on the application of directional antennas in multi-tier

Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000.

Digital Object Identifier 10.1109/ACCESS.2017.DOI

On the Application of DirectionalAntennas in Multi-tier Unmanned AerialVehicle NetworksJING ZHANG1, (Member, IEEE), HUAN XU1, LIN XIANG2, (Member, IEEE), AND JUN YANG.31School of Electronic Information and Communications, Huazhong University of Science and Technology, Wuhan 430074, Hubei, China (emails: {zhangjing,xuhuan1995}@hust.edu.cn)2Interdisciplinary Centre for Security, Reliability and Trust (SnT), University of Luxembourg, L-1855 Luxembourg, Luxembourg (email: [email protected])3School of Computer Science and Technology, Huazhong University of Science and Technology, Wuhan 430074, China (email: [email protected])

Corresponding author: Jun Yang (e-mail: [email protected]).

This work was supported in part by the Ministry of Science and Technology of China under Grant 2016YFE0119000, in part by theEuropean Research Council (ERC) project AGNOSTIC.

ABSTRACT This paper evaluates the performance of downlink information transmission in three-dimensional (3D) unmanned aerial vehicle (UAV) networks, where multi-tier UAVs of different types andflying altitudes employ directional antennas for communication with ground user equipments (UEs). Weintroduce a novel tractable antenna gain model, which is a nonlinear function of the elevation angle and thedirectivity factor, for directional antenna-based UAV communication. Since the transmission range of a UAVis limited by its antenna gain and the receiving threshold of the UEs, only UAVs located in a finite regionin each tier can successfully communicate with the UEs. The communication connectivity, associationprobability as well as coverage probability of the considered multi-tier UAV networks are derived for bothline-of-sight (LoS) and non-line-of-sight (NLoS) propagation scenarios. Our analytical results unveil that,for UAV networks employing directional antennas, a necessary tradeoff between connectivity and coverageprobability exists. Consequently, UAVs flying at low altitudes require a large elevation angle in order tosuccessfully serve the ground UEs. Moreover, by employing directional antennas an optimal directivityfactor exists for maximizing the coverage probability of the multi-tier UAV networks. Simulation resultsvalidate the analytical derivations and suggest the application of high-gain directional antennas to improvedownlink transmission in the multi-tier UAV networks.

INDEX TERMS directional antenna, multi-tier network, stochastic geometry, Unmanned aerial vehicle(UAV).

I. INTRODUCTIONUnmanned aerial vehicles (UAVs) have gained increasinginterest in both academia [1], [2] and industry [3], [4].By flying in the sky at moderate to high speeds, UAVscan provide flexible short-term services such as informationcollection over wireless, traffic surveillance, and disasterinformation dissemination to wherever demand occurs at alow cost [5]. An energy-efficient data collection method wasproposed for UAV-aided networks in [6], which can ensurefairness for ground sensors. For surveillance of multi-domainInternet-of-Things (IoT) devices, an approach based on linearinteger programming was proposed in [7] to minimize themaximum flying range of UAVs. In [8], joint optimization ofthe trajectory and scheduling of UAVs was investigated for aUAV-assisted emergency network, where UAVs are deployed

to re-establish communication between ground devices andsurviving BSs in the aftermath of natural disasters. Exploitingthe highly flexible and low-cost deployment of UAVs, severalworks have suggested employing UAVs as aerial base station-s (ABSs) to serve ground user equipments (UEs) directly andoffload traffic for terrestrial cellular networks. A novel three-dimensional (3D) ABS deployment was investigated in [9]for maximizing the number of UEs within ABSs’ coveragewhile fulfilling the quality-of-service (QoS) requirements ofUEs. The performance of a two-tier network consisting ofABSs and terrestrial cellular base stations (BSs) was ana-lyzed in [10], where ABSs are deployed to offload cellulartraffic in hotspot areas. A distributed algorithm was furtherproposed to minimize the average distance between UAVs

VOLUME 4, 2016 1

J. Zhang et al.: On the Application of Directional Antennas in Multi-tier Unmanned Aerial Vehicle Networks

and UEs without degrading the communication betweenUAVs and cellular BSs [11].

The existing works [9]–[11] have considered omni-directional antennas for UAV communication. However, asomni-directional antennas employ uniform antenna gains inall directions, the performance of UAV communication isseverely limited due to excess interference from neighboringUAVs and terrestrial nodes, especially at high altitudes withabundant line-of-sight (LoS) propagation [4]. To tackle thisissue, application of directional antennas for efficient UAVcommunications has recently gained tremendous attentiondue to the associated advantages of enhanced signal trans-mission, interference mitigation, and payload deployment.In particular, different from omni-directional antennas, UAVwith directional antenna generates highly directive beamswith strengthened signal power in the main lobe and re-duced power leakage in the side lobes. Consequently, di-rectional antennas with high antenna gain can improve thecommunication distance and the data rate in the downlinktransmission without increasing the power consumption ofthe UAVs. Moreover, due to the small footprint, the inter-ference to other UAVs and the terrestrial cellular systemis reduced by employing directional antenna. The resultinginterference mitigation capability can significantly enhancethe performance of UAV networks. Furthermore, due to thesize, weight, and power (SWAP) constraints of UAVs, anon-board deployment of large-scale antenna array is usuallydifficult, whereas directional antennas with large antennagain and flexible payload deployment provide a promising al-ternative for UAVs. Considering directional antennas in UAVnetworks, joint optimization of UAVs’ flying altitude and an-tenna beamwidth was investigated in [12] for maximizing thethroughput of downlink multicasting, downlink broadcasting,and uplink multiple access, respectively. A long-range broad-band aerial communication system using directional antennaswas proposed in [13], where a Wi-Fi infrastructure estab-lished in the air is exploited for real-time communication.Directional antenna has also been combined with millimeter-wave technique, for high-resolution 3D localization in [14],where multi-level beamforming with compressive sensingbased channel estimation is usually employed [15], [16].

Motivated by its huge potential, this paper investigatesthe application of directional antenna in UAV networkswith a focus on evaluating the network performance in thedownlink. Different from terrestrial cellular networks, UAVnetwork occupies a range of altitudes in the air and inherentlyhas a 3D network topology. This is usually captured by amulti-tier network model for performance evaluation, whereUAVs of a given tier keep flying at a certain altitude whilecommunicating with the ground UEs. Considering a multi-tier UAV network deployed atop terrestrial heterogeneousnetworks (HetNets), a cell management framework was pro-posed in [17] to improve the communication coverage andretransmission time for UEs in congested networks. In [18],the authors investigated the spectral efficiency of downlinkmulti-tier UAV networks and derived the optimal intensities

and altitudes for UAVs in different tiers. Assuming omni-directional antennas, the association probability, success-ful transmission probability, and area spectral efficiency ofmulti-tier UAV networks were analyzed in [19]. Despitethe fruitful development in the aforementioned works [17]–[19], a comprehensive performance evaluation for multi-tierUAV wireless networks employing directional antennas isstill lacking, which may be hindered by two potential chal-lenges. In particular, in the existing literature, especially formillimeter-wave communication, the gain of directional an-tenna was usually modeled as a flat-top antenna pattern withthe maximum antenna gain attained in the main lobe [20].Although this model is suitable for receivers at fixed locationand within short transmission distance, it is not applicablefor UAV communication networks. This is becauce the UAVflying in the sky usually changes its position and hence, theangle of arrival (AoA) and angle of departure (AoD), whichwill impact the antenna gain. In this case, the antenna gainmodel should capture the complicated channel variationsassociated with UAV communications along its flying trajec-tory, including channel conditions both line-of-sight (LoS)and non-line-of-sight (NLoS) propagation. Therefore, a noveltractable antenna gain model should be introduced for UAVcommunication networks.

On the other hand, the battery-powered UAVs usuallysuffer from severely limited energy supply. To reduce energyconsumption and prolong the lifetime, signal transmissionat UAVs has to respect a maximum transmit power budgetand, at the same time, ensure at least a minimum receivingpower at the ground UEs required to activate the receivingcircuits. Therefore, transmit power management at differentflying heights is crucial for UAVs and has been extensivelystudied in the literature while assuming omni-directionalantennas [21]–[23]. In [21], optimal power control for UAV-assisted networks serving underlaying D2D communicationwas investigated for minimizing the energy consumption andincreasing the battery’s service time. In space-air-groundthree-tier HetNets, the hovering altitude and transmit powerof UAVs were jointly optimized to reduce the cross-tierinterference [22]. In [23], joint trajectory and transmit poweroptimization was investigated for UAV communication tomaximize the average secrecy rate between UAV and groundUEs. Due to the signal enhancement and interference mit-igation enabled by directional antennas, which are furtheraffected by the 3D mobility of UAVs, the impact of transmitpower management on UAV communication, particularly atdifferent flight heights, needs to be newly investigated but hasnot been reported in the literature.

To address both challenges, in this paper, we proposea framework for modeling the downlink of UAV networkswhere, different from [9]–[11], [14]–[16], [18], UAVs areequipped with directional antennas. We assume that theground UEs are associated with the serving UAV that pro-vides the maximal receiving power while the UAVs in eachtier employ the same transmit power for communication. Thebeam shaped by directional antenna has complicated impact

2 VOLUME 4, 2016


on the user association in UAV networks. For example, UEsmay prefer a UAV located far away as serving ABS if theUAV is transmitting to the UEs in the main lobe. Moreover,for ground UEs located in the main lobe, directional antennasdeployed at UAVs will improve the connectivity probabilityas they can easily activate their circuits, i.e., satisfying thereceiving signal threshold. In contrast, for UEs located outof the main lobe, their connectivity probability reduces.Therefore, by employing directional antennas, there existsan interesting tradeoff between connectivity and coverageof UAV networks, whereas such tradeoff is unavailable forwireless networks employing omni-directional antennas. Inthis paper, we present a detailed analysis of the coverageand connectivity for the downlink of K-tier UAV networks.The stochastic geometry, which has been widely used foranalyzing cellular networks [24], UAV networks [25], andmillimeter-wave communication [26], is adopted in this pa-per to obtain closed-form results for performance evaluation.Our derivation results take into account the directivity ofantenna elements, the receiving threshold, and the trans-mit output power. We note that, due to the impact of itsantenna pattern, application of directional antenna leads tomuch more complicated performance analysis than in [9]–[11], [14]–[16], [18]. In [27], directional antennas with flat-top, sinc and cosine pattern functions are considered formillimeter-wave and cellular networks. However, the boundsfor the achievable transmission rate are obtained by utilizingthese approximate pattern functions, which fail to capture thedirectivity factor of directional antenna and its impact on theperformance of wireless networks. In this paper, we adopta novel tractable pattern function for directional antenna,which enables us to characterize the connectivity, associationprobability, and coverage probability of the considered UAVnetworks while capturing the impact of antenna directivityfactor.

The contributions of this paper are as follows:• We propose a tractable framework for modeling down-

link transmission employing directional antennas in K-tier UAV networks. The associated probability distribu-tion of communication distance between UEs and UAVsis analyzed taking into account the transmit output pow-er, flying height, and directional antenna pattern of theUAVs. We find that a maximal transmission height ofUAVs exists, independent of the directivity factor of di-rectional antennas, within which UAVs can successfullyconnect the UEs.

• The probabilities of connecting the UAV network andassociating with given tier of UAVs are both derivedin closed form. We show that, with the adopting ofdirectional antennas, ground UEs are prone to connectto UAVs flying at a large height.

• By investigating the coverage probability of downlinktransmission in K-tier UAV networks, we show thatUAV network equipped with directional antennas oflarge directivity factor can achieve much higher cover-age probability than that with omni-directional anten-

nas.The remainder of this paper is organized as follows. In

Section II, the system model of the considered multi-tierUAV networks employing directional antenna is presented.The probability distribution of communication distance forground UEs served by the considered UAV networks isderived in Section III, where the impact of transmit outputpower, receiving threshold, and directivity of antenna ele-ment is revealed. In Section IV, the connectivity and theassociation probabilities of the considered UAV networks areanalyzed, based on which the total coverage probability isfurther derived in Section V for LoS and NLoS transmissions.The derived results are validated via Monte Carlo simulationin Section VI, where the impact of maximal transmissiondistance, density and height of UAVs, and directivity factorof directional antenna on the downlink system performanceis revealed. Finally, Section VII concludes the paper.

II. SYSTEM MODELA. NETWORK MODELAs shown in Fig. 1, we consider downlink transmission ina 3D UAV network composing K tiers of ABSs. The ABSsin each tier are located at a given height but are randomlydistributed horizontally. Let (mi,k, hk) be the 3D locationof ABS i in tier k ∈ {1, · · · ,K}, where hk is the heightand mi,k denotes the horizontal coordinate. We assume thatthe horizontal locations of ABSs in tier k, denoted by Φk ={mi,k; i = 1, 2, 3 · · ·}, follow a homogeneous Poisson PointProcess (PPP) with density λk. The ABSs in tie k transmitsignals with output power Pk. The UEs are randomly dis-tributed on the ground with a height assumed to be zero. Thelocations of the UEs are modeled by a homogeneous PPPwith density λu, denoted as Φu = {xi}, which is indepen-dent of Φk. For a tractable analysis, we assume that the ABSsof each tier move only horizontally while providing wirelesscommunications in the considered multi-tier UAV networks.Since the speed of UAVs is relatively low, the locations ofthe ABSs in the air are considered to be fixed during thetransmission of a data packet, whereas the ABSs may flyhorizontally to different locations for the transmission ofmultiple data packets. Hence, the spatial distributions of theABSs under random horizontal movements can still be cap-tured by the PPP model. We assume that the channel fadingkeeps constant within a time slot, as commonly adopted forperformance analysis of UAV networks [28], [29].

The ground UEs are associated with the ABSs in a tierthat provide the strongest average receiving power. In thispaper, we aim to analyze the performance of the consideredmulti-tier UAV networks and would ignore the terrestrial BSswhich do not exist in the considered area or otherwise mayemploy orthogonal radio resources as the ABSs. Moreover,we focus on analyzing a typical UE U0 located at the originO and the typical cell C0, where the ground UEs withinC0, including the typical UE, are served by the same UAV.The derivation results of the typical UE can be extended toother UEs on the ground by applying the Palm theory [30].

VOLUME 4, 2016 3


LoS ABSNLoS ABS

tier k+1

tier k

...

...

x

y

z

o

FIGURE 1. llustration of the considered multi-tier UAV network model, wherethe ABSs serve the associated ground UEs within the coverage of theirdirectional beams.

We assume open access within the UAV network, wherebythe ground UEs are allowed to access ABSs in all tiers tomaximize the coverage probability. However, all downlinktransmission within the UAV network occupy the same spec-trum, whereby the UAVs in networks can interfere with eachother while serving the ground UEs.

B. DIRECTIONAL ANTENNAThe ABSs are equipped with directional antennas for com-munication with the ground UEs. In general, the antennagain of a directional antenna, denoted as G(ϕ,ψ), is a highlynonlinear function of the azimuth angle ϕ ∈ [−π, π] and theincidence angle ψ ∈ [0, π/2], which complicates the per-formance analysis. To simplify the derivations, in this paper,we consider conic directional antenna elements. The antennagain of a conic antenna is given as G(ϕ,ψ) = Aer cosm(ψ),where Aer is the maximal gain of antenna element and m isthe directivity factor dependent on the beam shape [31].

The directivity D of antenna element is obtained from thefollowing antenna equation [32]

D =4π∫ π

−π∫ π

2

0(G(ϕ,ψ)/Aer) sinψdψdϕ

. (1)

By substituting G(ϕ,ψ) into (1), the directivity of conicantenna is D = 2(m + 1), which only depents on thedirectivity factor. The considered conic antenna gain modelfacilitates us to evaluate the impact of the shaped beam onthe downlink performance of UAV networks. Fig. 2 showsthe normalized power pattern G(ϕ,ψ)/Aer for different m.

C. CHANNEL MODELThe ABSs with a large elevation angle usually have ahigh likelihood of establishing LoS communication with theground UEs [33]. As shown in Fig. 3, let θ be the elevation

FIGURE 2. The normalized power pattern versus the incidence angle fordifferent directivity factor m.

angle of the typical UAV in rad, which is the angle formedby the line from the UAV to the typical UE and the groundplane. In this paper, we assume that the ABSs always pointthe directional antennas toward the ground and, hence, wehave θ = π/2 − ψ. According to [33], the probability ofestablishing LoS communication from ABS at mi,k to theUE at O, denoted by PL (θ), is given as

PL (θ) =1

1 + ae−b(θ−a), (2)

where a and b are constants capturing the statistical proper-ties of the signal propagation environment. Consequently, theprobability of NLoS communication is given by PN (θ) =1− PL (θ).

Considering path loss and fading effect in the channelmodel, the receiving signal power at the typical UE is givenas

Pmi,k→o ,

{PkAer cosm

(π2− θ)GkL

−1L (mi,k, o), for LoS ,

PkAer cosm(π2− θ)HkL

−1N (mi,k, o), for NLoS ,

(3)

where Pk is the transmit power of the ABS located at mi,k

and Aer cosm(π2 − θ

)denotes the transmit antenna gain of

the ABS with incidence angle ψ = π2 − θ (Note that the

receiving antenna gain of ground UEs is assumed to be 1 in(3)). Moreover, L (mi,k, o) = ‖mi,k‖α models the path lossfor signal propagation from mi,k to o, which is a function ofthe Euclidean distance between mi,k and o, ‖mi,k‖, and thepath loss exponent α. We distinguish the path loss exponentsfor LoS and NLoS propagation by αL and αN , respectively,where αL ≤ αN usually holds. Moreover, the channel fadingfor LoS and NLoS propagation, denoted by Gk and Hk, aremodeled as Nakagami-m0 and Rayleigh random variables,respectively. Consequently, Gk ∼ Gamma (m0) follows theGamma distribution with scale parameter m0 while Hk ∼

4 VOLUME 4, 2016


Typical user

Ɵ

Ψ

u0

hk

mi,k

Ｏ

FIGURE 3. The footprint of a UAV with directional antenna.

exp (1) is exponentially distributed with unit mean power.For deriving the analytical results, the path loss exponent αLand αN are assumed to vary rarely across different tiers.

D. DOWNLINK SIGNAL TRANSMISSIONThe ABSs should utilize large enough transmit output powerto activate the receiver circuit at the UEs after combatingthe path loss and hence, the maximal allowable path lossbetween UAVs in tier k and the typical UE is given byLmax,k = Pk/ρc, where ρc is the receiving signal thresholdof the UEs. UAVs will fail to communicate with the UEswhen their distances exceed Lmax,k. This implies that onlya portion of UAVs can successfully connect the typical UE.Based on the PPP, the ABSs in tier k that can successfullyconnect the typical UE are uniformly distributed within a discbk (ok, rk) centered at ok = (0, 0, hk). The radius of the disc,rk, can be derived based on the channel model (3), as will berevealed in Section III.

We note that the transmit power Pk and the receivingsignal threshold ρc can jointly impact the performance ofthe considered multi-tier UAV networks. For example, themaximum distance between UAV and UEs, which impactsthe connectivity of UAV networks, will be limited by thereceiving signal threshold and the transmit power. Moreover,a higher transmit power Pk at UAVs will enable a large accessregion but leads to more interfering UAVs competing for theallocated spectrum. This reduces the signal-to-interference-plus-noise-ratio (SINR) and the coverage for UAV communi-cation. Therefore, the joint impact of the transmit power andthe receiving signal threshold on coverage and connectivityshould be investigated in detail, which is the aim for the restof this paper. The key notations used in this paper are listedin Table I.

TABLE 1. LIST OF KEY NOTATIONS

Symbols DefinitionsΦk, λk PPP of kth tier ABSs and densityρc Receiving signal threshold of UEsPk Transmit power of kth tier ABSs

αL, αN Path loss exponent for LoS and NLoS channelsG Nakagami-m0 fading of LoS channelH Reighlay fading of NLoS channelhk Height of kth tier ABSs

hLmax, hNmax Maximal height of UAV transmission under LoS

and NLoS communicationmLk ,m

Nk Maximal communication distance of UAVs in

tier k under LoS and NLoS communicationsbk (ok, rk) Disc with radius rk and center ok

III. COMMUNICATION DISTANCE DISTRIBUTIONIn this section, we first derive the maximal distance withinwhich the ABSs can successfully connect to the UEs. Thenwe characterize the probability distribution of the distancebetween the closest ABS in the k-th tier of the UAV networksand the ground UEs.

A. MAXIMAL COMMUNICATION DISTANCEFor the UAVs in tier k with height hk, the maximum com-munication distances to the typical UE U0 are calculated forLoS and NLoS propagation channels separately. When theABS located at mi,k transmits in LoS channels, the receivingsignal power at the typical UE is given as

Pr = PkAer cosm(π

2− θ)GkL

−1L (mi,k, o) . (4)

Since U0 is located at the origin, thereafter we simply denoteLL (x, o) and LN (x, o) as LL (x) and LN (x), respectively.In order to activate the receiver circuit, the average receivingpower Pr should exceed the receiving threshold ρc, i.e.,

PkAer

(hkmi,k

)m‖mi,k‖−αL ≥ ρc, (5)

where cos (π/2− θ) = hk/‖mi,k‖ according to Fig. 3.Therefore, the maximal communication distance for the

kth tier ABSs under LoS transmission is

mLk =

(PkAerh

mk

ρc

) 1αL+m

. (6)

Given the flying height of the ABSs in tier k, hk, the radiusof the disc of ABSs in tier k that can activate U0 is furtherderived as

rLk =

√(PkAerhmk

ρc

) 2αL+m

− h2k. (7)

Similarly, the maximal communication distance and radiusfor the kth tier ABSs under NLoS transmission are given as

mNk =

(PkAerh

mk

ρc

) 1αN+m

and rNk =

√(PkAerhmk

ρc

) 2αN+m

− h2k, (8)

VOLUME 4, 2016 5


respectively. Since αL ≤ αN , we have rLk ≥ rNk . That is,the ABSs implementing LoS and NLoS communications tothe UE may occupy different (though overlapping) regions.Therefore, the performance of LoS and NLoS communica-tions should be evaluated separately.

We note that, constrained by the maximal transmit powerof UAVs, Pmax, a maximal transmission height, hmax, alsoexists for the ABSs. If the height of UAVs in tier k exceedsthe maximal transmission height, i.e., hk > hmax , any UAVin this tier cannot successfully transmit signals to the typicalUE. Based on (7) and (8), the maximal transmission height intier k under LoS and NLoS communications can be obtainedas

hLmax =

(PmaxAer

ρc

)1/αL

and hNmax =

(PmaxAer

ρc

)1/αN

.

(9)

Note that the maximal transmission height is independentof the directivity factor of directional antenna, m. This isbecause the maximal transmission height appears when theABS is located at the center of disc bk. In this case, theelevation angle is θ = π/2 and the maximal antenna gainAer is achieved independent of the directivity factor m.B. DISTRIBUTION OF COMMUNICATION DISTANCERecall that, in tier k, only the ABSs located within discbk can successfully communicate with the typical UE. Thecommunication distance between the ABSs and the UE underLoS and NLoS communication, denoted by random variablesDLk andDN

k , have the following probability density functions(PDFs)

fDLk

(dk) =

{2dk

(rLk )2, ifhk ≤ dk ≤ mL

k ,

0, otherwise,(10)

and

fDNk

(dk) =

{2dk

(rNk )2, ifhk ≤ dk ≤ mN

k ,

0, otherwise,(11)

where rLk , rNk , mLk , and mN

k are given in (6)–(8). Note that(10) and (11) can be obtained similar to Lemmas 1 and 2in [28], which conclude that a given number of UAV nodesuniformly distributed within a finite region follow a binomialpoint process (BPP).Lemma 1: Based on the maximal transmission distance andthe uniform location distribution of UAVs in tier k, theaverage number of ABSs implementing LoS and NLoS com-munications are given as

E[nLk

]= 2πλk

∫ rLk

0

x

1 + ae−b

(arctan

(hkx

)−a

) dx, (12)

and

E[nNk

]= 2πλk

∫ rNk

0

xae−b

(arctan

(hkx

)−a

)

1 + ae−b

(arctan

(hkx

)−a

) dx. (13)

Proof: Recall that the UAV with elevation angle θ canimplement LoS communication with probability given in (2).The elevation angle can be expressed as θ = arctan

(hkd

),

where d is the distance between UAV in tier k and ok. The

number of UAVs in tier k implementing LoS communicationis then given by

E[nLk

]=

∫ 2π

0

∫ rLk

0

λkx

1 + ae−b

(arctan

(hkx

)−a

) dxdφk

= 2πλk

∫ rLk

0

x

1 + ae−b

(arctan

(hkx

)−a

) dx, (14)

where φk is the angle subtended by line from the ABS to okand the x-axis in disc bk, and is uniformly distributed within[0, 2π]. The UAVs failing to implement LoS communicationwill communicate under NLoS condition. The number ofUAVs in tier k implementing NLoS communication is

E[nNk

]=

∫ 2π

0

∫ rNk

0

λkx

(1− 1

1 + ae−b

(arctan

(hkx

)−a

))dxdφk

= 2πλk

∫ rNk

0

xae−b

(arctan

(hkx

)−a

)

1 + ae−b

(arctan

(hkx

)−a

) dx, (15)

which complete the proof.The typical UE is associated with the ABS providing the

maximal average receiving power. As all UAVs in tier kemploy the same transmit power Pk, the ABS in tier k havingthe closest distance will provide maximal average receivingpower for the typical UE. This motivates us to derive theprobability distribution of the closest distance between ABSsin tier k and the typical UE and the result is included in thefollowing lemma.Lemma 2: The PDF of the closest distance between the ABSin tier k and the typical UE under LoS transmission is

fXLk

(x) =

E[nLk](

1− x2−(hk)2

(rLk )2

)E[nLk]−1

· 2x

(rLk)2 , if hk ≤ x ≤ mL

k ,

0, otherwise,(16)

where E[nLk]

is the average number of UAVs in tier kimplementing LoS communication as given in (14).

Proof: For the ABSs in tier k having the closest distance,the cumulative distribution function (CDF) of the closestdistance to the typical UE is obtained as

FXLk

(x) = 1− P(XLk > x

)= 1− P

(min

{XLk,1, X

Lk,2, . . . , X

Lk,nL

k

}> x

)= 1−

[1− FXL

k(x)]E[nLk ]

. (17)

The PDF can obtain from the derivation of (17) as

fXLk

(x) =dFXL

k(x)

dx

= E[nLk

] [1− FXL

k(x)]E[nLk ]−1

fXLk

(x)

= E[nLk

](1− x2 − (hk)2

(rLk )2

)E[nLk ]−1

· 2x

(rLk )2 . (18)

6 VOLUME 4, 2016


Similarly, the PDF of the closest distance between theABSs in tier k and the typical UE under NLoS transmissioncan be obtained as

fXNk

(x) =

E[nNk](

1− x2−(hk)2

(rNk )2

)E[nNk ]−1

· 2x

(rNk )2, ifhk ≤ x ≤ mN

k ,

0, otherwise.(19)

IV. CONNECTIVITY AND ASSOCIATION PROBABILITIESOF UAV NETWORKS

In the considered UAV networks, the ground UE can connectwith an ABS provided its receiver circuit can be activated bythe ABS. Meanwhile, according to the association policy, theground UE will associate with the ABSs in any tier that pro-vide the maximum average receiving power. In this section,we will analyze the connection and association probabilitiesfor the considered multi-tier UAV networks.

A. CONNECTIVITY PROBABILITY OF UAV NETWORKS

Based on the analysis in Section III, the typical UE canconnect to the considered UAV networks if and only if thereexists k ∈ {1, · · · ,K} such that at least an ABS in tier k islocated within disc bk. On the contrary, for ABSs spatiallydistributed according to a PPP, the probability that ABSs intier k cannot connect with the typical UE, which is givenby P {Φi ∩ bi = φ,∀i = 1, · · · ,K}, can be characterized inLemma 3.Lemma 3: The probability that the typical UE cannot connectwith UAV networks is

P {Φi ∩ bi = φ,∀i = 1, · · · ,K} =

exp

−π K∑k=1

λk

(PkAerhmkm0ρc

) 2m+αL

Γ(

2αL+m

+m0

)Γ (m0)

−h2k

.

(20)

Proof: The typical UE cannot connect with the ABSsin tier k if and only if it can neither connect with the ABSsimplementing LoS nor NLoS communication, i.e.,

P{Φk ∩ bk=φ}=P{Φk ∩ bk,L=φ and Φk ∩ bk,N =φ} . (21)

Since bk,L and bk,N have the same center ok, bk,L andbk,N would overlap with each other. Moreover, as rNk ≤rLk , bk,L covers bk,N and hence, P {Φk ∩ bk = φ} =P {Φk ∩ bk,L=φ}. Consequently, the typical UE cannot con-nect with the ABSs if all ABSs in tier k are located outsidethe circle bk,L. Thus, the probability that the typical UEcannot connect with the ABSs implementing LoS commu-

nication in tier k is

P {Φk ∩ bk = φ}

= P(

maxmi,k∈Φk

PkAer cosm(π

2− θ)GkL

−1L (mi,k, o) < ρc

)= P

(PkAer

hmk

(r2 + h2k)m/2

Gk(r2 + h2

k

)−αL/2 < ρc

)

= P

(r2 >

(PkAerGkh

mk

ρc

) 2m+αL

− h2k

)

= exp

(−πλk

((PkAerh

mk

ρc

) 2m+αL

E[G

2m+αLk

]− h2

k

))

=exp

−πλk(PkAerhmk

m0ρc

) 2m+αL

Γ(

2αL+m

+m0

)Γ (m0)

−h2k

.

(22)

As the point processes of ABSs in different tiers are indepen-dent of each other, we have

P {Φi ∩ bi = φ, ∀i = 1, · · · ,K}

=

K∏k=1

P {Φk ∩ bk = φ}

=exp

−π K∑k=1

λk

(PkAerhmkm0ρc

) 2m+αL

Γ(

2αL+m

+m0

)Γ (m0)

−h2k

,

(23)

which complete the proof.Based on Lemma 3, the probability that the typical UE can

connect with the K-tier UAV networks is

1−exp

−π K∑k=1

λk

(PkAerhmkm0ρc

) 2m+αL

Γ(

2αL+m

+m0

)Γ (m0)

−h2k

.

(24)

From (24) we observe that the directivity of directionalantenna, m, highly impacts the connection probability of theground UEs. It can be further proved that the probabilityof UE connecting UAV networks decreases with m. Thisis because, by utilizing directional antennas, the radius rkof bk will decrease with m and more UAVs are locatedout of bk. Therefore, by employing directional antennas, theregion where UAVs can successfully connect with the UEwill reduce and less ground UEs can communicate with theUAV networks.

B. ASSOCIATION PROBABILITY OF UAV NETWORKS

When the typical UE can connect with the UAV networks, itwill associate with the serving ABS providing the maximalaverage receiving power. The probability that the typical UEis associated with the ABS in tier k under LoS commu-nication, denoted as P

(AL = k

), is given in the following

theorem.

VOLUME 4, 2016 7


Theorem 1: Under LoS communication, the typical UE willbe associated to the ABSs in tier k with a probability givenas

P(

AL = k)

=

∫ mLk

hk

K∏j=1,j 6=k

1−

(PjPk

) 2m+αL

(hjhk

) 2mm+αL r2 − h2

j(rLj)2

E[nLj ]

·K∏j=1

1−

(PjPk

) 2m+αN

(hjhk

) 2mm+αN r

2(m+αL)m+αN − h2

j(rNj)2

E[nNj ]

·E[nLk

](1− r

2 − (hk)2

(rLk )2

)E[nLk]−1

.2r

(rLk)2 dr. (25)

Proof: As the UAVs can implement both LoS and NLoScommunications, the associated ABS in tier k for servingthe typical UE has to satisfy two independent conditions.For association with the ABSs under LoS transmission, theserving ABS has to provide the maximal average receivingpower among all ABSs. Meanwhile, the receiving powerfrom the serving ABS has to exceed that from all ABSs underNLoS transmissions. Let R be the closest distance betweenthe UAVs in tier k and the typical UE. We have

P(

AL = k)

=

ER[PL(PLr,k(R) >max

j,j 6=kPLr,j

)·PL(PLr,k(R) > max

j=1,···,KPNr,j

)].

(26)

As the ABSs implement LoS and NLoS communicationsindependently, we can calculate (26) as

P(

AL = k)

= ER

K∏j=1,j 6=k

PL(PLr,k(R) > PLr,j

)·K∏j=1

PL(PLr,k(R) > PNr,j

)= ER

K∏j=1,j 6=k

PL(PkAer

(hkR

)mGkR

−αL>PjAer

(hjRj

)mGkR

−αLj

)

·K∏j=1

PL(PkAer

(hkR

)mGkR

−αL >PjAer

(hjRj

)mHkR

−αNj

)]

=

∫ mLk

hk

K∏j=1,j 6=k

PL

(Rj >

(PjPk

) 1m+αL

(hjhk

) mm+αL

r

)

·K∏j=1

PL

(Rj >

(PjPk

) 1m+αN

(hjhk

) mm+αN

rm+αLm+αN

)fR(r)dr

=

∫ mLk

hk

K∏j=1,j 6=k

1−

(PjPk

) 2m+αL

(hjhk

) 2mm+αL r2 − h2

j(rLj)2

E[nLj ]

·K∏j=1

1−

(PjPk

) 2m+αN

(hjhk

) 2mm+αN r


j(rNj)2

E[nNj ]

fR(r)dr,

(27)

where fR(r) has been obtained in (16). Theorem 1 can beproved by Substituting (16) into (27).

Under NLoS transmission, the probability of associationto the ABS in tier k can be similarly obtained as

P(

AN = k)

=

∫ mNk

hk

K∏j=1,j 6=k

1−

(PjPk

) 2m+αN

(hjhk

) 2mm+αN r2 − h2

j(rNj)2

E[nNj ]

·K∏j=1

1−

(PjPk

) 2m+αL

(hjhk

) 2mm+αL r

2(m+αN )m+αL − h2

j(rLj)2

E[nLj ]

·E[nNk

](1− r

2 − (hk)2

(rNk )2

)E[nNk ]−12r

(rNk )2 dr. (28)

LetXk be the distance between the typical UE and its servingABS when the typical UE is associated with the ABS in tierk. Recall that the ABSs which can connect to the typical UEmust be located within disc bk. Given the location distributionof ABSs in tier k, cf. Lemma 2, the PDF ofXk can be derivedin the following theorem.

Theorem 2: Under LoS transmission, the PDF of Xk, thedistance between the typical UE and its serving ABS in tierk, is given as

fXk (x) =

1

P(AL = k)

K∏j=1,j 6=k

1−

(PjPk

) 2m+αL

(hjhk

) 2mm+αL x2 −h2

j(rLj)2

E[nLj ]

·K∏j=1

1−

(PjPk

) 2m+αN

(hjhk

) 2mm+αN x


j(rNj)2

E[nNj ]

· E[nLk

](1− x2 − (hk)2

(rLk )2

)E[nLk ]−1

· 2x

(rLk )2 . (29)

Proof: Let Rk be the distance between the typical UEand its serving ABS. Under the condition that the associatedABS is located in tier k and has LoS communication with theUE, we have

P{RLk > x|ALk

}=

P{RLk > x,AL = k

}P (AL = k)

, (30)

8 VOLUME 4, 2016


with

P{RLk > x,AL = k

}=PL

(RLk > x,PLr,k (Rk) > max

j,j 6=kPLr,j and PLr,k (Rk) > max

j=1,···,KPNr,j

)=

∫ mLk

x

K∏j=1,j 6=k

PL

(Rj >

(PjPk

) 1m+αL

(hjhk

) mm+αL

r

)

·K∏j=1

PL

(Rj >

(PjPk

) 1m+αN

(hjhk

) mm+αN

rm+αLm+αN

)fRk (r)dr

=

∫ mLk

x

K∏j=1,j 6=k

1−

(PjPk

) 2m+αL

(hjhk

) 2mm+αL r2 − h2

j(rLj)2

E[nNj ]

·K∏j=1

1−

(PjPk

) 2m+αN

(hjhk

) 2mm+αN r


j(rNj)2

E[nLj ]

fRk (r)dr.

(31)

Moreover, substituting (25) into (30), we have

P {Xk > x} =

1

P(AL = k

)∫ mLkx

K∏j=1,j 6=k

1−

(PjPk

) 2m+αL

(hjhk

) 2mm+αL r2 −h2

j(rLj)2

E[nLj ]

·K∏j=1

1−

(PjPk

) 2m+αN

(hjhk

) 2mm+αN r


j(rNj)2

E[nNj ]

fRk (r)dr.

(32)

Finally, Theorem 2 can be proved be taking fXk(x) =d(1−P{Xk>x})

dx .Meanwhile, under the condition that the serving ABS with

LoS transmission is in tier k and has distance Rk to the UE,other ABSs in the UAV networks will interfere the desiredsignals due to the full spectrum reuse among the ABSs. Thereceiving interference power at the typical UE caused fromother interfering ABSs should be less than the receivingpower from the serving ABS. Define XI,j as the distancebetween the typical UE and an interfering ABS in tier j. ThePDF of XI,j can be described in the following lemma..Lemma 4: Given that the serving ABS is located in tier kat a distance of Rk, the PDF of the distance between thetypical UE and interfering ABS in tier j, XI,j , with LoStransmission is

f (xI,j > Rk) =

2xI,j

m2j−

(PjPk

) 2m+αL

(hjhk

) 2mm+αL R2

k

·(PjPk

) 1m+αL

(hjhk

) mm+αL , if Rk < mj ,

0, otherwise.(33)

Proof: As the distance between the typical UE and theserving ABS in tier k is Rk, the distance from the interferingABS in tier j to the UE should satisfy

PkAer (hk/R)m (Rk)−αL > PjAer

(hjXI,j

)m(XI,j)

−αL . (34)

We obtain that XI,j >(PjPk

) 1m+αL

(hjhk

) mm+αL Rk. Hence,

f (xI,j > Rk) =2xI,j

m2j −

(PjPk

) 2m+αN

(hjhk

) 2mm+αL Rk

2

, (35)

which completes the proof.Similarly, the PDF for the distance between the typical

UE and an interfering ABS in tier j, XI,j , with NLoStransmission is

f(xI,j >Rk)=

2xI,j

m2j−

(PjPk

) 2m+αN

(hjhk

) 2mm+αNR

2(m+αL)m+αNk

·(PjPk

) 1m+αN

(hjhk

) mm+αN , ifR

m+αLm+αNk <mj ,

0, otherwise.(36)

V. ANALYSIS OF COVERAGE PROBABILITYBased on the association probability in Section IV, we canfurther derive the coverage probability of the K-tier UAVnetworks. For this purpose, we first have to derive the Laplacetransform of the aggregated interference power caused fromthe interfering ABSs.

Based on the distance distribution of the serving ABS andinterfering ABSs, the total coverage probability of the K-tierUAV networks, Pc, can be calculated as

Pc =

n∑k=1

(PL(A = k)ALk + PN (A = k)ANk

), (37)

where PL(A = k) ( PN (A = k) ) is the coverage probabilityconditioning on that the serving ABS is in tier k and hasLoS (NLoS) transmission. ALk and ANk are the associationprobabilities that the serving ABS in tier k has LoS and NLoStransmissions, respectively.

Given that the distance from the serving ABS in tier k withLoS or NLoS transmission is Rk,o, the signal-to-interferenceratio (SIR) at the typical UE can be given as

SIRLk (Rk,o) =

PkAer(

hkRk,o

)mGrk,oR

−αLk,o

IL + IN, (38)

SIRNk (Rk,o) =

PkAer(

hkRk,o

)mHrk,oR

−αNk,o

IL + IN,

where IL and IN are aggregated interference powers frominterfering ABSs under LoS and NLoS transmissions, respec-tively.

Based on [34], the coverage probability PL(A = k) can bedefined as

PL(A = k) = ERk,o[P{

SIRLk (Rk,o) ≤ T

}], (39)

where T is the SIR threshold for downlink transmission.Moreover, given that the distance of the serving ABS isR, the Laplace transform of interference power with LoStransmission, IL, is given in the following Lemma.Lemma 5: Given that the serving ABS with LoS transmissionis located at a distance of R away from the typical UE, theLaplace transform of the interference power, IL, is given as

LLI (s|R) = ELIL [exp (−sIL) |R]ELIN [exp (−sIN ) |R] , (40)

VOLUME 4, 2016 9


where

ELIL [exp (−sIL) |R] =

K∏j=1,j 6=k

∫ mj(PjPk

) 1m+αL

(hjhk

) mm+αLR

(1 +

sPjAerhmj x−(αL+m)

m0

)−m0E[nLj ]

· 2x

m2j −R2

(PjPk

) 2m+αL

(hjhk

) 2mm+αL

dx

·∫ mk

R

(1 +

sPkAerhmj x−(αL+m)

m0

)−m0(E[nLk]−1)2x

m2k −R2

dx,

(41)

and

ELIN [exp (−sIN ) |R] =

K∏j=1

∫ mj(PjPk

) 1m+αN

(hjhk

) mm+αNR

(1

1 + sPjAerhmj x−(αN+m)

)E[nNj ]

· 2x

m2j −(PjPk

) 2m+αN (hj/hk)

2mm+αN R

2(m+αL)m+αN

dx. (42)

Proof: The Laplace transform of the interference power,IL + IN , can be derived as

LLI (s|R) = ELI [exp (−s (IL + IN )) |R]

= ELIL [exp (−sIL) |R]ELIN [exp (−sIN ) |R] . (43)

Herein, the Laplace transform of the interference powercaused by ABSs with LoS transmission is given in (44).Moreover, the interference power caused from ABSs withNLoS transmission has the following Laplace transform,

EIN [exp (−sIN ) |R]

= EIL

exp

−s K∑j=1

E[nNj ]∑i=1

Ii,j

|R

=EIL

K∏j=1

E[nNj ]∏i=1

exp

(−sPjAer

(hjui,j

)mHui,ju

−αNi,j

)|R

=

K∏j=1

EULN,j

,HLj

E[nNj ]∏i=1

exp(−SPjAerhmj Hui,ju

−(αN+m)i,j

)|R

=

K∏j=1

EULN,j

E[nNj ]∏i=1

EH[exp

(−sPjAerhmj Hu

−(αN+m)i,j

)]|R

=

K∏j=1

EULN,j

[ 1

1 + sPjAerhmj u−(αN+m)i,j

|R

]E[nNj ]

=

K∏j=1

∫ mj(PjPk

) 1m+αN

(hjhk

) mm+αNR

m+αLm+αN

[1

1 +sPjAerhmj x−(αN+m)

]E[nNj ]

· 2x

m2j −(PjPk

) 2m+αN

(hjhk

) 2mm+αN R

2(m+αL)m+αN

dx,

(45)

where ULN,j is the set of interfering UAVs implementingNLoS transmission in tier j. Substituting (44) and (45) into(43), we can obtain (40), which completes the proof.

Similarly, given that the serving ABS with NLoS transmis-sion has distanceR, the Laplace transform of the interferencepower, IL + IN , is

LNI (s|R) = ENIL [exp (−sIL) |R]ENIN [exp (−sIN ) |R] , (46)

where

ENIL [exp (−sIL) |R] =

K∏j=1

∫ mj(PjPk

) 1m+αL

(hjhk

) mm+αLR

m+αNm+αL

(1 +

sPjAerhmj x−αL−m

m0

)−m0E[nLj ]

· 2x

m2j −R

2(m+αN )m+αL

(PjPk

) 2m+αL

(hjhk

) 2mm+αL

dx, (47)

and

ENIN [exp (−sIN ) |R] =

K∏j=1,j 6=k

·∫ mj(PjPk

) 1m+αN

(hjhk

) mm+αNR

(1

1 + sPjAerhmj x−(αN+m)

)E[nNj ]

2x

m2j −R2

(PjPk

) 2m+αN

(hjhk

) 2mm+αN

dx

·∫ mk

R

(1

1 + sPkAerhmj x−(αN+m)

)(E[nNk ]−1) 2x

m2k −R2

dx.

(48)

Based on(40) and (46), the coverage probability of the K-tierUAV networks can be obtained in the following theorem.Theorem 3: The total coverage probability of theK-tier UAVnetwork, defined asPc =

∑nk=1

(PL(A = k)ALk + PN (A = k)ANk

),

cf. (37), can be obtained by substituting

PL(A = k) =

m0∑i=1

(−1)i+1

(m0

i

)∫ mkhk

LLI (s)|s=M fRLk|ALk

(r)dr,

(49)and

PN (A = k) =

∫ mk

hk

LNI (s)|s=P−1

kA−1er h

−mk

Rαl+m

k,oTfRL

k|ALk

(r)dr.

(50)whereM = iηP−1

k A−1er h

−mk RαL+m

k,o T . Moreover,ALk andANk aregiven in (22) and (25), respectively.

Proof: The coverage probability of the typical UE asso-ciated with the serving ABS under LoS transmission in tier kcan be calculated as

PL(A = k) = ERk,o[P{

SIRLk (Rk,o) ≥ T

}]=

∫ mk

hk

P{

SIRLk (r) ≥ T

}fRL

k(r)dr. (51)

10 VOLUME 4, 2016


ELIL [exp (−sIL) |R]

= EIL

exp

−s K∑j=1,j 6=k

E[nLj ]∑i=1

Ii,j +

E[nLk ]−1∑i=1

Ii,k

|R

= EIL

K∏j=1,j 6=k

E[nLj ]∏i=1

exp

(−sPjAer

(hjui,j

)mGuiju

−αLi,j

)·E[nLk ]−1∏i=1

exp

(−sPkAer

(hkui,j

)mGui,ku

−αLi,k

)|R

=

K∏j=1,j 6=k

EULj ,GLj

E[nLj ]∏i=1

exp(−sPjAerhmj Gui,ju

−(αL+m)i,j

)|R

·EULk,GLk

E[nLk ]−1∏i=1

exp(−sPkAerhmk Gui,ju

−(αL+m)i,k

)|R

=

K∏j=1,j 6=k

EULj

E[nLj ]∏i=1

EG[exp

(−SPjAerhmj Gu

−(αL+m)i,j

)]|R

·EULk

E[nLk ]−1∏i=1

EG[exp

(−SPkAerhmk Gu

−(αL+m)i,k

)]|R

=

K∏j=1,j 6=k

EULj

(1 +

sPjAerhmj u−(αL+m)i,j

m0

)−m0

|R

E[nLj ]EUL

k

(1 +

sPkAerhmk u−(αL+m)i,k

m0

)−m0

|R

E[nLk ]−1

=

K∏j=1,j 6=k

∫ mj(PjPk

) 1m+αL

(hjhk

) mm+aL R

[(1 +

sPmaxAerhmj x−(αL+m)

m0

)−m0]E[nLj ]

2x

m2j −(PjPk

) 2m+αL

(hjhk

) 2mm+αL R2

dx

·∫ mk

R

[(1 +

sPmaxAerhmj x−(αL+m)

m0

)−m0]E[nLk ]−1

2x

m2k −R2

dx.

(44)

where fRLk (r) is given in Lemma 3. Moreover, we have

P{

SIRLk (Rk,o) ≥ T

}= P

{Grk,o ≥ P

−1k A−1

er h−mk RαL+m

k,o T (IL + IN )}

(a)≈ 1−EI

[[1−exp

(−ηP−1

k A−1er h

−mk RαL+m

k,o T (IL + IN))]m0

]=

m0∑i=1

(m0

i

)(−1)i+1EI [exp(−sI)]|

s=iηP−1k

A−1kh−mk

RαL+m

k,oT.

(52)

where (a) follows from Lemma 6 in [34] and η =m0 (m0! )

(1/m0). Similarly, we have

P{

SIRNk (Rk,o) ≥ T

}= P

{Hrk,o ≥ P

−1k A−1

er h−mk RαL+m

k,o T (IL + IN )}

= EI[exp

(−P−1

k A−1er h

−mk RαL+m

k,o T (IL + IN ))]

= EI [exp(−sI)]|s=P−1

kA−1er h

−mk

RαL+m

k,oT. (53)

Based on Lemma 5 and (46), Theorem 3 can be thus proved.

VI. NUMERICAL AND SIMULATION RESULTSIn this section, we evaluate the performance of a two-tierUAV network, where the ABSs in tier 1 and 2 are located atheights h1 and h2, respectively. Unless otherwise specified,

the simulation parameters are set according to Table II. Wepresent both analytical and simulation results, where MonteCarlo simulations are employed to validate the analyticalresults obtained in Sections III-V. Thereby, the horizontallocations of UAVs and UEs are randomly generated in a largeplane and the height of UAVs in each tier is set based on theempirical data from Qualcomm [35]. We simulate 104 spatialrealizations for the locations of ground UEs and UAVs. Foreach spatial realization, the locations of the ground UEs andUAVs are fixed, which enables us to obtain the empiricalconnection and coverage probabilities. The final results aregathered by averaging over all simulation realizations. Wenote that Monte Carlo simulations have been widely adoptedto evaluate performance and validate analytical derivation forcellular networks and UAV networks [29], [36], [37].

A. CONNECTION PROBABILITY OF THE UAV NETWORK

Fig. 4 shows the connection probability of the consideredUAV network as a function of the height of the first-tierABSs when different receiving thresholds are employed atthe ground UEs. From Fig. 4 we observe that the Monto Carlosimulation results are in a good match with the analyticalresults, implying that the derivations in Section III are valid.As the height of the first-tier ABSs increases in the smallvalue regime, the connection probability increases quicklybefore it saturates. This result is due to the high directivity of

VOLUME 4, 2016 11


TABLE 2. SIMULATION PARAMETERS

Parameter Default ValuePmax 1Wρc −50dBmαL 2αN 2.8h1 100m [35]h2 2h1 [35]λ1 10/km2

λ2 λ1

m0 3m 6T 0.1Aer 5dB

FIGURE 4. Connection probability of the UAV network versus flying height ofthe first-tier ABSs for different receiving thresholds of the ground UEs.

directional antenna. In particular, the ABSs at a low heightwould transmit signal over the sidelobe of the generatedbeam, resulting in only small antenna gain. In this case, theconnection probability of the UAV network is small. As theABSs’s height increases, the connection probability of theUAV networks improve as more UEs receive signal from themain lobe of the UAVs such that the associated antenna gainincreases. On the other hand, we also observe from Fig. 4that, after the flying height exceeds a limit, e.g., 1300 m forreceiving threshold ρu=40 dBm, the connection probabilityquickly drops to zero. This is consistent with our derivationsof the maximum flying height in Section III. In particular,due to the receiving threshold of the typical UE and themaximum output power of the ABSs, only ABSs within agiven flying height can activate the receiving circuit at theUEs. Consequently, the region where ABSs can overcomethe path loss to a UE shrinks when the height of the ABSsincreases and further vanishes when the flying height ofthe ABSs exceeds hLmax. In the latter case, the connectionprobability of the UAV network decreases quickly.

Fig. 5 evaluates the connection probability of the consid-

FIGURE 5. Connection probability versus directivity factor of antenna elementfor different flying heights of the ABSs.

ered UAV network as a function of the antenna’s directivityfactor when the ABSs adopt different flying heights. FromFig. 5 we observed that, for directional antennas, the con-nection probability decreases with the antenna’s directivityfactor. This is because the main lobe of the generated direc-tional beam shrinks as the directivity factor increases. Conse-quently, a UE has to communicate with the ABS via the sidelobe with a high probability, which reduces the connectionprobability. In contrast, for omni-directional antennas withm=0, the connection probability is always close to 1 for theconsidered simulation setup. Moreover, by employing direc-tional antenna, it is interesting to observe that the ABSs flyingat a large height can provide a high connection probability,espcially when the directivity factor is large. This is because,as the flying height increases, only the ABSs close to the topof their served UEs can successfully connect with the UEs.Although a high directivity factor leads to a disc of smallthe radius, the large antenna gain enables ABSs to connectwith the UE at large height. This result implies that the UAVnetwork employing highly directive antennas is suitable to flyat a large height provided it is within the limit of the flyingmaximum height.

Fig. 6 shows the connectivity probability as a functionof the receiving threshold when the UAVs employ antennasof different directivity factors. From Fig. 6 we observe thatthe connection probability of the considered UAV networkdecreases with the receiving threshold of the UE. However,as the directivity factor of antenna elements increases, theconnection probability tends to decrease with the receivingthreshold at a slower rate. This is because the radius ofthe disc decreases with the directivity factor; consequently,fewer ABSs can successfully connect with the ground UEsusing their transmit output power. In the large regime ofthe receiving threshold, the connection probability of UAV

12 VOLUME 4, 2016


FIGURE 6. Connection probability versus receiving threshold of the groundUEs for antennas of different directivity factors.

network decreses sharply and approaches 0. In the latter case,the ground UE cannot connect to the UAV network. Thisis because the maximal transmission height decreases withthe receiving threshold. Consequently, more ABSs at largeflying heights will fail in signal transmission until, when thereceiving threshold of ground UEs is large enough, none ofthe ABSs can successfully connect with the ground UEs.

B. COVERAGE PROBABILITY OF THE UAV NETWORKFig. 7 shows the total coverage probability as a functionof the flying height of the first-tier ABSs when the groundUEs employ different receiving thresholds. From Fig. 7we observe that there exists an optimal flying height thatachieves the maximal coverage probability. This is becauseby employing directional antennas, the radius of disc bkincreases with the height of the ABSs such that more ABSscan serve the UEs. When the flying height of the ABSs is low(e.g. 100 m < h1 < 200 m), the radius of the disc is smallsuch that few ABSs can successfully activate the groundUE. As the flying height of the ABSs increases, the discof radius as well as the connection probability of the UAVnetwork improves. Nevertheless, as the number of interferingABSs also increases with the radius of the disc, the coverageprobability deteriorates. Therefore, a tradeoff between theconnection probability and the coverage probability existswhen employing directional antenna in UAV networks andthe maximal coverage probability of the UAV networks isachieved at the optimal flying height by balancing betweensignal enhancement and interference mitigation. From Fig. 7we also observed that the coverage probability decreasesquickly for large receiving thresholds of the ground UEs.This is because, by adopting large receiving thresholds, thedistances between the ground UEs and their serving ABSsdecrease. Consequently, the ABSs at a large flying heighthave to overcome the large path loss, which reduces the

FIGURE 7. Coverage probability versus flying height of the first-tier ABSs fordifferent received thresholds of the ground UEs.

FIGURE 8. Coverage probability versus SIR threshold of the UAV networks fordifferent densities of the ABSs.

coverage probability of the UAV network.Fig. 8 shows the coverage probability as a function of the

SIR threshold for different densities of UAVs. From Fig. 8,we observe that the coverage probability decreases with theSIR threshold. Similar results for a two-tier terrestrial cellularnetwork have been reported in [38]. We note that, in thesame SIR threshold regime, the coverage probability of theconsidered two-tier UAV network is always larger than thatof the two-tier cellular network considered in [38]. This isbecause the directional antenna provides additional antennagains to overcome the path loss and, at the same time, reducethe impact of interfering ABSs. Moreover, as the density ofthe ABSs increases, the coverage probability decreases dueto the increased number of interfering ABSs.

VOLUME 4, 2016 13


FIGURE 9. Coverage probability versus SIR threshold of the UAV network fordifferent heights of the ABSs.

FIGURE 10. Coverage probability versus directivity factor of directionalantenna for different path loss exponents of NLoS transmission.

Fig. 9 compares the coverage probability of the consideredtwo-tier UAV network with the baseline one-tier UAV net-work as considered in [28]. For both networks, the coverageprobability is evaluated as a function of the SIR thresholdfor deploying the UAVs at different flight heights, wherethe path loss exponent is αN = 2.5. It can be seen fromFig. 9 that the coverage probabilities of the considered UAVnetworks shows the same tendency, which decreases with theSIR threshold. However, the proposed two-tier UAV networkwith directional antenna always outperform the baseline UAVnetwork for the considered SIR thresholds.

Finally, Fig. 10 shows the coverage probability as a func-tion of the antenna’s directivity factor for different path loss

exponents under NLoS transmission. From Fig. 10 we ob-serve that, for directional antennas, the coverage probabilityincreases with the antenna’s directivity factor, as the antennagain of the serving ABSs enlarges. When the directivity fac-tor is high enough, the radius of the disc reduces. In this case,both the serving ABS and the interfering ABSs are locatedclose to the center of the disc and the number of interferingABSs reduces. Consequently, the SIR increases with thedirectivity factor of antennas and application of directionalantennas leads to much higher coverage probability than thatof omni-directional antennas. From Fig. 10 we also observethat, with a large pass loss exponent, the UAV network canobtain a high coverage probability. This result implies thatUAVs equipped with directional antenna can achieve highcoverage probability even for propagation scenarios of largepath loss.

VII. CONCLUSIONS AND FUTURE WORKThis paper developed a novel analytical framework for eval-uating the distance distribution, connectivity probability, andcoverage probability of K-tier UAV networks that employdirectional antennas. To facilitate a tractable performanceanalysis, we introduced a simple elevation angle based an-tenna pattern model to capture the antenna gain provided bydirectional antennas. It was revealed that the directivity fac-tor can highly impact the connection probability, especiallywhen the UAVs deployed at low flying heights have a largeelevation angle. However, the coverage probability of K-tier UAV networks can be enhanced by adopting directionalantennas as the interference power caused from other ABSsreduced. Both the analytical and simulation results showedthat the application of directional antennas for UAVs at largeflying heights can provide excess antenna gain to overcomethe propagation pass loss and, at the same time, mitigate theimpact of interfering UAVs. These results demonstrated thehuge potential of employing directional antennas to enhancethe performance of multi-tier UAV networks. In the future,the application of directional antenna for uplink communi-cation and the associated performance evaluation of multi-tier UAV networks in the uplink are interesting extensions ofthis work. Moreover, selecting the optimal directivity factorof directional antennas for given density and flying height ofUAVs is another compelling research direction.

REFERENCES[1] Y. Zeng, R. Zhang, and T. J. Lim, “Wireless communications with un-

manned aerial vehicles: opportunities and challenges,” IEEE Communica-tion Magazine, vol. 54, no. 5, pp. 36–42, May 2016.

[2] M. Mozaffari, W. Saad, M. Bennis, Y. Nam, and M. Debbah, “A tutorial onuavs for wireless networks: Applications, challenges, and open problems,”IEEE Communications Surveys Tutorials, pp. 1–1, 2019.

[3] 3GPP Technical Report 36.777. Technical specification group radio accessnetwork; Study on enhanced LTE support for aerial vehicles (Release 15).Dec. 2017.

[4] “Paving the path to 5G: Optimizing commercial ltenetworks for drone communication,” available online:https://www.qualcomm.com/news/onq/2016/09/06/paving-path-5goptimizing-commercial-lte-networks-drone-communication.

14 VOLUME 4, 2016


[5] A. Fotouhi, H. Qiang, M. Ding, M. Hassan, L. G. Giordano, A. Garcia-Rodriguez, and J. Yuan, “Survey on UAV cellular communications:Practical aspects, standardization advancements, regulation, and securitychallenges,” IEEE Communications Surveys Tutorials, pp. 1–1, 2019.

[6] A. E. A. A. Abdulla, Z. M. Fadlullah, H. Nishiyama, N. Kato, F. Ono, andR. Miura, “An optimal data collection technique for improved utility inuas-aided networks,” Proc. IEEE INFOCOM, pp. 736–744, April 2014.

[7] H. Kim and J. Ben-Othman, “A collision-free surveillance system usingsmart UAVs in multi domain IoT,” IEEE Communications Letters, vol. 22,no. 12, pp. 2587–2590, December 2018.

[8] N. Zhao, W. Lu, M. Sheng, Y. Chen, J. Tang, F. R. Yu, and K. Wong,“UAV-assisted emergency networks in disasters,” IEEE Wireless Commu-nications, vol. 26, no. 1, pp. 45–51, February 2019.

[9] M. Alzenad, A. El-Keyi, and H. Yanikomeroglu, “3-D placement of anunmanned aerial vehicle base station for maximum coverage of users withdifferent QoS requirements,” IEEE Wireless Communications Letters,vol. 7, no. 1, pp. 38–41, February 2018.

[10] X. Wang, H. Zhang, Y. Tian, and V. C. M. Leung, “Modeling and analysisof aerial base station-assisted cellular networks in finite areas under losand nlos propagation,” IEEE Transactions on Wireless Communications,vol. 17, no. 10, pp. 6985–7000, October 2018.

[11] A. V. Savkin and H. Huang, “Deployment of unmanned aerial vehicle basestations for optimal quality of coverage,” IEEE Wireless CommunicationsLetters, vol. 8, no. 1, pp. 321–324, February 2019.

[12] H. He, S. Zhang, Y. Zeng, and R. Zhang, “Joint altitude and beamwidthoptimization for UAV-enabled multiuser communications,” IEEE Commu-nications Letters, vol. 22, no. 2, pp. 344–347, February 2018.

[13] J. Chen, J. Xie, Y. Gu, S. Li, S. Fu, Y. Wan, and K. Lu, “Long-rangeand broadband aerial communication using directional antennas (ACDA):Design and implementation,” IEEE Transactions on Vehicular Technology,vol. 66, no. 12, pp. 10 793–10 805, December 2017.

[14] D. He, T. Bouras, X. Chen, W. Yu, Y. Zhang, and Y. Yang, “3-D spatialspectrum fusion indoor localization algorithm based on CSI-UCA smooth-ing technique,” IEEE Access, vol. 6, pp. 59 575–59 558, 2018.

[15] A. Abdelreheem, E. M. Mohamed, and H. Esmaiel, “Location-basedmillimeter wave multi-level beamforming using compressive sensing,”IEEE Communications Letters, vol. 22, no. 1, pp. 185–188, Jane 2018.

[16] A. Abdelreheem, E. M. Mohamed, and H. Esmaiel, “Adaptive location-based millimetre wave beamforming using compressive sensing basedchannel estimation,” IET Communications, vol. 13, no. 9, pp. 1287–1296,2019.

[17] I. Bor-Yaliniz and H. Yanikomeroglu, “The new frontier in RAN hetero-geneity: Multi-tier drone-cells,” IEEE Communications Magazine, vol. 54,no. 11, pp. 48–55, November 2016.

[18] S.Sekander, H.Tabassum, and E.Hossain, “Multi-tier drone architecturefor 5G/B5G cellular networks: Challenges, trends, and prospects,” IEEECommunications Magazine, vol. 56, no. 3, pp. 96–103, March 2018.

[19] D. Kim, J. Lee, and T. Q. S. Quek, “Multi-layer unmanned aerial vehiclenetworks: Modeling and performance analysis,” arXiV:1904.01167v1.

[20] K. Venugopal, M. C. Valenti, and R. W. Heath, “Device-to-device mil-limeter wave communications: Interference, coverage, rate, and finitetopologies,” IEEE Transactions on Wireless Communications, vol. 15,no. 9, pp. 6175–6188, September. 2016.

[21] M. M. Selim, M. Rihan, Y. Yang, L. Huang, Z. Quan, and J. Ma, “Onthe outage probability and power control of D2D underlaying noma uav-assisted networks,” IEEE Access, vol. 7, pp. 16 525–16 536, 2019.

[22] J. Wang, C. Jiang, Z. Wei, C. Pan, H. Zhang, and Y. Ren, “Joint uavhovering altitude and power control for space-air-ground iot networks,”IEEE Internet of Things Journal, vol. 6, no. 2, pp. 1741–1753, April 2019.

[23] G. Zhang, Q. Wu, M. Cui, and R. Zhang, “Securing uav communicationsvia joint trajectory and power control,” IEEE Transactions on WirelessCommunications, vol. 18, no. 2, pp. 1376–1389, Feb. 2019.

[24] J. Zhang, L. Xiang, D. W. K. Ng, M. Jo, and M. Chen, “Energy efficiencyevaluation of multi-tier cellular uplink transmission under maximum pow-er constraint,” IEEE Transactions on Wireless Communications, vol. 16,no. 11, pp. 7092–7107, November 2017.

[25] T. Hou, Y. Liu, Z. Song, X. Sun, and Y. Chen, “Multiple antenna aidedNOMA in UAV networks: A stochastic geometry approach,” IEEE Trans-actions on Communications, vol. 67, no. 2, pp. 1031–1044, Feb. 2019.

[26] J. Kim, J. Park, S. Kim, S. Kim, K. W. Sung, and K. S. Kim, “Millimeter-wave interference avoidance via building-aware associations,” IEEE Ac-cess, vol. 6, pp. 10 618–10 634, 2018.

[27] X. Yu, J. Zhang, M. Haenggi, and K. B. Letaief, “Coverage analysisfor millimeter wave networks: The impact of directional antenna arrays,”

IEEE Journal on Selected Areas in Communications, vol. 35, no. 7, pp.1498–1512, July 2017.

[28] V. V. Chetlur and H. S. Dhillon, “Downlink coverage analysis for a finite3-D wireless network of unmanned aerial vehicles,” IEEE Transactions onCommunications, vol. 65, no. 10, pp. 4543–4558, October 2017.

[29] P. K. Sharma and D. I. Kim, “Random 3D mobile UAV networks: Mo-bility modeling and coverage probability,” IEEE Transactions on WirelessCommunications, vol. 18, no. 5, pp. 2527–2538, May 2019.

[30] M. Haenggi, “Stochastic geometry for wireless networks,” Cambridge,U.K.: Cambridge Univ. Press, 2013.

[31] Z. Wang, P. S. Hall, J. R. Kelly, and P. Gardner, “Wideband frequency-domain and space-domain pattern reconfigurable circular antenna array,”IEEE Transactions on Antennas and Propagation, vol. 65, no. 10, pp.5179–5189, October 2017.

[32] J. Kraus and R. J. Marthefka, “Antennas: For all applications,” McGraw-Hill, New York, 2002.

[33] A. Al-Hourani, S. Kandeepan, and S. Lardner, “Optimal LAP altitudefor maximum coverage,” IEEE Wireless Communications Letters, vol. 3,no. 6, pp. 569–572, December 2014.

[34] T. Bai and R. W. Heath, “Coverage and rate analysis for millimeter-wave cellular networks,” IEEE Transactions on Wireless Communications,vol. 14, no. 2, pp. 1100–1114, February 2015.

[35] Qualcomm, LTE Unmanned Aircraft SystemsałTrial Report, vol. Qual-comm Technologies, Inc., May 2017.

[36] V. Naghshin, M. C. Reed, and N. Aboutorab, “Coverage analysis ofpacket multi-tier networks with asynchronous slots,” IEEE Transactionson Communications, vol. 65, no. 1, pp. 200–215, Jan. 2017.

[37] L. Xiang, X. Ge, C. Wang, F. Y. Li, and F. Reichert, “Energy efficiencyevaluation of cellular networks based on spatial distributions of traffic loadand power consumption,” IEEE Transactions on Wireless Communication-s, vol. 12, no. 3, pp. 961–973, March 2013.

[38] H. Jo, Y. J. Sang, P. Xia, and J. G. Andrews, “Heterogeneous cellularnetworks with flexible cell association: A comprehensive downlink s-inr analysis,” IEEE Transactions on Wireless Communications, vol. 11,no. 10, pp. 3484–3495, October 2012.

JING ZHANG received the Ph.D. degree fromHuazhong University of Science and Technology(HUST) in 2002 and 2010, respectively. He iscurrently an associate professor with HUST. Hiscurrent research interests include unmanned aerialvehicle communications, green communications,device-to-device communications, and millimeter-wave communications.

HUAN XU received his undergraduate degreefrom Wuhan University of Technology and is cur-rently pursuing a master’s degree at HuazhongUniversity of Science and Technology. His currentresearch interests include green communication,stochastic geometry and UAV wireless networkperformance analysis.

VOLUME 4, 2016 15


LIN XIANG (S’14, M’18) received the Bach-elor and Master degrees in electronics and in-formation engineering from Huazhong Universityof Science and Technology (HUST), China, in2009 and 2012, respectively, and the PhD degreefrom Friedrich-Alexander-University of Erlangen-Nuremberg (FAU), Germany, in 2018. From Aug.2010 to Feb. 2011, he was an exchange student atUniversity of Bologna, Italy, under support fromErasmus Mundus programme. He is currently a

research associate with the Interdisciplinary Centre for Security, Reliabilityand Trust (SnT), University of Luxembourg, Luxembourg. His researchinterests include resource allocation for wireless communication systems,performance analysis of wireless networks based on stochastic geometrytheory, and renewable energy integration and electric vehicle charging insmart grid. Dr. Xiang was a recipient of the Best Paper Award for IEEEGlobecom 2010. He was an Exemplary Reviewer of the IEEE TRANS-ACTIONS ON WIRELESS COMMUNICATIONS in 2018 and the IEEETRANSACTIONS ON COMMUNICATIONS in 2017 and 2018.

JUN YANG received Bachelor and Master degreein Software Engineering from Huazhong Univer-sity of Science and Technology (HUST), Chinain 2008 and 2011, respectively. Then, he got hisPh.D degree at School of Computer Science andTechnology, HUST, on June 2018. Currently, heworks as a postdoctoral fellow at Embedded andPervasive Computing (EPIC) Lab in School ofComputer Science and Technology, HUST. Hisresearch interests include cognitive computing,

software intelligence, Internet of Things, cloud computing and big dataanalytics, etc.

16 VOLUME 4, 2016

on the application of directional antennas in multi-tier

Documents