**3. System Model**

In this section, we first introduce the considered system model by defining the system and environmental input parameters. We then define the metrics of interest and outline the proposed methodology.

#### *3.1. Deployment Model and Metrics of Interest*

We assume deterministic Manhattan grid deployment with street width *l*, see Figure 1. The widths and lengths of building blocks are assumed to be *bw* and *bl*. The height of building blocks is assumed to be a random variable (RV), *HB*, with probability density function (pdf) *fHB* (*x*). We consider a certain zone of interest having *MV* and *MH* vertical and horizontal streets, respectively. We further assume that there are *N* ground-mounted mmWave BSs located on the streets leading to the spatial density of *N*/[*MV* ∗ (*l* + *bl*) ∗ *MH*(*l* + *bw*)] mmWave BSs per squared meter. On top of this, we assume that there are *M* rooftop-mounted mmWave BS randomly located on the building roofs.

**Figure 1.** Illustration of the considered deployment.

MmWave BSs are assigned to streets randomly, i.e., first a discrete uniformly distributed RV between 0 and *MH* + *MV* is used to determine the street index, and then the position of mmWave BS is determined by choosing *x* or *y* coordinate uniformly along the street width, excluding parts occupied by crossroads. Similarly, *NC*, *NC* < *N*, crossroadinstalled mmWave BSs are assigned to crossroads randomly using discrete RV uniformly distributed between 0 and *MVMH*. A particular location of mmWave BS on a crossroad is defined with respect to the left upper corner and is fully determined by the distances *lA*,<sup>1</sup> and *wA*,1. Similarly, we choose a particular position for BSs installed along the street at the distance *lA*,1, from the building. Note that in practice these BSs can be installed on lampposts, for example, and distances *lA*,1, and *wA*,<sup>1</sup> may coincide with the sidewalk width.

The UAV attitude is assumed to be constant, *hR*. We assume that UAV is in coverage of BS if there is a LoS path between UAV and BS and this path is less than a certain *r*. UAV is assumed to cross this region following a random line at the constant speed *vU*. We are interested in the UAV coverage probability–the probability that UAV is in coverage of at least one BS.

#### *3.2. Methodology at a Glance*

Instead of accounting for inherent dependencies between building positions and their shapes in regular urban deployments, we characterize LoS visibility regions in <sup>2</sup> located at the UAV flying altitude, *hR*, see Figure 1. Using these regions we then proceed by utilizing the tools of integral geometry to determine the probability that a random point in this plane is covered by at least one LoS visibility region immediately delivering the sought metrics of interest in a simple closed-form.

#### **4. UAV Blockage Analysis**

In this section, we develop our framework. We start by defining the so-called LoS visibility zones at the flying altitude of the UAV. Next, we utilize the integral geometry to specify the LoS probability for the ground deployment of mmWave BSs. Finally, we extend the methodology to account for rooftop-mounted mmWave BSs.

#### *4.1. Geometric Structure of LoS Zones*

We start by characterizing the LoS visibility zone induced by BR BS located along the street, see Figure 2a. As one may observe, this zone is of rectangular shape with sides that depend on (i) heights of buildings, *HB*,<sup>1</sup> and *HB*,2, (ii) maximum coverage of BS, *r*, and (iii) UAV altitude *hR*.

Observing Figure 3a, the length of the LoS visibility zone is

$$D = 2\sqrt{r^2 - (h\_R - h\_T)^2} \,\text{}\,\text{}\,\text{}$$

where *r* is the maximum communications distance,

$$r = \sqrt{10^{\frac{P\_A + G\_R + G\_T - N\_0 - S\_T - 32.4 - 20 \log\_{10} F\_C}{21}} - [h\_R - h\_T]^2},\tag{2}$$

where *ST* is the SNR threshold, *GT* and *GR* are the transmit and receive antenna gains, *PA* is the emitted power at mmWave BS, *N*0 is the thermal noise, *FC* is the carrier frequency.

The width of the LoS visibility zone, *L*, is an RV that is determined by building heights, *HB*,<sup>1</sup> and *HB*,2, where both have the same pdf *fHB* (*x*), see Figure 3b. Observe that angles *α*1 and *α*2 are given by

$$a\_i = \tan^{-1} \left( \frac{H\_{B,i} - h\_T}{l\_{A,i}} \right), i = 1, 2,\tag{3}$$

where *l*1, *l*2 are the distances to the buildings, see Figure 3b.

**Figure 2.** Two types of feasible LoS visibility zones in the considered scenario. (**a**) BS located along the street; (**b**) BS located at the crossroad.

**Figure 3.** Geometrical illustration of the sides of LoS visibility zone. (**a**) Length of the LoS visibility zone; (**b**) Width of the LoS visibility zone.

Further, using tan *βi* = *Li*/(*hR* − *hT*), *i* = 1, 2 and observing that angles *βi* are related to *αi* as *βi* = *π*/2 − *αi* we arrive at the following expressions for RVs *L*1 and *L*2

$$L\_i = (h\_R - h\_T) \tan\left(\frac{\pi}{2} - \tan^{-1}\left[\frac{H\_{B,i} - h\_T}{l\_{A,i}}\right]\right) = $$

$$= \frac{l\_{A,i}(h\_R - h\_T)}{H\_{B,i} - h\_T}, i = 1, 2. \tag{4}$$

One may now determine the mean area of the LoS visibility zone as

$$E[S\_B] = D \int\_0^\infty \int\_0^\infty f\_{H\_B}(\mathbf{x}) f\_{H\_B}(y) [L\_1(\mathbf{x}) + L\_2(y)] dx dy = $$

$$= 2 \sqrt{r^2 - (h\_R - h\_T)^2} E[L\_B] \, \tag{5}$$

where the mean length of the LoS visibility zone, *<sup>E</sup>*[*LB*], is provided by

$$\begin{split} E[L\_B] &= \int\_0^\infty \int\_0^\infty \left( \frac{l\_{A,i}(h\_R - h\_T)}{\mathbf{x} - h\_T} \right) \times \\ &\times \left( \frac{l\_{A,i}(h\_R - h\_T)}{\mathbf{y} - h\_T} \right) f\_{H\_B}(\mathbf{x}) f\_{H\_B}(\mathbf{y}) d\mathbf{x} dy\_\prime \end{split} \tag{6}$$

that can be evaluated in closed-form for a given distribution of the building height.

A simple ye<sup>t</sup> reliable approximation for (5) can be obtained by assuming the same random height of both buildings on the street, as it is usually the case in practice. In this case, the width of the blockage zone becomes

$$L\_B = L\_1 + L\_2 = \frac{(h\_T - h\_R)(l\_{A,1} + l\_{A,2})}{x - h\_T},\tag{7}$$

implying that (6) can be written as

$$E[L\_B] = \int\_0^\infty f\_{H\_B}(\mathbf{x}) \frac{(h\_T - h\_R)(l\_{A,1} + l\_{A,2})}{\mathbf{x} - hT} d\mathbf{x}.\tag{8}$$

For example, for *HB* having uniform distribution in (*<sup>A</sup>*, *B*) we have

$$E[L\_B] = \frac{(h\_T - h\_R)(l\_{A,1} + l\_{A,2})(\log[1 - \frac{A}{R\tau}] - \log[1 - \frac{B}{R\tau}])}{B - A}.\tag{9}$$

Similarly, the mean perimeter of the LoS visibility zone *B* is *E*[*LB*] = 2*D* + <sup>2</sup>*<sup>E</sup>*[*L*]. The LoS visibility zones induced by BS deployments on the crossroads can be found similarly. Indeed, as one may observe in Figure 2b, they consist of two overlapping LoS visibility zones forming a "cross". Individually, parameters of these two zones can be estimated as shown above.

#### *4.2. Blockage Probability with Grounded Infrastructure*

We are now in a position to evaluate blockage probability, *pB*, with ground-mounted BSs. The input parameters are the number of LoS visibility zones characterized by their mean areas and perimeters, *E*[*SB*] and *<sup>E</sup>*[*LB*], in <sup>2</sup> plane positioned at the UAV flying altitude *hR*.

To provide simple ye<sup>t</sup> accurate expression for blockage probability, we will rely upon the tools of integral geometry. Further, we need two fundamental notions of integral geometry. A curious reader is referred to [29] for a basic account of information and to [30] for modern developments in the field.

**Definition 1** (**Kinematic density, [29]**)**.** *Let K denote the group of motions of a set A in the plane. The kinematic density dA for the group of motions K in the plane for the set A is*

$$dA = d\mathbf{x} \wedge dy \wedge d\boldsymbol{\phi},\tag{10}$$

*where* ∧ *is the exterior product [31], x and y are Cartesian coordinates, φ is the rotation angle of A with respect to OX.*

**Definition 2** (**Kinematic measure, [29]**)**.** *The kinematic measure m of a set of group motions K on the plane is defined as the integral of the kinematic density dA over K, that is,*

$$m\_A = \int\_K dA = \int\_K d\mathbf{x} \wedge dy \wedge d\boldsymbol{\phi}.\tag{11}$$

Consider first a single mmWave BS in the area of interest *A* and let *B* define a LoS visibility zone. We are first interested in the probability *pC* that UAV, located at a randomly chosen point *P* in *A*, is in coverage of this BS, that is, it is located in *B*. Using conditional probability we may write

$$p\_{\mathbb{C}} = \frac{\Pr\{P \in A \cap B\}}{\Pr\{A \cap B \neq \emptyset\}},\tag{12}$$

where the probability that UAV location *P* belongs to the intersection area of two sets, *A* and *B*, is in the nominator, while the probability that these sets do intersect is in the denominator. thenotionofkinematic[29]

Using measure, we ge<sup>t</sup>

$$\Pr\{P \in A \cap B\} = m(A : P \in A \cap B\}),$$

$$\Pr\{A \cap B \neq \emptyset\} = m(A : A \cap A \neq \emptyset),\tag{13}$$

where the first expression is the kinematic measure of the set of motions of *A* such that *P* ∈ *A*, while the second one provides the measure of all motions of *A*, for which the intersection between *A* and *B* is non-zero.

Following [29], the first measure is

$$m\_{\bar{\jmath}}(P \in A \cap B \{\}) = \int\_{P \in B} f(\mathbf{x}, \mathbf{y}) d\mathbf{x} \wedge d\mathbf{y} \wedge d\boldsymbol{\phi},\tag{14}$$

where *f*(*<sup>x</sup>*, *y*) is the density of LoS visibility zone positions in *A*. Themeasureofallmotions of *A*,suchthat*A*∩*B*,is[29]

$$m\_{\vec{\wedge}}(A \cap B \neq 0) = \int\_{A \cap B \neq 0} f(\mathbf{x}, \mathbf{y}) d\mathbf{x} \wedge d\mathbf{y} \wedge d\boldsymbol{\phi}.\tag{15}$$

Finally, the sought probability is given by

$$p\_C = \frac{\int\_{P \in A \cap B} f(x, y) dx \wedge dy \wedge d\phi}{\int\_{A \cap B \neq \emptyset} f(x, y) dx \wedge dy \wedge d\phi} \, \tag{16}$$

and can be computed for a particular form of *A*, *B*, and *f*(*<sup>x</sup>*, *y*).

The numerator in (16) is computed as [29]

$$\begin{split} m\_j(P \in A \cap B \{ \} ) &= \int\_{P \in B} dx \wedge dy \wedge d\phi = \\ &= \int\_{P \in B} dx \wedge dy \int\_0^{2\pi} d\phi = 2\pi E[S\_B] \end{split} \tag{17}$$

where *E*[*SB*] is the mean area of LoS visibility zone provided in (5).

The measure of motions of *A* is such that *A* ∩ *B* = 0 is [29]

$$\begin{split} m\_{\hat{\mathbb{M}}}(A \cap B \neq 0) &= \int\_{A \cap B \neq 0} d\mathbf{x} \wedge d\mathbf{y} \wedge d\boldsymbol{\phi} = \\ &= 2\pi (S\_A + E[S\_B]) + L\_A E[L\_B], \end{split} \tag{18}$$

where *E*[*LB*] is the perimeter of LoS visibility zone, *SA* and *LA* are the area and the perimeter of *A*, given by

$$\begin{array}{l} S\_A = [M\_V \ast (l + b\_l) + b\_l] \ast [M\_H(l + b\_w) + b\_l], \\ L\_A = 2[M\_V \ast (l + b\_l) + b\_l] + 2[M\_H(l + b\_w) + b\_l]. \end{array} \tag{19}$$

Substituting (17), (18) into (12) we obtain

$$p\_{\mathbb{C}} = \frac{2\pi E[S\_B]}{2\pi (S\_A + E[S\_B]) + L\_A E[L\_B]}.\tag{20}$$

Recall that mmWave BSs are deployed randomly along the streets. When mmWave BS is deployed on the crossroad it creates two LoS visibility zones as illustrated in Figure 3b. Let *u* be the probability that mmWave BS is at the crossroad. This probability is found as the ratio of crossroad area to the overall area of streets as

$$u = \frac{M\_V M\_H l^2}{M\_H(l[(b\_W + l)M\_V + b\_W]) + M\_V(l[(b\_l + l)M\_H + b\_l]) - M\_V M\_H l^2}.\tag{21}$$

The mean number of LoS visibility zones of rectangular shape is then given by the mean of Binomial distribution with parameters *N* and *u* shifted by *N*, i.e., *N*(1 + *<sup>u</sup>*). Thus, the blockage probability can now be approximated as

$$p\_B = 1 - (1 - p\_C)^{N(1+u)},\tag{22}$$

where *pC* is provided in (20).

Substituting intermediate results and simplifying, we arrive at the closed-form expression for blockage probability in the presence of *N* ground-mounted mmWave BS as

$$p\_B = 1 - \left(1 - \frac{2\pi E[S\_B]}{2\pi (S\_A + E[S\_B]) + L\_A E[L\_B]}\right)^{\mathcal{N}\left(1 + \frac{M\_V M\_H l^2}{M\_H(l[(b\_W + l)M\_V + b\_W]) + M\_V(l[(b\_I + l)M\_H + b\_I]) - M\_V M\_H l^2}\right)},\tag{23}$$

where *SA* and *LA* are provided in (19), *E*[*SB*] and *E*[*LB*] are calculated using (5) and (6) for a given *fHB* (*x*).

#### *4.3. Blockage Probability with Rooftop-Mounted BSs*

The blockage probability heavily depends on the density of mmWave BSs, as well as on the heights of buildings. For some values of these input parameters, the blockage probability might be unacceptably high. In practical deployments, network operators may want to add additional dedicated mmWave BSs. Mounting these BSs on rooftops would allow for an unobstructed LoS of circular shape, drastically reducing blockage probability.

To assess joint deployment, one may apply the methodology developed in the previous section to rooftop mmWave BSs. The principal difference is that the LoS visibility zone is of circular form with radius *D*/2 as in (1) with *HB* replacing *hT*. However, as these mmWave BSs are now deployed on the roofs, *D* is a RV. Thus, we have

$$E[D] = \int\_0^\infty f\_{H\_B}(\mathbf{x}) 2\sqrt{r^2 - (h\_R - \mathbf{x})^2} d\mathbf{x}.\tag{24}$$

that can be evaluated for a given *fHB* (*x*), *x* < *hR*.

The blockage probability by *M* rooftop mmWave BSs is obtained similarly to (22). Finally, in the presence of *N* ground-mounted and *M* rooftop-mounted mmWave BSs the blockage probability is the product of individual blockage probabilities.
