Energy-Efficient Scheduling and Power Allocation in Downlink OFDMA Networks with Base Station Coordination

Transcription

1 1 Energy-Efficient Scheduing and Power Aocation in Downink OFDMA Networks with Base Station Coordination Luca Venturino, Meber, IEEE, Aessio Zappone, Meber, IEEE, Chiara Risi, and Stefano Buzzi, Senior Meber, IEEE arxiv: v1 [cs.it] 12 May 214 Abstract This paper addresses the probe of energy-efficient resource aocation in the downink of a ceuar OFDMA syste. Three definitions of the energy efficiency are considered for syste design, accounting for both the radiated and the circuit power. User scheduing and power aocation are optiized across a custer of coordinated base stations with a constraint on the axiu transit power either per subcarrier or per base station. The asyptotic noise-iited regie is discussed as a specia case. Resuts show that the axiization of the energy efficiency is approxiatey equivaent to the axiization of the spectra efficiency for sa vaues of the axiu transit power, whie there is a wide range of vaues of the axiu transit power for which a oderate reduction of the data rate provides a arge saving in ters of dissipated energy. Aso, the perforance gap aong the considered resource aocation strategies reduces as the out-of-custer interference increases. Index Ters Green counications, energy efficiency, resource aocation, scheduing, power contro, ceuar network, downink, base station coordination, OFDMA. I. INTRODUCTION Orthogona Frequency Division Mutipe Access OFDMA is the eading utiaccess technoogy in current wireess networks, ainy due to its abiity to cobat the effects of utipath fading [1]. In order to increase the capacity of OFDMA-based networks, attention has been devoted to the derivation of adaptive resource aocation schees, which take into account factors such as traffic oad, channe condition, and service quaity. In particuar, base station BS coordination has eerged as an effective strategy to itigate downink co-channe interference. Assuing that the data sybos are known ony by the serving BS, severa papers have shown that joint scheduing and power contro aong a set of coordinated BSs based on channe quaity easureents can greaty iprove the network su-rate [2] [1]. The work of Aessio Zappone has received funding fro the Geran Research Foundation DFG project CEMRIN, under grant ZA 747/1-2. The work of Stefano Buzzi and Chiara Risi has been funded by the European Union Seventh Fraework Prograe FP7/ under grant agreeent n Network of Exceence "TREND". This work was party presented at the 213 Future Networks and Mobie Suit, Lisbon, Juy 213, and at the 213 IEEE Internationa Syposiu on Persona, Indoor and Mobie Radio Counications, London, United Kingdo, Septeber 213. L. Venturino, C. Risi and S. Buzzi are with CNIT and DIEI, University of Cassino and Lazio Meridionae, Cassino FR, Itay e-ai: {.venturino,chiara.risi,buzzi}@unicas.it. A. Zappone is with the Dresden University of Technoogy, Counication Theory Laboratory, TU Dresden, Gerany e-ai: Aessio.Zappone@tu-dresden.de. On the other hand, environenta and econoic concerns require to aso account for the energy efficiency of a data network [11]. This topic has recenty gained big oentu and a nuber of specia issues, conferences, and research projects have been devoted to green counications in the ast few years: see, for instance, [12] [14], as the tip of the iceberg. Energy-aware design and panning is otivated by the fact that wireess networks are responsibe of a fraction between.2 and.4 percent of tota carbon dioxide eissions [15], and this vaue is expected to grow due to the ever-increasing nuber of subscribers. Indeed, energy efficiency wi be a key issue aso in future fifth-generation ceuar networks [16]. The biggest efforts to increase the energy efficiency of a wireess network are concentrated on the access network, since it consues the argest portion of energy [17]. Here, potentia soutions incude energy-saving agoriths for switching on and off BSs that are either inactive or very ighty oaded [18], energy-efficient hardware soutions [19], and energyefficient resource aocation agoriths. Focusing on this atter issue, in [2] [24], the energy efficiency is defined as the ratio between the throughput and the transit power, and the transit power eve axiizing the aount of data bits successfuy deivered to the receiver for each energy unit is derived. A ore genera definition of the energy efficiency is obtained when the circuit power dissipated to operate the devices is incuded as an additive constant at the denoinator. This approach has been considered in [25], where power contro for direct-sequence code division utipe access utiuser networks is tacked, in [26], where energy efficient counication in a singe-user utipe-input utipe-output MIMO syste is studied, and in [27], where power contro in reay-assisted wireess networks is considered. In [28], [29], severa odes for the circuit power consuption in wireess networks are eaborated. In [3], the tradeoff between energyand spectra-efficient transission in uticarrier systes is investigated. The papers [31], [32] focus on the upink of an OFDMA syste; the forer considers a singe-ce syste and derives ow-copexity scheduing and power contro strategies, whie the atter uses a gae-theoretic approach to derive decentraized resource aocation strategies for a utice OFDMA syste. With regard to the downink of an OFDMA syste, recent contributions incude [33], [34]. The paper [33] uses fractiona prograing to derive precoding coefficients, transit power, and user-subcarrier association for energy efficiency axiization in a uti-ce syste. The

2 2 paper [34], instead, investigates the tradeoff between energy efficiency and nuber of transit antennas. This paper considers the downink of a uti-ce OFDMA syste, where a nuber of BSs for a custer, share inforation on channe quaity easureents, and coaborate in order to perfor energy-efficient user scheduing and power contro on the sae radio spectru 1. The contributions of this work are suarized as foows. Three figures of erit reated to the energy efficiency of the coordinated BSs are considered, naey, the ratio between the su-rate and the power consuption, which is referred to as goba energy-efficiency GEE, the weighted su of the energy efficiencies achieved on each resource sot Su-EE, and the exponentiay-weighted product of the energy efficiencies achieved on each resource sot Prod-EE. These figures of erit capture different features of the considered counication syste, which we iustrate and discuss. Previous reated works have ainy focused on the axiization of GEE, but for different syste settings. To the best of our knowedge, the work which considers the scenario ost siiar to ours is [33]; however, whie [33] assues that users are associated to a BSs in the custer, a configuration usuay referred to as virtua or network MIMO, we consider a scenario wherein each user is associated to ony one BS. As to Su-EE and Prod-EE, they have been considered in non-cooperative gaes [32], [35], but not in the context of coordinated ceuar networks. We derive nove procedures aied at axiizing the above figures of erit with a constraint on the axiu transit power either per subcarrier or per base station: this is the ajor contribution of this work. GEE is optiized by soving a series of concave-convex fractiona reaxations, whie for Prod-EE a series of concave reaxations is considered. In both cases, the proposed procedures onotonicay converge to a soution which at east satisfies the first-order optiaity conditions of the origina probe. As to Su-EE, we propose an iterative ethod to sove the Karush Kuhn Tucker KKT conditions of the corresponding non-convex probe. For a figures of erit, we derive agoriths to copute a gobay-optia soution in the asyptotic noise-iited regie. Nuerica resuts indicate that the optiization of the considered figures of erit gives siiar perforance for ow vaues of the axiu transit power; in this case, axiizing the network energy efficiency is aso approxiativey equivaent to axiizing the network spectra efficiency. For arge vaues of the axiu transit power, a oderate reduction of the network spectra efficiency ay aow a significant energy saving; in this regie, Su-EE and Prod-EE aow to better contro the individua energy efficiency achieved by each BS than GEE, which is an attractive feature in heterogeneous networks. Aso, Prod-EE ensures a ore baanced use of the avaiabe subcarriers at the price of a ore severe oss 1 Our approach appies to both frequency- and tie-division dupexing. in ters of network spectra efficiency. The reainder of this paper is organized as foows. Section II contains the syste description and the probe foruation. Sections III, IV, and V contain the design of the agoriths axiizing GEE, Su-EE, and Prod-EE, respectivey. The nuerica resuts are presented in Section VI. Finay, concuding rearks are given in Section VII. II. SYSTEM DESCRIPTION We consider a custer of M coordinated BSs in the downink of an OFDMA network epoying N subcarriers and universa frequency reuse. Users and BSs are equipped with one receive and one transit antenna, respectivey. Each user is connected to ony one BS, which is seected based on ong-ter channe quaity easureents. We denote by B the non-epty set of users assigned to BS and assue that each BS serves at ost one user at a tie on each subcarrier. We consider an infinitey backogged traffic ode wherein each access point aways has data avaiabe for transission to a connected users. Aso, we assue that the channes reain constant during each transission frae, and that each user can accuratey estiate the channes fro the coordinated BSs to itsef and feedback the to its serving BS. A. Signa ode Assuing perfect synchronization, the discrete-tie baseband signa received by user s B on subcarrier n is y [n] s = H,sx [n] [n] + }{{} in-ce data H [n],s x[n] =1, } {{ } out-of-ce data + n [n] s. 1 }{{} noise In 1, H q,s [n] is the copex channe response between BS q and user s on subcarrier n, which incudes sa scae fading, arge scae fading and path attenuation [1], whie x [n] q is the copex sybo transitted by BS q on subcarrier n. The transitted sybos are odeed as independent rando variabes with zero ean and variance E{ x [n] 2 } = Finay, n [n] s is the additive noise received by user s, which is odeed as a circuary-syetric, Gaussian rando variabe with variance N s [n] /2 per rea diension. Different noise eves at each obie account for different eves of the out-of-custer interference and for different noise figures of the receivers. The signa-to-interference-pus-noise ratio SINR for BS on subcarrier n when serving user s B is SINR [n],s = 1 + G [n],s =1, G[n],s with G [n] q,s = H q,s [n] 2 /N s [n] ; aso, the corresponding achievabe inforation rate in bit/s is [36] [ ] R [n],s = B og SINR [n],s 3 where B is the bandwidth of each subcarrier. 2

3 3 B. Power ode Foowing [17], [28], [29], the consued power is odeed as the su of two ters, accounting for the power dissipated in the apifier and in the RF transit circuits, respectivey. The power dissipated in the apifier is expressed as γp, with p and γ 1 being the transit power output by the apifier and a scaing coefficient which accounts for the apifier and feeder osses. Instead, the power dissipated in the reaining circuit bocks is odeed as a constant ter θ >, which accounts for battery backup and for signa processing carried out in the ixer, frequency synthesizer, active fiters, and digita-to-anaog converter. Both θ and γ generay scae with the additiona osses incurred by the power suppy and/or the cooing equipent. Accordingy, the power consued by BS on subcarrier n is written as P c, [n] = θ [n] + γ [n]. 4 C. Energy-efficient resource aocation Let k, n B indicate the user schedued by BS on subcarrier n and define k [n] = k1, n,..., km, n T and k = vec{k [1]..., k [N] }. Aso, we define = 1,..., p[n] M T and p = vec{p [1],..., p [N] }. Syste optiization requires seecting k and p so to axiize a eaningfu figure of erit under soe physica constraint. In this work, we ai at axiizing the network energy efficiency. We consider three figures of erit, which encapsuate different aspects reated to the energy efficiency of the considered coordinated custer. The first one is the goba energy efficiency, defined as the ratio between the network su-rate and the network power consuption, i.e, GEEp, k = =1 =1 R [n],k,n θ [n] + γ [n]. 5 Another eaningfu figure of erit is the weighted su of the energy efficiencies across a subcarriers and BSs, i.e., Su-EEp, k =,k,n =1 θ [n] R [n],k,n + γ [n] where the weight w,s, [n] for = 1,..., M, n = 1,..., N and s B, ay account for the priority of the schedued users, the nature of the coordinated BSs, and the services assigned to each subcarrier. Differenty fro GEE, this figure of erit is we suited for heterogenous networks, as the weights can now be used to contro the energy efficiency achieved on a specific MN, then Su-EE is the arithetic ean of the energy efficiencies across a subcarriers and BSs. Finay, we consider the exponentiay-weighted product of the energy efficiencies across a subcarriers and BSs, i.e., subcarrier or BS. If we choose,s = 1 Prod-EEp, k = M N =1 θ [n] R[n],k,n + γ [n],k,n 6. 7 Due to its utipicative nature, axiization of 7 eads to a configuration where a subcarriers are aways used for transission by a BSs, which ay not be the case when GEE or Su-EE are considered. Therefore, axiizing Prod- EE eads to a ore baanced power aocation on the different subcarriers, aowing a siper design of the transit apifiers. Prod-EE aso grants the possibiity to tune the energy efficiency of each subcarrier through the choice of the weights. MN, Prod-EE is the geoetric ean of the energy efficiencies across a subcarriers and BSs. In the foowing, we present agoriths aied at axiizing the above figures of erit under per-bs or per-subcarrier power constraints. In keeping with a coon trend in the open iterature, we assue perfect channe state inforation and optiize the GEE, Su-EE, and Prod-EE based on instantaneous channes. An aternative approach, that is however out of the scope of this work, is to perfor resource aocation based on ong-ter variations of the channe [37], [38]. The proposed agoriths require to run in a centraized controer, which coects the channe easureents fro the coordinated BSs and outputs the scheduing and the transit power for each BS and subcarrier. This copexity overhead is expected to be affordabe with the currenty avaiabe technoogy. Indeed, with the advent of the software defined networking paradig and of the coud radio access network architecture, ceuar systes wi be ade of ight BSs perforing ony baseband to radio frequency conversion, whie neighboring BSs wi be connected via high-capacity inks to a centra unit perforing ost of the data processing [39], [4]. Ceary, our ethods we fit in this context. 2 Aso, BS coordination is usuay required ony for obie users that are at the edge of the ces, i.e., idway aong two or ore BSs; as a consequence, the nuber of obie terinas invoved ay be ony a sa fraction of the overa set of active users. For,s = 1 III. OPTIMIZATION OF GEE In this section, we study the axiization of 5. We first consider a per-bs power constraint and, then, we speciaize the resuts to the case of a per-subcarrier power constraint. With reference to the ore genera per-bs power constraint, the noise-iited scenario is aso addressed: considering this probe is interesting, since it eads to siper resource aocation agoriths that can be epoyed when the interce interference is weak and, hence, can be negected. A. Per-BS power constraint The probe to be soved is arg ax GEEp, k p,k s.t. p [n] P,ax,, k, n B,, n 2 Notice that the Coordinated Muti-Point CoMP transission has been recenty introduced in LTE-Advanced [41]; the CoMP transission invoves a higher degree of cooperation and inforation sharing aong BSs than the proposed resource aocation schees, yet it is feasibe. 8

4 4 where P,ax is the axiu power that can be radiated by BS. For any feasibe p, the optiization over k is separabe across BSs and subcarriers, and the soution is given by ˆk, n = arg ax R [n],s 9 s B for = 1,..., M and n = 1,... N. Next, observe that for any z and z, 3 the foowing inequaity hods [42]: og z α og 2 z + β 1 where α and β are defined as α = z 1 + z, β = og 21 + z z 1 + z og 2 z 11 and the bound is tight for z = z. As a consequence, for a given feasibe user seection k, the foowing ower bound to the objective function is obtained GEEp, k hp, k = where α [n] fp,k {}}{ [ ] B α [n] og 2 SINR [n],k,n + β [n] =1 and β [n] =1 θ [n] + γ [n] } {{ } gp 12 are approxiation constants coputed as in 11 for soe z [n] to be specified in the foowing. Consider now the transforation q [n] = n, and define q [n] = q [n] 1,..., q[n] M T, q = vec{q [1],..., q [N] }, and Q = {q R MN : N exp{q[n] } P,ax, }. We have the foowing resut, whose proof is reported in Appendix A. Lea 1. fexp{q}, k is a concave function of q with ax q Q fexp{q}, k. Aso, gexp{q} is a positive, convex function of q. Leveraging the above Lea, we proposed to sove 8 by iterativey optiizing the power aocation according to the ower bound in 12, coputing the best user seection according to 9, and tightening the bound in 12, as suarized in Agorith 1. As a consequence of Lea 1, the concave-convex fractiona probe in 13 can be soved using Dinkebach s procedure [43] outined in Agorith 2, 4 whie standard techniques can be used to sove the concave axiization in 14 [44]. As to Agorith 1, we have the foowing resut, whose proof is reported in Appendix B. Proposition 1. Agorith 1 onotonicay iproves the vaue of GEE at each iteration and converges. Aso, the soution obtained at convergence satisfies the KKT conditions for 8. Notice that the KKT conditions are first-order necessary conditions for any reative axiizer of 8, as the Sater s constraint quaification hods [45]. 3 We use the convention that og 2 = and og 2 =. 4 Here, the variabe ɛ denotes the required toerance, whie the indicator FLAG rues the exit fro the iterative repeat cyce. Siiar considerations appy to Agorith 4. Agorith 1 Proposed procedure to sove 8 1: Initiaize I ax and set i = 2: Initiaize p and copute k according to 9 3: repeat 4: Set z [n] = SINR [n],k,n and copute α[n] and β [n] as in 11, for = 1,..., M and n = 1,..., N 5: Update p by soving the foowing probe using Agorith 2 p = exp{q}: 6: Update k according to 9 7: Set i = i + 1 8: unti convergence or i = I ax arg ax hexp{q}, k 13 q Q Agorith 2 Dinkebach s procedure [43] to sove 13 1: Set ɛ >, π =, and FLAG = 2: repeat 3: Update q by soving the foowing concave axiization: arg ax fexp{q}, k πgexp{q} 14 q Q 4: if fexp{q}, k πgexp{q} < ɛ then 5: FLAG = 1 6: ese 7: Set π = fexp{q}, k/gexp{q} 8: end if 9: unti FLAG = 1 1 Noise-iited NL regie: Negecting the interce interference, GEE sipifies to GEE-NLp, k = f NLp,k {}}{ B og G [n],k,n =1 =1 θ [n] + γ [n] and the probe to be soved becoes arg ax GEE-NLp, k p,k s.t. P,ax,, k, n B,, n. 15 We now have the foowing resuts, whose proof is reported in Appendix C. Proposition 2. Agorith 3 onotonicay iproves the vaue of GEE-NL at each iteration, and provides a gobay optia soution to 15. Notice that the concave-inear fractiona probe 17 in Agorith 3 can be soved by using Agorith 4. Aso, the soution to the concave axiization 18 can be found fro the KKT conditions; in particuar, after standard anipuation

5 5 Agorith 3 Proposed procedure to sove 15 in the noise iited regie 1: Initiaize I ax and set i = 2: Initiaize p and copute k according to k, n = arg ax og G [n],s 16 s B for = 1,..., M and n = 1,..., N. 3: repeat 4: Update p by soving the foowing probe using Agorith 4: arg ax GEE-NLp, k p s.t. N p[n] 5: Update k according to 16 6: Set i = i + 1 7: unti convergence or i = I ax P,ax,,, n 17 Agorith 4 Dinkebach s procedure [43] to sove 17 1: Set ɛ >, π =, and FLAG = 2: repeat 3: Update p by soving the foowing concave axiization: arg ax f NLp, k πgp p s.t. N p[n] P,ax,,, n 4: if f NL p, k πgp < ɛ then 5: FLAG = 1 6: ese 7: Set π = f NL p, k/gp 8: end if 9: unti FLAG = 1 18 we obtain the foowing M waterfiing-ike probes one for each BS = ax, B/ n 2 πγ [n],k,n + λ P,ax 1 G [n],k,n for = 1,..., M, and the optia vaue of the non-negative Lagrange utipier λ can be derived by bisection search. B. Per-subcarrier power constraint The probe to be soved is { arg ax GEEp, k p,k s.t. P [n],ax, k, n B,, n. where P [n],ax is the axiu power that can be radiated by BS on subcarrier n. GEE is not a separabe function of p, whereby the above optiization probe does not decoupe across subcarriers. Luckiy enough, a derivations carried out in Section III-A can be repicated here with inor odifications. In particuar, Agoriths 1 and 2 reain the sae, except that the feasibe set Q in 13 and 14 ust be redefined as Q = {q R MN : exp{q [n] } P,ax, [n], n}. Interestingy, the soution to the concave axiization in 14 can now be coputed with a sipe iterative ethod. Indeed, consider the foowing KKT conditions for 14 [44]: d dq [n] λ [n] fexp{q}, k πgexp{q},, n λ [n] exp{q [n] } =,, n exp{q [n] } P,ax, [n], n λ [n] P,ax [n] exp{q [n] } =,, n 19 where λ [n] is the Lagrange utipier associated to the power constraint of BS on subcarrier n. After soe anipuations, the stationary condition in 19 can be recast as exp{q [n] } = α [n] B B j=1,j [ γ [n] π + λ [n] α [n] j G [n],kj,n 1 + M =1, j exp{q[n] n 2+ }G [n],kj,n ] 1. 2 Since the right hand side RHS of 2 is a standard interference function [46], the optia q can be obtained by starting fro any feasibe power aocation and iterativey soving the foowing fixed point equations: exp{q [n] } = in γ [n] π n 2+B j=1,j P [n],ax, α [n] B α [n] j G [n],kj,n 1 + M =1, j exp{q[n] for = 1,..., M and n = 1,..., N. IV. OPTIMIZATION OF SUM-EE }G [n],kj,n In this section, we study the axiization of 6 under a per-bs power constraint, i.e., arg ax Su-EEp, k p,k s.t. P,ax,, k, n B,, n. 21 Since Su-EE is a separabe function of the power variabes, the foowing resuts speciaize to a per-subcarrier power constraint in a straightforward anner; the corresponding detais

6 6 are oitted for brevity. Paraeing Section III, Probe 21 is aso studied in the noise-iited scenario. For a given feasibe p, the optiization over k is separabe across BSs and subcarriers, and the soution is given by ˆk, n = arg ax s B,sR [n],s 22 for = 1,..., M and n = 1,... N. On the other hand, for any feasibe user seection, the optia set of powers ust satisfy the foowing KKT conditions: d d Su-EEp, k + µ [n] λ =,, n 23a µ [n], λ,, n 23b,, n 23c P,ax, 23d µ [n] =,, n 23e λ P,ax =, 23f where λ and µ [n] are the Lagrange utipiers associated to the constraints on the axiu power radiated by BS and on the iniu power eve of BS on subcarrier n, respectivey. After standard agebraic anipuations, it can be shown that d d Su-EEp, k = B/ n 2Q [n],k,n G [n],k,n 1 + I [n],k,n + p[n] G [n],k,n where I [n],k,n = Q [n],k,n = M =1, θ [n] C [n],k,n = w[n] L [n],k,n = B n 2 w[n],k,n + γ [n],k,n γ[n] j=1,j whereby 23a can be rewritten as = C [n],k,n L[n],k,n G[n],k,n 24 R [n],k,n θ [n] + γ [n] Q [n] G [n] j,kj,n 1 + B/ n 2Q [n],k,n λ µ [n] + C [n],k,n + L[n],k,n Notice that I [n],k,n ,kj,n SINR[n] j,kj,n =1 G[n],kj,n 1 + I[n] 27,k,n G [n],k,n 28 is the co-channe interference for the user schedued by BS on sub-carrier n; Q [n],k,n. is an equivaent weight for the rate achieved by BS on sub-carrier n, scaed by the corresponding power consuption; C [n],k,n is a argina power cost paid by BS for transitting on subcarrier n to user k, n; finay, L [n],k,n accounts for the interference eakage to undesired receivers when serving user k, n. The stationary condition 28 indicates that, for any given set of schedued users k, BS shoud aocate ore power to subcarriers having arger equivaent weights and experiencing better channe conditions; aso, the taxation ters C [n],k,n and L[n],k,n ower the radiated power if the argina power price paid by BS to transit on subcarrier n to user k, n is arge and if this transission causes an excessive eakage to other co-channe users, respectivey. Inspired by the odified waterfiing ethods considered in [8], [47] for the weighted su-rate axiization, we now provide an iterative procedure to sove 22 and 23, which together are the first-order necessary conditions for the optia soutions to 21, as the Sater s constraint quaification hods [45]. Assue that soe feasibe p and k are given. Then, the equivaent weight Q [n],k,n, the argina power cost C [n],k,n, and the interference eakage L[n],k,n can be coputed fro 25, 26, and 27, respectivey, for = 1,..., M and n = 1,..., N. Then, each BS can copute the co-channe interference eves on each subcarrier fro 24 and update the corresponding radiated powers using 28; notice that the Lagrange utipiers ust be chosen so as to satisfy the power constraints 23c-23d and the corresponding copeentary sackness conditions 23e-23f, resuting in the foowing waterfiing-ike probes one for each BSs: = ax, B/ n 2Q λ + C [n] P,ax [n],k,n,k,n + L[n],k,n 1 + I [n],k,n G [n],k,n 29 for = 1,..., M, which in turn can be efficienty soved by bisection search. After updating the power eve, the schedued users can be recoputed according to 22, and the entire process can be iterated as suarized in Agorith 5 Deriving genera conditions under which Agorith 5 provaby converges sees intractabe. Nevertheess, shoud Agorith 5 converge to a feasibe soution, then the corresponding set of radiated powers, schedued users, and Lagrange utipiers satisfy by construction the KKT conditions in 22 and 23. Agorith 5 can be odified to enforce onotonic convergence by perforing the power update at ine 7 ony if the objective function is non decreased. 5 However, in this atter case, the soution at convergence is not guaranteed to siutaneousy satisfy 22 and In our experients in Section VI we have aways observed convergence of Agorith 5 without perforing this odification.

7 7 Agorith 5 Proposed procedure to sove 21 1: Initiaize I ax and set i = 2: Initiaize p and copute k according to 22 3: repeat 4: Copute Q [n],k,n, C[n],k,n and L [n],k,n, for = 1,..., M and n = 1,..., N, according to 25, 26 and 27, respectivey 5: for = 1 to M do 6: Copute I [1],k,1,..., I[N],k,N according to 24 7: Update p [1],..., p [N] according to 29 8: end for 9: Update k according to 22 1: i = i : unti convergence or i = I ax A. Insights into Agorith 5 Let I [n],k,n, Q[n],k,n, C[n],k,n, and L[n],k,n be preassigned and fixed, for = 1,..., M and n = 1,..., N. For a given set of schedued users k, a set of power eves which satisfy 23b-23f and 28 ust aso satisfy the KKT conditions of the foowing probes one for each BS: arg ax Q [n] p [1],...,p [N],k,n B og p[n] G [n],k,n 1 + I [n],k,n [ ] C [n],k,n + L[n],k,n s.t. P,ax, 3 for = 1,..., M. The first ter of the objective function in 3 is a weighted su of the rates achieved by BS on its subcarriers with weights Q [n],k,n I [n],k,n and interference eves. Aso, C[n],k,n p[n] are costs paid by BS to serve user k, n schedued on subcarrier n due to the power consuption i.e., the energy efficiency and the interference caused to other co-channe users, respectivey. Notice that 3 is a concave axiization; consequenty, the and L [n],k,n p[n] Lagrange utipier λ and the power eves p [1],..., p [N] coputed at ine 7 of Agorith 5 can aso be found by soving 3 with any convex optiization too [44]. B. Noise-iited regie Negecting the interce interference, Su-EE sipifies to Su-EE-NLp, k = B =1 og G [n],k,n,k,n θ [n] + γ [n] 31 which is a separabe function with respect to both p and k. For a fixed p, a soution to { arg ax Su-EE-NLp, k k s.t. k, n B,, n. is given by k, n = arg ax w,s [n] og G [n],s s B 32 for = 1,..., M and n = 1,..., N. Aso, for a fixed k, the reaxed probe {arg ax Su-EE-NLp, k p 33 s.t.,, n has a unique soution, say p, which is found by separatey axiizing each suand in the objective function [48]. We now give the foowing resut, whose proof is reported in Appendix D. Lea 2. Let ux = og 21 + ax with a and c positive x + c constants. The function u is concave in the region x x, where x is the unique soution to arg ax x ux. Using Lea 2, the power aocation probe in the noiseiited regie can be recast as a concave probe arg ax Su-EE-NLp, k p s.t. N p[n] P,ax,,, n 34 where p is the soution to 33. Consequenty, the optia resource aocation can be found by aternate axiization of 31 over the variabes k and p according to 32 and 34, respectivey. V. OPTIMIZATION OF PROD-EE In this section, we study the axiization of 7 under a per-bs power constraint. Since Prod-EE is separabe with respect to p, the foowing resuts can be speciaized to a persubcarrier power constraint in a straightforward anner. The probe to be soved is arg ax n Prod-EEp, k p,k s.t. P,ax,, k, n B,, n 35 where, without oss of optiaity, the objective function is the ogarith of 7. Notice first that a soution to 35 ust necessariy have R [n],k,n > for = 1,..., M and n = 1,..., N. Aso, for any feasibe p, the axiization with respect to k decoupes across BSs and subcarriers, yieding k, n = arg ax s B,s n θ [n] R [n],s + γ [n] 36 for = 1,..., M and n = 1,..., N. As to the optiization with respect to p, we consider the foowing ower bound to the objective function n Prod-EEp, k φp, k = n α[n] og 2 SINR [n] θ [n] + γ [n],k,n =1,k,n /B + β [n] 37

8 8 Agorith 6 Proposed procedure to sove 35 1: Initiaize I ax and set i = 2: Initiaize p and copute k according to 36 3: repeat 4: Set z [n] = SINR [n],k,n and copute α[n] and β [n] as in 11, for = 1,..., M and n = 1,..., N 5: Copute q as the soution to the concave probe 38 and update p = exp{q} 6: Update k according to 36 7: Set i = i + 1 [k] : unti convergence or i = I ax where α [n] and β [n] are coputed as in 11 for soe z [n] to be specified in the foowing. The above bound hods for a p such that the arguent of n is non-negative. Using the transforation p = exp{q}, the foowing reaxed power aocation probe is obtained: arg ax φexp{q}, k q s.t. e q[n] P,ax, α [n] G [n],k,n og 2 exp{q[n] } + β [n], 1+ G [n],k,n exp{q[n] } =1,, n. 38 Since og-su-exp is convex, the constrained axiization in 38 is concave and, hence, can be soved using standard techniques [44]. We now propose to sove 35 by iterativey optiizing the power aocation according to 38, coputing the best user seection according to 36, and tightening the bound in 37, as suarized in Agorith 6. The foowing convergence resut now hods; the proof is siiar to that of Proposition 1 and is oitted for brevity. Proposition 3. Agorith 6 onotonicay iproves the vaue of Prod-EE at each iteration and converges. Aso, the soution obtained at convergence satisfies the KKT conditions for 35. Notice again that, since the Sater s constraint quaification hods [45], the KKT conditions are first-order necessary conditions for any reative axiizer of 35. As to the noise iited regie, notice that the objective function in 35 sipifies to B og G [n],k,n,k,n n =1 θ [n] + γ [n] and a derivations in Section IV-B can be repicated here. VI. NUMERICAL RESULTS In this section, we study the syste perforance via Monte- Caro siuations. We consider the wireess ceuar network [k] Figure 1. Siuated ceuar network: BSs 1, 2, and 3 are coordinated, and users are randoy dropped in the grey area. in Figure 1. BSs 1, 2, and 3 coordinate their transission hence, M = 3 on N = 16 subcarriers with bandwidth B = 18 khz. Each BS serves three users, i.e., B = 3, which are unifory distributed in the grey area. As to the power ode, θ [n] 1 =.25 W, θ [n] 2 =.5 W, θ [n] 3 =.75 W, and γ [n] 1 = γ [n] 2 = γ [n] 3 = 3.8, for n = 1,..., 16, which are typica vaues for LTE systes [17]. On each subcarrier, we consider Rayeigh fading, Log-Nora shadowing with standard deviation 8 db, and the path-oss ode PLd = PL d /d 4, where d d is the distance in eters, and PL is the freespace attenuation at the reference distance d = 1 with a carrier frequency of 18 MHz [1]. Foowing [49], the noise variance N s [n] which accounts for the power of both the thera noise and the out-of-custer interference is odeed as N s [n] = F N B + P }{{} out PL thera noise j I 19 2 d d j,s 4 ξ [n] j,s } {{ } out-of-custer interference where F = 3 db is the noise figure of the receiver, N = 174 db/hz is the power spectra density of the thera noise, I = {4, 5,..., 27} is the set of uncoordinated BSs in Figure 1, is the average power radiated by the uncoordinated BSs on each subcarrier, d j,s is the distance fro BS j to user s, and ξ [n] j,s is the Log-Nora shadowing we assue that users ony track ong-ter interference eves fro uncoordinated BSs and, hence, short-ter fading is averaged out. Notice that = corresponds to the case in which the custer of coordinated BSs is isoated. The foowing anaysis refers to a per-subcarrier power constraint with P,ax [n] = P ax /N; a per-bs power constraint showed in our experients a siiar behavior and, hence, is not iustrated for brevity. Uness otherwise stated, the weights

9 9 15 = 15 P ax = 35 db 18 = 18 P ax = 35 db GEE [kbit/joue] Su EE [kbit/joue] P [db] ax GEE axiization Su EE axiization Prod EE axiization Su rate axiization Maxiu power [db] P [db] ax GEE axiization 2 Su EE axiization Prod EE axiization Su rate axiization Maxiu power [db] Figure 2. GEE vs P ax for = eft; GEE vs for P ax = 35 db right. Figure 3. Su-EE vs P ax for = eft; Su-EE vs for P ax = 35 db right.,k,n in 6 and 7 are set to 1/MN. Finay, a pots are obtained after averaging over 1 independent user drops. A. Ipeentation of the proposed agoriths A agoriths are initiaized by assuing that the BSs transit at the axiu power on each subcarrier. Moreover, etting f be the vaue of the objective function at iteration. The oop is stopped if f f 1 /f 1 < 1 4 or a axiu nuber of 5 iterations has been reached. The resource aocation strategies axiizing GEE, Su- EE, and Prod-EE are referred to as GEE-opt, Su-EE-opt, and Prod-EE-opt, respectivey. For the sake of coparison, we aso show the perforance obtained by transitting at the axiu power and by the resource aocation strategy in [8] axiizing the network su-rate, i.e., the nuerator of GEE in 5, which is referred to as su-rate-opt. Prod EE [kbit/joue] = P ax = 35 db GEE axiization Su EE axiization Prod EE axiization Su rate axiization Maxiu power B. Perforance resuts Figures 2 6 show GEE, Su-EE, Prod-EE, su-rate, and the power radiated by each BS, respectivey, for a considered resource aocations. In each figure, the subpot on the eft refers to an isoated custer, and the resuts are shown versus P ax ; the subpot on the right refers to a non-isoated custer, and the resuts are shown versus for P ax = 35 db. For an isoated custer, a soutions provide siiar perforance when P ax 1 db, since the radiated power consuption is negigibe with respect to the static power consuption and, aso, the cochanne interference is sa copared to the noise power. For arger vaues of P ax, instead, the considered figures of erit ead to different resource aocation strategies and, consequenty, syste perforance. In this regie, the su-rate-opt soution increases the network su-rate at the price of a heavy degradation in the syste energy efficiency, P ax [db] [db] Figure 4. Prod-EE vs P ax for = eft; Prod-EE vs for P ax = 35 db right. no atter which definition of energy efficiency is considered GEE, Su-EE, or Prod-EE. On the other hand, the GEEopt, Su-EE-opt, and Prod-EE-opt soutions exhibit a foor as P ax increases, since they do not use the excess avaiabe power to further increase the rate, as shown by Figure 6. For a non-isoated custer, the vaue of GEE, Su-EE, Prod- EE, and su-rate degrade for increasing vaues of the out-ofcuster interference, irrespectivey of the considered optiization criterion. Aso, the perforance gap aong the considered soutions reduces as increases, since the out-of-custer interference becoes doinant, aking coordinated resource

10 1 1 = 1 P ax = 35 db CDF.6.4 Su rate [kbit/s] GEE axiization.2 Su EE axiization Prod EE axiization [kbit/joue] x P [db] ax 3 2 GEE axiization 1 Su EE axiization Prod EE axiization Su rate axiization Maxiu power P [db] out STD [kbit/joue] P ax [db] Figure 5. Su-rate vs P ax for = eft; su-rate vs for P ax = 35 db right. Figure 7. Epirica CDF of the energy efficiency achieved on each subcarrier when P ax = 2 db and = top; standard deviation of the energy efficiency achieved on each subcarrier versus P ax when = botto. Radiated power per BS [db] = GEE axiization Su EE axiization Prod EE axiization Su rate axiization P ax = 35 db EE-opt aocation, thus confiring that Prod-EE axiization provides a ore baanced use of the avaiabe subcarriers. Finay, we study the convergence of the proposed agoriths. Figure 8 reports the vaue of GEE, Su-EE, and Prod- EE versus the nuber i of iterations of Agoriths 1, 5, and 6, respectivey. The upper pots refer to an isoated custer, whie the ower pots to a non-isoated custer. A agoriths reach a steady vaue in few iterations in a considered scenarios; aso, the convergence speed decreases for increasing vaues of P ax and for diinishing vaues of, i.e., when the syste perforance is iited by the co-channe interference generated by the other coordinated BSs P ax [db] [db] Figure 6. Per-BS radiated power vs P ax for = eft; per-bs radiated power vs for P ax = 35 db right. aocation ess and ess beneficia. It is interesting to notice that, in order to counteract the increased interference eve, GEE-opt, Su-EE-opt, and Prod-EE-opt soutions progressivey use a arger fraction of the avaiabe power. Figure 7 shows the epirica CDF of the energy efficiency achieved on each subcarrier for P ax = 2 db top and the standard deviation of the energy efficiency achieved on each subcarrier versus P ax botto, for the GEE-opt, Su-EE-opt, and Prod-EE-opt soutions; an isoated custer is considered. Resuts show that the energy efficiency achieved on the individua subcarriers is ess dispersed for the Prod- C. Infuence of the weights The weights in the definition of Su-EE and Prod-EE ay be used to give priority to specific subcarriers and/or BSs; this is an attractive feature, especiay in heterogeneous scenarios. As an exape, we consider the axiization of Su-EE with =.7, w[n] 2,s =.5, and 3,s =.3; b w[n] 1,s =.3, w[n] 2,s =.5, and w[n] 3,s =.7. In Figure 9, we consider the Su-EE-opt soution and report the average energy efficiency of each coordinated BS, i.e., two choices of the weights: a 1,s 1 N R [n],k,n γ [n] + θ [n] for = 1,..., M. An isoated custer is considered, and the resuts are potted versus P ax. Since θ [n] 1 =.25, θ [n] 2 =.5, and θ [n] 3 =.75, BS1 is the ost energy-efficient BS, whie BS3 is the ost energy-inefficient BS. In the first scenario, BS1 achieves an average energy-efficiency uch arger than that of other BSs, as it has the argest priority and the best energy efficiency. Instead, BS3 is extreey penaized, as it

11 11 [kbit/joue] [kbit/joue] GEE = P ax = 1 db P ax = 2 db P ax = 5 db 5 1 i GEE P ax = 35 db = 4 db = db = 4 db i Su EE = P ax = 1 db P ax = 2 db P ax = 5 db i Su EE P ax = 35 db = 4 db = db = 4 db i Prod EE = P ax = 1 db P ax = 2 db P ax = 5 db 5 1 i Prod EE P ax = 35 db = 4 db = db = 4 db Figure 8. GEE, Su-EE, and Prod-EE versus the nuber i of iterations of Agoriths 1, 5, and 6, respectivey. Top: P ax = 1, 2, 5 db and =. Botto: = 4,, 4 db and P ax = 35 db. Average energy efficiency [kbit/joue] BS1, w=[.7.5.3] BS2, w=[.7.5.3] BS3, w=[.7.5.3] BS1, w=[.3.5.7] BS2, w=[.3.5.7] BS3, w=[.3.5.7] θ 1 [n] =.25, θ2 [n] =.5, θ3 [n] = P ax [db] Figure 9. Average energy efficiency of each coordinated BS versus P ax. The Su-EE-opt soution and an isoated custer are considered. i If the axiu transit power scaes ineary with N, the syste perforance are ony arginay affected by the nuber of subcarriers. To be ore specific, et us first focus on GEE. When N scaes up, the syste su-rate proportionay increases, as ore subcarriers are avaiabe for transission and the average transit power per-subcarrier reains fixed. At the sae tie, the power consuption is aso increased, and the ratio between the su-rate and su-power reains substantiay unchanged. Siiary, the energy efficiency of each individua subcarrier reains approxiativey constant, and, consequenty, the arithetic and geoetric eans of the energy efficiencies across a subcarriers do not scae with N. In keeping with intuition, if the nuber of users in the coordinated custer is increased, the syste perforance tends to iprove due to the utiuser diversity gain [1]. Ceary, whie the network-wide perforance iproves, the average physica resources assigned to each user progressivey reduce. E. Discussion on agoriths copexity As seen fro Figure 8, the proposed agoriths converge in ony 5-15 iterations, depending on the operating scenario. For each ethod, the copexity of a singe iteration is ainy tied to the optiization of the transit power. In Agorith 1, the power update requires the soution of the fractiona progra 13. Severa equivaent ethods exist to tacke fractiona probes [29], and Agorith 1 is independent of the epoyed ethod. Here, we have resorted to Dinkebach s agorith, which is widey-used in the iterature. The Dinkebach s agorith has a super-inear convergence rate [29] and, in each iteration, ony requires the soution of a convex probe, which can be accopished in poynoia tie by eans of any convex prograing agoriths [44]. In Agorith 5, the power update just requires the coputation of the agebraic expressions in 28 and the soution to the waterfiing-ike probes in 29, which can be accopished in ogarithic tie through a bisection search. Finay, the power update in Agorith 6 requires the soution of a convex probe, which again is accopished in poynoia tie. Copexity generay grows with the nuber of coordinated BSs and active users. However, the nuber of coordinated BSs is usuay in the order of few units, since the advantages of cooperation with far-away BSs are argina. Moreover, coordination ay ony be perfored for ce-edge users, which typicay do not experience favorabe propagation conditions. has the worst energy efficiency and the saest priority. In the second scenario, a ore baanced resource aocation is obtained by assigning a higher priority to BS3 and a ower priority to BS1. Notice that the perforance of BS2 reains approxiativey unchanged, as its weights are kept fixed. D. Ipact of the nuber of subcarriers and users. Experients have been carried out to aso study the ipact of the nuber of subcarriers and users on the syste perforance; we offer here soe genera coents on the resuts, without incuding any detaied pot for the sake of brevity. VII. CONCLUSIONS We have studied the probe of resource aocation in the downink of an OFDMA network with base station coordination. Three figures of erit have been considered for syste design, naey, the ratio of the network su-rate to the network power consuption GEE, the weighted su of the energy efficiencies on each subcarrier Su-EE, and the exponentiay-weighted product of the energy efficiencies on each subcarrier Prod-EE. Agoriths for coordinated user scheduing and power aocation have been proposed, under a per-bs or per-subcarrier power constraint. In particuar, GEE is optiized by soving a series of concave-convex

12 12 fractiona reaxations, whie for Prod-EE a series of concave reaxations is considered; as to Su-EE, an iterative ethod to sove the Karush Kuhn Tucker conditions is proposed. For a figures of erit, agoriths to copute a gobay-optia soution are derived in the asyptotic noise-iited regie. It has been shown that Su-EE and Prod-EE provide ore degrees of freedo for syste design copared to the ore popuar GEE, as the corresponding weights ay be used to give priority to specific subcarriers and/or base stations. Aso, Prod-EE inherenty akes a ore baanced use of the avaiabe spectru, preventing the unpeasant situation where few subcarriers receive ost of the syste resources. A. Proof of Lea 1 APPENDIX Since the bound in 12 is tight at SINR [n],k,n = z[n], with z [n], we have ax fexp{q}, k B M q Q =1 og z [n]. Finay, concavity of fexp{q}, k and convexity of gexp{q} foow fro the fact that the og-su-exp function is convex [44]. B. Proof of Proposition 1 Let q = n p and k be the optiized vaues after iterations. Aso, et h be the ower bound in 12 when the approxiation constants are coputed according to p and k. Then, we have:... d GEEp, k a = h q, k b h q +1, k c GEEp +1, k d GEEp +1, k +1 a = h +1 q +1, k +1 b... where the equaity a is due to the fact that the reaxation 12 is tight at the current SINR vaues; the inequaity b is due to the fact that the Dinkebach s procedure coputes the gobay-optia soution to 13; the inequaity c foows fro 12; the inequaity d foows fro the fact that the user seection in 9 does not decrease the vaue of GEE. Since GEE is bounded above, the procedure ust converge. Next, the KKT conditions for 13 can be written as d dq [n] hexp{q}, k λ exp{q [n] } =,, n λ, N exp{q[n] } P,ax, λ P,ax N exp{q[n] } =, 39 where λ is the Lagrange utipier with respect to the power constraint of BS and d dq [n] hexp{q}, k = α [n] j=1, j 1 + B/ n 2 gexp{q} α [n] j exp{q [n] }G [n],kj,n =1, j exp{q [n] }G [n],kj,n γ [n] exp{q [n] hexp{q}, k } gexp{q}. Let q, k be the soution provided by Agorith 1 upon convergence. Then, there exists a set of utipiers λ such that the tripet q, k, λ siutaneousy satisfies 9 and 39. Finay, notice that the first-order optiaity conditions for 8 in the q-space are given by 9 and d dq [n] GEEexp{q}, k λ exp{q [n] } =,, n λ, N exp{q[n] } P,ax, λ P,ax N exp{q[n] } =, where d dq [n] GEEexp{q}, k = j=1, j 1 + B/ n 2 gexp{q} exp{q [n] }G [n],k,n =1 SINR [n] 1 + exp{q [n] }G [n],k,n j,kj,n exp{q[n] }G [n],kj,n =1 exp{q [n] }G [n],kj,n 4 γ [n] exp{q [n] GEEexp{q}, k }. gexp{q} The proof is copeted by noticing that the approxiation in 12 is exact at convergence and, therefore, the tripet q, k, λ aso soves 9 and 4. C. Proof of Proposition 2 Notice that GEE-NLp, k is a stricty pseudo-concave function of p, since it is the ratio between a stricty concave function and a inear function [48]. This ipies that the optia resource aocation strategy can be found by aternativey coputing the best k and p, as suarized in Agorith 3. D. Proof of Lea 2 Since u is stricty pseudo-concave, x is the unique soution of the equation dux dx ax + c = 1 + ax = og 21 + ax. 41 og 2 e

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