1402 IEEE COMMUNICATIONS LETTERS, VOL. 19, NO. 8, AUGUST 2015

Power Control for Interference Management andQoS Guarantee in Heterogeneous Networks

Tai Manh Ho, Student Member, IEEE, Nguyen H. Tran, Member, IEEE, Cuong T. Do,S. M. Ahsan Kazmi, Eui-Nam Huh, Member, IEEE, and Choong Seon Hong, Senior Member, IEEE

Abstract—We consider the sum-rate optimization problem withpower control for uplink transmission in a heterogeneous network(HetNet) consisting of a macrocell and multiple femtocells. The con-sidered problem includes the HetNet’s crucial constraints of bothcross-tier interference protection and user QoS in terms of outageprobability and average delay. We transform the original non-convex problem into a convex problem and develop a distributedalgorithm that can attain the global optimal transmit power val-ues. This algorithm, however, has heavy network overheads, whichmay lead to increased energy consumption for femtocell userequipment. We propose a new practical near-optimal distributedalgorithm that eliminates these network overheads. Numericalresults show that the schemes have nearly identical performance.

Index Terms—Power control, femtocell networks, interferencemanagement, optimization theory.

I. INTRODUCTION

S PECTRUM-SHARING between a macrocell and femto-cells in heterogeneous networks (HetNets) can signifi-

cantly reduce the network QoS due to cross-tier and co-tierinterference [1]. Consequently, the interference management(IM) in HetNets has become a popular research area. IM usingpower control (PC) to guarantee QoS for HetNets has beenextensively investigated in the literature [2]–[4].

In [2], PC schemes for two-tier femtocell networks have beenproposed. However, under the densely deployed scenario, theco-tier interference between femtocells cannot be neglected;and there is no global optimal PC strategy for the proposedschemes. In addition, that paper only considers the macro basestation (MBS) protection but does not guarantee the QoS forFUEs. The authors in [3] developed a hybrid PC scheme tosupport different kinds of users with differentiated QoS targets.However, the authors do not explicitly mention the MBS protec-tion mechanism in the case of high effective interference. Theauthors in [4] demonstrate increased performance by proposingan IM scheme that reduces the computation and overhead;however, the solution is sub-optimal and the network QoSguarantee is not considered.

In order to fill these gaps of previous works, we proposetwo distributed algorithms that can achieve: i) global optimaltransmit powers; ii) MBS interference protection; and iii) QoSguarantee for all FUEs. Our main contributions can be summa-rized as follows

• The first proposed algorithm can attain the global optimaltransmit powers using a dual-based algorithm. However,

Manuscript received March 2, 2015; accepted May 28, 2015. Date ofpublication June 12, 2015; date of current version August 10, 2015. This workwas supported by the Basic Science Research Program through the NationalResearch Foundation of Korea funded by the Ministry of Education under GrantNRF-2014R1A2A2A01005900. The associate editor coordinating the review ofthis paper and approving it for publication was J. Lee. (Corresponding author:Choong Seon Hong.)

The authors are with the Department of Computer Engineering, KyungHee University, Yongin 446-701, Korea (e-mail: hmtai@khu.ac.kr; nguyenth@khu.ac.kr; dtcuong@khu.ac.kr; ahsankazmi@khu.ac.kr; johnhuh@khu.ac.kr;cshong@khu.ac.kr).

Digital Object Identifier 10.1109/LCOMM.2015.2444844

Fig. 1. System model.

in order to update the dual variables and transmit powers,this scheme requires a large message exchange, whichincreases the network overhead and convergence time,resulting in additional energy consumption of FUEs.

• To alleviate the first algorithm’s overhead-induced draw-backs, we propose a second algorithm that achieves globaloptimal transmit powers of the approximation problemand is near-optimal but practically implementable.

• Simulation results show that the two algorithms havenearly identical performance and outperform the conven-tional works.

II. SYSTEM MODEL

A. Network Model

We consider a two-tier HetNet consisting a single centralMBS and a set of N femtocells implementing spectrum sharing.There is one femtocell access point (FAP) providing servicefor several FUEs in each femtocell. However, for analyticaltractability, we assume that at any given frequency band andtime period, there is only one active FUE in each femtocell.Even though we only consider complete channel sharing model,the partial channel sharing mode can be an interesting extensionin future.

We model the uplink transmission for a given time slot asshown in Fig. 1, where hn,n and hm,n are the channel powergains of the links between FUE n and FAP n and betweenFUE n and FAP m, respectively, and gn is the channel powergain between FUE n and the MBS. P = [P1, P2, . . . , PN] is thevector of power levels for FUEs, and Pmin

n ≤ Pn ≤ Pmaxn , ∀ n.

B. Problem Formulation

FUE Transmission Rate: The instantaneous signal-to-interference-noise-ratio (SINR) of FUE n at FAP n is

γn(P) = Pnhn,nFn,n∑m�=n Pmhn,mFn,m + σ 2

n, (1)

where Fn,m models the fast-fading channel from the FUE m toFAP n, and σ 2

n is the background noise at FAP n. The additionalinterference from the macrocell users can be absorbed into the

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HO et al.: POWER CONTROL FOR INTERFERENCE MANAGEMENT AND QoS GUARANTEE IN HetNets 1403

background noise. Employing a Rayleigh fading model, weassume Fn,m is i.i.d. exponentially distributed with unit mean.Over the considered time slot, hn,n and hm,n are assumedconstant [2]–[4]. The certainty-equivalent SINR [5] definedas follows:

γ̄n(P)= E[Pnhn,nFn,n]

E

[ ∑m�=n

Pmhn,mFn,m+σ 2

] = Pnhn,n∑m�=n

Pmhn,m+σ 2. (2)

The transmission rate Rn of FUE n can be modeled asRn(P) = W log(1 + Kγ̄n(P)), where W is the baseband band-width and K is a constant depending on modulation, codingscheme and bit-error rate (BER). Unless otherwise stated, weassume W = K = 1 without loss of generality.

The objective of the femtocell network is to maximize asystem-wide efficiency metric, e.g., the total system through-put. Here, we assume that system operates in a high SINRregime, i.e., SINR is much larger than 1; thus, the data ratecan be approximated as Rn = log(γ̄n(P)). This approximationis reasonable when the signal level is much higher than theinterference level. Therefore, in the high SINR regime, theaggregate data rate for the femtocell networks can be writtenas follows:

Rsum(P) =∑

n

Rn(P) = log

[∏n

(γ̄n(P))

]. (3)

FUE QoS: In wireless networks, one of the most importantQoS parameter for reliable communication is outage probabil-ity. A channel outage is declared (e.g., packets lost) when thereceived SINR falls below a given threshold SINR γ th, oftencomputed from the BER requirement. In this letter, we chooseRayleigh fading channel [5], [6] which can give a close-form ofoutage probability constraints to our problem as follows

Pr[γ̄n ≤ γ th

n

]=1 − exp

(− σ 2

n γ thn

Pnhn,n

) ∏m�=n

(1+γ th

nPmhn,m

Pnhn,n

)−1

.

(4)Another important QoS design consideration for wireless

communication is average delay. At the FUE n, the receivedpackets are first buffered in a queue and then transmitted at arate Rn. A FIFO queuing discipline is used here for simplicity.The packet arrival process of FUE n is assumed to be Poissonwith parameter λn and have an exponentially distributed lengthwith parameter πn. Using the model of an M/M/1 queue [7],the probability of FUE n having a backlog of Nn = k packets totransmit is well-known to be Pr[Nn = k] = (1 − ρ)ρk, whereρ = λn/πnRn(P), and the expected delay at FUE n is D̄n(P) =1/(πnRn(P)) − λn.

MBS Interference Protection: We assume that the maximumtolerable interference at MBS is I, i.e., the interference con-straint is used to assure that the aggregate interference from allFUEs to the MBS is less than I. Mathematically, this constraintcan be written as

∑n gnPn ≤ I.

We next consider the following sum-rate optimization prob-lem, taking into account constraints on maximum tolerableinterference, outage probability, and expected delay:

maximizeP∈P

Rsum(P)

subject toN∑

n=1

gnPn ≤ I,

D̄n(P) ≤ D̄maxn , ∀ n,

Pr[γ̄n ≤ γ th

n

]≤ ξn, ∀ n,

P = {Pn, n ∈ F |Pmin

n ≤ Pn ≤ Pmaxn

}, (5)

where D̄maxn and ξn ∈ (0, 1) are the upper bound of expected

delay and the outage probability threshold of FUE n, respec-tively. Problem (5) is generally a nonconvex problem. In thenext section, we first transform (5) into an equivalent convexproblem and then propose a distributed algorithm to achievethe global optimal transmit powers.

III. OPTIMAL ALGORITHM

A. Equivalent Convex Formulation

We define a new variable yn = log Pn and a new setP̂ = {yn, n ∈ F | log Pmin

n ≤ yn ≤ log Pmaxn }; thus, Pn = eyn . We

also introduce an auxiliary variable {Zn} to show that everyFUE n has the capability to estimate the interference eZn =∑

m�=n Pmhn,m. We transform problem (5) into the followingequivalent non-linear programming problem

minimizey∈P̂,Zn

N∑n=1

log

(e−yn

hn,n

(eZn + σ 2

n

))

s.t.N∑

n=1

gneyn − I ≤ 0,

log

(e−yn

hn,n

(eZn + σ 2

n

))− �n ≤ 0, ∀ n,

∑m�=n

log

(1+ eym−yn

γ thn hn,m

hn,n

)≤ log n(e

yn), ∀ n,

log

⎛⎝∑

m�=n

hn,meym

⎞⎠ − Zn ≤ 0, ∀ n, (6)

where �n = −1πn

(1

D̄maxn

+ λn

), and n(Pn) = exp(−σ 2

n γ thn /Pnhn,n)

1−ξn.

We further assume that γ thn and ξn are chosen such that there ex-

ist feasible points in problem (6). By employing admission con-trol or relaxing QoS constraints, we can cope with the issue ofnon-existent feasible points, which is reserved for future work.Problem (6) is a convex problem and at the optimal solution,the last constraint is active, i.e., log

(∑m�=n hn,meym

)= Zn.

The Lagrangian form of (6) can be decomposed into N sub-problems as follows:

L ({yn}, {Zn}, λ, μ, ν, ζ ) =N∑

n=1

Ln(yn, Zn, λ, μn, νn, ζn), (7)

where λ,μn, νn and ζn are Lagrange multipliers that representinterference price, delay price, outage price, and consistencyprice, respectively, and

Ln(yn, Zn, λ, μn, νn, ζn) = log

(e−yn

hn,n

(eZn + σ 2

n

))+ λgneyn

+ ζn log

⎛⎝∑

m�=n

hn,meym

⎞⎠− ζnZn + μn log

(e−yn

hn,n

(eZn + σ 2

n

))

+∑m�=n

νm log

(1 + eyn−ym

γ thm hm,n

hm,m

)− νn log n(e

yn). (8)

The dual problem is then given as

max D(λ, μ, ν, ζ )

s.t. λ,μ, ν, ζ ≥ 0, (9)

where D(λ, μ, ν, ζ ) = minyn,Zn

L({yn}, {Zn}, λ, μ, ν, ζ ) is the dual

function. Problem (6) is convex; hence, there exists a strictlyfeasible point so Slater’s condition holds, leading to strong

1404 IEEE COMMUNICATIONS LETTERS, VOL. 19, NO. 8, AUGUST 2015

duality [8]. This allows us to solve the primal (6) via the dual(9). The dual problem (9) can be solved using the sub-gradientmethod, which updates the Lagrange multipliers as follows:

λ(t+1) =[λ(t) + κ

(t)λ

(N∑

n=1

gnP(t)n − I

)]+, (10)

μ(t+1)n =

[μ(t)

n + κ(t)μ

(log

(eZ(t)

n + σ 2n

P(t)n hn,n

)− �n

)]+, (11)

ν(t+1)n =

⎡⎣ν(t)

n + κ(t)ν

⎛⎝∑

m�=n

log

(1 + γ th

nP(t)

m hn,m

P(t)n hn,n

)

− logn

(P(t)

n

) ⎞⎠⎤⎦

+, (12)

ζ (t+1)n =

⎡⎣ζ (t)

n + κ(t)ζ

⎛⎝log

⎛⎝∑

m�=n

P(t)m hn,m

⎞⎠ − Z(t)

n

⎞⎠⎤⎦

+, (13)

where κ(t)λ , κ

(t)μ , κ

(t)ν , and κ

(t)ζ are positive step sizes, and

[X]+ = max{X, 0}. Based on the KKT condition [8], the opti-mal transmit power {Pn} of each FUE n can be obtained through∂Ln(yn,Zn,λ,μn,νn,ζn)

∂yn= 0 as follows:

P(t+1)n =ey(t+1)

n =⎡⎢⎣ 1+μ

(t)n +ν

(t)n �

(t)n

σ 2n

log(1−ξn)

λ(t)gn+∑m�=n ν

(t)m

�(t)m hm,n

1+P(t)n hm,n�

(t)m

⎤⎥⎦

Pmaxn

Pminn

, (14)

where �(t)n = γ th

n /(P(t)n hn,n). The auxiliary variable Zn can be

achieved according to the KKT condition

eZ(t+1)n =

[ζ

(t)n σ 2

n

1 + μ(t)n − ζ

(t)n

]+. (15)

B. Optimal Distributed Algorithm

Based on above optimization analysis, we present the opti-mal distributed power control algorithm

Algorithm 1 Optimal Distributed Power Control

Initialization• Set t = 0, P(0)

n be any feasible point in feasible setPmin

n ≤ P(0)n ≤ Pmax

n ;• Set λ(0) ≥ 0, μ(0) ≥ 0, ν(0) ≥ 0, ζ (0) ≥ 0;• Set step size κ

(t)λ , κ

(t)μ , κ

(t)ν , κ

(t)ζ > 0;

Algorithm at FAP n1) Measure the interference

∑m�=n P(t)

m hn,m generated by

all other FUEs and the noise power level σ 2n ;

2) Calculate eZ(t+1)n according to (15);

3) Update the delay price μ(t+1)n and consistency price

ζ(t+1)n according to (11) and (13), respectively;

4) Transmit μ(t+1)n , hn,n to FUE n;

Algorithm at FUE n1) Estimate the channel gain gn and receive {gm, Pm}m�=n

to calculate the total interference at the MBS; Receiveμ

(t+1)n and ζ

(t+1)n , and estimate {hm,n}m�=n;

2) Update the interference price λ(t+1) and the outageprice ν

(t+1)n according to (10) and (12), respectively;

3) Receive {ν(t+1)m ,�

(t+1)m }m�=n, and then calculate the

power value according to (14);4) Broadcast gn, P(t+1)

n , ν(t+1)n , and �

(t+1)n ;

Remark 1:1) The convergence of Algorithm 1 can be proved using a

gradient-based standard technique [8].2) To update transmit power, each FUE n receives control

messages from other FUEs and estimates the channelgain between itself and other FAPs m, i.e., hm,n, using atraining sequence. In practical mobile wireless networks,hm,n are stochastic rather than deterministic, and path lossestimations can be inaccurate.

3) The delay price update (11) needs only local information.The outage price update (12) requires the FAP n to mea-sure interference separately from others FUEs P(t)

m hn,m orthe channel gain hn,m, in general, hn,m �= hm,n.

4) In Algorithm 1, the messages broadcast by FAPs andFUEs contain much individual information, causing in-cremental overhead and large convergence time.

IV. NEAR-OPTIMAL ALGORITHM

In this section, we eliminate the impractical issues of in-terference estimation and message passing in Algorithm 1 byproposing a near-optimal scheme.

In order to do that, we first relax the outage constraints of (6).We apply the upper and lower bounds on the outage probabilityderived in [5] to the outage constraint as Pr

[γn ≤ γ th

n

] ≤ 1 −exp

(− γ th

nγ̄n

)≤ ξn and γ th

nγ̄n+γ th

n≤ Pr

[γn ≤ γ th

n

] ≤ ξn, which cor-

respond to the SINR constraints γ̄n ≥ − γ thn

log(1−ξn)(upper bound)

and γ̄n ≥ γ thn

(1ξn

− 1)

(lower bound), respectively. Hence, the

outage constraints of problem (6) can be approximately re-placed by γ̄n ≤ ηn, where ηn is either of those two constants.

Next, we relax the interference constraints of (6). InAlgorithm 1, to achieve the interference information at MBS,each FUE has to broadcast the channel gain between itself andMBS with a transmit power value to all other FUEs. The broad-cast information increases the message exchange burden. Toovercome this drawback, we divide the maximum interferencelevel I into N equal parts. Each FUE assures that its interferenceto MBS does not exceed I/N.

Then, we have a new optimization problem

minimizey∈P̂,Zn

N∑n=1

log

(e−yn

hn,n

(eZn + σ 2

n

))s.t. gneyn ≤ I/N, ∀ n,

log

(e−yn

hn,n

(eZn + σ 2

n

))− �n ≤ 0, ∀ n,

−log γ̄n ≤ − log ηn, ∀ n,

log

⎛⎝∑

m�=n

hn,meym

⎞⎠ − Zn ≤ 0, ∀ n. (16)

This problem is also a convex problem. Using the same approachas in Section III, Lagrange multipliers are updated as follows

λ(t+1)n =

[λ(t)

n + κ(t)λ

(gnP(t)

n − I/N)]+

, (17)

ν(t+1)n =

[ν(t)

n + κ(t)ν

(log

(eZ(t)

n + σ 2n

P(t)n hn,n

)+ log ηn

)]+. (18)

Similarly, based on above approximation, we proposeAlgorithm 2 to obtain the global optimal solution for (16).

HO et al.: POWER CONTROL FOR INTERFERENCE MANAGEMENT AND QoS GUARANTEE IN HetNets 1405

Algorithm 2 Near-Optimal Distributed Power Control

InitializationAlgorithm at FAP n

1) Measure the interference∑

m�=n P(t)m hn,m;

2) Calculate

eZ(t+1)n =

[ζ

(t)n σ 2

n

1 + μ(t)n + ν

(t)n − ζ

(t)n

]+; (19)

3) Update μ(t+1)n , ν(t+1)

n , and ζ(t+1)n according to (11), (18),

and (13), respectively;Algorithm at FUE n

1) Estimate gn and update λ(t+1)n according to (17);

2) Receive μ(t+1)n , ν

(t+1)n , calculate the power value

P(t+1)n =

[1 + μ

(t)n + ν

(t)n

λ(t)n gn

]Pmaxn

Pminn

; (20)

Remark 2:

1) In Algorithm 2, each FAP n only needs to broadcast theconsistency price, and FUE does not need to broadcastany individual information.

2) The interference price (17) and outage price update (18)require only local information. The power update (20)only requires the consistency price from other FAPs butno individual information from other FUEs.

3) Algorithm 2 converges to the global optimal point of (16),which is an approximated problem of (6). Hence, theoptimal solution of (16) is considered as a near-optimalsolution to (6) (due to the tightness of the bounds).

V. NUMERICAL RESULTS

We consider a cellular network with one MBS with a cov-erage radius of r1 = 500 m and three femtocells distributedrandomly inside. The distances between the three FUEs andthe MBS are 50 m, 100 m, and 500 m, respectively. Thebaseband bandwidth W is set to 32 kHz, and we use K =−1.5/ log(5BER) with BER = 10−3 for MQAM modulation.Pmax and Pmin are set to 1 mW and 0 mW, respectively.Background noise is assumed to be 0.005 dB. The slow-fadingchannel gain is assumed to be h(d) = h0(

d100 )−4, where h0 is a

reference channel gain at a distance of 100 m. Packet traffic ateach FUE has intensity λn = 200 packets/s, and packet lengthis 100 bits [6]. The respective outage probability thresholds ξnof the three FUEs are [0.2, 0.1, 0.1]. The SINR thresholds γ th

nare [30, 20, 10] dB. We set the delay bounds as [0.01, 0.01,0.01] seconds. The maximum interference level I = 0.05 dB.

Fig. 2 illustrates the FUE power convergence of two al-gorithms. The numerical results show that the near-optimalscheme converges faster than the optimal scheme. We observethat the near-optimal algorithm converges close to optimalvalues. Moreover, FUE3 is the farthest from MBS; hence, it cantransmit with the highest power.

Fig. 3 shows the FUE throughput for different algorithms:target-SINR-tracking power control (TPC) or Foschini-Miljanicalgorithm [9], and opportunistic power control (OPC) [10].Our proposals outperform the TPC and OPC schemes and canachieve 14.7% and 2% higher throughput than that of TPCand OPC, respectively. This is due to the fact that in the TPCscheme, all users reach their target SINRs without optimizingtransmission power. In contrast, the OPC scheme results in avery different SINRs at the equilibrium where the strongest user(low interference) attains high SINR while the weak ones (high

Fig. 2. Power convergence: (a) O-DPC, (b) N-O-DPC.

Fig. 3. Aggregate throughput in comparison with other algorithms.

interference) achieve low SINRs. Therefore, the OPC schemetends to allocate more power to strong users to achieve highoverall throughput.

VI. CONCLUSION

We have designed two distributed algorithms to solvethe power control problem for interference management andnetwork QoS guarantee in two-tier heterogeneous networks.The first algorithm is the optimal scheme but is impractical.The second design with a small-size control message is anear-optimal scheme based on a tight bound approximationon outage probability and partitioning of the maximum inter-ference level. Numerical experiments show that the schemes’performances are almost indistinguishable and outperform theconventional works.

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