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Resource allocation with interference mitigation in OFDMA femtocells for cochannel deployment^{a}
EURASIP Journal on Wireless Communications and Networking volume 2012, Article number: 289 (2012)
Abstract
Femtocells have been considered as a promising technology to provide better indoor coverage and spatial reuse gains. However, the cochannel deployment of macrocells and femtocells is still facing challenges arising from potentially severe intercell interference. In this article, we investigate the uplink resource allocation problem of femtocells in cochannel deployment with macrocells. We first model the uplink power and subchannel allocation in femtocells as a noncooperative game, where intercell interference is taken into account in maximizing the femtocell capacity and uplink femtotomacro interference is alleviated by charging each femto user a price proportional to the interference that it causes to the macrocell. Based on the noncooperative game, we then devise a semidistributed algorithm for each femtocell to first assign subchannels to femto users and then allocate power to subchannels. Simulation results show that the proposed interferenceaware femtocell uplink resource allocation algorithm is able to provide improved capacities for not only femtocells, but also the macrocell, as well as comparable or even better tiered fairness in the twotier network, as compared with existing unpriced subchannel assignment algorithm and modified iterative water fillingbased power allocation algorithm.
Introduction
Nowadays above 50% of voice services and 70% of data traffics occur indoors [1]. Insufficient indoor coverage of macrocells has led to increasing interest in femtocells, which have been considered in major wireless communication standards such as 3GPP LTE/LTEAdvanced [2]. Dedicatedchannel deployment of femtocells, where femtocells and macrocells are assigned with different (or orthogonal) frequency bands, may not be preferred by operators due to the scarcity of spectrum resources and difficulties in implementation. While in cochannel deployment, where femtocells and macrocells share the same spectrum, crosstier interference could be severe [3], especially when femtocell base stations (FBSs) are deployed close to a macrocell base station (MBS) [4]. Due to the fundamental role of macrocells in providing blanket cellular coverage, their capacities and coverage should not be affected by cochannel deployment of femtocells.
Power control has widely been used to mitigate intercell interference in cochannel deployment of femtocells. For alleviating uplink interference caused by cochannel femto users to macrocells, a distributed femtocell power control algorithm is developed based on noncooperative game theory in [5], while in [6] femto users are priced for causing interference to macrocells in the power allocation based on a Stackelberg model. In [7], crosstier interference is mitigated through both openloop and closedloop uplink power controls. In [8], a distributed power control scheme is proposed based on a supermodular game.
A lot of work has also been done on subchannel allocation in cochannel deployment of femtocells. In [9], a hybrid frequency assignment scheme is proposed for femtocells deployed within coverage of a macrocell. In [10], distributed channel selection schemes are proposed for femtocells to avoid intercell interference, at the cost of reduced frequency reuse efficiency. In [11], a subchannel allocation algorithm based on a potential game model is proposed to mitigate both cotier and crosstier interferences.
Recently, several studies considering both power and subchannel allocation in femtocells have been reported. In [12], a joint power and subchannel allocation algorithm is proposed to maximize the total capacity of densely deployed femtocells, but the interference caused by femtocells to macrocells is not considered. In the collaborative resource allocation scheme [13], crosstier interference is approximated as additive white Gaussian noise (AWGN). In the Lagrangian dual decompositionbased resource allocation scheme [14], constraints on crosstier interference are used in power allocation, but subchannels are assigned randomly to femto users. In [15], a distributed downlink resource allocation scheme based on a potential game and convex optimization is proposed to increase the total capacity of macrocells and femtocells, but at the price of reduced femtocell capacity. In [16], the distributed power and subchannel allocation for cochannel deployed femtocells is modeled as a noncooperative game, for which a Nash Equilibrium is obtained based on a timesharing subchannel allocation, but the constraint on maximum femtouser transmit power is ignored in solving the noncooperative game.
In this article, we focus on the uplink power and subchannel allocation problem of orthogonal frequency division multiple access (OFDMA)based femtocells in cochannel deployment with macrocells. We first model the uplink power and subchannel allocation in femtocells as a noncooperative game, where intercell interference is taken into account in maximizing femtocell capacity and uplink interference from femto users to the macrocell is alleviated by charging each femto user a price proportional to the amount of interference that it causes to the macrocell. Based on the noncooperative game, we then devise a semidistributed algorithm for each femtocell to first assign subchannels to femto users and then allocate power to subchannels accordingly. Simulation comparisons with existing unpriced subchannel assignment and modified iterative water filling (MIWF)based power allocation algorithms show that the proposed interferenceaware femtocell uplink resource allocation algorithm is able to provide improved capacities for not only femtocells, but also the macrocell, as well as comparable or even better tiered fairness in a cochannel twotier network.
The rest of this article is organized as follows. The system model and problem formulation are presented in “System model and problem formulation” section. In “Interferenceaware resource allocation” section, the interferenceaware femtocell uplink resource allocation algorithm is proposed. Performance of the proposed algorithm is evaluated by simulations in “Simulation results and discussion” section. Finally, “Conclusion” section concludes the article.
System model and problem formulation
System model
As shown in Figure 1, we consider a twotier OFDMA network where K cochannel FBSs are randomly overlaid on a macrocell. We focus on resource allocation in the uplink of femtocells, that is, the subchannel assignment to femto users and the power allocation on subchannels in femtocells. Let M and F denote the numbers of active macro users camping on the macrocell and active femto users camping on each femtocell, respectively. Users are uniformly distributed in the coverage area of their serving cell. All femtocells are assumed to be closed access [17]. The OFDMA system has a bandwidth of B, which is divided into N subchannels. Channel fading on each subcarrier is assumed the same within a subchannel, but may vary across different subchannels. We assume that channel fading is composed of path loss and frequencyflat Rayleigh fading.
We denote $\left(\right)close="">{g}_{k,u,n}^{\text{MF}}$ and $\left(\right)close="">{g}_{j,k,u,n}^{\text{FF}}$ as the channel gains on subchannel n from femto user u in femtocell k to the MBS and FBS j, respectively, where j,k∈{1,2,…,K}, u∈{1,2,…,F}, and n∈{1,2,…,N}; denote $\left(\right)close="">{g}_{w,n}^{\text{M}}$ and $\left(\right)close="">{g}_{k,w,n}^{\text{FM}}$ as the channel gains on subchannel n from macro user w(∈{1,2,…,M}) to the MBS and FBS k, respectively; denote $\left(\right)close="">{p}_{k,u,n}^{\text{F}}$ and $\left(\right)close="">{p}_{w,n}^{\text{M}}$ as the transmit power levels on subchannel n of femto user u in femtocell k and macro user w, respectively. Then, we define $\left(\right)close="">{\mathit{P}}_{n}={\left[{p}_{k,u,n}^{\text{F}}\right]}_{K\times F}$ as the power allocation matrix of the K femtocells on subchannel n, and $\left(\right)close="">{\mathit{A}}_{n}={\left[{a}_{k,u,n}\right]}_{K\times F}$ as the subchannel assignment indication matrix for the K femtocells on subchannel n, where a_{k,u,n}=1 if subchannel n is assigned to femto user u in femtocell k, and a_{k,u,n}=0 otherwise.
The received signaltointerference and noise ratio (SINR) for femto user u on the n th subchannel in the k th femtocell is given by
where $\sum _{(j,v)\ne (k,u)}^{(K,F)}{a}_{j,v,n}{p}_{j,v,n}^{\text{F}}{g}_{k,j,v,n}^{\text{FF}}=\sum _{j=1}^{K}\sum _{v=1}^{F}{a}_{j,v,n}{p}_{j,v,n}^{\text{F}}$$\left(\right)close="">{g}_{k,j,v,n}^{\text{FF}}{a}_{k,u,n}{p}_{k,u,n}^{\text{F}}{g}_{k,k,u,n}^{\text{FF}}$ is the interference caused by other cochannel femtocells, $\left(\right)close="">{p}_{w,n}^{\text{M}}{g}_{k,w,n}^{\text{FM}}$ is the interference caused by the macrocell, and σ^{2}is the AWGN power.
The SINR for macro user w using the n th subchannel is given by
Based on Shannon’s capacity formula, the capacities on subchannel n of femto user u in femtocell k and macro user w are given, respectively, by
Problem formulation
The maximization of the total capacity of the K femtocells is formulated as follows.
where a femto user’s total transmit power is constrained by p^{max}, the power allocated to each subchannel is nonnegative, and each subchannel is assigned to no more than one user per femtocell.
It is assumed that the user assignment and power allocation can be performed independently for each subchannel, then the maximization of the total capacity of K femtocells is equivalent to the maximization of the total capacity of the K femtocells on one subchannel, and (5) and (6) can be simplified to
where $\left(\right)close="">{p}_{n}^{max}$ is the transmit power constraint on subchannel n for a femto user. Without loss of generality, we assume that $\left(\right)close="">{p}_{n}^{max}={p}^{max}/n$.
Interferenceaware resource allocation
In this section, we first model the uplink power and subchannel allocation problem in femtocells using a noncooperative game theory framework [18, 19], where a pricing scheme is imposed on femto users to mitigate the uplink interference caused by femto users to the macrocell. Then based on the noncooperative game framework, we propose a semidistributed algorithm for femtocells to assign subchannels to femto users assuming an arbitrary power allocation, and then optimize the power allocation on subchannels based on the obtained subchannel assignment.
A game theoretic framework
Based on the microeconomic theory [20], we model the femtocell uplink resource allocation problem as a femtocell noncooperative resource allocation game (FNRAG). The K femtocells are considered as selfish, rational players. Each of them tries to maximize its utility without considering the impact on other players. The FNRAG for subchannel n can be expressed as
where K={1,…,k,…,K},∀n∈{1,2,…,N} is the set of femtocells playing the game; {A_{ n }P_{ n }} is the strategy space of the players, with A_{ n }and P_{ n }being the subchannel assignment space and the power allocation strategy space, respectively; and $\left(\right)close="">{\mathit{\mu}}_{n}^{c}=\{{\mu}_{1,n}^{c},{\mu}_{2,n}^{c},\dots ,{\mu}_{K,n}^{c}\}$ is the set of net utility functions of the K players, in which
where $\alpha (\in {\mathbb{R}}^{+})$ (bps/W) is the pricing factor, and the price charged on a femto user is proportional to the uplink interference that it causes to the macrocell. If without the pricing part in the utility function, then a player will tend to maximize its utility by using the maximum transmit power, because $\left(\right)close="">{C}_{k,u,n}^{\text{F}}$ monotonically increases with $\left(\right)close="">{p}_{k,u,n}^{\text{F}}$ according to (1) and (3). This will lead to severe uplink interference to the macrocell.
Given the power and subchannel allocation in all other cochannel femtocells, the net utility function of femtocell k can be rewritten as
where $\left(\right)close="">{\mathit{p}}_{k,n}=\{{p}_{k,1,n}^{\text{F}},{p}_{k,2,n}^{\text{F}},\dots ,{p}_{k,F,n}^{\text{F}}\}$, a_{k,n}={a_{k,1,n},a_{k,2,n},…,a_{k,F,n}}, P_{−k,n} is the (K−1)×F matrix obtained by removing the k^{th}row from P_{ n }, A_{−k,n} is the (K−1)×F matrix obtained by removing the k th row from A_{ n }, and ${I}_{k,u,n}=\sum _{(j,v)\ne (k,u)}^{(K,F)}{a}_{j,v,n}{p}_{j,v,n}^{\text{F}}{g}_{k,j,v,n}^{\text{FF}}+{p}_{w,n}^{\text{M}}{g}_{k,w,n}^{\text{FM}}+{\sigma}^{2}$.
Definition 1
Given the uplink power allocation and subchannel assignment of all other cochannel femtocells, the best response of femtocell k is given by
In the following sections, in order to solve the noncooperative femtocell uplink resource allocation game in a semidistributed manner, we devise a semidistributed algorithm for each femtocell to first assign subchannels to its femto users for given power and subchannel allocation in all other femtocells and assuming an arbitrary power allocation of its own, and then optimize the power allocation based on the obtained subchannel assignment.
Interferenceaware subchannel allocation
In this section, we improve the subchannel allocation method in [21, 22] by pricing femto users according to their interference to macrocell in femtocell subchannel allocation.
Assuming that the k th row of the matrix P_{ n }contains an arbitrary power allocation of femtocell k on subchannel n, and given the power allocation and subchannel assignment of all other femtocells on subchannel n, then the best assignment of subchannel n in femtocell k is given by
According to (8), at most one element of a_{k,n}can take value of 1. Therefore, based on (11), the problem in (13) is equivalent to
where ${\xfb}_{k,n}$ is the best user for channel n in femtocell k to assign, and the assignment of subchannel n in femtocell k is indicated by ${\widehat{\mathit{a}}}_{k,n}=\{{\xe2}_{k,1,n},{\xe2}_{k,2,n},\dots ,{\xe2}_{k,F,n}\}$, where
In order to remove the dependence of the subchannel assignment on the assumed arbitrary power allocation, we let ${\widehat{\gamma}}_{k,n}=\underset{u}{max}\frac{{p}_{k,u,n}^{\text{F}}{g}_{k,k,u,n}^{\text{FF}}}{{I}_{k,u,n}}$, and then the transmit power of femto user u corresponding to ${\widehat{\gamma}}_{k,n}$ is given by $\frac{{\widehat{\gamma}}_{k,n}{I}_{k,u,n}}{{g}_{k,k,u,n}^{\text{FF}}}$. Accordingly, Equation (14) can be rewritten as
Interferenceaware power allocation
Once the uplink subchannel assignment has been determined by using (15) and (16) in each femtocell, the FNRAG in (9) can be reduced to a femtocell noncooperative power allocation game (FNPAG): $\left(\right)close="">{{\mathit{G}}_{n}}^{\prime}=\u3008\mathit{K},{\mathit{P}}_{n},{\mathit{\mu}}_{n}^{c}\u3009,\forall n$. Since subchannels have been assigned to specific femto users in each cell, we will drop the subscript u for simplicity hereafter.
Definition 2
Denote ${\widehat{\mathit{p}}}_{n}=\{{\widehat{p}}_{1,n}^{\text{F}},{\widehat{p}}_{2,n}^{\text{F}},\dots ,{\widehat{p}}_{K,n}^{\text{F}}\}$ as the optimal transmit power vector of the K cochannel femto users allocated to subchannel n under Nash Equilibrium in the FNPAG $\left(\right)close="">{{\mathit{G}}_{n}}^{\prime}$, if
where ${\widehat{\mathit{p}}}_{k,n}=\{{\widehat{p}}_{1,n}^{\text{F}},\dots ,{\widehat{p}}_{k1,n}^{\text{F}},{\widehat{p}}_{k+1,n}^{\text{F}},\dots ,{\widehat{p}}_{K,n}^{\text{F}}\}$ is the optimal transmit power vector of the K−1 cochannel femto users using subchannel n under Nash Equilibrium except for the cochannel femto user in femtocell k, and Nash Equilibrium is defined as the fixed points where no player can improve its utility by changing its strategy unilaterally [20].
Theorem 1
A Nash Equilibrium exists in the FNPAG: ${{\mathit{G}}_{n}}^{\prime}=\u3008\mathit{K},{\mathit{P}}_{n},{\mathit{\mu}}_{n}^{c}\u3009,\phantom{\rule{1em}{0ex}}\forall n$.
Proof
According to [20], a Nash Equilibrium exists in $\left(\right)close="">{{\mathit{G}}_{n}}^{\prime}$ if the following two conditions are satisfied: □

(1)
P _{ n }is nonempty, convex and compact in the finite Euclidean space ℜ ^{K×F}.

(2)
$\left(\right)close="">{\mathit{\mu}}_{n}^{c}$ is continuous and concave with respect to P _{ n }.
Since the power allocated on each subchannel is constrained between zero and the maximum power $\left(\right)close="">{p}_{n}^{max}$, the power allocation matrix P_{ n }is convex and compact, and condition (1) is satisfied.
For condition (2), it can be seen from (11) that $\left(\right)close="">{\mathit{\mu}}_{n}^{c}$ is continuous with respect to P_{ n }. To prove the quasiconcave property of (11), we take the derivative of (11) with respect to $\left(\right)close="">{p}_{k,n}^{\text{F}}$, and get
Taking the secondorder derivative of (11) with respect to $\left(\right)close="">{p}_{k,n}^{\text{F}}$ yields
Therefore, $\left(\right)close="">{\mu}_{k,n}^{c}$ is a quasiconcave function of $\left(\right)close="">{p}_{k,n}^{\text{F}}$. Since both conditions (1) and (2) hold, a Nash Equilibrium exists in the FNPAG. This completes the proof.
Lemma 1
The best response of femtocell k to the FNPAG $\left(\right)close="">{{\mathit{G}}_{n}}^{\prime}$ is given by
where $\left(\right)close="">{\left[x\right]}_{a}^{b}=min\{max\{a,x\},b\}$.
Proof
The ${\widehat{p}}_{k,n}^{\text{F}}$ in (21) is obtained by setting (18) to zero and solving the resulting equation for $\left(\right)close="">{p}_{k,n}^{\text{F}}$. □
Since (21) should be nonnegative, and the interference price factor α is nonnegative too, we get
Theorem 2
The FNPAG has a unique Nash Equilibrium.
Proof
It can be proved following similar proof in [5, 23]. □
Semidistributed implementation
Since only local information, such as uplink interference and channel gains seen by femto users, is needed for calculating (16) and (21), the interferenceaware femtocell uplink subchannel allocation scheme and power allocation scheme proposed in “Interferenceaware subchannel allocation” and “Interferenceaware power allocation” sections can be implemented in a distributed and semidistributed manners, respectively, as outlined in Algorithm 1.
Note that, $\left(\right)close="">{g}_{k,j,v,n}^{\text{FF}}$ and $\left(\right)close="">{g}_{k,w,n}^{\text{FM}}$ for the uplink can be estimated at femto user u in femtocell k by measuring the downlink channel gain of subchannel n from femtocell j and the macrocell, respectively, and utilizing the symmetry between uplink and downlink channels, or by using the site specific knowledge [5]. Furthermore, it can be assumed that there is a direct wire connection between an FBS and the MBS for the FBS to coordinate with the central MBS [4, 6], according to a candidate scheme proposed for 3GPP HeNB mobility enhancement [24].
Algorithm 1 can be implemented by each FBS, who only utilizing local information and limited interaction with MBS, therefore, Algorithm 1 is semidistributed and the practicability is guaranteed.
Algorithm 1 Semidistributed algorithm to solve FNRAG
1: FBS set: $\mathcal{K}=\{1,2,\dots ,K\}$; Femto user set per femtocell: $\mathcal{F}=\{1,2,\dots ,F\}$.
2: InterferenceAware Subchannel Allocation
3: Allocate the same power to each subchannel;
4: Femto user u in femtocell k measures $\left(\right)close="">{g}_{k,u,n}^{\text{MF}}$, $\left(\right)close="">{g}_{k,k,u,n}^{\text{FF}}$ and I_{k,u,n}, ∀k,u,n;
5: a_{k,u,n}=0, ∀k,u,n;
6: for each FBS do
7: Subchannel set: $\mathcal{N}=\{1,2,\dots ,N\}$
8: for u = 1 to F do
9: (a) find ${n}^{\ast}=arg\underset{n\in \mathcal{N}}{min}\frac{{g}_{k,u,n}^{\text{MF}}}{{g}_{k,k,u,n}^{\text{FF}}}{I}_{k,u,n}$;
10: (b) $\left(\right)close="">{a}_{k,u,{n}^{\ast}}=1$;
11: (c) $\mathcal{N}=\mathcal{N}\left\{{n}^{\ast}\right\}$;
12: end for
13: while$\mathcal{N}\ne \varphi $do
14: (a) find $({u}^{\ast},{n}^{\ast})=arg\underset{u\in \mathcal{F},n\in \mathcal{N}}{min}\frac{{g}_{k,u,n}^{\text{MF}}}{{g}_{k,k,u,n}^{\text{FF}}}{I}_{k,u,n}$;
15: (b) $\left(\right)close="">{a}_{k,{u}^{\ast},{n}^{\ast}}=1$;
16: (c) $\mathcal{N}=\mathcal{N}\left\{{n}^{\ast}\right\}$;
17: end while
18: end for
19: InterferenceAware Power Allocation
20: for each FBS do
21: for n=1 to N do
22: calculate (20) and (21);
23: end for
24: end for
Simulation results and discussion
In this section, we present simulation results to evaluate the performance of the proposed interferenceaware FNRAG algorithm, as compared with the unpriced suboptimal subchannel allocation (USSA) and MIWFbased power allocation algorithm [4, 25], which are outlined in Algorithm 2. Both the system capacity and the fairness between femtotier and macrotier are evaluated in the simulations.
Algorithm 2 USSA and MIWF algorithm
1: FBS set: $\mathcal{K}=\{1,2,\dots ,K\}$; Femto user set per femtocell: $\mathcal{F}=\{1,2,\dots F\}$.
2: USSA
3: Allocate the same power to each subchannel;
4: Femto user u in femtocell k measures $\left(\right)close="">{g}_{k,k,u,n}^{\text{FF}}$ and I_{k,u,n}, ∀k,u,n;
5: a_{k,u,n}=0, ∀k,u,n;
6: for k=1 to K do
7: Subchannel set: $\mathcal{N}=\{1,2,\dots ,N\}$
8: for u = 1 to F do
9: (a) find ${n}^{\ast}=arg\underset{n\in \mathcal{N}}{max}\frac{{g}_{k,k,u,n}^{\text{FF}}}{{I}_{k,u,n}}$;
10: (b) $\left(\right)close="">{a}_{k,u,{n}^{\ast}}=1$;
11: (c) $\mathcal{N}=\mathcal{N}\left\{{n}^{\ast}\right\}$;
12: end for
13: while$\mathcal{N}\ne \varphi $do
14: (a) find $({u}^{\ast},{n}^{\ast})=arg\underset{u\in \mathcal{F},\setminus \in \mathcal{N}}{max}\frac{{g}_{k,k,u,n}^{\text{FF}}}{{I}_{k,u,n}}$;
15: (b) $\left(\right)close="">{a}_{k,{u}^{\ast},{n}^{\ast}}=1$;
16: (c) $\mathcal{N}=\mathcal{N}\left\{{n}^{\ast}\right\}$;
17: end while
18: end for
19: MIWFBased Power Allocation
20: Implement the MIWF algorithm using bisection search [25].
In the simulations, the macrocell has a coverage radius of 500 m. Each femtocell has a coverage radius of 10 m. K FBSs and 50 macro users are randomly distributed in the macrocell coverage area. The minimum distance between the MBS and a macro user (or an FBS) is 50 m. The minimum distance between FBSs is 40 m. Femto users are uniformly distributed in the coverage area of their serving femtocell. Both macro and femtocells employ a carrier frequency of 2 GHz, B=10 MHz, and N=50. The AWGN variance is given by ${\sigma}^{2}=\frac{B}{N}{N}_{0}$, where N_{0}=−174 dBm/Hz. The Rayleighfading channel gains are modeled as unitmean exponentially distributed random variables. The average channel gain (including pathloss and antenna gains) for indoor femto user and outdoor macro user are modeled as λ d^{−4} and λ d^{−3}, respectively, where λ=2×10^{−4}[5]. Besides, α is selected as 4×10^{4}using the tryanderror method through simulations. The maximum transmit powers of a femto user and a macro user are set as 20 and 30 dBm, respectively.
Figure 2 shows the capacity of the macrocell when the number of femto users per femtocell increases from 1 to 6, for K = 20, 30, and 50. It can be observed that the proposed interferenceaware FNRAG algorithm outperforms the USSA and MIWFbased algorithm by up to a 23% increase in macrocell capacity. As the number of femtocells K increases, the advantage of the FNRAG algorithm becomes more noticeable, because the increased uplink interference caused by femtocells to the macrocell can be effectively mitigated by the pricing scheme imposed on femto users in the FNRAG algorithm, but not by the unpriced USSA and MIWFbased algorithm. As the number of femto users increases, the potential interferers will be more because of the cochannel deployed femtocells, but the number of available channels is constant. Therefore, the proposed algorithm will be more and more superior compared with the USSA and MIWFbased algorithm.
Figure 3 shows the total capacity of K femtocells and macrocell when the number of femto users per femtocell increases from 1 to 6, for K = 20, 30, and 50. We can see that the proposed FNRAG algorithm improves the total capacity of femtocells and macrocell over the USSA and MIWFbased algorithm by 5–10% when the number of femto users per femtocell is larger than 3. More gain is obtained as K increases, indicating that the proposed interferenceaware FNRAG algorithm can also effectively mitigate interference between neighboring femtocells, and hence is more applicable in dense deployment of cochannel femtocells than the USSA and MIWFbased algorithm. As the number of femto users increases, the cotier interference between femtocells is more severe, the interferenceaware subchannel assignment will be more effective in cotier interference mitigation.
In order to evaluate the fairness between the macro tier and femto tier, we use the tiered fairness index (TFI) [26], which is defined as
where $\left(\right)close="">{C}_{w}^{\text{M}}$ and $\left(\right)close="">{C}_{k,u}^{\text{F}}$ are the capacities of macro user w and femto user u in femtocell k, respectively.
Figure 4 compares the tiered fairness performance between the proposed FNRAG algorithm and the USSA and MIWFbased algorithm. It can be observed that the tiered fairness of the proposed FNRAG algorithm gets close to or becomes even better than that of the USSA and MIWFbased algorithm, as the number of femto users per femtocell goes beyond 3. This is because the proposed FNRAG algorithm alleviates the uplink interference generated by femto users to the macrocell by charging each femto user a price proportional to the interference that it causes to the macrocell, and the macrocell capacity and consequently the tiered fairness can be guaranteed. Since the USSA used in Algorithm 2 considers the fairness among femto users in each femtocell, the tiered fairness of Algorithm 2 is better than the proposed FNRAG algorithm when F is less than 3. The tiered fairness improves as K increases, because the spatial reuse gain increases with K.
Conclusion
In this article, we have proposed a semidistributed interferenceaware resource allocation algorithm for the uplink of cochannel deployed femtocells, based on a noncooperative game framework. Using the proposed algorithm, each femtocell can maximize its capacity through resource allocation, taking into account intercell interference reported by its femto users, and with uplink femtotomacro interference alleviated by a pricing scheme imposed on femto users. It has been shown through simulations that the proposed interferenceaware resource allocation algorithm is able to provide improved capacities of both macrocell and femtocells, together with comparable tiered fairness, as compared with the existing unpriced subchannel allocation and MIWFbased power allocation algorithm.
Endnote
^{a}This work has been partially presented at IEEE ICC 2012.
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Acknowledgements
The authors would like to thank Dr. David LópezPérez for his helpful discussions. This study was supported by the Scitech Projects of the Committee on Science and Technology of Beijing (D08080100620802, Z101101004310002), the National Natural Science Foundation of China (61101109), and the National Key Technology R&D Program of China (2010ZX0300300101, 2011ZX0300300201). This study was also partially supported by the UK EPSRC Grants EP/H020268/1, CASE/CNA/07/106, and EP/G042713/1.
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Keywords
 Nash Equilibrium
 Power Allocation
 Orthogonal Frequency Division Multiple Access
 Macrocell Base Station
 Macro User