On the Trade-offs between Coverage Radius, Altitude and Beamwidth for Practical UAV Deployments

Transcription

1 On the Trade-offs between Coverage Radius, Altitude and Beawidth for Pratial UAV Deployents Haneya Naee Qureshi, Student Meber, IEEE and Ali Iran, Senior Meber, IEEE Abstrat Current studies on Unanned Aerial Vehile UAV based ellular deployent onsider UAVs as aerial base stations for air-to-ground ouniation. However, they analyze UAV overage radius and altitude interplay while oitting or oversiplifying an iportant aspet of UAV deployent, i.e., effet of a realisti antenna pattern. This paper addresses the UAV deployent proble while using a realisti D diretional antenna odel. New trade-offs between UAV design spae diensions are revealed and analyzed in different senarios. The sensitivity of overage area to both antenna beawidth and height is opared. The analysis is extended to ultiple UAVs and a new paking shee is proposed for ultiple UAVs overage that offers several advantages opared to prior approahes. Index Ters Unanned aerial vehiles, aerial base stations, air-to-ground ouniation, D antenna odel, antenna beawidth I. INTRODUCTION The deand for ore diverse, flexible, aessible and resilient broadband servie with higher apaity and overage is on the rise. Soe of these requireents an be aoplished with UAVs ating as aerial base stations. This is beause of the several advantages UAV based ouniation offers suh as higher likelihood of line-of-sight LoS path and less satter and signal absorption as opared to terrestrial systes [], []. Moreover, the deand for inrease in apaity is leading towards deployent of sall ells in terrestrial networks, resulting in the need for higher ell ounts, leading to far larger nuber of ground sites. This akes the goal of attaining sealess overage over a wide geographial area through terrestrial systes unfeasible due to liited availability of suitable sites and loal regulations. This hallenge is likely to aggravate with advent of even saller Wave ells offering even ore sporadi overage []. Siilarly, satellite networks have their own liitations suh as high lateny, high propagation loss, liited orbit spae and high launhing osts [4]. On the ontrary, UAVs an be deployed quikly with uh ore flexibility to ove fro one point to another whih is a desirable feature for rapid, on-deand or eergeny ouniations [5]-[7]. UAVs an thus be seen as potential enablers to eet the several hallenges of next generation wireless systes by either funtioning as opleentary arhiteture with already existing ellular networks to opensate for ell overload during peak ties and eergeny situations [8],[9] or by serving as stand-alone arhiteture to provide new infrastruture, espeially in reote areas []-[]. In this doain, a new hybrid network arhiteture for ellular systes by leveraging the use of UAVs for data offloading is proposed in [9], []. Another signifiant appliation of UAVs is in the eerging Internet of Things IoT tehnology [4] and wireless sensor networks [5], where low altitude UAVs an provide a eans to ollet the IoT data fro devies with liited transit power and transit it to their intended reeivers. However, in order to fully reap the benefits of UAV based ouniation, optial design of UAVs deployent paraeters is of fundaental iportane. In this paper, we address the UAV deployent proble by analyzing the trade-offs between key syste design paraeters suh as height, antenna beawidth, and nuber of UAVs. By leveraging a ore realisti odel opared to prior studies on the topi, our analysis reveals several new insights and trade-offs between the design paraeters that reain unexplored in existing studies. A. Related Work Several studies have reently addressed UAV deployent for different servie requireents, ostly using altitude, transission power and nuber of UAVs as the only three deployent paraeters. For exaple, authors in [6] investigate the axiu overage and optial altitude assuing one UAV with no interferene. The optial altitude is estiated as a funtion of axiu allowed path loss and statistial paraeters of urban environent. However, this work is liited to a single UAV while using ean value of shadowing rather than its rando behavior and altitude as the only optiization paraeter to ontrol overage. In [7], authors deterine optial height for axiu overage for a single UAV based on overage probability and the inforation rate of users on the ground at a partiular UAV altitude. Authors in [8] extend the work in [6]-[7] to two UAVs, with and without interferene. Based on the path loss odels in [6] and [9], optial altitude is reported in [8] for both axiu overage and iniu required transit power. Continuing to analyze altitude versus overage radius relationship, in [], the sae tea of authors address the deployent proble with oexistene between UAV and under laid Devie-to-Devie DD ouniation networks. Apart fro overage area, other perforane indiators are also affeted by hanges in UAV height, suh as arrier to interferene ratio and handovers. Fousing on -wave band, authors in [] investigate overage versus arrier to interferene ratio patterns using an antenna pattern approxiated by a osine funtion raised to a power. Building upon the work in [], the effet of lateral displaeent of a UAV on interferene and handovers is studied in []. Authors in [] easure Reeive Signal Strength Indiator RSSI for three UAV based ellular networks using following odels:

2 Okuura-Hata, COST-Hata, and COST Walfish-Ikegai. It is reported that signal strengths derease faster with inrease in altitude. However, this work onsiders UAVs up to an altitude of 5 beause of their path loss odels onstraints. Fousing on just the altitude as a deployent paraeter, study in [4] investigates the altitude estiation of UAVs fro a ore pratial perspetive using easureents of the polarization of agneti field of low-frequeny radio signals. Authors in [5] estiate the relative attitude between two ouniating UAV nodes where the nodes are equipped with MIMO antenna arrays with diverse polarization. Other works that study UAV deployent fro the perspetive of optial altitude and overage radius inlude [6]-[8]. The proposed algorith in [6] finds the optial altitude of UAV based on the desired radius of overage in real tie. Another deployent odel is onsidered in [7], in whih the overage area is alulated nuerially by onsidering the altitude of UAV and the loation of both UAV and users in the horizontal diension. Results show that the size of the overage area is affeted by the environent. Ideas introdued in [7] are further elaborated in [8] leading to the onlusion that larger buildings require a higher UAV altitude. Additionally, [8] addresses network overage through a ognitive relay node network odel with a goal to enhane the perforane of standard relay nodes. Optial trajetory designs are studied in [9] by onsidering a onstant UAV height. Joint optiization of UAV s trajetory, as well as the bandwidth alloation and user partitioning between the UAV and ground base stations is analyzed in [9] and []. The authors in these studies ai to axiize the iniu throughput of all obile terinals in the ell and onsider orthogonal spetru sharing between the UAV and ground base stations. The fraework is extended to the spetru reuse ase and results show that the proposed hybrid network with optiized spetru sharing and ylial ultiple aess design signifiantly iproves the spatial throughput over terrestrial networks, while the spetru reuse shee an provide further throughput gains opared to orthogonal spetru sharing. Other UAV trajetory designs are onsidered in []-[]. Authors in [] ai to design the UAV trajetory to iniize its ission opletion tie, while ensuring that eah ground station suessfully reovers the file with a desired high probability. Optiization of ultiuser ouniation sheduling and assoiation jointly with the UAV s trajetory and power ontrol is addressed in []. Two other pratial types of UAV trajetories, naely irular flight and straight flight are onsidered in [] in order to haraterize the energy trade-off in ground-to-uav ouniation. Another interesting trade-off between throughput and delay fro the perspetive of trajetory optiization is studied in []. In this work, a new ylial ultiple aess shee is proposed to shedule the ouniations between the UAV and ground terinals in a ylial tie-division anner based on the flying UAV s position. Under this shee, the authors in [] reveal a fundaental trade-off between throughput and aess delay. UAV based relay network optiizations are studied in [4]- [6]. Copared with onventional stati relaying, authors in [6] onsider obile relaying, whih offers a new degree of freedo for perforane enhaneent via areful relay trajetory design. The authors in this work show that by optiizing the trajetory of the relay and power alloations adaptive to its indued hannel variation, obile relaying is able to ahieve signifiant throughput gains over the onventional stati relaying. Authors in [7] onsider the ipat of antenna power roll-off in G networks for fixed platfor height and propose a gain adjustent strategy for irular bea antennas. However, none of the aforeentioned studies [6]-[7] onsider the ipat of antenna gain pattern on the overage versus height trade-off. One reent study that takes into aount effet of diretional antenna onsiders joint altitude and beawidth optiization for UAV-enabled ultiuser ouniations [8]. In this study, users are partitioned into disjoint lusters and the UAV sequentially serves all lusters by hovering above the luster enters one by one. In [8], the authors onsider three ouniation odels: downlink ultiasting, downlink broadasting and uplink ultiple aess. One distinguishing feature of [8] is the onlusion that optial beawidth and height ritially depend on the ouniation odel onsidered. However, this study uses a step-wise antenna gain odel for analytial tratability and line-of-sight propagation onditions. Another reent study [9] that does onsider the effet of antenna also uses a step-wise antenna gain odel with only two possible values of antenna gain. Analysis inorporating realisti antenna odel has beoe ore iportant sine several studies are already onsidering ipleentations of diretional antenna in UAV-based ellular systes [4], suh as sart WiFi diretional antennas with servo otors [4]. While the UAV deployent proble has been investigated in a large nuber of reent studies as disussed above, to the best of authors knowledge, this is the first paper to study the optiization of UAV deployent design paraeters while using a realisti D diretional antenna odel in the syste. The analysis presented in this paper shows that the use of a realisti antenna pattern akes a trend shifting differene in the height versus overage trade-off and adds a new diension of beawidth to the UAV deployent design spae that reains unexained in earlier studies. B. Contributions and Organization The ontributions and organization of this paper an be suarized as follows: We develop a atheatial fraework for UAV deployent design while inorporating a realisti odel for a pratial diretional antenna. Current studies on UAV deployent either ignore the effet of D diretional antenna [6]-[5] or onsider an over-siplified odel for antenna gain [8][9]. Therefore, UAV deployent analysis presented in suh studies, yields results on optial height, overage radius and nuber of UAVs that ay not hold for real UAV deployents with pratial diretional antenna. We address this proble by using GPP defined D paraboli antenna pattern whose gain is realistially dependent on not only beawidth but also three diensional elevation angle. Setion II and III-A

3 We derive analytial expressions for overage haraterized by reeived signal strength RSS as a funtion of height, beawidth and overage radius. Setion II and III-B We present a atheatial fraework to quantitatively analyze trade-offs aong the following paraeters: i ell radius versus beawidth for varying heights, ii ell radius versus height for different beawidths and iii beawidth versus height for different overage radii. To the best of authors knowledge, this paper is first to investigate this interesting interplay aong the five key fators that define UAV based overage design spae: antenna beawidth and angular distane dependent gain, elevation angle dependent probability of line of sight, shadowing, free spae path loss and height. Setion IV- A We investigate the ipat of key UAV design paraeters on RSS and validate our derived expressions for probability density funtion PDF of RSS through siulations. Setion IV-B, IV-C and IV-D The proposed fraework is extended to a range of frequenies and environents. Setion IV-C Prior works on UAV deployent design [6], [7] [9] use UAV altitude as the only optiization paraeter to ontrol overage. Contrary to the findings fro these prior studies, based on our joint analysis of effet of beawidth and altitude on overage, we show that: There exists an optial beawidth for given height for axiu overage radius and vie versa. We also derive an expression for deterining optial beawidth/height for desired overage radius. Antenna beawidth is a ore pratial design paraeter to ontrol overage instead of UAV altitude. This is onluded by perforing oparative analysis of the two by quantifying the sensitivity of overage to both height and beawidth. Setion IV- E Contrary to what has been assued ipliitly or expliitly in prior studies, UAV altitude an not be optiized independent of antenna beawidth. In fat, both paraeters need to be optiized in tande with eah other to plan true overage. Coverage probability patterns with varying tilt angles and asyetrial beawidths are presented in Setion IV-F, whih highlight the apability of our derived equations and the underlying syste odel to extend the analysis to a wide range of senarios, suh as non-zero tilt angle and asyetrial beawidths. We also extend the analysis to ultiple UAVs. Soe reent studies have leveraged irle paking theory to deterine the nuber of UAVs needed to over a given area [9]. However, this approah has two aveats: It leaves signifiant overage holes when two or ore UAVs are used to over an area. The nuber of UAVs inrease draatially with inrease in required overage probability. To iruvent the probles posed by irle paking theory, we propose use of hexagonal paking and MS d! MS Referene Text axis axis for tilt for tilt ϕ tilt ϕ MS! a h UAV Fig. : Syste odel. r Referene axis Referene Text for aziuth axis for aziuth opare our results with that obtained by irle paking. This oparison identifies several further advantages of proposed approah. Setion IV-G Continuing our analysis on ultiple UAVs, we deterine the optial beawidth for different nuber of UAVs that yields axiu total overage for a target geographial area Setion IV-G. Results show that proposed ulti- UAV deployent fraework an eet sae overage requireents with less infrastruture nuber of UAVs opared to existing odel [9]. The key findings of the paper are onluded in Setion V. II. SYSTEM MODEL We onsider a syste odel illustrated in Fig.. The UAV resides at a height h and projets a ell with overage radius r when φ tilt = o. We define UAV overage area as a set of points in irle of radius r, where a obile station MS experienes a RSS, S r above a threshold, γ. Here r is easured fro the projetion of UAV on ground. In Fig., φ tilt is the tilt angle in degrees of the antenna ounted on UAV, φ MS is the vertial angle in degrees fro the referene axis for tilt to the MS. θ a is the angle of orientation of the antenna with respet to horizontal referene axis i.e., positive x-axis and θ MS is the angular distane of MS fro the horizontal referene axis. Utilizing the geoetry in Fig., the pereived antenna gain fro the UAV using a three diensional antenna odel reoended by GPP [4], at the loation of MS an be represented as in. Here B θ and B φ represent the horizontal half power beawidth with respet to θ diretion and the vertial half power beawidth with respet to φ diretion of the UAV antenna in degrees respetively while λ θ and λ φ represent the weighting fators for the bea pattern in both diretions respetively. G ax and A ax denote the axiu antenna gain in db at the boresight of the antenna and axiu attenuation at the sides and bak of boresight 9 respetively. G ax an be approxiated as log B φ B θ [4]. The air-to-ground hannel an be haraterized in ters of probabilities of LoS and non-line-of-sight NLoS senarios between the UAV and MS. Prior studies have used hannel odels proposed in [9], [44]-[45]. The hannel odels proposed in [9] and [44] are suited to only dense urban

4 4 φms φ tilt θms θ a Gφ MS, φ tilt, θ MS, θ a, B φ, B θ = λ φ G ax in, A ax + λ θ G ax in, A ax B φ TABLE I: Key sybol definitions. Sy. Units Definition Sy. Units Definition h height of UAV P l - probability of LoS senario r overage radius P n - probability of NLoS senario d UAV to MS distane /s speed of light X l db RV for loation variability in LoS senario X n db RV for loation variability in NLoS senario σ l db standard deviation of X l σ n db standard deviation of X n φ tilt o tilt angle of antenna P L ax db axiu path loss φ MS o vertial angle fro tilt referene axis to MS θ a o angle of antenna w.r.t horizontal referene axis f GHz frequeny X s db RV shadowing θ MS o angular distane of MS σ sn db standard deviation of S n B φ o db beawidth with respet to φ diretion B θ o db beawidth with respet to θ diretion µ n db ean of X n µ sh db ean of X s σ n db standard deviation of X n σ sh db standard deviation of X s λ φ - weighting fator for bea pattern in φ λ θ - weighting fator for bea pattern in θ diretion diretion R t k radius of desired geographial area ɛ - iniu overage probability G dbi antenna gain γ db reeived signal threshold R l db reeived signal strength in LoS senario R n db reeived signal strength in NLoS senario S n db produt of reeived signal strength in NLoS senario and probability of NLoS senario at any r: S n = P nr n S l db produt of reeived signal strength in LoS senario and probability of LoS senario at any r: S l = P l R l B o beawidth for irular bea pattern G ax db axiu antenna gain at boresight S r db reeived signal strength at any r: S db reeived signal strength inside a geographial S r = S l + S n region P ov - overage probability µ s - ean of S r A ax db axiu antenna attenuation at sides and bak of boresight σ sl σ s db db standard deviation of S l standard deviation of S r T db transit signal f S s - PDF of S N - nuber of UAVs f Sr s r - PDF of S r B θ and typial European ities, respetively. Moreover, hannel odels presented in [9] and [44] lak easureent based validation. On the other hand, the hannel odel in [45] not only provides a siulation based data for a diverse range of elevation angles, environents and frequenies, but also has been validated through extensive epirial easureents. Hene, we use the UAV hannel odel proposed in [45] to estiate the probability of LoS senario as follows:.j k P l φ MS =.j + 9 φ MS l n where j,...n are the set of epirial paraeters for different types of environents and are given in Table II. The angle, 9 φ MS = tan h r is the angle of elevation of the MS to the UAV. The probability of NLoS senario is then P l φ MS. In addition to free spae path loss, UAV-MS signal faes an elevation angle dependent shadowing. The ean and standard deviation for this shadowing an be odeled as [45]: µ sh = p µ + 9 φ MS q µ + t µ 9 φ MS σ sh = p σ + 9 φ MS q σ + t σ 9 φ MS 4 where p µ, q µ, t µ, p σ, q σ and t σ are paraeters obtained fro epirial easureents given in Table III. In Table III, the subsript v = {µ, σ}, is used to indiate that the paraeters are for ean and standard deviation, respetively. The RSS in LoS and NLoS senarios, as a funtion of path loss and antenna gain an now be represented as: 4πfd R l φ MS, φ tilt, θ MS, θ a, B φ, B θ, d = T log +Gφ MS, φ tilt, θ MS, θ a, B φ, B θ X l 5 4πfd R n φ MS, φ tilt, θ MS, θ a, B φ, B θ, d = T log +Gφ MS, φ tilt, θ MS, θ a, B φ, B θ X n X s 6 where T is transitted power, is the speed of light, f denotes the frequeny, d is the distane between UAV and MS and X s is shadow fading Gaussian N µ sh, σ sh rando variable RV with ean µ sh and standard deviation σ sh. X s is part of the reeived signal strength in NLoS senario only beause shadowing is a phenoenon assoiated exlusively with NLoS senario due to the presene of obstales in NLoS senario whih affet wave propagation [45]. For realisti syste level odeling of obile systes, rando oponents in db, X l and X n are added as environent dependent variables in LoS

5 5 TABLE II: Environent dependent paraeters for P l. Suburban Highrise urban j.6 5. k -.7 l n TABLE III: Frequeny dependent paraeters for shadowing. f GHz p v q v t v v = µ v = σ v = µ v = σ v = µ v = σ and NLoS senarios [45]. Note that in LoS senario, despite having a diret path between the UAV and MS, refletions fro satters in the surrounding of the MS an result in different signal strength at different MS loations even when the MS loations are at sae distane fro the UAV and have LoS. This loation dependent randoness in the reeived signal in LoS senario is aptured in the for of rando variable X l with log-noral distribution of ean zero. X l and X n are therefore, N, σ l and N, σ n RVs, where σ l and σ n denote the standard deviations, in db, of X l and X n respetively. III. UAV COVERAGE MODEL DEVELOPMENT A. Coverage Probability As desribed in Setion II, we define UAV overage area as a set of points in irle of radius r, where a MS experienes a RSS, S r above a threshold, γ. Then, the RSS at the boundary exeeds a ertain threshold, γ = T P L ax with a probability, P ov ɛ, where P L ax is the axiu allowable path loss. We define this overage probability, P ov as: P ov = P[S r γ] = P l φ MS P[R l φ MS, φ tilt, θ MS, θ a, B φ, B θ, d γ]+ P n φ MS P[R n φ MS, φ tilt, θ MS, θ a, B φ, B θ, d γ] 7 First, we alulate P[R n φ MS, φ tilt, θ MS, θ a, B φ, B θ, d γ] using 6 as in 8-, where is a result of opleentary uulative distribution funtion of a Gaussian rando variable and substituting γ = T P L ax in 9 and Gφ MS, φ tilt, θ MS, θ a, B φ, B θ is defined in. Without loss of generality, we assue X n and X s to be independent RVs and thus X n = X n + X s with ean µ n = µ sh and standard deviation, σ n = σ sh + σ n. Siilarly, we derive P[R l φ MS, φ tilt, θ MS, θ a, B φ, B θ, d γ], whih yields the following expression: P [R l φ MS, φ tilt, θ MS, θ a, B φ, B θ, d γ] = Q log 4πfd Gφ MS, φ tilt, θ MS, θ a, B φ, B θ P L ax π z σ l where Qz = exp u du. Moreover, fro ell geoetry in Fig. under the assuption: height of MS << h and the UAV is loated at oordinates x, y, z =,, h, where x and y are the artesian oordinates of MS on ground, φ MS and d an be expressed as: φ MS = tan x + y, d = h h + x + y The probability of overage at a partiular loation of user on ground an now be found by substituting, and in 7, whih yields the expression given in. However, sine UAVs are obile and an rapidly ove fro one loation to another to provide on-deand overage where needed, for ost pratial senarios, a irular overage pattern is onsidered, rather than tapering with antenna tilt. A speial ase of, in whih the horizontal and vertial antenna beawidths are syetri i.e., B φ = B θ = B and φ tilt = results in irular overage footprint of the reeived signal strength on ground. Note that with φ tilt =, the aziuth plane beoes perpendiular to the boresight and the seond part of is no longer appliable. In this senario, r = x + y, φ MS = tan r, d = h h + r 4 Further, G ax an be approxiated as log 9 B [4] and A ax an be ignored without ipating the required auray of this antenna odel that ainly onerns gain on and around the boresight. Applying these siplifiations to, substituting and 4 in - and then aking use of 7 yields the expression for P ov as a funtion of antenna beawidth, UAV height and overage radius in 5. B. Reeived Signal Strength One way to investigate the RSS on a partiular loation on ground for a given height and antenna beawidth for irular overage pattern is by evaluating the expeted value of S r over the rando variables X s, X l and X n as follows E[S r ] k = P l E[S r LoS] + P n E[S r NLoS] = P l E[R l R n ] + E[R n ].j k 4πfd =.j tan + h n µ sh log r l tan r + T h φtilt 9 + log B B 6

6 [ P [R n φ MS, φ tilt, θ MS, θ a, B φ, B θ, d γ] = P T + Gφ MS, φ tilt, θ MS, θ a, B φ, B θ log = = Q γ exp X n πσ n log 4πfd 4πfd T + Gφ MS, φ tilt, θ MS, θ a, B φ, B θ log + µ sh Gφ MS, φ tilt, θ MS, θ a, B φ, B θ P L ax σ sh + σ n σ n ] X n X s γ 4πfd µ n 6 8 dx n 9.j k log 4πfd Gφ MS, φ tilt, θ MS, θ a, B φ, B θ P L ax P ov φ MS, φ tilt, θ MS, θ a, d, B θ, B φ =.j n Q φms l σ l.j k.j + n Q log 4πfd + µ sh Gφ MS, φ tilt, θ MS, θ a, B φ, B θ P L ax + 9 φms l σ sh + σn log 4πf h Povr,.j k h, B =.j n Q +r log 9 tan B + h φ r tilt B P L ax + σ l + tan h r l.j k.j + n Q + tan h r l log 4πf h +r where k is a result of the law of total expetation and P l, R l and R n are funtions of r, h and B and defined in, 5 and 6 respetively. 6 is therefore obtained by substituting R l and R n fro 5 and 6 and then aking use of the relationship between P l φ MS and P n φ MS, i.e., P n φ MS = P l φ MS. However, R l is a rando variable due to the rando oponent, X l and R n is a rando variable due to the rando oponents, X n and X s. Therefore, RSS, an also be odeled as a rando variable. In order to derive an analytial expression for the PDF of RSS at any arbitrary ell loation, we express it as: S r h, r, B = S l h, r, B + S n h, r, B 7 where S l and S n are independent rando variables, given by S l h, r, B = P l φ MS R l h, r, B and S n h, r, B = P n φ MS R n h, r, B. Then, in order to derive an analytial expression for PDF of S r, we resort to transforations of rando variables and onvolution of the PDFs of S l and S n, resulting in the expression in 8. Coplete derivation of 8 is provided in Appendix. The noralized PDF of reeived signal strength inside a geographial region denoted by S by assuing that the UAV resides at oordinates x, y, z =,, h where h is height of UAV, an then be found as follows: f S s = f Sr s, x, ydxdy 9 A A in soe geographial region A that lies in the xy-plane. The integral in 9 an be solved through nuerial ethods. + µ sh log 9 B + σ sh + σn tan r h φ tilt B IV. NUMERICAL RESULTS AND ANALYSIS P L ax 5 A. Trade-offs between overage radius, beawidth and height Coverage Radius vs. Beawidth: As the height of UAV inreases for a partiular MS loation, φ MS in 4 dereases angle of elevation inreases, leading to an inrease in probability of LoS link in, derease in shadowing in 4 and inrease in free spae path loss as d inreases. While the effet of these fators on overage has been studied in earlier studies [6]-[7], the ipat of the fourth fator, antenna gain in onjuntion with these three fators reained unexained. Fig. shows that ipat of this fourth fator is so profound that at a given ell radius, it results in a height h, after whih the antenna gain trend with inreasing beawidth reverses. Fig. is plotted utilizing for φ tilt = at r = 5 under the assuptions stated in Setion III. The larger antenna gain at a given r is observed with inreased height beause, for the sae MS loation, φ MS in 4 dereases. For any two beawidths, B and B, h is the point of intersetion of gain versus height graphs illustrated in Fig. for r = 5. h an be alulated as follows: 9 tan r log B h = r tan B h 5 log 9 = log B B B B B B B tan r B h The overage radius versus beawidth trend for different heights in Fig. an now be analyzed in light of the aforeentioned fators. This figure is plotted by utilizing 5 for

7 7 f Sr s r = exp.j k s r.j+ tan h/r l + πσl n σl.j k tan h/r l +.j k tan h l + r n pµ+tan h/r qµ+tµ tan h/r n + + 4πf T +log.j k tan h/r l +.j k tan h l + r h +r log 9 B + n σn + pσ +tan h/r qσ +tσ tan h/r n σn + p σ+tan h r q σ+t σtan h r tan r/h B 8 Antenna Gain G [dbi] B= o B=4 o B=55 o B= o Height h [] Fig. : Antenna gain at r = 5 with varying heights for different beawidths. Coverage Radius r [] Beawidth B [ o ] h' h= h=5 h= h= h=5 h=5 h=7 h= h=5 h=5 Fig. : Coverage radius against beawidth for different heights. P L ax =5 db, φ tilt = o, ɛ =.8 and f = GHz in a suburban environent. The trend shift in Fig. is attributed to the following: as height inreases up to, the inrease in antenna gain and derease in shadowing offsets the inreased free spae path loss. As height inreases beyond, the inrease in antenna gain and derease in shadowing is overshadowed by the inrease in free spae loss. As a result, the overage radius inreases with beawidth, approahes to a axiu value and then starts to derease. Our analysis quantifies this axiu value of overage radius in relation to antenna gain, for instane, axiu overage radius of 5 at a beawidth of 55 o for h = 7 in Fig. an be attributed to the ourene of largest antenna gain at a beawidth of 55 o at a height of 7 in Fig.. Our UAV overage odel for the first tie shows the existene of optial beawidth for given height for axiu overage radius and vie versa, a trend that reains hidden in UAV overage odels presented in prior studies [6]-[9]. Coverage Radius vs. Height: Fig. 4 depits the relation of overage radius with height for different beawidths. Initially, as the height of the UAV inreases for sall beawidths, overage radius also inreases ontinuously. However, as beawidth inreases further, overage radius Radius r [] o 5 o 5 o 5 o 5 o 5 o 65 o 8 o o 4 o 7 o 5 o siulation Height h [] 4 Fig. 4: Coverage radius against height for different beawidths. Beawidth B [ o ] r= r= r=4 r= Height h [] 4 Fig. 5: Beawidth against height for different radii. versus height urves attain a paraboli shape. This is in ontrast to previous studies that suggest onotoni inrease of UAV overage radius with altitude [6] [9]. Thus, our analysis brings forth these new insights as it aptures the relative effet of all four fators that ipat the overage radius onurrently: angular distane dependent realisti nonlinear antenna odel, elevation angle dependent probability of line of sight, shadowing, and a easureent baked path loss odel. Height vs. Beawidth: There an be senarios where height of UAV is subjet to hanges due to fators beyond the syste designer s ontrol suh as weather, but the sae overage pattern has to be aintained. Our proposed odel provides a ehanis to address suh senarios by haraterizing the height versus beawidth relationship as shown in Fig. 5. For exaple, if a UAV is deployed to over r f = 4 with ɛ =.8 at a height of and if its height is hanged to 5, the UAV will need to adjust its beawidth fro 7 o to either 4 o or o in order to ontinue providing the sae overage. Fig. 5 also highlights the iportane of optiizing both beawidth and height in tande with eah other rather than independently as has been the ase in prior works [6] and [9].

8 8 Mean Reeived Signal Strength S [db] Radius r [] Height h [] Fig. 6: Mean reeived signal with varying height for B = 5 o. Mean Reeived Signal Strength S [db] Radius r [] 5 Beawidth B [ o ] Fig. 7: Mean reeived signal with varying beawidth at h = 5. B. Ipat of altitude, beawidth and radius on RSS In order to investigate the behavior of RSS in ontext with UAV design paraeters, Fig. 6 and Fig. 7 illustrate how the ean RSS varies with height and beawidth with inreasing ell radius r. In Fig. 6, ean RSS is plotted for B = 5 o, f = GHz, P t = 4 db and φ tilt = o in a suburban environent. As expeted, RSS dereases as r inreases for any fixed height. However, for any r, RSS initially inreases with height, reahes to a peak value and then dereases as height inreases further. The senario in Fig. 7 onsists of a UAV deployed at a height of 5. Sine a narrow beawidth an only over a sall overage area, the derease in RSS with inreasing radius is very rapid at low beawidths. Hene, the trend of RSS with beawidth is ore learly depited in Fig. 8a-8, whih are zooed-in plots of Fig. 7 for o B < 8 o. The ontinuous derease of overage radius with inrease in beawidth for r < 8 in Fig. 8a an be attributed to the ontinuous derease of antenna gain with radius in Fig. 9 for r < 8. As we approah loser to, the trend in antenna gain pattern starts to reverse for different beawidths, and we observe several interseting points fro < r < 5 in Fig. 9 and hene ean RSS in Fig. 8b attains a paraboli shape with inreasing beawidth in this range. At very large r, antenna gain trend reverses opletely and now it inreases with inreasing beawidths, whih is again in line with the RSS trend in Fig. 8. Note also that at very large overage radius, for exaple, at r = 4 in Fig. 8, RSS dereases very sharply fro 5 o to o as opared to derease in RSS fro o to 8 o. This is beause at 4 in Fig. 9, differene in gain between 5 o and o is quite high as opared to differene in gain between o and 8 o. C. Analysis for different frequenies and environents In Fig., we quantify the trade-off of overage radius with beawidth in hanging environents and at different frequenies. The figure is plotted for h =, P L ax = db for two extree environents, suburban and high rise urban. Fro Fig., we observe that in addition to overage radius dereasing sharply as frequeny inreases or environent beoes denser, the beawidth at whih this overage radius versus beawidth trend hanges is also lower in a ore denser environent or at a higher frequeny. For exaple, in high rise urban environent, derease in radius starts fro a beawidth of as low as o at. GHz, whereas for the sae frequeny, this derease does not start until o in ase of the suburban environent. Siilar observations an be ade by observing the effet of frequeny in the sae environent. This is not just beause of free spae path loss whih inreases with inreasing frequeny, but also beause of shadowing, whih inreases at higher frequenies for the sae environent [45]. In addition, the ipat of frequeny on overage radius redues as environent beoes denser. These observations ould play a valuable role for designing UAV based ellular systes at higher frequenies by utilizing the unused part of higher frequeny spetru suh as Wave. The analytial results are also orroborated with siulation results in Fig.. D. Validation of analysis with Monte Carlo Siulations The trade-offs between overage radius, beawidth and height presented in the preeding setions have already been verified through onte-arlo siulations as shown in Fig. 4 and Fig.. Next, we orroborate the results derived for RSS odel in 7-9 via siulations. Fig. and Fig. show PDF of RSS at two arbitrary points, loated at r = for UAVs deployed at height of h = and beawidths of B = 5 o and 5 o respetively. The RSS PDF obtained fro our derived analytial expression in 8 shows an exellent fit to siulation based RSS data. In Fig., users loated at r = reeive extreely low RSS sine a sall beawidth of 5 o an over only a sall area. In ontrast, when beawidth is inreased to 5 o, the sae users start to reeive a uh better overage i.e., between -85 to -65 db. We now extend our analysis fro RSS at any arbitrary point to RSS inside a geographial area of 8 8. Noralized histogras of RSS fro the siulation results and analytial PDFs fro 9 are in agreeent as shown in Fig. -4. Fro Fig. -4, we note that not only the range of RSS beoes narrower in a given area, but also the distribution of RSS approahes zero skewed Gaussian with either inreasing height or inreasing beawidth. E. Coparison of altitude and beawidth to ontrol overage Noting that both beawidth and height an be used for the sae purpose of ontrolling overage leads us towards oparing both of these paraeters by analyzing the sensitivity of overage radius to eah of these paraeters.

9 9 Mean Reeived Signal Strength S [db] Radius r [] 8 6 Beawidth B [ o ] Mean Reeived Signal Strength S [db] Radius r [] Beawidth B [ o ] Mean Reeived Signal Strength S [db] Radius r [] Beawidth B [ o ] a < r < 8 b 8 < r < < r < 5 Fig. 8: Mean reeived signal on ground with varying beawidths > o Antenna Gain G [dbi] B= o B=5 o B=8 o B= o Coverage radius r [] Fig. 9: Antenna gain at h = 5 with varying radius for different beawidths. Coverage Radius r [] Suburban, f=.ghz analytial Suburban, f=.ghz siulation Suburban, f=.5ghz analytial Suburban, f=.5ghz siulation Suburban, f=5.5ghz analytial Suburban, f=5.5ghz siulation Highrise Urban, f=.ghz analytial Highrise Urban, f=.ghz siulation Highrise Urban, f=.5ghz analytial Highrise Urban, f=.5ghz siulation Highrise Urban, f=5.5ghz analytial Highrise Urban, f=5.5ghz siulation Beawidth B [ o ] Fig. : Coverage radius against beawidth for varying frequeny and environent at h =. Probability Density f Sr s r.5..5 siulated data analytial expression Reeived Signal Strength S r [db] Fig. : PDF of RSS at r =, B = 5 o and h =. Fig. 5-6 show the gradient of radius with respet to beawidth obtained by differentiating 5 with respet to beawidth or deterining gradient of the urves in Fig. with respet to beawidth. By oparing the absolute values of r/ B in Fig. 5-6, we onlude that the hange in overage radius is ost sensitive to very high heights upto 5. Probability Density f Sr s r.5..5 siulated data analytial expression Reeived Signal Strength S r [db] Fig. : PDF of RSS at r =, B = 5 o and h =. Next, we analyze the rate of hange of radius with height r/ h in Fig. 7. Unlike Fig.5, the values of derivatives here follow a siilar pattern with hanging beawidths exept for beawidths < 5 o where r/ h is onstant. By oparing values of the derivative in Fig. 7 with Fig. 5-6 both oparisons range fro saples of heights fro between to 5 and beawidths fro o to 8 o, we note that ax r/ B >> ax r/ h. Nuerially, the axiu value of ax r/ B, - is alost 75 ties greater than the axiu value of ax r/ h, -7. This indiates that low beawidths lead to greatest hange derease in overage radius per unit beawidth as opared to hange in radius per unit height. This analysis an be leveraged to hoose between the height and beawidth or design appropriate obination of the two for optiizing overage. F. Coverage probability with varying tilt angles and asyetri beawidths Coverage analysis of senarios with varying tilt angles and asyetrial beawidths is presented in this setion. In Fig. 8, we opare the probability of overage using zero antenna tilt with a tilt angle of o. For non-zero tilt angles, the overage pattern fors an off-entered ellipse shape rather than a irle entered at the origin. The overage probability in ase of non-zero tilt angle is evaluated by using. Fig. 9 shows how the overage probability hanges with different values of B φ and B θ. The effet of hanging tilt values is depited in Fig.. It is observed that although a UAV with antenna tilt of 8 o overs ore area as opared to tilt angle of o, the axiu overage probability with φ tilt = 8 o is redued by half as opared to φ tilt = o.

10 -. siulation analytial expression Probability Density f Ss siulation analytial expression Probability Density f Ss Probability Density f Ss Reeived Signal Strength S [db] siulation analytial expression Reeived Signal Strength S [db] a h = Reeived Signal Strength S [db] b h = 5 h = siulation analytial expression Reeived Signal Strength S [db] Probability Density f Ss siulation analytial expression.4 Probability Density f Ss Probability Density f Ss Fig. : PDF of RSS on ground with hanging altitude of UAV for B = 5o Reeived Signal Strength S [db] a h = 5 siulation analytial expression Reeived Signal Strength RSL [db] b h = 5 h = 55 8 Derivative of Radius w.r.t Height r/ h Derivative of Radius w.r.t Beawidth r/ B [/o] Fig. 4: PDF of RSS on ground with hanging altitude of UAV for B = 5o h= h=5 h= h= h= B=o B=5o B=5o o B=5 B=5o B=5o B=65o o B=8 B=4 o B=7 o -5.5 Beawidth B [ ] Fig. 5: Gradient of overage radius with respet to beawidth. Derivative of Radius w.r.t Beawidth r/ B [/o].5 Height h [] o.5 4 Fig. 7: Gradient of overage radius with respet to height h=5 h=7 h= h=5 h= a φtilt = o 9 b φtilt = o, θa = 45o Fig. 8: Coverage probability at B = 5o and h = 5. o Beawidth B [ ] Fig. 6: Gradient of overage radius with respet to beawidth for different heights in low beawidth regie. G. Coverage Analysis with Multiple UAVs These figures highlight the apability of our derived equations and the underlying syste odel to extend the analysis to a wide range of senarios, suh as non-zero tilt angle and asyetrial beawidths. Coplete analysis of UAV syste design in suh extended senarios to provide an even ore flexible and on-deand ellular overage an be fous of a future study. Previous literature [9], utilizes irle paking theory to deterine the nuber of UAVs to ahieve a gain in overage probability in a ertain geographial area. In the irle paking proble, N idential irles ells are arranged inside a larger irle target area of radius Rt suh that the paking density is axiized and none of the irles overlap [46]. The radius of eah of the N irles that solves this proble is denoted by rax and one UAV provides overage to one sall ell ir-

11 a Bφ = o, Bθ = 7o b Bφ = 7o, Bθ = o Fig. 9: Coverage probability with asyetrial beawidths at φtilt = o, θa = 45o and h = 5. TABLE IV: Coparison between irular and hexagonal paking in ters of overage radius of eah UAV and axiu total overage. N rax fro [9] h rax C fro [9] Ch Rt Rt.5Rt.447Rt Rt.5Rt Rt.4Rt Rt.Rt Rt.86Rt Rt.Rt Rt.86Rt Rt.5Rt Rt.86Rt le. However, this approah towards deterining the needed nuber of UAVs for ahieving a overage level, has two ajor drawbaks. Firstly, signifiant gaps between irles or ells are inevitable when two or ore irles are used to over a given area. This is due to the inherent nature of irle paking theory, sine in order to over the target area opletely, N and rax. Seondly, the nuber of irles UAVs inrease rapidly with desired overage probability. We overoe both of these probles by introduing a UAV plaeent odel that siply uses hexagonal ell shapes instead of irle. This approah not only resolves aforeentioned probles but also leads to a better overage. To illustrate our approah, onsider the ase for N = in Fig.. If we onsider a hexagonal area with the longest distane fro enter to the edge denoted by Rt and the distane fro enter to the vertex of a hexagon by rh, then the axiu distane between any two farthest hexagons will be 4rh as shown in Fig. b. Our goal is to iniize this distane in order to axiize the paking density, thus leading to the arrangeent shown in Fig.. Here, the axiu distane between farthest hexagons is 4rh / < 4rh, leading to rh =.5Rt = rax. The axiu total overage in perentage for N = an then be alulated as follows: R t area overed by UAVs Ch = = = 75% total area to be overed Rt Siilar analysis is done for N = to and presented in Table IV, where C and Ch represents the axiu perentage of atual area overed out of total target area using irular and hexagonal paking respetively while rax and h rax represent the axiu possible radius of eah ell using the two approahes. We opare the results of our proposed approah with those fro [9] that utilizes irle paking to the sae effet. Fro our proposed approah, the iniu possible overage is 7 % for any nuber of UAVs, whereas with irle paking theory it drops to as low as 5% [9]. This has a diret ipat on the overage threshold requireent of the syste. It is highlighted in [9] that a.7 overage perforane is ipossible to ahieve with < N < 7 using irle paking approah. Our proposed hexagonal paking strategy, on the other hand, ensures that this overage perforane deand an be et with uh saller nuber of UAVs. We illustrate this by alulating the iniu nuber of UAVs required to over different geographial areas by utilizing Table IV for a overage threshold 7%, with a tolerane of ±% for =.8 over < h 5 and < B < 8. Fig. opares the resulting iniu nuber of UAVs obtained with hexagonal paking and irle paking fro [9]. For C = 7%, fro irle paking approah, we an over a desired area upto 4 k with, 7 or 8 UAVs. On the other hand, with our proposed hexagonal paking approah, we an over an area with a uh saller nuber of UAVs, i.e.,,, 4 or 5 UAVs. However, the nuber of UAVs required to serve ultiple users would also depend on transit power and bandwidth alloation of ultiple UAVs. Suh onsiderations are not a fous of this work. The reader is referred to three exellent works whih deal with axiizing the iniu average rate of users via joint bandwidth and transit power alloation [],[],[6] for detailed insight into these onsiderations. Another advantage that hexagonal paking offers is the relative salability of nuber of UAVs as the overage threshold hanges, whih in ase of irle paking inreases rapidly as overage threshold goes fro 5% to 8% as seen in Fig.. Next, we investigate the relationship between nuber of UAVs and beawidth of eah UAV. First, we find the axiu possible r for a given geographial area with radius Rt using Table IV as the nuber of UAVs vary and then the orresponding beawidth using Fig. fro our proposed odel. Fig. illustrates the results for a UAV deployed at a height of to over a target geographial area of Rt = 5 in a suburban environent. The overall dereasing trend between beawidth and nuber of UAVs for different overage requireents quantifies the intuitive observation that we an either over the sae area with a single UAV having a wide beawidth or with ultiple UAVs having narrow beawidths. For exaple, for a overage threshold of 6%, a target area of radius 5 an be overed either with UAVs, eah having a beawidth of o or with a single UAV having a beawidth of 5o. Thus proposed odel enables ore design options for a wireless syste designer with regards to onservation of infrastruture. Finally, in order to observe the trend of UAV altitude as the nuber of UAVs vary, the optial UAV altitude that yields axiu possible overage is plotted in Fig. 4 for different nuber of UAVs. The syste odel using binary antenna gain pattern proposed in earlier studies suh as [9] does not reflet

12 a φtilt = o b φtilt = 4o φtilt = 8o Fig. : Coverage probability with varying tilt angles at B = 4o and h =. r 4rh Rt Rt/ rh rh Rt b Max. distane between hexagons Hexagon paking a Cirle paking Fig. : Cirle paking vs hexagonal paking for N =. 9 Nuber of UAVs N 8 [9], C=7% [9], C=6% [9], C=5% proposed odel, C=5%, 6%, 7% Radius of desired area Rt [k] 4 Beawidth of eah UAV B [ o] Fig. : Miniu nuber of UAVs versus radius of desired area for different iniu overage thresholds. C=5% C=6% C=7% Radiu s of e ah U AV r ax [] Nuber of UAVs N Fig. : Optial beawidth for ultiple UAVs and orresponding radius of eah UAV. the role of beawidth in UAV altitude with inreasing nuber of UAVs. Hene, UAV altitude dereases onotonially as nuber of UAVs inreases. Fig. 4 shows this is not the ase when a pratial antenna gain pattern, as proposed in this study, is used. In the plot, one UAV overs axiu possible area for a ertain beawidth that an be found fro Fig. 4. For beawidths upto 5o, UAV altitude with nuber of UAVs follow the sae trend as in [9]. However, the trend hanges for higher beawidths. This trend is expliitly opared with the odel in [9] for a beawidth of o. This onludes that, ontrary to observation ade in prior studies with siple or no antenna odels, it is not neessary for UAV altitude to derease onotonially as the nuber of UAV inreases; in fat, it an also either inrease onotonially or behave as a obination of inreasing and dereasing altitude as the nuber of UAVs inrease. Presented analysis an also be exploited for interferene anageent in the presene of other aerial platfors sine it provides ultiple altitude options for UAV deployent, thus leading to ore flexible design options whih is iperative to the design of next generation ellular systes. Suh investigations of interferene using proposed odel an be fous of a future study. Note that in order to provide full overage of the onsidered area, signifiant overlaps will be inevitable. Several tehniques an be leveraged to optiize the nuber of UAVs needed for full overage, for exaple, [47] iniizes the nuber of UAVs needed to provide wireless overage for a group of distributed ground terinals, ensuring that eah ground terination is within the ouniation range of at least one UAV. The work in [47] analyzes the UAV plaeent proble under LoS onditions without antenna odel onsiderations and an be extended to inorporate other elevation angle dependent fators onsidered in this study. Deterining the optial overlap an be handled using tehniques suh as adaptive bandwidth alloation aong the UAVs as proposed in future work of [6]. For disussions related to other UAV deployent hallenges, suh as battery onsiderations, liited payload apaity, seurity and hostile weather onditions, the reader is referred to [], [48]. V. C ONCLUSION This paper provides a holisti analysis of the interplay between key UAV deployent paraeters: overage radius, height and beawidth while onsidering design spae diensions that reain unexplored in existing studies. It further provides a atheatial odel to estiate RSS at any distane fro boresight of antenna as a funtion of antenna beawidth and altitude. The analysis and results provides several new insights

13 Fig. 4: Altitude with varying no. of UAVs for different beawidths. that prior odels with no or siplified antenna, path loss, or shadowing odels do not reveal, suh as: UAV altitude or antenna beawidth does not have to neessarily inrease ontinuously for higher overage radius; ontrary to findings reported in soe prior studies, UAV overage radius does not neessarily inrease as altitude inreases; the iniu nuber of UAVs required to over a given area does not neessarily derease onotonially as UAV altitude inreases. These results allow us to deterine optial UAV paraeters for realisti deployent. Furtherore, based on the analysis of effet of beawidth and altitude on overage radius, it is found that antenna beawidth and altitude should be optiized siultaneously rather than independently as is the ase assued in previous works. It is also onluded that optiizing beawidth instead of height to ontrol overage ay be a ore pratial and that overage is ost sensitive to beawidths of less than 4 o. A hexagonal paking is proposed for solving overage optiization proble with ultiple UAVs. The advantage of proposed shee is that it leaves uh saller overage holes. Thus it an over a higher proportion of the given area with sae nuber of UAVs opared to irle paking and is salable in ters of nuber of UAVs with inreasing probability of overage. APPENDIX The probability density funtion of RSS at an arbitrary point in a ell is found by first deriving PDFs of reeived signal strengths in LoS and NLoS senarios using transforations of RVs. Thereafter, the theore for finding PDF of sus of independent rando variables fro [49] is applied. For the senario with irular overage pattern of the UAV, we an express the RV, S l h, r, B, fro 7 as: S l h, r, B = P l φ MS R l h, r, B [ 9 tan r = P l φ MS log B h φtilt B 4πfd ] + T log P l φ MS X l = P l φ MS A P l φ MS X l where P l is given in and A an be treated as a onstant for a UAV deployed at a fixed height and beawidth, given by: 9 A = T + log B tan r h φtilt B log 4πfd We proeed by first finding PDF of S l by applying transforations of rando variables as follows: F Sl s l = P S l s l = P P l φ MS A P l φ MS X l s l = P X l s l P l φ MS A P l φ MS sl P l φ MS A F Sl s l = F Xl 4 P l φ MS where F Sl s l and F Xl x l are the uulative distribution funtions CDFs of S l and X l respetively. Note that P l is a funtion of φ MS, whih is in turn a funtion of h and r. However, for opatness, this dependeny is oitted in subsequent analysis. Both sides of 4 are a funtion of s l and therefore, we differentiate both sides w.r.t s l in order to get the PDF: sl P l A d f Sl s l = f Xl P l d s l f Sl s l = P l f Xl sl P l A P l sl P l A P l 5 This allows us to find the PDF of S l based on the PDF of X l whih is N, σ n rando variable. Applying the transforation in 5 yields the following expression for f Sl s l : exp s l P l A P l σ l f Sl s l = 6 πpl σ l Following a siilar proedure, we then derive PDF of S n, whih yields following expression: exp sn [PnA µ sh] P n σn f Sn s n = +σ sh 7 πpn σ n + σsh We an now proeed to derive PDF of S r by perforing onvolution of 6 with 7 as in 8, where A in 8 is expanded in 9. Next, we perfor algebrai anipulation on 9 in to onvert the ters in it to C + D for in order to apply opleting the squares ethod. In order to oplete the square of, we define a new variable in the following anner: σ s = σs l + σs n where σs l = P l σ l and σs n = P n σn + σsh. Therefore, our PDF expression in 8 redues to: [ ] A f Sr s r = σ πσs π sl σ sn exp σ s σs l σsn σ s where A equals to. Squares are now opleted by olleting the appropriate ters as in 4. Finally, the exponent in an be broken

14 f Sr s r = [ exp s r s l [P n A µ sh ] π Pn σ n + σsh P n σn + σsh [ π π Pn σ n + σ sh P l σ l exp ] [ ] exp s l P l A π Pl σ l P l σ l ] = P l σl P n σn + σsh d s l 8 A = P l σl s r s l [P n A µ sh ] + P n σn + σsh sl P l A = P l σl s r + s l + [P n A µ sh ] s l s r s r [P n A µ sh ] + s l [P l A µ sh ] + P n σn + σsh s l + P l A s l P l A 9 [ A = s l P l σl + P n σn + σsh ] [ s l P l σ l s r P n A µ sh + P n σn + σsh ] Pl A + A = s l s l σ s l s r [P n A µ sh ] + σ s n P l A σ s A = s l σ s l s r [P n A µ sh ] + σs n P l A σs f Sr s r = A P l σ l s r + [P n A µ sh ] s r [P n A µ sh ] + P n σ n + σ sh [Pl A ] [ ] exp s r [P l A + P n A µ sh ] πσs σs + σ s l s r + [P n A µ sh ] s r [P n A µ sh ] + σ sn [P l A ] σs σ σs + sl s r [P n A µ sh ] + σs n Pl A σs σ sl s r [P n A µ sh ] + σ s n P l A π σ sl σ sn σ s exp s l σ d s l s s l r [P na µ sh ]+σ sn [P l A ] σs σs l σsn σ s 4 4 ds l } {{ } = 5 into a produt of two exponents as shown in 5. By noting that the integral in 5 is in fat a Gaussian distribution on S l and hene integrates to, substituting σ s fro, P l and P n fro leads to the expression of PDF of S r in 8. ACKNOWLEDGMENT This aterial is based upon work supported by the National Siene Foundation under Grant Nubers 6946, 55948, and 765. The stateents ade herein are solely the responsibility of the authors. For ore details, please visit: REFERENCES [] A. Iran and R. Tafazolli, Perforane & apaity of obile broadband WiMAX 8.6 e deployed via high altitude platfor, in IEEE European Wireless Conferene, 9, pp. 9. [] Y. Zeng, R. Zhang, and T. J. Li, Wireless ouniations with unanned aerial vehiles: opportunities and hallenges, IEEE Couniations Magazine, vol. 54, no. 5, pp. 6 4, 6. [] Y. Mao, Y. Luo, J. Zhang, and K. B. Letaief, Energy harvesting sall ell networks: feasibility, deployent, and operation, IEEE Couniations Magazine, vol. 5, no. 6, pp. 94, June 5. [4] G. Maral and M. Bousquet, Satellite Couniations Systes: Systes, Tehniques and Tehnology. John Wiley & Sons,. [5] K. Goez, T. Rasheed, L. Reynaud, and S. Kandeepan, On the perforane of aerial LTE base-stations for publi safety and eergeny reovery, in IEEE Globeo Workshops,, pp [6] Z. Shao, Y. Liu, Y. Wu, and L. Shen, A rapid and reliable disaster eergeny obile ouniation syste via aerial ad ho BS networks, in 7th International Conferene on Wireless Couniations, Networking and Mobile Coputing,, pp. 4. [7] W. H. Robinson and A. P. Lauf, Resilient and effiient MANET aerial ouniations for searh and resue appliations, in International Conferene on Coputing, Networking and Couniations ICNC,, pp [8] S. Kandeepan, K. Goez, L. Reynaud, and T. Rasheed, Aerialterrestrial ouniations: terrestrial ooperation and energy-effiient transissions to aerial base stations, IEEE Transations on Aerospae and Eletroni Systes, vol. 5, no. 4, pp , Otober 4. [9] J. Lyu, Y. Zeng, and R. Zhang, UAV-aided offloading for ellular hotspot, IEEE Transations on Wireless Couniations, 8. [] A. Iran, M. Shateri, and R. Tafazolli, On the oparison of perforane, apaity and eonois of terrestrial base station and high altitude platfor based deployent of 4G, in Proeedings of the 6th ACM Syposiu on Perforane Evaluation of Wireless Ad Ho, Sensor, and Ubiquitous Networks, 9, pp [] S. Rohde and C. Wietfeld, Interferene aware positioning of aerial relays for ell overload and outage opensation, in IEEE Vehiular Tehnology Conferene VTC Fall,, pp. 5. [] D. Grae, J. Thornton, G. Chen, G. P. White, and T. C. Tozer, Iproving the syste apaity of broadband servies using ultiple high-altitude platfors, IEEE Transations on Wireless Couniations, vol. 4, no., pp. 7 79, Marh 5. [] J. Lyu, Y. Zeng, and R. Zhang, Spetru sharing and ylial ultiple aess in UAV-aided ellular offloading, in GLOBECOM 7-7 IEEE Global Couniations Conferene, De 7, pp. 6. [4] N. H. Motlagh, T. Taleb, and O. Arouk, Low-altitude unanned aerial vehiles-based internet of things servies: Coprehensive survey and future perspetives, IEEE Internet of Things Journal, vol., no. 6, pp , 6. [5] C. Zhan, Y. Zeng, and R. Zhang, Energy-effiient data olletion in UAV enabled wireless sensor network, IEEE Wireless Couniations Letters, pp., 7. [6] A. Al-Hourani, S. Kandeepan, and S. Lardner, Optial LAP altitude for axiu overage, IEEE Wireless Couniations Letters, vol., no. 6, pp , 4. [7] A. Al-Hourani, S. Chandrasekharan, G. Kaandorp, W. Glenn, A. Jaalipour, and S. Kandeepan, Coverage and rate analysis of aerial base stations [letter], IEEE Transations on Aerospae and Eletroni Systes, vol. 5, no. 6, pp. 77 8, Deeber 6. [8] M. Mozaffari, W. Saad, M. Bennis, and M. Debbah, Drone sall ells in the louds: Design, deployent and perforane analysis, in IEEE Global Couniations Conferene GLOBECOM, 5, pp. 6. [9] F. Qixing, J. MGeehan, E. K. Taeh, and A. R. Nix, Path loss odels

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Tarnai, Upper bound of density for paking of equal irles in speial doains in the plane, Periodia Polytehnia. Civil Engineering, vol. 44, no., p.,. [47] J. Lyu, Y. Zeng, R. Zhang, and T. J. Li, Plaeent optiization of UAV-ounted obile base stations, IEEE Couniations Letters, vol., no., pp , 7. [48] L. Gupta, R. Jain, and G. Vaszkun, Survey of iportant issues in UAV ouniation networks, IEEE Couniations Surveys & Tutorials, vol. 8, no., pp. 5, 6. [49] C. M. Grinstead and J. L. Snell, Grinstead and Snell s introdution to probability. Chane Projet, 6. Haneya Naee Qureshi Haneya Naee Qureshi GSM 6 reeived her BS degree in Eletrial Engineering fro Lahore University of Manageent Sienes LUMS, Pakistan, in 6 and M.S. degree in Eletrial and Coputer Engineering fro the University of Oklahoa, USA in 7. She is urrently pursuing the Ph.D. degree in Eletrial and Coputer engineering with the University of Oklahoa, USA working in the Artifiial Intelligene AI for Networks Laboratory, where she is ontributing to several NSF-funded projets. Her urrent researh interests inlude network autoation and obination of ahine learning and analytis for future ellular systes. She has also been engaged in addressing hannel estiation and pilot ontaination proble in Massive MIMO TDD systes. Ali Iran Ali Iran M 5 reeived the B.S. degree in eletrial engineering fro the University of Engineering and Tehnology, Lahore, Pakistan, in 5 and the M.S. degree with Distintion in obile and satellite ouniations and the Ph.D. degree fro the University of Surrey, Guildford, U.K., in 7 and, respetively. He is an Assistant Professor in teleouniations with the University of Oklahoa, Tulsa, OK, USA, where he is the Founding Diretor of the Artifiial Intelligene AI for Networks Laboratory AI4Networks Researh Center and TurboRAN 5G Testbed. He has been leading several ultinational projets on Self Organizing Cellular Networks suh as QSON, for whih he has seured researh grants of over illion in last four years as the Lead Prinipal Investigator. He is urrently leading four NSF funded Projets on 5G aounting to over. illion. He has authored over 6 peerreviewed artiles and presented a nuber of tutorials at international forus, suh as the IEEE International Conferene on Couniations, the IEEE Wireless Couniations and Networking Conferene, the European Wireless Conferene, and the International Conferene on Cognitive Radio Oriented Wireless Networks, on his topis of interest. His researh interests inlude self-organizing networks, radio resoure anageent, and big-data analytis. He is an Assoiate Fellow of the Higher Eduation Aadey, U.K., and a eber of the Advisory Board to the Speial Tehnial Counity on Big Data of the IEEE Coputer Soiety.