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Theorem 1. The coverage probability for a ground user, located at a distance r ≤ h.tan(θB/2) from the projection of the UAV j on the desired area, is given by:

Pmin + LdB Pt G3dB + µLoS Pcov = PLoS,j Q − −  σLoS 

Pmin + LdB Pt G3dB + µNLoS + PNLoS,j Q − − , (7)  σNLoS 

where Pmin = 10 log βN + βI¯ is the minimum received power requirement (in dB) for a successful detection, N is Fig. 1: System model. the noise power, β is the signal-to-interference-plus-noise-ratio common approach is to consider the LoS and non-line-of- (SINR) threshold. I¯ is the mean interference power received sight (NLoS) links between the UAV and the ground users from the nearest UAV k which is given by: − − µLoS,k µNLoS,k n separately [10]. Each link has a specific probability of occur- ¯ 10 10 4πfcdk − I Ptg(ϕk) 10 PLoS,k + 10 PNLoS,k c . rence which depends on the elevation angle, environment, and ≈     Also, Q(.) is the Q function. relative location of the UAV and the users. Clearly, for NLoS links the shadowing and blockage loss is higher than the LoS Proof. The downlink coverage probability for a ground user links. Therefore, the received signal power from UAV j at a considering the mean value of interference between UAVs can be written as: user’s location can be given by [10]: Pr,j Pt + G3dB − LdB − ψLoS, LoS link, Pcov = P β = P [Pr,j (dB) Pmin] Pr,j (dB)= (2)  N + I¯ ≥  ≥ Pt + G3dB − LdB − ψNLoS, NLoS link, = PLoS,j P [Pr,j (LoS) Pmin]+ PNLoS,j P [Pr,j (NLoS) Pmin] ≥ ≥ where Pr,j is the received signal power, Pt is the UAV’s trans- (a) = PLoS,j P [ψLoS Pt + G3dB Pmin LdB ] mit power, and G3dB is the UAV antenna gain in dB. Also, LdB ≤ − − + PNLoS,j P [ψNLoS Pt + G3dB Pmin LdB] is the path loss which for the air-to-ground communication is: ≤ − − (b) Pmin + LdB Pt G3dB + µLoS 4πfcdj = PLoS,j Q − − LdB = 10nlog , (3) σ  c   LoS  Pmin + LdB Pt G3dB + µNLoS where fc is the carrier frequency, c is the speed of light, + PNLoS,j Q − − , (8)  σNLoS  dj is the distance between UAV j and a ground user, and 2 P n ≥ 2 is the path loss exponent. Also, ψLoS ∼ N(µLoS, σLoS) where [.] is the probability notation, and Pmin = 2 ¯ and ψNLoS ∼ N(µNLoS, σNLoS) are shadow fading with normal 10 log βN + βI . Clearly, due to the use of directional anten- distribution in dB scale for LoS and NLoS links. The mean nas, interference received from the nearest UAV k is dominant. and variance of the shadow fading for LoS and NLoS links Hence, I¯ can be written as: are (µ , σ2 ), and (µ , σ2 ). As shown in [10], the LoS LoS NLoS NLoS I¯ ≈ P E [P (LoS)] + P E [P (NLoS)] variance depends on the elevation angle and type of the LoS,k r,k NLoS,k r,k − − n µLoS µNLoS 4πf d − environment as follows: 10 10 c k = Ptg(ϕk) 10 PLoS,k + 10 PNLoS,k . σLoS(θj )= k1 exp(−k2θj ), (4) h i  c  E σNLoS(θj )= g1 exp(−g2θj ), (5) where [.] is the expected value over the received interference power. The mean interference is a reasonable approximation 1 where θj = sin− (h/dj ) is the elevation angle between the for the interference and leads to a tractable coverage probabil- UAV and the user, k1, k2, g1, and g2 are constant values which ity expression. In (8), (a) is a direct result of (2), and (b) comes depend on environment. Finally the LoS probability is given from the complementary cumulative distribution distribution γ by [10]: 180 function (CCDF) of a Gaussian random variable. Furthermore, PLoS,j = α θj − 15 , (6)  π  r ≤ h.tan(θB/2) implies that a user can be covered only if it is where α and γ are constant values reflecting the environment in the coverage range of a directional antenna with beamwidth  impact. Note that, the NLoS probability is PNLoS,j = 1 − θB. This proves the theorem. PLoS,j . Next, using this air-to-ground channel model, we derive Note that, Theorem 1 provides the coverage probability the downlink coverage probability and characterize an efficient for users located at any arbitrary range r. From Theorem scheme for deploying of multiple UAVs. 1, it is observed that changing the UAV’s altitude impacts the coverage by affecting several parameters including the III. OPTIMAL MULTI-UAV DEPLOYMENT distance between the UAV and users, LoS probability, and the feasible coverage radius (r ≤ h.tan( θB )). For instance, First, we find the coverage radius of each UAV in the 2 increasing a UAV’s altitude leads to a higher path loss and presence of interference from other UAVs. To this end, the LoS probability, as well as a higher feasible coverage radius. coverage probability of a single UAV needs to be derived. Clearly, in the presence of interference, the UAVs need to Then, we propose an efficient deployment strategy for M increase their transmit power in order to meet the coverage UAVs that maximizes the total coverage performance while requirements. Furthermore, as the number of UAVs increases, maximizing the coverage lifetime. the distance between the UAVs decreases, and hence, the 3 interference from the nearest UAV will increase. The coverage Table I: Covering a circular area with radius Rc using identical UAVs- the circle packing in a circle approach. radius of a UAV, ru, is the maximum range within which the Number of UAVs Coverage radius of each UAV Maximum total coverage probability that users are covered by the UAV is greater than 1 Rc 1 2 0.5R 0.5 a specified threshold (ǫ). Clearly, ru depends on the transmit c 3 0.464Rc 0.646 power, antenna beamwidth, ǫ, number of UAVs, and UAVs’ 4 0.413Rc 0.686 5 0.370Rc 0.685 locations. Thus, the coverage radius is given by: 6 0.333Rc 0.666 7 0.333Rc 0.778 ru = max{r|Pcov(r, Pt,θB) ≥ ε}. (9) 8 0.302Rc 0.733 9 0.275Rc 0.689 Now, consider the geographical area of interest which 10 0.261Rc 0.687 should be covered by multiple UAVs. The UAVs must be placed in a way to maximize the total coverage, and to avoid any overlapping in their coverage areas. Furthermore, while maximizing the total coverage, each UAV must use a minimum transmit power in order to maximize the coverage lifetime. The coverage lifetime is defined as the maximum time that the UAVs can provide coverage for the given area. The coverage Fig. 2: Packing problem in a circle with 3 . lifetime is maximized when the UAVs have the same transmit power (equal coverage radius). Therefore, assuming a circular For M = 3, all the circles tangent with each other if their coverage area for each UAV, the problem can be formulated centers are placed on the vertices of a equilateral . as follows: Hence, Figure 2 corresponds to the optimal placement. Then, 2 ru (~rj∗,h∗, ru∗ ) = argmax M.ru, (10) from Figure 2, x = cos(30o) and, i 1,...,M 2 √3R ∈{ } R = r + x = r 1+ → r = c ≈ 0.464R . c u u √3 u 2+√3 c st. ||~rj −~rk|| ≥ 2ru, j 6= k ∈{1, ..., M}, (11)   In our model, each circle corresponds to the coverage region (12) ||~rj + ru|| ≤ Rc, of each UAV, and maximizing packing density is related to ru ≤ h.tan(θB/2), (13) maximizing the coverage area with non-overlapping smaller circles. Therefore, given the radius of the desired area and the where M is the number of UAVs, Rc is the radius of the number of symmetric UAVs, we can determine the required desired geographical area, ~rj is the vector location of UAV j within the 2D plane of the desired area considering the center coverage radii of UAVs as well as their 3D locations which lead to the maximum coverage. Subsequently, based on the of the area as the origin, and ru is the maximum coverage radius of each UAV. Considering the optimization problem in required coverage range of each UAV (ru), the minimum (10), constraint (11) ensures that no coverage overlap occurs, transmit power of UAVs can be computed. Note that, the and (12) guarantees that UAVs do not cover outside of the UAV’s altitude should be adjusted based on the coverage radius and the antenna beamwidth by using h = ru . desired area. As a result, the potential interference between tan(θB/2) UAVs will be avoided, and also, users located outside the Next, as a function of the number of UAVs, we derive given area (undesired area) will not be affected by UAVs’ an upper bound for the altitude which guarantees the non- transmissions. overlapping condition between the UAVs’ coverage regions. Solving (10) is challenging due to the high number of un- Proposition 1. Given M UAVs, and Rc, the radius of the knowns and the nonlinear constraints. We model this problem desired area, an upper bound for the maximum UAVs’ altitude by exploiting the so-called circle packing problem [7]. In the for which the coverage overlap does not occur, is given by: circle packing problem, M circles should be arranged inside q R h ≤ m c , (14) a given surface such that the packing density is maximized θB (2 + qm) tan( 2 ) and none of the circles overlap. As an illustrative example, R Figure 2 shows the optimal packing of 3 equal circles inside a where qm is the maximum value of variable q ∈ that satisfies the following inequality: bigger circle. Also, Table I shows the radii of non-overlapping 2 π q√3+√4 q small circles which lead to the maximum packing density for −1 − + √12(1 M) 0. sin (q/2)  q  − ≥ a given circular area [7]. Clearly, the radius of each circle decreases as the number of circles increases. In Table I, the Proof. The maximum packing density (D) that can be total coverage represents the maximum portion of the desired achieved for a circle, using M equal smaller circles, is upper bounded by [7]: 2 area which can be covered by multiple circles. Note that, in Mqm D ≤ 2 . (15) general, the circle packing problem in a bounded area is known (2 + qm) 2 2 2 to be intractable [7]. In particular, it is not possible to find a Mru Mru Mqm Also, clearly, D = 2 , and hence, 2 ≤ 2 . general packing strategy that is optimal for any arbitrary M. Rc Rc (2+qm) qmRc θB ru Finally, considering ru ≤ , and tan( ) = , we Therefore, for each value of M a specific packing strategy (2+qm) 2 h qmRc  needs to be provided. As an example, we derive the optimal have h ≤ θ . (2+q ) tan( B ) packing method for M = 3. Consider a circular area with m 2 radius Rc. Clearly, the packing density is maximized if the Proposition 2, provides a necessary condition on the UAVs’ maximum distance between the farthest circles is minimized. altitude for avoiding an overlapping coverage. 4

1 1 Total coverage 6000 o Coverage lifetime θ = 100 0.9 0.8 B 5000 θ = 80o 0.8 B 0.6 4000 0.7 3000 0.4 0.6 2000 UAV altitude (m) 0.2 0.5 1000 1 2 3 4 5 6 7 8 9 10 Total coverage (Normalized)

0.4 0 Coverage lifetime (Normalized) Number of UAVs 1 2 3 4 5 6 7 8 9 10 Number of UAVs Fig. 4: UAV’s altitude versus number of UAVs. Fig. 3: Total coverage and coverage lifetime versus number of 11 UAVs for Rc = 5000 m. 10 Total coverage threshold=0.5 9 Total coverage threshold=0.6 IV. SIMULATION RESULTS AND ANALYSIS 8 Total coverage threshold=0.7 7 In our simulations, we consider the UAV-based communica- 6 tions over 2GHz carrier frequency (f = 2 GHz) in an urban 5 c 4 environment with α =0.6, γ =0.11, k1 = 10.39, k2 =0.05, 3

Number of UAVs 2 g1 = 29.06, g2 = 0.03, µLoS = 1dB, µNLoS = 20dB, 1 0 and n = 2.5 [10]. Moreover, using Theorem 1 and (9), the 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 coverage radius of each UAV is computed based on ǫ =0.80, Radius of desired area (km) β =5, and N = −120 dBm. Furthermore, here, the maximum Fig. 5: Number of required UAVs versus radius of desired area. coverage time duration of the UAVs (coverage lifetime) is V. CONCLUSIONS inversely proportional to the transmit power of UAVs. In this paper, we have studied the optimal deployment of Figure 3 shows the total coverage and the coverage lifetime multiple UAVs equipped with directional antennas used as as a function of the number of UAVs (M) for Rc = 5000m, aerial base stations. First, the downlink coverage probability o and θB = 80 . In this figure, the maximum achievable was derived based on the probabilistic LoS/NLoS links and coverage while maximizing the coverage lifetime is shown for considering the shadow fading. Next, given a desired geo- different number of UAVs. Clearly, by increasing the number graphical area which needs to be covered by multiple UAVs, of UAVs, the coverage lifetime increases due to the decrease an efficient deployment approach was proposed based on the in the transmit power of each UAV. In Figure 3, for the given circle packing theory that leads to a maximum coverage while area with a radius 5000m, a single UAV has the maximum each UAV uses a minimum transmit power. The results have coverage performance. However, in this case, the single UAV shown that, the optimal altitude and location of the UAVs can has a minimum coverage lifetime. Therefore, depending on the be determined based on the number of available UAVs and the size of the area, coverage and coverage lifetime requirements, gain/beamwidth of the directional antennas. an appropriate number of UAVs needs to be deployed. REFERENCES Figure 4 illustrates the optimal UAVs’ altitude versus the [1] M. Mozaffari, W. Saad, M. Bennis, and M. Debbah, “Drone small cells number of UAVs. Clearly, the altitude of UAVs should be de- in the clouds: Design, deployment and performance analysis,” in Proc. of creased as the number of UAVs increases. For higher number IEEE Global Communications Conference (GLOBECOM), San Diego, CA, USA, Dec. 2015. of UAVs, to avoid overlapping between the coverage regions of [2] E. P. De Freitas, T. Heimfarth, A. Vinel, F. R. Wagner, C. E. Pereira, and the UAVs, the coverage radius of UAVs must be decreased by T. Larsson, “Cooperation among wirelessly connected static and mobile reducing their height according to h = ru . As shown, sensor nodes for surveillance applications,” Sensors, vol. 13, no. 10, pp. tan(θB /2) 12 903–12 928, 2013. by doubling the number of UAVs from 3 to 6, the optimal [3] I. Bekmezci, O. K. Sahingoz, and S¸. Temel, “Flying ad-hoc networks altitude is reduced from 2000m to 1300m. Furthermore, the (FANETs): A survey,” Ad Hoc Networks, vol. 11, no. 3, pp. 1254–1270, optimal altitude is lower for higher antenna beamwidths. 2013. [4] M. Mozaffari, W. Saad, M. Bennis, and M. Debbah, “Unmanned aerial Figure 5 shows the minimum required number of UAVs in vehicle with underlaid device-to-device communications: performance order satisfy the coverage requirement of the given geograph- and tradeoffs,” IEEE Transactions on Wireless Communications, to ical area. In this figure, the coverage threshold corresponds appear, 2016. [5] D. Orfanus, E. P. de Freitas, and F. Eliassen, “Self-organization as a to the minimum portion of the given area which needs to be supporting paradigm for military UAV relay networks,” IEEE Commu- covered by the UAVs. This result is based on Pt = 35 dBm, nications Letters, vol. 20, no. 4, pp. 804–807, 2016. θ = 80o, and optimal altitudes subject to h < 5000m. [6] J. Kosmerl and A. Vilhar, “Base stations placement optimization in B wireless networks for emergency communications,” in Proc. of IEEE Interestingly, to satisfy at least 0.7 coverage requirement with International Conference on Communications (ICC), Sydney, Australia, a maximum coverage lifetime, either one UAV or more than June. 2014. 6 UAVs are required. In other words, for 1