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On Power and Load Coupling in Cellular Networksfor Energy OptimizationChin Keong Ho, Di Yuan, Lei Lei, and Sumei Sun

Abstract—We consider the problem of minimization of sumtransmission energy in cellular networks where coupling occursbetween cells due to mutual interference. The coupling rela-tion is characterized by the signal-to-interference-and-noise-ratio(SINR) coupling model. Both cell load and transmission power,where cell load measures the average level of resource usagein the cell, interact via the coupling model. The coupling isimplicitly characterized with load and power as the variablesof interest using two equivalent equations, namely, non-linearload coupling equation (NLCE) and non-linear power couplingequation (NPCE), respectively. By analyzing the NLCE andNPCE, we prove that operating at full load is optimal in mini-mizing sum energy, and provide an iterative power adjustmentalgorithm to obtain the corresponding optimal power solutionwith guaranteed convergence, where in each iteration a standardbisection search is employed. To obtain the algorithmic result, weuse the properties of the so-called standard interference function;the proof is non-standard because the NPCE cannot even beexpressed as a closed-form expression with power as the implicitvariable of interest. We present numerical results illustratingthe theoretical findings for a real-life and large-scale cellularnetwork, showing the advantage of our solution compared tothe conventional solution of deploying uniform power for basestations.

Index Terms—Cellular networks, energy minimization, loadcoupling, power coupling, power adjustment allocation, standardinterference function.

I. I NTRODUCTION

Data traffic is projected to grow at a compound annualgrowth rate of78% from 2011 to 2016 [1], fueled mainlyby multimedia mobile applications. This growth will lead torapidly rising energy cost [2]. In recent years, informationcommunication technology (ICT) has become the fifth largestindustry in power consumption [3]. In cellular networks, inparticular, base stations consume a significant fraction ofthetotal end-to-end energy [4], of which50%–80% of the powerconsumption is due to the power amplifiers [5], [6]. Thisobservation has motivated green communication techniquesfor cellular networks [7]–[14]. These technologies includeadaptive approaches such as switching off power amplifiers toprovide a tradeoff of energy efficiency and spectral efficiency[7], [8], selectively turning off base stations [9], as wellas en-ergy minimization approaches for relay systems [10], OFDMAsystems [11]–[13], and SC-FDMA systems [14]. Extensive

This paper is presented in part at the IEEE International Conference onCommunications, June 2014.

C. K. Ho and S. Sun are with the Institute for Infocomm Research,A*STAR, 1 Fusionopolis Way, #21-01 Connexis, Singapore 138632 (e-mail:{hock, sunsm}@i2r.a-star.edu.sg).

D. Yuan and L. Lei are with the Department of Science and Technology,Linkoping University, Sweden. (e-mail:{di.yuan, lei.lei}@liu.se)

survey of other saving-energy approaches are highlighted in[2], [15], [16].

In this paper, we focus on the important problem ofminimizing the sum energy used for transmission in cellularnetworks. Besides reducing the energy cost for transmission,minimizing the transmission energy may lead to selectionof power amplifiers with lower power rating, hence furtherreducing the overhead cost involved in turning on poweramplifiers.

In a cellular network where base stations are coupled dueto mutual interference, the problem of energy minimizationischallenging, as each cell has to serve a target amount of datato its set of users, so as to maintain an appropriate level ofservice experience, subject to the presence of the couplingrelation between cells. To tackle this energy minimizationproblem, we employ an analytical signal-to-interference-and-noise-ratio (SINR) model that takes into account the load ofeach cell [17]–[19], where a load of a cell translates intothe average level of usage of resource (e.g., resource unitsin OFDMA networks) in the cell. This load-coupling equationsystem has been shown to give a good approximation for morecomplicated load models that capture the dynamic nature ofarrivals and service periods of data flows in the network [20],especially at high data arrival rates. Further comparison ofother approximation models concluded that the load-coupledmodel is accurate yet tractable [21]. By using this tractablemodel, useful insights can then be developed for the design ofpractical cellular systems. In our recent works [22], we haveused the load coupling equation to maximize sum utility thatis an increasing function of the users’ rates.

Previous works [17]–[20], [22] using the load-couplingmodel all assume given and fixed transmission power. Fortransmission energy minimization, both power and load be-come variables and they interact in the coupling model,making the analysis more challenging. In fact, the couplingrelation between cell powers cannot be expressed in closedform even for given cell loads. The key aspects motivatingour theoretical and algorithmic investigations are as follows.First, is there an insightful characterization of the operatingpoint in terms of load that minimizes the sum transmissionenergy? Second, given a system operating point in load, whatare the properties of the coupling system in power? Third,even if power coupling cannot be expressed in closed form,is there some algorithm that converges to the power solutionfor given cell load?

Toward these ends, our contributions are as follows. Weshow that if full load is feasible, i.e., the users’ data require-ments can be satisfied, then operating at full load is optimal

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in minimizing sum transmission energy (Section IV-C, Theo-rem 1). If full load is not feasible, however, then no feasiblesolution exists (Section IV-C, Corollary 1). Thus, full load isnecessary and sufficient to achieve the minimum transmissionenergy. Moreover, the optimal power allocation for all basestations is unique (Section V-B, Theorem 2), and can benumerically computed based on an iterative algorithm that canbe implemented iteratively at each base station (Section V-D,Algorithm 1). To prove the algorithmic result, we make use ofthe properties of the so-called standard interference function[23]; the proof is however non-standard, because the functionof interest does not have a closed-form expression, and hencewe use an implicit method to verify its properties. We alsocharacterize the load region over all possible power alloca-tion given some minimum target data requirements (SectionV-C, Theorems 3–4). Finally, we obtain numerical results toillustrate the optimality of the full-load solution on a cellularnetwork based on a real-life scenario [24]. Compared withthe conventional solution where the uniform power is used forbase stations, we show the significant advantage of the power-optimal solution in terms of meeting user demand target andreducing the energy consumption.

The rest of the paper is organized as follows. Section II givesthe system model of the load-coupled network. Section IIIformulates the energy minimization problem. Section IV char-acterizes the optimality of full load, while Section V derivesproperties of the power-coupling system and an iterative powerallocation algorithm that achieves the power solution. Numer-ical results are given in Section VI. Section VII concludes thepaper.

Notations: We denote a column vector by a bold lower caseletter, saya, a matrix by a bold capital letter, sayA, and its(i, j)th element by its lower caseaij . We denote apositivematrix asA > 0 if aij > 0 for all i, j. Similarly, we denote anon-negativematrix asA ≥ 0 if aij ≥ 0 for all i, j. Similarconventions apply to vectors. Finally,0 and1 denote the all-zeros and all-ones vectors of suitable lengths.

II. SYSTEM MODEL

A. Preliminaries

We consider a cellular network consisting ofn base stationsthat interfere with each other due to resource reuse. We focuson the downlink communication scenario where base stationi ∈ N , {1, · · · , n} transmits with powerpi ≥ 0 per resourceunit (in time and frequency). We refer to celli interchangeablywith base stationi. For notational convenience, we collect allpower{pi} as vectorp ≥ 0.

We assume a given association of the users to the basestations. In this association, each base stationi serves oneunique group of users, denoted by setJi, where|Ji| ≥ 1. Userj ∈ Ji is served in celli at raterij that has to be at least a ratedemand ofdij,min ≥ 0 nats. Thus,dij,min relates to a quality-of-service (QoS) constraint. We collect all the rates as vector rand the corresponding minimum demands asdmin ≥ 0. Thus,a rate vector meets the QoS constraints ifr ≥ dmin.

B. Load Coupling

We first consider the load coupling model for the cellularnetwork. We denote byx = [x1, · · · , xn]T the load in thenetwork, where0 ≤ x ≤ 1. In LTE systems, the load can beinterpreted as the fraction of the time-frequency resources thatare scheduled to deliver data. We model the SINR of userjin cell i as [17]–[20]

SINRij(x,p) =pigij∑

k∈N\{i} pkgkjxk + σ2(1)

whereσ2 represents the noise power andgij is the channelpower gain from base stationi to userj; note thatgkj , k 6= i,represents the channel gain from the interfering base stations.The functionSINRij depends onxk for k 6= i, but not onxi;the dependence on the entire vectorx is maintained in (1) fornotational convenience. The SINR model (1) gives a goodapproximation of more complicated cellular network loadmodels [20]. Intuitively,xk can be interpreted as the likelihoodof receiving interference from cellk on all the resource units.Thus, the combined term(pkgkjxk) ∈ [0, pkgkj ] is interpretedas the average interference taken over time and frequency forall transmissions.

Given the SINR, we can transmit reliably at the maximumrate rij = B log(1 + SINRij) nat/s per resource block, whereB is the bandwidth of a resource unit andlog is the naturallogarithm. To deliver a rate ofrij nat for userj, the ith basestation thus requiresxij , rij/rij resource units. We assumethatM resource units are available. Thus, we get the load forcell i asxi =

∑j∈Ji

xij/M , i.e.,

xi =1

MB

∑

j∈Ji

rijlog (1 + SINRij(x,p))

, fi(x) (2)

for i ∈ N . Without loss of generality, we normalizerijby MB in (2) and so we setMB = 1. Let f(x) =[f1(x), · · · , fn(x)]T . In vector form, we obtain thenon-linearload coupling equation(NLCE)

NLCE : x = f(x; r,p) (3)

for 0 ≤ x ≤ 1, where we have made the dependence of theloadx on the rater and powerp explicit.

In the NLCE, the loadx appears in both sides of theequation and cannot be readily solved as a fixed-point solutionin closed form. Intuitively, this is because the loadxi for basestation i affects the loadxk of another base stationk 6= i,which would then in turn affect the loadxi. This difficulty inobtaining thex in the NLCE remains despite that the functionSINRij (and similarly functionfi) depends onxk for k 6= ibut not onxi.

We collect the QoS constraints asr ≥ dmin. Without loss ofgenerality, we assumedmin is strictly positive, as those userswith zero rate can be excluded from further consideration.Hence the power vector satisfiesp > 0 so as to serve all theusers. Consequently, the load must be strictly positive, i.e.,0 < x ≤ 1.

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III. E NERGY M INIMIZATION PROBLEM

Our objective is to minimize the sum transmission energygiven by

∑ni=1 xipi. We note that the product(xipi) measures

the transmission energy used by base stationi, because theloadxi reflects the normalized amount of resource units used(in time and frequency) while the powerpi is the amount ofenergy used per resource unit.

The energy minimization problem is given by ProblemP0.

P0 : minp>0,r>0,0<x≤1

xTp (4a)

s.t. x = f(x; r,p) (4b)

r ≥ dmin. (4c)

As was mentioned earlier, the power vectorp and rate vectorr vector are strictly positive to satisfy the non-trivial QoSconstraint. The load vectorx is in fact determined by theNLCE constraint (4b) and thus may be treated as an implicitvariable. The second constraint (4c) is imposed so that the rater satisfies the QoS constraint.

We denote an optimal solution to ProblemP0 asp⋆, r⋆ andthe corresponding load asx⋆ as determined by the NLCE. Akey challenge of ProblemP0 is that a positive solution pair(p, r) is considered feasible only if there exists a load suchthat (4b) holds. Whether this existence holds is not obviousdue to the non-linearity of the NLCE. As such, the convexityof the optimization problem cannot be readily established,andhence standard convex optimization techniques do not applyreadily.

IV. OPTIMALITY OF FULL LOAD

In Section IV-A and Section IV-B, we consider fundamentalproperties of rate and load, respectively, such that there existsa power satisfying the NLCE. To study the fundamentalproperties, we consider the existence of a load satisfyingx > 0. The additional constraint thatx ≤ 1 is taken intoaccount in Section IV-C, in which we prove the key result thatfull load, i.e.,x = 1, is a necessary and sufficient conditionfor the solution in ProblemP0 to be optimal.

A. Satisfiability of Rate

We first establish conditions on rate vectorr such that a loadx > 0 exists and satisfies the NLCE, possibly withx > 1.We denote the spectral radius of matrixA as ρ(A), definedas the absolute value of the largest eigenvalue ofA.

Lemma 1:For any powerp > 0, there exists a unique loadx > 0 satisfying the NLCE if and only if

ρ(Λ(r)) < 1 (5)

where the(i, k)th element ofΛ(r) is given by

λik =

{0, if i = k;∑

j∈Jigkjrij/gij , if i 6= k

(6)

which is a function ofr (but notp).Proof: Follows directly from [22, Theorem 1].

Due to Lemma 1, we say that the rate vectorr is satisfiableif ρ(Λ(r)) < 1. If r is not satisfiable, then there does not existany powerp > 0 that results in a load satisfying constraint

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

p1

p2

(a) Power region.

0 0.02 0.04 0.06 0.080

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

x1

x2

(b) Load region.

Fig. 1. Corresponding power region and load region satisfying the NLCE.

(4b). We note that even ifr is satisfiable, it is still possiblethat the load does not satisfyx ≤ 1 and hence violates itsupper bound. Thus, satisfiability is a necessary condition fora feasible solution to exist in ProblemP0, but it may not besufficient.

Henceforth, we assume that a rater is satisfiable; otherwiseno feasible solution exists in ProblemP0. Given p, we canthen numerically obtainx by the iterative algorithm for load(IAL) [22, Lemma 1], as follows. Specifically, starting froman arbitrary initial loadx0 > 0, define the output of theℓthalgorithm iteration as

xℓ = f(xℓ−1; r,p) (7)

for ℓ = 1, 2, · · · , L, whereL is the total number of iterations.ThenxL converges to the fixed-point solutionx of the NLCEasL goes to infinity. The IAL is derived using [23] by showingthat f is a so-called standard interference function, to bedefined in Section V-A.

B. Implementability of Load

Although Lemma 1 states that any given power vectorp

always corresponds to a load vectorx that satisfies the NLCE,the reverse is not true. To obtain some intuition why thisinverse mapping may fail, let us consider the special caseof n = 2 base stations with channel gaingij = 1 for alli, j, rate r = 1, and noise varianceσ2 = 1. We randomlychoose the powerp = [p1, p2]

T using a uniform distributionover 0 < pi ≤ 2, i = 1, 2, which is plotted in Fig. 1(a). Thecorresponding loadx = [x1, x2]

T obtained using the IAL isshown in Fig. 1(b). We see that indeed there is a load regionthat does not appear to correspond to any powerp > 0.

Given thatr is satisfiable, we say that a loadx is im-plementableif there exists powerp such that the NLCE issatisfied.

The following toy scenario shows that full load may notalways be implementable. In practice, this may occur duringpeak times in cellular hot spots, such as train stations, when

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mobile data cannot be sustained at high speeds even if alltime-spectrum resources are used no matter how power is set(cf. Lemma 1).

We assumen = 2 cells with one user per cell, each with rater = 2 nat, andx = 1. The channel gains from a base station tothe user it serves and the user it does not serve is set asg = 1and g′ = 1/3, respectively. Note that the rate is satisfiable

since we getΛ =

[0 g′r/g

g′r/g 0

]and henceρ(Λ) = 4/9

which satisfies (5). By symmetry, the power allocated for allcells must be the same withp1 = p2 = p and must thus satisfy(2), i.e., log(1 + gp/(g′p + σ2)) = r. For anyp ≥ 0, theleft hand side is upper bounded bylog(1 + gp/(g′p+ σ2)) ≤log(1+g/g′) = ln(4) = 1.39 nat, which is less thanr = 2 nat.Hence, regardless of the power allocation, (2) cannot hold andso full load is not implementable in this case.

C. Main Result: Full Load is Optimal

Our first main result is given by Theorem 1, which statesthat full load, if implementable, is optimal to minimize thesum energy in ProblemP0.

Lemma 2 given below is a key step to prove Theorem 1.Lemma 2:For ProblemP0, the optimal solution is such

that the load vector satisfiesx⋆ = 1.Proof: Note that the load satisfiesxi > 0 for all cell i

to satisfy non-trivial rate demands. Assume that at optimality,we have0 < x⋆ ≤ 1 where there exists at least one celli ∈ N with load 0 < x⋆i < 1 and powerp⋆i . With all otherpowerp⋆k and loadx⋆k fixed, k 6= i, we reduce the powerp⋆ito p′ = p⋆i − ǫ, ǫ > 0. Using (2), the corresponding loadx⋆istrictly increases tox′ = x⋆i +ǫ

′, ǫ′ > 0. We chooseǫ > 0 suchthat x′ ≤ 1. With this new power-load pair(p′, x′) for cell i,we claim that (see proofs below): (i) the objective functionisreduced, and (ii) the corresponding rate vectorr′ is such thatr′ ≥ r⋆, i.e., the NLCE constraint is satisfied sincer⋆ ≥ dmin.The two claims together imply thatx⋆ with 0 < x⋆i < 1 isnot optimal, independent of the actual celli. By contradiction,x⋆i = 1 for all i, i.e, x⋆ = 1.

We now prove the first claim. Denote the energy usedin cell i, as a function of its powerpi, as ei = xipi =∑

j∈Ji

rijpi

log(1+cijpi)where cij , gij/(

∑k∈N\{i} pkgkjxk +

σ2) does not depend onpi nor xi. Then

∂ei∂pi

=∑

j∈Ji

rij(1 + cijpi) log(1 + cijpi)− cijpi

log2(1 + cijpi)(1 + cijpi). (8)

It can be verified by calculus that the numerator of eachsummand is strictly increasing forpi ≥ 0. Since the numeratorequals zero atpi = 0, the numerator is strictly positive forpi > 0. Clearly the denominator is strictly positive forpi > 0.Thus, ∂ei

∂pi> 0. Hence, when the power for celli is decreased,

the energyei decreases. Thus, the objective function alsodecreases.

To prove the second claim, we first note that for celli, wehave constrained the new power-load pair(p′, x′) to satisfy(2). Thus, the new rate for celli, denoted byr′ij , j ∈ Ji, isthe same as the optimal rater⋆ij corresponding to the power-load pair(p⋆i , x

⋆i ). Next, we observe that the productx′p′ is

strictly smaller as compared tox⋆i p⋆i , according to the first

claim. Thus, for userj ∈ Jk in cell k 6= i, SINRkj(x) strictlyincreases. It follows that the NLCE for cellk is satisfied withthe same loadxk but with a larger rater′kj as compared tothe optimal rater⋆kj . In summary, we thus haver′ ≥ r⋆.

Theorem 1:Suppose full load, i.e.,x = 1, is imple-mentable. Then the optimal solution for ProblemP0 is asfollows: r⋆ = dmin, andp⋆ is such thatx⋆ = 1. The optimalpower vectorp⋆ is thus given implicitly by the NLCE as

1 = f(1;dmin,p⋆). (9)

Proof: The proof follows from Lemma 2 and Lemma 7given in the Appendix, which state thatx⋆ = 1 andr⋆ = dmin

are the optimal solutions, respectively. Substituting theoptimalsolutions into the NLCE results in (9).

From Theorem 1, serving the minimum required rate is opti-mal. This observation is intuitively reasonable as less resourcesare used and hence less energy is consumed. Interestingly,Theorem 1 states that having full load is optimal. This secondobservation is not as intuitive, since it is not immediatelyclear the effect of using higher load on both the sum energyand interference. This is because using a high load may leadto more interference to neighbouring cells, which may thenrequire other cells to use more energy to serve their users’rates. Mathematically, the reason can be attributed to the proofof Lemma 2, which shows that, as the power decreases, theenergy as well as the interference for each cell decreases, whileconcurrently the load increases. Thus, by using full load, theenergy is minimized.

Next, Corollary 1 provides a converse type of result to The-orem 1. The result follows from a theorem with a generalizedstatement, which we defer to Section V-B because the proofrequires the use of algorithmic notions for finding power givenload.

Corollary 1: If full load x = 1 is not implementable, thenthere is no other loadx ≤ 1 with x 6= 1 that is implementable.Thus, there is no feasible solution for ProblemP0.

Proof: The result follows as a special case of Theorem 3later in Section V-B.

Remark 1:Theorem 1 and Corollary 1 together thus showthat full load is both necessary and sufficient to achieve theminimum energy in ProblemP0.

Remark 2: It can be easily checked that Theorem 1 andCorollary 1 continue to hold even if we generalize theobjective function to any functionc(x1p1, x2p2, · · · , xnpn)that is increasing in each of its argument. For example,c(y1, · · · , yn) =

∑wiyi gives the weighted sum energy with

positive weights{wi, i = 1, · · · , n}.

V. OPTIMAL POWER SOLUTION

Although full load is optimal for ProblemP0, it is still notclear if the optimal powerp⋆ is unique and how to numericallycomputep⋆ in (9). Our second main result, Theorem 2,answers both questions, but in a more general setting. Namely,we provide theoretical and algorithmic results for findingpower p given arbitrary loadx that is implementable (notnecessarily all ones) and arbitrary rater that is satisfiable (notnecessarily equal todmin), so as to satisfy the NLCE.

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A. Standard Interference Function

Before we state the main result of the section, we recapthe standard interference function and the iterative algorithmintroduced in [23]. The algorithm shall be used to obtain theoptimal powerp⋆, and is also a key step in the proof of theimplementability of load.

Consider a functionI : Rn+ → R

n+. We denote the input as

p as we shall focus on using power as the input. We sayI(·)is a standard interference function if it satisfies the followingproperties for all input powerp ≥ 0 [23].

1) Positivity: I(p) > 0;2) Monotonicity: If p ≥ p′, thenI(p) ≥ I(p′).3) Scalability: For allα > 1, αI(p) > I(αp).Next, we consider both the synchronous and asynchronous

versions of theiterative algorithm for power(IAP), similar tothe two versions of iterative algorithm in [23]. IAP generatesa sequence of power vectors via multiple iterations. In eachiteration, the power vector produced amounts to evaluatingfunctionI(·) with the previous iterate as the input. As poweris a vector, when the calculation of one power elementis performed, there is a choice of whether or not to usethis updated power value in the function evaluation for theremaining power elements. These two choices lead to thesynchronous and asynchronous IAPs. We consider a specificform of asynchronous IAPs which will turn out to be usefulfor our proof of Theorem 3 later.

We assumeL iterations are performed in each case. For thesynchronousIAP, the entire power vector is updated in eachiteration. In contrast, for theasynchronousIAP, there areninner iterations for each (outer) iteration, and in each inneriteration, only one power element is updated.

• Synchronous IAP: Assume an arbitrary initial powergiven byp0 > 0. The output for iterationℓ = 1, · · · , L,is given by

pℓ = I(pℓ−1). (10)

Clearly, any power element ofpℓ is solely determined bypℓ−1.

• Asynchronous IAP: In each iterationℓ = 1, · · · , L, weperform n inner iterations. Assume an arbitrary initialpower given byp0

0 > 0. The output of theith inneriteration,i = 1, · · · , n, is given by

pℓi = I(pℓ−1i−1 ) (11)

wherepℓ−1i−1 represents the power vector containing the

most current elements after(ℓ − 1) outer iterations and(i−1) inner iterations (i.e., during theℓth iteration). AfterL (outer) iterations are fully completed, each withn inneriterations, we obtainpL

n as the final power vector solution.Lemma 3 demonstrates the use of the IAP algorithms to

obtain the unique fixed-point point solution.Lemma 3:Suppose a fixed-point solutionp exists forp =

I(p). If I is a standard interference function, then startingfrom any initial power vector, both the synchronous and asyn-chronous IAP algorithms converge to the fixed-point solutionp, which is unique.

Proof: We omit the proof which is found in [23, Theo-rems 2,4].

B. Main Result: Existence and Computation of Power Solution

Before proving the main result, we present and prove someproperties on how the elements of the power vector relateto each other in NLCE. The properties will then be used toestablish that the results in [23] with the notion of standardinterference function can be applied.

Let pi be the vector of length(n − 1) that contains allelements in vectorp except for elementpi. For example, ifp = [p1, p2, p3, · · · , pn], thenp2 = [p1, p3, · · · , pn]. Lemma 4shows that givenx andr, the dependency ofpi on pi (suchthat the NLCE holds) qualifies as a function, even if thefunction is not in closed form.

Lemma 4:Let p,x, r satisfy the NLCE, where the vectorsare strictly positive. Then there exists functionhi : Rn

++ →R

n++ satisfyingpi = hi(pi;x, r) for all i = 1, · · · , n. Writing

pi’s andhi’s in vector form, we getp = h(p;x, r).Proof: We fix x, r and drop these notations in the

function hi(·) for simplicity. To prove the existence of thefunction hi(·), we need to show that givenpi, there exists auniquepi for i = 1, · · · , n. First, we write the NLCE in (2)as

1 =∑

j∈Ji

aijlog (1 + pibij(pi, σ2))

, ηi(pi) (12)

where

aij , rij/xi (13)

bij(pi, σ2) ,

gij∑k∈N\{i} pkgkjxk + σ2

(14)

are both independent ofpi. We fix pi > 0 and σ2 ≥ 0. Itfollows that bij(pi, σ

2) > 0 and soηi(pi) > 0. Observe thatηi(pi) is a strictly decreasing function ofpi. Sinceηi(pi)→∞aspi → 0, andηi(pi)→ 0 aspi →∞, there exists a uniquepi > 0 such thatηi(pi) = 1, and thus satisfies (12). Hencethere exists a function of the formpi = hi(pi), for any i.

Remark 3:The functionhi(·) does not submit to a closed-form solution. For example, consider expressingpi in termsof pi in (12) where the number of summands is|Ji| > 1.Because each of them is non-linear inpi, the dependency ofpi on pi is not explicit.

Remark 4:Although hi(·) cannot be expressed in closedform, we can numerically obtain the outputpi of the functionhi given the inputpi. Equivalently, this means that we want toobtain the value ofpi such that (12) holds. This is computed,for example, by a bisection search onηi(pi) = 1, making useof the property thatηi(pi) is a strictly decreasing function.Specifically, we first choose an arbitrary but small powerp′

such thatηi(p′) > 1 and an arbitrary but large powerp′′ suchthat ηi(p′′) < 1. Next we use the new powerp = (p′ +p′′)/2 and evaluate ifηi(p) is greater or smaller than one,then replacep′ or p′′ by p, respectively. By performing thisprocedure iteratively, we have guaranteed convergence to thedesiredp that satisfiesηi(pi) = 1. This forms the basis for theproposed algorithm later in Section VI.

We observe thath(·) is to some extent similar tof(·) in theNLCE (3). From Remark 3, however, the functionh(·) cannotbe readily written as a closed-form expression. Thus, proving

6

properties related toh(·) is more challenging, as compared tothe case off(·) for which a closed-form solution is available.Nevertheless, Lemma 5 states thath(·) qualifies as a standardinterference function as defined in [23].

Lemma 5:Given loadx ≥ 0 and rater ≥ 0, h(p;x, r) isa standard interference function inp.

Proof: Henceforth we assume that loadx ≥ 0 andrate r ≥ 0 are given. For notational convenience, we dropthe dependence of these entities in the notation ofh(·). Weconsider an arbitraryi and refer toηi(pi), aij , bij as definedin (12), (13) and (14), respectively, throughout the proof.Forthis proof, it is useful to denote the functionηi(pi) explicitlyas ηi(pi, pi, σ

2) to ease the discussion. We prove each ofthe three properties required for standard interference functionbelow.

Positivity: From the proof of Lemma 4, there exists a uniquepi > 0 that satisfies (12), i.e.,hi(pi) > 0. This holds for alli, thush(p) > 0.

Monotonicity: From (12), we observe thatηi(pi, pi, σ2)

strictly increases aspi decreases, or as any element ofpi

increases. Hence, to satisfyηi(pi, pi, σ2) = 1, pi strictly

increases if any element ofpi increases. We note that anequivalent representation ofηi(pi, pi, σ

2) = 1 is pi = hi(pi).It follows thathi(pi) is increasing in any of the arguments.

Scalability: Let q1 = hi(pi) andq2 = hi(αpi), whereα >1. Observe that

ηi(q1, pi, σ2) = ηi(q2, αpi, σ

2) (15)

since both equal one according to (12). It is easy to checkthat q1bij(pi, σ

2) = αq1bij(αpi, ασ2). That is, multiplying

all the terms in the triplet(q1, pi, σ2) by a positive constant

still allows (12) to be satisfied. Thus we get from (15)

ηi(αq1, αpi, ασ2) = ηi(q2, αpi, σ

2). (16)

With the second argument inηi(y, ·, z) fixed, we note thatthe output of the function strictly decreases withy and strictlyincreases withz. By the equality in (16), it follows thatαq1 >q2 becauseασ2 > σ2. Taking into account of the definitionof q1 andq2, we have provedαhi(pi) > hi(αpi).

Using Lemma 4 and Lemma 5, we are ready to providethe main result, stating that NLCE can be expressed in analternative form with the power taken as the subject of interest.The proof is non-standard, because the relations among thepower elements do not submit to a closed form (Remark 3).Hence, it has been necessary to first establish that the relationbetween one power element and the others qualifies as afunction (Lemma 4). Next, we have used an implicit methodto prove thath(·) is indeed a standard interference function(Lemma 5).

Theorem 2:Given loadx and rater, the powerp thatsatisfies the NLCE can be represented equivalently in the formof a non-linearpowercoupling equation (NPCE) given by

NPCE : p = h(p;x, r) (17)

where h(·) is a standard interference function. Given thata solutionp exists, thenp is unique and can be obtainednumerically by the IAP.

Proof: Lemma 4 states the existence of the functionh(·), and hence allows us to obtain the NPCE. Lemma 5states thath(·) satisfies all the properties required for astandard interference function. The uniqueness and iterativecomputation ofp then follow from Lemma 3 with the standardinterference functionh(·).

Remark 5:So far we have assumed that there is no max-imum power constraint imposed for any element of powerp. If such power constraints are imposed, then a so-calledstandard constrained interference function defined in [23]canbe used instead to perform the IAP, in which the output of eachiteration is set to the maximum power constraint value, if thatreturned fromh is higher. This type of iteration converges toa unique fixed point [23, Corollary 1].

C. Characterization on Implementability of Load

Theorem 3 provides a monotonicity result for load im-plementability. We recall that a load vectorx is said to beimplementable if there exists powerp such that the NLCEholds.

Theorem 3:Consider two load vectors withx′ ≥ x andx′ 6= x. If x is implementable, thenx′ is implementable.Moreover, the respective corresponding powersp and p′

satisfyp′ < p.Proof: Supposex is implementable, i.e., there exists

powerp such that the NPCE (or equivalently the NLCE) holds.From Theorem 3,h(·) is a standard interference function. Weshall prove thatx′ is also implementable, i.e.,p′ exists.

Before we consider the general case ofx′ ≥ x, we firstfocus on the special case that strict inequality holds only forthe first element (with re-indexing if necessary), i.e.,x =[x1, x2, · · · , xn]T andx′ = [x′1, x2, · · · , xn]

T with x′1 > x1.We now use the asynchronous IAP (11) with loadx′, and weset the initial power asp0 = p. Our objective is to show thatthe power converges top′ that satisfies the NPCE withp′ < p.

Consider the asynchronous IAP (11) with outer iterationℓ = 1 and inner iterationi = 1, 2, · · · , n:

• For i = 1: Consider the NLCE for cell1 with the originalloadx and powerp:

x1 =∑

j∈J1

r1j

log(1 +

p1g1j∑k≥2

pkgkjxk+σ2

) . (18)

In the first iteration,x1 andp1 are updated by the actualload of interestx′1 and the iterated powerp11, respectively,with other load and power unchanged. Sincex′1 > x1, wemust havep11 < p1.From the proof in Lemma 2, the energye1 , p1x1with p1, x1 given by (18) satisfies∂e1/∂p1 > 0. Since∂e1/∂x1 = ∂e1/∂p1 ·∂p1/∂x1 and clearly∂p1/∂x1 < 0,we get∂e1/∂x1 < 0. Thus,p11x

′1 < p1x1.

• For i = 2: We shall show that the iterated power satisfiesp12 < p02 = p2. The NLCE for cell2 with the originalloadx and powerp can be written as:

x2 =∑

j∈J2

r2j

log(1 +

p2g2jp1g1jx1+

∑k≥3

pkgkjxk+σ2

)

7

Upon updating cell 2, we have updatedx1, p1 to thenewly iteratedx′1, p

11, respectively. Sincep11x

′1 < p1x1

as mentioned earlier,p12 < p2.• For i ≥ 3: For subsequent iterations, it can be shown

similarly that p1i < p0i = pi for i = 3, · · · , n. Thiscompletes the first outer iteration.

At this point, we getp1n < p. It can be similarly shown that

pℓ+1n < pℓ

n for ℓ > 1.For large number of iterationsL, the decreasing sequence

p0n,p

1n, · · · must converge sincepℓ

n ≥ 0 (i.e., it is boundedfrom below) for anyℓ due to the positivity of the standardinterference function. Thus, the power solution exists, i.e.,x′

is implementable.At convergence, we havelimL→∞ pL

n = p′ < p. Sofar we have assumed that only one element of the load isstrictly increased. In general, if more than one load element isincreased, repeating the argument sequentially for every suchelement proves that power exists and is decreased. Thus, ingeneralx′ is implementable forx′ ≥ x, wherep′ < p.

From Theorem 3, we also obtain the equivalent result thatx is not implementable ifx′ is not implementable.

The next theoretical characterization is on the imple-mentable load regionL over all non-negative power vectorsfor any given satisfiable rater, i.e., L , {x ≥ 0 : x =f(x; r,p),p ≥ 0}. Theorem 4 states that the boundary ofthis region is open. The norm‖ · ‖ in Theorem 4 can be anynorm, e.g., the 2-norm‖ · ‖2 or the maximum norm‖ · ‖∞.

Theorem 4:Suppose loadx is implementable with powerp and rater. Then there existsδ > 0, such that any loadvectorx′ with ‖x′−x‖ ≤ δ is implementable. Moreover, theimplementable load regionL is open.

Proof: Let p = βp with β > 1, and let the correspondingload satisfying the NLCE with rater be x. Note thatx exists,because the existence of load does not depend on power (cf.Lemma 1). By applying the IAL in (7) to obtainx (usingpower p) with the initial load set asx0 = x, it can be easilychecked that the load vector decreases in every iteration. Sincex > 0, the iterations must converge tox = limL→∞ xL < x.By Theorem 3, anyx′ ≥ x is implementable. Asx < x, thereis an implementable neighbourhood ofx. That is, there existsδ > 0, for which any load vectorx′ satisfying‖x′ − x‖ ≤ δis implementable. Since the result holds for anyx in L, itfollows thatL is open.

D. Algorithm for Optimal Power Vector

By Theorem 2, we can use the IAP to compute the optimalpowerp⋆ for (9) in Theorem 1 for any given implementableloadx⋆. We recall that to minimize the energy we setx⋆ = 1

(Theorem 1). To obtain the output of the functionh(·) in eachstep of the IAP, bisection search is able to determine the powerpi such thatηi(pi) = 1 (see Remark 4). Putting together thetheoretical insights results in the following formal algorithmicdescription (Algorithm 1) for computingp⋆.

Algorithm 1 solves the NPCE for givenx⋆, by iterativelyupdating the power vector and re-evaluating the resulting loadf(x⋆; r,p). The bulk of the algorithm starts at Line 2. The

Given:- target load vectorx⋆ = [x⋆1, x

⋆2, · · · , x

⋆n]

T

- rate vectorr such thatρ(Λ(r)) < 1- arbitrary initial power vectorp- toleranceǫ > 0Output : p⋆ with x⋆ = f(x⋆; r,p⋆)

1: Initialize x← f(x⋆; r,p).2: while ‖x− x⋆‖∞ > ǫ do3: for i = 1 : n do4: plefti ← ξ for any ξ such thatηi(ξ) > 15: prighti ← ψ for anyψ such thatηi(ψ) < 16: while |ηi(pi)− 1| > ǫ do7: if ηi(pi) ≤ 1 then8: prighti ← pi9: else if ηi(pi) > 1 then

10: plefti ← pi11: end if12: pi ← (plefti + prighti )/213: end while14: end for15: x← f(x⋆; r,p)16: end while17: p⋆ ← p, returnp⋆

Algorithm 1: IAP algorithm for computing optimal power.

outer loop terminates if the load vectorx has converged tox⋆. For each outer iteration, the inner loop is run starting atLine 3, for which the power vector for each celli is updated. Ineach update, the power range is first initialized to[ξ, ψ], whereξ < ψ, such thatηi(ξ) > 1 andηi(ψ) < 1. Since the functionηi(·) is a strictly decreasing function, the bisection search fromLines 7-12 ensures convergence to the unique solution forηi(pi) = 1, or equivalently, the value ofhi(pi;x, r). Loadre-evaluation is then carried out in Line 15.

VI. N UMERICAL EVALUATION

A. Simulation Setup

In this section, we provide numerical results to illustratethetheoretical findings. The simulations have been performed fora real-life based cellular network scenario, with publiclyavail-able data provided by the European MOMENTUM project[24]. The channel-gain data are derived from a path-loss modeland calibrated with real measurements of signal strength inthe network of a sub-area of Alexanderplatz in the city ofBerlin. The path-loss model takes into account the terrainand environment, pre-optimized antenna configuration (height,azimuth, mechanical tilt, electrical tilt); fast fading isnot partof the data made available. Further details are available in[24].

The scenario is illustrated in Fig. 2. The scenario has 50base station sites, sectorized into 148 cells. In Fig. 2, thereddots indicate base station sites and the green dots representthe location of users. Most of the sites have three sectors(cells) equipped with directional antennas. The blue shortlinesrepresent the antenna directions of the cells. The entire service

8

Fig. 2. Network layout and user distribution in an area of Alexanderplatz,Berlin. The units of the axes are in meters. Digital Map:c© OpenStreetMapcontributors, the map data is available under the Open Database License.

TABLE INETWORK AND SIMULATION PARAMETERS

Parameter ValueService area size 7500× 7500 m2

Pixel resolution 50× 50 m2

Number of sites 50Number of cells 148Number of pixels 22500Number of users 1480Thermal noise spectral density -145.1 dBm/HzTotal bandwidth per cell 4.5 MHzBandwidth per resource unit 180 kHzToleranceǫ in IAP 10

−5

Initial power vectorp in IAP 1 W

area of the Berlin network scenario is divided into 22500 pixelsas shown in Fig. 2. That is, each pixel represents a small squarearea, with resolution 50× 50 m2, for which signal propagationis considered uniform. Users located in the same pixel areassumed to have the same channel gains. In our simulations,each cell serves up to ten randomly distributed users in itsserving area as defined in the MOMENTUM data set. Thetotal bandwidth of each cell is 4.5 MHz. Following the LTEstandards, we use one resource block to represent a resourceunit with 180 kHz bandwidth each in the simulation. Networkand simulation parameters are summarized in Table I.

B. Results

Our objective is to numerically illustrate the relationshipamong the load, power, and sum transmission energy. First,we consider the use of uniform load withx = φ1 for various0 < φ ≤ 1, with φ = 1 being the case of full load.Given the load vectorx, the optimal power solutionp isthen obtained by using the IAP described by Algorithm 1.Next, for benchmarking, we consider the conventional schemethat employs uniform power allocationp = β1, β > 0. We

300 350 400 450 500 550 600 6500

20

40

60

80

100

Rate Demand (kbps)

Ene

rgy

Con

sum

ptio

n (J

)

uniform load=1uniform load=0.8uniform load=0.6uniform power

Fig. 3. Sum transmission energy with respect to user’s rate demand.

chooseβ that results in the minimum sum energy subject tothe constraint that the corresponding load satisfies0 ≤ x ≤ 1,as follows. From the proof of Lemma 2, the energy (given bythe product of load and power) for each cell strictly decreasesas the power strictly decreases. Thus, to minimize the sumenergy, we choose the smallestβ such thatx ≤ 1; this can beobtained by a bisection search starting with sufficiently smalland large values ofβ. For anyβ under consideration, the IALis used to obtain the load corresponding to the powerp = β1.

In the first numerical experiment, we consider the sumenergy for rate demandr = ξ1 with ξ being successivelyincreased, while keepingr satisfiable. Fig. 3 compares thesum energy for various uniform load levels, including full load,and that obtained by uniform power allocation. From Fig. 3,the sum energy for all cases appears to grow exponentiallyfast as the rate demand increases, approaching infinity asthe rate demand increases. The vertical dotted line in Fig. 3corresponds to the boundary when the rate demand is notsatisfiable, i.e.,ρ(Λ(r)) = 1, and hence represents the upperbound for which the system can support. This behaviouris consistent with Lemma 1. Deploying full load achievesthe smallest sum energy, in accordance with Lemma 2. Thereduction in sum energy is particularly evident in comparisonto the scheme of uniform power – the relative saving is90%or higher for the rate demand shown in Fig. 3. Conversely,for a fixed amount of sum energy, deploying full load andoptimizing the corresponding power allows for maximizingthe rate demand that can be served.

Next, we examine the energy consumption by progressivelyincreasing the uniform load for four rate demand levelsr = ξ1with ξ taking the values of 350 kbps, 450 kbps, 550 kbps and600 kbps. The results are shown in Fig. 4. We observe that thesum energy decreases monotonically by increasing the load.The reduction of sum energy appears to be exponentially fastin the low-load regime, but is much slower in the high-loadregime. In addition, the numerical results reinforce the fact

9

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

5

10

15

20

25

30

35

40

45

50

Load

Ene

rgy

Con

sum

ptio

n (J

)

rate demand=350 kbps rate demand=450 kbps rate demand=550 kbps rate demand=600 kbps

Fig. 4. Sum transmission energy with respect to cell load.

0 10 20 30 40 50 6010

−5

10−4

10−3

10−2

10−1

100

101

Number of Iterations

||x−x∗||2

rate demand=350 kbpsrate demand=450 kbps rate demand=550 kbps rate demand=600 kbps

Fig. 5. The evolution of the Euclidean distance between the iteratex andthe targetx⋆, given by the 2-norm‖x−x

⋆‖2, over iterations withx⋆= 1.

that some load vectors are not implementable. In particular,it is not always possible to obtain a power vectorp for aload vectorx = φ1 with very smallφ > 0. From Fig. 4, thesum energy surges to infinity when the load approaches somefixed (small) value, which suggests that, for anyr > 0, theload cannot become arbitrarily small, irrespective of power.

Furthermore, we numerically investigate the convergencebehavior of the IAP. The theoretical analysis on the con-vergence speed of the IAP depends on whetherh(·) furthersatisfies some property such as contractivity [25], for whichlinear rate of convergence can be shown to hold. In Fig. 5,we set the target load vectorx⋆ = 1 and the initial powervector p = 1 Watt, with rate demandr = ξ1 where ξ ∈{350, 450, 550, 600} kbps. The Euclidean distance betweenthe iteratex and targetx⋆ is given by the 2-norm‖x−x⋆‖2.The evolution of‖x−x⋆‖2 for the four different rate demand

1480 2960 4440 5920 7400

20

40

60

80

100

120

Number of Users

Req

uire

d Ite

ratio

ns

rate demand=50 kbps rate demand=100 kbps rate demand=150 kbps rate demand=200 kbps

Fig. 6. The number of iterations required to achieve convergence for differentnumber of users.

cases is illustrated in Fig. 5. We consider the algorithmconverged if the largest error between the load iterate andthe target load is less thanǫ = 10−5, i.e., if ‖x− x⋆‖∞ ≤ ǫ.For the four rate demand cases, convergence is reached after11, 19, 36 and 59 iterations, respectively. Given the size ofthe network (148 cells), the values are moderate. Also, wenotice that when the rate demand increases, more iterationsare required for convergence with a longer tail-off. This ismainly because a high rate demand means that, in general, theNPCE is operating in the high SINR regime. The amount ofprogress in load in an IAP iteration is mainly dependent on thedenominator in (3). For high SINR regime, the relative changein load is lesser due to the logarithm operator, thus slowingdown the progress. Moreover, the number of iterations dependson the initial power point. In general, fewer iterations arerequired if the starting power point is closer to the optimum.Note that no matter what the rate of convergence is, theconvergence of the IAP is guaranteed by Theorem 2. Theconvergence speed depends also on other factors, e.g., the scaleof the network (in terms of the number of BS and users), therate demand and the choice ofǫ. An explicit characterizationof this dependence is beyond the scope of this paper.

In case of the presence of some time constraints in apractical application, the IAP may be terminated before fullconvergence is reached. Thus, the capability of deliveringa load-feasible and close-to-convergent solution within fewiterations is of significance. It can be seen in Fig. 5 that amajority of the iterations is due to the tailing-off effect –the load vector is in fact close to the target value withinabout half of the iterations. For all the rate demand levels,convergence is in effect achieved in less than 20 iterations; thisis promising for the practical relevance of the proposed IAPscheme. Finally, to ensure that the load is strictly less than fullload for practical implementation, we may setx⋆ = (1− ǫ′)1with ǫ′ > ǫ.

Fig. 6 illustrates the the number of iterations required for

10

the load to converge, i.e., until‖x−x⋆‖∞ ≤ ǫ, with ǫ = 10−5.The total number of users is increased from the current1480(corresponding to10 users per cell) to7400 (correspondingto 50 users per cell). We have set the rate demand asr =ξ1 with ξ ∈ {50, 100, 150, 200} kbps, because the originalcase ofξ ∈ {350, 450, 550, 600} is no longer satisfiable for7400 users. From Fig. 6 , we see that the number of iterationsincreases as the number of users increases, and the rate ofincrease is higher if the number of users is large or the demandis high. Hence, for systems that support high data rate or largenumber of users, more computational resources are needed toimplement the algorithm.

VII. C ONCLUSION

We have obtained some fundamental properties for thecellular network modeled by a non-linear load coupling equa-tion (NLCE), from the perspective of minimizing the energyconsumption of all the base stations. To obtain analyticalresults on the optimality of full load, and the computation andexistence of the power allocation, we have investigated a dualto the NLCE, given by a non-linear power coupling equation(NPCE). Interestingly, although the NPCE cannot be statedin closed-form, we have obtained useful properties that areinstrumental in proving the analytical results. Our analyticalresults suggest that in load-coupled OFDMA networks or morespecifically LTE networks, the maximal use of bandwidthand time resources over power leads to the highest energyefficiency. In the literature, the maximal use of resources istypically suggested to maximize the network throughput; ourwork gives a similar conclusion but from a different andcomplementary approach of minimizing energy. To implementthe solution, the load and power solutions have to be computedand sent to all base stations for implementation. Hence, somelevel of coordination has to be set up in practice. In this paper,we have assumed the use of ideal power amplifier and that theusers’ associations to the base stations are given. The effects ofnon-linear power amplifier and the problem of user associationmay be considered as future work.

ACKNOWLEDGEMENTS

We would like to thank the anonymous reviewers for theirvaluable comments and suggestions. The work of the secondauthor has been supported by the Linkoping-Lund ExcellenceCenter in Information Technology (ELLIIT), Sweden. Thework of the third author has been supported by the ChineseScholarship Council (CSC) and the overseas PhD researchinternship scheme from Institute for Infocomm Research (I2R),A*STAR, Singapore.

APPENDIX

Lemma 6 (Theorem 2, [22]):Consider the NLCE (3) withpowerp fixed. Given the rate vectorsr′ and r with r′ ≥ r

andr′ 6= r, the corresponding load vectorsx′ andx satisfyx′ > x.

We omit the proof of Lemma 6, which is given in [22].Lemma 7:For ProblemP0, the optimal rate vector satisfies

r⋆ = dmin.

Proof: Suppose that at optimality, there exists at least onerate elementr⋆ij that is strictly greater than its corresponding(minimum) rate demanddij,min. Taking the power to be fixedasp⋆, if we decreaser⋆ij to dij,min, then the load will strictlydecrease while satisfying the constraint (4b) by Lemma 6.Thus, the objective function value decreases. This contradictsthe optimality ofr⋆ij . Thusr⋆ = dmin.

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