1

Distributed Resource Allocation for OFDMA

Femtocell Networks with Macrocell ProtectionHa Nguyen Vu and Long Bao Le

Abstract—We consider the joint subchannel allocation andpower control problem for OFDMA femtocell networks in thispaper. Specifically, we are interested in the fair resource sharingsolution for users in each femtocell that maximizes the sum minrate of all femtocells subject to protection constraints for theprioritized macro users. Toward this end, we describe the math-ematical formulation for the problem and present an optimalexhaustive search algorithm. Given the exponential complexityof the optimal exhaustive search algorithm, we then develop adistributed and low-complexity algorithm to solve the resourceallocation problem. We prove that the proposed algorithm con-verges. Finally, numerical results are presented to demonstratethe desirable performance of the proposed algorithms.

Index Terms—Femtocell networks, interference management,OFDMA, power control, subchannel assignment.

I. INTRODUCTION

Massive deployment of small cells such as femtocells is

one important solution to fundamentally improve the indoor

throughput and coverage performance of wireless cellular

networks [1, 2]. 4G and beyond cellular networks are based

on Orthogonal Frequency Division Multiple Access (OFDMA)

that provides flexibility in radio resource management and

robustness against adverse effects of multi-path fading [3].

Efficient deployment and operation of OFDMA femtocells,

however, demand to resolve many technical challenges which

range from resource allocation, synchronization [4–6] to pro-

tection of existing macrocells against cross-tier interference

from newly-deployed femtocells. Development of efficient

radio resource management techniques for femtocell networks,

which essentially requires to perform subchannel and power

allocation for femto user equipments (FUEs), is one of the

most critical research issues [7, 8].

There have been some existing works studying the resource

allocation problem for OFDMA-based femtocell networks [9–

12]. In [9], spectrum sharing and access control strategies

were proposed for femtocell networks by using the water-

filling algorithm and game theory technique. The authors of

[10] developed an adaptive femtocell interference management

algorithm comprising three control loops that run continuously

and separately at macro and femto base stations (MBS and

FBS). However, no explicit subchannel assignment was studied

and the maximum power constraint for each user was not

considered in this paper. There are other resource allocation

algorithms in the literature which aim to enhance the energy

efficiency for cognitive femtocells [11] or to achieve the user-

level fairness in open-access femtocells [12]. However, the

The authors are with INRS, University of Quebec, Montreal, QC, Canada.Emails: hanguyen,long.le@emt.inrs.ca.

works [11] and [12] do not provide QoS guarantees (e.g.,

in terms of target SINRs on subchannels) for users in both

network tiers.

In this paper, we study the resource allocation problem

for femtocell networks considering fairness for FUEs in each

femtocell, protection of MUEs, and total power constraints for

FUEs. We explicitly formulate the joint subchannel and power

allocation problem and present an exhaustive search algorithm

to determine its optimal solution. To overcome the exponential

complexity of the centralized exhaustive search algorithm, we

then develop a distributed and low-complexity resource alloca-

tion algorithm. The proposed algorithm iteratively updates the

subchannel assignments at each femtocell based on carefully

designed assignment weights and transmission powers for the

subchannels accordingly. We prove the convergence of the

proposed algorithm and analyze its complexity. Numerical re-

sults are presented to demonstrate the efficacy of the proposed

algorithm and its relative performance compared to the optimal

algorithm.

The remaining of this paper is organized as follows. We

describe the system model and the problem formulation in

Section II. In Section III, we present both optimal exhaustive

search and sub-optimal resource allocation algorithms. Numer-

ical results are presented in Section IV followed by conclusion

in Section V.

II. SYSTEM MODEL

We consider the uplink of the OFDMA-based macrocell-

femtocell network where users of both tiers share the spectrum

comprising N subchannels (SCs). We assume that there are

Mf FUEs served by (K − 1) FBSs, which are underlaid by

one macrocell serving Mm macro user equipments (MUEs).1

Let Mk be the set of user equipments (UEs) in the k-th cell,

i.e., they are served by BS k of the corresponding tier. For

convenience, let Mm , M1 represent the set of MUEs and

Mf , ∪Kk=2Mk = Mm + 1, ...,Mm +Mf denote the set

of all FUEs. In addition, let M and K be the sets of all UEs

and BSs, respectively. Then, we have M = Mm ∪ Mf =1, 2, ...,M and K = 1, 2, ...,K where BS 1 is assumed

to be the MBS.

We assume the fixed BS association for all UEs in the

network, i.e., each UE is served by a fixed BS in the cor-

responding cell. Also, let bi ∈ K denote the BS serving UE

i and let N = 1, 2, ..., N be the set of orthogonal SCs.

We assume there is no interference among transmissions on

1Extensions to the scenario with multiple macrocells is possible althoughwe are only interested in resource allocation for the femto tier in this paper.

978-1-4673-5939-9/13/$31.00 ©2013 IEEE978-1-4673-5939-9/13/$31.00 ©2013 IEEE

2013 IEEE Wireless Communications and Networking Conference (WCNC): MAC2013 IEEE Wireless Communications and Networking Conference (WCNC): MAC

440

2

different SCs. We consider a system with full frequency reuse

where all N SCs are allocated for UEs in all cells of either tier.

To describe the subchannel assignments (SA), let A ∈ RM×N

be the SA matrix for all M UEs over N SCs where

A(i, n) = ani =

1 if SC n is assigned for UE i

0 otherwise.(1)

Let pni represent the transmission power of UE i over SC n in

the uplink where pni ≥ 0. We impose the following constraints

on the total transmission powers

N∑

n=1

pni ≤ P max

i , i ∈ M (2)

where P max

i is the maximum transmission power of UE i.Similar to the SAs, we define P as an M×N power allocation

(PA) matrix where P(i, n) = pni . For convenience, we also

define partitions of SA and PA matrices A and P as follows.

In particular, let Ak,Pk ∈ R|Mk|×N represent the SA and

PA matrices for UEs in cell k over N channels, respectively.

We assume that a SC can be allocated to at most one UE in

any cell. Then, we have∑

i∈Mk

ani ≤ 1, ∀k ∈ K and ∀n ∈ N . (3)

Let hnij and ηni be the channel gain from UE j to BS i and

the noise power at BS i over SC n, respectively. Then, for a

given SA and PA solution, i.e., given A and P, the signal to

interference plus noise ratio (SINR) achieved BS bi due to the

transmission of UE i over the SC n can be written as

Γni (A,P) =

ani hnbiipni

∑

j /∈Mbianj h

nbij

pnj + ηnbi=

ani pni

Ini (A,P)(4)

where Ini (A,P) is the effective interference corresponding to

UE i on the SC n, which is defined as

Ini (A,P) ,

∑

j /∈Mbianj h

nbij

pnj + ηnbi

hnbii

. (5)

We assume that SC allocations for MUEs have been deter-

mined by a certain mechanism and fixed while PA over the

corresponding SCs for MUEs are updated to cope with the

cross-tier interference due to transmissions of FUEs. That

means A1 is fixed while we need to determine Ak, 2 ≤ k ≤ Kand the corresponding PAs. In addition, each MUE wishes

to maintain a predetermined target SINR γni for each of its

assigned SC n, (n ∈ N|a1n = 1). Specifically, we have the

following constraints for the MUEs

Γni (A,P) ≥ γn

i , if ani = 1, ∀i ∈ Mm. (6)

We further assume that each FUE also demands some

predetermined target SINRs for its assigned SCs. Also, we

define a normalized rate of one for any particular SC if the

underlying FUE can achieve its target SINR on this SC as

follows:

rni (A,P) =

0, if Γni (A,P) < γn

i ,

1, if Γni (A,P) ≥ γn

i .(7)

In general, the target SINRs for MUEs and FUEs on different

SCs can be different. However, to impose fairness among UEs

of each tier, we are interested in choosing the same target

SINRs for FUEs and MUEs, which are denoted as γf and

γm, respectively. In addition, the target SINR of FUEs γfwould be chosen larger than γm of MUEs since FUEs can

typically achieve higher transmission rate on each SC due to

its short distance to the associated FBS. Now, we define the

total normalized rate achieved by UE i for given matrices A

and P as

Ri(A,P) =

N∑

n=1

rni (A,P). (8)

To impose the max-min fairness for FUEs associated with the

same FBS, we define the following min-rate for femtocell k

R(k)(A,P) = mini∈Mk

Ri(A,P). (9)

We are now ready to state the resource allocation problem for

FUEs as follows:

max(A,P)

∑

2≤k≤K

R(k)(A,P) (10)

subject to constraints (2), (3), (6) (11)

where the min rate at femtocell k, i.e., R(k)(A,P), is given

in (9). Therefore, the objective of this resource allocation

problem is to maximize the total min rate of all femtocells

subject to user power constraints, SA constraints and protec-

tion constraints for MUEs.

III. OPTIMAL AND SUP-OPTIMAL ALGORITHMS

A. Feasibility of a Subchannel Assignment

The resource allocation problem (10)-(11) involves finding

the joint SA and PA solution. For a certain SA solution

presented by matrix A, we can indeed find its “best” PA

and verify its feasibility with respect to the corresponding

constraints in (11). In particular, we wish to maintain a

normalized rate of one on any SC allocated for any particular

FUE according to (7). In addition, we wish to maintain the

SINR constraints for MUEs in (6). In the following, we study

the feasibility of having a PA solution on a particular SC n to

maintain SINRs constraints Γni (A,P) ≥ γn

i for both MUEs

and FUEs who are allocated this SC.

Toward this end, let Sn = i ∈ M|ani 6= 0 = n1, ..., ncndenote the set of UEs who are assigned SC n and cn = |Sn| be

the number of elements in set Sn. Then, the SINR constraints

for UEs in set Sn are

Γni (A,P) ≥ γn

i , i ∈ Sn (12)

Using the SINR expression in (4), we rewrite these constraints

in matrix form as

(In −GnHn)pn ≥ gn (13)

where gn =[

gnn1, ..., gnncn

]Twith gni =

ηnbiγni

hnbii

, In is

cn × cn identity matrix, Gn = diagγnn1, ..., γn

ncn, pn =

441

3

[pnn1, ..., pnncn

]T and Hn is a cn × cn matrix which defined as

[

Hni,j

]

=

0, if j = i,hnbni

nj

hnbni

ni

, if j 6= i.(14)

According to the Perron-Frobenius theorem, there exits a non-

negative power solution for UEs over the SC n if and only

if the maximum eigenvalue of GnHn, i.e., spectrum radius

ρ(GnHn), is less than 1 [15]. In addition, the Pareto-optimal

power allocation for UEs in Sn (i.e., minimum power vector

in the element-wise sense) can be expressed as [15]

pn =

(In −GnHn)−1gn, |ρ(GnHn)| < 1,

+∞, otherwise.(15)

In addition, this Pareto-optimal PA can be achieved at equi-

librium by the well-known distributed Foschini-Miljanic [15]

power updates

pni (t+ 1) := pni (t)γni

Γni (t)

= Ini (t)γni (16)

where Γni (t) and Ini (t) are the SINR and effective interference

achieved by UE i in iteration t, respectively. Now, suppose

that for the SA solution corresponding to matrix A, we are

able to find finite PA vectors for all SCs that satisfy the

corresponding SINR constraints by using the centralized or

distributed mechanisms given in (15) and (16), respectively.

Then, it is clear that the underlying SA solution is feasible

if and only if finite PA solutions on all SCs exist (i.e., the

spectrum radius ρ(GnHn) is less than 1 as given in (15))

and satisfy the power constraints in (2). This pays the way

for developing optimal exhaustive search algorithm, which is

presented in the following, since the number of possible SAs

is finite.

B. Optimal Algorithm

1) Exhaustive Search Algorithm: Based on the results ob-

tained in section III-A, we can find the optimal solution for the

resource allocation problem of interest (10)-(11) by perform-

ing exhaustive search as follows. For a fixed and feasible A1,

let ΩA be the list of all potential SA solutions that satisfy

the “fairness condition” as∑

n∈N ani =∑

n∈N anj = τk for

all FUEs i, j ∈ Mk. Then, we sort the list ΩA in the

decreasing order of∑K

k=2 τk to obtain the sorted list Ω∗A.

Then, the feasibility of each SA solution in the list Ω∗Acan be verified as being presented in section III-A. Among all

feasible SA solutions, the feasible one achieving the highest

value of the objective function (10) and its corresponding

PA solution given in (15) give the optimal solution of the

optimization problem (10)-(11).

2) Complexity Analysis: The complexity of the exhaustive

search algorithm can be determined by calculating the number

of the elements in the list Ω∗A and the complexity involved

in the feasibility verification for each of them. The number of

the elements in Ω∗A, the number of potential SAs, is the

product of the number of potential SAs for all femtocells,

which satisfy the “fairness condition”. Therefore, the number

of potential SAs can be expressed as

T=K∏

k=2

σk∑

τk=0

Mk−1∏

i=0

CτkN−iτk

=K∏

k=2

σk∑

τk=0

N !

(τk!)Mk (N −Mkτk)!≈ O

(

(N !)(K−1))

,

(17)

where σk = ⌊N/Mk⌋ represents for the largest integer less

than or equal to N/Mk and Cnm = m!

n!(m−n)! denotes the

“m-choose-n” operation. According the section III-A, the

complexity of the feasibility verification for each potential

SA mainly depends on the eigenvalue calculation of the

corresponding matrix and solving the linear system to find

the PA solution for each SC. It requires O(K3) to calculate

the eigenvalues of GnHn [13] and O(K3) to obtain pn

by solving a system of linear equations [14]. Therefore,

the complexity of the optimal exhaustive search algorithm

is O(

K3 ×N × (N !)(K−1))

, which is exponential with the

number of SCs. This is quite expected since the underlying

resource allocation problem is a mixed integer problem, which

is NP-hard.

C. Sub-Optimal and Distributed Algorithm

To resolve the exponential complexity of the centralized

optimal algorithm presented in section III-B, we develop a

low-complexity resource allocation algorithm which can be

implemented in a distributed manner. As defined in (7) and (8),

the normalized rate of each FUE i is equal to the number of

assigned SCs given that the FUE can find feasible PA solutions

on its assigned SCs. In addition, we wish to maximize the

total min rate of all femtocells. Therefore, an efficient resource

allocation algorithm would attempt to assign the maximum and

equal number of SCs to each FUE in each femtocell and to

allocate the transmission powers for FUEs on these assigned

SCs so that they meet the SINR constraints in (7). In addition,

resource allocation for FUEs should be performed in such a

way that the SINR constraints for all MUEs given in (6) can

be maintained.

The proposed distributed resource allocation algorithm is

described in details in Algorithm 1. This algorithm employs

an iterative weight-based SA that is performed in parallel at all

femtocells. The weight for each SC and FUE pair is defined

as the multiplication of the estimated transmission power and

a scaling factor capturing the “quality” of the corresponding

assignment of the SC for the FUE. In addition, the scaling

factor is updated over iterations so that it becomes larger if

the corresponding assignment results in violations of SINR

constraints for FUEs and/or MUEs (i.e., constraints (7) and

(6)).

Let us first describe how each UE i in cell k estimates

the transmission power for SC n in each iteration l of the

algorithm. UE i estimates the effective interference on SC n

at the beginning of iteration l, denoted as In,(l)i , which is given

in (5). Then, it can calculate the required transmission power

on this channel using the Foschini-Miljanic power update rule

given in (16) as follows:

pn,mini = In,(l)i γn

i . (18)

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4

Then, FUE i in cell k update the assignment weight as wni =

χni p

n,mini where the scaling factor χn

i is defined as follows:

χni =

αni , if pn,mini ≤

P max

i

τk

αni θ

ni , if

P max

i

τk< pn,mini ≤ P max

i

αni δ

ni , if P max

i < pn,mini .

(19)

where τk denotes the current min rate at femtocell k; αni is a

factor, which is used to protect SINR constraints of MUEs

(i.e., it is increased if assigning SC n to FUE i tends to

result in violation of the SINR constraint for the corresponding

MUE on that SC); θni , δni are another factors which are set

higher if the assignment of SC n for FUE i tend to require

transmission power, which are larger than the average power

per SC (i.e.,P max

i

τk) and the maximum power budget (i.e., P max

i ),

respectively. We will set δni as δni = µθni where µ = 2 in

Algorithm 1.

Given the weights defined for each FUE i, femtocell k finds

the SA for its FUEs in each iteration by solving the following

optimization problem

minAk

∑

i∈Mk

∑

n∈Nani w

ni

s.t.∑

n∈N ani = τk, ∀i ∈ Mk.(20)

This problem aims to find an assignment matrix Ak for

which each FUE achieves a total normalized rate of τk and

the minimum total weight can be achieved. Therefore, by

assigning larger weight wni for bad SCs, the SA returned by

(20) would be an efficient solution.

The optimization problem (20) can be transformed into the

standard matching problem (say between “jobs” and “employ-

ees”) as follows. Suppose we create τk virtual “employees” for

each FUE in femtocell k then we can consider the matching

problem between τkMk virtual “employees” (virtual FUEs)

and N “jobs” (SCs). In particular, FUE i is equivalent to τkvirtual FUEs i1, ..., iτk. Let the edge vniu (u ∈ 1, ..., τk)

between SC n and virtual FUE iu represent the assignment of

that SC to the corresponding FUE. Then, the weight wniu

of the

edge vniu is equal to wni . After performing this transformation,

the SA solution of the problem (20) can be found by using

the well-known Hungarian algorithm (i.e., Algorithm 14.2.3

given in [16]). After running the Hungarian algorithm, the SA

for femtocell k is updated as follows. If there exits a virtual

FUE iu, u ∈ 1, ..., τk, being matched to SC n, then we have

ani = 1; otherwise, we set ani = 0. For further interpretation

of Algorithm 1, let W(l)k denote the total minimum weight

corresponding to the optimal solution of (20).

Main operations of Algorithm 1 can be summarized as

follows. In steps 3-9, MUEs calculate their transmission

powers based on current effective interference levels. Then,

if the power constraint of any MUE is violated, the MBS will

inform all FBSs which will find the FUE creating the largest

interference for the victim MUE and increase the factor αn∗

i

m∗

i

of this FUE by a factor of 2. In steps 11-18, each femtocell

which has any FUEs’s power constraints being violated in the

previous iteration solves the SA problem (20) with the current

weight values to update its SA solution and decreases its target

min rate τk by one if the value of the total weight W(l)k of

Algorithm 1 DISTRIBUTED RESOURCE ALLOCATION

1: Initialization

• Set p(0)i = 0 for all UE i, i ∈ Mf , feasible A1.

• Set τk = ⌊ N|Mk|

⌋ for all k ∈ 2, ...,K.

• Set αni = θni = 1, µ = 2, ∀i ∈ Mf , n ∈ N .

• Set k = 0, ∀k ∈ K. k > 2.

2: Start of Iteration l:3: Each UE i estimates I

n,(l)i and calculates pn,mini as in (18).

4: For the macrocell: Let βni =

∑

n∈N an,(l)i pn,mini /P max

i .

5: if βni ≤ 1 then

6: Set pn,(l+1)i = a

n,(l)i pn,mini , ∀n ∈ N .

7: else if βni > 1 then

8: Set pn,(l+1)i =

an,(l)i

pn,mini

βni

, ∀n∈N ; n∗i = argmax

n∈N ,cn>1an,(l)i

pn,mini ; m∗i = argmax

m∈Mf

an∗

im p

n∗

im h∗

1m; αn∗

i

m∗

i= 2α

n∗

i

m∗

iand

bm∗

i= 0.

9: end if

10: For each femtocell k ∈ 2, ...,K:

11: if k = 1 then

12: Keep A(l)k = A

(l−1)k

13: else if k = 0 then

14: Define weight edges wniu

between 1, ..., N and

∪i∈Mki1, ..., iτk, as in (19) and run Hungarian al-

gorithm with

wniu

to achieve W(l)k and update A

(l)k .

15: if W(l)k > V

∑

i∈MkP max

i then

16: Set τk := τk − 1.

17: end if

18: end if

19: Let βni =

∑

n∈N an,(l)i pn,mini /P max

i .

20: if βni ≤ 1 then

21: Set pn,(l+1)i = a

n,(l)i pn,mini , ∀n ∈ N and k,i = 1.

22: else if βni > 1 then

23: Set pn,(l+1)i =

an,(l)i

pn,mini

βni

, ∀n ∈ N ; n∗i = argmax

n∈Nan,(l)i

pn,mini ; θn∗

i

i = 2θn∗

i

i and k,i = 0.

24: end if

25: Set k =∏

i∈Mkk,i.

26: Let l := l + 1, return to Step 2 until convergence.

the problem (20) is greater than V∑

i∈MkP max

i where V is

the predetermined factor and V > 1. Finally, in steps 19-25

each FUE who has its SINR constraint satisfied will update

its transmission powers while any FUE who has its SINR

constraint violated will increase its θn∗

i

i parameter by a factor

of 2. Then, we return to step 2 until convergence. Algorithm

1 can be indeed implemented distributively. In fact, except

for steps 3-9, each BS of either tier can collaborate with its

associated UEs to conduct all required tasks in other steps.

Moreover, required tasks in steps 3-9 can be performed by

letting MBS and FBSs coordinate with each other over the

wired backhaul channel.

D. Convergence and Complexity Analysis of Algorithm 1

1) Convergence Analysis: The convergence of Algorithm 1

is stated in the following theorem.

443

5

−1000 −500 0 500 1000−1000

−800

−600

−400

−200

0

200

400

600

800

1000

−1000 −500 0 500 1000−1000

−800

−600

−400

−200

0

200

400

600

800

1000

BSs MUEs FUEs

macrocell

K=3, M1=5, M

2=M

3=2, N=10.

2 groups of 9femtocells

(b) Large Network

K=19, M1=16, M

2=...=M

19=4, N=64.

macrocell

(a) Small Network

2 femtocells

Fig. 1. Macrocell-femtocell networks used in simulation

Theorem 1. Algorithm 1 converges to a feasible solution

(A,P) of the problem (10)-(11) .

Proof: Let us consider the first scenario where there

exists an iteration after which the scaling factors χni for SA

weights given in (19) do not change. Then after this iteration,

the SA solution will be unchanged by Algorithm 1. Indeed,

Algorithm 1 simply updates transmission powers pn,mini for

all UEs on their assigned SCs by using the Foschini-Miljanic

power update rule. Therefore, according to the result in [15]

these power updates converge which implies the convergence

of our proposed algorithm.

Inversely, if the scaling factors χni are still changed over

iterations then we prove that the system will ultimately evolve

into the first scenario discussed above. First, it can be verified

that the power pn,mini given in (18) is lower-bounded by

ηnbi/hnbiiγni . Therefore, if the scaling factors χn

i keep increas-

ing over iterations then the total weight W(l)k returned by the

assignment problem (20) will increase over iterations as well.

Therefore, according steps 15-17 of Algorithm 1, there exist

some femtocells k which decrease their target min rate τk over

iterations. Since initial values of all τk are finite, this process

will terminate after a finite number of iterations. Then, the

system will be in the first scenario discussed above; therefore,

Algorithm 1 converges. This has completed the proof of the

theorem.

2) Complexity Analysis: It is observed that the major

complexity of Algorithm 1 is involved in solving the SA by

using the Hungarian method in step 14. According to Theorem

14.2.4. of [16], the complexity of the Hungarian algorithm is

O(N3). Therefore, the complexity of our proposed algorithm

is O(K ×N3) for each iteration. However, the local SAs can

be performed in parallel at all (K − 1) femtocells. Therefore,

the run-time complexity of our algorithm is O(N3) multiplied

by the number of required iterations, which is quite moderate

according to our simulation results (i.e., tens of iterations).

IV. NUMERICAL RESULTS

We present illustrative numerical results for two different

networks (with a small and large number of femtocells and

UEs) to evaluate the efficacy of our proposed algorithm. The

network setting and UE placement for our simulations are

illustrated in Fig. 1, where MUEs and FUEs are randomly lo-

cated inside circles of radii of r1 = 1000m and r2 = 30m, re-

10 15 20 25 30 354

5

6

7

8

9

10

Target SINR of FUEs (dB)

Tota

l fe

mto

cell

rat

e

Optimal

Supoptimal with V=2

Suboptimal with V=4

Fig. 2. Sum femtocell rate for small network with optimal and sup-optimalalgorithms where γm=10 dB, Wl=5 dB, P max

m =P max

f=0.01W .

0

2

4

6

τk

1 2 3 4 5 6 7 8 9 100

100

200

300

400

Iteration index

Aver

age

SIN

R

Macrocell M1

Femtocell M2

Femtocell M3

Fig. 3. Average SINRs and minimum number of SCs assigned for UEs incells where γm = 10 dB, γf = 25 dB, Wl = 5 dB, P max

m = P max

f=

0.01W .

spectively. The channel gains hnij are generated by considering

both Rayleigh fading, which is represented by an exponentially

distributed random variable with the mean value of one, and

the path loss Lij = Ailog10(dij)+Bi+Clog10(fc5 )+Wlnij ,

where dij is the distance from UE j to BS i; (Ai, Bi) are

set as (36, 40) and (35, 35) for MBS and FBSs, respectively;

C = 20, fc = 2.5 GHz; Wl is the wall-loss parameter, nij

is the number of walls between BS i and UE j. The noise

power is set as ηi = 10−13W , ∀i ∈ K. Each simulation result

is obtained by taking the average of 20 different runs where in

each run, UEs are randomly located and a feasible SA matrix

A1 is chosen so that each MUE is assigned NM1

channels.

In Fig. 2, we show the total femtocell rate versus the target

SINR γf of FUEs for the small network where the results

corresponding to both optimal and sup-optimal algorithms are

presented. As can be seen, for the low values of γf , the sup-

optimal algorithm can achieve almost the same rate as the

optimal one while for higher values of γf , the sup-optimal

achieve just slightly lower rate than that due to the optimal

one. Moreover, when we increase the value of parameter V , a

slightly better performance can be achieved. We then plot the

average SINRs over assigned SCs and the minimum number

of SCs assigned for FUEs in each femtocell over the iterations

for the small network in Fig. 3. This figure illustrates the

convergence of the proposed Algorithm 1 in terms of both

user SINR and number of assigned SCs per FUE.

444

6

10 15 20 25 30

50

100

150

200

250

300

Target SINR of FUEs (dB)

Tota

l fe

mto

cell

rat

e

SINRm

=10dB, Wl=5dB

SINRm

=14dB, Wl=5dB

SINRm

=18dB, Wl=5dB

SINRm

=10dB, Wl=13dB

SINRm

=14dB, Wl=13dB

SINRm

=18dB, Wl=13dB

Fig. 4. Sum femtocell rate versus the target SINR of FUEs for the largenetwork where P max

m = P max

f= 0.01W .

10−3

10−2

10−1

100

0

50

100

150

180

Maximum power constraint of each FUE (Pf

max)

Tota

l fe

mto

cell

rat

e

Target SINR of 16 MUEs = 10dB

Target SINR of 16 MUEs = 12dB

Target SINR of 32 MUEs = 10dB

Target SINR of 32 MUEs = 12dB

M1=16

M1=32

Fig. 5. Sum femtocell rate versus P max

ffor the large network where γf =

30 dB, Wl = 5 dB, P max

m = 0.01W .

10−3

10−2

10−1

100

30

40

50

60

70

80

Maximum power constraint of each MUE (Pm

max)

Tota

l fe

mto

cell

rat

e

Target SINR of 16 MUEs = 10dB

Target SINR of 16 MUEs = 14dB

Target SINR of 32 MUEs = 10dB

Target SINR of 32 MUEs = 14dB

M1=32

M1=16

Fig. 6. Sum femtocell rate versus P max

m for the large network where γf =

30 dB, Wl = 5 dB, P max

f= 0.01W .

Fig. 4 presents the total femtocell rate versus the target

SINR of FUEs which is achieved by running the sup-optimal

for the large network. This figure shows that the sum femtocell

rate decreases when the target SINRs of FUEs and/or MUEs

increases. This is due to the fact that when the target SINR

increases, UEs need more power to maintain the target SINRs,

which also produce more interference in the network. This

may force FUEs to achieve lower rates since the power

constraints are more likely to be violated. Moreover, the total

sum femtocell rate increases with the increase of wall loss

value Wl. This is because for the higher value of Wl, the wall

can better shield the interference among FUEs and MUEs,

which enables FUEs to utilize more SCs.

In Fig. 5 and 6, we illustrate the total femtocell rate versus

the maximum power of FUEs (P max

f ) and MUEs (P max

m ),

respectively for different target SINR of MUEs (the target

SINR of FUEs is 30dB). These figures show that the total

femtocell rate increases with the increases of maximum power

budget, P max

f or P max

m . In addition, when the number of MUEs

increases, the number of SCs assigned for each MUE (equalNM1

) decreases; however, the total femtocell rate increases

thanks to the better diversity gain offered by the macro tier.

V. CONCLUSION

In this paper, we develop distributed resource allocation

algorithm that maximizes the total min rate of femtocells while

ensuring user fairness and protecting QoS of MUEs. Numer-

ical results are then presented to demonstrate the efficacy of

the proposed algorithm.

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