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Energy-Efficient Resource Assignment and PowerAllocation in Heterogeneous Cloud Radio Access

NetworksMugen Peng,Senior Member, IEEE, Kecheng Zhang, Jiamo Jiang, Jiaheng Wang,Member, IEEE, and

Wenbo Wang,Member, IEEE

Abstract—Taking full advantages of both heterogeneous net-works (HetNets) and cloud access radio access networks (C-RANs), heterogeneous cloud radio access networks (H-CRANs)are presented to enhance both the spectral and energy efficiencies,where remote radio heads (RRHs) are mainly used to providehigh data rates for users with high quality of service (QoS)requirements, while the high power node (HPN) is deployed toguarantee the seamless coverage and serve users with low QoSrequirements. To mitigate the inter-tier interference and improveEE performances in H-CRANs, characterizing user associationwith RRH/HPN is considered in this paper, and the traditionalsoft fractional frequency reuse (S-FFR) is enhanced. Basedonthe RRH/HPN association constraint and the enhanced S-FFR,anenergy-efficient optimization problem with the resource assign-ment and power allocation for the orthogonal frequency divisionmultiple access (OFDMA) based H-CRANs is formulated as anon-convex objective function. To deal with the non-convexity, anequivalent convex feasibility problem is reformulated, and closed-form expressions for the energy-efficient resource allocationsolution to jointly allocate the resource block and transmitpower are derived by the Lagrange dual decomposition method.Simulation results confirm that the H-CRAN architecture andthe corresponding resource allocation solution can enhance theenergy efficiency significantly.

Index Terms—Heterogeneous cloud radio access network, 5G,green communication, fractional frequency reuse, resource

allocation

Copyright (c) 2015 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to pubs-permissions@ieee.org.

Mugen Peng (e-mail: pmg@bupt.edu.cn), Kecheng Zhang(e-mail: buptzkc@163.com), and Wenbo Wang (e-mail:wbwang@bupt.edu.cn) are with the Key Laboratory of UniversalWireless Communications for Ministry of Education, Beijing Universityof Posts and Telecommunications, Beijing, China. Jiamo Jiang (e-mail:jiamo.jiang@gmail.com) is with the Institute of CommunicationStandards Research, China Academy of Telecommunication Research ofMIIT, Beijing, China. Jiaheng Wang (e-mail:jhwang@seu.edu.cn) iswith the National Mobile Communications Research Laboratory, SoutheastUniversity, Nanjing, China.

The work of M. Peng, K. Zhang and W. Wang was supported in part bythe National Natural Science Foundation of China (Grant No.61222103),the National Basic Research Program of China (973 Program) (Grant No.2013CB336600, 2012CB316005), the State Major Science and Technol-ogy Special Projects (Grant No. 2013ZX03001001), the Beijing NaturalScience Foundation (Grant No. 4131003), and the Specialized ResearchFund for the Doctoral Program of Higher Education (SRFDP) (GrantNo.20120005140002). The work of J. Wang was supported by theNationalNatural Science Foundation of China under Grant 61201174, by the NaturalScience Foundation of Jiangsu Province under Grant BK2012325, and by theFundamental Research Funds for the Central Universities.

I. I NTRODUCTION

Cloud radio access networks (C-RANs) are by now rec-ognized to curtail the capital and operating expenditures,aswell as to provide a high transmission bit rate with fantasticenergy efficiency (EE) performances[1], [2]. The remoteradio heads (RRHs) operate as soft relay by compressingand forwarding the received signals from the mobile userequipment (UE) to the centralized base band unit (BBU) poolthrough the wire/wireless fronthaul links. To highlight theadvantages of C-RAN, the joint decompression and decodingschemes are executed in the BBU pool[3]. However, thenon-ideal fronthaul with limited capacity and long time delaydegrades performances of C-RANs. Furthermore, it is criticalto decouple the control and user planes in C-RANs, and RRHsare efficient to provide high capacity without consideringfunctions of control planes. How to alleviate the negativeinfluence of the constrained fronthaul on EE performances,and how to broadcast the control signallings to UEs withoutRRHs are still not straightforward in C-RANs[4].

Meanwhile, high power nodes (HPNs) (e.g., macro or microbase stations) existing in heterogeneous networks (HetNets)are still critical to guarantee the backward compatibilitywiththe traditional cellular networks and support the seamlesscoverage since low power nodes (LPNs) are mainly deployedto provide high bit rates in special zones[5]. Under helpof HPNs, the unnecessary handover can be avoided and thesynchronous constraints among LPNs can be alleviated. Ac-curately, although the HetNet is a good alternative to improveboth coverage and capacity simultaneously, there are tworemarkable challenges to block its commercial applications:i). The coordinated multi-point transmission and reception(CoMP) needs a huge number of signallings in backhaullinks to mitigate the inter-tier interferences between HPNsand LPNs, while the backhaul capacity is often constrained;ii). The ultra dense LPNs improve capacity with the costof consuming too much energy, which results in low EEperformances.

To overcome these aforementioned challenges in both C-RANs and HetNets, a new architecture and technology knownas the heterogeneous cloud radio access network (H-CRAN)is presented as a promising paradigm for future heterogeneousconverged networks[6]. The H-CRAN architecture shown inFig. 1 takes full advantages of both C-RANs and HetNets,where RRHs with low energy consumptions are cooperated

2

HPNRRH1

Massive

MIMO

Gateway

Data

Control

BBU Pool

RRHi

RRHL

Gateway

Data

UE1

UE2

Internet

Fig. 1. System architecture of the proposed H-CRANs

with each other in the BBU pool to achieve high cooperativegains. The BBU pool is interfaced to HPNs for coordinatingthe inter-tier interferences between RRHs and HPNs. Onlythe front radio frequency (RF) and simple symbol process-ing functionalities are configured in RRHs, while the otherimportant baseband physical processing and procedures ofthe upper layers are executed in the BBU pool. By contrast,the entire communication functionalities from the physical tonetwork layers are implemented in HPNs. The data and controlinterfaces between the BBU pool and HPNs are S1 and X2,respectively, which are inherited from definitions of the3rd

generation partnership project (3GPP) standards. Comparedwith the traditional C-RAN architecture in[1], H-CRANsalleviate the fronthaul constraints between RRHs and the BBUpool through incorporating HPNs. The control and broadcastfunctionalities are shifted from RRHs to HPNs, which alle-viates capacity and time delay constraints on the fronthauland supports the burst traffic efficiently. The adaptive sig-nalling/control mechanism between connection-oriented andconnectionless is supported in H-CRANs, which can achievesignificant overhead savings in the radio connection/release bymoving away from a pure connection-oriented mechanism.

In H-CRANs, the RRH/HPN association strategy is criticalfor improving EE performances, and main differences from thetraditional cell association techniques are twofold. First, thetransmit power of RRHs and HPNs is significantly different.Second, the inter-RRH interferences can be jointly coordinatedthrough the centralized cooperative processing in the BBUpool, while the inter-tier interferences between HPNs andRRHs are severe and difficult to mitigate. Consequently, itis not always efficient for UEs to be associated with neighborRRHs/HPNs via the strongest receiving power mechanism[7].As shown in Fig. 1, though obtaining the same receivingpower from RRH1 and HPN, both UE1 and UE2 prefer toassociate with RRH1 because lower transmit power is neededand more radio resources are allocated from RRH1 than thosefrom the HPN. Meanwhile, the association with RRHs candecrease energy consumptions by saving the massive use ofair condition.

Based on the aforementioned RRH/HPN association char-acteristics, the joint optimization solution for resourceblock

(RB) assignment and power allocation to maximize EE perfor-mances in the orthogonal frequency division multiple access(OFDMA) based H-CRANs is researched in this paper.

A. Related Work

OFDMA is a promising multi-access technique for exploit-ing channel variations in both frequency and time domainsto provide high data rates in the fourth generation (4G)and beyond cellular networks. To be backward compatiblewith 4G systems, the OFDMA is adopted in H-CRANs byassigning different RBs to different UEs. Recently, the radioresource allocation (RA) to maximize the spectral efficiency(SE) or meet diverse qualify-of-service (QoS) requirementsin OFDMA systems has attracted considerable attention[8] -[11]. The relay selection problem in a network with multipleUEs and multiple amplify-and-forward relays is investigated in[8], where an optimal relay selection scheme whose complex-ity is quadratic in the number of UEs and relays is presented.The authors in[9] propose an asymptotical resource allocationalgorithm via leveraging the cognitive radio (CR) techniquein open access OFDMA femtocell networks. The resourceoptimization for spectrum sharing with interference controlin CR systems is researched in[10], where the achievabletransmission rate of the secondary user over Rayleigh channelssubject to a peak power constraint at the secondary transmitterand an average interference power constraint at the primaryreceiver is maximized. In[11], the optimal power allocationfor minimizing the outage probability in point-to-point fadingchannels with the energy-harvesting constraints is investigated.

The EE performance metric has become a new design goaldue to the sharp increase of the carbon emission and operatingcost of wireless communication systems[12]. The EE-orientedradio resource allocation has been studied in various networks[13] - [16]. In [13], the distributed power allocation for multi-cell OFDMA networks taking both energy efficiency andinter-cell interference mitigation into account is investigated,where a bi-objective problem is formulated based on themulti-objective optimization theory. To maximize the averageEE performance of multiple UEs each with a transceiver ofconstant circuit power, the power allocation, RB allocationand relay selection are jointly optimized in[14]. The activenumber of sub-carriers and the number of bits allocated toeach RB at the source nodes are optimized to maximize EEperformances in[15], where the optimal solution turns outto be a bidirectional water-filling bit allocation to minimizethe overall transmit power. To maximize EE performancesunder constraints of total transmit power and interferencein CR systems, an optimal power allocation algorithm usingequivalent conversion is proposed in[16].

Intuitively, the inter-cell or inter-tier interference mitigationis the key to improve both SE and EE performances. Someadvanced algorithms in HetNets, such as cell association andfractional frequency reuse (FFR), have been proposed in[17]and[18], respectively. In[19], a network utility maximizationformulation with a proportional fairness objective is presented,where prices are updated in the dual domain via coordinatedescent. In[20], energy efficient cellular networks through the

3

employment of base station with sleep mode strategies as wellas small cells are researched, and the corresponding tradeoffissue is discussed as well.

To the best of our knowledge, there are lack of solutionsto maximize the EE performance in H-CRANs. Particularly,the RRH/HPN association strategy should be enhanced fromthe traditional strongest received power strategy. Furthermore,the radio resource allocation to achieve an optimal EE per-formance in H-CRANs is still not straightforward. To dealwith these problems, the joint optimization solution with theRB assignment and power allocation subject to the RRH/HPNassociation and interference mitigation should be investigated.

B. Main Contributions

The goal of this paper is to investigate the joint optimizationproblem with the RB assignment and power allocation subjectto the RRH/HPN association and inter-tier interference miti-gation to maximize EE performances in the OFDMA basedH-CRAN system. The EE performance optimization is highlychallenging because the energy-efficient resource allocation inH-CRANs is a non-convex objective problem. Different fromthe published radio resource optimization in HetNets and C-RANs, the characteristics of H-CRANs should be highlightedand modeled. To simplify the coordinated scheduling betweenRRHs and the HPN, an enhanced soft fractional frequencyreuse (S-FFR) scheme is presented to improve performancesof cell-center-zone UEs served by RRHs with individualspectrum frequency resources. The contributions of this papercan be summarized as follows:

• To overcome challenges in HetNets and C-RANs, H-CRANs are presented as cost-efficient potential solutionsto improve spectral and energy efficiencies, in whichRRHs are mainly used to provide high data rates for UEswith high QoS requirements in the hot spots, while HPNsare deployed to guarantee seamless coverage for UEs withlow QoS requirements.

• To mitigate the inter-tier interference between RHHs andHPNs, an enhanced S-FFR scheme is proposed, wherethe total frequency band is divided into two parts. Onlypartial spectral resources are shared by RRHs and HPNs,while the other is solely occupied by RRHs. The exclu-sive RBs are allocated to UEs with high rate-constrainedQoS requirements, while the shared RBs are allocated toUEs with low rate-constrained QoS requirements.

• An energy-efficient optimization problem with the RBassignment and power allocation under constraints of theinter-tier interference mitigation and RRH/HPN associ-ation is formulated as a non-convex objective function.To deal with this non-convexity, an equivalent convexfeasibility problem is reformulated, based on which aniterative algorithm consisting of both outer and inner loopoptimizations is proposed to achieve the global optimalsolution.

• We numerically evaluate EE performance gains of H-CRANs and the corresponding resource allocation opti-mization solution. Simulation results demonstrate that EEperformance gain of the H-CRAN architecture over the

traditional C-RAN/HetNet is significant. The proposediterative solution is converged, and its EE performancegain over the baseline algorithms is impressive.

The reminder of this paper is organized as follows. InSection II, we describe the system model of H-CRANs, theproposed enhanced S-FFR, and the problem formulation. Theoptimization framework is introduced in Section III. SectionIV provides simulations to verify the effectiveness of the pro-posed H-CRAN architecture and the corresponding solutions.Finally, we conclude the paper in Section V.

II. SYSTEM MODEL AND PROBLEM FORMULATION

The traditional S-FFR is considered as an efficient inter-celland inter-tier interference coordination technique, in which theservice area is partitioned into spatial subregions, and eachsubregion is assigned with different frequency sub-bands.Asshown in Fig. 2(a), the cell-edge-zone UEs do not interferewith cell-center-zone UEs, and the inter-cell interference canbe suppressed with an efficient channel allocation method[21].The HPN is mainly used to deliver the control signallings andguarantee the seamless coverage for UEs accessing the HPN(denoted by HUEs) with low QoS requirements. By contrast,the QoS requirement for UEs accessing RRH (denoted byRUEs) is often with a higher priority. Consequently, as shownin Fig. 2(b), an enhanced S-FFR scheme is proposed tomitigate the inter-tier interference between HPNs and RRHs,in which only partial radio resources are allocated to bothRUEs and HUEs with low QoS requirements, and the re-maining radio resources are allocated to RUEs with high QoSrequirements. In the proposed enhanced S-FFR, RUEs withlow QoS requirements share the same radio resources withHUEs, which is absolutely different from that in the traditionalS-FFR. If the traditional S-FFR is utilized in H-CRANs, onlythe cell-center-zone RUEs share the same radio resourceswith HUEs, which decreases the SE performance significantly.Further, it is challenging to judge whether UEs are located inthe cell-edge or cell-center zone for the traditional S-FFR.

These aforementioned problems are avoided in the proposedenhanced S-FFR, where only the QoS requirement shouldbe distinguished for RUEs. On the one hand, to avoid theinter-tier interference, the outband frequency is preferred touse according to standards of HetNets in 3GPP[22], whichsuggests that RBs for HPNs should be orthogonal with thosefor RRHs. On the other hand, to save the occupied frequencybands, the inband strategy is defined as well in 3GPP[22],which indicates that both RUEs and HUEs share the sameRBs even though the inter-tier interference is severe. To becompletely compatible with both inband and outband strate-gies in 3GPP, only two RB setsΩ1 andΩ2 are divided in thisproposed enhanced S-FFR scheme. Obviously, if locations ofRUEs could be known and the traffic volume in different zonesare clearly anticipated, more RB sets inΩ1 could be divided toachieve higher performance gains. The division of two RB setsin the proposed enhanced S-FFR is a good tradeoff betweenperformance gains and implementing complexity/flexibility.

The QoS requirement is treated as the minimum trans-mission rate in this paper, which is also called the rate-constrained QoS requirement. In this paper, the high and

4

(a) Traditional Soft FFR (b) Enhanced Soft FFR

HPNRRH

RRH

RRHRRH

For each cell

Fig. 2. Principle of the proposed enhanced soft fractional frequencyreuse scheme

low rate-constrained QoS requirements are denoted asηR andηER, respectively. For simplicity, it is assumed that there areN and M RUEs per RRH occupying the RB setsΩ1 andΩ2, respectively. In the OFDMA based downlink H-CRANs,there are totalK RBs (denoted asΩT ) with the bandwidthB0. TheseK RBs in ΩT are categorized as two types:Ω1

is only allocated to RUEs with high rate-constrained QoSrequirements, andΩ2 is allocated to RUEs and HUEs with lowrate-constrained QoS requirements. Since all signal processingfor different RRHs is executed on the BBU pool centrally,the inter-RRH interferences can be coordinated and the sameradio resources can be shared amongst RRHs. Hence, thechannel-to-interference-plus-noise ratio (CINR) for then-thRUE occupying thek-th RB can be divided into two parts:

σn,k =

dRnhRn,k/B0N0 , k ∈ Ω1

dRnhRn,k/(P

MdMn hMn,k +B0N0) , k ∈ Ω2

(1)wheredRn anddMn denote the path loss from the served RRHand the reference HPN to RUEn, respectively.hR

n,k andhMn,k represent the channel gain from the RRH and HPN to

RUE n on the k-th RB, respectively.PM =PM

max

Mis the

allowed transmit power allocated on each RB in HPN andPMmax denotes the maximum allowable transmit power of HPN.

N0 denotes the estimated power spectrum density (PSD) ofboth the sum of noise and weak inter-RRH interference (indBm/Hz).

The sum data rate for each RRH can be expressed as:

C(a,p) =

N+M∑

n=1

K∑

k=1

an,kB0log2(1 + σn,kpn,k), (2)

wheren ∈ 1, ..., N denotes the RUE allocated to the RBset Ω1, and n ∈ N + 1, ..., N + M denotes the RUEallocated to the RB setΩ2. The (N + M) × K matricesa = [an,k](N+M)×K

andp = [pn,k](N+M)×Krepresent the

feasible RB and power allocation policies, respectively.an,kis defined as the RB allocation indicator which can only be 1or 0, indicating whether thek-th RB is allocated to RUEn.pn,k denotes the transmit power allocated to RUEn on thek-th RB.

According to [12], the total power consumptionP (a,p)for H-CRANs mainly depends on the transmit power andcircuit power. When the power consumption for the fronthaulis considered, the total power consumption per RRH is written

as:

P (a,p) = ϕeff

N+M∑

n=1

K∑

k=1

an,kpn,k + PRc + Pbh, (3)

whereϕeff , PRc andPbh denote the efficiency of the power

amplifier, circuit power and power consumption of the fron-thaul link, respectively.

Similarly, the sum data rate for the HPN can be calculatedas:

CM (aM,pM) =

T∑

t=1

M∑

m=1

at,mB0log2(1 + σt,mpt,m), (4)

where t ∈ 1, ..., T denotes the HUE allocated to theRB set Ω2. The T × M matricesaM = [at,m]

T×Mand

pM =[

pmt,m]

T×Mrepresent the feasible RB and power

allocation policies for the HPN, respectively.at,m is definedas the RB allocation indicator which can only be 1 or 0,indicating whether them-th RB is allocated to the HUEt.pmt,m denotes the transmit power allocated to the HUEt onthe m-th RB. σt,m represents the CINR of thet-th HUE onthe m-th RB. The total power consumption of the HPN canbe given by

PM (aM,pM) = ϕMeff

T∑

t=1

M∑

m=1

at,mpMt,m + PMc + PMbh

, (5)

wherepMt,m denotes the transmission baseband power for theHPN when them-th RB is allocated to thet-th HUE, whichforms the transmit power vectorpM for all HUEs. ϕM

eff ,PMc andPMbh

denote the efficiency of power amplifier, thecircuit power and the power consumption of the backhaul linkbetween HPN and BBU pool, respectively. Considering thatthe HPN is mainly utilized to extend the coverage and providebasic services for HUEs, the same downlink transmit power fordifferent HUEs over different RBs is assumed aspMt,m = PM ,wherePM has been defined in (1).

The overall EE performance for the H-CRAN withL RRHsand1 HPN can be written as

γ =L ∗ C(a,p) + CM (aM,pM)

L ∗ P (a,p) + PM (aM,pM). (6)

For the dense RRH deployed H-CRAN,L is much largerthan 1. The inter-tier interference from RRHs to HUEs remainsconstant when the density of RRHs is sufficiently high, andhence the downlink SE and EE performances for the HPNcan be assumed to be stable. Therefore,C(a,p) is muchlarger thanCM (aM,pM), andPM (aM,pM) can be ignoredif L* P(a,p) is sufficiently large. WhenL is sufficiently large,the overall EE performance in (6) for the H-CRAN can beapproximated as:

γ ≈L ∗ C(a,p)

L ∗ P (a,p)=

C(a,p)

P (a,p). (7)

According to (7), the overall EE performance mainly de-pends on the EE optimization of each RRH. To make surethat HUEs meet the low rate-constrained QoS requirement, the

5

interference from RRHs should be constrained and not largerthan the predefined thresholdδ0. Consequently, to optimizedownlink EE performances, the core problem is converted tooptimize EE performance of each RRH with constraints onthe inter-tier interference to the HPN from RRHs when thedensity of RRHs is sufficiently high.

Problem 1 (Energy Efficiency Optimization): With theconstraints on the required QoS, inter-tier interference andmaximum transmit power allowance, the EE maximizationproblem in the downlink H-CRAN can be formulated as

maxa,p

C(a,p)

P (a,p)=

N+M∑

n=1

K∑

k=1

an,kB0log2(1 + σn,kpn,k)

ϕeff

N+M∑

n=1

K∑

k=1

an,kpn,k + PRc + Pbh

(8)

s.t.N+M∑

n=1

an,k = 1, an,k ∈ 0, 1, ∀k, (9)

K∑

k=1

Cn,k ≥ ηR, 1 ≤ n ≤ N, (10)

K∑

k=1

Cn,k ≥ ηER, N + 1 ≤ n ≤ N +M, (11)

N+M∑

n=N

an,kpn,kdR2Mk hR2M

k ≤ δ0, k ∈ ΩII, (12)

N+M∑

n=1

K∑

k=1

an,kpn,k ≤ PRmax, pn,k ≥ 0, ∀k, ∀n, (13)

whereCn,k = an,kB0log2(1 + σn,kpn,k) and the constraint(9) denotes the RB allocation limitation that each RB cannotbe allocated to more than one RUE at the same time. Theconstraints of (10) and (11) corresponding to the high andlow rate-constrained QoS requirements specify the minimumdata rate ofηR and ηER, respectively. (12) puts a limitationon pn,k to suppress the inter-tier interference from RRHs toHUEs that reuse the RBk ∈ ΩII. dR2M

k andhR2Mk represent

the corresponding path loss and channel gain on thek-th RBfrom the reference RRH to the interfering HUE, respectively.In (13), PR

max denotes the maximum transmit power of theRRH.

Based on the enhanced S-FFR scheme, the interference toHUE is constrained, and the SINR thresholdηHUE , whichis the minimum SINR requirement for decoding the signalof HUE successfully, is defined to represent the constraint of(12). When allocating them-th RB to thet-th HUE, the SINR(ηt,m) larger thanηHUE can be given by

ηt,m = PMdMmhMt,m/(L ∗ δ0 +B0N0) ≥ ηHUE . (14)

Obviously, the optimal RB allocation policya∗ and powerallocation policyp∗ with constraints of diverse QoS require-ments and variableηHUE to maximize the EE performance inProblem 1is a non-convex optimization problem due to formsof the objective function and the RB allocation constraint in(9), whose computing complexity increases exponentially withthe number of binary variables[23]. Intuitively, Problem 1is a mixed integer programming problem and the fractional

objective make it complicated and difficult to be solveddirectly with the classical convex optimization methods.

III. E NERGY-EFFICIENT RESOURCEALLOCATION

OPTIMIZATION

In this section, we propose an effective method to solveProblem 1in (8), where we first exploit the non-linear frac-tional programming for converting the objective function inProblem 1, upon which we then develop an efficient iterativealgorithm to solve this EE performance maximization problem.

A. Optimization Problem Reformulation

Since the objective function inProblem 1is classified as anon-linear fractional programm[24], the EE performance ofthe reference RRH can be defined as a non-negative variableγ in (7) with the optimal valueγ∗ =

C(a∗,p∗)P (a∗,p∗) .

Theorem 1 (Problem Equivalence):γ∗ is achieved if andonly if

maxa,p

C(a,p)− γ∗P (a,p) = C(a∗,p∗)− γ∗P (a∗,p∗) = 0,

(15)wherea,p is any feasible solution of Problem 1 to satisfythe constraints(9)-(13).

Proof: Please see Appendix A.Based on the optimal condition stated inTheorem 1,

Problem 1is equivalent toProblem 2as follows if we can findthe optimal valueγ∗. Althoughγ∗ cannot be obtained directly,an iterative algorithm (Algorithm 1 ) is proposed to updateγwhile ensuring that the corresponding solutiona,p remainsfeasible in each iteration. The convergence can be proved andthe optimal RA to solveProblem 2can be derived.

Problem 2 (Transformed Energy Efficiency Optimization):

maxa,p

C(a,p) − γ∗P (a,p), (16)

s.t. (9)− (13).

Note thatProblem 2is a tractable feasibility problem.Hence, the objective function of the fractional form in (8) is

transformed into the subtractive form. To design the efficientalgorithm for solvingProblem 2, we can define an equivalentfunctionF (γ) = max

a,pC(a,p)−γP (a,p) with the following

lemma.Lemma 1: For all feasiblea, p and γ, F (γ) is a strictly

monotonic decreasing function inγ, andF (γ) ≥ 0 .Proof: Please see Appendix B.Due to the constraint of (9), the feasible domain ofa is a

discrete and finite set consisting of all possible RB allocations,and thusF (γ) is generally a continuous but non-differentiablefunction with respect toγ.

B. Proposed Iterative Algorithm

Based onLemma 1, an iterative algorithm is proposedto solve the transformedProblem 2by updatingγ in eachiteration as the followingAlgorithm 1 .

The proposed iterativeAlgorithm 1 ensures thatγ increasesin each iteration. It can be observed that two nested loops

6

Algorithm 1 Energy-Efficient Resource Assignment andPower Allocation

1: Set the maximum number of iterationsImax, convergenceconditionεγ and the initial valueγ(1) = 0.

2: Set the iteration indexi = 1 and begin the iteration (OuterLoop).

3: for 1 ≤ i ≤ Imax

4: Solve the resource allocation problem withγ(i) (InnerLoop);

5: Obtaina(i),p(i), C(a(i),p(i)), P (a(i),p(i));6: if C(a(i),p(i))− γ(i)P (a(i),p(i)) < εγ then7: Seta∗,p∗=a(i),p(i) andγ∗=γ(i);8: break;9: else

10: Setγ(i+1) = C(a(i),p(i))P (a(i),p(i))

and i = i+ 1;11: end if12: end for

executed inAlgorithm 1 can achieve the optimal solution tomaximize EE performances. The outer loop updatesγ(i+1) ineach iteration with theC(a(i),p(i)) andP (a(i),p(i)) obtainedin the last iteration. In the inner loop, the optimal RB allocationpolicy a(i) and power allocation policyp(i) with a given valueof γ(i) are derived by solving the following inner-loopProblem3.

Problem 3(Resource Allocation Optimization in the InnerLoop):

maxa,p

C(a,p) − γ(i)P (a,p), (17)

s.t. (9)− (13),

whereγ(i) is updated by the last iteration in outer loop.Actually, with the help of the proposedAlgorithm 1 , the

solution toProblem 3is converged and the optimal solution ispresented. The global convergence has the following theorem.

Theorem 2 (Global Convergence):Algorithm 1 alwaysconverges to the global optimal solution of Problem 3.

Proof: Please see Appendix C.The optimization problem in (17) is non-convex and hard to

be solved directly. Generally speaking, if (17) can be solved bythe dual method, there exists a non-zero duality gap[25]. Theduality gap is defined as the difference between the optimalvalue of (17) (denoted byEE∗) and the optimal value of thedual problem for (17) (denoted byD∗). Fortunately, it can bedemonstrated that the duality gap between (17) and the dualproblem is nearly zero when the number of RBs is sufficientlylarge[26], which is illustrated as the following theorem.

Theorem 3 (Duality Gap): When the number of theresource block is sufficiently large, the duality gap between(17) and its dual problem is nearly zero, i.e.,D∗ −EE∗ ≈ 0holds.

Proof: Please see Appendix D.

C. Lagrange Dual Decomposition Method

Hence, according toTheorem 3, Problem 3in thei-th outerloop can be solved by the dual decomposition method. With

rearranging the constraints (10)–(13), the Lagrangian functionof the primal objective function is given by

L(a,p,β,λ, ν) =

N+M∑

n=1

K∑

k=1

an,kB0log2(1 + σn,kpn,k)

− γ(i)

(

ϕeff

N+M∑

n=1

K∑

k=1

an,kpn,k + PRc + Pbh

)

+

N∑

n=1

βn

[

K∑

k=1

an,kB0log2(1+σn,kpn,k)−ηR

]

+

N+M∑

n=N+1

βn

[

K∑

k=1

an,kB0log2(1+σn,kpn,k)−ηER

]

+

K∑

k=1

λk

(

δ0 −

N+M∑

n=1

an,kpn,kdR2Mk hR2M

k

)

+ ν

(

PRmax −

N+M∑

n=1

K∑

k=1

an,kpn,k

)

, (18)

whereβ = (β1, β2, . . . , βN+M ) 0 is the Lagrange multi-plier vector associated with the required minimum data rateconstraints (10) and (11).λ = (λ1, λ2, . . . , λK) 0 isthe Lagrange multiplier vector corresponding to the inter-tierinterference constraint (12) andλk = 0 for k ∈ ΩI. ν ≥ 0 isthe Lagrange multiplier for the total transmit power constraint(13). The operator 0 indicates that the elements of the vectorare all nonnegative.

The Lagrangian dual function can be expressed as:

g(β,λ, ν) = maxa,p

L(a,p,β,λ, ν)

= maxa,p

K∑

k=1

N+M∑

n=1

[

(βn + 1)an,kB0log2(1 + σn,kpn,k)

− γ(i)ϕeffan,kpn,k−λkan,kpn,kdR2Mk hR2M

k − νan,kpn,k

]

− γ(i)(PRc + Pbh)−

N∑

n=1

βnηR

−

N+M∑

n=N+1

βnηER +

K∑

k=1

λkδ0 + νPRmax

, (19)

and the dual optimization problem is reformulated as:

minβ,λ,ν

g(β,λ, ν), (20)

s.t. β 0,λ 0, ν ≥ 0.

It is obvious that the dual optimization problem is alwaysconvex. In particular, the Lagrangian functionL(a,p,β,λ, ν)is linear withβn, λk andν for any fixedan,k andpn,k, whilethe dual functiong(β,λ, ν) is the maximum of these linearfunctions. We use the dual decomposition method to solve thisdual problem, which is firstly decomposed intoK independentproblems as:

7

g(β,λ, ν) =K∑

k=1

gk(β,λ, ν)−γ(i)(PRc + Pbh)

−

N∑

n=1

βnηR−

N+M∑

n=N+1

βnηER +

K∑

k=1

λkδ0 +νPRmax,

(21)

where

gk(β,λ, ν)

= maxa,p

N+M∑

n=1

[

(1 + βn)an,kB0log2(1 + σn,kpn,k)

− γ(i)ϕeffan,kpn,k−λkan,kpn,kdR2Mk hR2M

k − νan,kpn,k

]

.

(22)

Supposed that thek-th RB is allocated to then-th UE, i.e.,an,k = 1, it is obvious that (22) is concave in terms ofpn,k.With the Karush-Kuhn-Tucker (KKT) condition, the optimalpower allocation is derived by

p∗n,k =

[

ω∗n,k −

1

σn,k

]+

, (23)

where [x]+ = maxx, 0, and the optimal water-filling levelω∗n,k is derived as

ω∗n,k =

B0(1 + βn)

ln 2(γ(i)ϕeff + λkdR2Mk hR2M

k + ν). (24)

Then, substituting the optimal power allocation obtained by(23) into the decomposed optimization problem (22), we canhave

gk(β,λ, ν)

= max1≤n≤N+M

(1+βn)B0

[

log2(ω∗n,kσn,k)

]+−

(γ(i)ϕeff+λkdR2Mk hR2M

k +ν)

[

ω∗n,k−

1

σn,k

]+

. (25)

With (24) and (25), the optimal RB allocation indicator forthe given dual variables can be expressed as:

a∗n,k =

1, n = arg max1≤n≤N+M

Hn,k,

0, otherwise,(26)

where

Hn,k =[

(1 + βn)log2(ω∗n,kσn,k)

]+

−(1 + βn)

ln 2

[

1−1

ω∗n,kσn,k

]+

. (27)

Then, the sub-gradient-based method[27] can be utilized tosolve the above dual problem and the sub-gradient of the dual

function can be written as

∇β(l+1)n =

K∑

k=1

C(l)n,k − ηR, 1 ≤ n ≤ N, (28)

∇β(l+1)n =

K∑

k=1

C(l)n,k − ηER, N + 1 ≤ n ≤ N +M, (29)

∇λ(l+1)k =

0, ∀k ∈ ΩI,

δ0 −N∑

n=1a(l)n,kp

(l)n,kd

R2Mk hR2M

k , ∀k ∈ ΩII,(30)

∇ν(l+1) = PRmax −

N∑

n=1

K∑

k=1

a(l)n,kp

(l)n,k, (31)

wherea(l)n,k and p(l)n,k represent the RB allocation and power

allocation which are derived by the dual variables of thel-thiteration, respectively andC(l)

n,k = a(l)n,kB0log2(1 + σn,kp

(l)n,k).

∇β(l+1)n , ∇λ

(l+1)k and ∇ν(l+1) denote the sub-gradient uti-

lized in the(l + 1)-th inner loop iteration. Hence, the updateequations for the dual variables in the(l + 1)-th iteration aregiven by

β(l+1)n =

[

β(l)n − ξ

(l+1)β ×∇β(l+1)

n

]+

,∀n, (32)

λ(l+1)k =

[

λ(l)k − ξ

(l+1)λ ×∇λ

(l+1)k

]+

, ∀k, (33)

ν(l+1) =[

ν(l) − ξ(l+1)ν ×∇ν(l+1)

]+

, (34)

whereξ(l+1)β , ξ(l+1)

λ andξ(l+1)ν are the positive step sizes.

IV. RESULTS AND DISCUSSIONS

In this section, the EE performance of the proposed H-CRAN and corresponding optimization solutions are evaluatedwith simulations. There areN = 10 RUEs located in eachRRH with the high rate-constrained QoS and allocated bythe orthogonal RB setΩ1. M is varied to denote the numberof RUEs with low rate-constrained QoS requirements. In thecase ofM = 0, there are no RUEs to share the sameradio resources with HUEs, and thus there are no inter-tierinterferences. Otherwise, there areM(> 0) RUEs with thelow rate-constrained QoS interfered by HPN. We assume thatdPn = 50 m, dMn = 450 m in the case of1 ≤ n ≤ N ; anddPn = 75 m, dMn = 375 m in the case ofN+1 ≤ n ≤ N+M .The distance between the reference RRH and HUEs who reusethe k-th RB is dR2M

k = 125 m. The number of total RBsis K = 25 and the system bandwidth is 5 MHz. The totaltransmit power of HPN is 43 dBm and equally allocated onall RBs. It is assumed that the path loss model is expressedas 31.5 + 40.0 ∗ log10(d) for the RRH-to-RUE link, and31.5+35.0 ∗ log10(d) for the HPN-to-RUE and RRH-to-HUElinks, whered denotes the distance between transmitter andreceiver in meters. The number of simulation snapshots is setat 1000. In all snapshots, the fast-fading coefficients are allgenerated as independently and identically distributed (i.i.d.)Rayleigh random variables with unit variances. The low andhigh rate-constrained QoS requirements are assumed to beηER = 64 kbit/s andηR = 128 kbit/s, respectively.

8

We assume the static circuit power consumption to bePRc = 0.1 W, and the power efficiency to beϕeff = 2 for

the power amplifiers of RRHs. For HPNs, they as assumed tobePM

c = 10.0 W andϕMeff = 4, respectively [28]. The power

consumptions for both RRHs and HPNs to receive the channelstate indication (CSI) and coordination signalling are takeninto account, i.e., the power consumptions of the fronthaullink Pbh, and the backhaul link between the HPN and theBBU pool Pbh are assumed to be 0.2 W.

A. H-CRAN Performance Comparisons

To evaluate EE performance gains of H-CRANs, the tradi-tional C-RAN, 2-tier HetNet, and 1-tier HPN scenarios arepresented as the baselines. Only one HPN and one RRHare considered in the H-CRAN scenario. Similarly, only twoHPNs are considered in the 1-tier HPN scenario, and one HPNand one Pico base station (PBS) are considered in the 2-tierHetNet scenario. For the 1-tier C-RAN scenario, two RRHsare considered, in which one RRH is used to replace the HPNin H-CRANs.

- 1-tier HPN: All UEs access the HPN, and the RB setΩ1 andΩ2 are allocated to the cell-center-zone and cell-edge-zone UEs, respectively. The optimal power and RBallocations are based on the classical water-filling and themaximum signal-to-interference-plus-noise ratio (SINR)scheduling algorithms, respectively. The number of cell-center-zone UEs is 10, and the number of cell-edge-zoneUEs with the low rate-constrained QoS requirement isvaried from 1 to 10.

- 2-tier overlaid HetNet: The orthogonal RB setΩ1 andΩ2 are allocated to the PBS and HPN, respectively. Thestatic circuit power consumption for the PBS isPP

c = 6.8W, and the power efficiency isϕP

eff = 4 for the poweramplifiers in PBS. Note that RBs allocated to the PBSand HPN are orthogonal with no inter-tier interferences.The number of RUEs served by the PBS keeps constantand is set at 10, and the number of HUEs served byHPN is varied. Note that UEs served by the HPN in thisscenario are denoted by HUEs with low rate-constrainedQoS requirements to be consistent with other baselines.

- 2-tier underlaid HetNet: All RBs in the setΩ1 andΩ2

are fully shared by UEs within the covering areas of thePBS and HPN. The optimal power and RB allocations areadopted to enhance EE performances[29]. The locationsand number of UEs served by the HPN and PBS are sameas those in the 2-tier overlaid HetNet scenario.

- 1-tier C-RAN: All RBs in the setΩ1 andΩ2 are fullyshared by RUEs. There are 10 cell-center-zone RUEs,and M cell-edge-zone RUEs in each RRH. Each RRHcovers the same area coverage of the PBS in the 2-tierHetNet scenario. The optimal power and RB allocationsare based on the classical water-filling and maximumSINR algorithms, respectively.

- 2-tier H-CRAN: The orthogonal RBs in setΩ1 are onlyallocated to RUEs, while RBs inΩ2 are shared bycell-edge-zone RUEs and HUEs, where the proposedoptimal solution subject to the interference mitigation

1 2 3 4 5 6 7 8 9 100.2

0.4

0.6

0.8

1

1.2

Number of UEs with low rate−constrained QoS

Ene

rgy

Effi

cien

cy (

bps/

Hz/

W)

1−tier HPN2−tier overlaid HetNet2−tier underlaid HetNet1−tier C−RAN2−tier H−CRAN

Fig. 3. Performance comparisons among 1-tier HPN, 2-tier HetNet,1-tier C-RAN, and 2-tier H-CRAN

and RRH/HPN association is used. The number of cell-center-zone RUEs keeps constant and is set at 10, andthe number of cell-edge-zone RUEs is varied to evaluateEE performances.

As shown in Fig. 3, EE performances are compared amongstthe 1-tier HPN, 2-tier underlaid HetNet, 2-tier overlaid HetNet,1-tier C-RAN, and 2-tier H-CRAN. The EE performancedecreases with the number of accessing UEs in the 1-tierHPN scenario because more powers are consumed to makecell-edge-zone UEs meet with the low rate-constrained QoSrequirements. The EE performance becomes better in the 2-tier HetNet than that in the 1-tier HPN because lower transmitpower is needed and higher transmission bit rate is achieved.Due to the spectrum reuse and the interference is alleviatedby the optimal solution in[29], the EE performance in the 2-tier underlaid HetNet scenario is better than that in the 2-tieroverlaid HetNet scenario. Furthermore, the EE performancein the 2-tier H-CRAN scenario is the best due to the gainsfrom both the H-CRAN architecture and the correspondingoptimal radio resource allocation solution. Note than the EEperformance of 1-tier C-RAN is a little worse than that of H-CRAN because it suffers from the coverage limitation whenRRH covers the area that the HPN serves in the 2-tier H-CRAN. Meanwhile, the EE performance of 1-tier C-RAN isbetter than that of both 1-tier HPN and 2-tire HetNet due to theadvantages of centralized cooperative processing and energyconsumption saving.

B. Convergence of the Proposed Iterative Algorithm

To evaluate EE performances of the proposed optimal re-source allocation solution (denoted by “optimal EE solution”),two algorithms are presented as the baselines. The first base-line algorithm is based on the fixed power allocation (denotedby “fixed power”), i.e., the same and fixed transmit power isset for different RBs, and the optimal power allocation derivedin (23) is not utilized. The second baseline algorithm is basedon the sequential RB allocation (denoted by “sequential RB”),where the RB is allocated to RUEs sequentially. The presented

9

0 2 4 6 8 100

0.5

1

1.5

Iteration Number

Ene

rgy

Effi

cien

cy (

bps/

Hz/

W)

Fixed Power, ηHUE

=0 dB

Sequential RB, ηHUE

=0 dB

Optimal EE, ηHUE

=0 dB

Fixed Power, ηHUE

= 20dB

Sequential RB, ηHUE

=20 dB

Optimal EE, ηHUE

= 20dB

Fig. 4. Convergence evolution under different optimization solutions

two baseline algorithms use the same constraints of (9)-(13)as the optimal EE solution does. There are 12 RRHs (i.e.,L= 12) uniformly-spaced around the reference HPN.

The EE performances of these three algorithms with dif-ferent allowed interference threshold under varied iterationnumbers are illustrated in Fig. 4. The number of RUEs withlow rate-constrained QoS requirement is assumed asM = 3. Itcan be generally observed that the plotted EE performances areconverged within 3 iteration numbers for different optimizationalgorithms. On the other hand, these two baseline algorithmsare converged more quickly than the proposal does becausethe proposal has a higher computing complexity. Furthermore,the maximum allowed inter-tier interference, which is definedas δ0 in (12), should be constrained to make the HUE workefficiently. To evaluateδ0 impacting on the EE performance,ηHUE described in (14) denoting the maximum allowed inter-tier interference is simulated. The largeηHUE indicates thatthe constraint ofδ0 should be controlled to a low level, whichsuggests to decrease the transmit power and even forbid the RBto be allocated to RUEs. Consequently, the EE performancewhenηHUE is 0 dB is better than that when it is 20 dB. TheEE performance decreases with the increasing SINR thresholdηHUE for these three algorithms.

C. Energy Efficiency Performances of the Proposed Solutions

In this part, key factors impacting on the EE performancesof the proposed radio resource allocation solution are evalu-ated. Since EE performances are closely related to the con-straints (12)–(13), the variablesηHUE andPR

max are two keyfactors to be evaluated in Fig. 5 and Fig. 6, respectively. Theratio ofΩ1 to ΩT impacting on EE performances is evaluatedin Fig. 7 to show performance gains of the enhanced S-FFR.

In Fig. 5, EE performances under the varied SINR thresh-olds of HUEsηHUE are compared among different algorithmswhen the maximum allowed transmit power of RRH is 20dBm or 30 dBm. WhenηHUE is not sufficiently large, the EEperformance almost keeps stable with the increasingηHUE

because the inter-tier interference is not severe due to adoptionof the enhanced S-FFR. However, the EE performance declinesfor these two baseline algorithms when the SINR thresholdηHUE is over 10 dB. While the proposed optimal EE solution

0 2 4 6 8 10 12 14 16 180.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

SINR Threshold of HUEs (ηHUE

) dB

Ene

rgy

Effi

cien

cy (

bps/

Hz/

W))

Fixed Power,PRmax

=30dBm

Sequential RB,PRmax

=30dBm

Optimal EE,PRmax

=30dBm

Fixed Power,PRmax

=20dBm

Sequential RB,PRmax

=20dBm

Optimal EE,PRmax

=20dBm

Fig. 5. EE performance comparisons under different SINR thresholdsof HUEs ηHUE

can sustain more inter-tier interferences, and the EE perfor-mance deteriorates afterηHUE is over 14 dB, indicating thatthe proposed solution can mitigate more inter-tier interferencesand provide higher bit rates for HUEs than the other twobaselines do. Summarily, the proposed optimal EE solutioncan achieve the best EE performance, while the “fixed power”algorithm has the worst EE performance, and the “sequentialRB” algorithm is in-between. Further, the EE performancesfor these three algorithms are strictly related to the maximumallowed transmit power of the RRH, and the EE performancewith PR

max = 30 dBm is better than that withPRmax = 20 dBm.

To further evaluate the impact of the maximum allowedtransmit power of RRHsPR

max on EE performances, Fig. 6compares the EE performance of different algorithms in termsof PR

max. In this simulation case,ηHUE is set at 0 dB, thenumber of RUEs with low rate-constrained QoS requirementsis set at 4, and the iteration number is set at 5. The maximumallowed transmit power of the HPN is set at 43 dBm and themaximum allowed transmit power of RRH varies within [14dBm, 36 dBm] with the step size of 2 dBm. WhenPR

max isnot large, EE performances increase almost linearly with therising PR

max for all three algorithms. WhenPRmax ≥ 22dBm,

both the SE performance and the total power consumptionincreases almost linearly with the risingPR

max. Therefore, theEE performance almost keeps stable. As shown in Fig. 6, EEperformances of the “sequential RB” algorithm is often betterthan those of the ”fixed power” algorithm due to the water-filling power allocation gains indicated in Eq. (23). Further,the proposed solution can achieve the best EE performancedue to gains of RB assignment and power allocation.

BesidesηHUE andPRmax, the ratio betweenΩ1 andΩ2 has

a significant impact on the EE performance of H-CRANs. TheEE performances of the H-CRAN under the enhanced S-FFRwith different ratios ofΩ1 to ΩT are evaluated in Fig. 7, wherethe number of RUEs with low and high rate constrained QoSrequirements are set atM = 5 and M = 10, respectively.Meanwhile, the number of total RBs isK = 25 and thesystem bandwidth is 5 MHz. Each UE is allocated by at leastone RB to guarantee the minimal QoS requirement. Fig. 7suggests that the EE performance increases almost linearly

10

14 16 18 20 22 24 26 28 30 32 34 360.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Maximum allowed transmit power (dBm)

Ene

rgy

Effi

cien

cy (

bps/

Hz/

W)

Fixed PowerSequential RBOptimal EE

Fig. 6. EE performance comparisons under different maximum allowedtransmit power of RRHs

0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8

0.6

0.8

1

1.2

1.4

1.6

The ratio of frequency resource allocated to Ω1

Ene

rgy

Effi

cien

cy (

bps/

HZ

/W)

PmaxR =18dBm

PmaxR =20dBm

PmaxR =22dBm

PmaxR =24dBm

Fig. 7. EE performance comparisons for different ratios ofΩ1 to ΩT

with the ratio ofΩ1 to ΩT . With more available exclusiveRBs for RRHs, there are fewer inter-tier interferences becausethe shared RBs specified for RUEs and HUEs become less.The increasing ratio ofΩ1 to ΩT results in the drasticallyincreasing of the expectation of SINRs for RUEs. Furthermore,since the inter-tier interference is mitigated by configuring fewshared RBs, the transmission power of RUEs increases due tothe proposed water-filling algorithm in Eq. (23), which furtherimproves overall EE performances of H-CRANs. Besides, theEE performance increases with the rising maximum allowedtransmit power of RRHPR

max, which verifies the simulationresults in Fig. 6 again. These results indicate that moreradio resources should be configured forΩ1 if only the EEperformance optimization is pursued. However, the fairness ofUEs should be considered jointly, and some necessary RBsshould be fixed for both HUEs and cell-edge-zone RUEs toguarantee the seamless coverage and successful delivery ofthecontrol signallings to all UEs in the real H-CRANs, which isa challenging open issue for the future research.

V. CONCLUSIONS

In this paper, the energy efficiency performance optimiza-tion for H-CRANs has been analyzed. In particular, theresource block assignment and power allocation subject to theinter-tier interference mitigation and the RRH/HPN associationhave been jointly optimized. To deal with the optimizationof resource allocations, a non-convex fractional programmingoptimization problem has been formulated, and the corre-sponding Lagrange dual decomposition method has been pro-posed. Simulation results have demonstrated that performancegains of H-CRANs over the traditional HetNet and C-RANare significant. Furthermore, the proposed optimal energy-efficient resource allocation solution outperforms the other twobaseline algorithms. To maximize EE performances further,the advanced S-FFR schemes with more RB sets should beresearched, and the corresponding optimal ratio of differentRB sets should be designed in the future.

APPENDIX APROOF OFTHEOREM 1

By following a similar approach presented in [30], we provethe Theorem 1 with two septated steps.

First, the sufficient condition ofTheorem 1 should beproved. We define the maximal EE performance ofProblem1 asγ∗ =

C(a∗,p∗)P (a∗,p∗) , wherea∗ andp∗ are the optimal RB and

power allocation policies, respectively. It is obvious that γ∗

holds:

γ∗ =C(a∗,p∗)

P (a∗,p∗)≥

C(a,p)

P (a,p), (35)

where a and p are the feasible RB and power allocationpolicies for solvingProblem 1. Then, according to (35), wecan derive the following formuals:

C(a,p) − γ∗P (a,p) ≤ 0,C(a∗,p∗)− γ∗P (a∗,p∗) = 0.

(36)

Consequently, we can conclude thatmaxa,p

C(a,p) −

γ∗P (a,p) = 0 and it is achievable by the optimal resourceallocation policiesa∗ and p∗. The sufficient condition isproved.

Second, the necessary condition should be proved. Supposedthat a∗ and p∗ are the optimal RB and power allocationpolicies of the transformed objective function, respectively, wecan haveC(a∗, p∗)− γ∗P (a∗, p∗) = 0. For any feasible RBand power allocation policiesa andp, they can be expressedas:

C(a,p)− γ∗P (a,p) ≤ C(a∗, p∗)− γ∗P (a∗, p∗) = 0. (37)

The above inequality can be derived as:

C(a,p)

P (a,p)≤ γ∗ and

C(a∗, p∗)

P (a∗, p∗)= γ∗. (38)

Therefore, the optimal resource allocation policiesa∗ andp∗ for the transformed objective function are also the opti-mal ones for the original objective function. The necessarycondition ofTheorem 1 is proved.

11

APPENDIX BPROOF OFL EMMA 1

We can define an equivalent function as:

F (γ) = maxa,p

C(a,p) − γP (a,p), (39)

and we assume thatγ1 andγ2 are the optimal value for thesetwo optimal RB allocation solutiona1,p1 and a2,p2,whereγ1 > γ2. Then,

F (γ2) = C(a2,p2)− γ2P (a2,p2)

> C(a1,p1)− γ2P (a1,p1)

> C(a1,p1)− γ1P (a1,p1) = F (γ1). (40)

Hence,F (γ) is a strictly monotonic decreasing function interms ofγ.

Meanwhile, leta′ and p′ be any feasible RB and powerallocation policies, respectively. Setγ′ = C(a′,p′)

P (a′,p′) , then

F (γ′) = maxa,p

C(a,p)− γ′P (a,p)

≥ C(a′,p′)− γ′P (a′,p′) = 0. (41)

Hence,F (γ) ≥ 0.

APPENDIX CPROOF OFTHEOREM 2

Supposed thatγ(i) andγ(i+1) represent the EE performanceof the reference RRH in thei-th and (i + 1)-th iteration ofthe outer loop, respectively, whereγ(i) > 0, and γ(i+1) >0, neither of them is the optimal valueγ∗. Meanwhile, theγ(i+1) is given byγ(i+1) = C(a(i),p(i))

P (a(i),p(i)), wherea(i) andp(i) are

the optimal RB and power solutions ofProblem 3in the i-thiteration of the outer loop, respectively. Note thatγ∗ is definedas the achieved maximum EE performance for all feasible RAsolutionsa,p, and thusγ(i+1) cannot be larger thanγ∗, e.g.,γ(i+1) ≤ γ∗. It has been proved thatF (γ) > 0 in Lemma 1whenγ is not the optimal valur to achieve the maximum EEperformance. Thus, theF (γ(i)) can be written as:

F (γ(i)) = C(a(i),p(i))− γ(i)P (a(i),p(i))

= P (a(i),p(i))(γ(i+1) − γ(i)) > 0. (42)

(42) indicates thatγ(i+1) > γ(i) due toP (a(i),p(i)) > 0,which suggests thatγ increases in each iteration of the outerloop in Algorithm 1 . According toLemma 1, the value ofF (γ) decreases with the increasing number of iterations dueto the incremental value ofγ.

On the other hand, it has been proved that the optimalconditionF (γ∗) = 0 holds in Theorem 1. The Algorithm1 ensures thatγ increases monotonically. When the updatedγincreases to the achievable maximum valueγ∗, Problem 2canbe solved withγ∗ andF (γ∗) = 0. Then, the global optimalsolutionsa∗ andp∗ are derived. We updateγ in the iterativealgorithm to find the optimal valueγ∗. It can be demonstratedthat F (γ) converges to zero when the number of iterationis sufficiently large and the optimal condition as stated inTheorem 1 is satisfied. Therefore, the global convergence ofAlgorithm 1 is proved.

APPENDIX DPROOF OFTHEOREM 3

We can rewrite Eq. (17) as:

C(a,p)− γ(i)P (a,p)

=N+M∑

n=1

K∑

k=1

an,kB0log2(1 + σn,kpn,k)

− γ(i)(ϕeff

N+M∑

n=1

K∑

k=1

an,kpn,k + PRc + Pbh)

=

K∑

k=1

(

N+M∑

n=1

an,kB0log2(1 + σn,kpn,k)

−

N+M∑

n=1

γ(i)ϕeffan,kpn,k −γ(i)

K

(

PRc + Pbh

)

)

. (43)

It is obvious that for the given RB allocation scheme, ifletting

fk (pn,k) =

N+M∑

n=1

an,kB0log2(1 + σn,kpn,k)

−N+M∑

n=1

γ(i)ϕeffan,kpn,k −γ(i)

K

(

PRc + Pbh

)

,

(44)

Eq. (43) can be written asC(a,p) − γ(i)P (a,p) =K∑

k=1

fk (pn,k), wherepn,k ∈ CN , and fk (·) : CN → R is

not necessarily convex. Similarly, the constraints (9)–(13) can

be expressed as the function ofpn,k withK∑

k=1

hk (pn,k) ≤ 0,

wherehk (·) : CN → RL, andL represents the number ofconstraints. Thus, Eq. (17) can be expressed as:

EE∗ = max

K∑

k=1

fk (pn,k), (45)

s.t.

K∑

k=1

hk (pn,k) ≤ 0, (46)

where0 ∈ RL. To prove the duality gap between Eq. (17) andthe optimal value of its dual problemD∗ is zero, a perturbationfunction v (H) is defined and can be written as:

v (H) = max

K∑

k=1

fk (pn,k), (47)

s.t.K∑

k=1

hk (pn,k) ≤ H , (48)

whereH ∈ RL is the perturbation vector.Following [25], if v (H) is a concave function ofH , the

duality gap betweenD∗ andEE∗ is zero. Therefore, to provethe concavity ofv (H), a time-sharing condition should bedemonstrated as follows.

Definition 1 (Time-sharing Condition): Letp1∗n,k andp2∗

n,k

be the optimal solutions of Eq. (47) tov (H1) and v (H2),

12

respectively. Eq. (45) satisfies the time-sharing condition if forany v (H1) and v (H2), there always exists a solutionp3

n,k

to meet:

∑Kk=1 hk

(

p3n,k

)

≤ αH1 + (1− α)H2, (49)∑K

k=1 fk

(

p3n,k

)

≥ αfk

(

p1∗n,k

)

+ (1− α) fk

(

p2∗n,k

)

,(50)

where0 ≤ α ≤ 1.Now we should prove thatv (H) is a concave func-

tion of H . For some α, it is easy to find H3 satis-fying H3 = αH1 + (1− α)H2. Let p1∗

n,k, p2∗n,k and

p3∗n,k be the optimal solutions with these constraints of

v (H1), v (H2), and v (H3), respectively. According to thedefinition of time-sharing condition, there exists ap3

n,k

satisfying∑K

k=1 hk

(

p3n,k

)

≤ αH1 + (1− α)H2 and∑K

k=1 fk

(

p3n,k

)

≥ αfk

(

p1∗n,k

)

+ (1− α) fk

(

p2∗n,k

)

. Since

p3∗n,k is the optimal solution tov (H3), it is obvious that∑K

k=1 fk

(

p3∗n,k

)

≥∑K

k=1 fk

(

p3n,k

)

≥ αfk

(

p1∗n,k

)

+

(1− α) fk

(

p2∗n,k

)

holds. Then, the concavity ofv (H) isproved.

Since v (H) is concave, Eq. (45) could be proved tosatisfy the time-sharing condition. The time-sharing condi-tion is always satisfied for the multi-carrier system whenthe number of carriers goes to infinity[26], such as theOFDMA based H-CRAN system in this paper. We can letp1n,k and p2

n,k be two feasible power allocation solutions.There areα percentages of the total carriers to be allocatedthe power withp1

n,k, while the remaining(1− α) percentagesof the total carriers are allocated the power withp2

n,k. ThenK∑

k=1

fk (pn,k) is a linear combination, which is expressed as

αfk

(

p1n,k

)

+ (1− α) fk

(

p2n,k

)

. Therefore, the constraintsare linear combinations, and it is proved that Eq. (45) satisfiesthe time-sharing condition. Consequently,v (H) is a concavefunction of H , and the duality gap betweenD∗ and EE∗

should be zero. This theorem is proved.

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