SPECIAL SECTION ON FUTURE NETWORKS: ARCHITECTURES, PROTOCOLS, AND APPLICATIONS

Received October 16, 2016, accepted November 24, 2016, date of publication December 5, 2016, date of current version January 4, 2017.

Digital Object Identifier 10.1109/ACCESS.2016.2635938

Cluster-Based Resource Allocation forSpectrum-Sharing Femtocell NetworksHAIBO ZHANG1,2, DINGDE JIANG3, (Member, IEEE), FANGWEI LI1, KAIJIAN LIU1,HOUBING SONG4, (Senior Member, IEEE), AND HUAIYU DAI2, (Fellow, IEEE)1Chongqing Key Lab of Mobile Communications Technology, Chongqing University of Posts and Telecommunications, Chongqing 400065, China2Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, NC 27695, USA3School of Computer Science and Engineering, Northeastern University, Shenyang 110819, China4Department of Electrical and Computer Engineering, West Virginia University, Montgomery, WV 25136, USA

Corresponding authors: H. Zhang (wdkyzl@gmail.com) and D. Jiang (jiangdd@mail.neu.edu.cn).

This work was supported in part by the National Nature Science Foundation of China under Grant 61301122, Grant 61271260, andGrant 61572231, in part by the Program for Changjiang Scholars and Innovative Research Team in the University of Ministry of Educationof China under Grant IRT16R72, in part by the Natural Science Foundation Project of Chongqing under Grant cstc2014jcyjA40052, and inpart by the Science and Technology Program of Shaanxi Province under Grant 2016KW-032.

ABSTRACT Femtocells in two-tier femto-macro networks can enhance indoor coverage and improve overallnetwork performance.Macro networksmay share spectrumwith overlaid femtocells so as to improve spectralefficiency. However, the deployment of femtocells also brings co-tier and cross-tier interferences, which willsignificantly degrade system performance. In order to solve this problem efficiently, we propose a distributedscheme to manage wireless resources in this heterogeneous networks. The feasible solution can be obtainedby dividing the problem into two sub-problems. First, we propose a femtocells clustering scheme, which usesa mathematical modeling idea based on LINGO, an optimization software that can solve the joint clusteringproblem for the femtocell access points (FAPs). The proposed branch-and-bound algorithm and the simplexalgorithm are used jointly to find the optimal solution by LINGO. The optimality of the proposed clusteringalgorithm is verified both theoretically and through simulations where the comparison with other algorithmsis made. Second, a novel algorithm is proposed to allocate sub-channels to the femtocell users (FUEs).Compared with other related schemes, the proposed channel-allocation algorithm can reduce the interferencemore effectively and achieve higher data-rate fairness among FUEs. Specifically, according to the situationthat the FUEs move in the room, the FUE mobility model is proposed to predict the change tendency ofpath loss values of the FUEs, which can guarantee the mobile service quality and improve system capacityeffectively. Finally, the power of the FAPs is adjusted dynamically through setting the interference thresholdto further improve the performance of the system.

INDEX TERMS Femtocells, clustering, resource allocation, branch-and-bound, the simplex algorithm,LINGO.

I. INTRODUCTIONWith the increase of requirements for indoor business, thenetwork managements of communication systems becomemore andmore complicated [1]–[4]. The indoor coverage andcommunication quality need to be significantly improved.One of the most promising solutions to improve the perfor-mance of the future wireless communication system is todeploy small-size cells extensively, which have been widelystudied in recent years [5]–[11]. For example, femtocell canbe deployed in the several scenarios, such as the cognitiveradio network, the cooperative networks. Compared with thetraditional macro cellular networks, the embedded femtocell

access points (FAPs) can expand coverage and improve sys-tem performance effectively.

Some exiting studies have been reported on femtocellinterference management and resource allocation [6]–[12].A graph-based scheme is applied to solve the sub-channelassignment and interference alignment problem in [12]. How-ever, the fairness among users isn’t considered. The modelsof open and closed femtocell services are applied to study theeconomic aspects of femtocell services with game theoreticmodels between providers and/or users in [13]. Althoughpart of the system performance is improved, the interferencecoordination becomes more complex. In the cognitive radio

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system, a resource allocation algorithm based on game theoryis proposed in [14]. Not only is the throughput increased,but the interference is also reduced. However, in these litera-tures, the clustering optimization [15], [16] is not taken intoaccount.

Many researchers focus on the joint consideration of theclustering optimization problem and the resource alloca-tion problem in the femtocell networks so as to reduce theco-tier interference and improve the system performancein [17]–[20]. For example, in an orthogonal frequency-division multiple-access (OFDMA) femtocell network serv-ing both quality of service (QoS)-constrained high-priorityand best-effort users, a new resource-allocation and admis-sion control algorithm is proposed based on clustering andtaking into account QoS requirements in [17]. In orderto mitigate inter-femtocell interference (IFI), the disjointIFI-minimizing clusters are formed by aMax k-Cut clusteringalgorithm according to the interference graph [18]. In [19],the authors present a new technique for jointly optimizingenergy consumption [21]–[24] and QoS in heterogeneouscellular networks employing fractional frequency reuse in themacrocell tier, network routing is also applied to maximizethe spectrum utilization. According to the clustered femtocellbase stations (FBSs) and predicted signal-to-interference-plus-noise ratio (SINR), the power control scheme is appliedto femtocell network in the downlink [20]. However, in theabove work, the optimal solution is very hard to obtainbecause of the characteristic of (non-deterministic polyno-mial (NP)-hard problem. Moreover, the optimal cluster sizeisn’t considered.

Considering joint clustering optimization and resourceallocation, a Semi-Definite Programming (SDP) based onrandom rounding algorithm is proposed in [25] based onCVX, a software package for specifying and solving convexprograms. Although the optimal cluster size has been takeninto account in this paper, it is not fast enough to obtainthe optimal solution for the clustering optimization problem.When the number of the FAPs increases, the algorithm in [25]may not be able to find the optimal solution effectively, andthere is no sufficient argument as to whether the clusteringoptimization problem has an optimal solution. In addition,their resource allocation algorithm doesn’t solve the problemof the average interference effectively. The data-rate fairnessissue is not taken into account either. In order to simulatethe practical application scene, the mobility of the femtocellusers (FUEs) should also be considered.

Considering the above issues, we propose a novel resourceallocation algorithm based on clustering for the closed sub-scriber group FAPs [25]. We formulate the problem of jointclustering optimization and resource allocation, which is aclassical NP-hard problem. It is very difficult to solve thisproblem. Without loss of generality, the problem is dividedinto two sub-problems, namely the clustering optimiza-tion problem and the resource allocation problem. Firstly,Femtocell Gateway (FGW) collects the information aboutFAPs. According to the clustering optimization algorithm, the

FAPs are assigned to different clusters to reduce interferenceeffectively. Then, a FAP is selected as the cluster head in eachcluster, which will be responsible for resource allocation inits own cluster. For the above optimization problem, we pro-pose a mathematical modeling idea based on LINGO, whichapplies the Branch-and-Bound algorithm and the SimplexAlgorithm to find the optimal solution.We theoretically provethat the solution obtained by our algorithm is the global opti-mal solution. The simulation results show that the proposedalgorithm can obtain the optimal solution for the clusteringoptimization problem in an efficient manner. In addition, weput forward a novel algorithm to solve the resource allocationproblem. Compared with other allocation algorithms, theproposed algorithm not only reduces interference but alsoimproves the data-rate fairness among FUEs. According tothe practical scenarios that the FUEs move in the room,the FUE mobility model is proposed. The tendency of pathloss values of the FUEs is predicted based on this model,which will further improve the continuity of all kinds ofdata services. The data rate requirements of mobile FUEs aremet more easily. In order to further enhance FUE data ratesand reduce interference between FUEs, a power allocationproblem is studied. Finally, power is adjusted dynamicallythrough setting the interference threshold to further improvethe performance of the system.

The rest of this paper is organized as follows. In Section 2,the system model is described. In Section 3, the optimizationproblem for the clustering is formulated. Detailed analysisis given to verify that the found solution by the proposedalgorithm is the global optimal solution. In Section 4, a novelresource allocation algorithm is proposed to reduce interfer-ence and improve fairness. In Section 5, numerical results fordifferent scenarios and topologies demonstrate the superiorityof the proposed schemes. Finally, Section 6 concludes thepaper.

II. SYSTEM MODELFig. 1 shows the topology of two-tier femto-macro networks,in which a large number of FBSs covering a small rangeare distributed in every room of every layer of big buildingswhich are in the coverage of the single overlay macrocellbase station (MBS). The macro users (MUEs) served by theoutdoorMBS are outdoors and indoorswhile the FUEs servedby FBSs are indoors. The channel propagation conditionsbetween FAPs and theirs FUEs are perfect. The channel gainincludes the path loss (PL) L, the shadow fading Ls and theantenna gain La. The total path loss between FAP and its FUEcan be expressed as PL = 15.3+37.6 log d+L+Ls−La. d isthe distance between FAP and the FUE.

In order to reduce the co-tier interference effectively, FAPsmay be grouped into different clusters. However, it is difficultto group FAPs according to their instantaneous channel gains.Therefore, the average channel gain is applied to reducethe complexity. As the distance between the FAP i and theFUE ki served by the FAP i is quite short, the channel gainbetween the FAP j and the FUE ki approximately equals to

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FIGURE 1. The model of the femto-macro system.

TABLE 1. System parameters

that between the FAP j and the FAP i on the sub-channel θ(i.e., gθj,ki ≈ gθj,i) [25]. Obviously, the channel gain matrixbetween the FAPs is symmetric, i.e., gθj,i = gθi,j. The systemparameters are listed in Tab. 1.

III. OPTIMIZATION PROBLEM FOR THE CLUSTERINGSimilar to [25], we also set up an undirected graphG1 = (V1,E1), V1 is the set of vertices representing theFAPs. E1 is the set of edges between two vertices.ω

+

i,j andω−

i,jare the non-negative link weights. The objective of clusteringoptimization problem is to group the vertices into sets ofsimilar vertices, where the degree to which they are similaris given by ω+i,j and the degree to which they are different isgiven by ω−i,j. That is to say, the objective is to find a partitionthat maximizes the sum of the weights ω+i,j of edges insidethe sets of partitions and the weights ω−i,j of edges betweenthe sets of partitions [25]. If there is serious interference

between two FAPs, the link weights ω+i,j between two FAPswill be large. Similarly, if the channel gain gθi,j is high betweentwo FAPs, the interference between them will be also verylarge. We observe that the channel gain and the path loss areinversely proportional, so we have a new idea that sets upω+i,j = 1/PL.

In order to mitigate the co-tier interference, the FAPswhich have the serious interference with each other may beplaced into the same cluster. Sub-channels should be assignedorthogonally to these FAPs in the same cluster and reusedbetween any two clusters. When the number of FAPs isincreased in a cluster, the reusability of sub-channels and thedata rate will decrease, so we should structure the clusteringoptimization problem reasonably. ω−i,j is a penalty term when

two FAPs are in different clusters. All in all, we need to finda method to maximize the objective function value in [25].In this paper, we assume that the value of ω−i,j is the samefor all FAPs (i.e.,ω−i,j = ω−ϕ ). Define ω

−ϕ = ω+i,j + 1,

1 ≤ i ≤ j ≤ F , ϕ = F2−F2 , 1 > 0, ∀ω−ϕ . The range of ω

−ϕ

can be expressed as (minω+i,j + 1 ≤ ω−ϕ ≤ maxω+i,j + 1).

As a result, the optimization problem for the clustering isshown as followed:

max∑i∈F

∑j∈F

ω+i,jxij + ω−ϕ (1− xi,j) (1)

C1 : xi,i = 1, ∀i ∈ F (2)

C2 : xi,j = xj,i, ∀i, j ∈ F (3)

C3 : xi,j + xj,k − xi,k ≤ 1,

∀i, j, k ∈ F , k > i, j 6= i, k (4)

C4 :∑j∈F

xi,j ≤ M , ∀i ∈ F (5)

C5 : xi,j ∈ {0, 1}, ∀i, j ∈ F (6)

where, xi,j is the FAP clustering indication factor. If FAP iand j are in the same cluster, xi,j = 1, otherwise, xi,j = 0.C1 indicates that any FAP is in the same cluster with itself.C2 shows that if FAP i and j are in the same cluster, then FAPj and i will be also in the same cluster. C3 indicates that ifFAP i and j are in the same cluster and FAP j and k are in thesame cluster, then FAP i and k are also in the same cluster.C4 states that the number of FAPs is not more than the totalnumberM of available sub-channels in a cluster.C5 indicateswhether FAP i and j are in the same cluster or not.

In order to prove the solution obtained by the proposedalgorithm based on the clustering optimization problem is theglobal optimal solution, according to [26] and [27], we givethe following definition:Definition 1: The vectors that satisfy all the constraint

conditions are the feasible solutions. The set consisting of allfeasible solutions is called as the feasible set or the feasibleregion.Definition 2: The feasible solution that makes the objective

function obtain the optimal value is called as the optimalsolution of linear programming.

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Definition 3: A is a coefficient matrix of constraint vari-able x. b is a column vector which is composed of the rightend of constraints. For the constraints Ax ≤ b, we assumethat the rank of the matrix A is m. P i (i = 1 ∼ n) is thei − th column vector of the matrix A. B is composed of them column vector of the A, i.e., B = (P i1P i2 . . .P im). If B isa nonsingular matrix, namely det B 6= 0, then B will be notonly a basematrix but also themaximum linearly independentsubset of the A.Definition 4: The solution that meets the constraint

conditions and in which the non-basic variable is zerois called as the basic solution. The basic solutions withnon-negative basic variables are called as basic feasiblesolutions.Definition 5: Suppose x1 x2..xk are the points in the feasi-

ble region R. If ∃λ1 λ2 . . . λk (λi ≥ 0, i = 1..k ,k∑i=1λi = 1),

which makes the equality x =k∑i=1λixi be established, x will

be called as a convex combination of x1 x2..xk .Definition 6: The basic feasible solution that makes the

objective function achieve the optimal is called as the basicoptimal solution.Definition 7:We assume thatU and V are two points in the

feasible region R. If ∀λ ∈ [0 1], ∃W (λ) = λU + (1 − λ)V ,which is also in the feasible region R, then the set of feasibleregion R will be called a convex set.

According to the above definition, we can get the followingconclusions:Corollary 1:According to the definition 3 and 4, ifAx ≤ b,

then the general necessary and sufficient conditions for x =( x1 x2 . . . xn ) to be the basic feasible solution are that thecolumn vector P i1P i2 . . .P ik of the matrix A correspondingto the base component xi1xi2 . . . xik of x is linearly indepen-dent.Corollary 2: According to the definition 4, if the unit

matrix E is a feasible solution, then they must be basicfeasible solutions.Proof: For the constraint condition C1, the main diagonal

elements of the unit matrix E are constant. The non-diagonalelements of the E are the basic or the non-basic variables,and their values are zero. Obviously, it also conforms to thedefinition 4. Therefore, the unit matrix E must be the basicfeasible solution.

Based on the above definitions and corollaries, we willpropose the following lemmas and proofs.Lemma 1: The feasible region about the cluster-

ing optimization problem is at the border of a convexset.

The optimal solution is n × n dimensions. Next, we willtake an example for the 3×3 dimensions matrix.Proof: According to the constraints C1 and C2, we

can construct any feasible solution, such as u =

(1u1u2; u11u3; u2u31), v = (1v1v2; v11v3; v2v31), u andv are the feasible solutions, so they meet the constraintsC3 and C4. Put u and v into the constraints C3 and C4,

there are: {u2 + u3 − u1 ≤ 1v2 + v3 − v1 ≤ 1

(7)

u1 + u2 ≤ M − 1u1 + u3 ≤ M − 1u3 + u2 ≤ M − 1v1 + v2 ≤ M − 1v1 + v3 ≤ M − 1v3 + v2 ≤ M − 1

(8)

For the definition 7, there is:

W(λ) = λu+ (1− λ)v

=

1λu1 + (1− λ)v1λu2 + (1− λ)v2λu1 + (1− λ)v11λu3 + (1− λ)v3λu2 + (1− λ)v2λu3 + (1− λ)v31

(9)

Put W (λ) into the constraint C3:

λu2 + (1− λ)v2 + λu3 + (1− λ)v3 − λu1 − (1− λ)v1= λ(u2 + u3 − u1)+ (1− λ)(v2 + v3 − v1) (10)

According to the equation (7), there is:

λu2 + (1− λ)v2 + λu3 + (1− λ)v3− λu1 − (1− λ)v1 ≤ λ+ (1− λ) = 1 (11)

Obviously,W (λ) meets the constraint C3.Similarly, bringW (λ) into the constraint C4, according to

the equation (8), we get:

λu1 + (1− λ)v1 + λu2 + (1− λ)v2 + 1= λ(u1 + u2)+ (1− λ)(v1 + v2)+ 1 ≤ M

λu1 + (1− λ)v1 + λu3 + (1− λ)v3 + 1= λ(u1 + u3)+ (1− λ)(v1 + v3)+ 1 ≤ M

λu3 + (1− λ)v3 + λu2 + (1− λ)v2 + 1= λ(u3 + u2)+ (1− λ)(v3 + v2)+ 1 ≤ M

(12)

Clearly, W (λ) meets the constraint C4. At the same time,W (λ) meets the constraints C1 and C2. C5 can be convertedto 0 ≤ xi,j ≤ 1. W (λ) meets the constraint of 0 ≤ xi,j ≤ 1.According to the definition 7, we can consider that the setof feasible solutions about the clustering optimization is aconvex set, then, C5 ∈ (0 ≤ xi,j ≤ 1) is at the limit point,namely, it is at the border. So the lemma is feasible. In thesame way, the matrix with n×n dimension can also be provedto be true.Lemma 2: If the set of the feasible solutions about the

clustering optimization problem is at the border of a convexset, then as long as the set of the feasible solutions is notempty, there must be the optimal solution, which is the limitpoint.Proof: We assume that x1,1x1,2 . . . xr,r represents all the

limit points on the feasible region R. All of the feasible

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solutions about the clustering optimization problem can beexpressed by the basic solutions, i.e.:

∀x ∈ R =r∑i=1

r∑j=1

λi,jxi,j, λi,j ≥ 0,r∑i=1

r∑j=1

λi,j = 1

(13)

The objective function value is:∑i

∑j

w+i,jxi,j+w−ϕ (1− xi,j) = s (14)

And

f ∗ = max s

= max∑i

∑j

w+i,jxi,j + w−ϕ (1− xi,j), 1 ≤ i, j ≤ r

(15)

So we can get:

s ≤r∑i=1

r∑j=1

λi,jf ∗ = f ∗ (16)

In summary, the above equations suggest that the objectivefunction value obtained in an arbitrary feasible solution is notmore than that in a limit point xε. Therefore, xε is the optimalsolution and the lemma is feasible.Lemma 3: It is the sufficiency and necessity condition that

the limit point is the basic feasible solution in the feasibleregion R.Proof: Each element can be considered as a variable.

Consequently, we convert the objective problem into thestandard form of linear programming. We denote xi,j asx1,1x1,2 . . . xn,n which corresponds to x1x2 . . . xn2 , accordingto the corresponding relationship, the objective function canbe expressed as the form of cTx, where c and x are columnvectors (i.e., n2 × 1 dimention). Similarly, the constraintcondition C3 can be transformed into Bx ≤ 1. The constraintcondition C4 can be transformed into Cx ≤ M . Finally, wewill get the expression:(

BC

)x ≤ b⇒ Ax ≤ b (17)

Necessity:We assume that x is the limit point, according tothe C1, x 6= 0. According to the corollary 1, if we can provethat the column vector P i1P i2 . . .P ik of A corresponding tothe base component xi,1xi,2 . . . xi,k of x is linearly indepen-dent, then the proposed proposition will be established.Proof:Assume the column vector P i1P i2 . . .P ik is linearly

dependent, the following equation holds:

λ1P i1 + λ2P i2 + . . .+ λkP ik = 0 (18)

For equation (18), there exists at least one non-zero vectorwith λ1λ2 . . . λk . We construct a y vector with the dimensionof n2, in which the component i1 i2 . . . ik corresponds tothese parameters λ1λ2 . . . λk respectively, the others are zero.

As y is not an empty set and the equation Ay = y1P i1+ . . .+ykP ik = 0 is feasible, we define:

α = min1≤t≤k

{xi,t|yt ||yt 6= 0

}(19)

Obviously, α > 0, for A(x ± ay) = Ax ± aAy = Ax ≤ b.So we can get two feasible solutions, which can be expressedas x1 = x + ay, x2 = x − ay, respectively. Therefore, x =12x1 +

12x2, for α > 0, y 6= 0. So, x 6= x1 6= x2, we can get

x2 through assigning a value to the parameter, i.e., λ = 12 . If

x can be expressed as a convex combination of the feasibleregion, which is contradictory to the definition of the pole,therefore, the necessity is proved.

Sufficiency:Let x be a basic feasible solution, x is not zero.Proof (Reduction to Absurdity): If x is not only a basic

feasible solution but also is not in the pole, then there aretwo distinct values x1 and x2 to meet the equation x =λx1 + (1 − λ)x2, λ ∈ (0, 1). They make the equation beestablished:{

Ax1 = x1,1P i1 + x1,2P i2 + . . . x1,kP ik = 0

Ax2 = x2,1P i1 + x2,2P i2 + . . . x2,kP ikD 0(20)

As x1 and x2 are the different feasible solutions, for theequation:

Ax1 − Ax2 = (x1,1 − x2,1)P i1 + . . .+ (x1,k − x2,k)P ik= 0 (21)

There is at least one non-zero coefficient, if P i1P i2 . . .P ikis linearly dependent, it will be contradictory to the fact thatP i1P i2 . . .P ik is linearly independent, so the fundamentalsolution must be in the pole. Thus, lemma 3 is accurate.Lemma 4:We assume that the rank of the constraint coeffi-

cient matrix A ism, the column vectors of the A are Non-zerovectors. If the feasible solutions exist, then these solutionsmust be fundamental solutions (or the pole values).Proof: Similarly, we convert the clustering optimization

problem into the standardized linear programming form bythe above method. Firstly, we assume x = (x1 x2 . . . xn2 )

T isthe feasible solution which satisfies the constraint conditionsAx ≤ b. In the same way, for the constraint condition C1,x has a non-zero element, the number of the positive com-ponents of x is k . They are x1 > 0, . . . , xk > 0, respectively,the rest components of x are zero. These positive componentscorresponding to the column vector of the A are P1 . . .Pk,respectively. According to the corollary 1, we need to provethey are linearly independent.

Assume these vectors P1 . . .Pk are linearly dependent,i.e., at least one parameter of λ1 λ2 . . . λk isn’t zero, so, theequation λ1P1 + λ2P2 + . . . + λkPk = 0 is established.Firstly, we imagine there is at least one parameter λi >0, 1 ≤ i ≤ k , then, we can construct a column vectorλ = (λ1 . . . λk , 0 . . . 0)T with the dimension of n2, therefore,the equation Aλ = λ1P1 + . . . + λkPk = 0 is established.∃λi > 0, we define α as α = min

1≤t≤k

{xt|λt ||λt 6= 0

}. Then

x − αλ ≥ 0,A(x − αλ) = Ax ≤ b. Clearly,x − αλ is

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TABLE 2. The simplex tableau based on LINGO:

a feasible solution. Now, the l-th component of the feasiblesolution x − αλ can be expressed as xl − αλl = 0, then,the new feasible solution can be shown as xα = x − αλ =(xl − αλl . . . x2− αλ2, 0, xl+1− αλl+1 . . . xk − αλk , 0 . . . 0).Compared with the solution x, the feasible solution xα is thelack of a positive component P l . The new column vectorobtained by removing P l is expressed as P1 P2 . . .Pk−1.If the new column vectors are linearly independent, thelemmawill be true. Otherwise, we need to continue to find thenew feasible solution by the above method until the columnvector is transformed into a unit matrix. According to thecorollary 2, it is clear that the unit matrix is basic feasiblesolution, so the lemma 4 can be proved.Lemma 5: We denote the basic optimal solution as x,

its check number with respect to non-basic variable is lessthan zero, and the variable xδ is a non-basic variable with anegative check number. There is xδ = 0 in all the optimalsolutions. Namely, the non-basic variables of the optimalsolutions are zero.Proof:The simplex tableau is set by Lingo in Table 2, where, bi ≥

0, i = 1 . . .m, cj ≤ 0, j = m+1, . . . n2, x = (b1 . . . bm0..0)T .Assume there are cj ≤ 0, j = m + 1, . . . n2, cδ < 0,

m + 1 ≤ δ ≤ n2 in the simplex tableau, if there is afeasible solution expressed as x = (x1 . . . xδ . . . xn2 )

T≥ 0, at

least one element will be more than zero (i.e.,xδ > 0), then,the objective function value of the clustering optimization

problem is z = z0 −n2∑

i=m+1cixi ≥ z0−cixi > z0. Obviously, x

is not the optimal solution, it is incompatible with the lemma,so the original proposition is true.

According to the literatures [26]–[28], we draw into thefollowing theorems:Theorem 1: In the case of the optimal solution, if the

check number of the non-basic variables xi is less than zero,the number of the optimal solutions will be the only one.If at least one check number of the non-basic variables xi iszero, the others are less than zero, a necessary and sufficientcondition for the only optimal solution is that the column

FIGURE 2. The flow chart of the check number.

FIGURE 3. The flow chart of an example.

vectors corresponding to all non-basic variables in which thecheck number is zero don’t include the positive component inthe simplex tableau.

The steps of obtaining the check number are expressed asbelow:

Firstly, we need to find the maximum linearly dependentsubset of the coefficient matrix A. Next, we need to makea judgment between the basic variables and the non-basicvariables. The rules of determining the loop is as follows:

We denote the non-basic variable as digital one (i.e., 1),these basic variables are digital two, three, etc. by analogyalong the direction of the arrow.

e.g.: In order to get the check number of the non-basicvariable xi, the loop can be expressed as below:The check number of the non-basic variable x11 is the sum

of these coefficients corresponding to the odd label variablesin the objective function minus the sum of these coefficientscorresponding to the even label variables in the objectivefunction. In the LINGO platform, the check number and thesimplex tableau are updated automatically by the simplexalgorithm.

In order to solve the above optimization problem, we pro-pose a novel strategy based on the LINGO platform [29].It can find the optimal solution through the branch-and-bound algorithm, which modifies the branch direction by thesimplex algorithm to avoid falling into a local optimum andgreatly improve the operation efficiency. The chart of theproposed strategy based on LINGO platform is expressed inFig. 4 as follows:

The specific steps of the proposed clustering optimiza-tion algorithm (named as Algorithm 1 in this paper) can bedescribed as follows:

Step 1) Transform the above optimization problem into aLINGO model;

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FIGURE 4. The framework of solving the clustering optimization problem.

FIGURE 5. The flow chart of the proposed clustering optimizationalgorithm(Algorithm 1).

Step 2) Apply the direct solution procedure to processequality constraint of the model;

Step 3) Recognize the type of the model on the LINGO (itis integer linear programming in this paper);

Step 4) Call the procedure of the Branch-and-Bound algo-rithm and the simplex algorithm, the procedure is describedas follows:

Firstly, we denote the current maximum objective functionvalue as c∗, the branch layer as p, the function value as f̄ inthe Fig. 5, the work-piece order corresponding to c∗ as p∗,the current node as p1, i.e., the sequence corresponding tothe current node which needs to be branched.

1) Initialization: Let p = 0, p1 = A, (i.e., empty set),c∗ = ∞, where the current node p1 is the root node.

2) Calculate sub-node: Each sub-node can be obtainedfrom the branch of the current node, then, we calculatethe lower bound l of each sub-node. Finally, these sub-nodes are sorted from small to large by the lower boundvalue. Update p (let p = p+ 1).

3) Establish the simplex tableau: Take each sub-node asthe initial value to establish the simplex tableau. It isproved that the optimal solution is in the boundaryand the check number corresponding to the non-basicvariables is the non-positive. So it is essential to cut offthe branches which deviate from the direction of theboundary and this condition which the check numberis the non-positive. Update p.

4) Update parameter: When all the current nodes aredetected, we need to update p, let p = p− 1. The nextprocess will skip to step 7, otherwise, we will markthe node b1 with the maximum lower bound value asQ in each sub-node of the current layer (there, it isthe p-th layer). The work-piece p2 of the p-th positioncorresponding to Q is added at the end of the node p1.Let the current node equal to Q. Then, the next processwill skip to step 5.

5) Loop 1: The above steps suggest that the current nodehas the maximum lower bound value in the co-tiernodes which have the same originating layer. If thelower bound value of the current node is not less thanc∗, it is not necessary to search the current node and theco-tier nodes which have the same originating layer.Therefore, the process which probes the originatingnodes of the current node is over. Update p (i.e., p =p − 1). The last one work-piece of p1 is removed.Finally, please skip to step 7, otherwise, skip to step 6.

6) Loop 2: Suppose p = n, we can get a suboptimal orderand make p∗ = p1. c∗ is the lower bound value in thecurrent node. We update p as p = p − 1. Please skipto step 7 after removing the last one work-piece of p1.Otherwise, skip to step 2.

7) Achieve optimal solution: If p 6= 0, please skip tostep 4 after removing the last one work-piece of p1.Otherwise, the whole process ends, so c∗ is the optimalvalue of the objective function and p∗ is the optimalsolution.

The above process is completed based on the LINGOplatform.

Next we will give a theoretical analysis about thesolution obtained by the proposed clustering optimizationAlgorithm 1.

First, according to the proofs from lemma 1, 2 and 4, weconclude that the proposed clustering optimization problemmust have an optimal solution.

Secondly, the solution obtained by the proposed Algo-rithm 1 meets the lemma 5 and the define 5, so it is the basicfeasible solution.

Then, according to the lemma 3, it is known that thesolution is also the limit point.

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Finally, according to the lemma 4 and theorem 1, it isverified theoretically that the clustering optimization solutionis the global optimal solution.

IV. THE PROBLEM OF RESOURCE ALLOCATIONSome technologies about resource allocation are used toman-age the limited wireless resource [30]–[33]. One of the keytechnologies in femtocell networks is the channel allocationtechnology, which can improve the FUEs experience effec-tively. Inmost exiting researches, the location of FUE is fixed.However, in practical applications, the random movementsof FUEs in the room will change the distance between FUEand FBS, which can affect the channel gain, and it may bereduce the quality of the mobile service and system capacityof femtocell networks. In another word, the FBS still usesthe location information I1 of the last time slot of a FUE incurrent time slot, however, the location information may havechanged in current time slot. For example, when the path lossPL2 of the new FUE location information I2 is greater than thepath loss PL1 of the original location information I1, if a sub-channel is being allocated to this FUE according to the pathloss PL1, then the communication may be interrupted. There-fore, sub-channel allocation based on the real-time channelgain is still a problem today. This paper proposes a newidea, which can predict the future location information of theFUE and compute PL2 according to the Gaussian distributionmodel. When PL2 > PL1, update PL1 = PL2. If PL2 < PL1,the channel condition of the new location will be changedbetter. In order to ensure real-time communication, PL1 stilltakes the original value. The channel gain is calculated byPL1to allocate sub-channels so as to ensure the communicationquality of the mobile FUEs.

Next, we will introduce the resources allocation and theFUEs mobile model.

(1) Sub-channels allocation problem is converted into amaximization data rate problem, the following questions areconstructed:

max0θki,i

∑i∈cl

N∑θ=1

0θki,i1f log2(1+ Pθki,iγ

θki,i) (22)

C1 :N∑θ=1

0θki,i1f log2(1+ Pθki,iγ

θki,i) ≥ φ

minki , ∀i (23)

C2 :N∑θ=1

0θki,iPθki,i ≤ p

maxi , ∀i (24)

C3 : 0θki,iPθki,ig

θkω,i ≤ ζ

θkω , ∀θ, i (25)

C4 : 0θki,iPθki,ig

θkj,i ≤ ζ

θkj , ∀θ, i ∈ Cl, j /∈ Cl (26)

C5 :∑i∈cl

0θki,i = 1, ∀θ (27)

where, 0θki,i ∈ {0, 1} is an indicator that takes the valueof 1 if sub-channel θ is allocated to the link between FUEki and FAP i, 0 otherwise. Pθki,i is the transmission powerof the FAP i on the sub-channel n. γ θki,i is the SINR of

the FUE ki in the FAP i on the sub-channel θ . It can be

defined as γ θki,i =gθki,i∑

j6=i,j∈F Pθkj,jgθki,i+Pθkω,ωg

θki,ω+N0

. PθMω,M is the

transmission power of the MUEMω served by MBSM . N0 isthe noise power. φi is the threshold of the minimum data raterequirement. C2 represents that the total transmission poweris not more than the maximum transmission power pmax

i . C3suggests that the interference between the MUE and FAP isnot more than the interference threshold ζ θkω . C4 suggeststhat the interference between the FUE and FAP is not morethan the interference threshold ζ θkj , Cl is the l-th FAP clusterobtained by Algorithm 1. C5 says that these sub-channels areallocated orthogonally in the same cluster.

The above problem about the resource allocation can be

simplified as ki(θ ) = argmaxi

(Pθki,iγ

θki,i

gθku,i), where ku is either a

neighboring MUE or an FUE whichever has higher channelgain to the target femtocell i [25], ki(θ ) denotes that sub-channel θ is assigned to FUE ki. Namely, the sub-channel is

allocated to the FUE with the maximal value ofPθki,iγ

θki,i

gθku,iin

each cluster.(2) The Gaussian distribution is adopted to simulate the

moving model of FUEs, first of all, the position of FUE inthe future is defined as:

(xt , yt ) ={xt−1 + vt1t cos(αt )+ nt−1yt−1 + vt1t sin(αt )+ nt−1

}(28)

Among them, (xt−1, yt−1) is the previous position of FUE,vt is themoving speed of FUE at the time t ,1t is the transitiontime interval, αt ∈ [0 2π ] is the moving direction of FUE atthe time t , nt−1 is the gaussian noise.The speed, direction and distance between FUE and FBS

are shown as follows:

vt = N (vt−1, β1t) (29)

αt = N (αt−1, 2π − a tan(√vt/2)1t) (30)

d =√x2t + y

2t (31)

where vt−1 and αt−1 are the previous speed and direction ofFUE respectively.N (µ, τ ) is Gaussian distribution,µ as aver-age, τ as the standard variance. β is the mobile accelerationfor FUE.

However, in [25], the resource allocation algorithm doesn’tsolve the problem of the average interference effectively. Theauthors don’t take into account the data-rate fairness andthe mobility of FUEs. In this paper, not only do we solvethe average interference effectively, but we also considerdata-rate fairness and the FUEs mobility. The proposed sub-channels allocation algorithm can be expressed as follows:

We give every FAP a serial number, then put them intothe different clusters by the Algorithm 1. We denote thetotal number of sub-channels as N and the total number ofFUEs as fue. We set up the allocation instruction matrixT = zeros(N ,fue) and the interference instruction matrixG = (fue, fue,N ). Initializes the location of FUE xt−1, yt−1and the direction αt−1. Calculate the path loss PL1 of the

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Algorithm 1 The Proposed Sub-Channels AllocationAlgorithm

1. Initialize T = zeros(N , fue),G = (fue, fue,N ) = 0,xt−1, yt−1 and αt−1.2. Calculate PL1. Predict PL2. If PL2 > PL1, updatePL1 = PL2.3. for θ = 1 : N

4. θ → ki, s.t. maxPθki,iγ

θki,i

gθku,i, T (θ, ki) = 1

5. end for6. while (j)7. if 0θki,iP

θki,ig

θkj,i ≥ ζ

θkj ,∀θ, i ∈ Cl, j /∈ Cl orφki < φmin

ki8. Make G(ki, kj, θ) = G(kj, ki, θ) = 1

9. θ → the other FUEs to makePθki,iγ

θki,i

gθku,imaximum

10. end if11. end while12. if θ can’t meet all of the FUEs in the cluster, remove θ13. Update T and G14. end if15. while (ki)16. Find out the set N ′ by G17. θ ′ ∈ N ′→ ki to make the data rate maximum18. if θ ′ ∈ kj, kj 6= ki, kj ∈ ψk and the number of thesub-channels of the FUE kj is more than 119. Delete θ ′ from the kj20. θ ′→ ki21. end if22. if there is only one sub-channel for the FUE kj23. Give up θ ′, delete θ ′ from N ′, update T24. end if25. end while26. while (ki ∈ ψk )27. Find out kj ∈ ψkby θ ∈ ki and θ ∈ kj28. if 0θki,iP

θki,ig

θkj,i ≥ ζ

θkj ,∀θ, i ∈ Cl, j /∈ Cl && N1 > 1

29. Delete θ from kj30. end if31. end while

current position. According to the formula (28-31), this paperpredicts the future position of FUE, and the correspondingpath loss PL2. If PL2 > PL1, update PL1 = PL2.

Sub-channels are assigned orthogonally in the same clusterand reused among different clusters. Firstly, we allocate allof the sub-channels to each FUE in a cluster. Secondly, weassume that FAPs send signals by the average transmissionpower (Pmax

i /F). The average allocated power will be real-located after the sub-channels are allocated. We choose the

sub-channels for the FUEs to makePθki,iγ

θki,i

gθku,imaximum in each

cluster, then update T (θ, ki) = 1. From the former clusters,we find out the FUE j who uses the same sub-channel withthe current cluster. We detect the FUE successively to judgewhether its channel quality is beyond the interference thresh-old by 0θki,iP

θki,ig

θkj,i ≥ ζ

θkj ,∀θ, i ∈ Cl, j /∈ Cl in current cluster,

if it does, we will make G(ki, kj, θ) = G(kj, ki, θ) = 1, and

continue to detect whether the achieved data rate φki of FUE imeets the minimum data rate requirement φmin

ki or not. If oneof two constraints cannot be met, we will allocate the sub-

channel n to another FUE to makePθki,iγ

θki,i

gθku,imaximum in the

current cluster, then we repeat it until all the constraints aremet. If all constraints are not met for all FUEs in the cluster,we will remove n, update T and G.

In order to guarantee the fairness among FUEs, we willadjust the sub-channel allocation result. There may be theFUEs who are not assigned a sub-channel. So, according tothe indication matrix T of channel assignment, we can detectwhether a current FUE has been assigned a sub-channel ornot. If the FUE ki is not assigned a sub-channel, we will findout the set N ′ of sub-channels on which the current FUE kihas interference with other FUEs by the instruction matrixG.We find out a sub-channel θ ′ from the set N ′ to make thedata rate of the FUE ki maximum. If the sub-channel θ ′ isallocated to another FUE kj in the current cluster ψk and thenumber of the sub-channels allocated to the FUE kj is morethan one, we will delete the sub-channel θ ′ from the set ofsub-channels allocated to the FUE kj, and allocate θ ′ to ki.If there is only one sub-channel for the FUE kj, we will giveup θ ′, and exclude θ ′ from the set N ′. Repeat it until all ofFUEs are assigned. Finally, update T .

To reduce the interference among clusters further, to beginfrom the first FUE in the current cluster, we find out theFUE kj from the former several clusters who uses the samesub-channel with the current FUE ki. We need to detectwhether the interference is less than the interference thresholdbetween the current FUE ki and the FUE kj. If it does not,we will distinguish whether the number N1 of sub-channelsallocated to the FUE kj is more than one by T . If it does, wewill delete the interference sub-channel from the kj. Finally,we repeat it until all the FUEs are tested in current cluster.

(3) Power allocation for FAPsConsidering the characteristic of random deployment and

limited power of FAPs, this paper designs a distributed powercontrol method which demands on low processing capacityfor FAPs, and sets lower interference threshold and calculatesthe interference among FAPs in the same cluster. If there isnot interference between FAP j and FAP i, nj = 0, otherwise,nj = 1. Power allocation is shown as follwing:

pj(t + 1) = qpj(t) (32)

q =

1− ρ, nj = 1,

qpj(t)pj(t)

SINRj > SINRminj

1+ ρ, nj = 0,qpj(t)pj(t)

SINRj < SINRmaxj

1, others(33)

SINRminj and SINRmax

j are the minimum and maximumSINR requirements of FAP j respectively. SINRj is the averageSINR for FAP j. pj(t) is the transmission power of FAP j atthe time t . ρ > 0 is the adjustment granularity about the

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TABLE 3. The simulation parameters.

transmission power. The transmission power of FAP j needsto meet:

pj(t + 1) = min(pj(t + 1), pmaxi ) (34)

V. SIMULATION RESULTSWe assume that there are 16 rooms for one layer in a build-ing, which are uniformly distributed in two scenarios withthe range of 40m×40m (1600m2) or 70m×70m (4900m2),so the area of each room is 100m2 or 306m2. The simula-tion parameters are set in Table 3. We implement the algo-rithms many times (1000-60000 times, see Table 4) undervarious topologies and scenarios, and the algorithms runiteratively each time until the optimal solution is found orit reaches the maximum iterations number (30000 in thispaper). ω−ϕ is a random value within the feasible range. Thesimulation results show no matter what value of ω−ϕ is, theproposed algorithm can always find out the global optimalsolution efficiently compared with those existing algorithmsso as to determine the maximum value of the objectivefunction.

The average data rate is defined

as∑F

i=1∑Nθ=1 0

θki,i1f log2(1+P

θki,iγ θki,i

)

F . The average interfer-ence at FUE kf on the sub-channel n is defined as∑F

i=1 (∑Nθ=1 (

∑Fj=1,j 6=i 0

θkj,j0θki,i

gθki,jPθkj,j

))

NF . The data rate fairness is

defined as FF =(∑K

ki=1φki )

2

K∑K

ki=1(φki )

2 , where φki denotes the data

rate of the FUE ki. K is the total number of FUEs.The others simulation parameters are shown in Tab. 3:Fig 6 shows each objective function value in 1000 times

runs under the case of 6, 8, 10, 12, 14, 16 FAPs. The globaloptimal solution can always be successfully found each timeunder all the cases by the proposed algorithm (red points) butlimited times by the SDP clustering algorithm in [25] (bluepoints) when the numbers of FAPs are 6, 8 10. However,the SDP clustering algorithm cannot find the global optimalsolution when the numbers of FAPs are 12, 14, 16. Thatis because the proposed algorithm simplifies the optimal

FIGURE 6. Objective function value vs. the number of counting.

constraint items and revises the direction of searching for theoptimal solution. So it can accurately narrow the scope ofthe solution space so as to find the optimal solution easily.However, the principle of the SDP clustering algorithm basedon the CVX platform is that the constraint of the originalproblem is flabby, so the scope of the solution space isexpanded. When the upper bound of the original problemis determined, the system will generate a group of randomnumbers obeying the normal distribution so as to search theoptimal solution by the Random Rounding algorithm, whicheasily falls into a local optimum value because of the numberrandomly generated. In addition, the latitude of the RandomRounding algorithm increases exponentiallywith the increaseof the number of FAPs. Therefore, with the increase of thenumber of FAPs, it is hard to obtain the global optimalsolution. In this paper, no matter how many the number ofFAPs is, the proposed algorithm will find the global optimalsolution efficiently.

As shown in Tab. 4, when the numbers of the FAPs are6, 8 and 10, the global optimal solution can be obtained bythe SDP clustering algorithm, but the efficiency is lower thanthat of the proposed algorithm. For example, when there are6 FAPs in the scope of 40m×40m, in 1000 times runs, thetimes of the global optimal solution obtained by the SDPclustering algorithms is 9. However, the proposed algorithmcan always find the optimal solution successfully under all thecases. Also, the global optimal solution can be first obtainedat the 78-th iteration by the SDP clustering algorithms whileat the 2-nd iteration by the proposed algorithm. When thenumber of the FAPs is 8, the global optimal solution is only

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TABLE 4. Comparison of the efficiency of the algorithms.

FIGURE 7. Average data rate vs. the number of FAPs.

obtained once (at the 15000-th iteration) in 30000 timesruns. Similarly, a global optimal solution is obtained once(at the 32496-th iteration) in 40000 times runs. However, theproposed algorithm first obtains the global optimal solutionat the 7-th and 5-th iteration respectively under the above twocases. Also, Tab. 4 shows that the time that the global optimalsolution is obtained by the proposed algorithm is much lessthan that by the SDP clustering algorithm, so the proposedalgorithm is more effective.

Fig. 7 shows the simulation scenario is that the FAPsare uniformly distributed in the scope of 40× 40m2, theinterference threshold is ζ nk = 10−11. We apply the Algo-rithm 3 of [5] to allocate the resources. Because both theSDP clustering algorithm and the proposed algorithm canfind the global optimal solution when the numbers of FAPsare 6-10, the average data rate of them is the same. However,the average data rate obtained by the proposed algorithm ishigher than the SDP clustering algorithm with the increaseof the number of FAPs. That is because SDP clusteringalgorithm can not find the global optimal solution whenthe scope of the solution space is expanded. In addition,under the condition of the uncoordinated scheme in [25], theaverage data rate is the lowest because of the existence ofmore interference. Also, when the number of FAPs increases,the average data rate will reduce, because the interferencesamong the adjacent FAPs are increased.

FIGURE 8. Average interference at an FUE vs. interference threshold.

Fig. 8 show the comparison results of several algorithmsunder two scenarios of 40×40m2 and 70×70m2, includ-ing the proposed clustering algorithm and HSA (HeuristicSub−channel Allocation algorithm in [25]), SDP Clusteringand HSA, the proposed Algorithm 2 (the proposed cluster-ing algorithm and resource allocation algorithm), and theproposed Algorithm 3 (the proposed clustering algorithm,resource allocation algorithm and power allocation algo-rithm). It’s shown that the average interferences at an FUEachieved by the proposed algorithms are lower than that bythe SDP clustering algorithm and HSA algorithm, whichindicates that the proposed clustering algorithm, resourceallocation algorithm and power allocation algorithm all canreduce the interference from inter-FAPs and intra-FAPs moreeffectively than the SDP clustering algorithm and HSA algo-rithm. It is worth mentioning that the average interferencesobtained by the proposed algorithms are much lower thanthat by the SDP clustering algorithm and HSA algorithm. Inaddition, Fig. 8 shows that the larger the distributed range is,the smaller the value of the interference between FUEs willbe in the case of the same number of FAPs. That is becausethe distance between two FUEs increases so that the path lossincreases. Also, the average interference of FUE increaseswith the increase of the interference threshold. However,

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FIGURE 9. The number of FBS with interference vs. interference threshold.

when the interference threshold is increased to the thresholdvalue, the average interference of FUE will not change. Thatis because the larger the interference threshold is, the morethe number of the valid sub-channel is, and the larger theaverage interference is. When the interference threshold isincreased to the threshold value, the interference between theFUEs is less than the interference threshold, so, we don’t needto consider the interference. When the interference thresholdcontinues to be improved, the allocation of the sub-channelswill not be changed. In addition, it is obvious that the inter-ference threshold value is 10−9(W ) in the range of 40×40m2

and 10−10(W ) in the range of 70×70m2. It shows that thelarger the distribution range of the FAPs is, the smaller theinterference threshold value will be.

Similarly, Fig.9 shows that the number of the FAPs whichcause interference by the proposed clustering algorithm andHSA algorithm is also less than that by the SDP clusteringalgorithm and HSA algorithm. So it also further validates thatthe solution obtained by the proposed clustering algorithmis the optimal solution. Also, the number of the FAPs withinterference by the proposed resource allocation algorithm isalso less than that by the HSA algorithm. It indicates thatthe proposed resource allocation algorithm can reduce theinterference effectively and enhance the experience of theFUEs. Furthermore, Fig. 9 shows that the lower the interfer-ence threshold is, the greater the number of the FAPs withinterference will be. If interference threshold is larger thanthe threshold value, the number of the FAPs with interferencewill be zero. We can also see that, the wider the distributedrange of the FAPs is, the fewer the number of FAPs withinterference will be in the same algorithm.

Fig. 10 shows that the data rate fairness of the FUEachieved by the proposed clustering algorithm is higher thanthat by the SDP clustering algorithm with the increase ofthe interference threshold, when the two algorithms bothuse the HSA algorithm to allocate sub-channels in thesame simulation scenario. It further explains the proposed

FIGURE 10. Data rate faireness vs. interference threshold.

clustering algorithm is superior to the SDP clustering algo-rithm. Simultaneously, the simulation results further con-firm the above theoretical proof that the optimal clusteringsolution can be found by the proposed clustering algorithm.The data rate fairness of the FUE achieved by the proposedAlgorithm 2 is always higher than that by the HSA algorithm,when they both use the proposed clustering algorithm tomakethe FAPs into clusters. It shows the proposed sub-channelsallocation algorithm not only can reduce the interference ofFUE, but also take into account the fairness of FUE. Sim-ilarly, Compared with the proposed Algorithm 2, the datarate fairness of the FUE by the proposed Algorithm 3 isimproved further. In addition, the values of the data rate fair-ness increase with the increase of the interference thresholdbut holding constant after reaching the interference thresholdvalue. The reason for this phenomenon is that the numberof the FUEs with interference decreases when the interfer-ence threshold increases, so the number of the available sub-channels increases. As a result, the opportunity that each FUEgets the sub-channel increases.

VI. CONCLUSIONThis paper investigates the interference management andresource allocation problems for two-tier heterogeneous net-works. First, we propose a mathematical modeling idea basedon LINGO which can efficiently solve the joint clusteringproblem for the FAPs. The Branch-and-Bound algorithmand the simplex algorithm are used synthetically to find theoptimal solution by LINGO. In addition, not only does thispaper theoretically prove that the solution obtained by theproposed clustering algorithm is the global optimal solution,but the simulation results have also showed that the proposedalgorithm can obtain the optimal solution compared withother algorithms based on the same clustering optimizationproblems and improve the efficiency of searching for theoptimal solution. Secondly, a FAP is selected as a cluster headthat is responsible for resource allocation among the FAPs

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in that cluster. Finally, we put forward a novel strategy forresource allocation. Compared with other related schemes,the proposed resource allocation algorithm can achievelower interference between FUEs and higher data-ratefairness.

VII. ACKNOWLEDGMENT(Haibo Zhang and Fangwei Li are co-first authors.)

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HAIBO ZHANG received the Ph.D. degreein signal and information processing from theBeijing University of Posts and Telecommunica-tions, Beijing, China, in 2013. From 2015 to 2016,he was a Visiting Scholar with the Departmentof Electrical and Computer Engineering, NorthCarolina State University, Raleigh, NC, USA.He is currently an Associate Professor withthe School of Communication and InformationEngineering, Chongqing University of Posts and

Telecommunications, Chongqing, China. His research interests include wire-less communication network, and wireless resource management.

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H. Zhang et al.: Cluster-Based Resource Allocation for Spectrum-Sharing Femtocell Networks

DINGDE JIANG (S’08–M’09) received the Ph.D.degree in communication and information systemsfrom the University of Electronic Science andTechnology of China, Chengdu, China, in 2009.From 2013 to 2014, he was a Visiting Scholar withthe Department of Computer Science and Engi-neering, University of Minnesota, Minneapolis,MN, USA. He is currently a Professor with theSchool of Computer Science and Engineering,Northeastern University, Shenyang, China. His

research focuses on network measurement, modeling and optimization,performance analysis, network management, network security in commu-nication networks, particularly in software-defined networks, information-centric networking, energy-efficient networks, and cognitive networks.His research was supported by the National Science Foundation of China, theProgram for New Century Excellent Talents in the University of the Ministryof Education of China, and so on. He currently serves as an Editor forone international journal. He has served as a Technical Program CommitteeMember for several international conferences. He has received the Best PaperAwards at International Conferences.

FANGWEI LI received the Ph.D. degree inelectromagnetic fields and waves from theUniversity of Electronic Science and Technologyof China, Chengdu, China, in 2003. He is currentlya Professor with the School of Communication andInformation Engineering, Chongqing Universityof Posts and Telecommunications, Chongqing,China. His current researches include wire-less communication, and mobile communicationsecurity technology.

KAIJIAN LIU received the master’s degreein physical electronics from the KunmingUniversity of Science and Technology, Kunming,China, in 2009. She is currently a Lecturer with theSchool of Optoelectronic Engineering, ChongqingUniversity of Posts and Telecommunications,Chongqing, China. Her research interests includewireless communication network, and wirelessresource management.

HOUBING SONG (M’12–SM’14) received thePh.D. degree in electrical engineering fromthe University of Virginia, Charlottesville, VA,in 2012.

In 2012, he joined the Department of Electricaland Computer Engineering, West Virginia Univer-sity, Montgomery, WV, where he is currently theGolden Bear Scholar, an Assistant Professor, andthe Founding Director with the Security and Opti-mization for Networked Globe Laboratory and the

West Virginia Center of Excellence for Cyber-Physical Systems sponsoredby West Virginia Higher Education Policy Commission. In 2007, he wasan Engineering Research Associate with the Texas A&M TransportationInstitute. He has authored over 100 articles. He is the Editor of fourbooks, including Smart Cities: Foundations, Principles and Applications,(Hoboken, NJ: Wiley, 2017), Security and Privacy in Cyber-Physical Sys-tems: Foundations, Principles and Applications, (Chichester, U.K.: Wiley,2017), Cyber-Physical Systems: Foundations, Principles and Applications,(Waltham, MA: Academic Press, 2016), and Industrial Internet of Things:Cybermanufacturing Systems, (Cham, Switzerland: Springer, 2016). Hisresearch interests include cyber-physical systems, Internet of Things, cloudcomputing, big data analytics, connected vehicle, wireless communicationsand networking, and optical communications and networking.

Dr. Song is a member of ACM. He was the very first recipient of theGolden Bear Scholar Award, the highest faculty research award at the WestVirginia University Institute of Technology, in 2016.

HUAIYU DAI (M’03–SM’09–F’17) received theB.E. and M.S. degrees in electrical engineeringfrom Tsinghua University, Beijing, China, in 1996and 1998, respectively, and the Ph.D. degree inelectrical engineering from Princeton University,Princeton, NJ, in 2002. He was with Bell Labs,Lucent Technologies, Holmdel, NJ, in 2000, andwith AT&T Labs-Research, Middletown, NJ, in2001. He is currently a Professor of Electrical andComputer Engineering with North Carolina State

University, Raleigh. His current research focuses on networked informationprocessing and crosslayer design in wireless networks, cognitive radionetworks, wireless security, associated information-theoretic and computation-theoretic analysis, communication systems and networks, advanced signalprocessing for digital communications, and communication theory and infor-mation theory. He received the Best Paper Awards at 2016 IEEE InfocomBigsecurity Workshop, 2016. He co-chaired the Signal Processing for Com-munications Symposium of the IEEE Globecom 2013, the CommunicationsTheory Symposium of the IEEE ICC 2014, and the Wireless Communica-tions Symposium of the IEEE Globecom 2014. He has served as an Editor ofthe IEEE Transactions on Communications, Signal Processing, andWirelessCommunications. He is also an Area Editor of wireless communicationsfor the IEEE Transactions on Communications. He co-edited two specialissues of EURASIP journals on distributed signal processing techniques forwireless sensor networks, and on multiuser information theory and relatedapplications, respectively.

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