Multi-Cell Flow-Level Performance ofTraffic-Adaptive Beamforming under Realistic

Spatial Traffic ConditionsHenrik Klessig, Maciej Soszka, and Gerhard Fettweis

Vodafone Chair Mobile Communications SystemsDresden University of Technology, Germany

Email: henrik.klessig, maciej.soszka, fettweis@tu-dresden.de

Abstract—Upcoming 5G wireless technology will enable vari-ous new applications and is going to support a massive amount ofdevices per cell. In this regard, ultra-dense small cell deploymentsare a means to cope with the need of extremely high data ratesof several tens of Gbps and with the increasing data trafficdemand. However, denser deployments may lead to more severeinter-cell interference, which is strongly connected to the actualspatial distribution of the mobile traffic demand. Traffic-adaptivebeamforming using phased antenna arrays can be an attractivesolution for concentrating capacity at desired traffic hot spotlocations while inter-cell interference is reduced. In this paper,we propose a flexible and holistic model, which describes flow-level performance of networks, which consist of base stationsequipped with phased antenna arrays, accurately and considersdynamic inter-cell interference. Moreover, we present a con-figurable spatial traffic model to generate data traffic mapswith various statistical properties. We use these traffic mapsto evaluate the performance of a traffic-adaptive beamformingalgorithm proposed and compare it the performance of a state-of-the-art antenna down-tilt algorithm.

Index Terms—beamforming; antenna array; phased array;network model; spatial traffic model; flow level modeling

I. INTRODUCTION

Future 5G cellular technology is expected to facilitate aplurality of new applications ranging from extensive contentprovision with data rates up to 10 Gbps to serving a massiveamount of devices (Internet of Things). In addition, applica-tions like autonomous driving call for solutions offering ultra-low latency and extremely high reliability. A densification inspace, e. g., by massive MIMO (multiply input multiple out-put), and a densification in spectrum, e. g. through millimeterwave communications, are the most favored answers to 5Grequirements with regard to desired data rates [1], [2]. In thiscontext, it is of high importance to consider more advancedinterference management solutions that are needed most forultra-dense deployments. However, a drawback of techniqueslike (e)ICIC ((enhanced) inter-cell interference coordination)for LTE (Long Term Evolution)-Advanced is that radio re-sources are wasted when muted to protect cell edge users.

An attractive alternative to the aforementioned methods isbeamforming using large-scale antenna arrays, which con-sist of several tens or even hundreds of antenna elements.

Such active antenna arrays allow to steer beams and formbeam patterns that are desirable for the predominant signalpropagation scenario and traffic distributions, and that reduceinterference among user groups or traffic hot spots. Further-more, considering design and steering aspects [3]–[5], phasedantenna arrays add considerable flexibility to the system byproviding the possibility of allocating capacity to locations,where it is actually needed. In addition, the design of suchantennas, in particular the number and arrangement of theirelements, enable controlling various characteristics like thebeamwidth or the number of sub-beams. Hence, large-scaleantenna arrays are expected to be an attractive solution forfuture 5G networks, especially for higher frequencies [6].

Unlike existing work, which uses channel models to showthe benefits of antenna arrays [7], [8], and which aims atproviding short time-scale beamforming algorithms [9], weconcentrate, in this paper, on the mathematical description oflarge-scale antenna array beam patterns and provide a modelthat allows to examine all flexibility in antenna array design.Moreover, we connect this model with a commonly used flowlevel model [10], [11], which considers the mutual coupling ofcells through dynamic inter-cell interference. Based thereon,we formulate an optimization problem to steer and formantenna array sub-beams at all base stations in a wirelessnetwork with the goal of "allocating" the transmit energyto spatial mobile data traffic hot spots. Thereby, we aim atreducing the negative impact of inter-cell interference on thedata rate and at achieving a certain degree of load balancingamong the cells. Hence, our approach targets evaluation andoptimization of network performance at longer time scales,say several minutes to a few hours. The usage of the modelsis illustrated by providing initial optimization results based ona realistic statistical modeling of spatial traffic distributions.

II. SYSTEM MODEL

In the following, we present a model for the evaluation oftraffic-adaptive beamforming algorithms. The model includesfour different aspects, namely, a general network and cellmodel, an antenna array beam pattern model, a queuing-theoretic tool for the derivation of key performance indicators,and a spatial data traffic model.978-1-4673-6540-6/15/$31.00 c© 2015 IEEE

AFi(ϑ, ϕ, θi, φi) =

Ne∑n=1

Ii,0 exp

j

(ψi,n (θi, φi) + 2πλ−1

(sinϑ

(xi,n cosϕ+ yi,n sinϕ

)+ zi,n cosϑ

))(5)

A. Network and Channel ModelWe consider a cellular network with N base stations (BS)

deployed in a region R ⊂ R2 and covering another sub-region L ⊂ R. Each cell i ∈ N := 1, . . . , N is defined asthe closed set Li ⊂ L of locations, where the correspondingBS serves the mobile users according to a user association rule.We assume that user equipments (UE) can only be associatedwith at most one BS, such that the association rule definesthe partition L1, . . . ,LN on L with

⋃i∈N Li = L and

Li ∩ Lj = ∅ for any i 6= j. Note that this model is rathergeneric. BSs can be of different type (macro to femto), mayoperate on different frequency bands, or may even have non-identical radio access technologies (multi-RAT). Moreover,the model does not make any assumption on deploymentstrategies, such that it can be applied to standard hexagonalmacro cell layouts, heterogeneous network, or to clustered,ultra-dense small cell networks. Without loss of generality, weconsider downlink transmission only. We calculate the receivepower pi(u) from BS i at location u ∈ R according to

pi(u, θi, φi) = ptx,i + Ψi(u)− Li(u)

+GBS,i(u, θi, φi) +GUE,i(u), (2)

where ptx,i, GBS,i(u, θi, φi), GUE,i(u), Ψi(u), and Li(u) de-note the maximum total transmit power of BS i, the antennagain experienced at location u from BS i, the UE’s antennagain, log-normal shadowing, and the pathloss, respectively.The quantities θi and φi will be introduced in the next section.Since we focus on providing a model for the BS antennaarray gain GBS,i, we assume isotropic antennas at the UEsfor simplicity, such that GUE,i(u) = 0 dBi.

B. Antenna Array Beam Pattern ModelIn this section, we present an approach to derive the phase

shifts of all antenna array elements based on sub-beams,which shall cover individual traffic hot spots. The approachis illustrated in Fig. 1.

1) Obtaining Phase Shifts: We aim at generating Ns sub-beams with an antenna array composed of Ne half-wave dipoleelements. W. l. o. g., we assume the same number of elementsfor each BS. Using the center of the antenna as the point oforigin, we define

(θi,k, φi,k

)as the desired spherical direction

(θi,k as polar and φi,k as azimuthal angle) of a sub-beam,which has the index k ∈ 1, . . . , Ns and belongs to BS i. Inorder to generate the directional sub-beam k, the signal phaseshift ψi,n,k of the nth antenna element has to be set to

ψi,n,k(θi,k, φi,k

)= −2πλ−1zi,n cos θi,k

− 2πλ−1(xi,n cosφi,k + yi,n sinφi,k

)sin θi,k, (3)

where(xi,n, yi,n, zi,n

)are the Cartesian coordinates of the nth

array element of BS i and λ is the carrier wavelength [12]. To

x

z

(xi,n, yi,n, zi,n)

y

φi,k

θi,k

x

45

y

ϕ

z

ϑf(ϑ, ϕ)

Figure 1. (Left) Desired sub-beam k of BS i with spherical direction(θi,k, φi,k

). (Right) Pattern of rotated half-wave dipole (red).

compute an overall resulting phase shift ψi,n for element n,we combine the sub-beam-specific phase shifts ψi,n,k using

ψi,n (θi, φi) = arg

Ns∑k=1

ai,n,k expjψi,n,k

(θi,k, φi,k

)(4)

where θi :=(θi,1, . . . , θi,k, . . . , θi,Ns

)Tis the vector of polar

angles, φi :=(φi,1, . . . , φi,k, . . . , φi,Ns

)Tis the vector of

azimuthal angles of a sub-beam, and j is the imaginary unit.The weighting factors ai,n,k can be used to prefer sub-beams.

2) Computing the Antenna Gain: In order to obtain theantenna gain of the entire antenna array, we have to considerboth, the pattern f of a single antenna element and the arrayfactor AFi of the array of BS i. We use the half-wave dipolemodel from [13] (p. 466, Eq. (10)), in which the dipole islocated in a distance h to an infinite flat sheet reflector. Aftersome coordinate system transformations, we obtain the patternof a single half-wave dipole, which is rotated by 45, by

f(ϑ, ϕ) =cos(π√2

4 g (ϑ, ϕ))

√1− 1

2g2(ϑ, ϕ)

sin(

2πhλ−1 sinϑ sinϕ),

(6)where g (ϑ, ϕ) = cosϑ+ sinϑ cosϕ, and (ϑ, ϕ) are polar andazimuthal coordinates with the center of the dipole being theorigin. Finally, using the array factor in Eq. (5) [12] - withIi,0 =

√ptx,i/(4πNe) - and normalizing yields the array gain

GBS,i(ϑ, ϕ, θi, φi) = 4π∣∣f (ϑ, ϕ)AFi (ϑ, ϕ, θi, φi)

∣∣2 p−1tx,i (7)

in spherical coordinates (with the antenna center as the origin).As a last step, the expression in Eq. (7) has to be transformedto fit the network model in Section II-A by considering BSheights and locations, as well as, UE heights. The result isthe antenna gain GBS,i(u, θi, φi) in dBi. Note that, by using auser association rule based on receive powers, the cell areas Lidepend also on the antennas’ beam configurations; however,in the remainder, we will still use Li for readability.

C. Flow-Level Model and Key Performance Indicators

In order to model mobile radio network performance prop-erly, the usage of so-called flow-level models [10] and networkinterference calculus [14] turned out to be very effective.Therefore, we adopt the flow-level model proposed in [11],which is summarized subsequently. It is assumed that dataflows arrive to the BS according to a Poisson process and thattheir size is exponentially distributed with mean Ω in bits. Letλ(u), u ∈ R, be the location-dependent flow arrival rate inflows per second per sq. km. The arrival rate λi with respectto BS i can be computed by λi (θ, φ) =

∫Li(θ,φ) λ(u) du. BSs

employ the Processor Sharing (PS) discipline.Let us consider a data flow arriving at location u ∈ Li,

meaning being associated with BS i. Assuming time-averagedinterference conditions with ηi being the ith element of thevector η ∈ [0, 1]

N and denoting the average BS utilization,we write for the signal-to-interference-plus-noise ratio (SINR)

γi(u, η, θ, φ) =pi (u, θi, φi)∑

j∈N\i ηjpj(u, θj , φj

)+N0

, (8)

where N0 is the noise power, and θ := (θ1, . . . , θN ) and φ :=(φ1, . . . , φN ) collect all sub-beam spherical directions. Usingthe achievable rate

ci (u, η, θ, φ) = aBmin

log2

1 + bγi (u, η, θ, φ)

, cmax

,

(9)with B, a, b, and cmax denoting the system bandwidth,bandwidth and SINR efficiencies, and the maximum rateachievable, respectively, the average rate provided by a BS,or in other words, the BS capacity, can be formulated as

Ci (η, θ, φ) =

[∫Li

δi (u, θ, φ)

ci (u, η, θ, φ)du

]−1. (10)

The term δi(u, θ, φ), u ∈ Li represents the normal-ized user distribution in cell i and can be computed us-ing δi(u, θ, φ) = λ(u)/λi (θ, φ). The aforementioned assump-tions on the arrival and service of data flows lead to anM/M/1/Ki PS queuing model, based on which the load ofBS i is given by ρi (η, θ, φ) := λi (θ, φ) Ω/Ci (η, θ, φ). Theparameter Ki ∈ N+ defines the maximum number of concur-rent data flows allowed, which leads to the facts that the BSblocks a data request with probability

Pb,i (η, θ, φ) =

((1− ρi) ρKii1− ρKi+1

i

)(η, θ, φ) , (11)

and that its resource utilization becomes

ηi (η, θ, φ) = fi (η, θ, φ) :=(ρi(1− Pb,i

))(η, θ, φ) . (12)

Eq. (12) can be easily solved numerically for the vector ηusing a fixed point iteration, see [11]. As a result, the meannumber ni of concurrently active flows in cell i is

ni (η, θ, φ) =

(ρi

1− ρi− (Ki + 1)ρKi+1

i

1− ρKi+1i

)(η, θ, φ) , (13)

from which, in turn, the expected data flow throughput acrosscell i can be derived as

Ri (θ, φ) =ηiCi (η, θ, φ)

ni (η, θ, φ). (14)

The expected flow throughput ri (u, θ, φ) at location u canbe obtained from Eq. (14) by substituting the cell capac-ity Ci (η, θ, φ) by the data rate ci (u, η, θ, φ). We will use theseexpressions to quantify the performance of the beamformingalgorithm presented in Section III.

D. Spatial Traffic Model

In order to quantify network performance properly, notonly temporal fluctuations of the traffic demand, but also, andmore importantly, its spatial distribution has to be modeledrealistically. More often than not, a uniform traffic distributionis unrealistic and using a (measured) traffic map from aspecific network like in [15] might be not sufficient enoughto characterize performance reliably. In [16], [17], it has beenshown that the "amplitude" Z of the data traffic density inbps per sq. km at some location u follows a log-normal distri-bution, that is Z(u) ∼ lnN (µ, σ2) represents a log-normalrandom field on R. More importantly, the authors in [16]characterize the log-normal random field by a correlationdistance dZ in addition to the location and shape parametersµ and σ. Let νZ(d) be the correlation function of Z(u) withd being the distance between any two locations u1 and u2.Then, the relation between the correlation function νX(d;α) ofthe corresponding zero-mean Gaussian random field X(u) :=lnZ(u)− µ and νZ(d) is [18]

νZ(d) =exp

(νX(d;α)σ2

)− 1

exp (σ2)− 1. (15)

Without loss of generality, we assume a correlation functionfor X(u) with exponential decay, i. e., νX(d;α) := e−αd (anyother parameterizable correlation function can be used). UsingEq. (15) and the fact that νZ(dZ) = 0.5, we have

α = − 1

dZln

(σ−2 ln

(0.5 exp(σ2) + 0.5

)). (16)

A realization z(u) ≡ λ(u)Ω of Z(u) can be obtained asfollows: First, the parameters µ and σ are computed fromgiven mean µZ = E [Z] and variance σ2

Z = Var [Z] of thedesired traffic density using µ = lnµZ − 0.5 ln

(σ2Z/µ

2Z + 1

)and σ =

√ln(σ2Z/µ

2Z + 1

), respectively. Second, a realiza-

tion x(u) of the zero-mean Gaussian random field X(u) withvariance σ2 and correlation function νX(d;α) is generatedwith standard methods, such as the Cholesky decomposition.For large areas and/or high spatial resolution, an iterativemethod that reduces computational complexity, like the oneproposed in [19], may be used. Note that, in order to considerthe desired correlation distance dZ of the log-normal randomfield when generating the Gaussian random field, we haveto insert the expression in Eq. (16) into νX(d;α). Finally,taking z(u) = exp

x(u) + µ

yields the log-normally dis-

tributed traffic density with the desired properties.

x coordinate in km0 0.2 0.4 0.6 0.8 1

yco

ordi

nate

inkm

0

0.2

0.4

0.6

0.8

1

(a) dZ = 60m, σZ = 50Mbps/km2

x coordinate in km0 0.2 0.4 0.6 0.8 1

yco

ordi

nate

inkm

0

0.2

0.4

0.6

0.8

1

(b) dZ = 60m, σZ = 150Mbps/km2

x coordinate in km0 0.2 0.4 0.6 0.8 1

yco

ordi

nate

inkm

0

0.2

0.4

0.6

0.8

1

(c) dZ = 120m, σZ = 50Mbps/km2

x coordinate in km0 0.2 0.4 0.6 0.8 1 z(u)

yco

ordi

nate

inkm

0

0.2

0.4

0.6

0.8

1

100

101

102

103

(d) dZ = 120m, σZ = 150Mbps/km2

Figure 2. Example traffic maps z(u) in Mbps/km2 for mean mZ = 100Mbps/km2 and different settings for dZ and σZ .

Fig. 2 illustrates generated traffic maps z(u) in Mbps/km2

for common mean µZ = 100 Mbps and different settings forthe correlation distance dZ and standard deviation σZ . Ascan be seen, the traffic distribution exhibits hot spots withhigher intensity, when the standard deviation σZ is increased.In addition, the hot spot size can be tuned by changing thecorrelation distance dZ . By tuning both parameters, σZ anddZ , the traffic can be adjusted to different regions, rangingfrom rural to urban for increasing σZ and decreasing dZ [16].

III. A TRAFFIC-ADAPTIVE BEAMFORMING ALGORITHM

We aim at improving the user throughput by a "spatialallocation" of transmit power to traffic hot spots throughbeamforming. However, enhancing the data rate at locationswith high data flow arrival rates may not be sufficient, sincedata flow dynamics are not considered well. Instead, we usethe α-fair utility from [20] and incorporate the average userthroughput per cell Ri. Thus, we have the utility

Φ (θ, φ) =

∑i∈N logRi (θ, φ) for α = 1,∑i∈N

(Ri(θ,φ))1−α

1−α otherwise.(17)

and the optimization problem

maximizeθ,φ

Φ (θ, φ) . (18)

Since Problem (18) is a non-convex optimization problem dueto the randomness of shadowing Ψi(u) and the traffic λ(u), weemploy a heuristic Nelder-Mead search method similar to [21].We extend this approach to a multi-cellular system by applyingthe algorithm to each cell individually and by iterating overall the cells similar to the method in [15].

IV. NUMERICAL RESULTS

In the following, we provide insights into potential opti-mization gains through traffic-adaptive beamforming.

A. Scenario and Benchmark

In order to illustrate the usage of the models proposedand to show basic results, we use an LTE-based macro-cellular network with an inter-site distance of 500 m. Thereare 57 cells deployed in a hexagonal layout. Performance isevaluated in the voronoi areas of the inner 19 cells only toavoid border effects and to consider inter-cell interference

Table ISCENARIO CONFIGURATION

Maximum transmit power ptx,i 49 dBmCarrier frequency [22] 2.0 GHzBandwidth B 20 MHzBandwidth efficiency a 0.63SINR efficiency b 0.4Path loss model for Li(u) [22] 128.1 dB + 37.6 log10(d/km) dB1

Shadowing correlation distance 20 mShadowing standard deviation [22] 10 dBShadowing inter-site correlation 0.5BS mounted on height [22] 32 mUE height [22] 1.5 mUE receiver sensitivity −120 dBmThermal noise −174 dBm/Hz1 The quantity d describes the distance from location u to BS i in km.

properly. We use rectangular phased arrays of size 12x12elements in a distance h = λ/4 in front of an infinite flatsheet reflector and the throughput-fair utility with α = 1.The distance between neighboring elements is λ/2. All otherrelevant scenario settings are summarized in Table I.

We compare the user throughput 5 %-iles achieved withthe traffic-adaptive beamforming algorithm ("bb Ns") withtwo benchmarks: The first is a standard 15-down-tilt 3GPPantenna pattern model [22] ("3GPP"). The second is a down-tilt-optimized network according to the algorithm describedin [15] with 3GPP antenna patterns and using the same utilityfunction as for the beamforming algorithm ("Powell"). Weillustrate our results obtained from 50 realizations of trafficmaps with various statistical settings by box plots.

B. Potential Optimization Gains

We find that, in all cases investigated, beamforming yieldsbetter results compared to the standard 3GPP configuration,as well as, to the benchmark from [15] ("Powell") - up to a5-fold increase in throughput. We also observe the tendencythat a higher number of sub-beams yields better networkperformance, which is a result of a more effective allocationof power to multiple traffic hot spots. From Fig. 3, we notethat a higher traffic standard deviation σZ slightly decreasesthe performance of both, the initial as well as the optimizednetwork. However, the mean traffic mZ and the correlationdistance dZ , i. e., the spatial spread of hot spots, strongly affectthe efficacy of the algorithms. From Figs. 4 and 5, we concludethat the beamforming algorithm is less robust to a higher meantraffic and larger hot spots, yet still superior to our benchmark.

3GPPPowell 1 bb 3 bb 5 bb 3GPPPowell 1 bb 3 bb 5 bb

Thr

ough

put

5%

-ile

inM

bps

0

2

4

6

8

10 σZ = 70Mbps/km2 σZ = 150Mbps/km2

Figure 3. Throughput 5 %-iles for different traffic standard deviations. mz =110Mbps/km2, dZ = 60m.

3GPPPowell 1 bb 3 bb 5 bb 3GPPPowell 1 bb 3 bb 5 bb

Thr

ough

put

5%

-ile

inM

bps

0

5

10

mZ = 90Mbps/km2

mZ = 150Mbps/km2

Figure 4. Throughput 5 %-iles for different traffic mean values. σz =110Mbps/km2, dZ = 60m.

V. CONCLUSIONS

In this paper, we have illustrated, how large-scale antennaarray technology and its performance can be modeled andevaluated accurately by connecting a flexible phased array an-tenna beam pattern model with a data flow level model, whichconsiders inter-cell interference. Further, the combined modelcan be adapted to millimeter wave beamforming technologyand, therefore, can be used as an evaluation framework for5G network concepts. By additionally using a realistic spatialtraffic model, we have shown that various traffic distributioncharacteristics have significant impact on the performance oflong-term beamforming algorithms, meaning beam adaptationto user groups or traffic hot spots. In particular, beamform-ing yields considerable performance gains in terms of userthroughput even at high traffic loads, as long as, traffic hotspots do not exhibit a too large spatial spread.

REFERENCES

[1] J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K.Soong, and J. C. Zhang, “What Will 5G Be?” IEEE Journal on SelectedAreas in Communications, vol. 32, no. 6, pp. 1065–1082, Jun. 2014.

[2] N. Bhushan, D. Malladi, R. Gilmore, D. Brenner, A. Damnjanovic,R. Sukhavasi, C. Patel, and S. Geirhofer, “Network densification: thedominant theme for wireless evolution into 5G,” IEEE CommunicationsMagazine, vol. 52, no. 2, pp. 82–89, Feb. 2014.

[3] P. Angeletti, G. Toso, and G. Ruggerini, “Array antennas with jointlyoptimized elements positions and dimensions part II: Planar circulararrays,” Antennas and Propagation, IEEE Transactions on, vol. 62, no. 4,pp. 1627–1639, April 2014.

[4] A. Massa, M. Donelli, F. De Natale, S. Caorsi, and A. Lommi, “Planarantenna array control with genetic algorithms and adaptive array theory,”Antennas and Propagation, IEEE Transactions on, vol. 52, no. 11, pp.2919–2924, Nov 2004.

[5] M.-S. Lee, “A low-complexity planar antenna array for wirelesscommunication applications: Performance analysis and beamforming,”in Global Telecommunications Conference, 2009. GLOBECOM 2009.IEEE, Nov 2009, pp. 1–6.

3GPPPowell 1 bb 3 bb 5 bb 3GPPPowell 1 bb 3 bb 5 bb

Thr

ough

put

5%

-ile

inM

bps

0

5

10

15 dZ = 30m dZ = 120m

Figure 5. Throughput 5 %-iles for different traffic correlation distances.mz = 110Mbps/km2, σZ = 110Mbps/km2.

[6] S. Han, C.-L. I, Z. Xu, and C. Rowell, “Large-scale antenna systemswith hybrid analog and digital beamforming for millimeter wave 5G,”Communications Magazine, IEEE, vol. 53, no. 1, pp. 186–194, January2015.

[7] Y.-S. Cheng and C.-H. Chen, “A novel 3D beamforming schemefor LTE-Advanced system,” in Network Operations and ManagementSymposium (APNOMS), 2014 16th Asia-Pacific, Sept 2014, pp. 1–6.

[8] C. Waldschmidt, S. Schulteis, and W. Wiesbeck, “Complete RF systemmodel for analysis of compact MIMO arrays,” Vehicular Technology,IEEE Transactions on, vol. 53, no. 3, pp. 579–586, May 2004.

[9] A. Tall, Z. Altman, and E. Altman, “Multilevel beamforming for highdata rate communication in 5G networks,” ArXiv e-prints, Apr. 2015.

[10] J. Roberts, “Traffic theory and the Internet,” IEEE CommunicationsMagazine, vol. 39, no. 1, pp. 94–99, 2001.

[11] H. Klessig, A. Fehske, and G. Fettweis, “Admission control ininterference-coupled wireless data networks: A queuing theory-basednetwork model,” in 2014 12th International Symposium on Modelingand Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt),May 2014, pp. 151–158.

[12] A. J. Fenn, “Phased Array Antennas: An Introduction,” in AdaptiveAntennas and Phased Arrays for Radar and Communications. ArtechHouse, 2007, ch. Phased Arr.

[13] J. D. Krause, Antennas. TATA McGRAW-HILL Edition, Secondedition, New Delphi, 1997.

[14] R. L. G. Cavalcante, S. Stanczak, M. Schubert, A. Eisenblaetter, andU. Tuerke, “Toward Energy-Efficient 5G Wireless CommunicationsTechnologies: Tools for decoupling the scaling of networks from thegrowth of operating power,” IEEE Signal Processing Magazine, vol. 31,no. 6, pp. 24–34, Nov. 2014.

[15] A. J. Fehske, H. Klessig, J. Voigt, and G. P. Fettweis, “ConcurrentLoad-Aware Adjustment of User Association and Antenna Tilts inSelf-Organizing Radio Networks,” IEEE Transactions on VehicularTechnology, vol. 62, no. 5, pp. 1974–1988, Jun. 2013.

[16] D. Lee, S. Zhou, X. Zhong, Z. Niu, X. Zhou, and H. Zhang, “Spatialmodeling of the traffic density in cellular networks,” IEEE WirelessCommunications, vol. 21, no. 1, pp. 80–88, Feb. 2014.

[17] H. Klessig, V. Suryaprakash, O. Blume, A. Fehske, and G. Fettweis,“A Framework Enabling Spatial Analysis of Mobile Traffic Hot Spots,”IEEE Wireless Communications Letters, vol. 3, no. 5, pp. 537–540, Oct.2014.

[18] E. Vanmarcke, Random fields: analysis and synthesis. World Scientific,2010.

[19] H. Claussen, “Efficient modelling of channel maps with correlatedshadow fading in mobile radio systems,” in Personal, Indoor and MobileRadio Communications, 2005. PIMRC 2005. IEEE 16th InternationalSymposium on, vol. 1, Sept 2005, pp. 512–516.

[20] H. Kim, G. de Veciana, X. Yang, and M. Venkatachalam, “DistributedAlpha-Optimal User Association and Cell Load Balancing in WirelessNetworks,” IEEE/ACM Transactions on Networking, vol. 20, no. 1, pp.177–190, Feb. 2012.

[21] M. Soszka, S. Berger, A. Fehske, M. Simsek, B. Butkiewicz, andG. Fettweis, “Coverage and capacity optimization in cellular radionetworks with advanced antennas,” in WSA 2015; 19th InternationalITG Workshop on Smart Antennas; Proceedings of, March 2015, pp.1–6.

[22] Technical Specification Group Radio Access Network, “TR 36.814v9.0.0 - Evolved Universal Terrestrial Radio Access (E-UTRA) - Furtheradvancements for E-UTRA: Physical layer aspects,” 3rd GenerationPartnership Project, Tech. Rep., 2010.