Connectivity and Blockage Effects in Millimeter-Wave Air-To-Everything Networks

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1 1 Connetivity and Blokage Effets in Millimeter-Wave Air-To-Everything Networks Kaifeng Han, Kaibin Huang and Robert W. Heath Jr. arxiv: v1 [s.it] 1 Aug 2018 Abstrat Millimeter-wave (mmwave) offers high data rate and bandwidth for air-to-everything (A2X) ommuniations inluding air-to-air, air-to-ground, and air-to-tower. MmWave ommuniation in the A2X network is sensitive to buildings blokage effets. In this paper, we propose an analytial framework to define and haraterise the onnetivity for an aerial aess point (AAP) by jointly using stohasti geometry and random shape theory. The buildings are modelled as a Boolean line-segment proess with fixed height. The bloking area for an arbitrary building is derived and minimized by optimizing the altitude of AAP. A lower bound on the onnetivity probability is derived as a funtion of the altitude of AAP and different parameters of users and buildings inluding their densities, sizes, and heights. Our study yields guidelines on pratial mmwave A2X networks deployment. Index Terms A2X ommuniations, mmwave networks, blokage effets, network onnetivity, stohasti geometry, random shape theory. I. INTRODUCTION Air-to-everything (A2X) ommuniation an leverage aerial aess points (AAPs) mounted on unmanned aerial vehiles (UAVs) to provide seamless wireless onnetivity to various types of users [1] (see Fig. 1). Millimeter-wave (MmWave) ommuniation is one way to provide high data rates for aerial platforms [2]. Unfortunately, mmwave ommuniation is sensitive to building blokages [3], whih are widely expeted in urban deployments of AAPs. In this paper, we define and haraterize the onnetivity for an AAP, using tools from stohasti geometry and random shape theory. K. Han and K. Huang are with the Dept. of EEE at The University of Hong Kong, Hong Kong ( {kfhan, haungkb}@eee.hku.hk). R. W. Heath, Jr. is with The University of Texas at Austin, Austin, TX USA ( rheath@utexas.edu). The orresponding author is R. W. Heath, Jr.

2 2 Aerial aess point Tower-user Airborne-user H a 3D building H u H u H b o Ground-user! ` H u Figure 1: An illustration of the A2X ommuniation network. A typial (entral) AAP provides wireless onnetivity to different types of users, inluding AAP onnets with ground-users (e.g., mobile) via air-to-ground (A2G), tower-users (e.g., base station, BS) via air-to-tower (A2T), and airborne-users (e.g., UAV) via air-to-air (A2A) ommuniations. The altitude of AAP is denoted by H a, and the height of users are denoted by H u. The 3D buildings are modelled as a Boolean line-segment proess with fixed height H b. Leveraging UAVs as AAPs has been studied in reent literature [4] [10]. A single-uav network was proposed in [4], where the network overage was maximized by optimizing the UAVs altitudes. The overage performane an also be maximized via optimizing the plaement of UAVs [5]. In [6], the overage probability of a finite 3D multi-uav network was alulated via a stohasti geometri approah. Both network overage and the sum-rate of a hybrid A2G-D2D network were investigated in [7]. An analytial framework that UAV uses ground-bs for wireless bakhaul was proposed in [8] with providing the analysis for suess probability of establishing a bakhaul link as well as bakhaul data rate. In [9], the multiple-input multiple-output (MIMO) non-orthogonal multiple aess (NOMA) tehniques were used in UAV network and the outage probability and ergodi rate of network were studied based on a stohasti geometry model. In [4], [7], [8], the blokage effets were haraterized by a statistial model where the link-level line-of-sight (LOS) probability is approximated as a simple sigmoid funtion. The parameters of sigmoid funtion are determined the by buildings density, sizes, and heights distribution. The model is unsuitable for mmwave A2X networks sine it fails to apture the fat that multiple nearby links ould be simultaneously bloked by the same building and does not onsider the diversity in user types (e.g., their different heights). In [10], a mathematial framework was proposed for studying mmwave A2A networks, in whih multiple aerial-users are equipped with antenna arrays. Bloking effets were not inluded sine A2A senario was assumed to be

3 3 well above the blokages. In this paper, we develop an analytial framework for haraterizing the blokage effets and onnetivity of a mmwave A2X network overed by a single AAP. The 3D buildings are modelled as a Boolean line-segment proess with fixed height. Given an arbitrary building, the orresponding bloking area is derived as a funtion of altitude of AAP, users and buildings parameters in luding their density, sizes, and heights. Based on the model, the AAP overage area is maximized (or equivalently the bloking area minimized) by optimizing the altitude of AAP. Furthermore, both upper and lower bounds on the bloking area and a suboptimal result of AAP s altitude are derived in losed-form. Finally, the spatial average onnetivity probability of a typial A2X network is obtained, whih may be maximized by optimizing the AAP s altitude. II. SYSTEM MODEL AND PERFORMANCE METRIC Consider a mmwave A2X network as illustrated in Fig. 1. In this letter, we fous on the downlink ommuniation from a typial low-altitude AAP to users with different heights. A. Channel Model between AAP and Users The mmwave hannel between the AAP and the different types of users is assumed to be LOS or bloked by a building. For simpliity, we assume the non-los (NLOS) signals are ompletely bloked due to severe propagation loss from penetration and limited refletion, diffration, or sattering [3]. For the LOS ase, the hannel is assumed to have path-loss without smallsale fading [11]. We assume perfet 3D beam alignment between AAP and users for maximal diretivity [10]. For the path-loss model, we assume the referene distane is 1m. The AAP transmission with power P and propagation distane r is attenuated modelled as r α where α is the path-loss exponent [11]. Let σ 2 be the thermal noise power normalized by the transmit power P. The orresponding signal-to-noise ratio (SNR) reeived at user is defined as P r = Gr α σ 2 where G denotes the beamforming. We assume that the user is onneted to the AAP if the reeive SNR exeeds a given threshold γ. We say that the AAP has a maximal overage ( ) 1 G α sphere with the radius R max =. The 2D projetion of the overage sphere of the AAP σ 2 γ into the plane with user s height H u forms a disk with the radius Λ H = Rmax 2 (H a H u ) 2, alled the effiient overage disk and denoted by O(Λ H ). The user is onneted to the AAP if its 2D loation is inside the effiient overage disk and the link between the user and the AAP is LOS. For higher users heights, i.e., larger H u, the effiient overage disk is larger. Let the

4 4 enter of effiient overage disk, i.e., the 2D projetion of AAP s loation, be the origin denoted by o R 2. B. 3D Building Model A 3D building model is adopted to haraterize blokage effets where buildings are modelled as the Boolean line-segment proess with the same fixed height H b for tratability [12], [13]. Adding randomness to the buildings height will be left to future work. Speifially, buildings are approximated as line segments with random length on the 2D plane. Although the buildings have polygon shapes in pratie, we are interested in their 1D intersetions with the ommuniation links and thus assuming their shape as lines is a reasonable approximation. The enter loations of the line-segments are modelled as a homogeneous PPP Φ = {x} on R 2 plane with density λ b. The lengths {l} and orientations {ω} of blokage line-segments are independent identially distributed random variables. Let f L (l) be the distribution of l and let f Θ (ω) be that of ω. The Boolean line-segments model an be extended to other models as disussed in Remark 2. C. Connetivity and Performane Metris We assume that all the users an be simultaneously onneted to the A2X network if they are in AAP s overage sphere and the link between user and AAP is LOS without being bloked by any building. Consider an arbitrary building whose 2D line-segment enter is loated at x R 2. The building results in a blokaging area S b (x) where the links between users and AAP are fully bloked by the building (see Fig. 2). To measure the network performane, we define the spatial average onnetivity probability, denoted by p, as the spatial average fration of the A2X network that is onnetable at any time [14]. The p is mathematially expressed as x {Φ O(Λ p = 1 E H )} S b(x), (1) O(Λ H ) where O(Λ H ) = πλ H denotes the size of O(Λ H ). III. ANALYSIS FOR NETWORK CONNECTIVITY A. Size of Bloking Area We begin by alulating the size of bloking area S b (x) for an arbitrary building whose 2D line-segment enter loated at x. We first fix the length l and ω of the typial building. Let

5 5 S u t h! q H d S v S b ` x d x o w s p d L Figure 2: 2D projetion of an arbitrary building modelled by a line-segment pq. Some geometrial relations are desribed as follows. d S = o q, d L = o p, Λ H = o u = o s, ω = qxh, θ = qop, and hxo = π/2. The grey area overed by qpsvt is the bloking area S b (x) and the area overed by qop (blue area) and tvu (green area) is the overage area. Speifially, the blue area overed by tvu denotes the overage S due to the fat that higher altitude of AAP an over more LOS area. d x be the distane between x and o. Let d S be the minimal (shortest) distane between o and line-segment (2D projetion of building) and d L be the maximal (longest) distane (see the lines oq and op in Fig. 2). If the AAP s altitude does not exeed building s height, i.e., H a H b, the size of bloking area S b (x) (see the gray area overed by qpsvt in Fig. 2) is alulated by ( ) 1 2 [θλ2 H d d 2 Sd L sin θ], where θ = aros x 1 4 l2 d S d L and 1 d S = 4 l2 + d 2 x d x l sin ω, [ ] (2) 1 d L = min Λ H, 4 l2 + d 2 x + d x l sin ω. If H a > H b, the bloking area S b (x) ould be further redued sine the AAP overs more area via LoS links due to the benefit of higher altitude. Compared with the overage area of AAP when H a H b, we define this additional overage area due to H a > H b as the overage, denoted by S (x) (see the area overed by tvu (green area) in Fig. 2). Based on the basi geometri alulation, S (x) is alulated as where θ follows. ( = aros S (x) = 1 2 ( d S os β 1 H b Hu Ha Hu Λ H ) ) θ 0 (d S os β) 2 os 2 (ϕ + β) dϕ + θ Λ 2 H, (3) and β = artan ( ) os θ d S d L. Then, S sin θ b (x) is alulated as

6 6 Lemma 1 (Size of S b (x)). The bloking area is S b (x) = 1 2 ( θλ 2 H d S d L sin θ ) 1 (H a > H b ) S (x), (4) where 1( ) denotes the indiator funtion and S (x) is given in (3). The alulations follow from geometry, the detailed proof is omitted due to limited spae. To simplify the result in Lemma 1 and obtain more insights therein, both upper and lower bounds of S (x) are derived. By assuming the distane between any point on building s line-segment and o has the same value d L or d S given in (2), the lower and upper bounds of S (x), denoted by S ( ) (x) and S(+) (x), are derived as S ( ) (x) = θ 2 Λ 2 H ( d L 1 H b H u H a H u ) 2 +, (5) and S (+) (x) is obtained by replaing d L in S ( ) (x) with d S. The result for the bounds of S (x) is summarized as follows. Lemma 2 (Bounds of S (x)). The overage S (x) an be upper or lower bounded as follows. S ( ) where Λ H is speified in Lemma 1 and [A] + = max[0, A]. (x) S (x) S (+) (x), (6) The bounds for S an be treated as the bounds for S b via substituting (6) into (4): S ( ) b (x) S b (x) S (+) b (x), where S ( ) b (x) = 1 2 (θλ2 H d Sd L sin θ) 1 (H a > H b ) S (+) obtained via replaing S (+) (x) in S( ) b (x) with S ( ) (x). (x) and S(+) b (x) is Remark 1 (Optimal Altitude of AAP). A larger AAP s altitude H a an effetively inrease the overage (LoS) area, while shrinking the radius of effetive overage disk. We haraterize ( ) this 2 behavior by optimizing S ( ) (x) to obtain a suboptimal solution for H a. When < Λ 2 H, we have d L 1 H b Hu Ha Hu Ha = arg max S ( ) (x) = ( d 2 L (H b H u ) ) H b. (7) H a Substituting H a into (4) gives the suboptimal solution of S b (x). Remark 2. Extending the urrent building model to any model that eah building has a random size in 2D projetion, suh as retangle [3] or disk, follows a similar analytial struture. The

7 7 main differene is that the area of buildings should be inluded into bloking area S b. Also, S needs to be realulated based on different building s shape. For instane, if the 2D projetion of a building is modelled as a disk with radius diameter l (i.e., ylinder in 3D), the blokaging area is realulated as S b = θ 2 Λ2 H ( d x l l2 (θ + π) ) 1 (H a > H b ) S (x), where S (x) ( is lower bounded by S ( ) [Λ (x) = θ 2H ( ] 2 + dx + 1l)) H b Hu Ha Hu B. Network Connetivity Probability In this setion, we alulate the onnetivity probability defined in (1). Notie that the spatial orrelation between different buildings exsits suh as the bloking areas of multiple buildings may overlap with eah other. For analytial tratablility, we ignore the spatial orrelation of buildings due to overlap in the blokaging area of multiple buildings. This assumption is aurate when density of buildings is not very high, whih has been validated in [3]. We derive a lower bound of p by jointly using Campbell s theorem, random shape theory, with the results given in Lemmas 1 and 2. Theorem 1 (Connetivity Probability of AAP). The onnetivity probability p is lower bounded by p ( ) where Ω H = as p ( ) = 1 πλ bθ Λ 2 H L Θ Λ 2 H 0 F(r, l, ω)rdrf Θ (ω)dωf L (l)dl, (8) F(r, l, ω) = θ [ Λ 2 H 1 (H a > H b ) [ Λ 2 H (d L + Ω H ) 2] ] d2 L sin θ, (9) ( ) 1, 1 H b H u H a H u and Λ 2 H is speified in Lemma 1. Proof: See Appendix A. Remark 3. The lower bound p ( ) beomes tighter when density of buildings, i.e., λ b, beomes smaller. This is beause sparsely deployed buildings result in less spatial orrelation. Remark 4. Based on the disussion in Remark 1 and expression of F(r, l, ω), the onnetivity probability p an also be maximizing by optimizing the APP s altitude H u.

8 Upper bound Exat value Lower bound Coverage Gain (m 2 ) Altitude of AAP (m) Figure 3: The effet of the AAP s altitude on overage S. The overage is shown to be a onave funtion of AAP s altitude. The parameters of building are set as {d x, l, ω} = {25m, 6m, π/4}. The upper and lower bounds are plotted based on (6). It is observed that S is well bounded and the lower bound beomes tighter when AAP s altitude is small and upper bound beomes tighter when AAP s altitude is large Simulation Lower bound Connetivity Probability Density of Buildings (1/m 2 ) #10-4 Figure 4: The effet of building density on onnetivity probability p. The exat value of p is plotted via Monte Carlo simulation. The lower bound p ( ) is plotted based on (8). It is observed that onnetivity probability dereases with building density and the lowered bound beomes tighter when building density is small. IV. SIMULATION RESULTS In this setion, we validate the analytial results via Monte Carlo simulation. The radius of maximal overage sphere is R max = 100m. The height of building is H b = 30m and that of user is H u = 2m. The density of buildings is λ b = m 2. The length l and orientation ω of building s line-segments follow independently and uniformly distributions. Speifially, l is uniformly distributed in (0, 15m] and ω is uniformly distributed in (0, π].

9 9 Fig. 3 shows the overage S alulated via (3) and its bounds S (+), S( ) alulated via Lemma 2 versus the altitude of AAP H a. It is observed that S is well bounded by S (+) and S ( ). The lower bound beomes tighter when H a is small and the upper bound beomes tighter when H a is large. This agrees with the intuition beause larger or smaller altitude of AAP results in larger or smaller overage, respetively, whih makes the bound tighter. Moreover, S is maximized via optimizing H a, whih onfirms the disussion given in Remark 1. In Fig. 4, we validate the lower bound of onnetivity probability p, i.e., p ( ), given in Theorem 1 by omparing it with the exat value via Monte Carlo simulation. It is observed that both p and p ( ) derease with building density λ b. More importantly, p ( ) when buildings are sparsely deployed, whih aligns with the disussion in Remark 3. beomes tighter V. CONCLUSIONS AND FUTURE WORK In this letter, we propose an analytial framework to define and haraterize the onnetivity in a mmwave A2X network. Based on the blokage model that buildings are modeled by the Boolean line-segment proess with fixed height, we alulate the blokage area due to an arbitrary building and the onnetivity probability of an AAP. Moreover, the AAP s altitude an be optimized to maximize the overage area as well as the network onnetivity. Future work will fous on studying the effets of spatial orrelation of buildings on network onnetivity and modelling a A2X network inluding multi-aap s onnetions. VI. ACKNOWLEDGMENTS This work was supported in part by the National Siene Foundation under Grant No. ECCS and Hong Kong Researh Grants Counil under the Grants and APPENDIX A. Proof of Theorem 1 By omitting the spatial orrelations between {S b (x)}, onnetivity probability p defined in (1) is lowered bounded as follows. x {Φ O(Λ p 1 E H )} S b(x) O(Λ H ) (a) = 1 πλ bθ Λ 2 H L Θ Λ 2 H 0 S b (r)rdrf Θ (ω)dωf L (l)dl. (10)

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