CHAPTER 4

RESOURCE ALLOCATION FOR CLUSTERED SMALL CELLS IN TWO-TIER OFDMA NETWORKS

4.1 INTRODUCTION

As we have discussed before, in an OFDMA-based two-tier cellular network with spectrum sharing among small cells and macrocells, co-tier interference and CTI significantly affect network performance [1]. Hence, interference mitigation techniques need to be developed to manage the radio resources of small cells in order to achieve the QoS requirements of all the users. In this chapter, we study the problem of sub-channel and power allocation in a dense deployment of femtocells in an OFDMA-based two-tier network with constraints on co-tier interference and CTI as well as minimum data rate requirements. We formulate the problem of joint sub-channel and power allocation for a clustered femtocell network with interference constraints on co-tier interference and CTI as well as data rate constraints as an optimization problem. The optimization problem aims at maximizing the sum-rate of the femtocells in a cluster of cooperating femtocells. This problem turns out to be a Mixed Integer Non-Linear Program (MINLP). Therefore, to solve this problem sub-optimally, different approaches are proposed. The first approach works by reformulating the MINLP problem as a convex one by using the time sharing idea, where sub-channels can be time shared among femtocells. Joint sub-channel and power allocation using the reformulated problem gives an upper bound to the original problem. In the other approaches, joint sub-channel and power allocation is done in two phases. In the first phase, given power allocation, sub-channels are allocated to femtocells. For sub-channel allocation, different schemes are proposed. The first scheme uses Linear Programming (LP) and the time sharing concept. This approach also gives an upper bound to the optimal solution. Other schemes that preserve the integrality constraint (i.e., a sub-channel is allocated to only one femtocell) are proposed. Those schemes are Branch and Bound (BnB), LP with rounding, Lagrangian Relaxation, and a heuristic scheme. Knowing the sub-channel allocation, in the second phase, we perform power allocation. The power allocation problem is a non-linear convex problem that can be efficiently solved by interior point method. We compare the proposed schemes in terms of closeness to the optimal solution (which is obtained by an exhaustive search) and complexity. The obtained results show that the proposed schemes have performance that is close to that of the optimal scheme and have reduced complexity. In addition, we study the effect of clustering (i.e., placing femtocells into organized disjoint groups, where co-tier interference is eliminated) on the femtocell network performance. We compare different cluster configurations to a distributed scheme (i.e., without clustering) and a centralized scheme (i.e., with all femtocells in one cluster). Results show that in a dense deployment of femtocells with tight interference constraints, clustering is a very efficient technique.

In Section 4.2, a review of related work is presented. The system model and assumptions are discussed in Section 4.3. In Section 4.4, the main problem for joint sub-channel and power allocation is formulated. Section 4.5 reformulates joint sub-channel and power allocation as a convex problem. Section 4.6 then presents the different proposed schemes for sub-channel allocation. Section 4.7 discusses the power allocation step. Finally, Section 4.8 presents the numerical results before we conclude the chapter in Section 4.9.

4.2 RELATED WORK

4.2.1 Clustering and Coalition Formations of Femtocells

In [2–5], the authors propose a coalition formation games where, femtocells cooperate by forming coalitions. A coalition here is a group of femtocells organized by a coalition level scheduler such that interference within a coalition is eliminated by time-division duplex (TDD) mode of operation. Therefore, femtocells in a coalition can use the same sub-channels but on orthogonal time slots. However, there is inter coalition interference. Interference in this work is considered among femtocells only due to the assumption of split spectrum operation with macrocells. Each femtocell has four sub-channels (independent of the coalition size). In forming the coalitions, there is a cost in terms of transmission power that prevents forming coalitions with distant femtocells, and hence grand coalitions. Each femtocell evaluates the options of joining available coalitions in the form of iterations. From the results, it was shown that the number of iterations increases exponentially as the number of femtocells increases. Obtaining the optimal coalition structure is very challenging, and hence the authors proposed an algorithm that is based on recursive core to obtain the coalitions in a distributed manner.

In [6], the authors propose a scheme for macro and femtocell interference mitigation. In their work, femtocells exist in groups in the form of grid. Each group of FAPs is managed by a femtocell controller; hence, we have a centralized operation. Interference exists between FAPs and macrocells. To reduce the amount of interference introduced to the MUEs, the FAPs rely on cognitive capabilities and sensing, where they employ both overlay and underlay modes of operation. Each of the FAPs and MUEs requires at most one sub-channel. No power control is considered for the macrocell but power control is considered for the FAPs. Femtocells are grouped into clusters, where a cluster is a group of co-channel FAPs that use the same sub-channel. Femtocells suffering from strong mutual interference are placed in different clusters. Formation of the clusters is done by using a graph approach relying on the max k-cut problem. A sub-optimal algorithm is proposed to place the femtocells in the clusters. Then each cluster is assigned a sub-channel using graph colouring. The only aspect considered by the authors here to maximize the throughput is by minimizing the co-tier interference. After that power control is performed for femtocells.

In [7], the authors formulate and solve the problem of femtocell deployment in a two-tier cellular network. Given a certain number of femtocells, the target is how to optimally deploy them to serve a set of house holds (HH) with the minimum data rate requirements and set their transmission powers. The number of femtocells is not enough to have a femtocell installed in every HH. Therefore, one femtocell will be installed in an HH that will act as a cluster head (CH) and will serve neighboring HHs acting as cluster members (CMs) in a TDMA fashion. The CH with its CMs form a cluster, where interference is eliminated through TDMA operation. The authors divide the joint optimization problem at hand into two sub-problems, namely, the cluster formation sub-problem and the power control subproblem. In their work, the number of clusters is known a priori and the maximum number of HHs that a FAP can serve is known as well. Therefore, starting with some initial coalition structure, the power control sub-problem is solved to obtain the social welfare. After that, using the simulated annealing algorithm, another coalition structure is proposed for which a new power control sub-problem is solved. The new social welfare is compared to that of the previous coalition structure. Iterations proceed up to the best coalition structure. A shared spectrum operation is assumed where macrocell and femtocells operate on the same channel. Hence, no channel allocation is considered and closed access FAPs are assumed.

4.2.2 Resource Allocation in Clustered Femtocells

In [8], the authors propose a joint power control and resource allocation algorithm in an OFDMA femtocell network, where femtocells are grouped into disjoint clusters. However, the authors only consider orthogonal channel assignment between femtocells and macrocells; hence, no CTI exists. In addition, optimal selection of the cluster size is not discussed. In [9], the authors propose a dynamic clustering based sub-band allocation scheme in a dense femtocell environment. However, the authors only consider frequency allocation. Also, they treat the interference from the macrocell network as additive white Gaussian noise (AWGN). The authors in [10] propose a coalition formation game approach to form cooperative groups among interfered femtocells by sharing the radio spectrum.

In [11], a DL power control method is proposed to mitigate interference in a macrocell–femtocell network. The QoS for both MUE and FUE is guaranteed by limiting interference from femtocells to nearby MUEs. A centralized and a distributed approach for power control are proposed and compared. However, no frequency allocation is considered.

In [12], a joint power control and resource allocation scheme for a co-channel femtocell–macrocell network is proposed with QoS guarantees for both MUE and FUE. Both centralized and distributed approaches are introduced.

In [13], a joint power and sub-channel allocation scheme is proposed to maximize the system capacity for indoor dense mobile communication systems. The authors assume that the femtocells are densely deployed and propose a centralized resource allocation framework. The authors prove that the optimal power allocation in such an environment has a special form known as binary power allocation, where for every sub-channel, a single femtocell only loads power. However, in [11–13], the effect of clustering is not studied for the femtocell resource allocation problem. In addition, [13] considers the interference from macrocell base stations (MBSs) as additive white Gaussian noise (AWGN).

A resource allocation and admission control algorithm, called QoS-based femtocell resource allocation (Q-FCRA), is proposed in [8], which is based on clustering and taking into account QoS constrained of high priority user and best effort users. The Q-FCRA algorithm is comprised of three main phases: (i) Cluster formation, (ii) Intra-cluster resource allocation, and (iii) Inter-cluster resource contention resolution. When powered on, a FAP listens to its surrounding transmissions and gathers information through measurements collected from users attached to it or via a receiver function within the FAP. Based on this information, the FAP computes the number of interfering femtocells and transmits it along with its Physical Cell Identity (PCI) to each of them. The FAP with the highest interference degree among its one-hop neighbors is selected as a CH. After partitioning the femtocell network into clusters, resources are jointly allocated to all FAPs within each cluster taking into account QoS requirements of attached users. The resource allocation problem is formulated as a multi-objective optimization problem. The objective is to find the optimal resource allocation of a set of tiles in each FAP to deliver user data, while minimizing the interference among the FAPs and at the same time providing QoS guarantees for high priority users as well as maximizing the throughput for best- effort users.

4.3 SYSTEM MODEL AND ASSUMPTIONS

We consider a multi-tier wireless network where femtocells are deployed in a dense manner to cover an indoor area and are overlaid by a single macrocell. In such an environment, FUEs will have very good channel conditions to their femtocells. Signals received from outdoor macrocells, however, are highly attenuated. We denote by kf a user served by femtocell f. The MUEs exist indoor as well and are served by macrocell m, where we denote by km a user served by macrocell m. We denote by F the set of femtocells where, and by the macrocell. Let be the set of indices of the sub-channels in the system and N=|N|. We assume channel states of sub-carriers are the same within a sub-channel n of bandwidth . Define gni,j as the channel gain between user i and BS j on sub-channel n. In our model, channel gain includes path-loss, log-normal shadowing, and Rayleigh fading.

The unit power SINR of an FUE kf served by femtocell f on sub-channel n is as follows:

(4.1)

where pni,j is the power allocated to the link between user i and BS j on sub-channel n and No is the noise power.

Closed access femtocells are assumed where the access is restricted to registered UEs only. Femtocells and the macrocell operate in a shared spectrum environment. Femtocells have unique Cell-IDs. All the FUEs are capable of performing interference measurements and reporting them to the FAPs. Femtocells are equipped with GPS devices to obtain the accurate estimate of femtocell locations so that femtocell clusters can be formed. Communication among femtocells is possible via air or backhaul.

The femtocells are divided into disjoint clusters. The idea behind clustering is to divide the joint sub-channel and power allocation problem in the femtocell network into smaller sub-problems. We define as the set of clusters of femtocells. A cluster is the lth set of femtocells such that , , , and . Within each cluster, one femtocell is elected as a CH and it takes the responsibility of performing sub-channel and power allocation within the cluster. Clustering is simply done by grouping femtocells closely located to each other. Therefore, it is basically done based on distance. Note that the entire set of sub-channels is available to each cluster and within a cluster, no two femtocells transmit simultaneously on the same sub-channel. Therefore, no co-tier interference exists within a cluster.

Various configurations are available for cluster sizes. One extreme is to have a grand cluster, where all femtocells are in one cluster. One benefit of this configuration is that it eliminates co-tier interference among femtocells. However, in addition to the huge burden imposed on the CH performing resource allocation, the share of each femtocell in the available spectrum is small. We refer to this scheme as the centralized scheme. Another extreme is to have femtocells acting independently, that is, no clustering. In this case, each femtocell has the entire spectrum available and the complexity of resource allocation is the lowest. However, co-tier interference is severe in this case. We refer to this scheme as the distributed scheme.

4.4 JOINT SUB-CHANNEL AND POWER ALLOCATION IN FEMTOCELL CLUSTERS

For each femtocell cluster cl, we require to maximize the sum-rate of all femtocells within the cluster, given that there are interference constraints for co-channel FUEs in neighboring clusters as well as co-channel MUEs. In addition, we have a data rate requirement for each femtocell. The CH within each cluster cl solves the following optimization problem:

(4.2)

subject to

(4.3)

(4.4)

(4.5)

(4.6)

(4.7)

where is an indicator if sub-channel n is allocated to the link between user kf and femtocell f, is the power assigned to the link between them.

In the optimization problem (4.2), the sum-rate maximization is subject to a data rate requirement Rf and total power budget Pfmax for each femtocell f as indicated in (4.3) and (4.4), respectively. We have interference constraints for co-channel MUEs served by macrocell m and FUEs served by femtocells j in a neighboring cluster as given in (4.5) and (4.6), respectively. The reference user concept [14] is applied here, where the interference constraints are for co-channel MUEs and FUEs having the highest channel gain to the target femtocell f. The interference constraints in (4.5) and (4.6) can be further simplified to a single constraint by choosing one of them having the higher channel gain to the target femtocell f. In this way, the interference constraints can be rewritten as follows:

(4.8)

A user k now is either a co-channel MUE or an FUE whichever has higher channel gain to the target femtocell f. Finally, (4.7) is the exclusion constraint indicating that sub-channel n can be used in one femtocell only.

Since this problem is an MINLP, obtaining its optimal solution will be prohibitive. Therefore, several approaches will be proposed to solve it. One approach will be to reformulate the problem as a convex one by using the idea of time sharing. The other approaches will solve the problem in two phases:

• Phase 1: Given power allocation, we can perform sub-channel allocation. We assume the initial power on each sub-channel as the minimum of either or . The idea is to keep power as uniform as possible and at the same time not to violate the interference constraints. For Phase 1, several schemes with varying performance and complexity will be proposed.
• Phase 2: Given sub-channel allocation, we perform power allocation.

It is worth mentioning that, the CHs perform resource allocation in parallel. The FUEs in a cluster measure the interference levels in the previous time slot, and report them to their FAPs which, in turn, pass them to the CH. Based on those measurements, the CH performs resource allocation in the current time slot [15].

4.5 JOINT SUB-CHANNEL AND POWER ALLOCATION USING CONVEX REFORMULATION

A common approach in the literature is to relax the constraint that sub-channels can be used by one femtocell. Thus is reinterpreted as the sharing factor of femtocell f for sub-channel n. In addition, we define a new variable . Then becomes the actual transmitted power [16–18]. For each cluster cl, problem in (4.2) can thus be reformulated as follows:

(4.9)

subject to

(4.10)

(4.11)

(4.12)

(4.13)

(4.14)

(4.15)

where

In (4.9), is not allowed to be zero since the objective function in (4.9) is not defined for . However, the objective function approaches close to zero when is arbitrarily small. Hence, the nature of the objective function remains the same [16]. A nice property of the optimization problem in (4.9) is that it is convex. The objective function is concave, inequality constraint in (4.10) is convex and all the remaining inequality and equality constraints are affine. Therefore, this problem can be efficiently solved by the interior point method [19]. It is worth mentioning that this approach gives an upper bound solution to (4.2).

4.6 SUB-CHANNEL ALLOCATION

In this section, knowing power allocation, we perform sub-channel allocation using various schemes of different performance and complexity.

4.6.1 Branch and Bound

For given power allocation, we have an ILP that can be optimally solved using the BnB technique. We have the following optimization problem:

(4.16)

subject to

(4.17)

(4.18)

where

BnB is guaranteed to find the optimal sub-channel allocation but its complexity in the worst case is as high as that of exhaustive search which is O(FN).

4.6.2 Linear Programming

By relaxing the integrality constraint on to be taking any value [0, 1], we have a Linear Program (LP) with an objective function value ZLP-TS that can be solved using simplex or interior point method. In this way, can now be defined as a time sharing factor. It indicates the amount of time for which sub-channel n is allocated to user kf in femtocell f. The solution obtained in this way is an upper bound solution to the optimal sub-channel allocation problem.

To retrieve the integrality property again for the LP solution, we assign sub-channel n to user kf in femtocell f according to:

(4.19)

where kf(n) denotes sub-carrier n allocated to user kf in femtocell f. We denote the resulting objective function value as ZLP and call this technique as LP with rounding. Using the interior point method, the complexity of sub-channel allocation using LP is O(F3N3).

4.6.3 Lagrangian Relaxation

The idea behind Lagrangian relaxation is that some optimization problems can be easily solved if some set of annoying constraints are removed (dualized). By doing so, we can solve an easier version of the original problem [20].

In our sub-channel allocation problem, if we dualize the data rate requirement constraints, we have the following optimization problem:

(4.20)

subject to

(4.21)

where , uf is a non-negative Lagrange multiplier and

(4.22)

The function ZLG(uf) is the Lagrange dual function.

This problem is an assignment problem that can be solved, for a given multiplier uf, using a greedy approach, where for each sub-channel n, we allocate it to user kf in femtocell f according to the following:

(4.23)

The complexity of this greedy approach is O(FN). Now, to obtain the optimal multiplier uf, we have the following optimization problem:

(4.24)

subject to

(4.25)

To obtain the values of the Lagrange multipliers uf, we shall use the sub-gradient method. For a multiplier uf we have

(4.26)

where [x]+=max(x, 0), and t(y) is the step size obtained in the yth iteration using the following expression:

(4.27)

where , and is the objective value of the best known feasible solution to (4.16). Iterations proceed until the multipliers converge or we reach a maximum number of iterations.

4.6.4 Heuristic Scheme

In this section, we propose a low complexity heuristic scheme for sub-channel allocation. If we reformulate the objective function and data rate constraint in (4.2) and (4.3) to , we will have expressions that are equivalent to the original ones for binary values of [21]. If we further relax to take any value [0, 1], we have an NLP that is convex in [19].

Now, for the optimization problem (4.2), if we apply KKT conditions, we obtain the following formula for sub-channel allocation among femtocells:

(4.28)

where , , , and are Lagrange multipliers associated with data rate constraints, total power budget, interference constraints and exclusion constraints, respectively. Therefore, we can deduce that a sub-channel n is better be allocated to a user kf in femtocell f having good SINR value and having low channel gain gnk,f to a reference user (hence, low interference to this co-channel user).

We can propose a heuristic by constructing the following ratio for each femtocell f on sub-channel n: . A sub-channel n is allocated to the link between user kf and femtocell f based on the following criterion:

(4.29)

We denote the resulting objective function value by ZHEUR. The complexity of this approach is O(FN). This approach, however, has lower complexity than that of the Lagrangian relaxation approach because it does not involve the iterations of the sub-gradient method.

4.6.5 Feasibility Guarantee Algorithm

A problem with the proposed sub-channel allocation schemes in Sections 4.6.2 (LP with rounding),4.6.3 and 4.6.4 is that we can have some femtocells with no sub-channels allocated, leading to an infeasible solution to the original problem. Hence, to restore feasibility back, an algorithm called Feasibility Guarantee Algorithm (FGA) is proposed.

The idea of the algorithm is for the unsatisfied femtocells to choose the best sub-channel from the set of allocated sub-channels to the most satisfied femtocells in a cluster cl. In the algorithm, we denote by Nf, the set of sub-channels allocated to femtocell f. Γ is the set of sub-channel allocation indicators for all femtocells . S and U are the set of satisfied femtocells and unsatisfied femtocells, respectively. Iterations are repeated until all unsatisfied femtocells are allocated a sub-channel. Based on the fact that the FUE is very close to its femtocell, and hence has a very good channel condition, the power allocation step can satisfy its data rate requirement.

Algorithm 4.1   FGA for cl

4.7 POWER ALLOCATION

Given sub-channel allocation, we can perform power allocation in an optimal manner. We have the following optimization problem:

(4.30)

subject to

(4.31)

(4.32)

(4.33)

where

The optimization is a non-linear convex problem that can be efficiently solved by the interior point method [19].

4.8 PERFORMANCE EVALUATION

4.8.1 Parameters

We consider 10 femtocells deployed in an indoor area of dimensions to constitute a dense environment and they are within the coverage area of a macrocell. The FUEs and MUEs exist indoor. Each femtocell has a single FUE and the MUEs are served by their macrocell. The communication links between BSs and UEs are affected by path-loss, shadowing, and Rayleigh fading. Indoor femto channel models for urban deployment are used [22]. For path-loss between macrocell m and UE in the indoor area, we have . For path-loss between femtocell and its FUE, we have and for path-loss between femtocell and another UE, we have . For interference measurements, the worst-case initial condition is assumed for all femtocells in the network, where all femtocells transmit with uniform power on all sub-channels. The simulation parameters are shown in Table 4.1.

Table 4.1 Parameters

Parameter	Value
Carrier frequency	2 GHz
Number of femtocells	10
Maximum femtocell transmission power	20 mW
FUEs per femtocell	1
Number of sub-channels	10
Sub-channel bandwidth	180 KHz
Noise power No	–174 dBm/Hz + 10log10()
Macrocell radius	200 m
Standard deviation for log-normal
shadowing in macrocell	6 dB
Standard deviation for log-normal
shadowing in femtocell	4 dB
Outdoor wall loss Low	20 dB
Penetration loss of the inner walls qLiw	15 dB

We study the network performance in terms of average achieved data rate and average transmission power percentage for different cluster sizes. Beside the distributed scheme and the centralized scheme, we consider semi-distributed schemes with clusters of two (2 femtocells per cluster) and clusters of five (5 femtocells per cluster). Note that we only consider equal-sized clusters.

To assess the proposed sub-channel allocation schemes, the optimal solution is obtained by an exhaustive search. For each cluster, all possible combinations of sub-channel allocations are tried for all femtocells, and then the transmission power is allocated in an optimal fashion. The sub-channel allocation combined with the corresponding power allocation yielding the highest sum-rate of the femtocells is chosen as the optimal solution. The sub-channel allocation using the exhaustive search has a complexity of O(FN).

4.8.2 Numerical Results

Average achieved data rate versus interference threshold: We study the average achieved data rate for all femtocells versus the interference threshold for different cluster sizes. Different MUE densities are considered as well. Sub-channel allocation is performed using LP with time sharing.

In Figure 4.1(a), we study the average achieved data rate for all femtocells versus the interference threshold for the FUEs and the MUE. We have a single MUE. We observe that at loose interference thresholds, the distributed scheme has the highest average data rate. Though it has the highest amount of co-tier interference, each femtocell is allowed to use all the available sub-channels. As the interference constraints become tighter, the benefit of cooperation through clustering starts to appear with the cluster size of two offering the highest average data rate. At very tight interference thresholds, different clustering configurations and the centralized one have similar performance, which is better than that of the distributed scheme. For the centralized scheme, although the share of each femtocell in the transmission bandwidth is the lowest, it has eliminated interference among the femtocells.

Figure 4.1(b) shows similar results but with multiple MUEs. The achieved data rates for all of the schemes generally decrease due to the increased number of MUEs. From Figures 4.1(a) and (b), we can conclude that cooperation among femtocells is generally effective in improving the femtocell performance.

Average transmission power percentage versus interference threshold: We study the average percentage of femtocell transmission powers defined as versus the interference threshold for different cluster sizes. Different MUE densities are considered as well. Sub-channel allocation is performed using LP with time sharing.

Figure 4.2(a) shows the percentage of average transmission power for all femtocells considering a single MUE. As the interference threshold becomes tighter, the transmission power of all femtocells decreases. Since the centralized scheme has the MUE only to take care of, it is allowed to transmit at the maximum amount of power. On the other hand, the distributed scheme has the least allowed transmission power, since any femtocell has to take care of interference with all its neighbours.

Figure 4.2(b) shows the percentage of average transmission power for all femtocells considering multiple MUEs. Since we have a larger number of MUEs, all configurations have similar performance of decreased power with tighter interference thresholds. A generic femtocell in the distributed scheme and in the clustering scheme with cluster size of two has a higher number of sub-channels to use, and hence a higher opportunity to transmit more power.

Comparison among the proposed schemes: Figures 4.3 and 44.4 show the average achieved data rate versus the interference threshold for the clustering scheme with cluster size of two and five, respectively. The performances of the different schemes are shown for comparison. The optimal scheme is found for the cluster size of two only. From these two figures we can observe that the LP with time sharing and the convex approach have the highest average data rate since both give an upper bound to the optimal solution. Since the convex approach performs sub-channel and power allocation jointly, it has a higher data rate than that of the LP with time sharing approach. It is worth mentioning that both the schemes, however, solve a different problem from the original one. The BnB technique comes next. For the clustering scheme with cluster size of two, the BnB solution coincides with the optimal solution. Finally, we have the LP technique with rounding, Lagrangian Relaxation, and heuristic schemes, where they all have similar performance which is near optimal with the benefit of reduced complexity.

From the performance evaluation results, we have the following conclusion: in a dense deployment scenario with tight interference constraints, cooperation and coordination among femtocells is beneficial. The MINLP problem of sub-channel and power allocation in femtocell clusters can be solved with efficient schemes of reduced complexity and near optimal performance.

4.9 SUMMARY AND FUTURE RESEARCH DIRECTIONS

We have investigated the effect of clustering of densely deployed femtocells into cooperative groups on the system performance. At tight interference constraints, which will be used to protect co-channel FUEs and MUEs, clustering has been shown to be an efficient technique. The problem of sub-channel and power allocation in femtocell clusters turns out to be an MINLP. A reformulation has been done for the problem to transform the MINLP problem into a convex one which gives an upper bound to the optimal solution. Efficient suboptimal schemes offering near optimal solution and having lower complexity have been proposed as well.

Extension of this work can include formulating clustering as an optimization problem to obtain the optimal cluster size. Also, resource allocation for MUEs, can be considered to have a complete framework for a two-tier network. The effects of the channel gain uncertainties and backhaul constraints on the resource allocation in a clustered femtocell network can be investigated as well.

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