12
4158 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 64, NO. 9, SEPTEMBER 2015 Sum-Rate Maximization for Multicell OFDMA Systems Sung-Yeon Kim, Jeong-Ahn Kwon, and Jang-Won Lee, Senior Member, IEEE Abstract—Recently, joint subchannel allocation and (transmis- sion) power control problems for multicell orthogonal frequency- division multiple access (OFDMA) systems have been actively studied. However, since the problems are notoriously difficult and complex, only heuristic approaches are mainly used to study them, instead of the optimal approach for achieving maximum system capacity. In this paper, we study this problem from the viewpoint of optimal subchannel allocation and power control, aiming at maximizing the sum rate of the multicell OFDMA system. By using a monotonic optimization approach, we develop an algorithm for the optimal subchannel allocation and power control that achieves the maximum sum rate of the system. In addition, we also develop an algorithm that provides both upper and lower bounds on the maximum sum rate of the system with lower computational com- plexity. To evaluate the tightness of the upper and lower bounds, we also study the conditions when the two bounds are close to each other so that they can be good approximations to the maximum sum rate of the system. Through numerical results, we show that the bounds provide good approximations to the maximum sum rate of the multicell OFDMA system in most cases. Index Terms—Multicell systems, orthogonal frequency-division multiple access (OFDMA), power control, subchannel allocation, sum-rate maximization. I. I NTRODUCTION O RTHOGONAL frequency-division multiple access (OFDMA) is a deployed multiple-access scheme for diverse wireless systems, and efficient resource allocation for the OFDMA system is one of the most important research issues over the last decade. In particular, in the OFDMA system, subchannel allocation and (transmission) power control can be jointly considered to improve the system performance, and this problem has been extensively studied with various system models and various approaches. In [1]–[9], joint subchannel allocation and power control problems for the single-cell OFDMA system are studied. In Manuscript received August 6, 2013; revised May 14, 2014 and July 28, 2014; accepted October 8, 2014. Date of publication October 16, 2014; date of current version September 15, 2015. This work was supported in part by the Mid-career Researcher Program through the National Research Foundation under Grant 2013R1A2A2A01069053 funded by the Ministry of Science, ICT, and Future Planning (MSIP), Korea, and in part by the ICT R&D Program of MSIP/IITP, under Grant 14-823-04-002, Development of LTE-A Based Small Base-Station Technology Supporting Single RF Multiple Streams. This paper was presented in part at the 11th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks, Tsukuba Science City, Japan, May 13–17, 2013, and the Sixth International Conference on Ubiquitous and Future Networks, Shanghai, China, July 8–11, 2014. The review of this paper was coordinated by Prof. R. Jantti. The authors are with the Department of Electrical and Electronic Engineer- ing, Yonsei University, Seoul 120-749, Korea. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TVT.2014.2363476 the single-cell OFDMA system, since there is no intra- and intercell interferences, we do not have to consider interference among users, which makes solving resource allocation prob- lems relatively simple. Hence, the optimal resource allocation that achieves the maximum system capacity can be obtained in the single-cell OFDMA system. However, if we consider a multicell environment, which is a more realistic scenario, despite of no intracell interference by using OFDMA, we should deal with intercell interference among users in different cells, which makes the structure of the problem for the multicell system totally different from that for the single-cell system. In fact, the problem for the multicell system is much more difficult and complex than that for the single-cell system. Moreover, in general, it is difficult to directly apply resource allocation algorithms for the single-cell OFDMA system to the multicell OFDMA system. Recently, various joint subchannel allocation and power control problems in the multicell OFDMA system have been studied in [10]–[28]. In [10]–[22], a joint subchannel alloca- tion and power control problem is divided into a subchannel allocation subproblem and a power control subproblem, and heuristic algorithms to solve the two subproblems are proposed. Although various heuristics to solve the problem are used, they do not guarantee achieving the optimality of algorithms that provides maximum system capacity. In [23]–[26], game- theoretic approaches are used to jointly consider subchannel allocation and power control in the multicell OFDMA system. However, in general, game-theoretic approaches do not also guarantee achieving the optimal resource allocation that pro- vides maximum system capacity. In [27] and [28], an algorithm that achieves the optimal joint subchannel allocation and power control in the two-cell OFDMA system is developed. However, the approach in [27] and [28] is tailored only to the two-cell system, and it is hard to generalize to the general multicell OFDMA system. Despite the aforementioned extensive studies, to our best knowledge, the optimal solution for the joint subchannel allo- cation and power control problem in the multicell OFDMA sys- tem and its maximum system capacity have not been achieved yet. Hence, in the previous works for the multicell OFDMA system, the performance of the proposed algorithm is compared with that of relatively simple heuristic algorithms, such as random subchannel allocation, fixed subchannel allocation, and equal power control, without showing how well its performance approximates that of the optimal algorithm. In this paper, we study a joint subchannel allocation and power control problem that aims at maximizing the sum rate of the downlink in the multicell OFDMA system. This problem is 0018-9545 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

Sum-Rate Maximization for Multicell OFDMA Systems

  • Upload
    others

  • View
    6

  • Download
    0

Embed Size (px)

Citation preview

Page 1: Sum-Rate Maximization for Multicell OFDMA Systems

4158 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 64, NO. 9, SEPTEMBER 2015

Sum-Rate Maximization forMulticell OFDMA Systems

Sung-Yeon Kim, Jeong-Ahn Kwon, and Jang-Won Lee, Senior Member, IEEE

Abstract—Recently, joint subchannel allocation and (transmis-sion) power control problems for multicell orthogonal frequency-division multiple access (OFDMA) systems have been activelystudied. However, since the problems are notoriously difficult andcomplex, only heuristic approaches are mainly used to study them,instead of the optimal approach for achieving maximum systemcapacity. In this paper, we study this problem from the viewpointof optimal subchannel allocation and power control, aiming atmaximizing the sum rate of the multicell OFDMA system. By usinga monotonic optimization approach, we develop an algorithm forthe optimal subchannel allocation and power control that achievesthe maximum sum rate of the system. In addition, we also developan algorithm that provides both upper and lower bounds on themaximum sum rate of the system with lower computational com-plexity. To evaluate the tightness of the upper and lower bounds,we also study the conditions when the two bounds are close to eachother so that they can be good approximations to the maximumsum rate of the system. Through numerical results, we show thatthe bounds provide good approximations to the maximum sumrate of the multicell OFDMA system in most cases.

Index Terms—Multicell systems, orthogonal frequency-divisionmultiple access (OFDMA), power control, subchannel allocation,sum-rate maximization.

I. INTRODUCTION

O RTHOGONAL frequency-division multiple access(OFDMA) is a deployed multiple-access scheme for

diverse wireless systems, and efficient resource allocation forthe OFDMA system is one of the most important researchissues over the last decade. In particular, in the OFDMA system,subchannel allocation and (transmission) power control canbe jointly considered to improve the system performance, andthis problem has been extensively studied with various systemmodels and various approaches.

In [1]–[9], joint subchannel allocation and power controlproblems for the single-cell OFDMA system are studied. In

Manuscript received August 6, 2013; revised May 14, 2014 and July 28,2014; accepted October 8, 2014. Date of publication October 16, 2014; dateof current version September 15, 2015. This work was supported in part bythe Mid-career Researcher Program through the National Research Foundationunder Grant 2013R1A2A2A01069053 funded by the Ministry of Science, ICT,and Future Planning (MSIP), Korea, and in part by the ICT R&D Program ofMSIP/IITP, under Grant 14-823-04-002, Development of LTE-A Based SmallBase-Station Technology Supporting Single RF Multiple Streams. This paperwas presented in part at the 11th International Symposium on Modeling andOptimization in Mobile, Ad Hoc, and Wireless Networks, Tsukuba ScienceCity, Japan, May 13–17, 2013, and the Sixth International Conference onUbiquitous and Future Networks, Shanghai, China, July 8–11, 2014. Thereview of this paper was coordinated by Prof. R. Jantti.

The authors are with the Department of Electrical and Electronic Engineer-ing, Yonsei University, Seoul 120-749, Korea.

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TVT.2014.2363476

the single-cell OFDMA system, since there is no intra- andintercell interferences, we do not have to consider interferenceamong users, which makes solving resource allocation prob-lems relatively simple. Hence, the optimal resource allocationthat achieves the maximum system capacity can be obtainedin the single-cell OFDMA system. However, if we considera multicell environment, which is a more realistic scenario,despite of no intracell interference by using OFDMA, weshould deal with intercell interference among users in differentcells, which makes the structure of the problem for the multicellsystem totally different from that for the single-cell system. Infact, the problem for the multicell system is much more difficultand complex than that for the single-cell system. Moreover,in general, it is difficult to directly apply resource allocationalgorithms for the single-cell OFDMA system to the multicellOFDMA system.

Recently, various joint subchannel allocation and powercontrol problems in the multicell OFDMA system have beenstudied in [10]–[28]. In [10]–[22], a joint subchannel alloca-tion and power control problem is divided into a subchannelallocation subproblem and a power control subproblem, andheuristic algorithms to solve the two subproblems are proposed.Although various heuristics to solve the problem are used,they do not guarantee achieving the optimality of algorithmsthat provides maximum system capacity. In [23]–[26], game-theoretic approaches are used to jointly consider subchannelallocation and power control in the multicell OFDMA system.However, in general, game-theoretic approaches do not alsoguarantee achieving the optimal resource allocation that pro-vides maximum system capacity. In [27] and [28], an algorithmthat achieves the optimal joint subchannel allocation and powercontrol in the two-cell OFDMA system is developed. However,the approach in [27] and [28] is tailored only to the two-cellsystem, and it is hard to generalize to the general multicellOFDMA system.

Despite the aforementioned extensive studies, to our bestknowledge, the optimal solution for the joint subchannel allo-cation and power control problem in the multicell OFDMA sys-tem and its maximum system capacity have not been achievedyet. Hence, in the previous works for the multicell OFDMAsystem, the performance of the proposed algorithm is comparedwith that of relatively simple heuristic algorithms, such asrandom subchannel allocation, fixed subchannel allocation, andequal power control, without showing how well its performanceapproximates that of the optimal algorithm.

In this paper, we study a joint subchannel allocation andpower control problem that aims at maximizing the sum rate ofthe downlink in the multicell OFDMA system. This problem is

0018-9545 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

Page 2: Sum-Rate Maximization for Multicell OFDMA Systems

KIM et al.: SUM-RATE MAXIMIZATION FOR MULTICELL OFDMA SYSTEMS 4159

a coupled mixed-integer nonconvex problem, and thus, in gen-eral, it is highly difficult to obtain its optimal solution. However,we will show that we can convert the problem into a monotonicoptimization problem [29]–[31] from which we can developthe optimal subchannel allocation and power control solutionthat maximizes the sum rate of the system. This monotonicoptimization approach is successfully used to obtain the optimalpower control (and scheduling) in wireless networks with asingle channel [32], [33]. However, the OFDMA system is amultichannel system in which we should deal with not onlypower control for subchannels but also subchannel allocationto users. Moreover, power control for subchannels are coupledthrough the total transmission power constraint at the basestation (BS). Hence, the algorithms and proofs in [32] and[33] cannot be directly used for our problem. In addition todeveloping the optimal algorithm, we also develop an algorithmthat provides the upper and lower bounds on the maximum sumrate of the system with lower computational complexity thanthe optimal algorithm and study the tightness of the bounds.

Compared with the previous works for resource allocationin the multicell OFDMA system, the main contributions of ourwork can be summarized as follows: 1) To our best knowledge,our work is the first that develops an optimal algorithm forjoint subchannel allocation and power control that achievesthe maximum sum rate of the system; 2) we also develop analgorithm that provides upper and lower bounds on the max-imum sum rate of the system with lower computational com-plexity than the optimal algorithm and show that the boundsare tight in practice; 3) although our algorithms might not bepractically implementable due to some limitations, such as theircomputational complexity and signaling overheads, they can beuseful tools for a benchmark to evaluate the performance ofvarious heuristic algorithms that already have been developedor that will be developed in the future. For example, in [21],a problem that is similar to ours is studied, providing upperand lower bounds on the maximum sum rate of the systemand the approximated solution assuming the high signal-to-interference-plus-noise ratio (SINR) environment. Hence, wewill compare the performances of our algorithms with those ofthe algorithms in [21], which provides the relative accuracy ofsolutions obtained in [21] compared with our solutions.

The rest of this paper is organized as follows. In Section II,we describe the system model and formulate the optimizationproblem. In Section III, we develop an optimal algorithm forjoint subchannel allocation and power control that maximizesthe sum rate of the system. In Section IV, we develop an algo-rithm that provides upper and lower bounds on the maximumsum rate of the system and study their tightness. We providenumerical results in Section V and finally conclude this paperin Section VI.

II. SYSTEM MODEL AND PROBLEM

In this paper, we consider a downlink in a multicell OFDMAsystem with K BSs and Mk mobile stations (MSs) for each BSk. We denote the set of BSs as K = {1, 2, . . . ,K} and the setof MSs that communicate with BS k as Mk = {1, 2, . . . ,Mk}.Each BS has L subchannels whose set is denoted as L =

{1, 2, . . . , L}. We define the subchannel allocation indicatoral(k,m) as

al(k,m)=

{1, if BS k allocates its subchannel l to MS m0, otherwise

(1)

and let a = (al(k,m))∀k∈K, ∀m∈Mk ∀ l∈L. We assume that each

BS can assign each subchannel to, at most, one MS in its cell,which is represented by∑

m∈Mk

al(k,m) ≤ 1 ∀ k ∈ K ∀ l ∈ L. (2)

We denote the transmission power of BS k at subchannel l as plkand assume that each BS operates under the total transmissionpower constraint, i.e., pmax, as∑

l∈Lplk ≤ pmax ∀ k ∈ K. (3)

To model the achieved transmission rate of a link throughwhich BS k communicates with MS m at subchannel l, we willuse the Shannon capacity. We first define the received SINR ofthe link as

γl(k,m)(p) =

gl(k,m)plk∑

n∈K,n�=k

gl(n,m)pln +N0

where gl(n,m) is the channel gain between BS n and MS m at

subchannel l, N0 is the noise power, and p = (plk)∀k∈K, ∀l∈L.Then, the achieved transmission rate of the link through whichBS k communicates with MS m at subchannel l is obtained as

rl(k,m)(a,p) = al(k,m) log(

1 + γl(k,m)(p)

). (4)

By using it, the achieved sum rate of MS m that communicateswith BS k is obtained as

Rk,m(a,p) =∑l∈L

rl(k,m)(ap)

=∑l∈L

al(k,m) log(

1 + γl(k,m)(p)

).

Finally, the achieved sum rate of the system is obtained as

ψ(a,p) =∑k∈K

∑m∈Mk

Rk,m(a,p)

=∑k∈K

∑m∈Mk

∑l∈L

al(k,m) log(

1 + γl(k,m)(p)

). (5)

In this paper, we want to maximize the sum rate of the systemin (5) by optimizing subchannel allocation for each user andtransmission power control for each subchannel considering theconstraints for subchannel allocation in (1) and (2) and that fortransmission power in (3). Hence, the optimization problem isformulated as

(P1) maximizea,p

ψ(a,p)

subject to a ∈ A,p ∈ P

Page 3: Sum-Rate Maximization for Multicell OFDMA Systems

4160 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 64, NO. 9, SEPTEMBER 2015

where

A =

{a

∣∣∣∣∣∑

m∈Mk

al(k,m) ≤ 1 ∀ k ∈ K ∀ l ∈ L

al(k,m) ∈ {0, 1} ∀ k ∈ K ∀m ∈ Mk ∀ l ∈ L

}

P =

{p

∣∣∣∣∣∑l∈L

plk ≤ pmax ∀ k ∈ K,

plk ≥ 0 ∀ k ∈ K ∀ l ∈ L

}.

Note that problem (P1) is a coupled mixed-integer nonlinearnonconvex problem, which is, in general, difficult to solve.

III. OPTIMAL ALGORITHM

To solve problem (P1), here, we develop an algorithm that iscalled a joint subchannel allocation and power control basedon polyblock outer approximation (JSPPA) algorithm. Thisalgorithm is based on the algorithm for monotonic optimization[29]–[33]. We first introduce some mathematical preliminariesand then present the algorithm.

A. Mathematical Preliminaries [29]–[31]

We first introduce some preliminary definitions and proper-ties for monotonic optimization.

Definition 1 (Smallest Element): For a vector a, min(a) isthe smallest element of vector a.

Definition 2 (Dominant): For any two vectors x, x′ ∈ Rn,we write x � x′ if xi ≤ x′

i ∀ i = 1, . . . , n, and say that x′

dominates x. In another case, for a scalar y and a vectorz ∈ Rn, we write y � z if y ≤ zi ∀ i = 1, . . . , n, and we writey z if y ≥ zi ∀ i = 1, . . . , n.

Definition 3 (Hadamard Product): For any two vectors x,x′ ∈ Rn, the Hadamard product x ◦ x′ is a vector whose (i, j)element (x ◦ x′)(i,j) is given by (x)(i,j) × (x′)(i,j).

Definition 4 (Normal): A set G ⊂ Rn+ is said to be normal,

if for any two vectors, x, x′ ∈ Rn+ such that x � x′, if x′ ∈ G,

then x ∈ G.Proposition 1: The intersection and the union of normal sets

are still normal sets.Definition 5 (Reverse Normal): A set H ⊂ Rn

+ is said to bereverse normal, if for any two vectors, x, x′ ∈ Rn

+ such thatx � x′, if x ∈ H, then x′ ∈ H.

Definition 6 (Increasing Function): A function f :Rn−→Ris said to be increasing function on Rn

+, if for any two vectors,x, x′ ∈ Rn

+ such that x � x′, f(x) ≤ f(x′).Definition 7 (Monotonic Optimization): A monotonic opti-

mization problem is a class of optimization problems that havethe following formulation:

maximize f(x)

subject to x ∈ G ∩H

where the domain G is a nonempty normal set, the domainH is a closed reverse normal set, and the function f(x) is anincreasing function.

B. JSPPA Algorithm

We now show that problem (P1) can be represented as amonotonic optimization problem. To this end, by using the factthat al(k,m) can be only either 0 or 1, we first reformulate theachieved transmission rate of a link in (4) as

rl(k,m)(ap) = al(k,m) log(

1 + γl(k,m)(p)

)

= log(

1 + al(k,m)γl(k,m)(p)

)∀ k ∈ K ∀m ∈ Mk ∀ l ∈ L.

We now define a new variable zl(k,m) that denotes the achievedSINR plus one for MS m that communicates with BS k atsubchannel l, i.e.,

zl(k,m) = 1 + al(k,m)γl(k,m)(p) ∀ k ∈ K ∀m ∈ Mk ∀ l ∈ L

and we let z = (zl(k,m))∀k∈K, ∀m∈Mk, ∀l∈L. We call z a vertex.

By using zl(k,m)’s, problem (P1) can be reformulated as thefollowing equivalent optimization problem:

(P2) maximizez

f(z) =∑k∈K

∑m∈Mk

∑l∈L

log(zl(k,m)

)

subject to z ∈ Z

where

Z =

{z

∣∣∣∣ 1 ≤ zl(k,m) ≤ 1 + al(k,m)γl(k,m)(p)

∀ k ∈ K ∀m ∈ Mk ∀ l ∈ L ∀a ∈ A ∀p ∈ P

}.

We now show that problem (P2) is a monotonic optimizationproblem.

Proposition 2: Problem (P2) is a monotonic optimizationproblem.

Proof: We define the hyperrectangle for each subchannelallocation a ∈ A and each transmission power p ∈ P as

Z(a,p) = {z |1 � z � 1+ a ◦ γ(p)}

where γ(p) = (γl(k,m)(p))∀k∈K ∀m∈Mk ∀ l∈L

. Then, the feasi-

ble set Z can be represented as

Z =⋃

a∈A,p∈PZ(a,p).

Since Z(a,p) for any given a and p is normal, by Proposition 1,the feasible set Z is also normal. Furthermore, since theobjective function of problem (P2) is an increasing function,problem (P2) is a monotonic optimization problem. �

Since problem (P2) is the monotonic optimization problem,we can solve it by using Algorithm 1, which is called apolyblock outer approximation algorithm for monotonic opti-mization [29]–[33]. We now describe how Algorithm 1 brieflyworks; see [29]–[33] for details.

Page 4: Sum-Rate Maximization for Multicell OFDMA Systems

KIM et al.: SUM-RATE MAXIMIZATION FOR MULTICELL OFDMA SYSTEMS 4161

Algorithm 1 Polyblock outer approximation algorithm [32]

1: Initialization: Set the initial vertex set T0 as

T0 = {z|z = 1+ a ◦ σ ∀a ∈ A}

where

σl(k,m) =

gl(k,m)pmax

N0∀ k ∈ K ∀m ∈ Mk ∀ l ∈ L

and set n = 0.2: repeat3: In the set Tn, select the vertex zn that maximizes the

objective function of the problem as

zn = argmaxz∈Tn

{f(z) =

∑k∈K

∑m∈Mk

∑l∈L

log(zl(k,m)

)}.

4: For selected zn, by using the projection algorithm,i.e., Algorithm 2, obtain (λ∗,p∗)

(λ∗,p∗) = Algorithm 2(zn).

5: Calculate the point πZ(zn) as

πZ(zn) = λ∗(zn − 1) + 1.

6: Update the vertex set as

Tn+1 = (Tn − {zn}) ∪ T †n

where

T †n =

{yl(k,m)

∣∣∣∣yl(k,m)=zn−(zn−πZ(zn)) ◦ el(k,m)

∀ k ∈ K ∀m ∈ Mk ∀ l ∈ L

}

and el(k,m) is a vector whose elements are all zeros exceptthat the (k,m, l)th element is equal to one.

7: n ← n+ 1.8: until (‖zn − πZ(zn)‖)/(‖zn‖) ≤ ε.9: Let z∗ = πZ(zn).

Since the objective function in problem (P2) is a monotonicincreasing function, its optimal solution is achieved at theboundary of its feasible region Z . Hence, the polyblock outerapproximation algorithm, which solves monotonic optimiza-tion, iteratively finds the optimal vertex z∗ at the boundary ofthe feasible regionZ . To find the optimal vertex z∗, Algorithm 1starts with an infeasible vertex that is located outside of thefeasible region Z . In Algorithm 1, we first define the maximumachievable SINR for MS m that communicates with BS kat subchannel l as σl

(k,m) = (gl(k,m)pmax)/N0, and let σ =

(σl(k,m))∀k∈K ∀m∈Mk ∀ l∈L

. By using σ, we set the initial vertex

set T0 as

T0 = {z|z = 1+ a ◦ σ ∀a ∈ A}.

Since each vertex in the initial vertex set consists of themaximum achievable SINR for each MS at each subchannelassuming that there is no interference and the maximum trans-mission power is allocated, it is an infeasible vertex that islocated outside of the feasible region Z . To find the optimal

(feasible) vertex, which is at the boundary of the feasible region,the algorithm iteratively finds feasible vertices starting from theinitial vertex set T0.

Algorithm 2 Projection algorithm [32], [34]

1: Initialization:2: Define

νl(k,m)(p)= gl(k,m)plk ∀ k ∈ K ∀m ∈ M(k) ∀ l ∈ L

μl(k,m)(p)=

∑n∈K,n�=k

gl(n,m)pln +N0,

∀ k ∈ K ∀m ∈ M(k) ∀ l ∈ L.

3: Let ν(p) = (νl(k,m)(p))∀k∈K ∀m∈M(k) ∀ l∈Land μ(p) =

(μl(k,m)(p))∀k∈K ∀m∈M(k) ∀ l∈L

.

4: For given zn from Algorithm 1, define

δ(λ,p) = min (ν(p)− λ (μ(p) ◦ (zn − 1))) .

5: Select arbitrary p0 ∈ P , and set λ0 = 1 and t = 0.6: repeat7: For given pt

λt+1 = min(ν(pt) ◦ (μ(pt) ◦ (zn − 1))−1

).

8: For given λt+1

pt+1 = argmaxp∈P

δ(λt+1,p).

9: t ← t+ 1.10: until δ(λt,pt) ≤ 0.11: Let (λ∗,p∗) = (λt,pt).

The operation in each iteration can be summarized as thefollowing three steps. In the first step, in the current vertexset, we select a vertex that maximizes the objective function ofproblem (P2) as shown in line 3. Note that the selected vertex,i.e., zn, may not be feasible.

In the second step, from the selected vertex, which is ob-tained in the first step, we find a feasible vertex that is locatedat the boundary of the feasible region as shown in lines 4and 5. We first define the projection of the selected vertex asa feasible vertex that is located on the crossing point of thefeasible region’s boundary and the line from the selected vertexto the origin 1. To obtain the projection of vertex zn, we firstobtain the ratio of the distance from vertex zn to the boundaryof the feasible region to that from vertex zn to the projectionorigin 1, λ∗, by solving the following problem:

λ∗ = max {ω |ω(zn − 1) + 1 ∈ Z }

= max{ω∣∣0 � ω � γ(p) ◦ (zn − 1)−1 ∀p ∈ P

}= max

p∈Pmin

(γ(p) ◦ (zn − 1)−1

)

Page 5: Sum-Rate Maximization for Multicell OFDMA Systems

4162 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 64, NO. 9, SEPTEMBER 2015

where the first equality indicates the definition of λ∗, andthe second equality is followed by the definition of Z . Thegiven problem, i.e., maxp∈P min(γ(p) ◦ (zn − 1)−1), is thetype of a generalized linear fractional programming problem,which can be solved by using a Dinkelbach-type algorithm, i.e.,Algorithm 2 (projection algorithm) [32], [34]. Note that γ(p)can be represented as γ(p) = ν(p) ◦ μ(p)−1 by definitions ofν(p) and μ(p) as shown in lines 2 and 3 of Algorithm 2.Then, for given p, the smallest element of vector (γ(p) ◦(zn − 1)−1), i.e., min(γ(p) ◦ (zn − 1)−1), can be easily cal-culated as shown in line 7 of Algorithm 2, and, for given λ,p that maximizes (γ(p) ◦ (zn − 1)−1) is calculated in line 8of Algorithm 2. In the Dinkelbach-type algorithm, instead of afactional function, i.e., (γ(p) ◦ (zn − 1)−1), its approximatedfunction, i.e., δ(λ,p), that is defined in line 4 of Algorithm 2is used. Since the problem in line 8 is linear programmingwith the approximated function, it is easier to solve the prob-lem in line 8 than to solve the problem with the originalfractional function. The given two calculations are iterativelyperformed, and finally, the projection algorithm producesλ∗ = maxp∈P min(γ(p) ◦ (zn − 1)−1). The convergence ofAlgorithm 2 is proven in [34, Th. 25]. With λ∗ obtained inAlgorithm 2, the projection of the selected vertex is calculatedas in line 5 of Algorithm 1.

In the last step, the vertex set is updated by using the vertexobtained in the first step, i.e., zn, and the corresponding pro-jected feasible vertex obtained in the second step, i.e., πZ(zn).A new vertex is obtained by replacing the element of theinfeasible vertex by the element of the feasible vertex as shownin line 6. The given three steps are iteratively performed, andfinally, the algorithm converges to the optimal vertex z∗ ofproblem (P2). The convergence of the algorithm is proved in[30, Th. 1].

From the optimal vertex z∗, we can obtain the optimal sub-channel and transmission power allocation. Since each elementof the optimal vertex z∗ represents the SINR plus one for eachMS at each subchannel, if an element of the optimal vertex isequal to one, i.e., if the SINR corresponding to that elementis zero, then it indicates that the corresponding subchannel isnot allocated to the corresponding MS. On the other hand, if anelement of the optimal vertex is larger than one, i.e., if the SINRcorresponding to that element is positive, then it indicates thatthe corresponding subchannel is allocated to the correspondingMS. Hence, from the optimal vertex z∗, we can obtain theoptimal subchannel allocation a∗ as

al∗(k,m) =

{1, zl∗(k,m) > 1

0, zl∗(k,m) = 1∀ k ∈ K ∀m ∈ Mk ∀ l ∈ L.

Note that since the optimal vertex z∗ is feasible, i.e., z∗ ∈ Z ,it is obvious that the optimal subchannel allocation is alsofeasible, i.e., a∗ ∈ A.

When we obtain the optimal vertex z∗ in lines 5 and 9, i.e.,z∗ = πZ(zn) = λ∗(zn − 1) + 1, we use the result of the pro-jection algorithm, (λ∗,p∗). Since (λ∗,p∗) makes the optimalvertex z∗, p∗ is the transmission power that corresponds tothe optimal vertex, i.e., the optimal transmission power. Hence,from the optimal vertexz∗ achieved by using Algorithms 1 and 2,

we can obtain the optimal subchannel allocation a∗ and theoptimal transmission power p∗ of problem (P1). The overalldescription of the JSPPA algorithm is provided in Algorithm 3.

Algorithm 3 JSPPA Algorithm

1: Reformulate problem (P1) as monotonic optimizationproblem (P2).

2: Solve problem (P2) by using Algorithm 1, and obtain theoptimal vertex z∗.

3: Obtain the optimal subchannel allocation from z∗ as

al∗(k,m)=

{1, zl∗(k,m)>1

0, zl∗(k,m)=1∀ k∈K ∀m∈M(k) ∀ l∈L.

4: Obtain the optimal transmission power p∗ that corre-sponds to z∗ in Algorithm 2.

IV. UPPER AND LOWER BOUNDS

Although we are able to develop the optimal algorithm thatprovides the maximum sum rate of the system, its computa-tional complexity is very high. In fact, as we will discuss laterin this section, its complexity exponentially increases as thenumber of subchannel increases. Hence, here, we develop analgorithm that provides the upper and lower bounds on themaximum sum rate of the system with lower computationalcomplexity than the optimal algorithm. We then study the prop-erties of the achieved bounds and compare the computationalcomplexity of the algorithm for the bounds and that of theoptimal algorithm.

A. Algorithm for Upper and Lower Bounds

Here, we consider the following modified problem, which isobtained by relaxing the total transmission power constraint (3)in problem (P1) but limiting the transmission power for eachsubchannel instead, i.e.,

(P3) maximizea,p

ψ(a,p)

subject to a ∈ A,p ∈ P̃

where

P̃ ={p∣∣0 ≤ plk ≤ pmax ∀ k ∈ K ∀ l ∈ L

}.

We can reformulate problem (P3) as a monotonic optimizationproblem in a similar way as problem (P2) in the previoussection and obtain its optimal solution by using the JSPPAalgorithm. We denote the optimal solution of problem (P3) as(ao,po).

Since problem (P3) is obtained by relaxing the total trans-mission power constraint (3) from problem (P1), we caneasily show that the maximum sum rate achieved by solving

Page 6: Sum-Rate Maximization for Multicell OFDMA Systems

KIM et al.: SUM-RATE MAXIMIZATION FOR MULTICELL OFDMA SYSTEMS 4163

problem (P3), ψ(ao,po), is larger than or equal to that achievedby solving problem (P1), ψ(a∗,p∗), i.e.,

ψ(a∗,p∗) ≤ ψ(ao,po).

Hence, ψ(ao,po) can be the upper bound of ψ(a∗,p∗).However, the solution (ao,po) may not be a feasible solution

of problem (P1), since it may not satisfy the total transmissionpower constraint (3), although it always satisfies all otherconstraints in problem (P1). To obtain a feasible solution thatsatisfies all the constraints in problem (P1) from (ao,po), wefirst define the largest total transmission power ρ(po) amongthose of BSs when the solution (ao,po) is applied as

ρ(po) = max

{pok

∣∣∣∣∣pok =∑l∈L

plok ∀ k ∈ K}. (6)

We then define (a†,p†) as

a† = ao and p† =

(pmax

ρ(po)

)po. (7)

We can easily show that (a†,p†) is a feasible solution ofproblem (P1) that satisfies all the constraints in problem (P1)including the total transmission power constraint (3). Note thatalthough (a†,p†) is a feasible solution, it may not be an optimalsolution of problem (P1). Hence, (a†,p†) can provide the lowerbound of the maximum sum rate achieved by problem (P1), i.e.,

ψ(a†,p†) ≤ ψ(a∗,p∗).

We now have the following theorem, which indicates that(ao,po) provides the upper bound of the maximum sum ratein the multicell OFDMA system, whereas (a†,p†) provides itslower bound.

Theorem 1:

ψ(a†,p†) ≤ ψ(a∗,p∗) ≤ ψ(ao,po).

B. Properties of Upper and Lower Bounds

Here, we study several properties of upper and lower boundsobtained in the previous section. Specifically, we will provideconditions with which upper and lower bounds are close toeach other. In other words, we will show when both boundsprovide good approximations to the maximum sum rate ofthe multicell OFDMA system. Recall that (a∗,p∗) denotes theoptimal solution of problem (P1), and (a†,p†) and (ao,po)denote solutions for its lower and upper bounds, respectively.In addition, ψ(a∗,p∗) denotes the maximum sum rate, andψ(a†,p†) and ψ(ao,po) denote its lower and upper bounds,respectively.

We first define the total interference level of MS m thatcommunicates with BS k at subchannel l as

I l(k,m)(p) =∑

n∈K,n�=k

gl(n,m)pln

and, in addition, define

Imin(p)=min{I l(k,m)(p) ∀ k ∈ K ∀m ∈ M(k) ∀ l ∈ L

}.

Hence, Imin(p) is the smallest interference level under trans-mission power p.

We first provide following three lemmas.Lemma 1: In po, at least one element is equal to pmax.

Proof: Assume that all elements in po are strictly less thanpmax. Then, we can find a constant α > 1 such that p′ = αpo,and all elements in p′ are less than or equal to pmax, whichimplies that (ao,p′) is still a feasible solution of problem (P3),and

γl(k,m)(p

′) > γl(k,m)(p

o) ∀ k ∈ K ∀m ∈ M(k) ∀ l ∈ L.

This implies that

ψ(ao,p′) > ψ(ao,po)

which contradicts that (ao,po) is the optimal solution of prob-lem (P3), and the proof is completed. �

Lemma 2:

pmax ≤ ρ(po) ≤ L× pmax

where L is the number of subchannels, and ρ(po) is definedin (6).

Proof: By the definition of ρ(po) in (6) and Lemma 1, thegiven result can be immediately obtained. �

Lemma 3: Imin(po)/N0 → ∞ if and only if Imin(p

†)/N0 → ∞.

Proof: By Lemma 2 and (7), we have

1Lpo ≤ p† ≤ po (8)

and thus

1L

Imin(po)

N0≤ Imin(p

†)

N0≤ Imin(p

o)

N0

which completes the proof. �We now have one of our main results. The following theorem

says that if the interference power dominates the noise power,the upper and lower bounds are very close to each other, andthus, they can be good approximations to the maximum sumrate of the multicell OFDMA system.

Theorem 2:

limImin(po)/N0→∞

ψ(a†,p†) = limImin(po)/N0→∞

ψ(a∗,p∗)

= limImin(po)/N0→∞

ψ(ao,po).

Proof: By Theorem 1, we obtain

ψ(a†,p†) ≤ ψ(a∗,p∗) ≤ ψ(ao,po)

and thus, we only have to show

limImin(po)/N0→∞

ψ(a†,p†) = limImin(po)/N0→∞

ψ(ao,po).

In addition, since a† = ao, we only have to show that

limImin(po)/N0→∞

γl(k,m)(p

†) = limImin(po)/N0→∞

γl(k,m)(p

o)

∀ k ∈ K ∀m ∈ M(k) ∀ l ∈ L.

Page 7: Sum-Rate Maximization for Multicell OFDMA Systems

4164 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 64, NO. 9, SEPTEMBER 2015

First

limImin(po)/N0→∞

γl(k,m)(p

o) = limImin(po)/N0→∞

gl(k,m)plok

I l(k,m)(po)+N0

= limImin(po)/N0→∞

gl(k,m)plok

I l(k,m)(po)

= limImin(po)/N0→∞

pmax

ρ(po)gl(k,m)p

lok

pmax

ρ(po)Il(k,m)(p

o).

Then, by (7) and Lemma 3, we have

limImin(po)/N0→∞

pmax

ρ(po)gl(k,m)p

lok

pmax

ρ(po)Il(k,m)(p

o)

= limImin(p†)/N0→∞

gl(k,m)pl†k

I l(k,m)(p†)

= limImin(p†)/N0→∞

gl(k,m)pl†k

I l(k,m)(p†) +N0

= limImin(po)/N0→∞

γl(k,m)(p

†).

Hence

limImin(po)/N0→∞

γl(k,m)(p

†) = limImin(po)/N0→∞

γl(k,m)(p

o)

and this is true for ∀ k ∈ K ∀m ∈ M(k) ∀ l ∈ L, which com-pletes the proof. �

From the given theorem, we can say that if the interferencepower is much larger than the noise power, i.e.,

Imin(p†) � N0 and Imin(p

o) � N0

we have

ψ(a†,p†) ≈ ψ(a∗,p∗) ≈ ψ(ao,po)

and thus, both solutions (a†,p†) and (ao,po) from prob-lem (P3) can be good approximations to the optimal solution(a∗,p∗) of problem (P1).

In addition, we can also show that if the system is operatedin a high-SINR environment, the upper and lower bounds canbe very close to each other, and thus, they can be good approx-imations to the maximum sum rate of the multicell OFDMAsystem. We first define

γmin(p) = min

{γl(k,m)(p)

∣∣∣∣ γl(k,m)(p) > 0 ∀ k ∈ K∀m ∈ M(k) ∀ l ∈ L

}.

Hence, γmin(p) is the smallest positive SINR under transmis-sion power p. We now show the following lemma.

Lemma 4: γmin(po) → ∞ if and only if γmin(p

†) → ∞.Proof: We only have to show that γl

(k,m)(po) → ∞ if and

only if γl(k,m)(p

†) → ∞ ∀ k ∈ K ∀m ∈ Mk ∀ l ∈ L. By (7)

and (8), there exists an α such that p† = αpo and 1/L ≤ α ≤ 1.Hence

γl(k,m)(p

o) ≥ γl(k,m)(p

†)

≥ γl(k,m)

(po

L

)

=gl(k,m)p

lok /L∑

n∈K,n�=k

gl(n,m)plon /L+N0

≥ 1L

gl(k,m)plok∑

n∈K,n�=k

gl(n,m)plon +N0

=1Lγl(k,m)(p

o)

and thus

1Lγl(k,m)(p

o) ≤ γl(k,m)(p

†) ≤ γl(k,m)(p

o). (9)

This is true for ∀ k ∈ K ∀m ∈ Mk ∀ l ∈ L, which completesthe proof. �

Then, we have the following result.Theorem 3:

limγmin(p†)→∞

ψ(a∗,p∗)

ψ(a†,p†)= 1 and lim

γmin(po)→∞

ψ(a∗,p∗)

ψ(ao,po)= 1.

Proof: By Theorem 1, we only have to show that

limγmin(po)→∞

ψ(ao,po)

ψ(a†,p†)= 1.

In addition, since ψ(ao,po) ≥ ψ(a†,p†), we only have toshow that

limγmin(po)→∞

ψ(ao,po)

ψ(a†,p†)≤ 1.

Since the Shannon capacity log(1 + γ) is an increasing andconcave function of γ, we have

log(

1 + γl(k,m)(p

o))− log

(1 + γl

(k,m)(p†))

≤ d

dγ(log(1+γ))

∣∣∣γ=γl(k,m)

(p†)

(γl(k,m)(p

o)−γl(k,m)(p

†))

=1

1 + γl(k,m)(p

†)

(γl(k,m)(p

o)− γl(k,m)(p

†)).

Then, by (9), we have

log(

1 + γl(k,m)(p

o))

≤ log(

1 + γl(k,m)(p

†))+

(L− 1)γl(k,m)(p

†)

1 + γl(k,m)(p

†).

Page 8: Sum-Rate Maximization for Multicell OFDMA Systems

KIM et al.: SUM-RATE MAXIMIZATION FOR MULTICELL OFDMA SYSTEMS 4165

TABLE ISIMULATION SCENARIOS

By using the given inequality, Lemma 4, and (7), i.e., a† = ao,we have

limγmin(po)→∞

ψ(ao,po)

ψ(a†,p†)

= limγmin(po)→∞

∑k∈K

∑m∈Mk

∑l∈L

alo(k,m) log(

1 + γl(k,m)(p

o))

∑k∈K

∑m∈Mk

∑l∈L

al†(k,m) log(

1 + γl(k,m)(p

†))

≤ limγmin(po)→∞

∑k∈K

∑m∈Mk

∑l∈L

al†(k,m) log(

1 + γl(k,m)(p

†))

∑k∈K

∑m∈Mk

∑l∈L

al†(k,m) log(

1 + γl(k,m)(p

†))

+ limγmin(p†)→∞

∑k∈K

∑m∈Mk

∑l∈L

al†(k,m)

(L−1)γl(k,m)

(p†)

1+γl(k,m)

(p†)∑k∈K

∑m∈Mk

∑l∈L

al†(k,m) log(

1+γl(k,m)(p

†))

= 1

which completes the proof. �From the given theorem, we can say that if the system is oper-

ated in a high-SINR environment, i.e.,γmin(po) is large, we have

ψ(a∗,p∗)

ψ(a†,p†)≈ 1 and

ψ(a∗,p∗)

ψ(ao,po)≈ 1

and thus, both solutions (a†,p†) and (ao,po) from prob-lem (P3) can be good approximations to the optimal solution(a∗,p∗) of problem (P1).

C. Computational Complexity Comparison

Here, we compare the computational complexity of the opti-mal algorithm developed in the previous section with that of thealgorithm for the upper and lower bounds in this section. Notethat both algorithms use the JSPPA algorithm, which is an iter-ative algorithm. Hence, to calculate its computational complex-ity accurately, we need to know the number of iterations untilit converges, which is, in general, hard to estimate. However,note that the JSPPA algorithm (i.e., Algorithm 1) starts with theinitial vertex set, iteratively updates the vertex set, and finds thevertex maximizing the objective function in the updated vertexset. Furthermore, the computational complexity of finding thevertex that maximizes the objective function in the vertex setdepends on the size of the vertex set, and the vertex set isupdated from the initial vertex set. Hence, the computationalcomplexity of the JSPPA algorithm (i.e., Algorithm 1) stronglydepends on the number of elements in the initial vertex set,and here, we compare the computational complexity of the twoalgorithms focusing on the number of elements in their initialvertex set.

In the optimal algorithm [i.e., the JSPPA algorithm for prob-lem (P1)], the number of vertices of the initial vertex set can beobtained as

|T0| =(∏

k∈K(Mk + 1)

)L

.

Hence, it exponentially increases as the number of subchannelsincreases. However, in the algorithm for the upper and lowerbounds [i.e., the JSPPA algorithm for problem (P3)], since thetotal transmission power constraint is relaxed, problem (P3) canbe separated as L subproblems, each of which corresponds toeach subchannel. Hence, the number of vertices of the initialvertex set can be obtained as

|T0| =(L∏k∈K

(Mk + 1)

)

and thus, it linearly increases as the number of subchannelincreases.

In addition, as previously mentioned, we may regard prob-lem (P3) as L parallel subproblems of problem (P1). Hence,although we are not able to provide the number of requirediterations for each algorithm analytically, we can easily inferthat the number of required iterations for the convergence ofthe algorithm for upper and lower bounds is much smaller thanthat of the optimal algorithm. From this inference and the givenanalysis, we can conclude that the computational complexity ofthe algorithm for upper and lower bounds is much lower thanthat of the optimal algorithm.

V. NUMERICAL RESULTS

Here, we provide numerical results to evaluate the per-formances of our algorithms considering three scenarios assummarized in Table I. In the first scenario, we compare theperformance of the optimal algorithm with those of the algo-rithms for upper and lower bounds. In this scenario, due to thehigh computational complexity of the optimal algorithm, weconsider a simple system that consists of three BSs, four MSswith the fixed position for each BS, and five subchannels. In thesecond scenario, to support theoretical results for the boundsin the previous section more clearly, we consider a relativelysimple system that consists of a one-tier seven-cell system, fourMSs with the fixed position for each BS, and higher numberof subchannels (i.e., 50 and 100). In the third scenario, to showthe tightness of our proposed bounds in more general situations,we consider a more general system that consists of a one-tierseven-cell system, more MSs (i.e., 2, 4, and 10) with the randomposition, and higher number of subchannels (i.e., 50 and 100).In addition to evaluating the performances of our algorithms,

Page 9: Sum-Rate Maximization for Multicell OFDMA Systems

4166 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 64, NO. 9, SEPTEMBER 2015

Fig. 1. Simple three-cell system.

TABLE IISYSTEM PARAMETERS

here, we also compare the performances of our algorithms withthose of the algorithms in [21], showing that our algorithmsprovide more accurate results.

We first compare the maximum sum rate of the system withits upper and lower bounds that are obtained by our algorithms.As mentioned earlier, due to the high computational complexityof the optimal algorithm, we consider a simple system in whichthere are three BSs and four MSs for each BS, i.e., the firstscenario, as shown in Fig. 1. We set the intersite distance to be1.5 km, the total bandwidth to be 1 MHz, and the bandwidth ofeach subchannel to be 200 KHz. In addition, based on [35], weset the path loss of a link to be 128 + 37.6 log(d), where d isthe link distance in kilometers, the fading model to be Rayleighfading, the noise power spectral density to be −174 dBm/Hz,and the noise figure to be 10 dB. Parameters and their valuesfor the system are summarized in Table II.

In Figs. 2 and 3, we provide the maximum sum rate ofthe system, i.e., ψ(a∗,p∗), and its upper and lower bounds,i.e., ψ(ao,po) and ψ(a†,p†), according to the maximum totaltransmission power pmax and the distance between the BSand its MS, respectively. We also plot the maximum sum rateachieved in [21], which is obtained with assuming the high-SINR environment. First of all, we can see that these figuresvalidate Theorem 1, i.e., ψ(a†,p†) ≤ ψ(a∗,p∗) ≤ ψ(ao,po).These figures also show that both upper and lower bounds arevery close to the maximum sum rate of the system, whichimplies that they are tight. In addition, we can see that theresults in [21] cannot achieve the maximum sum rate. In par-ticular, since the results in [21] are obtained assuming the high-SINR environment, as the maximum total transmission powerdecreases (in Fig. 2) and as the distance between the BS and itsMS increases (in Fig. 3), their accuracy gets worse.

Fig. 2. Maximum sum rate and its upper and lower bounds according to themaximum total transmission power.

Fig. 3. Maximum sum rate and its upper and lower bounds according to thedistance between the BS and its MS.

Fig. 4. One-tier seven-cell system.

We now consider a one-tier seven-cell system in which eachBS is located in the center of each cell and four MSs arelocated in each cell, as shown in Fig. 4. We set the totalbandwidth to be 10 MHz, and other parameters are set tobe the same as in Table II. In this system, due to the highcomputational complexity, we do not provide the results of theoptimal algorithm, and we focus on showing the tightness of its

Page 10: Sum-Rate Maximization for Multicell OFDMA Systems

KIM et al.: SUM-RATE MAXIMIZATION FOR MULTICELL OFDMA SYSTEMS 4167

TABLE IIIRATIO OF THE SMALLEST INTERFERENCE POWER TO THE NOISE POWER

ACCORDING TO THE MAXIMUM TOTAL TRANSMISSION POWER

Fig. 5. Normalized difference according to the maximum total transmissionpower.

upper and lower bounds. To this end, we define the normalizeddifference, i.e., ξ(ψ(ao,po), ψ(a†,p†)), between the upper andlower bounds of the maximum sum rate of the system as theratio of the difference between the two bounds to the lowerbound as follows:

ξ(ψ(ao,po), ψ(a†,p†)

)=

ψ(ao,po)− ψ(a†,p†)

ψ(a†,p†).

In Table III, we provide the ratio of the smallest inter-ference power to the noise power (RSIN), i.e., Imin(p)/N0,according to the maximum total transmission power pmax atthe BS with fixing the distance between the BS and its MSto be 375 m. This table shows that as the maximum totaltransmission power increases, the RSIN also increases. Hence,from Theorem 2, we can expect that as the maximum totaltransmission power increases, the difference between upper andlower bounds decreases. In Fig. 5, we provide the normalizeddifference between our upper and lower bounds and that be-tween upper and lower bounds in [21] varying the maximumtotal transmission power pmax. As expected, as the maximumtotal transmission power increases, the normalized difference ofour bounds gets smaller, which verifies Theorem 2. We can seethat the normalized differences of our bounds are very small inmost cases, which implies that our bounds are tight and providegood approximations to the maximum sum rate of the system.In addition, we can also see that our bounds are tighter thanthose in [21].

In Table IV, we provide the smallest positive SINR (SP-SINR), i.e., γmin(p), according to the distance between theBS and its MS with fixing the maximum total transmissionpower at the BS to be 43 dBm. This table shows that as thedistance between the BS and its MS gets closer, the SP-SINRgets larger. Hence, from Theorem 3, we can expect that asthe distance gets closer, the normalized difference gets smaller.In Fig. 6, we provide the normalized difference of our upper

TABLE IVSMALLEST POSITIVE SINR ACCORDING TO

THE DISTANCE BETWEEN BS AND MS

Fig. 6. Normalized difference according to the distance between the BS andits MS.

Fig. 7. Normalized difference according to the maximum total transmissionpower with varying the number of subchannels.

and lower bounds and that of upper and lower bounds in [21]varying the distance between the BS and its MS. As expected,as the distance gets closer, the normalized difference of ourbounds gets smaller, which verifies Theorem 3. In addition,as in the previous case, the normalized differences of ourbounds are small in most cases, and they are smaller thanthose of the algorithms in [21]. Hence, this figure also indicatesthat our upper and lower bounds are tight and provide goodapproximations to the maximum sum rate of the system.

Due to the relationship between our upper and lower boundsin (6) and (7), we may expect that the gap between the twobounds increases as the number of subchannels increases. InFig. 7, we show the effect of the number of subchannelson the tightness of the bounds. We provide the normalizeddifferences of our upper and lower bounds for the cases whenthe numbers of subchannels are 50 and 100, varying the maxi-mum total transmission power with fixing the distance betweenthe BS and its MS to be 375 m. In practical systems, e.g.,LTE, the maximum available number of subchannels is 100when the bandwidth of the system is 20 MHz. For the case whenthe bandwidth is 20 MHz (when the number of subchannels

Page 11: Sum-Rate Maximization for Multicell OFDMA Systems

4168 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 64, NO. 9, SEPTEMBER 2015

Fig. 8. Normalized difference according to the distance between the BS andits MS with varying the number of subchannels.

Fig. 9. Normalized difference according to the maximum total transmissionpower with varying the number of MSs per cell and the number of subchannels.

is 100), we set the maximum total transmission power to betwice that in the case when the bandwidth is 10 MHz (whenthe number of subchannels is 50). As shown in Fig. 7, as thenumber of subchannels increases, we indicate that the normal-ized difference increases. However, our normalized differencedoes not significantly increase although the maximum numberof subchannels is applied. In Fig. 8, we provide the normalizeddifferences of our upper and lower bounds for the given twocases, varying the distance between the BS and its MS withfixing the maximum total transmission power to be 43 dBmwhen the number of subchannel is 50 and to be 46 dBm whenthe number of subchannel is 100. As shown in the previouscase, our normalized difference does not significantly increaseeven when the maximum number of subchannels is applied.

To provide a more general scenario in the one-tier seven-cell system, we now consider a one-tier seven-cell system inwhich each BS is located in the center of each cell and MSsare randomly deployed on each cell. In Fig. 9, we providethe normalized difference according to the maximum totaltransmission power pmax with varying the number of MSs percell and the number of subchannels. As shown in Fig. 9, thenormalized difference is small in most cases, which indicatesthat our bounds are tight and provide good approximationsto the maximum sum rate of the system even in the generalscenario. Hence, we believe that our proposed upper and lowerbounds can be a good guideline for most practical systems.

VI. CONCLUSION

In this paper, we have studied a joint subchannel allocationand power control problem for the downlink in the multicellOFDMA system. Through a monotonic optimization approach,we developed an algorithm for optimal subchannel allocationand power control that maximizes the sum rate of the system.In addition, we also developed an algorithm that provides theupper and lower bounds of the maximum sum rate of the systemwith lower computational complexity. Numerical results showthat our upper and lower bounds are tight in most cases, partic-ularly in an interference-dominated or a high-SINR scenario.

Although our algorithm for the bounds has lower computa-tional complexity than the optimal algorithm, we admit that itstill has high computational complexity to be implemented inthe practical system. However, we believe that our algorithmscan be useful tools for a benchmark to evaluate the efficiency ofother heuristic algorithms, as we did in the numerical results.

REFERENCES

[1] G. Song and Y. Li, “Cross-layer optimization for OFDM wire-less networks—Part I: Theoretical framework,” IEEE Trans. WirelessCommun., vol. 4, no. 2, pp. 614–624, Mar. 2005.

[2] G. Song and Y. Li, “Cross-layer optimization for OFDM wirelessnetworks—Part II: Algorithm development,” IEEE Trans. WirelessCommun., vol. 4, no. 2, pp. 625–634, Mar. 2005.

[3] K. Seong, M. Mohseni, and J. Cioffi, “Optimal resource allocation forOFDMA downlink systems,” in Proc. IEEE ISIT , 2006, pp. 1394–1398.

[4] H. Zhu and J. Wang, “Chunk-based resource allocation in OFDMAsystems—Part I: Chunk allocation,” IEEE Trans. Commun., vol. 57, no. 9,pp. 2734–2744, Sep. 2009.

[5] H. Zhu and J. Wang, “Chunk-based resource allocation in OFDMAsystems—Part II: Joint chunk, power and bit allocation,” IEEE Trans.Commun., vol. 60, no. 2, pp. 499–509, Feb. 2012.

[6] Y.-B. Lin, T.-H. Chiu, and Y. Su, “Optimal and near-optimal resourceallocation algorithms for OFDMA networks,” IEEE Trans. WirelessCommun., vol. 8, no. 8, pp. 4066–4077, Aug. 2009.

[7] B.-G. Kim and J.-W. Lee, “Joint opportunistic subchannel and power sched-uling for relay-based OFDMA networks with scheduling at relay stations,”IEEE Trans. Veh. Technol., vol. 59, no. 5, pp. 2138–2148, Jun. 2010.

[8] B.-G. Kim and J.-W. Lee, “Opportunistic resource scheduling forOFDMA networks with network coding at relay stations,” IEEE Trans.Wireless Commun., vol. 11, no. 1, pp. 210–221, Jan. 2012.

[9] J.-A. Kwon and J.-W. Lee, “Opportunistic scheduling for an OFDMAsystem with multi-class services,” Wireless Commun. Mobile Comput.,vol. 12, no. 12, pp. 1104–1114, Aug. 2012.

[10] L. Venturino, N. Prasad, and X. Wang, “Coordinated scheduling andpower allocation in downlink multicell OFDMA networks,” IEEE Trans.Veh. Technol., vol. 58, no. 6, pp. 2835–2848, Jul. 2009.

[11] S. V. Hanly, L. L. H. Andrew, and T. Thanabalasingham, “Dynamic alloca-tion of subcarriers and transmit powers in an OFDMA cellular network,”IEEE Trans. Inf. Theory, vol. 55, no. 12, pp. 5445–5462, Dec. 2009.

[12] R. Y. Chang, Z. Tao, J. Zhang, and C.-C. J. Kuo, “Multicell OFDMAdownlink resource allocation using a graphic framework,” IEEE Trans.Veh. Technol., vol. 58, no. 7, pp. 3494–3507, Sep. 2009.

[13] M. Pischella and J. Belfiore, “Weighted sum throughput maximization inmulticell OFDMA networks,” IEEE Trans. Veh. Technol., vol. 59, no. 2,pp. 896–905, Feb. 2010.

[14] Y. Jia, Y. Wang, and J. Lu, “Utility-based resource allocation in a multi-cell OFDMA system with base station cooperation,” in Proc. IEEEWCNC, 2010, pp. 1–6.

[15] Y. Hua, Q. Zhang, and Z. Niu, “Resource allocation in multi-cell OFDMA-based relay networks,” in Proc. IEEE INFOCOM, 2010, pp. 1–9.

[16] T. Wang and L. Vandendorpe, “Iterative resource allocation for maximiz-ing weighted sum min-rate in downlink cellular OFDMA systems,” IEEETrans. Signal Process., vol. 59, no. 1, pp. 223–234, Jan. 2011.

[17] K. W. Choi, E. Hossain, and D. I. Kim, “Downlink subchannel and powerallocation in multi-cell OFDMA cognitive radio networks,” IEEE Trans.Wireless Commun., vol. 10, no. 7, pp. 2259–2271, Jul. 2011.

Page 12: Sum-Rate Maximization for Multicell OFDMA Systems

KIM et al.: SUM-RATE MAXIMIZATION FOR MULTICELL OFDMA SYSTEMS 4169

[18] D. W. K. Ng and R. Schober, “Resource allocation and scheduling inmulti-cell OFDMA systems with decode-and-forward relaying,” IEEETrans. Wireless Commun., vol. 10, no. 7, pp. 2246–2258, Jul. 2011.

[19] M. Dong, Q. Yang, F. Fu, and K. S. Kwak, “Joint power and subchannelallocation in relay aided multi-cell OFDMA networks,” in Proc. IEEEISCIT , 2011, pp. 548–552.

[20] M. Moretti, A. Todini, A. Baiocchi, and G. Dainelli, “A layered archi-tecture for fair resource allocation in multicellular multicarrier systems,”IEEE Trans. Veh. Technol., vol. 60, no. 4, pp. 1788–1798, May 2011.

[21] H. Tabassum, Z. Dawy, and M. S. Alouini, “Resource allocation via sum-rate maximization in the uplink of multi-cell OFDMA networks,” WirelessCommun. Mobile Comput., vol. 11, no. 12, pp. 1528–1539, Dec. 2011.

[22] M. Fallgren, “An optimization approach to joint cell, channel and powerallocation in multicell relay networks,” IEEE Trans. Wireless Commun.,vol. 11, no. 8, pp. 2868–2875, Aug. 2012.

[23] Z. Han, Z. Ji, and K. Liu, “Non-cooperative resource competition gameby virtual referee in multi-cell OFDMA networks,” IEEE J. Sel. AreasCommun., vol. 25, no. 6, pp. 1079–1090, Aug. 2007.

[24] Z. Liang, Y. Chew, and C. Ko, “On the modeling of a non-cooperativemulticell OFDMA resource allocation game with integer bit-loading,” inProc. IEEE GLOBECOM, 2009, pp. 1–6.

[25] Q. D. La, Y. H. Chew, and B. H. Soong, “Performance analysis of down-link multi-cell OFDMA systems based on potential game,” IEEE Trans.Wireless Commun., vol. 11, no. 9, pp. 3358–3367, Sep. 2012.

[26] S. Buzzi, G. Colavolpe, D. Saturnino, and A. Zappone, “Potential gamesfor energy-efficient power control and subcarrier allocation in uplinkmulticell OFDMA systems,” IEEE J. Sel. Topics Signal Process., vol. 6,no. 2, pp. 89–103, Apr. 2012.

[27] N. Ksairi, P. Bianchi, P. Ciblat, and W. Hachem, “Resource allocation fordownlink cellular OFDMA systems—Part I: Optimal allocation,” IEEETrans. Signal Process., vol. 58, no. 2, pp. 720–734, Feb. 2010.

[28] N. Ksairi, P. Bianchi, P. Ciblat, and W. Hachem, “Resource allocationfor downlink cellular OFDMA systems—Part II: Practical algorithms andoptimal reuse factor,” IEEE Trans. Signal Process., vol. 58, no. 2, pp. 735–749, Feb. 2010.

[29] H. Tuy, “Normal sets, polyblocks and monotonic optimization,” VietnamJ. Math., vol. 27, no. 4, pp. 277–300, 1999.

[30] H. Tuy, “Monotonic optimization: Problems and solution approaches,”SIAM J. Optim., vol. 11, no. 2, pp. 464–494, 2000.

[31] Y. J. Zhang, L. Qian, and J. Huang, “Monotonic optimization in com-munication and networking systems,” Found. Trends Netw., vol. 7, no. 1,pp. 1–75, 2012.

[32] L. P. Qian, Y. J. Zhang, and J. Huang, “MAPEL: Achieving global op-timality for a non-convex wireless power control problem,” IEEE Trans.Wireless Commun., vol. 8, no. 3, pp. 1553–1563, Mar. 2009.

[33] L. P. Qian and Y. J. Zhang, “S-MAPEL: Monotonic optimization fornon-convex joint power control and scheduling problems,” IEEE Trans.Wireless Commun., vol. 9, no. 5, pp. 1708–1719, May 2010.

[34] J. B. G. Frenk and S. Schaible, “Fractional programming,” in Hand-book of Generalized Convexity and Generalized Monotonicity, vol. 76.New York, NY, USA: Springer-Verlag, 2005, pp. 335–386.

[35] “Physical layer aspects for E-UTRA,” 3rd Generation Partnership Project,Sophia-Antipolis, France, Tech. Rep. 25.814, 2006.

Sung-Yeon Kim received the B.S. degree fromYonsei University, Seoul, Korea, in 2007, where heis currently working toward the joint M.S. and Ph.D.degrees, all in electrical and electronic engineering.

His research interests include cooperative commu-nication, performance analysis, and optimization incommunication networks.

Jeong-Ahn Kwon received the B.S. and M.S. de-grees in electrical and electronic engineering fromYonsei University, Seoul, Korea, in 2006 and 2008,respectively.

His research areas include opportunistic schedul-ing, resource allocation, optimization, and small-celland heterogeneous networks.

Jang-Won Lee (SM’12) received the B.S. degreein electronic engineering from Yonsei University,Seoul, Korea, in 1994; the M.S. degree in electri-cal engineering from Korea Advanced Institute ofScience and Technology, Daejeon, Korea, in 1996;and the Ph.D. degree in electrical and computerengineering from Purdue University, West Lafayette,IN, USA, in 2004.

During 1997–1998, he was with Dacom R&DCenter, Daejeon. During 2004–2005, he was a Post-doctoral Research Associate with the Department

of Electrical Engineering, Princeton University, Princeton, NJ, USA. SinceSeptember 2005, he has been with the School of Electrical and ElectronicEngineering, Yonsei University, where he is currently an Associate Professor.His research interests include resource allocation, quality-of-service and pricingissues, optimization, and performance analysis in communication networks andsmart grids.