ONE of the emerging technologies towards enabling Fifth

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1 Eergy ad Spectrum Efficiecy Trade-off for Gree Small Cell Networs Haris Pervaiz, Leila usavia ad Qiag Ni School of Computig ad Commuicatios, IfoLab 21, Lacaster Uiversity, Lacaster, UK {h.pervaiz, l.musavia, Abstract Gree Small Cell Networs aim at achievig high rates ad low powers by offloadig users with low sigal-to-oiseratios from macrocell to the pico base statio. I this wor, we propose to joitly optimise eergy efficiecy EE ad spectrum efficiecy SE such that the etwor providers ca dyamically tue the trade-off parameter for differet desig requiremets. This paper formulates the EE-SE trade-off as a multi-objective optimisatio problem OP i the upli of multi-user twotier Orthogoal Frequecy Divisio ultiplexig Heterogeeous Networs. Usig the weighted sum method, the OP ca be trasformed ito a sigle-objective optimisatio problem SOP. The proposed EE ad SE trade-off optimisatio problem is strictly quasi-cocave. Hece, usig Dual Decompositio approach, we derive the uique optimal solutio. Numerical results demostrate the effectiveess of the proposed approach ad illustrate the fudametal tradeoff betwee EE ad SE for differet tradeoff parameters such as imum trasmissio power ad circuit power. Idex Terms HetNets, Gree Commuicatios, Eergy ad Spectrum Efficiecy, Resource Allocatio, Small Cells. I. INTRODUCTION ONE of the emergig techologies towards eablig Fifth Geeratio 5G is heterogeeous etwors HetNets which iclude Gree Small Cell Networs cosistig of lowpower base statio BS, e.g., microcells, picocells, ad femtocells, overlaid withi the macrocell geographical area, deployed by either users or etwor operators who share the same spectrum with the macrocells [1]. The purpose of HetNets is to allow user equipmets UEs to access small cells eve though the UEs are withi the coverage of macrocell. The deploymet of small cells has a great potetial to improve the spatial reuse of radio resources ad also to ehace the eergy efficiecy EE of the etwor [2] [3]. I traditioal etwors, the spectrum efficiecy SE metric is cosidered the mai performace idicator which measures how efficietly the frequecy resources are utilized regardless of the efficiet power cosumptio. O the other had, EE is emergig as oe of the ey performace idicators for the ext geeratio wireless commuicatios systems. The motivatio behid EE arises due to the curret eergy cost payable by operators for ruig their access etwors as a sigificat factor of their operatioal expeditures OPEX. Hece, gree etworig paradigm, which focuses o reducig eergy cosumptio by brigig the BS closer to the UEs [4], motivates usig HetNets for higher EE. This wor was partially supported by the UK EPSRC uder Grat umber EP/K011693/1 ad the EU FP7 uder Grat Number PIRSES-GA Curretly, most of EE gais i HetNets are achieved with sacrificig SE [4]. We ote that the user lyig withi the coverage area of heterogeeous etwor ca efficietly utilise its resources i order to either improve its achievable EE or SE. I this tred, the EE-imisatio problem i a upli of HetNets is aalytically solved for a sigle user case uder miimum target rate ad imum trasmissio power costraits i [5]. A joit badwidth ad power allocatio scheme to optimise EE for a set of users withi the heterogeeous etwors is proposed i [6]. This scheme is implemeted for the multi-user system i a distributed maer i [6]. A joit BS associatio ad power cotrol scheme which itet to satisfy the user s target sigal-to-iterferece-plus-oise ratio SINR for the upli of a large-scale HetNets is proposed i [7]. A efficiet power allocatio scheme to ivestigate the power cosumptio ad EE without degradig the etwor throughput i Log Term Evolutio LTE HetNets is proposed i [8]. I [9], a distributed o-cooperative game is proposed to improve the system EE i the dowli trasmissio of HetNets. I this wor, the BSs autoomously choose their optimal trasmissio strategies while balacig the load amog themselves ad satisfyig the users quality-of-service requiremets. Cosiderig that simply imisig either EE or SE does ot utilise the resources efficietly, there is a icreasig attetio for 5G etwors to joitly optimise the two coflictig objectives, i.e., EE ad SE. However, most of the curret literature maily focus o the EE-SE tradeoff i the dowli trasmissio scheme of traditioal Orthogoal Frequecy Divisio odulatio Access OFDA based macrocell oly systems for example, [10] ad [11]. Accordig to the best of our owledge, there is o wor o joit EE-SE tradeoff i the HetNets cosiderig multiuser multi-carrier systems. I this wor, we address a EE- SE tradeoff resource allocatio techique i a upli HetNet as a multi-objective optimizatio problem OP to simultaeously imise both EE ad SE cosiderig imum iput power costrait. We trasform the formulated OP ito a sigle-objective optimizatio problem SOP usig weighted sum method. Provig that the EE-SE tradeoff SOP is strictly quasi-cocave with respect to the trasmit power, we derive the optimal solutio. By exploitig the fractioal programmig cocept, the formulated SOP ca be trasformed ito a equivalet subtractive form which is tractable. Numerical results demostrate the impact of imum trasmit power, the chael-to-oise ratio, the circuit power ad the tradeoff parameter o EE-SE tradeoff.

2 II. SYSTE ODEL We cosider a upli two-tier HetNet composed of oe macrocell overlaid with oe pico BS with total umber of users N ad total umber of subacrriers K. We deote the idex set of all subcarriers as = {1,,K}, the set of all users as = {1,,N} ad the set of etwors as m = {1,,}. We further assume that the chael state iformatio CSI correspodig to each subcarrier is perfectly ow to the UEs trasmitters. Further, we cosider a orthogoal subcarrier selectio scheme which assigs each subcarrier exclusively to either pico BS PB or macrocell c at ay time such that K PB Kc = where K c ad K PB idicate the set of subcarriers assiged to the macrocell ad pico BS, respectively. Assume σ PB ad σ c deote the subcarrier allocatio idices for pico BS ad macrocell, ad the assigmet of subcarriers to the users as well. Particularly, whe subcarrier K PB, for = {1,,K}, is allocated to user, for = {1,,N}, = 1, ad otherwise, σ PB = 0. Similarly, if the is allocated to user, σ c = 1, ad otherwise, σ c = 0. The istataeous rate achieved o each subcarrier by user for macrocell ad pico BS ca be hece writte as: the σ PB subcarrier K c r c r PB = σ c Bc log 2 1+γ c = σ PB BPB log 2 1+γ PB p c p PB, K c, 1a, K PB, 1b where p PB ad pc idicate the power allocated to the subcarrier for user i the pico BS ad macrocell, respectively. Similarly, the rate of user usig subcarrier choosig macrocell or pico BS is represeted by r c ad rpb, respectively. Each etwor m {c, PB} has its ow badwidth equally divided amog its subcarriers. B c ad B PB represet the subcarrier spacig i macrocell ad pico BS, respectively. γ c ad γ PB represet the chael-to-oise-ratio CNR of user o subcarrier i the macrocell ad pico BS, respectively, ad ca be defied as: γ c = h c 2 σ c ρ2 c PLc, γ PB = h PB 2 σ PB ρ2 PB PLPB, 2a 2b where h c ad hpb represet the chael amplitude gai for user o subcarrier K c ad K PB, respectively. The distace-based path loss i macrocell ad pico BS are deoted by PL c ad PL PB. The oise power i subcarrier for macrocell ad pico BS are respectively give byρ 2 c = Bc N 0 ad ρ 2 PB = BPB N 0, where N 0 is the oise spectrum desity. For simplicity, we assume that a set of available etwors are ow i two-tier HetNets. I practice, the trasmissio power available at user, P, is limited to a imum threshold, i.e., which ca be formulated as: P P P, = {1,,N}, P = K p m, m. 3a 3b I a upli trasmissio sceario, multiple users trasmit data towards a BS so each commuicatio li betwee user ad BS itroduces a idividual circuit power P C. Sice the circuit power is related to the UE hadsets, we assume P c C = P PB C = P C. Hece, the overall power cosumptio model ad the trasmissio power i a upli of HetNets are modelled as below: P = ǫ 0 p m +N P C, 4 m K N where N represets the total umber of active users ad ǫ 0 is a iverse of power amplifier efficiecy. Eergy Efficiecy η EE is defied as the amout of data trasferred per uit eergy cosumed by the system usually measured i b/j ad is defied as: η EE = R P = ǫ 0 m K N m K N p m r m +N P C, 5 where R deotes the total achievable data rate. η EE is strictly quasi-cocave with respect to trasmissio power P T [11]. Hece, there exists oe ad oly oe optimal solutio that imises η EE which strictly icreases with P T [ ] 0,Pη EE while strictly decreases with P T [ Pη EE, ]. SE η SE, o the other had, is a measure that reflects the efficiet utilizatio of the available spectrum i terms of throughput ad it is commoly expressed i uits of b/s/hz. η SE is strictly icreasig with trasmissio power P T, is cocave i P T, ad ca be defied as: η SE = R B = m K N K a K m r m B m K m, 6 where B deotes the total occupied badwidth ad K a is the umber of active subcarriers. It is usually ot always possible to imise both EE ad SE simultaeously. It is also worthwhile to metio that i most of the power regios, the power allocatio strategies to icrease these metrics are coflictig approaches. I detail, EE ad SE both icrease with trasmissio power P T util it reaches the eergy-efficiet trasmissio power P T = Pη EE. After this poit, EE decreases with a icrease i SE. These fact motivate us to dyamically tue the EE ad SE trade-off depedet o the available resources, i terms of badwidth ad the trasmissio power. I the followig sectios, we propose a eergy-efficiet user associatio scheme i which the user associates to the BS with the imum achievable EE. Uique associatio of users with the macrocell or pico BS is assumed [1]. Specifically, each user ca oly be associated with oe BS. ore detail ca be foud i Sectio IV. III. PROBLE FORULATION OF EE-SE TRADEOFF Our goal is to optimise EE ad SE simultaeously. I this sectio, we formulate EE ad SE trade-off with the imum iput power costrait i a upli trasmissio scheme of Two- Tier HetNets. We formulate the EE-SE trade-off as a OP accordig to σ m,pm η EE ad σ m,pm η SE. 7

3 I order to maitai the balace betwee EE ad SE i the cosidered OP, we trasform the optimisatio problem usig ormalised factors ad θ SE such that EE ad SE are i the similar scale. Usig the weighted sum method [12], we covert the OP i 7 ito a SOP defied by η, yieldig σ m,pm s.t. m=1=1 m=1 α η EE +1 αθ SE η SE K 8a 0 α 1 8b p m P,. 8c σ m 1,,. p m 0,σm {0,1},,, m. 8e 8d Here, 8a represets the EE-SE tradeoff optimisatio problem ad α is the tradeoff parameter such that 0 α 1. 8a ca further be simplified to 1 α η = η EE + θ SE η SE. 9 σ m α,pm 1 α I 9, we replace with β which ca be from 0 to. α We further simplify 9 as η = η θse η SE = η EE +β. 10 σ m,pm The imisatio problem 8a 8e is a iteger combiatorial fractioal programmig problem ad is geerally NPhard. For better tractability, we first relax the iteger variables, σ m {0,1} ito cotiuous variables, σm [0,1]. After some mathematical maipulatios, the modified optimisatio problem for 8a 8e ca be writte as η = s.t. σ m,pm K m=1=1 η EE 1+β θ SEP, 11a B p m P,. 11b N σ m 1,. β 0,p m 0, σm [0,1],,, m. 11d 11c Note, the uit of η is b/j. η EE is quasi-cocave i P T ad R is strictly cocave i P T. Hece, η is cotiuously differetiable ad quasi-cocave with respect to the trasmissio power P T. 1 As metioed i [13], ay optimisatio problem i fractioal form ca be trasformed ito a equivalet optimisatio problem i subtractive form. Hece, the o-liear fractioal optimizatio problem i 11a ca be trasformed ito the parameterized fuctio as show i 12. The costraits i 1 Due to the space limitatio, the proof is omitted. 11c 11d are later cosidered by dual decompositio method such that each subcarrier ca be exclusively assiged to a sigle user ad the o-egative optimal powers are computed. The optimal solutio ca be determied by fidig the root to the Uη as show i 12 usig various root fidig methods [14]. U η = σ m,pm η N K =1 r m N P C +ǫ 0 1+β θ SEP B N K p m =1 12 From 12, it implies that U η strictly decreases with respect to η. It also imply that η,u η > 0 ad η,u η < 0. From 12, it is quite obvious that Uη > 0, whe η 0. I this wor, we will solve 12 for η > 0. IV. EE AND SE TRADE-OFF RESOURCE ALLOCATION SCHEE The solutio to EE-SE tradeoff optimisatio problem is formulated as a iterative two-layer solutio combiig Dielbach type method outer layer ad Lagragia dual decompositio approach ier layer. This process is repeated util both procedures coverge to a optimum value. We have proposed a iterative Dielbach type method as a outer layer solutio to fid a optimal solutio to 12 by determiig a root to U η = 0. At a iteratio i 1, the value of η is iitialised ad the U η is solved for a give value of η, i.e., η i 1, ad the optimal power p i 1 is computed usig dual decompositio approach i.e., ier layer solutio. The optimal power computed i iteratioi 1 ca be used to update the value ofη for iteratio i. This process is repeated util covergece. The pseudo code for the Dielbach method is show i Algorithm-I. We utilise the dual decompositio approach [15] to solve U η = 0 i each iteratio of Dielbach type method. It is show that the dual-compositio approach has lower computatioal complexity ad the duality gap for o-covex optimisatio approaches to zero for sufficietly large umber of subcarriers [14]. I order to apply dual decompositio method, we first eed to fid the Lagragia fuctio of 12. Usig stadard optimisatio methods proposed i [14], the Lagragia fuctio of 12 ca be writte as: Lp m,µ = η + N K r m =1 ε 0 N K =1 N µ =1 P 1+β τ EE τ SE p m +N P C K p m m=1=1, 13 where τ EE = P ad τ SE = B θ SE. Followig 13, the Lagragia dual fuctio correspodig to problem 12 is gµ = σ m,pm The correspodig dual problem to 12 is mi µ gµ Lp m,µ. 14 s.t. µ 0, 15

4 where gµ is the dual fuctio give as K N gµ = g µ ηnp C + µ P, 16 =1 ad g µ is defied by g µ = σ,p ηε 0 N N p m The dual problem ca be give by: mi µ 0 σ m,pm =1 r m 1+β τ EE N Lp m,µ. τ SE µ p m. 17 The dual problem ca be decomposed ito two layers amely as lower layer ad master layer. I the lower layer, K subproblems are solved i parallel to compute the power ad subcarrier allocatio o each subcarrier K for the give values of µ ad η. I the master layer, the Lagragia multipliers are updated usig subgradiet method. For fixed set of Lagrage multipliers ad a give parameter η, the power for user o subcarrier ca be computed by taig the derivative of 17 with respect to p m as follows: g µ = p m B m l2 1+β τee τ SE γ m 1+γ m pm µ +ηε 0 18 Algorithm-I: Iterative EE ad SE Tradeoff Algorithm:- Iitialize iter = umber of iteratios = 10, = imum acceptable tolerace = 10 3, Set i=1 ad ηi = 0, While U η < i < iter do Solve 12 for a give value of ηi usig Algorithm-II. N K m=1 =1 =1 Update ηi+1 = rm 1+β τ EE τ SE Update i = i+1 ed While Output: [η] N P C+ǫ 0 m=1 N =1 By applyig the KKT coditios, we get Lp m,µ > 0, p m = p m = 0, 0 < p m < 0, p m = 0 Hece, p m = 0, otherwise. = p K =1 pm < p + B m 1+β τ EE τ SE l2µ +ηε 0 1, if σ m γ m = where x + = 0,x. The optimal solutio of 11a ca the be expressed as p m = mi p m,p. The dual variable µ must satisfy the KKT coditios i order to be optimal. Each subcarrier is allocated to the correspodig user which imises 20. Therefore, a feasible subcarrier assigmet matrix is give as: σ m = σ m { 1, if,m, = arg m, η m, 0, otherwise. 20 where = 1 idicates that the subcarrier is assiged to user associated with etwor m ad η m = B m log 2 1+γ m pm ǫ 0p m +PC. To miimise the dual fuctio gµ, the subgradiet method [14] ca be used to update the dual variable µ. The, we ca update the Lagrage multiplier µ accordig to + K µ i+1 = µ i P si p m. 21 i m=1=1 Here, i is the iteratio umber ad s i is the costat size of the step. The Lagragia multipliers are updated accordigly util the covergece is achieved idicatig that the dual optimal poit is achieved. The subgradiet update is guarateed to coverge to optimal µ as log as s i is chose to be sufficietly small [14]. A commo practice is to choose square summable step sizes i cotrast to absolute step sizes [15]. I this paper, we have used s i = 0.1 i as a step size. V. SIULATION RESULTS We cosider a two-tier HetNets eviromet with a sigle macrocell with 500 m radius overlaid with a pico BS with a radius of 50 m. The badwidth of each subcarrier is 30 Hz. The imum trasmissio power of users cosidered i the simulatio vary from 200 mw to 500 mw, respectively, whereas the value of circuit power of users is set fixed to P C =100 mw. We assume that the users are uiformly distributed withi the simulated sceario. The path-loss model for macrocell ad pico BS are give as PLdB = 34+40log 10 d ad PLdB = log 10 d [1], where d is the distace of user from the BS i m, ad therefore, PL c = 10 PLc db/10 ad PL PB = 10 PLPB db/10. The oise spectrum desity is assumed to be N 0 = 174dBm/Hz. I this wor, the power amplifier efficiecy is assumed as 38%, i.e., ǫ 0 = The imum trasmissio power for all users are same, hece, P will be referred to as P. The ormalizatio factors used i our wor are assumed to be = ǫ 0 P + P C ad θ SE =. All the simulatio results preseted are B m m K m averaged over 10,000 chael realizatios. The covergece of Algorithm I ad II for a give imum upli trasmissio power of P = 0.2 W is give i Fig. 1a ad Fig. 1b which show that Algorithms I ad II coverge to optimal values withi 4 ad 83 iteratios, respectively. Fig. 2 aalyses the imum achievable η versus varyig P for differet values of β. Fig. 2 reveals that η icreases with a icrease i β, whereas η first icreases with P. The after particular value of P, it starts decreasig due to the cosidered ratio, i.e., τ EE = P, i the optimisatio problem. For smaller values of P, the achievable η icreases whe P icreases. Furthermore, for higher values of P, the achievable η decreases with P.

5 Algorithm-II: Joit User associatio, Subcarrier ad Power Allocatio Iput: [η,β,ǫ 0,γ m ] Step 1: Iitialize i = 0, p m = 0,µi = 0.01, for = 1,,N, = 1,,K,m = 1,,. Step 2: For = 1 : K For = 1 : N Calculate p m accordig to 19. ed For Obtai the user associatio ad sub-carrier assigmet accordig to 20 respectively. ed For Step 3: i=i+1 Update µ i+1 accordig to 21. Step 4: Repeat steps 2 ad 3 util µ i+1 are coverged. Output: [ p m ], σm η b/j/hz Number of Iteratios a Algorithm I Power W x Number of Iteratios b Algorithm II Fig. 1: Covergece of Proposed Algorithms I & II. Fig. 3 shows the plots for ratio of optimal average trasmit power ad P versus weighted coefficiet β. I Fig. 3, Pη EE deotes the optimal trasmit power that imises EE or η at β = 0, whereas Pη deotes the proposed optimal trasmit power level that imises η at ay give value of β > 0. 2 It ca be see that the optimal trasmit power Pη mootoically icreases with β. Fig. 3 shows that at β close to 8.8, Pη coverges to the imum trasmissio power P = 0.2 W, whereas specifically whe P = 0.5 W, Pη coverges to the imum trasmissio power at β close to 12. This happes due to the fact that whe P icreases, the ormalisig factor θse decreases, which i tur results i reducig the impact of the tradeoff parameter β. This is a importat observatio idicatig that to achieve imum SE for higher values of 2 For clarity purpose, it is metioed that η = η. η b/j/hz P W Fig. 2: η versus P for differet values of β. P, a higher value of β eeds to be chose whe compared to a smaller value of P. Fig. 4 shows the plots for imum achievable EE ad SE at the optimal tradeoff trasmit power values as previously show i Fig. 3 versus β. It shows that SE is o-decreasig with respect to β, whereas EE is o-icreasig with β. Whe β is small, i.e., β = 0, the tradeoff solutio imise EE, whereas SE is imised whe β is large, i.e., β. Furthermore, both EE ad SE become costat as the trasmissio power approaches to P whe β is close to 8.8 i case of P = 0.2 W ad β 12.6 i case of P = 0.5 W. This pheomea justifies the fact that icreasig β gives more weightage to SE, ad therefore, more trasmit power is cosumed, ad i tur, higher SE ca be achieved. For example, for the case of P = 0.2 W ad required EE level of 120 b/j/hz, the optimal β = 3, which results i achievable SE of 15 b/s/hz. Similalry, for the requiremet to achieve average SE of 18 b/s/hz, the the optimal β = 10, which results i achievable EE of 61 b/j/hz. Ituitioally, we ca say that EE is always imised at β = 0 whereas SE is imised at differet values of β, which deped o the imum trasmissio power. We also study the impact of trasmissio power budget ratio to the imum available trasmissio power o the EE ad SE tradeoff. For example, the miimum achievable EE is 61 b/j/hz for P = 0.2 W ad drops to 34 b/j/hz for P = 0.5 W. Similarly, the imum SE is 18.1 b/s/hz for P = 0.2 W ad icreases to 19.5 b/s/hz for P = 0.5 W. This idicates that more power ca be saved by lowerig the imum trasmissio power which provides a good metric for gree commuicatios. The tradeoff betwee EE ad SE for various ormalised circuit power cosumptio values, i.e., w = PC P at β = 10, is show i Fig. 5. We observe that EE ad SE cotradicts each other whe the trasmit power is higher tha Pη EE. A small loss i EE ca result i a sigificat gai i SE. O the other had, both EE ad SE icrease whe the trasmit power is lower tha Pη EE. For w = 0, the EE-SE tradeoff curve is liear ad a icrease i w causes reductio i the EE. From Fig. 5, it is evidet that for P η > Pη EE, there is always a tradeoff betwee EE ad SE o matter how the parameter w chages. The lower the value of w, the flatter is the EE-SE tradeoff curve. β=0 β=1 β=3 β=5

6 Relative optimal power = P T / P P η * at P = 0.2 W P η * at P = 0.5 W * P ηee at P =0.2 W * P at P =0.5 W ηee Weighted coefficiet β Fig. 3: Optimal trasmit power versus weighted coefficiet β with P = 0.5 W, P C = 0.1 W, ad B m = 30 Hz. EEb/J/Hz EE at P =0.2 W EE at P =0.5W SE at P =0.2 W SE at P =0.5W Weightig coefficiet β Fig. 4: EE ad SE versus weighted coefficiet β for various values of P with P C = 0.1 W ad B m = 30 Hz. EEb/J/Hz w=0.25 w=0.375 w=0.5 w= SEb/s/Hz Fig. 5: EE versus SE at β = 10 for various ratios of P C P. SEb/s/Hz VI. CONCLUSIONS I this paper, the multi objective problem of simultaeously imizig EE ad SE of a upli of a two-tier OFDAbased HetNets with imum iput power costrait is solved. At first, the problem is coverted ito a SOP ad the is solved usig a two layer optimisatio approach i which the outer layer is solved by Dielbach method as show i Algorithm- I whereas the ier layer is solved usig LDD approach as show i Algorithm-II. Due to the quasi-cocavity ature of the proposed approach, the global optimal solutio is derived usig LDD. From the simulatio results, we ca obtai two mai observatios. Firstly, SE is imised at differet values of tradeoff factor β depedig o the imum trasmissio power. Secodly, the proposed tradeoff factor β ca help savig power by lowerig the operatioal power. The tradeoff performace, η, is a icreasig fuctio of trasmissio power for smaller values of P, whereas η is a decreasig fuctio of trasmissio power for higher values of P. REFERENCES [1] Q. Ye, B. Rog, Y. Che,. Al-Shalash, C. Caramais, ad J. Adrews, User associatio for load balacig i heterogeeous cellular etwors, IEEE Tras. o Wireless Commu.,, vol. 12, o. 6, pp , Jue [2] C. S. Che, F. Baccelli, ad L. Roullet, Joit optimizatio of radio resources i small ad macro cell etwors, i Proc. IEEE Vehicu. Tech. Cof. VTC-Sprig, Budapest, Hugary, ay 2011, pp [3] C. Zaraovitis ad Q. Ni, Eergy efficiet desigs for commuicatio systems: Resolutios o iverse resource allocatio priciples, IEEE Commu. Letters,, vol. 17, o. 12, pp , December [4] H. Pervaiz, L. usavia, ad Q. Ni, Joit user associatio ad eergyefficiet resource allocatio with miimum-rate costraits i two-tier hetets, i Proc. IEEE It. Sym. o Persoal Idoor ad obile Radio Commuicatios PIRC, Sept. 2013, pp [5] O. Galiia, S. Adreev, A. Turliov, ad Y. Koucheryavy, Optimizig eergy efficiecy of a multi-radio mobile device i heterogeeous beyod-4g etwors, Performace Evaluatio, vol. 78, o. 0, pp , [Olie]. Available: [6]. Ismail, A. Gamage, W. Zhuag, ad X. She, Eergy efficiet upli resource allocatio i a heterogeeous wireless medium, i Proc. IEEE It. Cof. o Commuicatios ICC, Jue 2014, pp [7] L. P. Qia, C. Qia, Y. Wu, ad Q. Che, Power cotrolled system reveue imizatio i large-scale heterogeeous cellular etwors, i Proc. IEEE It. Cof. o Commuicatios ICC, Jue 2014, pp [8] G. Araiti, J. Cosmas, A. Iera, A. Loiacoo, A. oliaro, ad A. Orsio, Power cosumptio model usig gree policies i heterogeeous etwors, i Proc. IEEE It. Sym. o Broadbad ultimedia Systems ad Broadcastig BSB, Jue 2014, pp [9] S. Samaraoo,. Beis, W. Saad, ad. Latva-aho, Opportuistic sleep mode strategies i wireless small cell etwors, i Proc. IEEE It. Cof. o Commuicatios ICC, Jue 2014, pp [10] S. Khaurel, L. usavia, ad T. Le-Ngoc, Trade-off betwee spectral ad eergy efficiecies i a fadig commuicatio li, i Proc. IEEE Vehicu. Tech. Cof. VTC-Sprig, Jue 2013, pp [11] C. Xiog, G. Y. Li, S. Zhag, Y. Che, ad S. Xu, Eergy-ad spectralefficiecy tradeoff i dowli OFDA etwors, IEEE Tras. o Wireless Commu.,, vol. 10, o. 11, pp , [12] R. T. arler ad J. S. Arora, Survey of multi-objective optimizatio methods for egieerig, Structural ad multidiscipliary optimizatio, vol. 26, o. 6, pp , [13] C. He, G. Li, F.-C. Zheg, ad X. You, Eergy-efficiet resource allocatio i OFD systems with distributed ateas, IEEE Tras. o Vehicu. Tech.,, vol. 63, o. 3, pp , arch [14] S. Boyd ad L. Vadeberghe, Covex optimizatio, Cambridge Uiversity Press, Cambridge, UK, [15] D. P. Palomar ad. Chiag, A tutorial o decompositio methods for etwor utility imizatio, IEEE Joural o Selected Areas i Commu.,, vol. 24, o. 8, pp , 2006.