5. **Markov structure and more:** In general, the Markov structure described in Section 6 and Section 9 for the handover process should allow one to derive many structural properties of the process, e.g. CLT, LLN and large deviations, for all the models defined above.

6. **Other distance metrics:** It would be interesting to determine the handover frequency for a general class of dynamic tessellations, where, instead of the Euclidean metric, one uses other metrics, for example: $l_p$-distances, Manhattan distance, supremum distance. This can be considered as a more abstract mathematical problem.

7. **Other dynamical tessellations:** The dynamical versions of the Voronoi tessellations defined in [22] would be important extensions of this article.

8. **Other handover protocols:** One can define the handover in terms of SINR based association rule and answer same questions in this case.

9. **Multi-hop association rule:** In the architecture of communication networks involving satellites, a user is served by an anchor station via multiple satellites connected through inter-satellite-links. Change of one of the connections: user to satellite, satellite to satellite or satellite to anchor station, leads to handovers of different types. Analyzing the geometric and statistical properties of this handover process in this practical setup and for all the models described above, is an important goal of the line of research.

10. **Other types of dynamics:** In this article we have looked at the dynamics of the stations according to a simple way-point model, where the motion is without change of direction and speed. One can consider other classes of dynamics for the stations according to, for example, more general random way-point models, jump processes, Brownian motion, diffusion processes etc., as long as the displacement theorem holds true for the underlying point process of stations. This gives rise to a rich classes of dynamical tessellations where a similar analysis would be of interest.

## 12. TABLE OF NOTATION

For the ease of reading, we provide a table of notation, abbreviations and their meaning, that have been used at various places in the article.

<table><caption>TABLE 12.1. List of Notations</caption><thead><tr><th>Symbol</th><th>Page</th><th>Description</th></tr></thead><tbody><tr><td>|x|</td><td>3</td><td>Euclidean norm ||x||<sub>2</sub> of x ∈ ℝ<sup>2</sup></td></tr><tr><td>B<sub>r</sub>(x)</td><td>4</td><td>Open ball centered at x and radius r</td></tr><tr><td>H<sup>+</sup></td><td>8</td><td>Upper half plane ℝ × ℝ<sup>+</sup></td></tr><tr><td>Q<sub>s</sub></td><td>28</td><td>{(t, h) ∈ H<sup>+</sup> : t ≥ s}, the right quadrant on the right of the vertical line t = s on H<sup>+</sup></td></tr><tr><td>L<sub>X</sub>(γ)</td><td>26</td><td>Laplace transform of a random variable X for parameter γ ≥ 0</td></tr><tr><td>X ≤<sub>Lt</sub> Y</td><td>27</td><td>L<sub>X</sub>(γ) ≤ L<sub>Y</sub>(γ), i.e., Laplace transforms order between two random variables X and Y, for any γ ≥ 0</td></tr><tr><td>Φ<sub>λ</sub> or Φ̃</td><td>3</td><td>Homogeneous Poisson point process (PPP) on ℝ<sup>2</sup></td></tr><tr><td>Φ̃<sub>λ</sub> or Φ̃</td><td>8</td><td>Marked Poisson point process on ℝ<sup>2</sup></td></tr><tr><td>m<sub>Ψ</sub></td><td>3</td><td>Intensity measure of any point process Ψ</td></tr><tr><td>Ψ|<sub>A</sub></td><td>17</td><td>Point process Ψ restricted to a set A</td></tr><tr><td>Supp(Ψ)</td><td>25</td><td>Support of a point process Ψ</td></tr><tr><td>P<sub>Ψ</sub></td><td>3</td><td>Probability distribution of a Point process Ψ</td></tr><tr><td>E<sub>Ψ</sub></td><td>9</td><td>Expectation with respect to P<sub>Ψ</sub></td></tr><tr><td>L<sub>Ψ</sub>(f)</td><td>29</td><td>Laplace transform of a function f w.r.t. a point process Ψ</td></tr><tr><td>P<sup>0</sup><sub>Ψ</sub></td><td>5</td><td>Palm probability distribution of a Point process Ψ</td></tr><tr><td>E<sup>0</sup><sub>Ψ</sub></td><td>9</td><td>Palm expectation of Ψ</td></tr><tr><td>L<sup>0</sup><sub>Ψ</sub>(f)</td><td>29</td><td>Laplace transform of a function f w.r.t. a point process Ψ under the Palm probability measure P<sup>0</sup><sub>Ψ</sub></td></tr><tr><td>H</td><td>8</td><td>Head point process (HPP)</td></tr><tr><td>C<sub>(R,α)</sub></td><td>10</td><td>Radial bird closed set corresponding to the station given by (R, α)</td></tr></tbody></table>

*Continued to next page*
<table><caption>Table (12.1) (continued from previous page)</caption><thead><tr><th>Symbol</th><th>Page</th><th>Description</th></tr></thead><tbody><tr><td>C<sub>(t,h)</sub></td><td>15</td><td>Radial bird with head point at (t, h) ∈ H<sup>+</sup></td></tr><tr><td>H<sub>c</sub></td><td>12</td><td>Marked head point process (MHPP), with closed-set valued marks</td></tr><tr><td>P<sub>c</sub></td><td>11</td><td>Radial bird particle process (RBPP)</td></tr><tr><td>L<sub>e</sub></td><td>13</td><td>Lower envelope in single-speed</td></tr><tr><td>L<sup>+</sup><sub>e</sub></td><td>15</td><td>The open region above the lower envelope L<sub>e</sub> in single-speed</td></tr><tr><td>Ψ<sup>x,x'</sup></td><td>17</td><td>Two-point Palm version of a Point process Ψ with two points x, x'</td></tr><tr><td>or Ψ + δ<sub>x</sub> + δ<sub>x'</sub></td><td></td><td></td></tr><tr><td>P<sup>x,x'</sup><sub>Ψ</sub></td><td>17</td><td>Two-point Palm probability measure of Ψ with two points x, x'</td></tr><tr><td>E<sup>x,x'</sup><sub>Ψ</sub></td><td>18</td><td>Two-point Palm expectation with two points x, x'</td></tr><tr><td>Ψ<sup>n,≠</sup></td><td>18</td><td>Factorial power of order n of a point process Ψ</td></tr><tr><td>U<sup>x</sup><sub>r</sub></td><td>13</td><td>Upper half-ball of radius r, centered at (x, 0)</td></tr><tr><td>V<sub>λ</sub> or V</td><td>4, 16</td><td>Handover point process (HoPP)</td></tr><tr><td>L<sub>s</sub></td><td>21</td><td>The vertical line t = s</td></tr><tr><td>η<sub>s</sub></td><td>21</td><td>point process on R<sup>+</sup> for intersection points of all the birds with L<sub>s</sub></td></tr><tr><td>λ<sub>V</sub></td><td>4, 16</td><td>Intensity of the handover point process V or the handover frequency</td></tr><tr><td>H<sub>V</sub></td><td>25</td><td>The point process of visible heads</td></tr><tr><td>A<sub>V</sub></td><td>25</td><td>The point process of abscissas of visible head point from H<sub>V</sub></td></tr><tr><td>RMHP</td><td></td><td>Right-most head point</td></tr><tr><td>LMHP</td><td></td><td>Left-most head point</td></tr><tr><td>R</td><td>22</td><td>PP for abscissas of RMHP for a handover</td></tr><tr><td>L</td><td></td><td>PP for abscissa of LMHP for a handover</td></tr><tr><td>v<sub>[n]</sub></td><td>50</td><td>Set of speeds {v<sub>1</sub>, v<sub>2</sub>, ..., v<sub>n</sub>}</td></tr><tr><td>L<sub>e</sub>(v<sub>1</sub>, v<sub>2</sub>), L<sub>e</sub>(v<sub>[n]</sub>)</td><td>36, 50</td><td>Lower envelope in two-speed and n-speed case, respectively</td></tr><tr><td>L<sup>+</sup><sub>e</sub>(v<sub>1</sub>, v<sub>2</sub>), L<sup>+</sup><sub>e</sub>(v<sub>[n]</sub>)</td><td>36, 50</td><td>The open region above the lower envelope L<sub>e</sub>(v<sub>1</sub>, v<sub>2</sub>) and L<sub>e</sub>(v<sub>[n]</sub>) in two and multi-speed case, respectively</td></tr><tr><td>(ˆs, ˆh)</td><td>23, 67</td><td>≡ (ˆs(t<sub>1</sub>, h<sub>1</sub>, t<sub>2</sub>, h<sub>2</sub>), ˆh(t<sub>1</sub>, h<sub>1</sub>, t<sub>2</sub>, h<sub>2</sub>)), intersection point of two birds of same type at (t<sub>1</sub>, h<sub>1</sub>), (t<sub>2</sub>, h<sub>2</sub>)</td></tr><tr><td>(ˆs<sub>k</sub>, ˆh<sub>k</sub>)</td><td>67</td><td>≡ (ˆs<sub>k</sub>(t<sub>1</sub>, h<sub>1</sub>, t<sub>2</sub>, h<sub>2</sub>), ˆh<sub>k</sub>(t<sub>1</sub>, h<sub>1</sub>, t<sub>2</sub>, h<sub>2</sub>)), k-th intersection point of two birds of different type at (t<sub>1</sub>, h<sub>1</sub>) and (t<sub>2</sub>, h<sub>2</sub>), for k = 1, 2</td></tr><tr><td>E<sup>s,v</sup><sub>u</sub></td><td>15</td><td>Upper half-ellipse given by v<sup>2</sup>(t - s)<sup>2</sup> + h<sup>2</sup> = u<sup>2</sup>, for speed v, in (t, h)-coordinate system</td></tr><tr><td>H<sup>l</sup></td><td>35</td><td>HPP with head points of type l</td></tr><tr><td>C<sup>l</sup><sub>(t,h)</sub></td><td>35</td><td>Radial bird of type l with head at (t, h) ∈ H<sup>+</sup></td></tr><tr><td>H<sup>l</sup> + δ<sub>(t<sub>1</sub>,h<sub>1</sub>)</sub> + δ<sub>(t<sub>2</sub>,h<sub>2</sub>)</sub></td><td>45</td><td>Two-point Palm version of Σ<sub>i∈[n]</sub> H<sup>i</sup>, (t<sub>1</sub>, h<sub>1</sub>), (t<sub>2</sub>, h<sub>2</sub>) of type l</td></tr><tr><td>+ Σ<sub>i≠l</sub> H<sup>i</sup></td><td></td><td></td></tr><tr><td>P<sup>l</sup><sub>H</sub>(t<sub>1</sub>,h<sub>1</sub>), (t<sub>2</sub>,h<sub>2</sub>)</td><td>45</td><td>P<sup>(t<sub>1</sub>,h<sub>1</sub>),(t<sub>2</sub>,h<sub>2</sub>)</sup> ⊗ Ω<sub>i≠l</sub> P<sup>H<sup>i</sup></sup>, the two-point Palm probability measure for H<sup>l</sup> + δ<sub>(t<sub>1</sub>,h<sub>1</sub>)</sub> + δ<sub>(t<sub>2</sub>,h<sub>2</sub>)</sub> + Σ<sub>i≠l</sub> H<sup>i</sup></td></tr><tr><td>H<sup>l</sup> + δ<sub>(t<sub>1</sub>,h<sub>1</sub>)</sub> + H<sup>r</sup> + δ<sub>(t<sub>2</sub>,h<sub>2</sub>)</sub></td><td>45</td><td>Two-point Palm version of Σ<sub>i∈[n]</sub> H<sup>i</sup>, (t<sub>1</sub>, h<sub>1</sub>) and (t<sub>2</sub>, h<sub>2</sub>) of type l and r, respectively</td></tr><tr><td>+ Σ<sub>i≠l,r</sub> H<sup>i</sup></td><td></td><td></td></tr><tr><td>P<sup>(l,r)</sup><sub>H</sub>(t<sub>1</sub>,h<sub>1</sub>), (t<sub>2</sub>,h<sub>2</sub>)</td><td>48, 52</td><td>P<sup>(t<sub>1</sub>,h<sub>1</sub>)</sup> ⊗ P<sup>(t<sub>2</sub>,h<sub>2</sub>)</sup> ⊗ Ω<sub>i≠l,r</sub> P<sup>H<sup>i</sup></sup>, the two-point Palm probability measure for H<sup>l</sup> + δ<sub>(t<sub>1</sub>,h<sub>1</sub>)</sub> + H<sup>r</sup> + δ<sub>(t<sub>2</sub>,h<sub>2</sub>)</sub> + Σ<sub>i≠l,r</sub> H<sup>i</sup></td></tr><tr><td>V<sub>l</sub></td><td>43</td><td>HoPP of pure type l</td></tr><tr><td>R<sub>l</sub></td><td>44</td><td>PP for abscissas of RMHP for pure handovers of type l</td></tr><tr><td>L<sub>l</sub></td><td></td><td>PP for abscissas of LMHP for pure handovers of type l</td></tr><tr><td>(<sup>k</sup><sub>l,r</sub>)</td><td>43</td><td>The type of handover due to k-th intersection of radial birds of type l and r, with l ≠ r and the head point of type l is on the left of the head point of type r, where l, r, k ∈ {1, 2}</td></tr><tr><td>V<sup>(k)</sup><sub>l,r</sub> for k = 1, 2</td><td>43</td><td>HoPP of type (<sup>k</sup><sub>l,r</sub>)</td></tr></tbody></table>

Continued to next page
<table><caption>**Table (12.1) (continued from previous page)**</caption><thead><tr><th>Symbol</th><th>Page</th><th>Description</th></tr></thead><tbody><tr><td>V</td><td>44</td><td>:= &Sigma;<sub>l &isin; {1,2}</sub> V<sub>l</sub> + &Sigma;<sub>r &ne; l, k &isin; {1,2}</sub> V<sub>l,r</sub><sup>(k)</sup></td></tr><tr><td>&Lambda;<sub>l,r</sub><sup>(k)</sup> for k = 1, 2</td><td>44</td><td>Intensity of the HoPP V<sub>l,r</sub><sup>(k)</sup> of type (k) (l,r)</td></tr><tr><td>&lambda;<sub>V</sub></td><td>44</td><td>:= &Sigma;<sub>l &isin; {1,2}</sub> &Lambda;<sub>l</sub> + &Sigma;<sub>r &ne; l, k &isin; {1,2}</sub> &Lambda;<sub>l,r</sub><sup>(k)</sup>, total handover frequency</td></tr><tr><td>R<sub>l,r</sub><sup>(k)</sup> for k = 1, 2</td><td>44</td><td>PP for abscissas of RMHP for mixed handover of type (k) (l,r)</td></tr><tr><td>L<sub>l,r</sub><sup>(k)</sup> for k = 1, 2</td><td>47</td><td>PP for abscissas of LMHP for mixed handovers of type (k) (l,r)</td></tr><tr><td>[<sup>q</sup>&frasl;<sub>i,j</sub>]</td><td>62</td><td>Types of handovers, with new serving station is of type j, the previous serving station is of type i and q = 1 or 2, depending on the head point corresponding to the new serving station is on left or right</td></tr><tr><td>W<sub>i</sub></td><td>64</td><td>The version of V<sub>l</sub>, where (1) &rarr; [<sup>2</sup>&frasl;<sub>i,i</sub>]</td></tr><tr><td>L<sub>i</sub></td><td>64</td><td>The version of &Lambda;<sub>l</sub>, where (1) &rarr; [<sup>2</sup>&frasl;<sub>i,i</sub>]</td></tr><tr><td>W<sub>i,j</sub><sup>(q)</sup></td><td>64</td><td>The version of V<sub>l,r</sub><sup>(k)</sup>, where (k) &rarr; [<sup>q</sup>&frasl;<sub>i,j</sub>]</td></tr><tr><td>L<sub>i,j</sub><sup>(q)</sup></td><td>64</td><td>The version of &Lambda;<sub>l,r</sub><sup>(k)</sup>, where (k) &rarr; [<sup>q</sup>&frasl;<sub>i,j</sub>]</td></tr><tr><td>W</td><td>64</td><td>:= &Sigma;<sub>i &isin; {1,2}</sub> W<sub>i</sub> + &Sigma;<sub>i &ne; j, q &isin; {1,2}</sub> W<sub>i,j</sub><sup>(q)</sup> &equiv; V</td></tr><tr><td>&lambda;<sub>W</sub></td><td>64</td><td>&equiv; &lambda;<sub>V</sub></td></tr><tr><td>[<sup>q, q'</sup>&frasl;<sub>i, j, m</sub>]</td><td>65</td><td>Type of pair of consecutive handovers, where the next and the upcoming serving stations are of type j, m, respectively and the previous serving station is of type i. q = 1 or 2, depending on the head point corresponding to the next serving station is on the left or right that of previous one. q' = 1 or 2, depending on the head point corresponding to the upcoming serving station is on the left or right that of next</td></tr></tbody></table>

# APPENDIX A.

## A.1. Displacement theorem for the Poisson point process.
For some $t > 0$, define the independent random displacement map $D_t: \mathbb{R}^2 \times (-\pi, \pi] \to \mathbb{R}^2$ such that for any $x \in \mathbb{R}^2$ and direction $\theta \in (-\pi, \pi]$,

$$D_t(x, \theta) := x + v_x t,$$

where $v_x := (v \cos \theta, v \sin \theta)$, is the velocity vector of the atom at $x$. The displacements are all i.i.d. but all equal in norm for every atom and fixed $t$. For any $t > 0$, the displacement kernel $\kappa_t: \mathbb{R}^2 \times (-\pi, \pi] \times \mathcal{B}(\mathbb{R}^2) \to \mathbb{R}^+$, is defined as

$$\kappa_t(x, \theta, B) := \mathbb{P}(D_t(x, \theta) \in B | \tilde{\Phi}) = \mathbb{P}(x + v_x t \in B | \tilde{\Phi}) = \mathbb{1}_{\{x+v_x t \in B\}}, \quad (1)$$

for any $B \in \mathcal{B}(\mathbb{R}^2)$, initial location $x \in \mathbb{R}^2$ and direction of motion $\theta \in (-\pi, \pi]$, where $\mathcal{B}(\mathbb{R}^2)$ is the Borel $\sigma$-algebra on $\mathbb{R}^2$. Since $\kappa_t(x, \theta, \mathbb{R}^2) = 1$, the measure $\kappa_t(x, \theta, \cdot)$ is a probability kernel for any $x \in \mathbb{R}^2$ and $\theta \in (-\pi, \pi]$, see the [3, definition 14.D.1].

**Proposition A.1.** For any $t > 0$, the following statements are true:

(i) The displacement kernel $\kappa_t$ preserves the Lebesgue measure.

(ii) The point process $\tilde{\Phi}^t$ has the same distribution as $\Phi$.

*Proof of Proposition A.1(i).* For the first part, let $B$ be a Borel subset of $\mathbb{R}^2$, $x \in \mathbb{R}^2$ and $\theta \in (-\pi, \pi]$. The displacement kernel is $\kappa_t(x, \theta, B) = \mathbb{1}_{\{x+v_x t \in B\}}$. In turn, the intensity measure of the point process
$\tilde{\Phi}^t$ becomes

$$m_{\tilde{\Phi}^t}(B) = \mathbb{E}[\tilde{\Phi}^t(B)] = \mathbb{E}\left[\mathbb{E}\left[\sum_{i \in \mathbb{N}} \mathbb{1}_{\{Y_i \in B\}} | \tilde{\Phi}\right]\right] = \mathbb{E}\left[\sum_{i \in \mathbb{N}} \mathbb{E}\left[\mathbb{1}_{\{Y_i \in B\}} | \tilde{\Phi}\right]\right] = \mathbb{E}\left[\sum_{i \in \mathbb{N}} \kappa_t(X_i, \Theta_i, B)\right]. \quad (2)$$

Applying the Campbell-Mecke formula in (2), we get

$$\mathbb{E}\left[\sum_{i \in \mathbb{N}} \kappa_t(X_i, \Theta_i, B)\right] = \mathbb{E}\left[\int_{\mathbb{R}^2 \times (-\pi, \pi]} \kappa_t(x, \theta, B) m_{\tilde{\Phi}}(dx, d\theta)\right] = \int_{\mathbb{R}^2} \int_{-\pi}^{\pi} \kappa_t(x, \theta, B) \frac{\lambda}{2\pi} dx d\theta. \quad (3)$$

Using the displacement kernel from (1) the last term in (3) equals to

$$\frac{\lambda}{2\pi} \int_{\mathbb{R}^2} \int_{-\pi}^{\pi} \mathbb{1}_{\{x+v_x t \in B\}} dx \, d\theta = \frac{\lambda}{2\pi} \int_{-\pi}^{\pi} \int_{\mathbb{R}^2} \mathbb{1}_{\{x \in B - v_x t\}} dx \, d\theta = \frac{\lambda}{2\pi} \int_{-\pi}^{\pi} |B - v_x t| d\theta = \lambda |B|. \quad (4)$$

Thus the intensity measure $m_{\tilde{\Phi}^t}$ of the point process $\tilde{\Phi}^t$ also has Lebesgue density. $\square$

**Proof of Proposition A.1 (ii).** For the second part, since the displacements are i.i.d., by the displacement theorem [3, Theorem 2.2.17], the point process $\tilde{\Phi}^t$ is Poisson. This can be seen through the Laplace transform of the point process $\tilde{\Phi}^t$. Let us enumerate the points of $\Phi$ and $\Phi^t$ respectively as $\{X_i\}_{i \in \mathbb{N}}$ and $\{Y_i\}_{i \in \mathbb{N}}$ respectively. Let $f: \mathbb{R}^2 \to \mathbb{R}^+$ be a measurable function. Then the Laplace transform of the function $f$ with respect to $\tilde{\Phi}^t$ is

$$\mathcal{L}_{\tilde{\Phi}^t}(f) := \mathbb{E}\left[\exp\left(-\int_{\mathbb{R}^2} f \, d\tilde{\Phi}^t\right)\right] = \mathbb{E}\left[\exp\left(-\sum_{i \in \mathbb{N}} f(Y_i)\right)\right] = \mathbb{E}\left[\prod_{i \in \mathbb{N}} e^{-f(Y_i)}\right]. \quad (5)$$

Conditioning with respect to the point process $\tilde{\Phi}$ for last term in (5), we have

$$\mathbb{E}\left[\prod_{i \in \mathbb{N}} e^{-f(Y_i)}\right] = \mathbb{E}\left[\mathbb{E}\left[\prod_{i \in \mathbb{N}} e^{-f(Y_i)} | \tilde{\Phi}\right]\right] = \mathbb{E}\left[\prod_{i \in \mathbb{N}} \int_{\mathbb{R}^2} e^{-f(y)} \kappa_t(X_i, \Theta_i, dy)\right]. \quad (6)$$

From (1) we write, $\kappa_t(X_i, \Theta_i, dy) = 1_{\{X_i+V_i t \in dy\}}$. Let us now define a measurable function $g: \mathbb{R}^2 \times (-\pi, \pi] \to \mathbb{R}^+$ using the inner integral in (6),

$$g(x, \theta) = -\log \int_{\mathbb{R}^2} e^{-f(y)} \kappa_t(x, \theta, dy). \quad (7)$$

Using the function $g$ from (7) in (6) we obtain the Laplace transform as

$$\mathcal{L}_{\tilde{\Phi}^t}(f) = \mathbb{E}\left[\exp\left(-\sum_{i \in \mathbb{N}} g(X_i, \Theta_i)\right)\right] = \mathbb{E}\left[\exp\left(-\int_{\mathbb{R}^2 \times (-\pi, \pi]} g d\tilde{\Phi}\right)\right] = \mathcal{L}_{\tilde{\Phi}}(g). \quad (8)$$

Evaluating the Laplace transform $\mathcal{L}_{\tilde{\Phi}}(g)$ of the function $g$ with respect to the marked Poisson point process $\tilde{\Phi}$, we obtain

$$\begin{align*}
\mathcal{L}_{\tilde{\Phi}}(g) &= \exp\left(-\int_{\mathbb{R}^2} \int_{-\pi}^{\pi} (1 - e^{-g(x, \theta)}) m_{\tilde{\Phi}}(dx, d\theta)\right) \\
&\stackrel{(7)}{=} \exp\left(-\int_{\mathbb{R}^2} \int_{-\pi}^{\pi} \left(1 - \int_{\mathbb{R}^2} e^{-f(y)} \mathbb{1}_{\{x+v_x t \in dy\}}\right) m_{\tilde{\Phi}}(dx, d\theta)\right) \\
&\stackrel{(Prop. A.1(i))}{=} \exp\left(-\int_{\mathbb{R}^2} \int_{-\pi}^{\pi} \int_{\mathbb{R}^2} (1 - e^{-f(y)}) \mathbb{1}_{\{x+v_x t \in dy\}} m_{\tilde{\Phi}}(dx, d\theta)\right) \\
&\stackrel{(Fubini)}{=} \exp\left(-\int_{\mathbb{R}^2} \left(1 - e^{-f(y)}\right) \int_{\mathbb{R}^2} \int_{-\pi}^{\pi} \mathbb{1}_{\{x+v_x t \in dy\}} m_{\tilde{\Phi}}(dx, d\theta)\right) \\
&\stackrel{(4)}{=} \exp\left(-\int_{\mathbb{R}^2} \left(1 - e^{-f(y)}\right) m_{\tilde{\Phi}^t}(dy)\right) \\
&\stackrel{(Prop. A.1(i))}{=} \exp\left(-\int_{\mathbb{R}^2} \left(1 - e^{-f(y)}\right) \lambda dy\right) = \mathcal{L}_{\Phi}(f), \quad (9)
\end{align*}$$
since from part (i), we get that $m_{\tilde{\Phi}t}$ has Lebesgue density. We obtain from (8) and (9) that $\mathcal{L}_{\tilde{\Phi}t}(f) = \mathcal{L}_{\Phi}(f)$ and hence by the characterization of the Poisson point process, the point process $\tilde{\Phi}t$ corresponding to the new locations is also Poisson. $\square$

**A.2.** *Proof of Lemma 5.13.* Fix $\hat{h}_0 > 0$ and suppose there are two radial birds with head at $(t_1, h_1)$ and $(t_2, h_2)$, such that $t_2 \le t_1$. The radial birds $C_{(t_1,h_1)}$ and $C_{(t_1,h_1)}$ intersect at $(0, \hat{h}_0)$ and $(t_2, h_2)$ lies anywhere on the part of the boundary $U_{\hat{h}_0}^0$ lying on the left of the vertical line $t = t_1$. We prove the result in two cases: $t_1 \ge 0$ and $t_1 < 0$.

**Case 1.** $t \ge 0$. Consider the set $U_{\hat{h}_t}^t \setminus U_{\hat{h}_0}^0$, where $\hat{h}_t := ((t-t_1)^2 + h_1^2)^{\frac{1}{2}}$, for some fixed $(t_1, h_1) \in \partial U_{\hat{h}_0}^0$. Let $Q_{t_1} := \{(t,h) \in \mathbb{H}^+ : t \ge t_1\}$, be the region on the right of the vertical line $t = t_1$ in $\mathbb{H}^+$.

<figure><img src="image_54.png" /><figcaption>FIGURE A.1. For $0 < \tau_1 < \tau_2$, the half balls in color teal and red correspond to $\tau_1$ and $\tau_2$, respectively. Here $U_{\hat{h}_{\tau_1}}^t \cap Q_{t_1} \subsetneq U_{\hat{h}_{\tau_2}}^t \cap Q_{t_1}$. The blue and purple half balls are for reference.</figcaption></figure>

Define $R_t := U_{\hat{h}_t}^t \cap Q_{t_1}$. Our claim holds if and only if the same holds for the collection $\{R_t\}_{t>0}$, since for each $t > 0$, $R_t = (U_{\hat{h}_t}^t \setminus U_{\hat{h}_0}^0) \cup (U_{\hat{h}_0}^0 \cap Q_{t_1})$ and $U_{\hat{h}_t}^t \setminus U_{\hat{h}_0}^0 \subset R_t$. Let $0 < \tau_1 < \tau_2$. The proof will be complete if we can prove that $R_{\tau_1} \subsetneq R_{\tau_2}$. Observe that $\partial U_{\hat{h}_{\tau_1}}^{\tau_1}$ and $\partial U_{\hat{h}_{\tau_2}}^{\tau_2}$ intersects at $(t_1, h_1)$, and they intersect each other just once. Using properties of two non-concentric and intersecting circles, for all $t' < t_1$, $(\hat{h}_{\tau_1}^2 - (t' - \tau_1)^2)^{\frac{1}{2}} > (\hat{h}_{\tau_2}^2 - (t' - \tau_2)^2)^{\frac{1}{2}}$ and for all $t' > t_1$, $(\hat{h}_{\tau_1}^2 - (t' - \tau_1)^2)^{\frac{1}{2}} < (\hat{h}_{\tau_2}^2 - (t' - \tau_2)^2)^{\frac{1}{2}}$, irrespective of the values of $\hat{h}_{\tau_1}$ and $\hat{h}_{\tau_2}$. This in particular implies $U_{\hat{h}_{\tau_1}}^t \cap Q_{t_1} \subsetneq U_{\hat{h}_{\tau_2}}^t \cap Q_{t_1}$, i.e., $R_{\tau_1} \subsetneq R_{\tau_2}$, see Figure A.1.

<figure><img src="image_55.png" /><figcaption>FIGURE A.2. For $0 < \tau_1 < \tau_2$, the half balls in color teal and red correspond to $\tau_1$ and $\tau_2$, respectively. Here $U_{\hat{h}_{\tau_1}}^t \cap Q_{t_1} \subsetneq U_{\hat{h}_{\tau_2}}^t \cap Q_{t_1}$. The purple half ball is just a reference with respect to the next handover.</figcaption></figure>

**Case 2.** $t_1 < 0$. One can prove the claim for the case $t_1 < 0$ similarly, since the intersection point corresponding to the handover at time 0, lies on the increasing wing of the radial bird $C_{(t_1,h_1)}$, as depicted in Figure A.2. By the increasing wing property as in Remark 5.14, for all $t \ge t_1$, the extra region $U_{\hat{h}_t}^t \setminus U_{\hat{h}_0}^0$ created by subsequent half-balls, is increasing. $\square$
### A.3. Proof of Proposition 8.16.
We first provide the justification for the claim for the three type of handovers at time 0, for $\tau_n = 1$, i.e., when the station that takes over at time 0 is of type 1. This corresponds to intersections of types 1, $(\frac{1}{2,1})$ or $[\frac{2}{2,1}]$ and $(\frac{1}{1,2})$ or $[\frac{1}{2,1}]$, respectively. Let us start the proof of Part (i) for these cases.

**Proof of Proposition 8.16, Part (i), typical handover type $(\frac{1}{2,1})$ or equivalently $[\frac{2}{2,1}]$.** Suppose the handover happens at a distance $\hat{h}_0$, which is given by the first intersection of two radial birds with head points $(t_n, h_n) = (t_1, h_1)$ of type $\tau_n = 1$ and $(t_p, h_p) = (t_2, h_2)$ of type $\tau_p = 2$, such that $t_1 \ge t_2$. By construction, $(t_1, h_1)$ lies on $\partial E_{\hat{h}_0}^{0,v_1}$ and $(t_2, h_2)$ lies on $\partial E_{\hat{h}_0}^{0,v_2}$. In fact, for any $(t_1, h_1)$ lying on $\partial E_{\hat{h}_0}^{0,v_1}$ and $(t_2, h_2)$ lying on $\partial E_{\hat{h}_0}^{0,v_2}$, we get the same handover. Given $\hat{h}_0, t_1, h_1$, define $\hat{h}_1(t) := (v_1^2(t-t_1)^2 + h_1^2)^{\frac{1}{2}}$ and consider the two collections $\{E_{\hat{h}_1(t)}^{t,v_1}\}_{t \ge 0}$ and $\{E_{\hat{h}_1(t)}^{t,v_2}\}_{t \ge 0}$, of upper half ellipses $E_{\hat{h}_1(t)}^{t,v_1}$ and $E_{\hat{h}_1(t)}^{t,v_2}$ with their top point lying on the radial bird $C_{(t_1,h_1)}^1$.

**Case 1.** $\{E_{\hat{h}_1(t)}^{t,v_1} \setminus E_{\hat{h}_0}^{0,v_1}\}_{t \ge 0}$. It can be proved, using the techniques applied for the single-speed case, that the collection $\{E_{\hat{h}_1(t)}^{t,v_1} \setminus E_{\hat{h}_0}^{0,v_1}\}_{t \ge 0}$ is non-decreasing in $t$, since the boundaries $\partial E_{\hat{h}_1(t)}^{t,v_1}$ pass through $(t_1, h_1)$, for all $t$, see Picture 1 of Figure A.3.

<figure><img src="image_56.png" /></figure>

(1) The collection $\{E_{\hat{h}_1(t)}^{t,v_1} \setminus E_{\hat{h}_0}^{0,v_1}\}_{t \ge 0}$ is non-decreasing, since all the boundaries pass through $(t_1, h_1)$.

<figure><img src="image_57.png" /></figure>

(2) Existence of $0 \le \tilde{t} \le t_1$, given $(t_1, h_1)$. The collection $\{E_{\hat{h}_1(t)}^{t,v_2} \setminus E_{\hat{h}_0}^{0,v_2}\}_{t \ge 0}$ is non-decreasing, for all $t \in [\tilde{t}, t_1]$.

<caption>FIGURE A.3. Monotonicity property of the collection $\{E_{\hat{h}_1(t)}^{t,v_l} \setminus E_{\hat{h}_0}^{0,v_l}\}_{t \ge 0}$.</caption>

**Case 2.** $\{E_{\hat{h}_1(t)}^{t,v_2} \setminus E_{\hat{h}_0}^{0,v_2}\}_{t \ge 0}$. We prove this case in two parts. In the first part, for $t \in [t_1, \infty)$, using the increasing wing property, as described in Remark 8.17, $\{E_{\hat{h}_1(t)}^{t,v_2} \setminus E_{\hat{h}_0}^{0,v_2}\}_{t \ge t_1}$ is non-decreasing.

We consider now the part where $t \in [0, t_1]$, which corresponds to the decreasing wing of the radial bird $C_{(t_1,h_1)}^1$. We first claim that, given $(t_1, h_1)$, there exists $\tilde{t} > 0$ such that, for all $t < \tilde{t}$, $E_{\hat{h}(t)}^{t,v_2} \subset E_{\hat{h}_0}^{0,v_2}$ and for all $t \ge \tilde{t}$, $E_{\hat{h}(t)}^{t,v_2} \cap (E_{\hat{h}_0}^{0,v_2})^c \ne \emptyset$. Indeed, the critical value $\tilde{t}$ is given by the non-trivial solution of the equation $t + \hat{h}_t/v_2 = \hat{h}_0/v_2$, which is either 0 or $\frac{2}{v_1^2-v_2^2}(v_1^2t_1 - v_2\hat{h}_0)$. Define,

$$ \tilde{t} = \begin{cases} 0, & \text{if } v_1^2 t_1 \le v_2 \hat{h}_0, \\ \frac{2}{v_1^2 - v_2^2} (v_1^2 t_1 - v_2 \hat{h}_0), & \text{if } v_1^2 t_1 > v_2 \hat{h}_0. \end{cases} \qquad (10) $$

Given $v_1, v_2, \hat{h}_0$, define

$$ t^* := \begin{cases} 0, & \text{if } v_1^2 t_1 \le v_2 \hat{h}_0, \\ \frac{2v_2 \hat{h}_0}{v_1^2 + v_2^2}, & \text{if } v_1^2 t_1 > v_2 \hat{h}_0. \end{cases} \qquad (11) $$
As a side remark, observe that $t^*$ is given by an intersection of the line segment joining the two points $(0, \hat{h}_0)$ and $(\hat{h}_0/v_2, 0)$ and $\partial E_{\hat{h}_0}^{0,v_1}$, which are $(0, \hat{h}_0)$ and $(\frac{2v_2\hat{h}_0}{v_1^2+v_2^2}, \frac{\hat{h}_0(v_2^2-v_2^2)}{v_1^2+v_2^2})$.

**Subcase 2.1.** $v_1^2 t_1 > v_2 \hat{h}_0$. In this case $\tilde{t} = \frac{2}{v_1^2-v_2^2}(v_1^2 t_1 - v_2 \hat{h}_0)$ and $t^* = \frac{2v_2 \hat{h}_0}{v_1^2+v_2^2}$. One can show from (10) that, for all $(t_1, h_1) \in \partial E_{\hat{h}_0}^{0,v_1}$ such that, for all $t_1 \in [0, t^*]$, we have $\tilde{t} \le t_1$ and for all $t_1 \in (t^*, \hat{h}_0/v_1]$, we have $\tilde{t} > t_1$. Indeed, if $t_1 \in (t^*, \hat{h}_0/v_1]$, then there does not exist any radial bird of type 2 with head point lying outside $E_{\hat{h}_0}^{0,v_2}$ that intersects the radial bird $C_{(t_1,h_1)}^1$, twice on its decreasing wing. We perform further analysis by dividing in two cases: $(t_1, h_1)$ being such that $t_1 \in (t^*, \hat{h}_0/v_1]$ or $t_1 \in [0, t^*]$.

**Subsubcase 2.1.1.** $t_1 \in (t^*, \hat{h}_0/v_1]$. In this case, by definition of $t^*$, we have $\tilde{t} > t_1$ and by definition of $\tilde{t}$, $E_{\hat{h}(t)}^{t,v_2} \subset E_{\hat{h}_0}^{0,v_2}$, for all $t < \tilde{t}$ and in particular for all $t < t_1$. Also, since $\tilde{t} > t_1$, for $t_1 \in (t^*, \hat{h}_0/v_1]$, $\tilde{t}$ corresponds to the increasing wing of the bird $C_{(t_1,h_1)}^1$. Hence, using the increasing wing property, as described in Remark 8.17, the collection of open sets $\{E_{\hat{h}_1(t)}^{t,v_2} \setminus E_{\hat{h}_0}^{0,v_2}\}_{t \ge t_1}$ is non-decreasing, and so is $\{E_{\hat{h}_1(t)}^{t,v_2} \setminus E_{\hat{h}_0}^{0,v_2}\}_{t \ge \tilde{t}}$ in particular. Also by definition of $\tilde{t}$, $E_{\hat{h}_1(t)}^{t,v_2} \setminus E_{\hat{h}_0}^{0,v_2} = \emptyset$, for all $t \in [0, \tilde{t}]$. As a whole, the collection $\{E_{\hat{h}_1(t)}^{t,v_2} \setminus E_{\hat{h}_0}^{0,v_2}\}_{t \ge 0}$ is non-decreasing.

**Subsubcase 2.1.2.** $t_1 \in [0, t^*]$. Here again, by definition of $t^*$, we have $\tilde{t} < t_1$. We are interested in understanding the behavior of the collection $\{E_{\hat{h}_1(t)}^{t,v_2} \setminus E_{\hat{h}_0}^{0,v_2}\}_{t \in [\tilde{t}, t_1]}$. It only concerns the interval $[\tilde{t}, t_1]$, because $\{E_{\hat{h}_1(t)}^{t,v_2} \setminus E_{\hat{h}_0}^{0,v_2}\}_{t \ge t_1}$ is anyway non-decreasing using the increasing wing property, as in Remark 8.17 and $E_{\hat{h}_1(t)}^{t,v_2} \setminus E_{\hat{h}_0}^{0,v_2} = \emptyset$, for $t \in [0, \tilde{t}]$. Let us present a general argument for the monotonicity property of interest for the interval $[\tilde{t}, t_1]$.

Let us take a time point $\tau \in [\tilde{t}, t_1)$ and choose $\varepsilon > 0$ arbitrarily small. The boundaries of $\partial E_{\hat{h}_1(\tau)}^{\tau,v_2}$ and $\partial E_{\hat{h}_1(\tau+\varepsilon)}^{\tau+\varepsilon,v_2}$ are given by the equations

$$v_2^2(t - \tau)^2 + h^2 = \hat{h}_1(\tau)^2, \quad v_2^2(t - \tau - \varepsilon)^2 + h^2 = \hat{h}_1(\tau + \varepsilon)^2, \qquad (12)$$

respectively, where

$$\hat{h}_1(\tau) = (v_1^2(\tau - t_1)^2 + h_1^2)^{\frac{1}{2}}, \quad \hat{h}_1(\tau + \varepsilon) = (v_1^2(\tau + \varepsilon - t_1)^2 + h_1^2)^{\frac{1}{2}}. \qquad (13)$$

Suppose $(\bar{t}, \bar{h}) = (\bar{t}(\tau, \varepsilon), \bar{h}(\tau, \varepsilon)) \in \mathbb{H}^+$ is an intersection point, if it exists, of the two ellipses in (12). We have the following proposition:

**Proposition A.2.** For any $\tau$ and $\varepsilon > 0$, with $\tilde{t} \le \tau < t_1$, the following holds true:

(a). $(\bar{t}, \bar{h})$ exists and $(\bar{t}, \bar{h}) \in E_{\hat{h}_0}^{0,v_2}$,

(b). $E_{\hat{h}_1(\tau)}^{\tau,v_2} \setminus E_{\hat{h}_0}^{0,v_2} \subset E_{\hat{h}_1(\tau+\varepsilon)}^{\tau+\varepsilon,v_2} \setminus E_{\hat{h}_0}^{0,v_2}$.

By Proposition A.2, the collection of sets $\{E_{\hat{h}_1(t)}^{t,v_2} \setminus E_{\hat{h}_0}^{0,v_2}\}_{t \in [\tilde{t}, t_1]}$ is non-decreasing, in particular. Finally, the sequence $\{E_{\hat{h}_1(t)}^{t,v_2} \setminus E_{\hat{h}_0}^{0,v_2}\}_{t \ge 0}$ is non-decreasing.

**Subcase 2.2.** $v_1^2 t_1 \le v_2 \hat{h}_0$. Then $t^* = 0$, and $\tilde{t} = 0$ as well. By definition of $\tilde{t}$, $E_{\hat{h}_1(\tau)}^{\tau,v_2} \cap (E_{\hat{h}_0}^{0,v_2})^c \ne \emptyset$, for all $\tau \ge 0$. We establish the claim using the same steps of Subsubcase 2.1.2, along with the help of Proposition A.2, with $\tilde{t} = 0$. We do not provide the complete proof of Proposition A.2, with $\tilde{t} = 0$. As a matter of fact, we just show that $(\hat{h}_1(0))' \ge -v_2$. The last inequality holds true, since from (17), we
<figure><img src="image_58.png" /><figcaption>FIGURE A.4. For any $\tau \in [\tilde{t}, t_1)$ and the intersection point $(\tilde{t}, \bar{h})$ of the two boundaries $\partial E_{\hat{h}_1(\tau)}^{\tau, v_2}$ and $\partial E_{\hat{h}_1(\tau+\varepsilon)}^{\tau+\varepsilon, v_2}$, belongs to $E_{\hat{h}_0}^{0, v_2}$.</figcaption></figure>

have

$$ (\hat{h}_1(0))' = -\frac{v_1^2 t_1}{\hat{h}_0} \geq -v_2, \qquad (14) $$

using the assumption $v_1^2 t_1 \leq v_2 \hat{h}_0$ of this Subcase 2.2.

This completes the proof of Part (i) of Proposition 8.16, when the typical handover is of type $\begin{bmatrix} 1 \\ 2,1 \end{bmatrix}$ or equivalently $\begin{bmatrix} 2 \\ 2,1 \end{bmatrix}$. $\square$

*Proof of Proposition A.2, Part (a).* We first prove that the two boundaries $\partial E_{\hat{h}_1(\tau)}^{\tau, v_2}$ and $\partial E_{\hat{h}_1(\tau+\varepsilon)}^{\tau+\varepsilon, v_2}$ intersect for any $0 \le \tau < t_1$ and arbitrarily small $\varepsilon > 0$. For this, it is enough to show that, $\hat{h}_1(\tau+\varepsilon) \le \hat{h}_1(\tau)$ and

$$ \tau + \varepsilon + \frac{1}{v_2} \hat{h}_1(\tau + \varepsilon) \ge \tau + \frac{1}{v_2} \hat{h}_1(\tau) \quad \text{i.e.,} \quad \frac{\hat{h}_1(\tau + \varepsilon) - \hat{h}_1(\tau)}{\varepsilon} \ge -v_2, \qquad (15) $$

i.e., the right most point $(\tau + \varepsilon + \frac{1}{v_2} \hat{h}_1(\tau + \varepsilon), 0)$ of the upper half-ellipse $E_{\hat{h}_1(\tau+\varepsilon)}^{\tau+\varepsilon, v_2}$, is on the right of the right-most point $(\tau + \frac{1}{v_2} \hat{h}_1(\tau), 0)$ of the upper half-ellipse $E_{\hat{h}_1(\tau)}^{\tau, v_2}$. The first inequality $\hat{h}_1(\tau + \varepsilon) \le \hat{h}_1(\tau)$ is true since, $\tau < t_1$, which correspond to the decreasing wing of $C_{(t_1, h_1)}^1$. The second inequality (15) is true if we can show that $(\hat{h}_1(\tau))' \ge -v_2$ for all $0 \le \tau < t_1$, where

$$ (\hat{h}_1(\tau))' = \frac{v_1^2(\tau - t_1)}{\hat{h}_1(\tau)}. \qquad (16) $$

If we can show that $(\hat{h}_1(\tilde{t}))' \ge -v_2$, then, by the property of the decreasing wing of the radial bird $C_{(t_1, h_1)}^1$, we have $(\hat{h}_1(\tau))' > (\hat{h}_1(\tilde{t}))' \ge -v_2$, for all $\tau \in [\tilde{t}, t_1)$. Indeed, using the equality $\tilde{t} + \hat{h}_1(\tilde{t})/v_2 = \hat{h}_0/v_2$, we get

$$ (\hat{h}_1(\tilde{t}))' = \frac{v_1^2(\tilde{t} - t_1)}{\hat{h}_1(\tilde{t})} = \frac{v_1^2(\tilde{t} - t_1)}{\hat{h}_0 - v_2\tilde{t}}. \qquad (17) $$

We have $\frac{v_1^2(\tilde{t}-t_1)}{\hat{h}_0-v_2\tilde{t}} \ge -v_2$, since

$$ v_1^2(\tilde{t} - t_1) + v_2(\hat{h}_0 - v_2\tilde{t}) = (v_1^2 - v_2^2) \left( \tilde{t} - \frac{1}{v_1^2 - v_2^2} (v_1^2 t_1 - v_2 \hat{h}_0) \right) \stackrel{(10)}{=} v_1^2 t_1 - v_2 \hat{h}_0 \ge 0. \qquad (18) $$

This proves the inequality (15) and as a result shows the existence of the intersection between the two boundaries $\partial E_{\hat{h}_1(\tau)}^{\tau, v_2}$ and $\partial E_{\hat{h}_1(\tau+\varepsilon)}^{\tau+\varepsilon, v_2}$.
We find the coordinate of the intersection $(\bar{t}, \bar{h})$ of the two ellipses using the governing equations in (12), where

$$ \bar{t} = \frac{1}{2} \left[ 2\tau + \varepsilon + \frac{\hat{h}_1^2(\tau) - \hat{h}_1^2(\tau + \varepsilon)}{v_2^2 \varepsilon} \right], \quad \bar{h}^2 = \hat{h}_1^2(\tau) - v_2^2(\bar{t} - \tau)^2 = \hat{h}_1^2(\tau) - \frac{v_2^2}{4} \left[ \varepsilon + \frac{\hat{h}_1^2(\tau) - \hat{h}_1^2(\tau + \varepsilon)}{v_2^2 \varepsilon} \right]^2. \quad (19) $$

Let us first prove that for any $\tau$, with $\tilde{t} \le \tau < t_1$, $(\bar{t}, \bar{h}) \in E_{\hat{h}_0}^{0,v_2}$, i.e., $0 < v_2^2 \bar{t}^2 + \bar{h}^2 < \hat{h}_0^2$. Using the expressions of $\bar{t}, \bar{h}^2$ from (19) we get that,

$$ \begin{aligned} v_2^2 \bar{t}^2 + \bar{h}^2 &= \frac{v_2^2}{4} \left[ 2\tau + \varepsilon + \frac{\hat{h}_1^2(\tau) - \hat{h}_1^2(\tau + \varepsilon)}{v_2^2 \varepsilon} \right]^2 + \hat{h}_1^2(\tau) - \frac{v_2^2}{4} \left[ \varepsilon + \frac{\hat{h}_1^2(\tau) - \hat{h}_1^2(\tau + \varepsilon)}{v_2^2 \varepsilon} \right]^2 \\ &= v_2^2 \tau \left[ \tau + \varepsilon + \frac{\hat{h}_1^2(\tau) - \hat{h}_1^2(\tau + \varepsilon)}{v_2^2 \varepsilon} \right] + \hat{h}_1^2(\tau) \quad (20) \\ &= v_2^2 \tau (\tau + \varepsilon) + \frac{\tau + \varepsilon}{\varepsilon} \hat{h}_1^2(\tau) - \frac{\tau}{\varepsilon} \hat{h}_1^2(\tau + \varepsilon). \quad (21) \end{aligned} $$

Now, due to the fact that $(\tau, \hat{h}_1(\tau)), (\tau + \varepsilon, \hat{h}_1(\tau + \varepsilon)) \in C_{(t_1, h_1)}^1$, using the expressions for $\hat{h}_1(\tau)$ and $\hat{h}_1(\tau + \varepsilon)$ from (13) to the last expression in (21), we obtain that

$$ \begin{aligned} v_2^2 \bar{t}^2 + \bar{h}^2 &= v_2^2 \tau (\tau + \varepsilon) + \frac{\tau + \varepsilon}{\varepsilon} [v_1^2 (\tau - t_1)^2 + h_1^2] - \frac{\tau}{\varepsilon} [v_1^2 (\tau + \varepsilon - t_1)^2 + h_1^2] \\ &= v_2^2 \tau (\tau + \varepsilon) + h_1^2 + v_1^2 (t_1^2 - \tau (\tau + \varepsilon)) \\ &= h_1^2 + v_1^2 t_1^2 - (v_1^2 - v_2^2) \tau (\tau + \varepsilon) = \hat{h}_0^2 - (v_1^2 - v_2^2) \tau (\tau + \varepsilon), \quad (22) \end{aligned} $$

since $h_1^2 + v_1^2 t_1^2 = \hat{h}_0^2$. The quantity in the last step of (22) is less than $\hat{h}_0^2$, since $v_1 > v_2$ and greater than 0, which is clear from (20), since $\tau < t_1$ and $\varepsilon > 0$ is arbitrarily small. This implies $(\bar{t}, \bar{h}) \in E_{\hat{h}_0}^{0,v_2}$, see Figure A.4. $\square$

**Proof of Proposition A.2, Part (b).** Let us take $\tau \in [\tilde{t}, t_1)$ and $\varepsilon > 0$ arbitrarily small. Then $(\bar{t}, \bar{h})$ being the intersection of $\partial E_{\hat{h}_1(\tau_1)}^{\tau, v_2}$ and $\partial E_{\hat{h}_1(\tau+\varepsilon)}^{\tau+\varepsilon, v_2}$, implies that for all $t' < \bar{t}$, we have $v_2^2(t' - \tau)^2 + h^2 < v_2^2(t' - \tau - \varepsilon)^2 + h^2$ and for all $t' > \bar{t}$, we have $v_2^2(t' - \tau)^2 + h^2 > v_2^2(t' - \tau - \varepsilon)^2 + h^2$, using the curves as in (12) of the boundaries, see Figure A.4. This along with the fact that $(\bar{t}, \bar{h}) \in E_{\hat{h}_0}^{0,v_2}$, allows one to conclude that $E_{\hat{h}_1(\tau)}^{\tau, v_2} \setminus E_{\hat{h}_0}^{0, v_2} \subset E_{\hat{h}_1(\tau+\varepsilon)}^{\tau+\varepsilon, v_2} \setminus E_{\hat{h}_0}^{0, v_2}$. This justifies the claim. $\square$

<figure><img src="image_59.png" /></figure>

(1) The case where typical handover is of type 1 or equivalently $[\begin{matrix} 1 \\ 1,2 \end{matrix}]$. The subsequent half ellipses corresponding to both the speeds $v_1, v_2$, have their top point lying on $C_{(t_1, h_1)}^1$.

<figure><img src="image_60.png" /></figure>

(2) The case where typical handover is of type $(\begin{matrix} 1 \\ 1,2 \end{matrix})$ or $[\begin{matrix} 1 \\ 2,1 \end{matrix}]$. The subsequent half ellipses corresponding to both the speeds $v_1, v_2$, have their top point lying on $C_{(t_2, h_2)}^1$.

<caption>FIGURE A.5. The other two cases of typical handovers being of type 1 and type $(\begin{matrix} 1 \\ 1,2 \end{matrix})$ or $[\begin{matrix} 1 \\ 2,1 \end{matrix}]$.</caption>

**Proof of Proposition 8.16, Part (i), typical handover type 1 or $[\begin{matrix} 1 \\ 1,1 \end{matrix}]$ and $(\begin{matrix} 1 \\ 1,2 \end{matrix})$ or $[\begin{matrix} 1 \\ 2,1 \end{matrix}]$.** Our claim for this part in the case of typical handovers of type 1 or equivalently $[\begin{matrix} 1 \\ 1,1 \end{matrix}]$ and $(\begin{matrix} 1 \\ 1,2 \end{matrix})$ or equivalently
½,1], can be proved similarly, since the proof only concerns the type of the radial bird, say $C_{(t_1,h_1)}^1$, corresponding to the station that takes over the service after the handover at the intersection, see Picture (1) of Figure A.5 and $C_{(t_2,h_2)}^1$, corresponding to Picture (2) of Figure A.5 respectively. The sequence of upper half-ellipses that have their top point lying on $C_{(t_1,h_1)}^1$ and $C_{(t_2,h_2)}^1$, respectively. □

**Remark A.3.** The converse of the argument used in the proof of Proposition A.2, part (b), can be proved to be false in general case, using a counter example depicted in Picture 2 of Figure A.3. Indeed, there exists $t > t_1$ such that the boundaries $\partial E_{\hat{h}_1(t)}^{t,v_2}$, $\partial E_{\hat{h}_1(t+\epsilon)}^{t+\epsilon,v_2}$, with the top point lying on the increasing wing of $C_{(t_1,h_1)}^1$, intersect each other outside of $E_{\hat{h}_0}^{0,v_2}$ or never, but still we have $E_{\hat{h}_1(t)}^{t,v_2} \setminus E_{\hat{h}_0}^{0,v_2} \subset E_{\hat{h}_1(t+\epsilon)}^{t+\epsilon,v_2} \setminus E_{\hat{h}_0}^{0,v_2}$.

*Proof of Proposition 8.16, Part (ii).* This scenario appears when at the typical handover a station of type 2 takes over, which correspond to one of the types: $2, \binom{2}{1,2}$ or $\left[\frac{2}{1,2}\right]$ and $\left(\frac{2}{2,1}\right)$ or $\left[\frac{1}{2,1}\right]$, respectively.

**Case 1.** Type $\binom{2}{1,2}$ or equivalently $\left[\frac{2}{1,2}\right]$. Here $(t_n, h_n) = (t_1, h_1)$, which is of type 2. Define $\hat{h}_2(t) := (v_2^2(t-t_1)^2 + h_1^2)^{\frac{1}{2}}$. In this case, the boundaries $\partial E_{\hat{h}_2(t)}^{t,v_2}$ of the sets in the collection $\{E_{\hat{h}_2(t)}^{t,v_2}\}_{t \ge 0}$, passes through the head point $(t_1, h_1)$, see Picture (1) of Figure A.6. By applying the same argument of single speed case, Lemma 5.13, we can show that $\{E_{\hat{h}_2(t)}^{t,v_2} \setminus E_{\hat{h}_0}^{0,v_2}\}_{t \ge 0}$ is non-decreasing.

<figure><img src="image_61.png" /></figure>

(1) The collection $\{E_{\hat{h}_2(t)}^{t,v_2} \setminus E_{\hat{h}_0}^{0,v_2}\}_{t \ge 0}$ is non-decreasing in t.

<figure><img src="image_62.png" /></figure>

(2) The collection $\{E_{\hat{h}_2(t)}^{t,v_1} \setminus E_{\hat{h}_0}^{0,v_1}\}_{t \ge 0}$ is non-monotonic in t, as the sequence of sets are not comparable.

FIGURE A.6. Sequence of sets $\{E_{\hat{h}_2(t)}^{t,v_l} \setminus E_{\hat{h}_0}^{0,v_l}\}_{t \ge 0}$, for $l = 1, 2$.

The other collection $\{E_{\hat{h}_2(t)}^{t,v_1} \setminus E_{\hat{h}_0}^{0,v_1}\}_{t \ge 0}$ corresponding to the speed $v_1$, can be shown to be not always monotonic, by creating artificial example depicted in Picture (2) of Figure A.6.

**Case 2.** Type 2 or equivalently $\left[\frac{1}{2,2}\right]$. This case can be proved similarly to Case 1, since the subsequent half-ellipses corresponding to different speeds $v_1, v_2$ are such that their top point lies on the radial bird $C_{(t_1,h_1)}^2$, see Picture (1) of Figure A.7.

**Case 3.** Type $\binom{2}{2,1}$ or equivalently $\left[\frac{1}{1,2}\right]$. In this case the subsequent half-ellipses are such that their top point lies on the increasing wing of the radial bird $C_{(t_2,h_2)}^2$, see Picture (2) of Figure A.7. The proof of monotonicity of $\{E_{\hat{h}_2(t)}^{t,v_2} \setminus E_{\hat{h}_0}^{0,v_2}\}_{t \ge 0}$ follows when using the increasing wing property, as in Remark 8.17. The non-monotonicity of $\{E_{\hat{h}_2(t)}^{t,v_1} \setminus E_{\hat{h}_0}^{0,v_1}\}_{t \ge 0}$ can be shown using a similar counter example in Case 1. □
<figure><img src="image_63.png" /></figure>

(1) The case of typical handover is of type 2 or equivalently $[ \begin{smallmatrix} 1 \\ 2,2 \end{smallmatrix} ]$. The subsequent half-ellipses corresponding to speeds $v_1, v_2$ are such that their top point lies on the radial bird $C^2_{(t_1,h_1)}$.

<figure><img src="image_64.png" /></figure>

(2) The case of typical handover is of type $(\begin{smallmatrix} 2 \\ 2,1 \end{smallmatrix})$ or $[ \begin{smallmatrix} 1 \\ 1,2 \end{smallmatrix} ]$. The subsequent half-ellipses corresponding to speeds $v_1, v_2$ are such that their top point lies on the increasing wing of the radial bird $C^2_{(t_2,h_2)}$.

FIGURE A.7. The other two cases of typical handovers being of type 2 or equivalently $[ \begin{smallmatrix} 1 \\ 2,2 \end{smallmatrix} ]$ and type $(\begin{smallmatrix} 2 \\ 2,1 \end{smallmatrix})$ or equivalently $[ \begin{smallmatrix} 1 \\ 1,2 \end{smallmatrix} ]$.

### A.4. Proof of Lemma 5.18.
Suppose $s, s' \in \mathbb{R}$ with $s \le s'$ and $h, h' \in \mathbb{R}^+$ for which $U_h^s \cap U_{h'}^{s'} \ne \emptyset$ and $\exists h_1 \ge 0$ such that $\partial^* U_h^s \cap \partial^* U_{h'}^{s'} = \{(0, h_1)\}$. Observe that the area of the region $U_h^s \cup U_{h'}^{s'}$ (see Figure 5.5) is

$$ |U_{h'}^{s'} \cup U_h^s| = \begin{cases} |U_h^s| + |U_{h'}^{s'} \setminus U_h^s|, & \text{if } s < s' < 0, \\ |U_{h'}^{s'}| + |U_h^s| - |U_{h'}^{s'} \cap U_h^s|, & \text{if } s < 0 < s', \\ |U_{h'}^{s'}| + |U_h^s \setminus U_{h'}^{s'}|, & \text{if } 0 < s < s'. \end{cases} \quad (23) $$

In the first case $s < s' < 0$, we have

$$ \begin{aligned} |U_{h'}^{s'} \setminus U_h^s| &= \int_0^{h_1} \int_{s+(h^2-y^2)^{\frac{1}{2}}}^{s'+(h'^2-y^2)^{\frac{1}{2}}} \mathrm{d}x \mathrm{d}y \\ &= h_1(s' - s) + \int_0^{h_1} \left[ (h'^2 - y^2)^{\frac{1}{2}} - (h^2 - y^2)^{\frac{1}{2}} \right] \mathrm{d}y \\ &= h_1(s' - s) + \frac{1}{2} \left[ h_1 (h'^2 - h_1^2)^{\frac{1}{2}} + h'^2 \arcsin(h_1/h') \right] - \frac{1}{2} \left[ h_1 (h^2 - h_1^2)^{\frac{1}{2}} + h^2 \arcsin(h_1/h) \right] \\ &= h_1(s' - s) + \frac{1}{2} h_1 (|s' - s|) + \frac{1}{2} \left[ h'^2 \arcsin(h_1/h') - h^2 \arcsin(h_1/h) \right] \\ &= \frac{1}{2} h_1 (s' - s) + \frac{1}{2} F_1(h_1, h, h'), \end{aligned} \quad (24) $$

where,

$$ F_1(h_1, h, h') := h'^2 \arcsin(h_1/h') - h^2 \arcsin(h_1/h). \quad (25) $$

In the second case $s < 0 < s'$, we have

$$ \begin{aligned} |U_{h'}^{s'} \cap U_h^s| &= \int_0^{h_1} \int_{s'-(h'^2-y^2)^{\frac{1}{2}}}^{s+(h^2-y^2)^{\frac{1}{2}}} \mathrm{d}x \mathrm{d}y \\ &= -h_1(s' - s) + \int_0^{h_1} \left[ (h'^2 - y^2)^{\frac{1}{2}} + (h^2 - y^2)^{\frac{1}{2}} \right] \mathrm{d}y \\ &= -h_1(s' - s) + \frac{1}{2} \left[ h_1 (h'^2 - h_1^2)^{\frac{1}{2}} + h'^2 \arcsin(h_1/h') \right] + \frac{1}{2} \left[ h_1 (h^2 - h_1^2)^{\frac{1}{2}} + h^2 \arcsin(h_1/h) \right] \\ &= -h_1(s' - s) + \frac{1}{2} h_1 (|s' - s|) + \frac{1}{2} \left[ h'^2 \arcsin(h_1/h') + h^2 \arcsin(h_1/h) \right] \\ &= -\frac{1}{2} h_1 (s' - s) + \frac{1}{2} F_2(h_1, h, h'), \end{aligned} \quad (26) $$

where,

$$ F_2(h_1, h, h') := h'^2 \arcsin(h_1/h') + h^2 \arcsin(h_1/h). \quad (27) $$
In the last case $0 < s < s'$, similarly we have

$$
\begin{align*}
|U_h^s \setminus U_{h'}^{s'}| &= \int_0^{h_1} \int_{s-(h^2-y^2)^{\frac{1}{2}}}^{s'-(h'^2-y^2)^{\frac{1}{2}}} dx dy \\
&= h_1(s' - s) + \int_0^{h_1} \left[ (h^2 - y^2)^{\frac{1}{2}} - (h'^2 - y^2)^{\frac{1}{2}} \right] dy \\
&= h_1(s' - s) + \frac{1}{2} \left[ h_1(h^2 - h_1^2)^{\frac{1}{2}} + h^2 \arcsin(h_1/h) \right] - \frac{1}{2} \left[ h_1(h'^2 - h_1^2)^{\frac{1}{2}} + h'^2 \arcsin(h_1/h') \right] \\
&= h_1(s' - s) + \frac{1}{2} h_1(|s| - |s'|) + \frac{1}{2} \left[ h^2 \arcsin(h_1/h) - h'^2 \arcsin(h_1/h') \right] \\
&= \frac{1}{2} h_1(s' - s) - \frac{1}{2} F_1(h_1, h, h'), \tag{28}
\end{align*}
$$

where the function $F_1$ is defined in (25). Considering all the cases for $s, s'$, we get the result by combining the expressions for $|U_{h'}^s \setminus U_h^s|, |U_h^s \cap U_{h'}^{s'}|$ and $|U_h^s \setminus U_{h'}^{s'}|$ respectively from (24), (26) and (28) to (23). $\square$

### A.5. Area of the regions $E_{\ell_1}^{s_1, v} \cup E_{\ell_2}^{s_2, v}$.
The area of union of half ellipses are computed using elementary geometry in the following Lemma. We compute the area of the region $E_{\ell_1}^{s_1, v} \cup E_{\ell_2}^{s_2, v}$ for general speed $v$, also as preparation of the multi-speed case. For any speed $v$, two half-ellipses $E_{\ell_1}^{s_1, v}, E_{\ell_2}^{s_2, v}$ are given by the equations

$$v^2(t - s_1)^2 + h^2 = \ell_1^2 \text{ and } v^2(t - s_2)^2 + h^2 = \ell_2^2. \tag{29}$$

Let the intersection point in $\mathbb{H}^+$, of the two ellipses in (29) be $(t(v), h(v))$, which is given by

$$t(v) := \frac{1}{2(s_2 - s_1)} \left[ (\ell_1^2 - \ell_2^2)/v^2 - s_1^2 + s_2^2 \right] \quad \text{and} \quad h(v) := (\ell_1^2 - v^2(t(v) - s_1)^2)^{1/2}. \tag{30}$$

**Lemma A.4.** Let $v > 0$ and $(s_1, \ell_1)$, $(s_2, \ell_2)$ be two points such that $s_1 < s_2$. Suppose the ellipses in (29) intersects at $(t(v), h(v))$. Then the area of the region $E_{\ell_1}^{s_1, v} \cup E_{\ell_2}^{s_2, v}$ is given by

$$|E_{\ell_1}^{s_1, v} \cup E_{\ell_2}^{s_2, v}| = \begin{cases} \frac{1}{2v} [\pi \ell_1^2 + vh(v)(s_2 - s_1) + F_1(h(v), \ell_1, \ell_2)], & \text{if } s_1 < s_2 < t(v), \\ \frac{1}{2v} [\pi \ell_2^2 + \pi \ell_1^2 + vh(v)(s_2 - s_1) - F_2(h(v), \ell_1, \ell_2)], & \text{if } s_1 < t(v) < s_2, \\ \frac{1}{2v} [\pi \ell_2^2 + vh(v)(s_2 - s_1) - F_1(h(v), \ell_1, \ell_2)], & \text{if } t(v) < s_1 < s_2, \end{cases} \tag{31}$$

where $t(v), h(v)$ as in (30) and the functions $F_1, F_2$ are as in (25) and (27).

<figure><img src="image_65.png" /><figcaption>**FIGURE A.8.** Given $(s_1, \ell_1)$ and $(s_2, \ell_2)$, the area of the union of two half-ellipses should be different for different speed and depending on $t(v) < s_1 < s_2$ or $s_1 < t(v) < s_2$ or $s_1 < s_2 < t(v)$.</figcaption></figure>

*Proof.* The area of the region $E_{\ell_1}^{s_1, v} \cup E_{\ell_2}^{s_2, v}$ is given by

$$|E_{\ell_1}^{s_1, v} \cup E_{\ell_2}^{s_2, v}| = \begin{cases} |E_{\ell_1}^{s_1, v}| + |E_{\ell_2}^{s_2, v} \setminus E_{\ell_1}^{s_1, v}|, & \text{if } s_1 < s_2 < t(v), \\ |E_{\ell_1}^{s_1, v}| + |E_{\ell_2}^{s_2, v}| - |E_{\ell_1}^{s_1, v} \cap E_{\ell_2}^{s_2, v}|, & \text{if } s_1 < t(v) < s_2, \\ |E_{\ell_2}^{s_2, v}| + |E_{\ell_1}^{s_1, v} \setminus E_{\ell_2}^{s_2, v}|, & \text{if } t(v) < s_1 < s_2. \end{cases} \tag{32}$$
In the first case $s_1 < s_2 < t(v)$, we have

$$
\begin{align*}
|E_{\ell_2}^{s_2, v} \setminus E_{\ell_1}^{s_1, v}| &= \int_0^{h(v)} \int_{s_1 + \frac{1}{v}(\ell_1^2 - h^2)^{\frac{1}{2}}}^{s_2 + \frac{1}{v}(\ell_2^2 - h^2)^{\frac{1}{2}}} dt \, dh \\
&= h(v)(s_2 - s_1) + \frac{1}{v} \int_0^{h(v)} \left( (\ell_2^2 - h^2)^{\frac{1}{2}} - (\ell_1^2 - h^2)^{\frac{1}{2}} \right) dh \\
&= h(v)(s_2 - s_1) + \frac{1}{2v} \left[ h(v) \left( \ell_2^2 - h(v)^2 \right)^{\frac{1}{2}} - h(v) \left( \ell_1^2 - h(v)^2 \right)^{\frac{1}{2}} \right] \\
&\quad + \frac{1}{2v} \left[ \ell_2^2 \arcsin(h(v)/\ell_2) - \ell_1^2 \arcsin(h(v)/\ell_1) \right] \\
&\stackrel{29}{=} h(v)(s_2 - s_1) + \frac{1}{2v} [vh(v)|t(v) - s_2| - vh(v)|t(v) - s_1| + F_1(h(v), \ell_1, \ell_2)] \\
&= h(v)(s_2 - s_1) - \frac{1}{2}h(v)(s_2 - s_1) + \frac{1}{2v}F_1(h(v), \ell_1, \ell_2) \\
&= \frac{1}{2}h(v)(s_2 - s_1) + \frac{1}{2v}F_1(h(v), \ell_1, \ell_2), \tag{33}
\end{align*}
$$

where $F_1(h(v), \ell_1, \ell_2)$ is defined in (25). Let us now consider the second case, namely $s_1 < t(v) < s_2$.
Then $|E_{\ell_1}^{s_1, v} \cup E_{\ell_2}^{s_2, v}| = |E_{\ell_1}^{s_1, v}| + |E_{\ell_2}^{s_2, v}| - |E_{\ell_1}^{s_1, v} \cap E_{\ell_2}^{s_2, v}|$, where the area of the common region is

$$
\begin{align*}
|E_{\ell_1}^{s_1, v} \cap E_{\ell_2}^{s_2, v}| &= \int_0^{h(v)} \int_{s_2 - \frac{1}{v}(\ell_2^2 - h^2)^{\frac{1}{2}}}^{s_1 + \frac{1}{v}(\ell_1^2 - h^2)^{\frac{1}{2}}} dt \, dh \\
&= h(v)(s_1 - s_2) + \frac{1}{2v} \left[ h(v) \left( \ell_1^2 - h(v)^2 \right)^{\frac{1}{2}} + h(v) \left( \ell_2^2 - h(v)^2 \right)^{\frac{1}{2}} \right] \\
&\quad + \frac{1}{2v} \left[ \ell_1^2 \arcsin(h(v)/\ell_1) + \ell_2^2 \arcsin(h(v)/\ell_2) \right] \\
&= h(v)(s_1 - s_2) + \frac{1}{2v} \left[ h(v) \left( \ell_2^2 - h(v)^2 \right)^{\frac{1}{2}} + h(v) \left( \ell_1^2 - h(v)^2 \right)^{\frac{1}{2}} \right] + \frac{1}{2v} F_2(h(v), \ell_1, \ell_2) \\
&\stackrel{(29)}{=} h(v)(s_1 - s_2) + \frac{1}{2}h(v) [|t(v) - s_2| + |t(v) - s_1|] + \frac{1}{2v} F_2(h(v), \ell_1, \ell_2) \\
&= -\frac{1}{2}h(v)(s_2 - s_1) + \frac{1}{2v} F_2(h(v), \ell_1, \ell_2), \tag{34}
\end{align*}
$$

since $|t(v) - s_2| + |t(v) - s_1| = s_2 - s_1$, since $s_1 \le t(v) \le s_2$, from the second picture of Figure A.8. Above $F_2(h(v), \ell_1, \ell_2)$ is as defined in (27). In the last case $t(v) < s_1 < s_2$. Then we have $|E_{\ell_1}^{s_1, v} \cup E_{\ell_2}^{s_2, v}| = |E_{\ell_2}^{s_2, v}| + |E_{\ell_1}^{s_1, v} \setminus E_{\ell_2}^{s_2, v}|$. Thus

$$
\begin{align*}
|E_{\ell_1}^{s_1, v} \setminus E_{\ell_2}^{s_2, v}| &= \int_0^{h(v)} \int_{s_1 - \frac{1}{v}(\ell_1^2 - h^2)^{\frac{1}{2}}}^{s_2 - \frac{1}{v}(\ell_2^2 - h^2)^{\frac{1}{2}}} dt \, dh \\
&= h(v)(s_2 - s_1) + \frac{1}{v} \int_0^{h(v)} \left( (\ell_1^2 - h^2)^{\frac{1}{2}} - (\ell_2^2 - h^2)^{\frac{1}{2}} \right) dh \\
&= h(v)(s_2 - s_1) + \frac{1}{2v} \left[ h(v) \left( \ell_1^2 - h(v)^2 \right)^{\frac{1}{2}} - h(v) \left( \ell_2^2 - h(v)^2 \right)^{\frac{1}{2}} \right] \\
&\quad + \frac{1}{2v} \left[ \ell_1^2 \arcsin(h(v)/\ell_1) - \ell_2^2 \arcsin(h(v)/\ell_2) \right] \\
&\stackrel{29}{=} h(v)(s_2 - s_1) + \frac{1}{2v} [vh(v)|t(v) - s_1| - vh(v)|t(v) - s_2| - F_1(h(v), \ell_1, \ell_2)] \\
&= h(v)(s_2 - s_1) + \frac{1}{2}h(v)(s_1 - s_2) - \frac{1}{2v}F_1(h(v), \ell_1, \ell_2) \\
&= \frac{1}{2}h(v)(s_2 - s_1) - \frac{1}{2v}F_1(h(v), \ell_1, \ell_2), \tag{35}
\end{align*}
$$

since $|t(v) - s_1| - |t(v) - s_2| = s_1 - s_2$, as $t(v) < s_1 < s_2$, see the first picture of Figure A.8. Above $F_1(h(v), \ell_1, \ell_2)$ is as defined in (25). Thus we have the result. $\square$
ACKNOWLEDGEMENT

The authors would like to thank Bartłomiej Blaszczyszyn, Raphaël Lachièze-Rey and Gourab Ghatak for many important discussions and for providing many references. The authors are thankful to the ERC-NEMO grant, under the European Union’s Horizon 2020 research and innovation program, grant agreement number 788851 to INRIA Paris. This research was funded in part by the France 2030 BPI “5G NTN mmWave” project to Télécom Paris.

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(FB) MATHNET, INRIA PARIS, 48 RUE BARRAULT, 75013 PARIS, FRANCE, AND INFRES, TÉLÉCOM PARIS, 19, PLACE MARGUERITE PEREY, 91123 PALAISEAU, FRANCE

Email address: francois.baccelli@inria.fr

(SKJ) INFRES, TÉLÉCOM PARIS, 19, PLACE MARGUERITE PEREY, 91123 PALAISEAU, FRANCE, AND MATHNET, INRIA PARIS, 48 RUE BARRAULT, 75013 PARIS, FRANCE

Email address: sanjoy.jhawar@telecom-paris.fr, sanjoy-kumar.jhawar@inria.fr