{{k}_{l}}\left(S\right)$ for some k_{ l }, thus$\sum _{l\in \mathcal{L}}\sum _{{k}_{l}\in {\mathcal{N}}_{l}}$$\sum _{S\in \mathbb{S}}{P}^{\prime}\left(S\right){U}_{{k}_{l}}\left(S\right)\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}>\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\sum _{l\in \mathcal{L}}\sum _{{k}_{l}\in {\mathcal{N}}_{l}}\sum _{S\in \mathbb{S}}{P}^{\ast}\left(S\right){U}_{{k}_{l}}\left(S\right)$, which contradicts the fact that P^{∗} is the optimal solution. The proof is completed. □
Distributed learning algorithm for joint strategy selection
Algorithm description
In this section, we present a distributed learning algorithm which always leads to the set of correlated equilibria. From the result, each player can independently determine its own cooperative strategy. Concretely, the proposed algorithm is based on the noregret procedure of [29]. In this procedure, players may depart from their current play with probabilities that are proportional to measures of regret for not having used other strategies in the past.
 (1)
Initialization: At the initial time n = 1, each player initializes its strategy arbitrarily
 (2)
Iterative update process: For n = 1,2,3,…
Utility update
For all$l\in \mathcal{L}$, each player${k}_{l}\in {\mathcal{K}}_{l}$ calculates the utility of the current strategy${S}_{{k}_{l}}\in {\Omega}_{{k}_{l}}$ and the utility for choosing the different strategy${S}_{{k}_{l}}^{\prime}\in {\Omega}_{{k}_{l}}$.
Regret value update
where${R}_{{k}_{l}}^{n}({S}_{{k}_{l}},{S}_{{k}_{l}}^{\prime})$ represents the average regret value at time n for not having played, every time that${}_{S}^{{k}_{l}}$ was played in the past, the different strategy${S}_{{k}_{l}}^{\prime}$.
Transition probability update
where μ is a normalization factor which is chosen to ensure the probabilities are nonnegative.
Strategy update
At the period n + 1, k_{ l } updates its decision strategy according to the transition probability distribution.
In the proposed algorithm, each player does not need to be concerned about the individual strategies and utilities of other players, global network structure, etc. Each one just needs to know the effect of other players on its individual utility function. In addition, each player views its current actual strategy as a reference point, and makes a decision for next period according to propensities to depart from it. However, the change should bring the improvement in individual utility, relative to the current choice.
Remark 3
The implementation of the proposed algorithm needs the history of play${H}^{n}={\left({S}^{{}^{\tau}}\right)}_{\tau =1}^{n}\in \prod _{\tau =1}^{n}\mathbb{S}$ given. And the BSs take the responsibility naturally and expediently, thus the cooperative strategy is obtained. As Hart and MasColell observe in[28] that, “there is a natural coordination device: the common history, observed by all players.”
Convergence analysis
where${e}_{{S}^{n+1}}=[0,0,\dots ,1,0,\dots ,0]$ denotes the$\left\left.\mathbb{S}\right\right.$ dimensional unit vector with the one in the position of S^{n + 1}.
Theorem 3
If every player follows the proposed algorithm, the empirical distributions of play z^{ n }converge almost surely as$n\to \infty $ to the set of correlated equilibria of our game. That z^{ n }converges to the set of correlated equilibria has been proved in many works, such as[24, 28, 34]. Here, we only provide a brief sketch of these proofs:
 (1)
Huang and Krishnamuthy [24] prove convergence indirectly by proving an inequality which is originated from the Blackwell’s sufficient condition for approachability
 (2)
In [28], the proof is based on a recursive formula for the distance of the vector of regrets to the negative orthant. In particular, by adopting multiperiod recursion where a large “block” of periods is combined together instead of oneperiod recursion, the conditions of Blackwell’s approachability theorem are proved
 (3)
In [34], the proof relies on a stochastic averaging theory. Due to the set theoretic nature of the correlated equilibira, the convergence analysis is carried out through a differential inclusion, which is the set theoretic extension of a differential equation.
Computational complexity analysis
At each iteration, each player k_{ l }needs to keep a record of the utility of choosing the current strategy and the utilities for changing to the other strategies. In addition, the proposed algorithm requires one table lookup, not more than n + KL additions and KL + 1 multiplication to update the regret value, and one comparison to choose the next strategy. And similar to the analysis in[35, 36], the complexity of our algorithm only depends on the number of player’s strategies, that is,$O\left(\left{\Omega}_{{k}_{l}}\right\right)$.
Remark 4
The information exchange among the BSs only includes the history play of the players in other cells. Thus, the signaling overhead is very low.
Simulation results and analysis
We initialize the game with a random user assignment for each player. The players will take action to search for improvement in utility value by looking for the best response strategy after observing the opponent’s action.
Conclusion
In this work, we have presented a distributed subcarrier allocation approach with limited BS coordination for multicell OFDMA systems. The goal is to maximize the performance by controlling the cochannel interference at the same time. Concretely, we model a joint strategy selection game in a novel point of view that each subcarrier performs as a game player to choose the most satisfying user, which guarantees the fairness from the perspective of the subcarriers and focus on the implementation of the set of correlated equilibria to analyze the outcome of the proposed game. Moreover, since any change of resource allocation in a specific cell will affect the performance of the nearby cells and the outcomes of individual optimization might not always be as good as those of system optimization, joint resource allocation via BS coordination is considered. Then, we develop a novel distributed subcarrier allocation algorithm based on noregret procedure to learn the correlated equilibrium, which demands less information exchange and computational complexity. The simulation results show that the proposed algorithm achieves good performance, such as quick convergence, large interference mitigation, evident capacity improvement, and good fairness. Further study could be focused on both the power and subcarrier allocation simultaneously to achieve a higher overall throughput of the system.
Declarations
Acknowledgements
This study was supported by the National Natural Science Foundation of Jiangsu Province (No. BK2010101), the National Natural Science Foundation of China (No. 61001107) and the Major National Science & Technology Specific Project (No. 2010ZX0300600204).
Authors’ Affiliations
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