Resource Allocation with Fairness Consideration inOFDMA-based Relay Networks

Zhihua Tang, Guo Wei

Wireless Information Network Lab, University of Science and Technology of ChinaP.O.BOX 4, Hefei, Anhui, P.R.China, 230027

Abstract—This paper considers a broadband cellular orthogo-nal frequency division multiple access (OFDMA) system withrelay nodes operating in decode-and-forward (DF) mode. Anoptimization problem for relay selection, subcarrier assignmentand power allocation that maximizes the downlink capacity withfairness consideration among users is formulated for such asystem. Since the problem cannot be solved easily in direct forms,we make continuous relaxation and solve the dual problem of therelaxed problem using a subgradient method. An integer-valuedsolution is then obtained from the result of the subgradientmethod. Numerical results show that our proposed algorithmsolves the problem with superior capacity performance and lowoutage probability.

I. INTRODUCTION

The ever-growing wireless multimedia services demandhigh-speed and reliable communication over wireless chan-nels. Orthogonal frequency division multiplexing (OFDM)technique has been regarded as a solution to provide highspeed multimedia transmission [1]. In OFDM systems, thetotal channel bandwidth is divided into narrowband subcarriersthat can be considered as parallel independent channels. There-fore a high-rate data stream is obtained via multiple parallellow-rate data streams. This process combats intersymbol in-terference (ISI), which is the main problem in wideband trans-mission over multipath fading channels. Orthogonal frequencydivision multiple access (OFDMA) can be considered as anextension of OFDM to multiuser scenarios, in which a subsetof subcarriers is assigned exclusively to each user. An intrinsicadvantage of OFDMA over other multiple access methodsis the capability to exploit the frequency selectivity enabledmultiuser diversity by adaptive resource allocation. Therefore,resource management in the OFDMA systems, including avariety of subcarrier assignment, power allocation, and bit-loading algorithms, which can provide quality of service (QoS)guarantees, has drawn enormous attention in recent years [2],[3].

Relay networks have attracted extensive attention recentlydue to their potential to increase coverage area and channelcapacity. The resource allocation for OFDM-based relay net-works has already been studied [4]–[9]. In [4] and [5], theauthors focus on the single source-destination pair throughsingle relay scenario. In [6]–[8], the authors extend the studyto multirelay and multiuser scenarios. Both [6] and [7] aim

Supported by the National Basic Research Program of China (973 Program),2007CB310602.

to maximize the downlink throughput of networks. In [6], theoverall optimization is divided into a combinatorial optimiza-tion for joint subcarrier assignment and relay selection anda convex optimization for power allocation. In [7], the opti-mal subcarrier allocation problem with predetermined powerassignment for each subcarrier and the optimal joint powerand subcarrier allocation problem are formulated and solvedusing convex optimization. However, the above algorithmsoptimize the system throughput without consideration of theinstantaneous fairness among users. In [8], the author presentsa resource allocation scheme with fairness consideration. Itassumes that one user can get the data directly transmittedfrom the source or relayed through the relay station at the sametime. However, adaptive power allocation is not applied at therelays, which can further improve the performance. In [9], theresource allocation for multiple source nodes, multiple relaynodes, and a single destination node is studied. The optimalsource, relay and subcarrier allocation problem with fairnessconstraint on relay nodes is formulated as a binary integerprogramming problem and tackled using a graph theoreticalapproach.

In this paper, we are interested in maximizing the down-link capacity of OFDMA-based relay networks with fairnessconsideration among users. The fairness here means that agiven rate target is instantaneously considered for all usersduring resource allocation. Instead of solving this problemdirectly, we make continuous relaxation, as did in [2], torecast it as a concave optimization problem, and solve thedual problem using a subgradient method [10]. From thesolution of the subgradient method, we obtain an integer-valued solution, which is a near-optimal solution to the originalunrelaxed problem. We compare our proposed algorithm witha heuristic method based on nearest RS selection, two-stepgreedy subcarrier assignment, and equal power allocation.Numerical results show a remarkable capacity increment andthe low outage probability of our proposed algorithm.

The organization of this paper is as follows. In Section II,the system model is given and the capacity maximization prob-lem with fairness consideration among users is formulated.In Section III, we make continuous relaxation and solve theproblem from the result of a subgradient method. Simulationresults are discussed in Section IV. Finally, conclusions aredrawn in Section V.

978-1-4244-2948-6/09/$25.00 ©2009 IEEE

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.

The first time slot

The second time slot

BS

UE 1

UE 2

UE M-1

UE M

RS 1

RS 2

RS K

1R

0R

Fig. 1. The downlink OFDMA-based relay networks

II. SYSTEM MODEL AND PROBLEM FORMULATION

A. OFDMA-based Relay NetworksWe consider the same cellular system as in [7], shown in

Fig. 1. A base station (BS) is located at the center and Kfixed RSs are located uniformly in the inner bound, operatingin decode-and-forward (DF) mode [11]. The radii of the innerand outer bounds of this region are R1 and R0, respectively.M user equipments (UEs) are uniformly distributed in theboundary region. It is assumed that the UE can receive signalfrom the RS but not from the BS because of the distance orthe obstacles as discussed in [5]. This means that the downlinkcommunication covers two equal time slots. Each RS receivesand decodes the data transmitted from BS in the first time slot(perfect decoding is assumed) and then relays the data to UEsin the following time slot. The overall bandwidth B is dividedinto N subcarriers for OFDM transmission.

Let xnm be the transmitted signal from BS on subcarrier n

for UE m. Let ynsk and yn

km denote the received signals at RSk in the first slot and at UE m in the second slot, respectively.We then have the following relationships:

ynsk =

√pn

skhnskxn

m + nnk , (1)

ynkm =

√pn

kmhnkmxn

m + nnm, (2)

where pnsk and pn

km denote the power used to transmit to RSk by BS and to UE m by RS k on subcarrier n, respectively.hn

sk and hnkm denote the channel gains between BS and RS k,

RS k and UE m on subcarrier n. nnk and nn

m are independentand identically distributed (i.i.d.) complex Gaussian noise withi.i.d. real and imaginary parts, both having variance N0/2. Wefurther assume that the channels are slowly-varying, and thusthe channels between BS and RSs, RSs and UEs, can be es-timated at RSs and UEs, respectively. The channel estimationresults are fed back to BS for centralized processing.

Based on the Shannon formula, the subcarrier capacitybetween BS and RS k on subcarrier n is given as

fnsk(pn

sk) =B

2Nlog2

(1 +

pnsk|hn

sk|2

BN0/N

). (3)

Similarly, the subcarrier capacity between RS k and UE m is

fnkm(pn

km) =B

2Nlog2

(1 +

pnkm|hn

km|2

BN0/N

). (4)

There are three questions to be answered in the resourceallocation for such networks. The first one is how to pairRS to UE so that the former one is assisting the latter one’sreception. The second one is how to assign subcarriers to thesepairs. The third one is how to allocate power in BS and RSs.The three questions cannot be separated from each other. Forsimplicity, we assume that each RS uses the same subcarrierto relay information it received from the same subcarrier, i.e.,the paired RS and UE receive on the same subcarrier. As aconsequence, fn

sk(pnsk) = fn

km(pnkm). It is easy to obtain that

pnsk =

|hnkm|2

|hnsk|2

pnkm. (5)

So the power allocated in BS can be uniquely determined bythe power allocated in RSs, and vice versa. We further assumethat each subcarrier is allocated to at most one RS-UE pair.

B. Problem Formulation

Let ρnkm denote the binary indicator with ρn

km = 1 in-dicating the assignment of subcarrier n to UE m and RSk pair, and ρn

km = 0, otherwise. By the introduction ofρn

km, the RS-UE pairing and subcarrier assignment questionscan be jointly represented. Our objective is to maximize thedownlink capacity with fairness consideration among UEs.The optimization problem can be formulated as follows:

max(ρ,p)

N∑n=1

K∑k=1

M∑m=1

ρnkmfn

km(pnkm) (6)

s.t. ρnkm ∈ {0, 1},∀k,m, n; (7a)

pnkm ≥ 0,∀k,m, n; (7b)K∑

k=1

M∑m=1

ρnkm≤1,∀n; (7c)

N∑n=1

K∑k=1

M∑m=1

ρnkm

|hnkm|2

|hnsk|2

pnkm ≤ Pmax

BS ; (7d)

N∑n=1

M∑m=1

ρnkm · pn

km ≤ Pmaxk ,∀k; (7e)

N∑n=1

K∑k=1

ρnkmfn

km(pnkm) ≥ R̄m,∀m, (7f)

where ρ and p are the arrays of indicator variable and powerallocation, whose elements are ρn

km and pnkm, respectively.

PmaxBS and Pmax

k are the total transmission power of BS andRS k, respectively. R̄m is the target rate of UE m. Constraint(7c) follows from the assumption that each subcarrier isallocated to at most one RS-UE pair. (7d) and (7e) correspondto the power constraints of BS and RSs, respectively. (7f)denotes the required rate of each UE, reflecting the fairnessconsideration.

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.

Problem (6) is not concave with respect to (ρnkm, pn

km).Thus, obtaining a globally optimal solution is rather difficult.In the following section, (6) is reformulated as a concavemaximization problem by introducing a set of new variables.Then a subgradient method is proposed to solve the resourceallocation problem.

III. RESOURCE ALLOCATION

A. Dual Problem

To make the problem (6) tractable, ρnkm is relaxed to be a

real number in [0, 1]. This continuous relaxation permits timesharing of each subcarrier. We take similar approach as in[2] to reformulate (6) to a concave optimization problem. Wedefine sn

km = ρnkm · pn

km and rewrite (6) as

max(ρ,s)

N∑n=1

K∑k=1

M∑m=1

ρnkmfn

km

( snkm

ρnkm

)(8)

s.t. 0 ≤ ρnkm ≤ 1,∀k,m, n (9a)

snkm ≥ 0,∀k,m, n (9b)K∑

k=1

M∑m=1

ρnkm≤1,∀n; (9c)

N∑n=1

K∑k=1

M∑m=1

|hnkm|2

|hnsk|2

snkm ≤ Pmax

BS ; (9d)

N∑n=1

M∑m=1

snkm ≤ Pmax

k ,∀k; (9e)

N∑n=1

K∑k=1

ρnkmfn

km

( snkm

ρnkm

)≥ R̄m. (9f)

According to (4), the function fnkm(·) is concave. Then

the function ρnkmfn

km

(sn

km

ρnkm

)is also concave with respect to

(ρnkm, sn

km) because it is the perspective function of fnkm(sn

km)[12]. So the objective function in (8) is a concave function.Moreover, constraints (9a)-(9f) are all convex. Hence, problem(8) is a concave maximization problem over a convex set. Theduality gap is zero, and the solution of the dual problem isequal to that of the primal problem.

For the formulation of the dual problem, we first define the

Lagrangian function as follows:

L(ρ, s,η, κ, λ, µ) =N∑

n=1

K∑k=1

M∑m=1

ρnkmfn

km

( snkm

ρnkm

)+

N∑n=1

ηn

(1 −

K∑k=1

M∑m=1

ρk,n

)

+ κ

(Pmax

BS −N∑

n=1

K∑k=1

M∑m=1

|hnkm|2

|hnsk|2

snkm

)

+K∑

k=1

λk

(Pmax

k −N∑

n=1

M∑m=1

snkm

)

+M∑

m=1

µm

( N∑n=1

K∑k=1

ρnkmfn

km

( snkm

ρnkm

)− R̄m

),

(10)

where η = [η1, . . . , ηN ], κ = [κ], λ = [λ1, . . . , λK ] andµ = [µ1, . . . , µM ] are the vectors of Lagrangian multipliersintroduced by (9c)-(9f), respectively. Then, the dual objectivefunction problem is computed as

g(κ, λ, µ) ={

maxρ,s,η L(ρ, s, η, κ, λ, µ)s.t. (9a) − (9c) (11)

and the dual problem is given as

min(κ,λ,µ)

g(κ,λ, µ)

s.t. κ º 0, λ º 0, µ º 0,(12)

where “º” represents the component-wise inequality.

B. Proposed Method

Since the dual problem is always convex, it is guaranteedthat the gradient-type algorithms converge to the global opti-mum. For (12), we can use a subgradient method. It is hard todifferentiate with κ, λ, µ. At the first step of this method, weinitialize κ(0), λ(0), µ(0), and given κ(i), λ(i), µ(i), where i isthe iteration index, compute the dual objective function (11).After differentiating with respect to ρn

km and snkm, respectively,

we obtain the necessary conditions for the optimal solution,ρn∗

km and sn∗km. Specifically, if ρn

km 6= 0, we have

∂L(ρ, s, η, κ, λ, µ)∂ρn

km

= (1 + µm)[fn

km

( snkm

ρnkm

)− sn

km

ρnkm

fn′

km

( snkm

ρnkm

)]− ηn{

= 0, 0 < ρnkm < 1,

> 0, ρnkm = 1,

(13)

∂L(ρ, s, η, κ, λ, µ)∂sn

km

= (1 + µm)fn′

km

( snkm

ρnkm

)− κ

|hnkm|2

|hnsk|2

− λk{= 0, sn

km > 0,< 0, sn

km = 0.(14)

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.

Here, fn′

km(·) is the derivative of fnkm(·). From (14), we can

deduce that

snkm

ρnkm

=

0, if fn′−1

km

((κ|hn

km|2|hn

sk|2 + λk

)/(1 + µm

))< 0,

fn′−1km

((κ|hn

km|2|hn

sk|2 + λk

)/(1 + µm

)), otherwise,

(15)where fn−1

km (·) is the inverse function to fnkm(·). From (13),

we can conclude that

Hnkm(κ, λ, µ)

= (1 + µm)[fn

km

( snkm

ρnkm

)− sn

km

ρnkm

fn′

km

( snkm

ρnkm

)]{

= ηn, 0 < ρnkm < 1,

> ηn, ρnkm = 1.

(16)

The continuous relaxation generally yields a fractionalvalued solution, and we should round ρn

km to 0 or 1 to getan integer-valued solution. Since the constraint (9c) must besatisfied, we find from (16) that for each n, only the RS-UE pair (k,m) with the biggest Hn

km(κ, λ, µ) can use thatsubcarrier. It follows that

ρn∗k′m′ = 1, ρn∗

km = 0, for all k 6= k′, m 6= m′, (17)

where(k′,m′) = arg max

(k,m)Hn

km(κ,λ,µ). (18)

Hnkm(κ, λ, µ) can be obtained by substituting (15) into (16).

Hence, for a fixed set of κ, λ, µ, we can determine (k′, m′)for each n, thus ρn∗

km and sn∗km are obtained.

Substituting ρn∗km and sn∗

km into (11), κ(i), λ(i) and µ(i) canbe updated using a subgradient method as follows:

κ(i+1) =[κ(i) − t

(i)1

(Pmax

BS −N∑

n=1

K∑k=1

M∑m=1

|hnkm|2

|hnsk|2

snkm

)]+

,

λ(i+1)k =

[λ

(i)k − t

(i)2

(Pmax

k −N∑

n=1

M∑m=1

snkm

)]+

,

k = 1, . . . ,K,

µ(i+1)m =

[µ(i)

m − t(i)3

( N∑n=1

K∑k=1

ρnkmfn

km(sn

km

ρnkm

) − R̄m

)]+

,

m = 1, . . . ,M, (19)

where t(i)1 , t

(i)2 , t

(i)3 are sequences of step size designed

properly [10]. Then, with this subgradient method, we canobtain a near-optimal solution of (12) with a sufficientlylarge number of iterations, Imax. Note that ρn

km obtained incomputing the dual objective function is always integer-valued,that is, there is no time-sharing.

Our proposed resource allocation method can be summa-rized in Table I.

IV. SIMULATION RESULTS

The simulation parameters are listed in Table II, whichmainly refer to [7].

TABLE IPROPOSED RESOURCE ALLOCATION METHOD

1) Initialize κ(0), λ(0), µ(0), i = 0, Mout = 0.2) While (i < Imax)

a) Calculate Hnkm(κ, λ, µ), ∀k, m, n, using (15) and (16).

b) Determine (k′, m′) for each n. ρn∗km and sn∗

km are obtainedaccording to (17) and (15), respectively.

c) Update κ(i), λ(i), µ(i) according to (19), i = i + 1.

3) For each m, if∑N

n=1

∑K

k=1ρn∗

kmfnkm(

sn∗km

ρn∗km

) < R̄m,Mout = Mout + 1.

4) If Mout ≥ 1, an outage event happens, else the objective∑N

n=1

∑K

k=1

∑M

m=1ρn∗

kmfnkm

(sn∗

kmρn∗

km

)is recorded.

TABLE IISIMULATION PARAMETERS

Parameter ValueCell radius R0 1 kmInner cell radius R1 0.75 kmPropagation model BS-RS:128.1+28.8log10(R)(path-loss, in dB) RS-UE:128.1+37.6log10(R)

(R is the distance in kilometers)Small-scale fading model 6-path Rayleigh fading with

exponential power delay profile and10 µs delay spread

Normalized doppler spread 0.01RS antenna gain 10 dBUE antenna gain 0 dBUE noise figure 9 dBThermal noise density -174 dBm/HzBS and RS Tx power 24 dBmChannel bandwidth B 1.25 MHzNumber of subcarriers N 64Number of UE M 8

Step sizes t(i)1 , t

(i)2 , t

(i)3 0.1/

√i

With fixed power budgets at BS and RSs, the target ratemay not be satisfied if the channel is in deep fading. An effec-tive resource allocation algorithm with fairness considerationshould be the one that ensures the target rate to be satisfiedwith high probability even for UEs in deep fading. To evaluatesuch effectiveness, we define an outage probability as

pout = Pr(Mout ≥ 1), (20)

where Mout is the number of UEs whose achievable rates arebelow the target at a particular time snapshot.

Numerical results are averaged over 100 scenarios. In eachscenario, all UEs are relocated and 105 successive channelrealizations are implemented. Each realization is evaluatedusing the subgradient method with 103 iterations.

The target rate for each user is set to 4 bits/symbol. In thefollowing, we show the performance of the proposed algorithmon average capacity (bits/symbol) and outage probability byvarying the number of RSs. For comparison, we take aheuristic algorithm, NearRS+GreedySA+EquPA, as a perfor-mance benchmark. NearRS represents that each UE selectsa nearest RS to relay its subcarriers. GreedySA representsthe greedy subcarrier allocation. It is performed in two steps:Initial subcarrier allocation that considers per-user fairness;

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.

1 2 3 4 5 6 7 80

50

100

150

200

250

Number of RSs

Avera

ge C

apacity (

bits/s

ym

bol)

Proposed algorithm

NearRS+GreedySA+EquPA

Fig. 2. Average capacity comparison over different number of RSs

1 2 3 4 5 6 7 810

-5

10-4

10-3

10-2

10-1

100

Number of RSs

Outa

ge P

robabili

ty

Proposed algorithm

NearRS+GreedySA+EquPA

Fig. 3. Outage probability comparison over different number of RSs

and residual subcarrier allocation that further increases thecapacity. EquPA means the power of BS is equally allocatedto its transmitted subcarriers. Then, the power allocation ofRSs can be determined according to (5).

Fig.2 shows the average capacity comparison over differentnumber of RSs when the number of UEs is fixed to 8. Theaverage capacities of both algorithms increase significantlywith the number of RSs due to cooperative diversity. Ourproposed algorithm outperforms NearRS+GreedySA+EquPAfor all number of RSs by exploiting the available RSs andpower more efficiently. The gain is more than 100% when thenumber of RSs is not more than 6.

In Fig.3, the outage probabilities for both algorithms overdifferent number of RSs are plotted. Our proposed algorithmoffers smaller outage probability than the reference algorithmand decreases rapidly over the number of RSs. When thenumber of RSs is more than 6, the outage probability isbelow 10−4, demonstrating the effectiveness of our proposedalgorithm.

V. CONCLUSION

In this paper, we show that the capacity of a broadbandcellular OFDMA system can be remarkably increased byusing RSs and sophisticated resource allocation. In particular,we formulate a capacity maximization problem with fairnessconsideration among users. Solving this problem is compu-tationally prohibitive when the number of variables is large.We relax the problem as a concave optimization problem, byallowing time-sharing usage of each subcarrier by multipleRS-UE pairs. Then, we formulate the dual problem of therelaxed problem and solve it using a subgradient method. Fromthe result of the subgradient method, we obtain an integer-valued solution, which is the near-optimal solution to theoriginal unrelaxed problem. Numerical results show that ourproposed algorithm solves the problem with superior capacityperformance and low outage probability.

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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2009 proceedings.