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TOWARD ENERGY-EFFICIENT 5G WIRELESS COMMUNICATIONS TECHN OLOGIES

Renato L. G. Cavalcante†, Sławomir Stanczak†⋆, Martin Schubert‡, Andreas Eisenblatter∗, Ulrich Turke∗

† Fraunhofer Heinrich Hertz Institute, Berlin, Germany.⋆ Technical University of Berlin, Germany‡ Huawei Technologies - European Research Center, Munich, Germany∗ atesio GmbH, Berlin, Germany

The densification and expansion of wireless networkspose new challenges on energy efficiency. With a drastic in-crease of infrastructure nodes (e.g. ultra-dense deployment ofsmall cells), the total energy consumption may easily exceedan acceptable level. While most studies focus on the energyradiated by the antennas, the bigger part of the total energybudget is actually consumed by the hardware (e.g., coolersand circuit energy consumption). The ability to shutdowninfrastructure nodes (or parts of it) or to adapt the transmis-sion strategy according to the traffic will therefore becomean important design aspect of energy-efficient wireless ar-chitectures. Network infrastructure should be regarded asaresource that can be occupied or released on demand, andthe modeling and optimization of such systems are highlynontrivial problems. In particular, elements of the networkinfrastructure should be released by taking into account trafficforecasts to avoid losing the required coverage and capacity.However, even if traffic profiles were perfectly known, thedetermination of the elements to be released is complicatedby the potential interference coupling between active ele-ments and the sheer size of the optimization problems indense networks.

INTRODUCTION

Due to the compelling need for broadband mobile accessto the Internet, over the past decade there has been a dra-matic growth in demand for wireless access worldwide. Thisgrowth is expected to continue in the years to come, driven byan increasing interest in various wireless services and noveltypes of machine-to-machine (M2M) and device-to-device(D2D) communications. The vision is to create the so-called“Internet-of-Things” by integrating billions of sensors and

Copyright (c) 2014 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must be ob-tained by sending a request to pubs-permissions@ieee.org.The work waspartially supported by the European Commission within the FP7 ProjectsGreenNets (grant agreement n. 286822) and by the German Federal Min-istry of Economics and Technology (BMWi) within the projectComGreen(grant number 01ME11010). Part of this work has also been performed in theframework of the FP7 project ICT-317669 METIS, which is partly funded bythe European Union. The authors would like to acknowledge the contribu-tions of their colleagues in METIS, although the views expressed are thoseof the authors and do not necessarily represent the project.

actuators into physical objects and connecting them to thenetwork via wireless connections. Requiring no human in-volvement, such communications may exceed any existinglimits on information dissemination, leading to a data explo-sion of unprecedented magnitude.

The above vision can only be brought to reality if weintroduce major changes to the way current cellular networksare designed and operated. The need for such changes canbe partially justified by the results in the landmark study byGupta and Kumar [1]. This study strongly suggests that tra-ditional large-scale networks (i.e., networks of spatially andtemporarily independent sources, arbitrarily located sourceand destination nodes, and arbitrary traffic demands) in-evitably face the problem of asymptotically vanishing per-user throughputs whenever the following restrictions hold: i)nodes are stationary, ii) interference is treated as noise [2],and iii) the network operates without any underlying infras-tructure such as the presence of base stations (i.e., the infor-mation needs to be carried from node to node in a multi-hopfashion). As an immediate consequence, to overcome thisfundamental limit, we must drop at least one of the restric-tions when designing wireless networks in order to provideadditional dimensions for network optimization.

The first restriction of the study in [1] has been liftedin [3], where the authors analyze an infrastructure-less net-work with mobile nodes and interference treated as noise.It has been shown that mobility can stabilize the through-put at the cost of delay. Further studies reveal and charac-terize a fundamental trade-off between throughput and delay(e.g., [4]). From these studies, we conclude that mobility maybe an important ingredient in enhancing the performance offuture wireless networks, but, due to strict delay constraintsof many wireless applications, it cannot be the ultimate solu-tion to the problem.

By dropping the second restriction, we can make nodesexploit, shape, or reject interference through advanced multi-user transmission and reception techniques. Indeed, the stud-ies in [5], [6] have shown that the throughput in large-scaleinfrastructure-less networks can be stabilized by resortingto cooperation and other multi-user communications strate-gies that are derived from information-theoretic results onbroadcast, multiple-access, and relay channels. An additionalperformance-enhancing approach falling into the class of

interference-shaping techniques is interference alignment [7].However, these interference-mitigating methods alone can-not deliver the promised gains in practice because of, forexample, the lack of channel state information and the lackof perfect synchronization in real systems. Therefore, a fixednetwork infrastructure, which is of vital importance to currentnetworks, is also envisioned to play a crucial role in futuresystems.

In particular, networks with densely deployed infrastruc-ture nodes are one of the main pillars in the current 5G dis-cussion to enhance the throughput of cellular networks at rel-atively low operational costs [8]. The vision is to have smalland low cost base stations to form small cells and to pro-vide Internet access by using short-distance links [9]. Thisvision is partially motivated by the analysis in [1]; in fact,the study in [10] shows that the per-user throughput can beimproved significantly even when interference is treated asnoise, provided that the density of infrastructure nodes growssufficiently fast with the number of users.

One of the main challenges that may limit the acceptabledensity of future networks is the high capital and operationalcosts. In particular, a large part of operational costs is directlyrelated to the energy consumed for transmission and for op-eration of the network infrastructure [11]. We argue in thisarticle that future 5G wireless communications technologiesneed to be energy-efficient to reduce the total cost per trans-mitted bit, thereby providing cost-effective, affordablewire-less bandwidth.

In the text that follows, we give an overview of fundamen-tal aspects of energy saving mechanisms. Using information-theoretic arguments, we show that, without any additional en-ergy saving measures, current approaches aiming at reducingthe (radiated) energy-per-bit may, in fact, increase thetotalenergy consumption because of, for example, the energy re-quired to power hardware and to perform signal processingtasks. Therefore, novel energy saving schemes taking theseoften ignored sources of energy consumption into account arerequired. We discuss the state-of-the-art and challenges to beaddressed in the development of 5G systems. Particular fo-cus is given to approaches that save energy by switching offhardware or by adapting the transmission strategy. In moredetail, if properly exploited, spatial and temporal fluctuationsof traffic in real networks create opportunities to switch offhardware. We show machine learning and statistical tools thatcan be used to predict the periods of low traffic, and we alsoprovide some observations on the evolution of traffic patternsin current networks. These observations may be used to guidethe selection of prior knowledge to be incorporated into traf-fic forecast tools for future networks. Even with good traf-fic forecast tools, determining network elements that can beswitched off is not a trivial task because of the possible inter-ference coupling among active network elements. We there-fore review general tools that can be used to study a plethoraof interference models in a unified way. These tools are not

restricted to the domain of wireless communication, and wepoint out some simple and useful (and perhaps novel) resultsthat can greatly simplify the analysis of coupled systems. Thetraffic and interference models are then used to pose the (typ-ically intractable) energy saving optimization problems,andwe present heuristics that i) have a strong analytical justifi-cation and ii) have the potential to scale to problems of hugedimensions, as required in dense networks.

ENERGY EFFICIENCY IN WIRELESS NETWORKS

In the literature, there are different notions of energy effi-ciency, and selecting a suitable definition is a multi-facetedproblem with profound theoretical and practical implications.

In communication systems with finite energy constraints,it is natural to relate the energy efficiency to the amount ofenergy that we need to transmit a finite number of bits sub-ject to a given error probability. This fact has led researchersto consider the capacity per unit energy (or the capacity offinite-energy channels in bits), where the energy of the code-words is kept finite as the code length tends to infinity [12](see also the discussion in [13, p.15]). This notion, whichmeasures the maximum rate per unit energy(bit per secondper Joule),is difficult to handle with the framework of clas-sical multi-user information theory. Therefore, information-theoretic studies have typically considered the notion of en-ergy per one bit(power divided by data rate),which is de-fined as the amount of energy that is required for reliable(i.e., asymptotically error-free) communication of one bit ofinformation at some rate [14]. We emphasize thatthe two no-tions are not equivalentbecause, if the number of bits tendsto infinity at some rate, and the energy per bit is fixed, thenthe total energy used for transmission tends to infinity [13].In particular, the notion of energy per bit, which commonlyonly considers the energy radiated by antenna, is often usedto show that recent techniques such as advanced multi-usercommunication, massive MIMO, interference alignment, andnetwork coding are energy-efficient solutions. However, asthe following example shows, when other sources of energyconsumption (such as hardware and signal processing opera-tions) are taken into account, then the energy savings providedby these solutions may not be so clear. For convenience, inthe text that follows, we use the general termradiated energyto refer to the energy radiated by antennas, and we reservethe termoperating energyto refer to the remaining sources ofenergy consumption.

Example 1. The study in [15] investigates the impact ofin-frastructure nodes (base stations, access points, etc.)on thethroughput scaling. Inspired by the operation of current prac-tical systems, where the interference is treated as noise andthe transmission is considered successful if a given signal-to-interference-plus-noise (SINR) ratio is attained, the authorspropose a communication scheme for random networks (i.e.,

wireless devices placed randomly in an uniform and inde-pendent manner, whereas infrastructure nodes are placed ar-bitrarily in a predefined manner, independent of the place-ment of the wireless devices) that achieves the throughput

scaling ofΘ(mn

), provided thatm(n) ∈ ω

(√n

logn

)and

m(n) ∈ O(

nlogn

). Here,m denotes the number of base sta-

tions, andn is the number of users. It can be shown [16, De-liverable D4.2] that, for this scheme and this scaling of basestations, by choosingm(n) = ( n

logn)b with b ∈ (1

2, 1], the

radiated energy per information bit diminishes to zero as thenumber of users tends to infinity. In contrast, the operatingenergy consumed per information bit Eb(n) increases at the

order of Eb(n) ∈ Θ(n(log n

n

)b).

The above example reveals that, although the transmit en-ergy per bit may vanish as the number of nodes increases inthe considered scenario, the operating energy per transmit-ted information bit increases without bound (in the best caseasΘ(logn)). Therefore, in highly dense networks (m andnlarge), the energy consumed by hardware is dominant, so wedraw the following important conclusion:

Fact 1. Advanced multi-user communication strategies suchas cooperative (relaying) techniques, massive MIMO, inter-ference alignment, and network coding are highly promis-ing approaches to push the performance of wireless networkswith respect to throughput, delay, and error probability toor-ders of magnitude beyond the performance limits of contem-porary cellular networks. These technologies can decreasethe transmit energy per bit, but, alone, they do not reduce sig-nificantly the operating energy. On the contrary, they maylead to a significant increase of the operating energy. This isdue to the increased number of antennas (and the accompany-ing hardware), and the energy-expensive and time-consumingsignal processing algorithms.

One of the major design principles to reduce the consump-tion of the radiated energy and the operating energy is to ad-just the capacity of the network to the demand. In particular,in delay-sensitive applications, one option is to devise load-dependent algorithms that deactivate network elements in acoordinated manner in order to provide the desired coverageand throughput performance at any given point in time. Theenergy saving capabilities of these algorithms is limited bya fundamental trade-off between energy efficiency and delayconstraints, which, to a large extent, remains an open prob-lem [17]. We can take this idea one step further and con-sider that the algorithms are also able to choose the most suit-able multi-user communication strategy (e.g., network cod-ing, MIMO technique, etc.) for the active network elements.Signal processing tools that can be used to develop and tosupport these load-adaptive schemes are the topic of the re-maining sections.

TRAFFIC PATTERNS

In current communication networks, traffic typically followsa roughly periodic pattern; traffic is high during day and lowat night, with some local variations depending on whetherwe consider industrial, commercial, residential, or ruralareas.These spatial and temporal fluctuations create opportunities tosave energy by switching off unnecessary network elements.To exploit this approach to its fullest potential, we need re-liable forecasts of local traffic, a task that calls for machinelearning algorithms and statistical tools.

In general, the prediction power of learning algorithmsimproves as we increase the amount of available prior infor-mation, and one of the most natural assumptions to use intraffic forecasts is the rough periodicity of the time series.However, especially in future networks, assumptions of thistype have to be used carefully for two main reasons. First,in current networks much of the traffic is generated by users,so the patterns are correlated to those of the human activity.For instance, users are more likely to watch streaming videosduring day or evening than very late at night. In contrast, infuture networks, it is envisioned that machine-to-machineanddevice-to-device communications will be ubiquitous, and,forthe devices requesting such services, the time of the day whendata is sent may be irrelevant, or it may be even desirable tosend data when other users are less active (e.g., noncriticalsoftware updates or backups could be performed while usersare sleeping to avoid unnecessary congestion in the network).Second, current studies showing the coarse periodicity of traf-fic typically consider traffic aggregated over regions of thenetwork, but at a local scale (individual cells of the network)the patterns many not necessarily follow the global trend tooclosely. This last observation has already been noticed in his-torical data coming from a real network of a large Europeancity [16, Deliverable D6.2], where the time series of key per-formance indicators (KPIs) related to data traffic are less reg-ular than those related to voice calls. In Fig. 1 we show syn-thetic voice and data traffic with similar statistical propertiesto those found in a typical cell of the above-mentioned realnetwork. Note that, when data traffic is considered, bursts oftraffic are frequently observed in periods when voice trafficispredictably low. The practical implication of these observa-tions is that, especially at a local scale, forecasting algorithmsshould take into account the type of the service. The clearperiodicity of current traffic patterns may not be necessarilypresent in future systems. In particular, prediction algorithmsfor data traffic should use statistically robust methods becauseof the irregularity of the time series, whereas prediction al-gorithms for voice traffic may be able to safely assume thecoarse periodicity observed in Fig. 1(a).

To produce forecasts of time series strongly related tovoice calls, we can use approaches similar to those used forelectrical load forecasting because electrical load in powergrids and voice calls in cellular networks present similar pe-

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Fig. 1: Normalized traffic as a function of time. (a) Synthetic voice traffic in a single cell of a global system for mobilecommunications (GSM) network. (b) Synthetic data traffic ina single cell of a universal mobile telecommunications system(UMTS) network. (c) Forecast for voice traffic obtained withthe framework of Gaussian processes (“std” stands for standarddeviation) — three weeks of data are used for training.

riodicity. This knowledge and other contextual information,such as the presence of holidays and major events, can beeasily incorporated into algorithms based on Gaussian pro-cesses (GPs) [18] and other kernel-based methods [16, De-liverable D4.1]. The main idea is to choose a covariance orkernel function based on the observed features of the time se-ries under consideration. In particular, the framework of GPsis a promising approach that enables us to specify those fea-tures merely in general terms, and confidence intervals forthe predictions are readily available. For example, we assumethat the time series in Fig. 1(a) has a clear periodic pattern, sowe can specify a periodic covariance function and leave theperiod as a free parameter (hyperparameter) to be selected bymaximizing a merit function with a very natural probabilisticinterpretation [18, Ch. 5] (e.g., the marginal log-likelihood).This example is not particularly interesting because the pe-riod can be easily obtained by simply looking at the data, butthe approach can be straightforwardly applied to capture moresubtle features such as small variations of traffic according tothe day of the week. To this end, we can design “atomic”covariance functions, each of which responsible for captur-ing one desired feature of the time series. Then we constructthe final covariance function by combining the atomic func-tions with operations that preserve covariance functions,andwe compute the hyperparameters by maximizing the marginallog-likelihood. In Fig. 1(c) we show results for the predictionof voice traffic obtained by following the general guidelinesfor the selection of the covariance function outlined in [18,Ch. 5.4.3]. For this result, we use synthetic data, but the samealgorithm provides good forecasts for most cells of a real net-work. However, if the time series contain too many bursts oftraffic, such as that depicted in Fig. 1(b), the traditional frame-work of GPs may have bad generalization properties. In suchcases, robust statistical tools are required, and we now reviewpossible approaches.

It has been observed that, in the short/medium term (up toa couple of months), samples of traffic-related KPIs for either

working days or holidays can be assumed to come from i.i.d.random variables for most cells of a real network [16, Deliver-able D6.2] if they are spaced by multiples of 24h (indicationof the validity of this assumption can be obtained with theturning point test [19]). As a result, we can use simple robusttools based on order statistics, such as tolerance intervals [19],to obtain knowledge about upper bounds for traffic at anygiven hour of the day. More precisely, letX1:n ≤ . . . ≤ Xn:n

be the sorted values of the i.i.d. random variablesX1, . . . , Xn

corresponding to traffic measurementsx1, . . . , xn for a givenhour of the day. Denote the cumulative distribution functionof the random variables byF (x) and the inverse cumulativedistribution function byF−1(p) = sup{x | F (x) ≤ p},wherep ∈ ]0, 1[. Then, for a fixed quantilep, the follow-

ing holds [20]:P (F (Xk:n) > p) = 1 −∑n

i=k

(ni

)pi (1 −

p)n−i, which is an exact value that does not depend on thedistribution of the random variables. If this probability is suf-ficiently low andp sufficiently high, we can configure the cellor base station to serve at most traffic valueXk:n. Similarstrikingly simple and exact results exist to other types of inter-vals (e.g., prediction intervals), and good agreement betweenthe theoretical and empirical results has been obtained withreal network data [16, Deliverable D6.2].

A major limitation of the above robust approaches is thatthey are unable to detect trends and to capture correlationsbetween samples from consecutive hours of the day (samplesfrom different hours are analyzed independently). Correla-tions and trends can be captured by robust machine learningtools such as those described in [21]. To improve further theestimates, we can also try to exploit temporal and spatial cor-relations among cells. Extensions of this type should con-sider carefully the computational complexity because of thelarge number of network elements in future networks. Unfor-tunately, obtaining data sets for research from real networks isdifficult, a fact that has limited the literature on this importanttopic.

NETWORK INTERFERENCE CALCULUS

Even with good traffic forecasts, identifying the best actionto save energy still remains a difficult problem because ofthe possible interference coupling among active network el-ements. Therefore, it comes as no surprise that energy savingalgorithms also need at least a rough estimate of the interfer-ence patterns, and we now turn the attention to some basicresults on interference calculus [22,23], a general mathemati-cal framework that unifies many interference models in wire-less systems. The presentation is heavily based on the studyin [22], which shows algorithms for power-control in CDMAsystems. From a mathematical perspective, these algorithmssolve general fixed point problems, so they have been used inapplications different from that originally envisioned. We ex-pect these algorithms based on interference calculus to playan important role in the analysis of future systems owing tothe generality of the framework.

Standard Interference Functions

In the following discussionR+ denotes the set of non-negative reals,R++ is the set of strictly positive reals, and1is a vector of ones. Inequalities involving vectors should beunderstood as element-wise inequalities. Interference func-tions are defined as follows (for convenience, our definitionsare slightly different from that originally stated in [22]):

Definition 1. A functionI : RM+ → R++ is said to be a

standard interference function if the following axioms hold:1. (Scalability)αI(x) > I(αx) for all x ∈ R

M+ and all

α > 1.2. (Monotonicity)I(x1) ≥ I(x2) if x1 ≥ x2.GivenM standard interference functionsIi : R

M+ →

R++, i = 1, . . . ,M , we call the mappingJ : RM+ → R

M++

withJ (x) := [I1(x), . . . , IM (x)]T a “standard interferencemapping” or simply “interference mapping.”

Checking whether a given function is a standard interfer-ence function by using the definition is not necessarily easy.Fortunately, the following proposition shows that a large classof functions frequently used to model wireless systems arestandard interference functions. We note that the simple resultshown below has been explicitly mentioned in the nonpublicreport [16, D52] (see also [24] and the references therein).

Proposition 1. Concave functionsI : RM+ → R++ are stan-

dard interference functions.

The usefulness of Proposition 1 lies in the fact that thereexist many simple and well-known techniques to identify con-cave and convex functions [25]. The converse of Proposi-tion 1 does not hold in general. However, we can expand theclass of functions that can be easily identified with the helpofProposition 1 by using the following operations that preserveinterference functions/mappings.

Fact 2. (Operations that preserve standard interference func-tions/mappings [22,23]:)

1. Standard interference functions are closed under finiteaddition and multiplication by strictly positive constants. Forinstance, ifI1 : RM

+ → R++ andI2 : RM+ → R++ are stan-

dard interference functions, thenI ′(x) = α1I1(x)+α2I2(x)for α1, α2 > 0 is a standard interference function.

2. If Ii : RM+ → R++ (i ∈ {1, . . . , N}, N ∈ N) are stan-

dard interference functions, thenI ′(x) := mini∈{1,...,N} Ii(x)andI ′′(x) := maxi∈{1,...,N} Ii(x) are standard interferencefunctions.

3. Standard interference mappings are closed under fi-nite composition. For example, ifJ1 : R

M+ → R

M++ and

J2 : RM+ → R

M++ are standard interference mappings, then

J ′(x) = J1(J2(x)) is a standard interference mapping.

We are mostly interested in studying fixed points of stan-dard interference mappings, and the following result can beused for this purpose. When interference calculus is used toinvestigate the performance of communication systems, thefixed points describe, for example, the load or interferenceexperienced by network elements.

Fact 3. (Selected properties of standard interference map-pings [22]:) Let Ii : R

M+ → R++ be a standard interfer-

ence function for everyi ∈ {1, . . . ,M}, and consider thecorresponding mappingJ : RM

+ → RM++ given byJ (x) :=

[I1(x), . . . , IM (x)]T . Then the following holds:

1. If the mappingJ has a fixed point (i.e.,∅ 6= Fix(J ) :={x ∈ R

M+ | x = J (x)}), then the fixed point is unique.

2. The mappingJ has a fixed point if and only if thereexistsx′ ∈ R

M++ satisfyingJ (x′) ≤ x′.

3. If J has a fixed point, then the sequence{xn}n∈N gen-erated byxn+1 = J (xn) satisfies the following:

i) For an arbitrary vectorx0 ∈ RM+ , the sequence

{xn}n∈N converges to the fixed pointx⋆ ∈ Fix(J ).

ii) If x0 = 0, then the sequence{xn}n∈N is monotonouslyincreasing; i.e.,xn+1 ≥ xn.

iii) If J (x0) ≤ x0, then the sequence{xn}n∈N ismonotonously decreasing; i.e.,xn+1 ≤ xn.

Remark 1. Suppose that, in addition to being an interfer-ence mapping,J : RM

+ → RM++ is also upper bounded; i.e.,

J (x) ≤ B 1 for some fixedB ∈ R and everyx ∈ RM+ .

In this case, Facts. 3.1 and 3.2 guarantee the existence ofa unique fixed pointx⋆, and Fact 3.3 suggests the follow-ing iterative procedure to compute the fixed point. Producein parallel two sequences{x′

n+1 = J (x′n)} and{x′′

n+1 =J (x′′

n)}, wherex′0 = 0 andx′′

0 = B 1. Fact 3 shows thatx′n ≤ x⋆ ≤ x′′

n for everyn ∈ N and that both sequences{x′

n} and{x′′n} converge tox⋆. Furthermore,x′

n+1 ≥ x′n

andx′′n+1 ≤ x′′

n. We can therefore stop the algorithm at iter-ationn whenever‖x′

n − x′′n‖∞ ≤ ǫ is satisfied, whereǫ > 0

is the desired precision. The algorithm is guaranteed to ter-minate with a finite number of iterations. In other words,x′

n

andx′′n can serve as lower and upper bounds, respectively,

for x⋆.

We now relate these results to load estimation in multi-carrier systems with fast link adaptation (as envisioned in5Gsystems). The couplings of the models shown below havebeen originally studied on a case-by-case basis, but recentlytheir connection to interference calculus has been indepen-dently established in [16, Deliverable D5.2] [24].

Load estimation in wireless networks

The model described below is based on the discussion in [26],and this or similar models have been used for various networkoptimization tasks for many years. Later we use this model togain insight onto the challenges associated with the formula-tion of energy saving optimization problems.

In more detail, we focus on a cellular radio network withM base stations (or cells) andN test points (an abstract con-cept to represent demand of users in a given region). We de-note the set of base stations byM = {1, 2, ...,M} and the setof test points byN = {1, 2, ..., N}. The quality-of-service(QoS) requirement of each test pointj ∈ N is representedby a minimum amount of datadj ∈ R++ that needs to besent during a unit of time. We denote byX ∈ {0, 1}M×N

the assignment matrix; the component of itsith row andjthcolumn takes the valuexi,j = 1 if test point j is assignedto base stationi or the valuexi,j = 0 otherwise (we alsoassume that each base station serves at least one test point).The power gain between base stationi and test pointj is de-noted bygi,j ∈ R+. Each base stationi ∈ M transmitswith fixed power spectral density per minimum resource unit(e.g., resource blocks in multi-carrier systems, time slots inTDMA systems, etc.) in scheduling, which we denote byPi ∈ R++. The vector of load of the base stations is givenby ρ := [ρ1, . . . , ρM ]T ∈ R

M+ , whereρi ∈ R+ is the load

at base stationi. In words, the loadρi is defined as the ratiobetween the number of resource units requested by test pointsserved by base stationi ∈ M and the numberK of resourceunits available in the system. We can obtain an estimate ofρ

by solving the following system of nonlinear equations [26]:

ρ1 = I1(ρ), · · · , ρM = IM (ρ), (1)

where

Ii(ρ) :=∑

j∈N

dj xi,j

Kωi,j(ρ), (2)

σ2 is the noise power, and

ωi,j(ρ) := B log2

1 +Pi gi,j

η(∑

l∈M\{i} Pl gl,j ρl + σ2

)

(3)

is the the spectral efficiency (i.e., the effective bit rate per re-source unit) of the link connecting base stationi to test pointj. In (3),ωi,j depends on the effective bandwidth per resourceunit B and the SINR scaling factorη, which are parametersthat are typically fitted from simulations of measurements ofthe actual spectral efficiencyωi,j as a function ofρ for a par-ticular network configuration (choice of schedulers, systembandwidth, MIMO transmission scheme, etc.). We refer thereader to [27] for additional details. Intuitively, each term inthe sum in (2) is the fraction of resource units, relative to thetotal number of resource units K, that the test pointj requestsfrom base stationi if data ratedj is desired.

A positive vectorρ⋆ = [ρ⋆1, . . . , ρ⋆M ]T is a solution of the

system of nonlinear equations in (1) if and only ifρ⋆ is a fixedpoint ofJ (ρ) := [I1(ρ), . . . , IM (ρ)]T (i.e.,ρ⋆ ∈ Fix(J )).Once a solution is obtained (assuming that it exists and isunique) we can verify whether the network configuration cansupport the traffic demand by checking whetherρ⋆i ≤ 1 foreveryi ∈ {1, . . . ,M}; i.e., base stations do not use more re-source units than available, in which case we say that the net-work configuration is feasible. As a result, answering ques-tions regarding uniqueness and existence of a fixed point ofJ (and also iterative methods to compute fixed points) is cru-cial to verify feasibility of networks. To this end, we can usethe result in Proposition 1 and Fact 3. More precisely, wenote thatIi(ρ) is a standard interference function because itis positive and concave (see Proposition 1). Concavity ofIifollows from simple facts [25]: i)f(x) := 1/ log2(1 + 1/x)is a concave function on the domainR++, ii) composition ofconcave functions with affine transformations preserves con-cavity, and iii) the set of concave functions is closed underaddition and multiplication by strictly positive real numbers.(See [26] for an alternative argument.) By identifyingJ asa standard interference mapping, we can now use the knownresults in Fact 3 to compute fixed points, answer questions re-garding existence and uniqueness of fixed points, etc. Notethat some previous studies have not identified the functionsIias standard interference functions. The advantage of workingwith the framework of interference calculus (in addition toshowing that many existing results are direct consequencesofFact 3) becomes clear when we extend the interference cou-pling model in (1) by using simple operations that preserveinterference mappings. By doing so, we can analyze morecomplex communication systems than those described above(see also [24] [16, Deliverable D5.2] for more details on theexamples).

Example 2. Assume that the spectral efficiencyωi,j in (3)is upper bounded byω, which is a natural assumption inreal systems. In such cases, we can evaluate the feasi-bility of a network by computing the fixed point of themappingJ ′(ρ) := [I ′1(ρ), . . . , I

′M (ρ)], where I ′i(ρ) :=

∑j∈N max

{dj xi,j

Kωi,j(ρ),dj xi,j

Kω

}. Note that we can easily

verify that I ′i is a standard interference function by using

Fact 2.

Example 3. Suppose that the system considered in Example2 is not feasible; i.e., the interference mappingJ ′ either doesnot have a fixed point or, if a fixed point exists, some of itscomponents have value strictly larger than one (in which caseat least one of the base stations require more resource unitsthan available in the system). In such cases, the system maystill work with overloaded base stations dropping users (i.e.,only a fraction of the traffic of the overloaded base stationsis served), but the fixed point ofJ ′ is not useful to indicatethe load in non-overloaded base stations. To capture this fea-ture of real systems, we can impose limits on the maximumpossible load by using the interference mappingJ ′′(ρ) :=[I ′′1 (ρ), . . . , I

′′M (ρ)], whereI ′′i (ρ) := min {I ′i(ρ), 1}. We

have already seen that each functionI ′i is a standard inter-ference function, so Fact 2 shows thatJ ′′ is also a standardinterference function. By noticing thatJ ′′(ρ) ≤ 1 for everyρ ∈ R

M+ , Fact 3.2 shows thatJ ′′ is guaranteed to have a

fixed point, which can be computed with the scheme with thenon-heuristic stopping criterion in Remark 1.

The aforementioned interference coupling models are alsouseful to highlight limitations of interference calculus,whichshould be addressed in future extensions of the framework.As discussed in the introduction, future systems will be com-posed of combinations of many energy efficiency transmis-sion schemes. In particular, when network elements are co-operative or apply interference-exploiting methods, extend-ing the above models while remaining under the frameworkof interference calculus is difficult. For example, althoughthe model in (3) can be adjusted to account for some lim-ited advanced communication strategies (presence of intelli-gent schedulers, MIMO transmitters, etc.), it is an approxi-mation that can be too crude in future systems where coop-eration among network elements will be taken to completelynew levels. We refer the reader to [23] for recent advances inthe field.

ALGORITHMS FOR ENERGY SAVINGS

We now turn the attention to algorithms that have the objec-tive of switching off as many network elements as possiblewhile satisfying constraints such as coverage and data raterequirements. These algorithms typically use as an input thetraffic forecasts and interference models described in previoussections. To avoid notational clutter, we assume for the mo-ment that all network elements consume the same amount ofenergy and that the radiated energy is negligible. In particular,the latter assumption is an acceptable approximation in cur-rent cellular systems [11]. All assumptions are later droppedto take into account heterogeneous systems with hardwaremore energy efficient than that available today.

We associate network elements of a communication sys-tem with a vectorx = [x1, . . . , xM ]T ∈ {0, 1}M , where

M is the number of network elements and each variablexi ∈{0, 1} takes the value one if network elementi is active, or thevalue zero otherwise. ByX we denote the set of configura-tionsx ∈ {0, 1}M satisfying some required constraints (e.g.,capacity constraints). The mathematical problem that energy-saving algorithms try to solve can be typically described asavariation of the following combinatorial problem:

minimize ‖x‖0 (4)

subject to x ∈ X ,

x ∈ {0, 1}M (5)

where‖x‖0 is the function, informally calledl0-norm (‖ · ‖0does not satisfy all axioms of norms), which returns the num-ber of nonzero elements of the vectorx. The above energyproblems are typically NP-hard, so we cannot expect to solvethem both fast and optimally, especially when consideringthe densification of future networks, which will lead to prob-lems of huge dimensions. If optimality is desired, we canuse a branch and bound algorithm, but solving even fairlysmall problems often takes a very long time [28]. As a result,with the densification of the networks, recent studies [29–32]have focused on fast, but suboptimal heuristics, and many ofthem [28–30,32] build upon theoretically sound methods thataim at solving discrete optimization problems by using thel1-norm as a proxy of thel0-norm. The reason for this approx-imation is that thel1-norm is a (convex) sparsity promotingnorm [25, 28, 33], and convex problems are typically easierto solve than nonconvex problems. (We can also interpret thel1-norm as the convex envelope of thel0-norm for an appro-priate domain [25].) By also relaxing nonconvex constraintsto convex constraints, many efficient optimization techniquesbecome available [25, 32]. In particular, we often replace thediscrete constraint in (5) by

x ∈ [0, 1]M . (6)

The solution of the resulting convex optimization problem(assuming thatX is also a convex set) can then guide otherheuristics to make the final hard decisions on the active set ofnetwork elements. This approach is used in, for example, thestudy in [32], which considers an antenna selection problemin energy limited point-to-point communication systems withconstraints given in terms of the required channel capacity.

In the compressive sensing community, problems with thel0-norm in the objective are increasingly being solved witha method called reweightedl1-norm [33], which in recentyears are finding applications in cellular communication sys-tems [29, 30]. We first review the majorization-minimization(MM) algorithm in order to then explain these ideas.

The MM algorithm [33, 34]: The discussion here followsclosely that in the studies in [33,34]. Suppose that the objec-tive is to minimize a functionf : X → R, whereX ⊂ R

M .Unless the optimization problem has a very special structurethat can be exploited, such as convexity, finding an optimal

solutionx⋆ (provided that one exists) is intractable in gen-eral. To devise a suboptimal approach, assume that we areable to construct a functiong : X × X → R, hereafter calledmajorizing function, satisfying the following properties:

f(x) ≤ g(x,y), ∀x,y ∈ X (7)

and

f(x) = g(x,x), ∀x ∈ X . (8)

Then, starting fromx0 ∈ X , the MM algorithm produces asequence{xn} ⊂ X by

xn+1 ∈ arg minx∈X

g(x,xn). (9)

We can verify that the sequence{f(xn)} is monotonouslydecreasing:f(xn+1) = g(xn+1,xn+1) ≤ g(xn+1,xn) ≤g(xn,xn) = f(xn), where the equalities follow from (8),and the first and second inequalities follow from (7) and (9),respectively. As a result,f(xn) → c ∈ R for somec ≥f(x⋆) asn → ∞, where we assume thatx⋆ is a solutionof the original optimization problem. In practice, we stop thealgorithm when we observe no progress in the objective value,and we note that convergence of{f(xn)} does not in generalimply the convergence of the sequence{xn}.

The main challenge in applying the MM algorithm is tofind a majorizing functiong such that the iteration in (9) canbe implemented efficiently. Fortunately, in some special casesof practical interest, we can construct a majorizing functioneasily. For example, iff can be decomposed asf(x) =f1(x) + f2(x), wheref1 is a differentiable concave functionandf2 is a convex function, then we can use

g(x,y) = f1(y) +∇f1(y)T (x− y) + f2(x) (10)

as the majorizing function, where∇f1(y) is the gradient off1 aty (if f1 is not differentiable, we can replace the gradientby an arbitrary subgradient). With this choice, the optimiza-tion problem in (9) becomes a convex optimization problemprovided that the setX is convex, hence it can be solved withefficient methods [25, 32]. We can verify the validity of theproperty in (7) from the first-order characterization of con-cave functions [25].

Let us turn the attention to the optimization problem in(4)-(5) with the constraint in (5) replaced by that in (6). Itiswell know that thel0-norm satisfies the following [33,34]:

‖x‖0 = limǫ→0

M∑

i=1

log(ǫ+ |xi|)− log(ǫ)

log(1 + ǫ−1), (11)

which, if each componentxi of the vectorx = [x1, . . . , xM ]T

is constrained to be non-negative as in (6), suggests the useofthe functionfǫ(x) =

∑M

i=1(log(ǫ + xi) − log(ǫ))/ log(1 +ǫ−1) with a small design parameterǫ > 0 as a approximationof thel0-norm (other choices are possible [33]). Furthermore,note that the functionfǫ(x) is concave, so the MM algorithmcan be used to generate a sequence of vectors{xn} with de-creasing objective value as discussed above.

Energy saving optimization problems in cellular networks

We now exemplify how to apply the above ideas to energysaving problems in cellular networks. (These problems alsohighlight the need to study interference coupling in energyefficient networks.) To this end, consider the notation intro-duced in the previous section on interference calculus, andassume that, for each test pointj ∈ N , we have an estimateof the traffic demand per unit of time, which, as before, wedenote bydj > 0. This estimate can be obtained with learn-ing algorithms. We assume that the operating energy of eachbase station (which here can also designate a cell, a radio unit,etc.) i ∈ M is given byci > 0, and the energy related to ra-diation is approximated by a functionfi : [0, 1] → R+ of theloadρi ∈ [0, 1]. To detect base stations that can be switchedoff, we may try to solve the following problem:

min.

M∑

i=1

ci|ρi|0 +M∑

i=1

fi(ρi) (12)

s.t. ρi =

N∑

j=1

djKωi,j(ρ)

xi,j , i ∈ M (13)

M∑

i=1

xi,j = 1 j ∈ N

xi,j ∈ {0, 1} i ∈ M, j ∈ N (14)

ρi ∈ [0, 1] i ∈ M (15)

where{xi,j}i∈M,j∈N (assignment variables) and{ρi}i∈M

(load at base stations) are the optimization variables, and|x|0is the function that takes the value 0 ifx = 0 or the value 1otherwise. By solving problem (12)-(15), base stations withno load (i.e., those withρi = 0) can be deactivated. In theproblem, we assume that the constraints are feasible. If not,we can add slack variables and an additionall0 or l1 penaltynorm to the merit function, but, for brevity, we do not considersuch extensions here. One of the main complications to theabove problem is the interference coupling appearing in thenonconvex constraint in (13). We have already seen in pre-vious sections that computing interference or load, even withfixed network configurations, is a nontrivial task. Therefore,it is common in the literature to fix the value of the spectralefficiency of the link connecting base stationi to test pointjto some constantωi,j, which is obtained by considering theworst-case interference or an average case. Unfortunately,even with this simplification, the resulting problem is diffi-cult to solve because it is a generalization of the standard bin-packing problem, which is NP-hard. To obtain a fast heuristic,we can use the MM algorithm as follows. First, as discussedin the previous section, we relax the discrete constraint in(14)to consider continuous valuesxi,j ∈ [0, 1]. (In particular,this modification is natural when base stations are allowed toserve only a fraction of the traffic requested at test points.)Then, by also using the approximation discussed below (11),

we obtain the following optimization problem, which can beefficiently addressed with the majorization-minimizational-gorithm if fi(ρi) is a convex or concave function (see the dis-cussion above (10)), and we note that a linear function is oftenconsidered a good approximation of the energy related to ra-diation [11]:

min.M∑

i=1

cilog(ǫ+ ρi)− log(ǫ)

log(1 + ǫ−1)+

M∑

i=1

fi(ρi) (16)

s.t. 0 ≤ ρi =

N∑

j=1

djKωi,j

xi,j ≤ 1, i ∈ M (17)

M∑

i=1

xi,j = 1, j ∈ N

xi,j ∈ [0, 1] i ∈ M, j ∈ N (18)

A solution of this modified problem provides a good indi-cation of which base stations to switch off (ρi close to zero),a result that can guide other heuristics to make the final harddecisions. Empirical evidence suggests that these algorithmsbased on convex optimization are competitive, in terms ofenergy savings, against heuristics that deal with the noncon-vex problems directly. We also note that convex optimizationtechniques can naturally consider the energy related to radi-ation, and they can exploit the rich structure of the problemsto decrease the computational effort (e.g., many assignmentsare impossible owing to the distances involved). We referthe reader to [29] for a comparison of the above MM-basedmethod against the cell zooming approach in [31]. In Fig. 2we compare the solution obtained with 10 iterations of theMM algorithm against the optimal solution of the problem in(12)-(15) with the constraints in (13) replaced by the worst-case constraints in (17). To obtain discrete values for eachassignmentxi,j after applying the MM algorithm to the prob-lem in (16)-(18), we use the heuristic outlined in [29]. Alloptimization problems have been solved with CPLEX, a stan-dard commercial solver. The simulated network mimics anLTE system with 100 base stations where the radiated energyis neglected, which is done to simplify the process of obtain-ing optimal solutions. The bandwidth of each base station isset to 100 MHz, and the data rate requirement of each testpoint/user is128 kbps. All other parameters are exactly thesame as the network simulated in [29]. We observe in Fig. 2that, although we obviously loose optimality by applying theMM algorithm, the time to obtain a network with fairly lowenergy consumption grows slowly as we increase the numberof test points, which stands in sharp contrast to the algorithmthat obtains optimal solutions. For a variation of the aboveideas to the problem of energy efficiency of small cell net-works with best effort users, we refer the reader to [30]. Theseresults indicate that techniques based on convex optimization(which, unlike discrete heuristics, can easily deal with the en-

ergy related to radiation) have great potential to scale to verylarge problems.

SUMMARY AND OUTLOOK

We have showed that many communication schemes aimingat reducing the transmit energy per bit may, in fact, increasethe total energy consumption. The analysis and developmentof wireless communication systems have traditionally con-sidered only the energy radiated by antennas, but neglectedthe energy required for operating the network. By meansof information-theoretic arguments, we showed that the lattercannot be neglected when considering dense deployments ofbase stations. One way of saving on operating power is to ad-just the network to the demand by switching off unnecessarynetwork elements. We devised algorithms to select repeatedlya suitable subset of the network’s base station.

These tools for energy saving need to be refined and ex-tended to consider, for example, mobility, temporal trafficprofiles, and the high level of cooperation among network el-ements of future systems. In particular, many issues remainopen concerning the presented energy saving optimization ap-proach based on majorization-minimization: i) Under whichconditions does the sequence produced by the majorization-minimization algorithm converge in this particular applicationdomain? To the best of the authors’ knowledge, MM-basedalgorithms have not been formally shown to converge evenwhen applied to problems in compressive sensing. ii) Howcan we use realistic load estimates in the optimization prob-lems (not average or worst-case estimates), since the spectralefficiency is a function of the assignments{xi,j}i∈M,j∈N?Integrating the results on interference calculus into the en-ergy saving problems may be a direction for future research.iii) How are temporal traffic patterns and mobility exploited tosave energy? This topic is particularly important for content-centric networking, where we have the additional option ofcaching content with the intent to save energy by consideringcurrent and future channel conditions. iv) How are distributedversions of the optimization algorithms implemented? In-formation exchange among network elements may consumewireless resources, but this fact is not considered in most opti-mization models. v) How is the flexibility of choosing differ-ent communication strategies (e.g., cooperative transmissionschemes) added while keeping the energy optimization prob-lems tractable? Adding this level of flexibility complicates theoptimization problems. As illustrated above, even the simplertask of computing load in the presence of cooperative systemsis already difficult.

In summary, our work shows that dense and ultra densedeployments of future 5G networks will hardly become real-ity unless the scaling of base stations can be decoupled fromthe growth of operating power. Switching off network ele-ments when they are not required is one way to do so.

0 200 400 600 800 100010

20

30

40

50

60

70

Num

ber

of a

ctiv

e ba

se s

tatio

ns

Number of test points

MM AlgorithmOptimal solution

(a)0 200 400 600 800 1000

0

10

20

30

40

50

Com

puta

tiona

l tim

e [s

]

Number of test points

MM AlgorithmOptimal solution

(b)

Fig. 2: Comparison between the optimal solution of the discrete problem with the MM heuristic in a system with 100 basestations. We obtain the error bars corresponding to 95% confidence intervals by estimating the error in the mean from100realizations of the simulations. (a) Average number of active base stations as a function of the number test points/users. (b)Average computational time as function of the number of testpoints/users.

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