where $\hat{h}_k \equiv \hat{h}_k(t, h_1, h_2)$ is derived in (7.22) for both $k = 1, 2$. The total handover frequency is

$$ \lambda_V = \frac{\sqrt{\lambda}}{\pi} (v_1 + v_2) + 2\Lambda_{2,1}^{(1)} + 2\Lambda_{2,1}^{(2)}. $$

**7.2. Multi-speed case.** Our techniques naturally lift to the finite set of speeds case, except for a small adjustment required to quantify the effect of stations of all the different speeds on handovers of the same or mixed types. Suppose the set of speeds is $v_{[n]} := \{v_i\}_{i \in [n]}$, with the order $v_1 > v_2 > \cdots > v_n$ and initial locations of stations are independent Poisson point processes $\tilde{\Phi}^l = \sum_{i \in \mathbb{N}} \delta_{(R_i^{(l)}, \alpha_i^{(l)})}$, for $l \in [n]$.

**7.2.1. Head point process and radial bird particle process.** For different speeds $v_l$ with $l \in [n]$, consider $n$ different head point processes for $l \in [n]$,

$$ \mathcal{H}^l := \sum_{i \in \mathbb{N}} \delta_{(T_i^{(l)}, H_i^{(l)})}, \quad \text{where} \quad (T_i^{(l)}, H_i^{(l)}) := \left( -\frac{R_i^{(l)}}{v_l} \cos \alpha_i^{(l)}, R_i^{(l)} |\sin \alpha_i^{(l)}| \right). $$

The following result is similar to Lemma 7.1.

**Lemma 7.24.** For each $l \in [n]$, $\mathcal{H}^l$ is a Poisson point process on $\mathbb{H}^+$ with intensity measure $v_l$ where $dv_l := v_l dt \otimes 2\lambda_l dh$. For $l \in [n]$, the point processes $\mathcal{H}^l$, are independent.

We shall write $\mathcal{H} = \sum_{l \in [n]} \mathcal{H}^l$, the superposition of $n$ independent Poisson point processes $\{\mathcal{H}^l\}_{l \in [n]}$. We write the law of the head point process $\mathcal{H} = \sum_{l \in [n]} \mathcal{H}^l$, as a product law $\mathbb{P}_{\mathcal{H}} := \bigotimes_{l \in [n]} \mathbb{P}_{\mathcal{H}^l}$, since $\mathcal{H}^l$'s are independent among each other. The radial bird particle process can be constructed as the sum $\mathcal{P}_c(v_{[n]}) := \sum_{l \in [n]} \mathcal{P}_c(v_l)$, where,

$$ \mathcal{P}_c(v_l) = \sum_i \delta_{C_i^l}, $$

where the support is the set of closed subsets of $\mathbb{H}^+$ of the form (7.3). The Poisson structure and time-stationarity of $\mathcal{P}_c(v_{[n]})$ is also preserved in this case, as seen in Lemma 7.2 for the two-speed case. In view of Remark 4.12, we can equivalently consider $\mathcal{P}_c(v_{[n]})$ as the marked head point process $\mathcal{H}_c(v_{[n]}) = \sum_{l \in [n]} \mathcal{H}_c(v_l)$ with closed set valued marks, where

$$ \mathcal{H}_c(v_l) = \sum_i \delta_{(T_i^{(l)}, H_i^{(l)}), C_i^l}. $$

**7.2.2. Lower envelope process.** The lower envelope process is a stochastic process on $\mathbb{H}^+$, is defined as $\{(t, L_n(t))\}_{t \in \mathbb{R}}$, where

$$ L_n(t) := \min_{l \in [n]} \inf_{i \in \mathbb{N}} f_{v_l} \left( (R_i^{(l)}, \alpha_i^{(l)}), t \right). $$

The lower envelope is also time-stationary in view of Lemma 7.3 for two-speed case. The lower envelope is defined as a random closed subset of $\mathbb{H}^+$ as

$$ \mathcal{L}_e(v_{[n]}) := \{(t, L_n(t)) : t \in \mathbb{R}\}. $$

Let us define the open region above the lower envelope $\mathcal{L}_e(v_{[n]})$ as

$$ \mathcal{L}_e^+(v_{[n]}) := \{(t, h) \in \mathbb{H}^+ : h > L_n(t)\}. \quad (7.53) $$

In the multi-speed case, we have a *half-ellipse condition*, analogous to Lemma 7.4, the characterization for a point to be on the lower envelope $\mathcal{L}_e(v_{[n]})$. We shall state the result without proof.

**Lemma 7.25 (Half-ellipse condition).** For any point $(s, u) \in \cup_{k \in [n]} \cup_{i \in \mathbb{N}} C_i^l$ the following holds true:

(i). $(s, u) \in \mathcal{L}_e(v_{[n]})$ if and only if $\mathcal{H}^l(E_u^{s,v_l}) = 0$ for all $l \in [n]$;

(ii). the probability of such an event is

$$ \mathbb{P}((s, u) \in \mathcal{L}_e(v_{[n]})) = e^{-\lambda \pi u^2}, \quad (7.54) $$

where $\lambda := \sum_{l \in [n]} \lambda_l$.
### 7.2.3. The handover point process and its intensity.
In view of the construction of the handover point process in the two-speed case in Subsection 7.1.7, we can also define the handover point process in the multi-speed case using the multi-speed radial bird particle process $\mathcal{H}_c(v_{[n]})$. By Lemma 7.25, for the intersection point $(\hat{s}, \hat{h})$ of two radial birds with their heads at $(t, h)$, $(t', h')$ give rise to a handover if and only if $\mathcal{H}^l(E_{\hat{h}}^{\hat{s}, v_l}) = 0$, for all $l \in [n]$.

Formally, taking inspiration from the two-speed case in Subsection 7.1.7, we can construct different handover point processes $\mathcal{V}_l$ corresponding pure handovers of type $l$, for $l \in [n]$, as in (7.25), and $\mathcal{V}_{l,r}^{(k)}$ corresponding to mixed handovers of type $(l,r)$, for $l \neq r \in [n]$, $k \in \{1, 2\}$, as in (7.26), (7.27). Here, we recall that this type of mixed handover is due to the $k$-th intersection of the radial birds of types $l, r$ such that $l \neq r$ and the head point of the former is on the left of that of the latter.

Based on all possible types of pure and mixed intersections, while respecting the intersection criterion determined in Lemma 7.5 and Lemma 7.9, the handover point process $\mathcal{V}$, similar to (7.28), is defined as

$$ \mathcal{V} := \sum_{l \in [n]} \mathcal{V}_l + \sum_{l \neq r \in [n], k=1,2} \mathcal{V}_{l,r}^{(k)}. \quad (7.55) $$

In the following, we just state the result about the time stationarity of the handover point process $\mathcal{V}$, which is inherited from the stationarity of the head point process $\mathcal{H}$.

**Lemma 7.26 (Time-stationarity of $\mathcal{V}$).** In the multi-speed case, the handover point process $\mathcal{V}$ is stationary with respect to time.

The construction of the point processes $\mathcal{R}_l$ for $l \in [n]$ and $\mathcal{R}_{l,r}^{(k)}$ for $l \neq r \in [n]$, $k \in \{1, 2\}$, corresponding to the right most head points is also the same and so is the existence of the bijections as in Lemma 7.18.

**Lemma 7.27.** (i). For all $l \in [n]$, there exists a bijection $\beta_l$ between $\mathcal{V}_l$ and $\mathcal{R}_l$.

(ii). For all $l \neq r \in [n]$, $k \in \{1, 2\}$, there exists a bijection $\beta_{l,r}^{(k)}$ between $\mathcal{V}_{l,r}^{(k)}$ and $\mathcal{R}_{l,m}^{(k)}$.

We obtain the equality of intensities as a corollary:

**Corollary 7.28.** (i). For $l \in [n]$, the intensity of the point processes $\mathcal{V}_l$ and $\mathcal{R}_l$ are equal, $\lambda_{\mathcal{R}_l} = \lambda_{\mathcal{V}_l} = \Lambda_l$.

(ii). For all $l \neq r \in [n]$, $k \in \{1, 2\}$, the intensity of the point processes $\mathcal{V}_{l,r}^{(k)}$ and $\mathcal{R}_{l,r}^{(k)}$ are equal, $\lambda_{\mathcal{R}_{l,r}^{(k)}} = \lambda_{\mathcal{V}_{l,r}^{(k)}} = \Lambda_{l,r}^{(k)}$.

In the multi-speed scenario, the handover frequency can be written as a sum of pure and mixed handovers

$$ \lambda_{\mathcal{V}} = \sum_{l \in [n]} \Lambda_l + \sum_{l \neq r \in [n], k \in \{1, 2\}} \Lambda_{l,r}^{(k)}, $$

as described in the following result which generalizes Theorem 4.26 and Theorem 7.20.

**Theorem 7.29.** The handover frequency in multi-speed case with set of speeds $v_{[n]}$ is given by:

(1) The pure handover frequencies are

$$ \Lambda_i = \frac{4v_i\sqrt{\lambda}}{\pi} \left(\frac{\lambda_i}{\lambda}\right)^2, \quad \text{for } i \in [n]. \quad (7.56) $$

(2) For any pair $i \neq j \in [n]$, the mixed handover frequencies satisfies, $\Lambda_{i,j}^{(1)} = \Lambda_{j,i}^{(2)}$ and $\Lambda_{i,j}^{(2)} = \Lambda_{i,j}^{(1)}$.

(3) For any pair $j < i \in [n]$ and $k \in \{1, 2\}$,

$$ \Lambda_{i,j}^{(k)} = 4\lambda_i \lambda_j v_i v_j \int_0^\infty \left[ \int_0^{h_j} \int_{t^*}^\infty e^{-\lambda \pi \hat{h}_k^2(t,h_j,h_i)} dt \, dh_i + \int_{h_j}^\infty \int_0^\infty e^{-\lambda \pi \hat{h}_k^2(t,h_j,h_i)} dt \, dh_i \right] dh_j, \quad (7.57) $$
where

$$ \hat{h}_k^2(t, h_j, h_i) = v_j^2 \left( \frac{v_i^2 t + (-1)^k \sqrt{\Delta_{i,j}}}{v_j^2 - v_i^2} \right)^2 + h_j^2 \quad \text{and} \quad \Delta_{i,j} = v_j^2 v_i^2 t^2 - (h_j^2 - h_i^2)(v_j^2 - v_i^2), $$

both of them as a function of $t, h_j, h_i$. The quantity

$$ t^* \equiv t^*(h_j, h_i) := \frac{1}{v_j v_i} (h_j^2 - h_i^2)^{\frac{1}{2}} (v_j^2 - v_i^2)^{\frac{1}{2}}, $$

is the non-negative solution of $\Delta_{i,j} = 0$.

**Proof of Theorem 7.29, part (1).** In the first part, the intensity of pure handovers is computed similar to (7.34) in the proof of two-speed case, as

$$ \Lambda_i = \frac{4v_i \sqrt{\lambda}}{\pi} \left(\frac{\lambda_i}{\lambda}\right)^2, \qquad (7.58) $$

for each $i \in [n]$, where $\lambda = \sum_{i \in [n]} \lambda_i$. $\square$

**Proof of Theorem 7.29, part (2).** Let $i > j \in [n]$ and assume that $v_j > v_i$. We are interested in the handover between two stations with speed $v_j, v_i$, which we call mixed handovers. For such mixed handovers, by the symmetry argument as in the two-speed case in the proof of Part (ii) Theorem 7.20, we have that $\Lambda_{i,j}^{(1)} = \Lambda_{j,i}^{(2)}$ and $\Lambda_{i,j}^{(2)} = \Lambda_{j,i}^{(1)}$ for all $j < i \in [n]$. $\square$

**Proof of Theorem 7.29, part (3).** We just evaluate $\Lambda_{i,j}^{(k)}$ for $j < i \in [n]$ and $k = 1, 2$ for handovers of type $\binom{k}{i,j}$. Let us straight away start from Equation (7.42) for the two-speed case in Theorem 7.20 and write the mixed handover frequency $\Lambda_{i,j}^{(k)}$ as an integral by applying the Campbell-Mecke formula for the factorial moment measure of order 2 for the sum $\mathcal{H} := \sum_{i \in [n]} \mathcal{H}^i$, as

$$ \Lambda_{i,j}^{(k)} = 4\lambda_i \lambda_j v_i v_j \int_0^1 \int_0^\infty \int_0^\infty \int_{D_\ell(t_j, h_j, h_i)} \mathbb{E}_{\mathcal{H}}^{(i,j)} \left[ \mathbb{1}_{A_{i,j}^{(k)}} (\mathcal{H}^i + \delta_{(t_i, h_i)} + \mathcal{H}^j + \delta_{(t_j, h_j)} + \sum_{m \neq i, j} \mathcal{H}^m) \right] dt_i dh_i dh_j dt_j. \qquad (7.59) $$

Here $A_{i,j}^{(k)}$ is the event that there is a handover of type $\binom{k}{i,j}$ at $(\hat{s}_{i,j}^{(k)}, \hat{h}_{i,j}^{(k)})$, for $k = 1, 2$, due to the intersection of the radial birds that have their heads at $(t_j, h_j)$ and $(t_i, h_i)$, with $t_i < t_j$. Above, we took the expectation inside the integral under the two-point Palm probability measure

$$ \mathbb{P}_{\mathcal{H}}^{(i,j),(t_i,h_i),(t_j,h_j)} := \mathbb{P}_{\mathcal{H}^i} \otimes \mathbb{P}_{\mathcal{H}^j} \otimes \bigotimes_{m \neq i,j} \mathbb{P}_{\mathcal{H}^m}. $$

Let us evaluate the term $\Lambda_{i,j}^{(k)}$ similarly to the two-speed case as

$$ \Lambda_{i,j}^{(k)} = 4\lambda_j \lambda_i v_j v_i \int_0^1 \int_0^\infty \int_0^{h_j} \int_{-\infty}^{t_r-t^*} \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{A_{i,j}^{(k)}} \left( \mathcal{H}^i + \delta_{(t_i, h_i)} + \mathcal{H}^j + \delta_{(t_j, h_j)} + \sum_{m \neq i, j} \mathcal{H}^m \right) \right] dt_i dh_i dh_j dt_j \\ + 4\lambda_j \lambda_i v_j v_i \int_0^1 \int_0^\infty \int_{h_j}^\infty \int_{-\infty}^{t_j} \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{A_{i,j}^{(k)}} \left( \mathcal{H}^i + \delta_{(t_i, h_i)} + \mathcal{H}^j + \delta_{(t_j, h_j)} + \sum_{m \neq i, j} \mathcal{H}^m \right) \right] dt_i dh_i dh_j dt_j, \qquad (7.60) $$

where $t^*$ is defined similarly to (7.15):

$$ t^* \equiv t^*(h_j, h_i) := \frac{1}{v_j v_i} (h_j^2 - h_i^2)^{\frac{1}{2}} (v_j^2 - v_i^2)^{\frac{1}{2}}. \qquad (7.61) $$

The quantity $t^*$ is well defined, since $v_j > v_i$. In (7.60), we have used the fact that $t_i \le t_j - t^*$ if $h_i < h_j$ and $t_i \le t_j$, for $h_i \ge h_j$.
<figure><img src="image_36.png" /><figcaption>FIGURE 7.10. We consider $(\hat{s}_1, \hat{h}_1) = (\hat{s}_{i,j}^{(1)}, \hat{h}_{i,j}^{(1)})$, $(\hat{s}_2, \hat{h}_2) = (\hat{s}_{i,j}^{(2)}, \hat{h}_{i,j}^{(2)})$ for simplicity. The mixed intersection $(\hat{s}_1, \hat{h}_1)$ represents a handover if the regions $E_{\hat{h}_1}^{\hat{s}_1, v_i}$ are empty of heads from $\mathcal{H}^i$ for all $i \in [n]$. The red and blue ellipses are for the speeds $v_j$ and $v_i$, respectively and the green and black ones correspond to other speeds $v \neq v_i, v_j$.</figcaption></figure>

Under the two-point Palm probability measure $\mathbb{P}_{\mathcal{H}}^{(i,j),(t_i,h_i),(t_j,h_j)}$, we have

$$ \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{A_{i,j}^{(k)}} \left( \mathcal{H}^j + \delta_{(t_j,h_j)} + \mathcal{H}^i + \delta_{(t_i,h_i)} + \sum_{m \neq i,j} \mathcal{H}^m \right) \right] = \mathbb{P}_{\mathcal{H}}^{(i,j),(t_i,h_i),(t_j,h_j)}(A_{i,j}^{(k)}). \quad (7.62) $$

Applying the Slivnyak-Mecke theorem and using Figure 7.10, the term in (7.62) evaluates to

$$ \prod_{m \in [n]} \mathbb{P}_{\mathcal{H}^m} \left( \mathcal{H}^m (E_{\hat{h}_k}^{\hat{s}_k, v_m}) = 0 \right) = \prod_{m \in [n]} e^{-2\lambda_m v_m \frac{\pi \hat{h}_k^2}{2v_m}} = e^{-\pi \hat{h}_k^2 \sum_{m \in [n]} \lambda_m} = e^{-\pi \lambda \hat{h}_k^2}, \quad (7.63) $$

since $\sum_{m \in [n]} \lambda_m = \lambda$, where $(\hat{s}_k, \hat{h}_k) = (\hat{s}_k(t_i, h_i, t_j, h_j), \hat{h}_k(t_i, h_i, t_j, h_j)) \equiv (\hat{s}_k(t_j - t_i, h_i, h_j), \hat{h}_k(t_j - t_i, h_i, h_j))$, the k-th intersection point of the radial birds at $(t_i, h_i)$, $(t_j, h_j)$. For any pair $i, j \in [n]$, with $j < i$ and $k = 1, 2$, the mixed handover frequency is

$$ \begin{aligned} \Lambda_{i,j}^{(k)} = {}& 4\lambda_i\lambda_j v_i v_j \int_0^1 \int_0^\infty \int_0^{h_j} \int_{-\infty}^{t_j-t^*} e^{-\pi\lambda\hat{h}_k^2(t_j-t_i,h_j,h_i)} \, dt_i \, dh_i \, dh_j \, dt_j \\ & + 4\lambda_i\lambda_j v_i v_j \int_0^1 \int_0^\infty \int_{h_j}^\infty \int_{-\infty}^{t_i} e^{-\pi\lambda\hat{h}_k^2(t_j-t_i,h_j,h_i)} \, dt_i \, dh_i \, dh_j \, dt_j \\ = {}& 4\lambda_i\lambda_j v_i v_j \int_0^\infty \left[ \int_0^{h_j} \int_{t^*}^\infty e^{-\pi\lambda\hat{h}_k^2(t,h_j,h_i)} \, dt \, dh_i + \int_{h_j}^\infty \int_0^\infty e^{-\pi\lambda\hat{h}_k^2(t,h_j,h_i)} \, dt \, dh_i \right] dh_j, \end{aligned} \quad (7.64) $$

which is the required expression for $\Lambda_{i,j}^{(k)}$, where $t^* \equiv t^*(h_j, h_i) := \frac{1}{v_j v_i} (h_j^2 - h_i^2)^{\frac{1}{2}} (v_j^2 - v_i^2)^{\frac{1}{2}}$. From (7.58) and (7.64) and part (2), we get the value of total handover frequency

$$ \lambda_v = \sum_{i \in [n]} \Lambda_i + 2 \sum_{j<i \in [n], k=1,2} \Lambda_{i,j}^{(k)}. $$

This completes the proof of Theorem 7.29. $\square$

## 8. HANDOVER PALM DISTRIBUTION AND APPLICATIONS: TWO-SPEED CASE

In this section, we focus again on the two-speed case. We first determine the Palm probability measure with respect to handovers with the goal to determine the Palm probability distribution of inter-handover times. An identical analysis allows one to come up with a result similar to the mass transport principle (Lemma 5.1) for the two-speed case. In these initial steps, we derive the results in the two-speed case, as a first step towards the multi-speed case.

### 8.1. Palm probability with respect to handovers.
Without loss of generality, let us assume that $v_1 > v_2$. Like in the single-speed case, we consider that the reference probability space is $(\Omega, \mathcal{F}, \mathbb{P})$,
with $(\Omega, \mathcal{F})$ the canonical space of head point process on $\mathbb{H}^+$. Each $\omega \in \Omega$ is a realization of the point process $\mathcal{H} = \sum_{l \in \{1,2\}} \mathcal{H}^l$. We equip this measure space with the shift $\{\theta_t\}_{t \in \mathbb{R}}$, along the time axis, defined as

$$ \theta_t(\mathcal{H}) = \theta_t(\mathcal{H}^1) + \theta_t(\mathcal{H}^2) = \sum_i \delta_{(T_i^{(1)} - t, H_i^{(1)})} + \sum_i \delta_{(T_i^{(2)} - t, H_i^{(2)})}. \quad (8.1) $$

The law $\mathbb{P}$ of the head point process $\mathcal{H}$ is left invariant by this shift and the latter is also ergodic for $\mathbb{P}$. The handover point process $\mathcal{V}$ is constructed as in Subsection 7.1.7, more precisely as in (7.28) using the pure handover point processes $\mathcal{V}_l$, for $l = 1, 2$, and the mixed handover point processes $\mathcal{V}_{l,m}^{(k)}$, for $l \neq m, k \in \{1, 2\}$. The stationarity, or in other words $\theta_t$-compatibility of $\mathcal{V}$, is established in Lemma 7.17, and $\mathcal{V}$ has a positive and finite intensity from Theorem 7.20. We denote the Palm probability w.r.t. $\mathcal{V}$ on the probability space $(\Omega, \mathcal{F})$, as $\mathbb{P}_{\mathcal{V}}^0$. In what follows, we describe a variant of the mass transport principle, Lemma 5.1, for the two-speed case.

Recall the point processes $\mathcal{R}_l$, for $l \in \{1, 2\}$, and $\mathcal{R}_{l,r}^{(k)}$, for $l \neq r, k \in \{1, 2\}$, corresponding to the rightmost head points for different types of handovers. The same mass transport principle, Lemma 5.1, under the bijections defined in Lemma 7.18 applies to all the individual Palm probability measures, stated in the following.

**Lemma 8.1.** For all non-negative measurable functions $f$ on $(\Omega, \mathcal{F})$, we have:

$$ (i). \text{ For all } l \in \{1,2\}, \qquad \mathbb{E}_{\mathcal{V}_l}^0[f(\mathcal{H})] = \mathbb{E}_{\mathcal{R}_l}^0[f(\theta_{\beta_l(0)}(\mathcal{H}))], \quad (8.2) $$

where $\beta_l$ is the bijection between the point processes $\mathcal{V}_l$ and $\mathcal{R}_l$, corresponding to handovers of pure type $l$.

$$ (ii). \text{ For all } l \neq r, k \in \{1,2\}, \qquad \mathbb{E}_{\mathcal{V}_{l,r}^{(k)}}^0[f(\mathcal{H})] = \mathbb{E}_{\mathcal{R}_{l,r}^{(k)}}^0[f(\theta_{\beta_{l,r}^{(k)}(0)}(\mathcal{H}))], \quad (8.3) $$

where $\beta_{l,r}^{(k)}$ is the bijection between the point processes $\mathcal{V}_{l,r}^{(k)}$ and $\mathcal{R}_{l,r}^{(k)}$, corresponding to handovers of mixed type $\binom{k}{l,r}$.

Hence, we have the following decomposition of $\mathbb{E}_{\mathcal{V}}^0[f(\mathcal{H})]$, for any non-negative measurable function $f$ on $(\Omega, \mathcal{F})$, based on these different types of handovers using the Palm probability measure for the sum of multi-type independent point processes (cf. [4, Subsection 1.6, p. 36]),

**Theorem 8.2.** For all non-negative measurable functions $f$ on $(\Omega, \mathcal{F})$,

$$ \mathbb{E}_{\mathcal{V}}^0[f(\mathcal{H})] = \sum_{l \in \{1,2\}} \frac{\Lambda_l}{\lambda_{\mathcal{V}}} \mathbb{E}_{\mathcal{V}_l}^0[f(\mathcal{H})] + \sum_{l \neq r, k \in \{1,2\}} \frac{\Lambda_{l,r}^{(k)}}{\lambda_{\mathcal{V}}} \mathbb{E}_{\mathcal{V}_{l,r}^{(k)}}^0[f(\mathcal{H})], \quad (8.4) $$

in terms of the pure and mixed Palm probability measures $\mathbb{E}_{\mathcal{V}_l}^0$, for $l \in \{1, 2\}$, and $\mathbb{E}_{\mathcal{V}_{l,r}^{(k)}}^0$, for $l \neq r, k \in \{1, 2\}$.

Using the Poisson structure of $\mathcal{H}$, the pure handover Palm probability measures $\mathbb{P}_{\mathcal{V}_l}^0$, for $l = 1, 2$, are expressed as follows, similarly to Theorem 5.4:

**Lemma 8.3.** For $l = 1, 2$ and for all non-negative measurable functions $f$ on $(\Omega, \mathcal{F})$,

$$ = \frac{4\lambda_l^2 v_l^2}{\Lambda_l} \int_{(\mathbb{R}^+)^3} \mathbb{E}_{\mathcal{H}} \left[ f \left( \theta_{\hat{s}(0,h,-t',h')} (\mathcal{H}^l + \delta_{(0,h)} + \delta_{(-t',h')} + \mathcal{H}^m) \right) \prod_{i=1,2} \mathbb{1}_{\mathcal{H}^i} \left( E_{\hat{h}(0,h,-t',h')}^{\hat{s}(0,h,-t',h')}, v_i \right) \right] dt' dh' dh, $$

where, $m \neq l$ and $(\hat{s}(t, h, t', h'), \hat{h}(t, h, t', h'))$ are the coordinates of the intersection point of the two radial birds with some head points $(t, h)$ and $(t', h')$.
As a general principle, for any pair of head points $(t_1, h_1)$, $(t_2, h_2) \in \mathbb{H}^+$, the intersection point of the associated radial birds of the same type is denoted by $(\hat{s}(t_1, h_1, t_2, h_2), \hat{h}(t_1, h_1, t_2, h_2))$. The intersection points are given by (7.12) and (7.22) if the corresponding birds are of different types and we denote them by $(\hat{s}_k(t_1, h_1, t_2, h_2), \hat{h}_k(t_1, h_1, t_2, h_2))$ for $k = 1, 2$. The proof of Lemma 8.3 follows the same steps as those of Lemma 5.3 and of part (ii) of Lemma 8.1, except for the extra condition on multiple half-ellipses being empty of head points corresponding to different speeds. Having said this, we skip the proof of the last result and move towards the more interesting case of mixed Palm probability measures $\mathbb{P}_{\mathcal{V}_{l,r}^{(k)}}^0$, for $l, r, k \in \{1, 2\}$ with $l \neq r$, for which we provide more details in the following:

**Lemma 8.4.** For $l \neq r, k = 1, 2$ and for all non-negative measurable functions $f$ on $(\Omega, \mathcal{F})$,

$$ 
\begin{aligned}
& \frac{\Lambda_{l,r}^{(k)}}{4\lambda_1\lambda_2v_1v_2} \mathbb{E}_{\mathcal{V}_{l,r}^{(k)}}^0 [f(\mathcal{H})] \\
&= \int_0^\infty \int_0^h \int_{t^*}^\infty \mathbb{E}_{\mathcal{H}} \left[ f \left( \theta_{\hat{s}_k(0,h,-t',h')} (\mathcal{H}^r + \delta_{(0,h)} + \mathcal{H}^l + \delta_{(-t',h'})) \prod_{i=1,2} \mathbb{1}_{\mathcal{H}^i} \left( E_{\hat{h}_k(0,h,-t',h')}^{\hat{s}_k(0,h,-t',h')}, v_i \right) \right) =0 \right] dt' dh' dh \\
&\quad + \int_0^\infty \int_h^\infty \int_0^\infty \mathbb{E}_{\mathcal{H}} \left[ f \left( \theta_{\hat{s}_k(0,h,-t',h')} (\mathcal{H}^r + \delta_{(0,h)} + \mathcal{H}^l + \delta_{(-t',h'})) \prod_{i=1,2} \mathbb{1}_{\mathcal{H}^i} \left( E_{\hat{h}_k(0,h,-t',h')}^{\hat{s}_k(0,h,-t',h')}, v_i \right) \right) =0 \right] dt' dh' dh,
\end{aligned}
$$
where, $(\hat{s}_k(0, h, t', h'), \hat{h}_k(0, h, t', h'))$ are the coordinates of the k-th intersection of the two radial birds with heads $(0, h)$ and $(t', h')$ of type $r$ and $l$, respectively, $t^* = \frac{1}{v_1v_2}(h^2 - h'^2)^{\frac{1}{2}}(v_1^2 - v_2^2)^{\frac{1}{2}}$.

**Remark 8.5.** Because of the integration range, the expression on the right integral does not include the region where there is no intersection of the radial birds with their heads at $(0, h)$ and $(-t', h')$.

*Proof of Lemma 8.4, l = 2, r = 1 and k = 1, 2.* Let us first derive the result for $l = 2, r = 1$ and $k = 1, 2$, i.e., for $\mathbb{E}_{\mathcal{V}_{2,1}^{(k)}}^0 [f(\mathcal{H})]$. The proof follows the same steps as those of the proof of Lemma 5.3, under some adaptation. Suppose $(T_i^1, H_i^1)$ and $(T_j^2, H_j^2)$ in the support of $\mathcal{H}^1, \mathcal{H}^2$, respectively, such that $T_j^2 \in D_\ell(T_i^1, H_i^1, H_j^2)$. Let $(\hat{S}_{2,1}^{(k)}, \hat{H}_{2,1}^{(k)})$, in short written as $(\hat{S}^{(k)}, \hat{H}^{(k)})$, denote the coordinates of the k-th intersection of the two radial birds with these heads. By (8.3) and the definition of $\mathbb{P}_{\mathcal{R}_{2,1}^{(k)}}^0$,

$$ 
\begin{aligned}
\mathbb{E}_{\mathcal{V}_{2,1}^{(k)}}^0 [f(\mathcal{H})] &= \mathbb{E}_{\mathcal{R}_{2,1}^{(k)}}^0 \left[ f(\theta_{\beta_{2,1}^{(k)}(0)}(\mathcal{H})) \right] = \frac{1}{\Lambda_{2,1}^{(k)}} \mathbb{E} \left[ \sum_{T_i^1 \in \mathcal{R}_{2,1}^{(k)} : 0 \le T_i^1 \le 1} f \left( \theta_{\beta_{2,1}^{(k)}(0)} \circ \theta_{T_i^1}(\mathcal{H}) \right) \right] \\
&= \frac{1}{\Lambda_{2,1}^{(k)}} \mathbb{E} \left[ \sum_{T_i^1 \in \mathcal{R}_{2,1}^{(k)} : 0 \le T_i^1 \le 1} f \left( \theta_{T_i^1 + \beta_{2,1}^{(k)}(T_i^1)}(\mathcal{H}) \right) \right], \quad (8.5)
\end{aligned}
$$

using the fact that $\theta_{\beta_{2,1}^{(k)}(0)} \circ \theta_{T_i^1} = \theta_{T_i^1 + \beta_{2,1}^{(k)}(T_i^1)}$. The expectation in (8.5) can be evaluated as

$$ 
\mathbb{E} \left[ \sum_{(T_i^1, H_i^1) \in \mathcal{H}^1 : 0 \le T_i^1 \le 1} \mathbb{1}_{\exists (T_j^2, H_j^2) \in \mathcal{H}^2 : T_j^2 \in D_\ell(T_i^1, H_i^1, H_j^2)} f \left( \theta_{T_i^1 + \beta_{2,1}^{(k)}(T_i^1)}(\mathcal{H}) \right) \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{H}^{(k)}}^{\hat{s}^{(k)}, v_l} \right) = 0 \right] \quad (8.6)
$$

$$ = \mathbb{E} \left[ \sum_{(T_i^1, H_i^1) \in \mathcal{H}^1 : 0 \le T_i^1 \le 1} \sum_{(T_j^2, H_j^2) \in \mathcal{H}^2 : T_j^2 \in D_\ell(T_i^1, H_i^1, H_j^2)} f \left( \theta_{T_i^1 + \beta_{2,1}^{(k)}(T_i^1)}(\mathcal{H}) \right) \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{H}^{(k)}}^{\hat{s}^{(k)}, v_l} \right) = 0 \right] $$

$$ = \mathbb{E} \left[ \sum_{(T_i^1, H_i^1) \in \mathcal{H}^1 : 0 \le T_i^1 \le 1} \sum_{(T_j^2, H_j^2) \in \mathcal{H}^2 : T_j^2 \in D_\ell(T_i^1, H_i^1, H_j^2)} f \left( \theta_{\hat{S}^{(k)}}(\mathcal{H}) \right) \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{H}^{(k)}}^{\hat{s}^{(k)}, v_l} \right) = 0 \right], \quad (8.7) $$
since $T_i^1 + \beta_{2,1}^{(k)}(T_i^1) = \hat{S}^{(k)}$. Applying the Campbell-Mecke formula for the factorial power of order 2 based on the two-point Palm probability measure $\mathbb{P}_{\mathcal{H}}^{(t,h),(t',h')} = \mathbb{P}_{\mathcal{H}^1}^{(t,h)} \otimes \mathbb{P}_{\mathcal{H}^2}^{(t',h')}$, we can evaluate the expectation in (8.7) as

$$4\lambda_1\lambda_2 v_1 v_2 \int_0^1 \int_0^\infty \int_0^\infty \int_{D_\ell(t,h,h')} \mathbb{E}_{\mathcal{H}}^{(t,h),(t',h')} \left[ f\left(\theta_{\hat{s}_k(t,h,t',h')}(\mathcal{H})\right) \times \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l}\left(E_{\hat{h}_k(t,h,t',h')}(v_l)\right) = 0 \right] dt' dh' dh dt \quad (8.8)$$

$$= 4\lambda_1\lambda_2 v_1 v_2 \int_0^1 \int_0^\infty \int_0^\infty \int_{D_\ell(t,h,h')} \mathbb{E}_{\mathcal{H}} \left[ f\left(\theta_{\hat{s}_k(t,h,t',h')}(\mathcal{H}^1 + \delta_{(t,h)} + \mathcal{H}^2 + \delta_{(t',h')})\right) \times \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l}\left(E_{\hat{h}_k(t,h,t',h')}(v_l)\right) = 0 \right] dt' dh' dh dt, \quad (8.9)$$

where the set $D_\ell(t, h, h')$ is defined as in (7.8). In case $h < h'$, we have $D_\ell(t, h, h') = (-\infty, t]$ but in case $h \ge h'$, we have $D_\ell(t, h, h') = (-\infty, t - t^*)$, where $t^*$ is as defined in (7.15). This simplifies the multiple integral in (8.9) to

$$\int_0^1 \int_0^\infty \int_0^h \int_{-\infty}^{t-t^*} \mathbb{E}_{\mathcal{H}} \left[ f\left(\theta_{\hat{s}_k(t,h,t',h')}(\mathcal{H}^1 + \delta_{(t,h)} + \mathcal{H}^2 + \delta_{(t',h')})\right) \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l}\left(E_{\hat{h}_k(t,h,t',h')}(v_l)\right) = 0 \right] dt' dh' dh dt \\ + \int_0^1 \int_0^\infty \int_h^\infty \int_{-\infty}^t \mathbb{E}_{\mathcal{H}} \left[ f\left(\theta_{\hat{s}_k(t,h,t',h')}(\mathcal{H}^1 + \delta_{(t,h)} + \mathcal{H}^2 + \delta_{(t',h')})\right) \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l}\left(E_{\hat{h}_k(t,h,t',h')}(v_l)\right) = 0 \right] dt' dh' dh dt,$$

which can later be transformed to

$$\int_0^\infty \int_0^h \int_{t^*}^\infty \mathbb{E}_{\mathcal{H}} \left[ f\left(\theta_{\hat{s}_k(0,h,-t',h')}(\mathcal{H}^1 + \delta_{(0,h)} + \mathcal{H}^2 + \delta_{(-t',h')})\right) \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l}\left(E_{\hat{h}_k(0,h,-t',h')}(v_l)\right) = 0 \right] dt' dh' dh \\ + \int_0^\infty \int_h^\infty \int_0^\infty \mathbb{E}_{\mathcal{H}} \left[ f\left(\theta_{\hat{s}_k(0,h,-t',h')}(\mathcal{H}^1 + \delta_{(0,h)} + \mathcal{H}^2 + \delta_{(-t',h')})\right) \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l}\left(E_{\hat{h}_k(0,h,-t',h')}(v_l)\right) = 0 \right] dt' dh' dh, \quad (8.10)$$

by making the change of variable $t - t'$ to $t'$, where $t^* = \frac{1}{v_1 v_2} (h^2 - h'^2)^{\frac{1}{2}} (v_1^2 - v_2^2)^{\frac{1}{2}}$. We obtain the result for $l = 2, r = 1$ and $k = 1, 2$, by combining the three equations (8.7), (8.9) and (8.10). $\square$

Proof of Lemma 8.4, $l = 1, r = 2$ and $k = 1, 2$. Here we derive the result, similarly to above for $l = 2, r = 1$ and $k = 1, 2$, i.e., for $\mathbb{E}_{\mathcal{V}_{1,2}^{(k)}}[f(\mathcal{H})]$. Suppose $(T_i^2, H_i^2)$ and $(T_j^1, H_j^1)$ in the support of $\mathcal{H}^2, \mathcal{H}^1$, respectively, such that $T_j^1 \in D_\ell'(T_i^2, H_i^2, H_j^1)$. The k-th intersection point of the two radial birds with these heads is $(\hat{S}_{1,2}^{(k)}, \hat{H}_{1,2}^{(k)})$, in short written as $(\hat{S}^{(k)}, \hat{H}^{(k)})$. By (8.3) and the definition of $\mathbb{P}_{\mathcal{R}_{1,2}^{(k)}}^0$ we have

$$\mathbb{E}_{\mathcal{V}_{1,2}^{(k)}}^0 [f(\mathcal{H})] = \mathbb{E}_{\mathcal{R}_{1,2}^{(k)}}^0 \left[ f(\theta_{\beta_{1,2}^{(k)}(0)}(\mathcal{H})) \right] = \frac{1}{\Lambda_{1,2}^{(k)}} \mathbb{E} \left[ \sum_{T_i^2 \in \mathcal{R}_{1,2}^{(k)} : 0 \le T_i^2 \le 1} f \left( \theta_{\beta_{1,2}^{(k)}(0)} \circ \theta_{T_i^2}(\mathcal{H}) \right) \right] \\ = \frac{1}{\Lambda_{1,2}^{(k)}} \mathbb{E} \left[ \sum_{T_i^2 \in \mathcal{R}_{1,2}^{(k)} : 0 \le T_i^2 \le 1} f \left( \theta_{T_i^2 + \beta_{1,2}^{(k)}(T_i^2)}(\mathcal{H}) \right) \right], \quad (8.11)$$

using the fact that $\theta_{\beta_{1,2}^{(k)}(0)} \circ \theta_{T_i^2} = \theta_{T_i^2 + \beta_{1,2}^{(k)}(T_i^2)}$. The last expectation in (8.11) is equal to
$$
\begin{align}
& \mathbb{E} \left[ \sum_{T_i^2 \in \mathcal{R}_{1,2}^{(k)} : 0 \le T_i^2 \le 1} f \left( \theta_{T_i^2 + \beta_{1,2}^{(k)}(T_i^2)}(\mathcal{H}) \right) \right] \nonumber \\
&= \mathbb{E} \left[ \sum_{(T_i^2, H_i^2) \in \mathcal{H}^2 : 0 \le T_i^2 \le 1} \mathbb{1}_{\exists(T_j^1, H_j^1) \in \mathcal{H}^1 : T_j^1 \in D'_l(T_i^2, H_i^2, H_j^1)} f \left( \theta_{T_i^2 + \beta_{1,2}^{(k)}(T_i^2)}(\mathcal{H}) \right) \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l \left( E_{\hat{H}^l(k)}^{\hat{S}^{(k)}, v_l} \right) = 0} \right] \nonumber \\
&= \mathbb{E} \left[ \sum_{(T_i^2, H_i^2) \in \mathcal{H}^2 : 0 \le T_i^2 \le 1} \sum_{(T_j^1, H_j^1) \in \mathcal{H}^1 : T_j^1 \in D'_l(T_i^2, H_i^2, H_j^1)} f \left( \theta_{T_i^2 + \beta_{1,2}^{(k)}(T_i^2)}(\mathcal{H}) \right) \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l \left( E_{\hat{H}^l(k)}^{\hat{S}^{(k)}, v_l} \right) = 0} \right] \nonumber \\
&= \mathbb{E} \left[ \sum_{(T_i^2, H_i^2) \in \mathcal{H}^2 : 0 \le T_i^2 \le 1} \sum_{(T_j^1, H_j^1) \in \mathcal{H}^1 : T_j^1 \in D'_l(T_i^2, H_i^2, H_j^1)} f \left( \theta_{\hat{S}^{(k)}}(\mathcal{H}) \right) \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l \left( E_{\hat{H}^l(k)}^{\hat{S}^{(k)}, v_l} \right) = 0} \right], \tag{8.12}
\end{align}
$$

where $T_i^2 + \beta_{1,2}^{(k)}(T_i^2) = \hat{S}^{(k)}$. Applying the Campbell-Mecke formula for the factorial power of order 2 based on the two-point Palm probability measure $\mathbb{P}_{\mathcal{H}}^{(t,h),(t',h')} = \mathbb{P}_{\mathcal{H}^2}^{(t,h)} \otimes \mathbb{P}_{\mathcal{H}^1}^{(t',h')}$, the expectation in (8.12) evaluates to

$$
4\lambda_1\lambda_2 v_1 v_2 \int_0^1 \int_0^\infty \int_0^\infty \int_{D'_\ell(t,h,h')} \mathbb{E}_{\mathcal{H}} \left[ f \left( \theta_{\hat{s}_k(t,h,t',h')} (\mathcal{H}^2 + \delta_{(t,h)} + \mathcal{H}^1 + \delta_{(t',h')}) \right) \right. \\
\left. \qquad \times \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l \left( E_{\hat{h}_k(t,h,t',h')}^{\hat{s}_k(t,h,t',h'),v_l} \right) = 0} \right] dt' dh' dh dt, \quad (8.13)
$$

similarly to (8.9), where the set $D'_\ell(t, h, h')$ is defined as in (7.9). The set $D'_\ell(t, h, h') = (-\infty, t]$ when $h < h'$ but $D'_\ell(t, h, h') = (-\infty, t - t^*)$, when $h \ge h'$, where $t^*$ is as defined in (7.15). This leads to a simplification of the multiple integral in (8.13) to

$$
\begin{align}
& \int_0^1 \int_0^\infty \int_0^h \int_{-\infty}^{t-t^*} \mathbb{E}_{\mathcal{H}} \left[ f \left( \theta_{\hat{s}_k(t,h,t',h')} (\mathcal{H}^2 + \delta_{(t,h)} + \mathcal{H}^1 + \delta_{(t',h')}) \right) \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l \left( E_{\hat{h}_k(t,h,t',h')}^{\hat{s}_k(t,h,t',h'),v_l} \right) = 0} \right] dt' dh' dh dt \nonumber \\
& \qquad + \int_0^1 \int_0^\infty \int_h^\infty \int_{-\infty}^t \mathbb{E}_{\mathcal{H}} \left[ f \left( \theta_{\hat{s}_k(t,h,t',h')} (\mathcal{H}^2 + \delta_{(t,h)} + \mathcal{H}^1 + \delta_{(t',h')}) \right) \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l \left( E_{\hat{h}_k(t,h,t',h')}^{\hat{s}_k(t,h,t',h'),v_l} \right) = 0} \right] dt' dh' dh dt \nonumber \\
&= \int_0^\infty \int_0^h \int_{t^*}^\infty \mathbb{E}_{\mathcal{H}} \left[ f \left( \theta_{\hat{s}_k(0,h,-t',h')} (\mathcal{H}^2 + \delta_{(0,h)} + \mathcal{H}^1 + \delta_{(-t',h')}) \right) \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l \left( E_{\hat{h}_k(0,h,-t',h')}^{\hat{s}_k(0,h,-t',h'),v_l} \right) = 0} \right] dt' dh' dh \nonumber \\
& \qquad + \int_0^\infty \int_h^\infty \int_0^\infty \mathbb{E}_{\mathcal{H}} \left[ f \left( \theta_{\hat{s}_k(0,h,-t',h')} (\mathcal{H}^2 + \delta_{(0,h)} + \mathcal{H}^1 + \delta_{(-t',h')}) \right) \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l \left( E_{\hat{h}_k(0,h,-t',h')}^{\hat{s}_k(0,h,-t',h'),v_l} \right) = 0} \right] dt' dh' dh, \tag{8.14}
\end{align}
$$

by making the change of variable $t-t'$ to $t'$, where $t^* = \frac{1}{v_1 v_2} (h^2 - h'^2)^{\frac{1}{2}} (v_1^2 - v_2^2)^{\frac{1}{2}}$. The result is obtained in this case by combining the three equations (8.14), (8.13) and (8.12). $\square$

**Remark 8.6.** For the unit function, i.e., $f(\mathcal{H}) = 1$, we can check that $E_{\mathcal{V}}^0[f(\mathcal{H})] = 1$. Indeed, using Lemma 8.3 and part (1) of Theorem 7.29 we have

$$
\begin{align}
\mathbb{E}_{v_l}^0[f(\mathcal{H})] &= \frac{4\lambda_l^2 v_l^2}{\Lambda_l} \int_{(\mathbb{R}^+)^3} \mathbb{E}_{\mathcal{H}} \left[ \prod_{i=1,2} \mathbb{1}_{\mathcal{H}^i \left( E_{\hat{h}_k(0,h,-t',h')}^{\hat{s}_k(0,h,-t',h'),v_l} \right)} \right] dt' dh' dh \nonumber \\
&= \frac{4\lambda_l^2 v_l^2}{\Lambda_l} \int_{(\mathbb{R}^+)^3} e^{-\lambda\pi\hat{h}^2(0,h,-t',h')} dt' dh' dh = \frac{4v_l\sqrt{\lambda}}{\pi\Lambda_l} \left(\frac{\lambda_l}{\lambda}\right)^2 = 1. \tag{8.15}
\end{align}
$$
Using Lemma 8.4, we also have

$$
\begin{align*}
\mathbb{E}_{\mathcal{V}_{l,r}^{(k)}}[f(\mathcal{H})] &= \frac{4\lambda_1\lambda_2 v_1 v_2}{\Lambda_{l,r}^{(k)}} \int_0^\infty \int_0^h \int_{t^*}^0 \mathbb{E}_{\mathcal{H}} \left[ \prod_{i=1,2} \mathbb{1}_{\mathcal{H}^i} \left( E_{\hat{h}_k(0,h,-t',h')}(\tilde{s}_k(0,h,-t',h'),v_i) \right) \right] dt' dh' dh \\
&\quad + \frac{4\lambda_1\lambda_2 v_1 v_2}{\Lambda_{l,r}^{(k)}} \int_0^\infty \int_h^\infty \int_0^\infty \mathbb{E}_{\mathcal{H}} \left[ \prod_{i=1,2} \mathbb{1}_{\mathcal{H}^i} \left( E_{\hat{h}_k(0,h,-t',h')}(\tilde{s}_k(0,h,-t',h'),v_i) \right) \right] dt' dh' dh \\
&= \frac{4\lambda_1\lambda_2 v_1 v_2}{\Lambda_{l,r}^{(k)}} \int_0^\infty \left[ \int_0^h \int_{t^*}^\infty e^{-\lambda\pi\hat{h}_k^2(0,h,-t',h')} dt' dh' + \int_h^\infty \int_0^\infty e^{-\lambda\pi\hat{h}_k^2(0,h,-t',h')} dt' dh' \right] dh \\
&= \frac{1}{\Lambda_{l,r}^{(k)}} \Lambda_{l,r}^{(k)} = 1, \tag{8.16}
\end{align*}
$$

from the integral in (7.47). Thus using Theorem 8.2, Lemma 8.1 and (7.48), we have

$$
E_{\mathcal{V}}^{0}[f(\mathcal{H})] = \frac{1}{\lambda_{\mathcal{V}}} \sum_{l \in \{1,2\}} \Lambda_{l} + \frac{1}{\lambda_{\mathcal{V}}} \sum_{l,r, k \in \{1,2\}: l \neq r} \Lambda_{l,r}^{(k)} = 1.
$$

8.1.1. Decomposition of $\mathbb{E}_{\mathcal{V}}^0[f(\mathcal{H})]$ for symmetric functions $f$. The time reversal argument also provides symmetric relations among the Palm probability measures $\mathbb{E}_{\mathcal{V}_{l,r}^{(k)}}[f(\mathcal{H})]$, for pairs of values of $l \neq r, k \in \{1,2\}$. It is described in the forth-coming result, under a necessary condition that the function $f$ is symmetric, which we define as follows. Define $-\mathcal{H} := \sum_i \delta_{(-T_i, H_i)}$. We say that a function $f: \Omega \to \mathbb{R}^+$ is said to be symmetric, if $f(-\mathcal{H}) = f(\mathcal{H})$.

**Lemma 8.7.** For all non-negative measurable and symmetric functions $f$

$$
\mathbb{E}_{\mathcal{V}_{1,2}^{(1)}}[f(\mathcal{H})] = \mathbb{E}_{\mathcal{V}_{2,1}^{(2)}}[f(\mathcal{H})] \quad \text{and} \quad \mathbb{E}_{\mathcal{V}_{1,2}^{(2)}}[f(\mathcal{H})] = \mathbb{E}_{\mathcal{V}_{2,1}^{(1)}}[f(\mathcal{H})]. \qquad (8.17)
$$

Proof. Let $\mathcal{L}_{1,2}^{(k)}$ be the point process made of the abscissas of the left most head point corresponding
to handovers of type $\binom{k}{1,2}$. Then, there exist bijections $\tilde{\beta}_{1,2}^{(k)}$ between $\mathcal{V}_{1,2}^{(k)}$ and $\mathcal{L}_{1,2}^{(k)}$, defined similarly to
$\beta_{1,2}^{(k)}$, for $k=1,2$, as in Part (ii) of Lemma 7.18. Under the symmetry assumption on the measurable
function $f$, we have the following relations:

$$
\mathbb{E}_{\mathcal{V}_{1,2}^{(1)}}[f(\mathcal{H})] = \mathbb{E}_{\mathcal{L}_{1,2}^{(1)}}\left[f(\theta_{\tilde{\beta}_{1,2}^{(1)}(0)}(\mathcal{H}))\right] = \mathbb{E}_{\mathcal{R}_{2,1}^{(2)}}\left[f(\theta_{\beta_{2,1}^{(2)}(0)}(\mathcal{H}))\right] = \mathbb{E}_{\mathcal{V}_{2,1}^{(2)}}[f(\mathcal{H})], \quad (8.18)
$$

$$
\mathbb{E}_{\mathcal{V}_{1,2}^{(2)}}[f(\mathcal{H})] = \mathbb{E}_{\mathcal{L}_{1,2}^{(2)}}\left[f(\theta_{\tilde{\beta}_{1,2}^{(2)}(0)}(\mathcal{H}))\right] = \mathbb{E}_{\mathcal{R}_{2,1}^{(1)}}\left[f(\theta_{\beta_{2,1}^{(1)}(0)}(\mathcal{H}))\right] = \mathbb{E}_{\mathcal{V}_{2,1}^{(1)}}[f(\mathcal{H})]. \quad (8.19)
$$

In both relations, the third equality can be justified through a time-reversal argument, similarly to
(7.41, proved in the following lemma. This completes the proof. $\square$

**Lemma 8.8.** For all non-negative measurable and symmetric functions $f$, we have

$$
\mathbb{E}_{\mathcal{L}_{1,2}^{(1)}} \left[ f(\theta_{\tilde{\beta}_{1,2}^{(1)}(0)}(\mathcal{H})) \right] = \mathbb{E}_{\mathcal{R}_{2,1}^{(2)}} \left[ f(\theta_{\beta_{2,1}^{(2)}(0)}(\mathcal{H})) \right], \qquad (8.20)
$$

$$
\mathbb{E}_{\mathcal{L}_{1,2}^{(2)}} \left[ f(\theta_{\tilde{\beta}_{1,2}^{(2)}(0)}(\mathcal{H})) \right] = \mathbb{E}_{\mathcal{R}_{2,1}^{(1)}} \left[ f(\theta_{\beta_{2,1}^{(1)}(0)}(\mathcal{H})) \right]. \quad (8.21)
$$
Proof. For the equality in (8.20), we can write similarly to (8.5),

$$
\begin{align}
\mathbb{E}_{\mathcal{L}_{1,2}^{(1)}} [f(\theta_{\tilde{\beta}_{1,2}^{(1)}(0)}(\mathcal{H}))] &= \frac{1}{\Lambda_{1,2}^{(1)}} \mathbb{E} \left[ \sum_{T_i^1 \in \mathcal{L}_{1,2}^{(1)} : 0 \le T_i^1 \le 1} f \left( \theta_{\tilde{\beta}_{1,2}^{(1)}(0)} \circ \theta_{T_i^1}(\mathcal{H}) \right) \right] \nonumber \\
&= \frac{1}{\Lambda_{1,2}^{(1)}} \mathbb{E} \left[ \sum_{T_i^1 \in \mathcal{L}_{1,2}^{(1)} : 0 \le T_i^1 \le 1} f \left( \theta_{T_i^1 + \tilde{\beta}_{1,2}^{(1)}(T_i^1)}(\mathcal{H}) \right) \right], \tag{8.22}
\end{align}
$$

using the fact that $\theta_{\tilde{\beta}_{1,2}^{(1)}(0)} \circ \theta_{T_i^1} = \theta_{T_i^1+\tilde{\beta}_{1,2}^{(1)}(T_i^1)}$. Thus from (8.22) we have

$$
\begin{align}
& \mathbb{E} \left[ \sum_{T_i^1 \in \mathcal{L}_{1,2}^{(1)} : 0 \le T_i^1 \le 1} f \left( \theta_{T_i^1 + \tilde{\beta}_{1,2}^{(1)}(T_i^1)} (\mathcal{H}) \right) \right] \nonumber \\
&= \mathbb{E} \left[ \sum_{(T_i^1, H_i^1) \in \mathcal{H}^1 : 0 \le T_i^1 \le 1} \mathbb{1}_{\exists(T_j^2, H_j^2) \in \mathcal{H}^2 : T_j^2 \in D_r(T_i^1, H_i^1, H_j^2)} f \left( \theta_{T_i^1 + \tilde{\beta}_{1,2}^{(1)}(T_i^1)} (\mathcal{H}) \right) \prod_{l=1,2} \mathbb{1}_{H^l \left( E_{\hat{H}^{(1)}}^{\hat{S}^{(1)},v_l} \right) = 0} \right] \nonumber \\
&= \mathbb{E} \left[ \sum_{(T_i^1, H_i^1) \in \mathcal{H}^1 : 0 \le T_i^1 \le 1} \sum_{(T_j^2, H_j^2) \in \mathcal{H}^2 : T_j^2 \in D_r(T_i^1, H_i^1, H_j^2)} f \left( \theta_{T_i^1 + \tilde{\beta}_{1,2}^{(1)}(T_i^1)} (\mathcal{H}) \right) \prod_{l=1,2} \mathbb{1}_{H^l \left( E_{\hat{H}^{(1)}}^{\hat{S}^{(1)},v_l} \right) = 0} \right]. \tag{8.23}
\end{align}
$$

Using the fact that $T_i^1 + \tilde{\beta}_{1,2}^{(1)}(T_i^1) = \hat{S}^{(1)}$ we have (8.23) that

$$
\begin{align}
& \mathbb{E} \left[ \sum_{T_i^1 \in \mathcal{L}_{1,2}^{(1)} : 0 \le T_i^1 \le 1} f \left( \theta_{T_i^1 + \tilde{\beta}_{1,2}^{(1)}(T_i^1)} (\mathcal{H}) \right) \right] \nonumber \\
&= \mathbb{E} \left[ \sum_{(T_i^1, H_i^1) \in \mathcal{H}^1 : 0 \le T_i^1 \le 1} \sum_{(T_j^2, H_j^2) \in \mathcal{H}^2 : T_j^2 \in D_r(T_i^1, H_i^1, H_j^2)} f \left( \theta_{\hat{S}^{(1)}(\mathcal{H})} \right) \prod_{l=1,2} \mathbb{1}_{H^l \left( E_{\hat{H}^{(1)}}^{\hat{S}^{(1)},v_l} \right) = 0} \right] \nonumber \\
&= \mathbb{E} \left[ \sum_{(T_i^1, H_i^1) \in \mathcal{H}^1 : 0 \le T_i^1 \le 1} \sum_{(T_j^2, H_j^2) \in \mathcal{H}^2 : T_j^2 \in D_\ell(T_i^1, H_i^1, H_j^2)} f \left( \theta_{\hat{S}^{(2)}(\mathcal{H})} \right) \prod_{l=1,2} \mathbb{1}_{H^l \left( E_{\hat{H}^{(2)}}^{\hat{S}^{(2)},v_l} \right) = 0} \right] \tag{8.24} \\
& \hspace{2em} \stackrel{(8.7),(8.5)}{=} \Lambda_{2,1}^{(2)} \mathbb{E}_{\mathcal{R}_{2,1}^{(2)}}^0 \left[ f(\theta_{\beta_{2,1}^{(2)}(0)}(\mathcal{H})) \right], \nonumber
\end{align}
$$

where $(\tilde{S}^{(2)}, \tilde{H}^{(2)})$ is the intersection of type $(\tilde{\beta}_{2,1}^{(2)})$. In the second step in (8.24), after counting backwards in time, we have used the symmetry of the function $f$, i.e., $f(\mathcal{H}) = f(-\mathcal{H})$. As seen before in Part (ii) of Theorem 7.20, we have $\Lambda_{1,2}^{(1)} = \Lambda_{2,1}^{(2)}$. We obtain the result using the last fact and combining (8.22) and (8.24). The equality in (8.21) can be proved similarly. $\square$

Under this symmetry we can re-write the decomposition by folding the symmetric terms for mixed handovers in (8.4), as

$$
\mathbb{E}_{\mathcal{V}}^{0}[f(\mathcal{H})] = \sum_{l \in \{1,2\}} \frac{\Lambda_l}{\lambda_{\mathcal{V}}} \mathbb{E}_{\mathcal{V}_l}^{0}[f(\mathcal{H})] + 2 \sum_{r<l,k \in \{1,2\}} \frac{\Lambda_{l,r}^{(k)}}{\lambda_{\mathcal{V}}} \mathbb{E}_{\mathcal{V}_{l,m}^{(k)}}[f(\mathcal{H})]. \quad (8.25)
$$

8.2. Typical handover distance in the two-speed case. Let $\hat{H}$ be the random variable for the distance of the typical handover. Based on different type of typical handovers, we have following result about the decomposition of the Laplace transform of $\hat{H}^2$ under the Palm probability measure $\mathbb{P}_{\mathcal{V}}^0$.
**Lemma 8.9.** under the Palm probability measure $\mathbb{P}_V^0$ of handovers, for any parameter $\gamma \ge 0$, the Laplace transform of $\hat{H}^2$ is

$$ \mathcal{L}_{\hat{H}^2}^{0}(\gamma) = \frac{1}{\lambda_V} \frac{4\sqrt{\lambda}}{\pi} \left(1 + \frac{\gamma}{\lambda\pi}\right)^{-3/2} \sum_{l \in \{1,2\}} v_l \left(\frac{\lambda_l}{\lambda}\right)^2 + \frac{8\lambda_1\lambda_2 v_1 v_2}{\lambda_V} \sum_{k \in \{1,2\}} \eta_{2,1}^{(k)}(\gamma), \quad (8.26) $$

where, for $k = 1, 2$, $\eta_{2,1}^{(k)}(\gamma)$ is determined in (8.30) and (8.31), respectively.

*Proof of Lemma 8.9.* We apply the decomposition of the Palm probability measure with respect to handovers as in Theorem 8.2, with the help of Lemma 8.3, Lemma 8.4, for the symmetric function $f(\mathcal{H}) = e^{-\gamma \hat{H}^2}$. The decomposition is according to the different types of handovers. Let $\hat{H}_l$ is the typical handover distance corresponding to the handover of pure type $l$, for $l \in \{1, 2\}$. For $l \ne r, k \in \{1, 2\}$, $\hat{H}_{l,r}^{(k)}$ denotes the typical handover distance due to the typical handover of type $\binom{k}{l,r}$. Observation 7.14 supports the fact that

$$ \hat{H}_{2,1}^{(1)} \stackrel{d}{=} \hat{H}_{1,2}^{(2)} \text{ and } \hat{H}_{2,1}^{(2)} \stackrel{d}{=} \hat{H}_{1,2}^{(1)}. \quad (8.27) $$

Then using the symmetry in (8.27) and the formula (8.25), the Laplace transform of $\hat{H}^2$ under the Palm probability measure $\mathbb{P}_V^0$ is

$$ \mathbb{E}_{\mathcal{V}}^{0} \left[ e^{-\gamma \hat{H}^2} \right] = \sum_{l \in \{1,2\}} \frac{\Lambda_l}{\lambda_{\mathcal{V}}} \mathbb{E}_{\mathcal{V}_l}^{0} \left[ e^{-\gamma \hat{H}_l^2} \right] + 2 \sum_{k \in \{1,2\}} \frac{\Lambda_{2,1}^{(k)}}{\lambda_{\mathcal{V}}} \mathbb{E}_{\mathcal{V}_{2,1}^{k}}^{0} \left[ e^{-\gamma (\hat{H}_{2,1}^{(k)})^2} \right]. \quad (8.28) $$

for any parameter $\gamma \ge 0$. Using calculations similar to those in Equation (7.34) and Remark 5.7, we can obtain that

$$ \mathbb{E}_{\mathcal{V}_l}^{0} \left[ e^{-\gamma \hat{H}_l^2} \right] = \frac{1}{\Lambda_l} \left( \frac{\lambda_l}{\lambda} \right)^2 \frac{4v_l \sqrt{\lambda}}{\pi} \left( 1 + \frac{\gamma}{\lambda\pi} \right)^{-3/2} = \left( 1 + \frac{\gamma}{\lambda\pi} \right)^{-3/2}, \quad (8.29) $$

since $\Lambda_l = \left(\frac{\lambda_l}{\lambda}\right)^2 \frac{4v_l\sqrt{\lambda}}{\pi}$. Thus we can say that, for the pure handover of type $l$, the handover distance $\hat{H}_l$ follows a Nakagami distribution with parameters $(\frac{3}{2}, \frac{3}{2\lambda\pi})$, for each $l = 1, 2$, see Lemma 5.6. In what follows, we evaluate the individual Laplace transforms $\mathbb{E}_{\mathcal{V}_{2,1}^{(1)}}[e^{-\gamma(\hat{H}_{2,1}^{(1)})^2}]$ and $\mathbb{E}_{\mathcal{V}_{2,1}^{(2)}}[e^{-\gamma(\hat{H}_{2,1}^{(2)})^2}]$, which is enough due to the symmetry in (8.27). Using the formula in Lemma 8.4, we get

$$ \begin{aligned} & \frac{\Lambda_{2,1}^{(1)}}{4\lambda_1\lambda_2 v_1 v_2} \mathbb{E}_{\mathcal{V}_{2,1}^{(1)}}[e^{-\gamma(\hat{H}_{2,1}^{(1)})^2}] \\ &= \int_0^\infty \int_0^h \int_{t^*}^\infty \mathbb{E}_{\mathcal{H}} \left[ e^{-\gamma \hat{h}_1^2 (0,h, -t', h')} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}_1 (0,h, -t', h')}^{\hat{s}_1 (0,h, -t', h')}, v_l \right) \right] =0 \right] dt' dh' dh \\ & \quad + \int_0^\infty \int_h^\infty \int_0^\infty \mathbb{E}_{\mathcal{H}} \left[ e^{-\gamma \hat{h}_1^2 (0,h, -t', h')} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}_1 (0,h, -t', h')}^{\hat{s}_1 (0,h, -t', h')}, v_l \right) \right] =0 \right] dt' dh' dh \\ &= \int_0^\infty \int_0^h \int_{t^*}^\infty e^{-(\gamma+\lambda\pi)\hat{h}_1^2 (0,h, -t', h')} dt' dh' dh + \int_0^\infty \int_h^\infty \int_0^\infty e^{-(\gamma+\lambda\pi)\hat{h}_1^2 (0,h, -t', h')} dt' dh' dh \\ &:= \eta_{2,1}^{(1)}(\gamma), \end{aligned} \quad (8.30) $$

where $t^* = \frac{1}{v_1 v_2} (h^2 - h'^2)^{\frac{1}{2}} (v_1^2 - v_2^2)^{\frac{1}{2}}$, given $h, h'$. Similarly, we can rewrite the other term as

$$ \begin{aligned} & \frac{\Lambda_{2,1}^{(2)}}{4\lambda_1\lambda_2 v_1 v_2} \mathbb{E}_{\mathcal{V}_{2,1}^{(2)}}[e^{-\gamma(\hat{H}_{2,1}^{(2)})^2}] \\ &= \int_0^\infty \int_0^h \int_{t^*}^\infty e^{-(\gamma+\lambda\pi)\hat{h}_2^2 (0,h, -t', h')} dt' dh' dh + \int_0^\infty \int_h^\infty \int_0^\infty e^{-(\gamma+\lambda\pi)\hat{h}_2^2 (0,h, -t', h')} dt' dh' dh \\ &:= \eta_{2,1}^{(2)}(\gamma), \end{aligned} \quad (8.31) $$
where $t^* = \frac{1}{v_1 v_2} (h^2 - h'^2)^{\frac{1}{2}} (v_1^2 - v_2^2)^{\frac{1}{2}}$, given $h, h'$. Observe that, taking $\gamma = 0$ in (8.30) and (8.31), and later using (7.64) we have

$$4\lambda_1\lambda_2 v_1 v_2 \eta_{2,1}^{(k)}(0) = \Lambda_{2,1}^{(k)}, \qquad (8.32)$$

for $k = 1, 2$. The final expression for $\mathcal{L}_{\hat{H}^2}^0(\gamma)$ can be obtained by substituting the resultant from (8.29), (8.30) and (8.31) to (8.28) as

$$\begin{aligned} \mathcal{L}_{\hat{H}^2}^0(\gamma) &= \sum_{l \in \{1,2\}} \frac{\Lambda_l}{\lambda_v} \left(1 + \frac{\gamma}{\lambda\pi}\right)^{-3/2} + 2 \sum_{k \in \{1,2\}} \frac{\Lambda_{2,1}^{(k)}}{\lambda_v} \frac{4\lambda_1\lambda_2 v_1 v_2}{\Lambda_{2,1}^{(k)}} \eta_{2,1}^{(k)}(\gamma) \\ &= \frac{1}{\lambda_v} \frac{4\sqrt{\lambda}}{\pi} \left(1 + \frac{\gamma}{\lambda\pi}\right)^{-3/2} \sum_{l \in \{1,2\}} v_l \left(\frac{\lambda_l}{\lambda}\right)^2 + \frac{8\lambda_1\lambda_2 v_1 v_2}{\lambda_v} \sum_{k \in \{1,2\}} \eta_{2,1}^{(k)}(\gamma), \end{aligned} \qquad (8.33)$$

where $\Lambda_l = \left(\frac{\lambda_l}{\lambda}\right)^2 \frac{4v_l\sqrt{\lambda}}{\pi}$, for $l = 1, 2$ and $\eta_{2,1}^{(k)}(\gamma)$ determined in (8.30) and (8.31), for $k = 1, 2$. $\square$

**Remark 8.10** ($\mathcal{L}_{\hat{H}^2}^0(0) = 1$). We can perform a sanity check that $\mathcal{L}_{\hat{H}^2}^0(0) = 1$. Indeed, by taking $\gamma = 0$ in (8.33), we have,

$$\mathcal{L}_{\hat{H}^2}^0(0) = \frac{1}{\lambda_v} \frac{4\sqrt{\lambda}}{\pi} \sum_{l \in \{1,2\}} v_l \left(\frac{\lambda_l}{\lambda}\right)^2 + \frac{8\lambda_1\lambda_2 v_1 v_2}{\lambda_v} \sum_{k \in \{1,2\}} \eta_{2,1}^{(k)}(0) = \frac{1}{\lambda_v} \sum_{l \in \{1,2\}} \Lambda_l + \frac{2}{\lambda_v} \sum_{k \in \{1,2\}} \Lambda_{2,1}^{(k)} = 1,$$

where we have used the fact that $4\lambda_1\lambda_2 v_1 v_2 \eta_{2,1}^{(k)}(0) = \Lambda_{2,1}^{(k)}$ from (8.32), for $k = 1, 2$, $\Lambda_l = \frac{4\sqrt{\lambda}}{\pi} v_l \left(\frac{\lambda_l}{\lambda}\right)^2$, for $l = 1, 2$ and the total handover frequency derived in (7.48).

**Remark 8.11** (Single-speed case). Using the last result, we can recover the Laplace transform of $\hat{H}^2$ under the Palm probability measure $\mathbb{P}_V^0$ in the single-speed case, which is $\mathcal{L}_{\hat{H}^2}^0(\gamma) = (1 + \frac{\gamma}{\lambda\pi})^{-3/2}$, for any speed $v$, as shown in Lemma 5.6. Take $v_1 = v_2 = v$ and $\lambda_1 = p\lambda$ and $\lambda_2 = (1-p)\lambda$ with $p \in [0, 1]$. In (8.26), there is contribution from just one of the pair of terms $(\eta_{2,1}^{(1)}(\gamma), \Lambda_{2,1}^{(1)})$ and $(\eta_{2,1}^{(2)}(\gamma), \Lambda_{2,1}^{(2)})$, corresponding to the mixed handovers, since there is just one intersection between any two radial birds, almost surely. Let us denote the surviving term as $(\eta_{2,1}(\gamma), \Lambda_{2,1})$, where $\eta_{2,1}(\gamma)$ can be determined from one of (8.30) and (8.31). Then from (8.26), we have

$$\begin{aligned} \mathcal{L}_{\hat{H}^2}^0(\gamma) &= \frac{1}{\lambda_v} \frac{4\sqrt{\lambda}}{\pi} \left(1 + \frac{\gamma}{\lambda\pi}\right)^{-3/2} v(p^2 + (1-p)^2) + \frac{8\lambda^2 v^2}{\lambda_v} p(1-p)\eta_{2,1}(\gamma) \\ &= \frac{1}{\lambda_v} \frac{4\sqrt{\lambda}}{\pi} \left(1 + \frac{\gamma}{\lambda\pi}\right)^{-3/2} v(p^2 + (1-p)^2) + \frac{2}{\lambda_v} p(1-p) \frac{4v\sqrt{\lambda}}{\pi} \left(1 + \frac{\gamma}{\lambda\pi}\right)^{-3/2} \\ &= \frac{1}{\lambda_v} \frac{4v\sqrt{\lambda}}{\pi} \left(1 + \frac{\gamma}{\lambda\pi}\right)^{-3/2} (p^2 + (1-p)^2 + 2p(1-p)) = \left(1 + \frac{\gamma}{\lambda\pi}\right)^{-3/2}, \end{aligned} \qquad (8.34)$$

since $\lambda_v = \frac{4v\sqrt{\lambda}}{\pi}$, in the single-speed case and $4\lambda^2 v^2 \eta_{2,1}^{(k)}(\gamma) = \frac{4v\sqrt{\lambda}}{\pi} (1 + \frac{\gamma}{\lambda\pi})^{-3/2}$, as in (8.29), using the fact that $t^* = 0$ in one of (8.30) or (8.31).

### 8.3. Distribution of inter-handover time: two-speed case.
The entire subsection is devoted to our final result about the Palm probability distribution of the inter-handover time $T$. It is based on an extension of the ideas in the proof of Theorem 5.17, with the required adaptation due to the existence of different types of handovers at a typical time. Under the Palm probability measure $\mathbb{P}_V^0$ of handovers, the inter-handover time is the duration between the handover at time 0 and the next handover. We denote the inter-handover time by $T$. Before moving to the Palm probability distribution of the inter-handover time, we refer the reader to Subsection 7.1.4 and Subsection 7.1.5 for the geometric interpretation of the intersection criteria (Subsection 7.1.3) and their applications. This enables us to properly manage the rise in the complexity resulting from the introduction of more than one speed.

#### 8.3.1. Inter-handover time.
Suppose for any $s \in \mathbb{R}$, $T_1(s)$ denotes the first time a handover happens after time $s$. To formally define the inter-handover time, suppose there is a handover at time 0 and at distance $\hat{h}_0$. For ease of usage, we introduce an alternative notation for handover types. We denote
the head points of the radial birds corresponding to the station that takes over the service and leaves the service, by $(t_n, h_n)$ and $(t_p, h_p)$, respectively. Here the subscript $n$ stands for *next* and $p$ stands for *previous*. We use the notation $\tau_n = \text{type}(n)$, for the type of the next head point, and $\tau_p = \text{type}(p)$, for the type of the previous head point. Let us define

$$ q := \begin{cases} 1 & \text{if } t_n \ge t_p, \\ 2 & \text{if } t_n < t_p. \end{cases} \qquad (8.35) $$

Then we define $\begin{bmatrix} q \\ \tau_p, \tau_n \end{bmatrix}$ to be a class or type of a handover where the next serving station is of type $\tau_n$, the previous one is of type $\tau_n$ and $q = 1$ or 2, depending on whether $t_n \ge t_p$ or $t_n < t_p$, respectively. The correspondence between the notation used so far and the new one is given below. Consider the pure handovers of type $l$ as $\begin{pmatrix} 1 \\ l,l \end{pmatrix}$, for $l = 1, 2$. Then the bijection between the two notations is given in Table 8.1.

<table><caption>TABLE 8.1. Table of correspondence between old and new notations $\begin{pmatrix} k \\ l,r \end{pmatrix}$ and $\begin{bmatrix} q \\ \tau_p, \tau_n \end{bmatrix}$. Here HoPP stands for handover point process.</caption><thead><tr><th rowspan="2"></th><th colspan="2">Old notation</th><th colspan="3">New notation</th></tr><tr><th>type</th><th>HoPP</th><th>type</th><th>τ<sub>n</sub></th><th>HoPP</th></tr></thead><tbody><tr><td rowspan="2">Pure handovers</td><td>(<sup>1</sup>/<sub>1,1</sub>)</td><td>V<sub>1</sub></td><td>[[<sup>1</sup>/<sub>1,1</sub>]]</td><td>τ<sub>n</sub> = 1</td><td>W<sub>1</sub></td></tr><tr><td>(<sup>1</sup>/<sub>2,2</sub>)</td><td>V<sub>2</sub></td><td>[[<sup>1</sup>/<sub>2,2</sub>]]</td><td>τ<sub>n</sub> = 2</td><td>W<sub>2</sub></td></tr><tr><td rowspan="4">Mixed handovers</td><td>(<sup>1</sup>/<sub>1,2</sub>)</td><td>V<sub>1,2</sub><sup>(1)</sup></td><td>[[<sup>2</sup>/<sub>2,1</sub>]]</td><td>τ<sub>n</sub> = 1</td><td>W<sub>2,1</sub><sup>(2)</sup></td></tr><tr><td>(<sup>2</sup>/<sub>1,2</sub>)</td><td>V<sub>1,2</sub><sup>(2)</sup></td><td>[[<sup>1</sup>/<sub>1,2</sub>]]</td><td>τ<sub>n</sub> = 2</td><td>W<sub>1,2</sub><sup>(1)</sup></td></tr><tr><td>(<sup>1</sup>/<sub>2,1</sub>)</td><td>V<sub>2,1</sub><sup>(1)</sup></td><td>[[<sup>1</sup>/<sub>2,1</sub>]]</td><td>τ<sub>n</sub> = 1</td><td>W<sub>2,1</sub><sup>(1)</sup></td></tr><tr><td>(<sup>2</sup>/<sub>2,1</sub>)</td><td>V<sub>2,1</sub><sup>(2)</sup></td><td>[[<sup>2</sup>/<sub>1,2</sub>]]</td><td>τ<sub>n</sub> = 2</td><td>W<sub>1,2</sub><sup>(2)</sup></td></tr></tbody></table>

Suppose the typical handover is created by the intersection of the radial birds $C_{(t_n,h_n)}^{\tau_n}$ and $C_{(t_p,h_p)}^{\tau_p}$, for some head points $(t_n, h_n)$ and $(t_p, h_p)$ of type $\tau_n$ and $\tau_p$, respectively, for some $\tau_n, \tau_p \in \{1, 2\}$. Note that $\tau_n, \tau_p$ can be equal. Beware that, unlike in Subsection 7.1.4, we do not fix here the type of the head points $(t_n, h_n)$ and $(t_p, h_p)$. The handover at time 0 is pure and of type $\begin{bmatrix} 1 \\ \tau_p, \tau_n \end{bmatrix}$, if $\tau_p = \tau_n$. If $\tau_p \neq \tau_n$, these two birds can lead to a handover of any of the types $\begin{bmatrix} 1 \\ \tau_p, \tau_n \end{bmatrix}$, $\begin{bmatrix} 2 \\ \tau_p, \tau_n \end{bmatrix}$, for $\tau_p \neq \tau_n \in \{1, 2\}$. More precisely, for $\tau_p \neq \tau_n$ with $\tau_n = 1$ or 2, if $t_n < t_p$, then the handover is mixed and can be of type $\begin{bmatrix} 2 \\ 2,1 \end{bmatrix}$ or $\begin{bmatrix} 2 \\ 1,2 \end{bmatrix}$, and if $t_n \ge t_p$, then it can be of type $\begin{bmatrix} 1 \\ 1,2 \end{bmatrix}$ or $\begin{bmatrix} 1 \\ 2,1 \end{bmatrix}$.

From now onward, we only use the new notational system of $\begin{bmatrix} q \\ \tau_n, \tau_p \end{bmatrix}$ for pure and mixed handovers. We write $v_{\tau_n}$ and $v_{\tau_p}$ in short as $v_n$ and $v_p$, respectively and write $C_{(t_n,h_n)}^n$ and $C_{(t_p,h_p)}^p$ in short for $C_{(t_n,h_n)}^{\tau_n}$ and $C_{(t_p,h_p)}^{\tau_p}$, respectively. Consider the set $\{\hat{h}_n(t)\}_{t \ge 0}$, where $\hat{h}_n(t) := (v_n^2(t-t_n)^2 + h_n^2)^{\frac{1}{2}}$, denotes the height of the radial bird $C_{(t_n,h_n)}^n$ at time $t$. Essentially, the curve $\hat{h}_n(\cdot)$, as a function of time, represents the curve of the radial bird $C_{(t_n,h_n)}^{\tau_n}$, which takes over. Observe that $\hat{h}_n(0) = (v_n^2 t_n^2 + h_n^2)^{\frac{1}{2}} := \hat{h}_0$.
For $m \in \{1, 2\}$ such that $m \neq \tau_n$, let $\{E_{\hat{h}_n(t)}^{t, v_n}\}_{t \ge 0}$ and $\{E_{\hat{h}_n(t)}^{t, v_m}\}_{t \ge 0}$, be the two collections of open half-ellipses of heights $\{\hat{h}_n(t)\}_{t \ge 0}$. For $t \ge 0$, define the extra regions

$$S_t^n := \left( \bigcup_{t' \in [0, t]} E_{\hat{h}_n(t')}^{t', v_n} \setminus E_{\hat{h}_0}^{0, v_n} \right)^{\circ} \quad \text{and} \quad S_t^m := \left( \bigcup_{t' \in [0, t]} E_{\hat{h}_n(t')}^{t', v_m} \setminus E_{\hat{h}_0}^{0, v_m} \right)^{\circ}. \qquad (8.36)$$

Due to their definition as unions of sets, we have:

**Lemma 8.12.** For $m \neq \tau_n$, the collections of open sets $\{S_t^n\}_{t \ge 0}$ and $\{S_t^m\}_{t \ge 0}$ are both non-decreasing in t.

**Definition 8.13** (Inter-handover time). Let $m \in \{1, 2\}$ be such that $m \neq \tau_n$. Then, the time of the next handover is defined as

$$T := \inf \left\{ s > 0 : \forall t \in (0, s], \mathcal{H}^n(S_t^n) = 0, \mathcal{H}^m(S_t^m) = 0 \text{ and } \mathcal{H}^n(\overline{S_s^n}) = 2 \text{ or } \mathcal{H}^m(\overline{S_s^m}) = 1 \right\}, \quad (8.37)$$
which coincides with $T_1(0)$, the time point of the first handover happens after time 0.

Here is the rationale for the definition of the inter-handover time $T = T_1(0)$, essentially under the Palm distribution. As a consequence of the increasing property of the collections $\{S_t^n\}_{t \ge 0}$ and $\{S_t^m\}_{t \ge 0}$, presented in Lemma 8.12, we can conclude that $\mathcal{H}^n(S_{T_1(0)}^n) = 0$ implies $\mathcal{H}^n(S_t^n) = 0$, and $\mathcal{H}^m(S_{T_1(0)}^m) = 0$ implies $\mathcal{H}^m(S_t^m) = 0$, for all $t \in [0, T_1(0)]$. On the other hand, by the definition of $T_1(0)$ in (8.37), the condition that $\mathcal{H}^n(\overline{S_{T_1(0)}^n}) = 2$ or $\mathcal{H}^m(\overline{S_{T_1(0)}^m}) = 1$ essentially implies that there exists a head point $(t_u, h_u)$, where $u$ stands for upcoming, of type $\tau_u = \tau_n$ or $m$ such that $C_{(t_n, h_n)}^n$ and $C_{(t_u, h_u)}^m$ intersect at $(T_1(0), \hat{h}_{T_1(0)})$, $\mathcal{H}^n(S_{T_1(0)}^n) = 0$ and $\mathcal{H}^m(S_{T_1(0)}^m) = 0$, so that a handover happens at $T_1(0)$, and in addition to that, it is the first handover after 0. Also, at $t = T_1(0)$, either $\mathcal{H}^n(\overline{S_{T_1(0)}^n}) = 2$ or $\mathcal{H}^m(\overline{S_{T_1(0)}^m}) = 1$, depending on whether $(t_u, h_u)$ is of type $\tau_u = \tau_n$ or type $\tau_u = m$.

The point $(t_u, h_u)$ can again be the previously considered head point $(t_p, h_p)$ in certain situations, where $\tau_n \neq \tau_p$, in which case we have $\tau_u = \tau_p = m$ and the condition $\mathcal{H}^m(\overline{S_{T_1(0)}^m}) = 1$, in the definition of $T_1(0)$, is anyway met. This situation appears when the typical handover is given by the first intersection of two radial birds of different types and the next handover is given by the second intersection of the same pair. We refer the reader to Figure 8.8 and Figure 8.11 in the later part of this article, for better understanding of this scenario. Thus we also have the following corollary of Lemma 8.12, similar to Corollary 5.16, for the two-speed case:

**Corollary 8.14.** For the time $s$ meeting the stopping condition in the definition of $T_1(0)$ in (8.37), there exists an a.s. unique point $(t_u, h_u) \in \mathbb{H}^+$, such that the radial bird with a head at this point, intersects the radial bird $C_{(t_n, h_n)}^n$ and produces the handover at time $T_1(0)$.

**Remark 8.15.** The specific sub-domain of $\mathbb{H}^+$ where to encounter the head point in question above, will be determined in the individual cases of the Palm distribution, depending on the type of the handover. We call these sub-domains as **unexplored regions**.

**8.3.2. Monotonicity properties of the collection** $\{E_{\hat{h}_n(t)}^{t, v_n} \setminus E_{\hat{h}_0}^{0, v_n}\}_{t \ge 0}$. Recall the setting of Subsection 8.3.1, where there exists a handover at time 0 such that a station corresponding to type $\tau_n$ takes over as a serving station after time 0. In the following, we discuss the monotonicity properties of the collections $\{E_{\hat{h}_n(t)}^{t, v_n} \setminus E_{\hat{h}_0}^{0, v_n}\}_{t \ge 0}$ and $\{E_{\hat{h}_n(t)}^{t, v_m} \setminus E_{\hat{h}_0}^{0, v_m}\}_{t \ge 0}$ for $m \neq \tau_n$, under all circumstances.

**Proposition 8.16.** Under the foregoing assumption, we have the following monotonicity properties:

(i). If $\tau_n = 1$, both collections of sets $\{E_{\hat{h}_1(t)}^{t, v_1} \setminus E_{\hat{h}_0}^{0, v_1}\}_{t \ge 0}$ and $\{E_{\hat{h}_1(t)}^{t, v_2} \setminus E_{\hat{h}_0}^{0, v_2}\}_{t \ge 0}$, where $\hat{h}_1(t) := (v_1^2(t - t_n)^2 + h_n^2)^{\frac{1}{2}}$, are non-decreasing in t.
(ii). If $\tau_n = 2$, the collection of sets $\{E_{\hat{h}_2(t)}^{t,v_2} \setminus E_{\hat{h}_0}^{0,v_2}\}_{t \ge 0}$, where $\hat{h}_2(t) := (v_2^2(t-t_n)^2 + h_n^2)^{\frac{1}{2}}$, is non-decreasing, but the collection of sets $\{E_{\hat{h}_2(t)}^{t,v_1} \setminus E_{\hat{h}_0}^{0,v_1}\}_{t \ge 0}$ is not always monotonic in t.

We provide a proof of this monotonicity in Appendix A.3, with the help of the increasing wing property discussed below for the two-speed case, similar to Remark 5.14 for the single speed case.

**Remark 8.17 (Increasing wing property).** Under the Palm distribution of handovers, let $(0, \hat{h}_0) \in C_{(t_n,h_n)}^m$. Define $\hat{h}_n(t) := (v_n^2(t-t_n)^2 + h_n^2)^{\frac{1}{2}}$. In case $\tau_n = 1$, both collections of sets $\{E_{\hat{h}_1(t)}^{t,v_1} \setminus E_{\hat{h}_0}^{0,v_1}\}_{t \ge t_1}$ and $\{E_{\hat{h}_1(t)}^{t,v_2} \setminus E_{\hat{h}_0}^{0,v_2}\}_{t \ge t_1}$, are increasing, since for any $t \ge t_1$ and the point $(t, \hat{h}_1(t))$ lies on the increasing wing of the radial bird $C_{(t_1,h_1)}^1$, see Picture (1) of Figure 8.1. For $\tau_n = 2$, the collection $\{E_{\hat{h}_2(t)}^{t,v_2} \setminus E_{\hat{h}_0}^{0,v_2}\}_{t \ge t_1}$ is increasing, whereas the collection $\{E_{\hat{h}_2(t)}^{t,v_1} \setminus E_{\hat{h}_0}^{0,v_1}\}_{t \ge t_1}$ is non-monotonic, see Picture (2) of Figure 8.1. Indeed, for any $\varepsilon > 0$ arbitrarily small, and $t \ge t_n$, we have $E_{\hat{h}_2(t)}^{t,v_1} \setminus E_{\hat{h}_0}^{0,v_1} \not\subset E_{\hat{h}_2(t+\varepsilon)}^{t+\varepsilon,v_1} \setminus E_{\hat{h}_0}^{0,v_1}$.

<figure><img src="image_37.png" /><figcaption>(1) For $\tau_n = 1$ and $t \ge t_1$, the regions $E_{\hat{h}_1(t)}^{t,v_1} \setminus E_{\hat{h}_0}^{0,v_1}$, for $l = 1, 2$, are increasing. We just draw two sets of half-ellipses at time $t = \hat{s}_1, \hat{s}_2$, where $h_1(\hat{s}_1) = \hat{h}_1$ and $h_1(\hat{s}_2) = \hat{h}_2$.</figcaption></figure>

<figure><img src="image_38.png" /><figcaption>(2) For $\tau_n = 2$ and $t \ge t_2$, the regions $E_{\hat{h}_2(t)}^{t,v_2} \setminus E_{\hat{h}_0}^{0,v_2}$ are increasing, where as $E_{\hat{h}_2(t)}^{t,v_1} \setminus E_{\hat{h}_0}^{0,v_1}$ is non-monotonic. We just draw two sets of half-ellipses at time $t = \hat{s}_1, \hat{s}_2$, where $h_2(\hat{s}_1) = \hat{h}_1$ and $h_2(\hat{s}_2) = \hat{h}_2$.</figcaption></figure>

<caption>**FIGURE 8.1.** For $(t_n, h_n)$ of type $\tau_n = 1$ or 2 and for $t \ge t_n$, the pairs of upper half-ellipses $(E_{\hat{h}_1(t)}^{t,v_1}, E_{\hat{h}_1(t)}^{t,v_2})$ or $(E_{\hat{h}_2(t)}^{t,v_1}, E_{\hat{h}_2(t)}^{t,v_2})$, respectively, benefit from the increasing wing property.</caption>

### 8.3.3. Palm distribution of inter-handover time.
The following theorem determines the Laplace transform of $T$, which is equal to $T_1(0)$, under the Palm probability distribution $P_V^0$ in terms of the head point process $\mathcal{H}$. Using this alternative nomenclature for handover types, for pure handovers of type $i$ or equivalently $[\begin{smallmatrix} 1 \\ i, i \end{smallmatrix}]$, we denote the pure handover point processes and their intensities as $W_i$ and $L_i$, for $i = 1, 2$, respectively. Similarly, we denote the mixed handover point processes and their intensities as $W_{i,j}^{(q)}$ and $L_{i,j}^{(q)}$, with $\tau_p = i$ and $\tau_n = j \neq i$, respectively, for mixed handovers of type $[\begin{smallmatrix} q \\ i, j \end{smallmatrix}]$. We refer to Table 8.1 for the bijection between the two sets of notations. We also write the entire handover point process as

$$ \mathcal{V} \equiv \mathcal{W} := \sum_{i \in \{1,2\}} W_i + \sum_{i \neq j, q \in \{1,2\}} W_{i,j}^{(q)}, $$

and $\lambda_W = \lambda_V$.

**Theorem 8.18.** Let $\rho \ge 0$. In the two-speed case, under the Palm probability measure $P_V^0$ of handovers, the Laplace transform of inter-handover time $T$ is given by,

$$ \mathbb{E}_{\mathcal{W}}^0[e^{-\rho T}] = \sum_{i \in \{1,2\}} \frac{L_i}{\lambda_{\mathcal{W}}} \mathbb{E}_{W_i}^0[e^{-\rho T}] + \sum_{i \neq j, q \in \{1,2\}} \frac{L_{i,j}^{(q)}}{\lambda_{\mathcal{W}}} \mathbb{E}_{W_{i,j}^{(q)}}^0[e^{-\rho T}]. \quad (8.38) $$
Proof of Theorem 8.18. The proof is based on the application of Theorem 8.2 to the decomposition of the Palm probability measure. To get the result, we apply the decomposition (8.4) to the function $f(H) = e^{-\rho T}$, for any $\rho \ge 0$. The terms $\mathbb{E}_{W_i}^0[e^{-\rho T}]$, for $i \in \{1, 2\}$, correspond to the individual Palm probability measures for pure handovers of type $[\begin{smallmatrix} 1 \\ i, i \end{smallmatrix}]$ and $\mathbb{E}_{W_{i,j}}^0[e^{-\rho T}]$, for $i \neq j, q \in \{1, 2\}$, correspond to the individual Palm probability measures for mixed handovers of type $[\begin{smallmatrix} q \\ i, j \end{smallmatrix}]$, which are determined in the following lemmas. $\square$

**Lemma 8.19.** For any $\rho \ge 0$, under the Palm probability measure of pure handovers of type $[\begin{smallmatrix} 1 \\ i, i \end{smallmatrix}]$ with $\tau_p = \tau_n = i$, the Laplace transform of $T$ is

$$ \mathbb{E}_{W_i}^0[e^{-\rho T}] = \frac{4\lambda_i^2 v_i^2}{L_i} \int_{(\mathbb{R}^+)^3} \zeta_{i,i}(h, t', h') dt' dh' dh, \quad (8.39) $$

for $i = 1, 2$, where the function $\zeta_{1,1}(h, t', h')$ is determined in (8.49) and $\zeta_{2,2}(h, t', h')$, in (8.60).

**Lemma 8.20.** For $i \neq j, q \in \{1, 2\}$, under the Palm probability measure of mixed handovers of type $[\begin{smallmatrix} q \\ i, j \end{smallmatrix}]$, the Laplace transform of $T$ is

$$ \mathbb{E}_{W_{i,j}}^0[e^{-\rho T}] = \frac{4\lambda_1\lambda_2 v_1 v_2}{L_{i,j}^{(q)}} \xi_{i,j}^q(\rho, v_i, v_j), \quad (8.40) $$

where the functions $\xi_{i,j}^{(q)}(\rho, v_i, v_j)$ are determined in (8.69), (8.78), (8.86) and (8.97), respectively, for the four different cases for $i \neq j, q \in \{1, 2\}$.

**Remark 8.21.** In the expressions of the mixed Palm probability measures, the functions $\xi_{i,j}^{(q)}(\rho, v_i, v_j)$ for $i \neq j, q \in \{1, 2\}$, are also functions of the underlying parameters $\lambda_1, \lambda_2$. We skip the dependence in $\lambda_1, \lambda_2$ for brevity.

The derivation of $\mathbb{E}_{W_i}^0[e^{-\rho T}]$ with $i \in \{1, 2\}$, $\tau_p = \tau_n = i$, can be found in Subsection 8.3.5 for the case $[\begin{smallmatrix} 1 \\ 1, 1 \end{smallmatrix}]$ and Subsection 8.3.6 for the case $[\begin{smallmatrix} 1 \\ 2, 2 \end{smallmatrix}]$. We derive expressions for $\mathbb{E}_{W_{i,j}}^0[e^{-\rho T}]$ with $i \neq j, q \in \{1, 2\}$ in Subsection 8.3.7 for the case $[\begin{smallmatrix} 1 \\ 2, 1 \end{smallmatrix}]$, Subsection 8.3.8 for $[\begin{smallmatrix} 2 \\ 1, 2 \end{smallmatrix}]$, Subsection 8.3.9 for $[\begin{smallmatrix} 2 \\ 2, 1 \end{smallmatrix}]$, and Subsection 8.3.10 for $[\begin{smallmatrix} 1 \\ 1, 2 \end{smallmatrix}]$, respectively.

### 8.3.4. Handover pair type.
As a general principle, we denote by $[\begin{smallmatrix} q, q' \\ \tau_p, \tau_n, \tau_u \end{smallmatrix}]$ the type of all pairs of consecutive handovers with $[\begin{smallmatrix} q \\ \tau_p, \tau_n \end{smallmatrix}]$ the type of the first handover, $\tau_u$ the type of the radial bird corresponding to the upcoming station and

$$ q' := \begin{cases} 1 & \text{if } t_u \ge t_n, \\ 2 & \text{if } t_u < t_n. \end{cases} \quad (8.41) $$

In this composite notation $[\begin{smallmatrix} q, q' \\ \tau_p, \tau_n, \tau_u \end{smallmatrix}]$ of a pair of consecutive handovers, $[\begin{smallmatrix} q' \\ \tau_n, \tau_u \end{smallmatrix}]$ denotes the type of the next handover. This notation encodes the information about the types and relative positions of three head points involved in two consecutive handovers. Table 8.2 below describes all possible pairs of consecutive handovers, that can be produced from each of the type of handovers. This table also summarizes the strategy to sort out the Palm probability distribution of inter-handover time, depending on the type of initial handover and the type of the radial bird producing the next handover. There can be a scenario where the same station of type $\tau_p$, that was a serving station previously, becomes the upcoming serving station again. We denote the pair type in this situation as $[\begin{smallmatrix} q, q' \\ \tau_p, \tau_n, \tau_p^* \end{smallmatrix}]$, where we use $\cdot$* to denote this reappearance of the same station corresponding to $\tau_p$, as the upcoming serving station.

**Remark 8.22.** Table 8.3, representing the adjacency matrix, summarises the possible transitions among the handover types. We use 1 to denote the possible transitions and 0 to denote the impossible transitions. We use $\cdot$* to denote that the transition can happen in two different ways, including the one with reappearance in our context. One can also obtain the transition diagram based on the adjacency matrix among the handover types from Table 8.3, see Figure 8.2.
<table><caption>TABLE 8.2. Table of pair types $[q, q']_{\tau_p, \tau_n, \tau_u}$, associated to each individual type $[q]_{\tau_p, \tau_n}$. The special case: $[q, q']_{\tau_p, \tau_n, \tau_p^*}$ denotes the reappearance phenomenon where the previous bird coincides with the upcoming bird.</caption><thead><tr><th></th><th>Type</th><th colspan="2">Pair, τ<sub>u</sub> = 1</th><th colspan="2">Pair, τ<sub>u</sub> = 2</th><th></th></tr><tr><th></th><th>-</th><th>-</th><th>-</th><th>-</th><th>Reappearance</th><th></th></tr></thead><tbody><tr><td rowspan="2">Pure<br>handovers<br>[<sup>q</sup>/<sub>τ<sub>p</sub>, τ<sub>n</sub></sub>], τ<sub>p</sub> = τ<sub>n</sub></td><td>[<sup>1</sup>/<sub>1,1</sub>]</td><td>[<sup>1,1</sup>/<sub>1,1,1</sub>]</td><td>-</td><td>[<sup>1,1</sup>/<sub>1,1,2</sub>]</td><td>[<sup>1,2</sup>/<sub>1,1,2</sub>]</td><td>-</td></tr><tr><td>[<sup>1</sup>/<sub>2,2</sub>]</td><td>[<sup>1,1</sup>/<sub>2,2,1</sub>]</td><td>[<sup>1,2</sup>/<sub>2,2,1</sub>]</td><td>[<sup>1,1</sup>/<sub>2,2,2</sub>]</td><td>-</td><td>-</td></tr><tr><td rowspan="4">Mixed<br>handovers<br>[<sup>q</sup>/<sub>τ<sub>p</sub>, τ<sub>n</sub></sub>], τ<sub>p</sub> ≠ τ<sub>n</sub></td><td>[<sup>1</sup>/<sub>2,1</sub>]</td><td>[<sup>1,1</sup>/<sub>2,1,1</sub>]</td><td>-</td><td>[<sup>1,2</sup>/<sub>2,1,2</sub>]</td><td>[<sup>1,1</sup>/<sub>2,1,2</sub>]</td><td>[<sup>1,2</sup>/<sub>2,1,2*</sub>]</td></tr><tr><td>[<sup>2</sup>/<sub>1,2</sub>]</td><td>[<sup>2,1</sup>/<sub>1,2,1</sub>]</td><td>-</td><td>[<sup>2,1</sup>/<sub>1,2,2</sub>]</td><td>-</td><td>-</td></tr><tr><td>[<sup>2</sup>/<sub>2,1</sub>]</td><td>[<sup>2,1</sup>/<sub>2,1,1</sub>]</td><td>-</td><td>[<sup>2,1</sup>/<sub>2,1,2</sub>]</td><td>-</td><td>[<sup>2,1</sup>/<sub>2,1,2*</sub>]</td></tr><tr><td>[<sup>1</sup>/<sub>1,2</sub>]</td><td>[<sup>1,2</sup>/<sub>1,2,1</sub>]</td><td>[<sup>1,1</sup>/<sub>1,2,1</sub>]</td><td>[<sup>1,1</sup>/<sub>1,2,2</sub>]</td><td>-</td><td>-</td></tr></tbody></table>

<table><caption>TABLE 8.3. All possible transitions among the handover types, where the transitions are from each type on the left column to each type on the top row.</caption><thead><tr><th>Types</th><th><sup>1</sup>/<sub>1,1</sub></th><th><sup>1</sup>/<sub>2,2</sub></th><th><sup>1</sup>/<sub>1,2</sub></th><th><sup>1</sup>/<sub>2,1</sub></th><th><sup>2</sup>/<sub>1,2</sub></th><th><sup>2</sup>/<sub>2,1</sub></th></tr></thead><tbody><tr><th><sup>1</sup>/<sub>1,1</sub></th><td>1</td><td>0</td><td>1</td><td>0</td><td>1</td><td>0</td></tr><tr><th><sup>1</sup>/<sub>2,2</sub></th><td>0</td><td>1</td><td>0</td><td>1</td><td>0</td><td>1</td></tr><tr><th><sup>1</sup>/<sub>1,2</sub></th><td>0</td><td>1</td><td>0</td><td>1</td><td>0</td><td>1</td></tr><tr><th><sup>1</sup>/<sub>2,1</sub></th><td>1</td><td>0</td><td>1</td><td>0</td><td>1*</td><td>0</td></tr><tr><th><sup>2</sup>/<sub>1,2</sub></th><td>0</td><td>1</td><td>0</td><td>1</td><td>0</td><td>0</td></tr><tr><th><sup>2</sup>/<sub>2,1</sub></th><td>1</td><td>0</td><td>1*</td><td>0</td><td>0</td><td>0</td></tr></tbody></table>

<figure><img src="image_39.png" /><figcaption>FIGURE 8.2. Transition diagram for all possible transitions among the handover types, where * is used on two of the directed edges, to denote that the transition can happen in two possible ways.</figcaption></figure>
**Proofs of Lemma 8.19 and Lemma 8.20:** In determining the individual Palm expectations in (8.38), through Lemma 8.19 and Lemma 8.20, we frequently use the coordinates of the pure and mixed intersections of the radial birds. Suppose $(t_1, h_1)$, $(t_2, h_2) \in \mathbb{H}^+$, are two head points of same or different types. From now onward the subscript used in the notation of the head point, does not correspond to its type, unlike in Subsection 8.3.1. The intersection point is denoted by $(\hat{s}(t_1, h_1, t_2, h_2), \hat{h}(t_1, h_1, t_2, h_2))$, when the two birds are of the same type. On the other hand, when the two birds are of different types, the intersection points are denoted by $(\hat{s}_1(t_1, h_1, t_2, h_2), \hat{h}_1(t_1, h_1, t_2, h_2))$ and $(\hat{s}_2(t_1, h_1, t_2, h_2), \hat{h}_2(t_1, h_1, t_2, h_2))$. In the following, we may use slightly different notations, whenever necessary in the proofs.

**8.3.5.** *Proof of Lemma 8.19, type (1,1) or equivalently $\begin{bmatrix} 1 \\ 1,1 \end{bmatrix}$.*

*Proof.* Using Lemma 8.3 for the first term in (8.38) with $\tau_n = 1 = \tau_p$ and $q = 1$, we have

$$ \mathbb{E}_{W_1}^0 [e^{-\rho T}] = \frac{4\lambda_1^2 v_1^2}{L_1} \int_{(\mathbb{R}^+)^3} \mathbb{E}_{\mathcal{H}} \left[ e^{-\rho(T \circ \theta_{\hat{s}})} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} (E_{\hat{h}}^{\hat{s},v_l}) = 0 \right] dt' dh' dh, \quad (8.42) $$

where we consider two radial birds of type $\tau_n = 1 = \tau_p$ at $(t_n, h_n) = (0, h)$ and $(t_p, h_p) = (-t', h')$ and where, in short, we wrote $(\hat{s}(0, h, -t', h'), \hat{h}(0, h, -t', h')) = (\hat{s}, \hat{h})$. The station corresponding to the head point $(0, h)$ is the serving station after time $\hat{s}$, since $\tau_n = 1$ and $q = 1$. Since $(0, h)$ is of type 1, we define $\hat{h}_1(t) := (v_1^2 t^2 + h^2)^{\frac{1}{2}}$ and write $\hat{h}_1(t) = \hat{h}_t$, within this proof for brevity. Observe that $T \circ \theta_{\hat{s}} = T_1(\hat{s}) - \hat{s}$, where $T_1(\hat{s})$ is the time until the first handover after time $\hat{s}$. The inner expectation in (8.42) can be evaluated by looking for the next handover given by the intersection of the radial bird with its head at $(0, h)$ and a third radial bird with its head at, say $(T_j, H_j)$, in view of Corollary 8.14, which is either of type 1 or type 2, as depicted in Figure 8.3.

**Case 1.** $(T_j, H_j)$ is of type 1. In this case $(T_j, H_j) \in \mathcal{H}^1$ is outside the region $E_{\hat{h}}^{\hat{s},v_1}$. Necessarily, $T_j \ge 0$. Indeed, if $T_j < 0$, then the intersection of the birds with head at $(0, h)$ and $(T_j, H_j)$, either produces a handover before time $\hat{s}$ or lies in $\mathcal{L}_e^+(v_1, v_2)$, the open region above the joint lower envelope, defined in (7.5). Thus $T_j \ge 0$ and the type of pair of consecutive handovers is $\begin{bmatrix} 1,1 \\ 1,1,1 \end{bmatrix}$. The upcoming head point $(T_j, H_j)$ of type 1 can be found in the unexplored region defined as $Q_0 \setminus E_{\hat{h}}^{\hat{s},v_1}$. Let $(\hat{s}(0, h, T_j, H_j), \hat{h}(0, h, T_j, H_j))$ be the unique intersection between the two birds and, in this case, $T_1(\hat{s}) := \hat{s}(0, h, T_j, H_j)$. For this intersection to give that the next handover takes place immediately after time $\hat{s}$, the region $(\bigcup_{t \in [\hat{s}, \hat{s}(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}}^{\hat{s},v_l})^\circ$, beyond $E_{\hat{h}}^{\hat{s},v_l}$, must have no point of $\mathcal{H}^l$ for $l = 1, 2$. The required region is constructed as a union, using the increasing sets as in Lemma 8.12 and the Definition 8.13, due to the potential lack of monotonicity among the individual sets in the union. This corresponds to the first term in (8.43) and is also depicted in Picture (1) of Figure 8.3.

**Case 2.** $(T_j, H_j)$ is of type 2. In this case $(T_j, H_j) \in \mathcal{H}^2$, is outside $E_{\hat{h}}^{\hat{s},v_2}$ and $T_j$ can be positive or negative. In this case, the unexplored region is $\mathbb{H}^+ \setminus E_{\hat{h}}^{\hat{s},v_2}$ for the upcoming head point of type 2. Then by part (i) of Lemma 7.12, applied to $(s, u) = (\hat{s}, \hat{h})$, it is true that the radial birds with head points $(T_j, H_j)$ and $(0, h)$ intersect each other, even though the point $(T_j, H_j)$ is far on the left or right of $(0, h)$. The type of the pair of handovers is $\begin{bmatrix} 1,2 \\ 1,1,2 \end{bmatrix}$ or $\begin{bmatrix} 1,1 \\ 1,1,2 \end{bmatrix}$, respectively, depending on whether $T_j < 0$ or $T_j \ge 0$. Let $(\hat{s}_1(0, h, T_j, H_j), \hat{h}_1(0, h, T_j, H_j))$ and $(\hat{s}_2(0, h, T_j, H_j), \hat{h}_2(0, h, T_j, H_j))$, be the two intersection points, as illustrated in Picture (2) of Figure 8.3. By part (ii) of Lemma 7.12, we have $\hat{s}_1 \le \hat{s} \le \hat{s}_2$ and, by Observation 7.11, we also have $\hat{s}_1(0, h, T_j, H_j) \le 0 \le \hat{s}_2(0, h, T_j, H_j)$. The first intersection point cannot correspond to a future handover, since $\hat{s}_1(0, h, T_j, H_j) \le \hat{s}$, see Picture (2) of Figure 8.3. The only possibility for a future handover is hence $(\hat{s}_2(0, h, T_j, H_j), \hat{h}_2(0, h, T_j, H_j))$, and in this case, $T_1(\hat{s}) := \hat{s}_2(0, h, T_j, H_j)$. Observe that, using Lemma 8.12, for this intersection to be the next handover, we must have the extra region $(\bigcup_{t \in [\hat{s}, \hat{s}_2(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}}^{\hat{s},v_l})^\circ$ empty of points of $\mathcal{H}^l$, for
l = 1, 2. The union is due to the Definition 8.13, and therein, a potential lack of monotonicity among the individual sets in the union. We have used this in the second and third terms in (8.43).

Based on the last discussions on Case 1 and Case 2, and with the help of Corollary 8.14, we can decompose the inner expectation in (8.42) as

$$
\begin{align}
& \mathbb{E}_{\mathcal{H}} \left[ e^{-\rho(T_1(\hat{s}) - \hat{s})} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} (E_{\hat{h}}^{\hat{s}, v_l}) = 0 \right] \nonumber \\
&= \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\exists(T_j, H_j) \in \mathcal{H}^1 : T_j \ge 0, \cap_{l=1,2} \left\{ \mathcal{H}^l \left( \bigcup_{t \in [\hat{s}, \hat{s}(0, h, T_j, H_j)] E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}}^{\hat{s}, v_l} \right)^{\circ} = 0 \right\}} e^{-\rho(\hat{s}(0, h, T_j, H_j) - \hat{s})} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} (E_{\hat{h}}^{\hat{s}, v_l}) = 0 \right] \nonumber \\
&\quad + \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\exists(T_j, H_j) \in \mathcal{H}^2 : T_j < 0, \cap_{l=1,2} \left\{ \mathcal{H}^l \left( \bigcup_{t \in [\hat{s}, \hat{s}_2(0, h, T_j, H_j)] E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}}^{\hat{s}, v_l} \right)^{\circ} = 0 \right\}} e^{-\rho(\hat{s}_2(0, h, T_j, H_j) - \hat{s})} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} (E_{\hat{h}}^{\hat{s}, v_l}) = 0 \right] \nonumber \\
&\quad + \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\exists(T_j, H_j) \in \mathcal{H}^2 : T_j \ge 0, \cap_{l=1,2} \left\{ \mathcal{H}^l \left( \bigcup_{t \in [\hat{s}, \hat{s}_2(0, h, T_j, H_j)] E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}}^{\hat{s}, v_l} \right)^{\circ} = 0 \right\}} e^{-\rho(\hat{s}_2(0, h, T_j, H_j) - \hat{s})} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} (E_{\hat{h}}^{\hat{s}, v_l}) = 0 \right]. \tag{8.43}
\end{align}
$$

Using the fact that there is a unique point satisfying the condition in each case, the last expression can be written as

$$
\begin{align}
& \mathbb{E}_{\mathcal{H}} \left[ \sum_{(T_j, H_j) \in \mathcal{H}^1 : T_j \ge 0} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( \bigcup_{t \in [\hat{s}, \hat{s}(0, h, T_j, H_j)] E_{\hat{h}_t}^{t, v_l} \cup E_{\hat{h}}^{\hat{s}, v_l} \right) = 0 \right] e^{-\rho(\hat{s}(0, h, T_j, H_j) - \hat{s})} \nonumber \\
&\quad + \mathbb{E}_{\mathcal{H}} \left[ \sum_{(T_j, H_j) \in \mathcal{H}^2 : T_j < 0} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( \bigcup_{t \in [\hat{s}, \hat{s}_2(0, h, T_j, H_j)] E_{\hat{h}_t}^{t, v_l} \cup E_{\hat{h}}^{\hat{s}, v_l} \right) = 0 \right] e^{-\rho(\hat{s}_2(0, h, T_j, H_j) - \hat{s})} \nonumber \\
&\quad + \mathbb{E}_{\mathcal{H}} \left[ \sum_{(T_j, H_j) \in \mathcal{H}^2 : T_j \ge 0} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( \bigcup_{t \in [\hat{s}, \hat{s}_2(0, h, T_j, H_j)] E_{\hat{h}_t}^{t, v_l} \cup E_{\hat{h}}^{\hat{s}, v_l} \right) = 0 \right] e^{-\rho(\hat{s}_2(0, h, T_j, H_j) - \hat{s})} \right] . \tag{8.44}
\end{align}
$$

Using part (i) of Proposition 8.16, the increasing property of the sequence of regions $\{E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}}^{\hat{s}, v_l}\}_{t \ge \hat{s}}$

<figure><img src="image_40.png" /></figure>

(1) For the intersections $(\hat{s}, \hat{h})$ and $(\hat{s}', \hat{h}')$ to represent two consecutive handovers, the regions $E_{\hat{h}}^{\hat{s}, v_1} \cup E_{\hat{h}'}^{\hat{s}', v_1}$ and $E_{\hat{h}}^{\hat{s}, v_2} \cup E_{\hat{h}'}^{\hat{s}', v_2}$ must have no points from $\mathcal{H}^1$ and $\mathcal{H}^2$, respectively. Here $(\hat{s}', \hat{h}')$ is a realization of $(\hat{s}(0, h, T_j, H_j), \hat{h}(0, h, T_j, H_j))$.

<figure><img src="image_41.png" /></figure>

(2) For the intersections $(\hat{s}, \hat{h})$ and $(\hat{s}_2, \hat{h}_2)$ to represent two consecutive handovers, the regions $E_{\hat{h}}^{\hat{s}, v_1} \cup E_{\hat{h}_2}^{\hat{s}', v_1}$ and $E_{\hat{h}}^{\hat{s}, v_2} \cup E_{\hat{h}_2}^{\hat{s}', v_2}$ must have no points from $\mathcal{H}^1$ and $\mathcal{H}^2$, respectively. Here $(\hat{s}_2, \hat{h}_2)$ is a realization of $(\hat{s}_2(0, h, T_j, H_j), \hat{h}_2(0, h, T_j, H_j))$.

FIGURE 8.3. The two cases in (8.44), i.e., the next handover can be given by a bird at $(T_j, H_j) = (t_1, h_1)$ of type 1 or 2, with $(t_1, h_1)$ lying outside $E_{\hat{h}}^{\hat{s}, v_1}$ or $E_{\hat{h}}^{\hat{s}, v_2}$, respectively.
we have, for both $l = 1, 2$

$$ \bigcup_{t \in [\hat{s}, \hat{s}(0,h,T_j,H_j)]} E_{\hat{h}_t}^{t,v_l} \cup E_{\hat{h}}^{\hat{s},v_l} = E_{\hat{h}(0,h,T_j,H_j)}^{\hat{s}(0,h,T_j,H_j),v_l} \cup E_{\hat{h}}^{\hat{s},v_l} \quad \text{and} \quad \bigcup_{t \in [\hat{s}, \hat{s}_2(0,h,T_j,H_j)]} E_{\hat{h}_t}^{t,v_l} \cup E_{\hat{h}}^{\hat{s},v_l} = E_{\hat{h}(0,h,T_j,H_j)}^{\hat{s}(0,h,T_j,H_j),v_l} \cup E_{\hat{h}}^{\hat{s},v_l}, $$

depending on whether $(T_j, H_j)$ is of type 1 or type 2. Hence, (8.44) can be written as

$$ \begin{aligned} & \mathbb{E}_{\mathcal{H}} \left[ \sum_{(T_j, H_j) \in \mathcal{H}^1 : T_j \ge 0} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}(0,h,T_j,H_j)}^{\hat{s}(0,h,T_j,H_j),v_l} \cup E_{\hat{h}}^{\hat{s},v_l} \right) \right] = 0 \\ & + \mathbb{E}_{\mathcal{H}} \left[ \sum_{(T_j, H_j) \in \mathcal{H}^2 : T_j < 0} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}_2(0,h,T_j,H_j)}^{\hat{s}_2(0,h,T_j,H_j),v_l} \cup E_{\hat{h}}^{\hat{s},v_l} \right) \right] = 0 \\ & + \mathbb{E}_{\mathcal{H}} \left[ \sum_{(T_j, H_j) \in \mathcal{H}^2 : T_j \ge 0} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}_2(0,h,T_j,H_j)}^{\hat{s}_2(0,h,T_j,H_j),v_l} \cup E_{\hat{h}}^{\hat{s},v_l} \right) \right] = 0 \\ & := \zeta \begin{bmatrix} 1, 1 \\ 1, 1, 1 \end{bmatrix} + \zeta \begin{bmatrix} 1, 2 \\ 1, 1, 2 \end{bmatrix} + \zeta \begin{bmatrix} 1, 1 \\ 1, 1, 2 \end{bmatrix}. \end{aligned} \quad (8.45) $$

We formally define the notation $\zeta[\tau_p, \tau_n, \tau_u]^{q, q'}$, adopting here a general convention to be used for the rest of the article. Suppose the typical handover is given by the head points $(t_p, h_p), (t_n, h_n)$. Then $\zeta[\tau_p, \tau_n, \tau_u]^{q, q'}$ is a function of $\rho, h_n, |t_p - t_n|, h_p$, i.e., $\zeta[\tau_p, \tau_n, \tau_u]^{q, q'} = \zeta[\tau_p, \tau_n, \tau_u](\rho, h_n, |t_p - t_n|, h_p)$. The function $\zeta[\tau_p, \tau_n, \tau_u]^{q, q'}(\rho, h_n, |t_p - t_n|, h_n)$ is the conditional Laplace transform of the inter-handover time $T$ and the event that the upcoming handover is of type $[\tau_n, \tau_u]^{q'}$, given that the typical handover type is $[\tau_p, \tau_n, \tau_u]^{q, q'}(0, h_n, |t_p - t_n|, h_p)$ is the conditional probability of the event that the head point corresponding to the upcoming station is of type $\tau_u$ and the corresponding head point is on the right or left side of $(t_n, h_n)$ depending on whether $q' = 1$ or 2. In the current proof, since $[\tau_p, \tau_n]^{q} = [1, 1]$, we have $\zeta[1, 1, \tau_u]^{q'}$ as functions of $\rho, h, t', h'$, for different values of $\tau_u$ and $q'$, as appeared in (8.45), for which we evaluate the individual terms.

Applying the Campbell-Mecke formula, the first term in (8.45) equals

$$ \begin{aligned} & \zeta \begin{bmatrix} 1, 1 \\ 1, 1, 1 \end{bmatrix} \\ & = 2\lambda_1 v_1 \int_0^{\hat{s}+\hat{h}/v_1} \int_{(\hat{h}^2-v_1^2(t_1-\hat{s})^2)^{1/2}}^{\infty} \mathbb{E}_{\mathcal{H}} \left[ e^{-\rho(\hat{s}(0,h,t_1,h_1)-\hat{s})} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}(0,h,t_1,h_1)}^{\hat{s}(0,h,t_1,h_1),v_l} \cup E_{\hat{h}}^{\hat{s},v_l} \right) \right] dh_1 dt_1 \\ & + 2\lambda_1 v_1 \int_{\hat{s}+\hat{h}/v_1}^{\infty} \int_0^{\infty} \mathbb{E}_{\mathcal{H}} \left[ e^{-\rho(\hat{s}(0,h,t_1,h_1)-\hat{s})} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}(0,h,t_1,h_1)}^{\hat{s}(0,h,t_1,h_1),v_l} \cup E_{\hat{h}}^{\hat{s},v_l} \right) \right] dh_1 dt_1, \end{aligned} \quad (8.46) $$

since the point $(t_1, h_1)$ must be such that $t_1 \ge 0$ and must lie outside the region $E_{\hat{h}}^{\hat{s},v_1}$. We write in short $(\hat{s}', \hat{h}')$ for $(\hat{s}(0,h,t_1,h_1), \hat{h}(0,h,t_1,h_1))$. In the last expression, we have separated the integral in two parts, $[0, \hat{s} + \hat{h}/v_1] \times \mathbb{R}^+ \setminus E_{\hat{h}}^{\hat{s},v_1}$ and $(\hat{s} + \hat{h}/v_1, \infty) \times \mathbb{R}^+$.

In short, we write $(\hat{s}_2, \hat{h}_2)$ for $(\hat{s}_2(0,h,t_1,h_1), \hat{h}_2(0,h,t_1,h_1))$ and get a representation of the second term in (8.45) as

$$ \begin{aligned} & \zeta \begin{bmatrix} 1, 2 \\ 1, 1, 2 \end{bmatrix} = 2\lambda_2 v_2 \int_{-\infty}^{\hat{s}-\hat{h}/v_2} \int_0^{\infty} \mathbb{E}_{\mathcal{H}} \left[ e^{-\rho(\hat{s}_2-\hat{s})} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}_2}^{\hat{s}_2,v_l} \cup E_{\hat{h}}^{\hat{s},v_l} \right) \right] dh_1 dt_1 \\ & + 2\lambda_2 v_2 \int_{\hat{s}-\hat{h}/v_2}^{0} \int_{(\hat{h}^2-v_2^2(t_1-\hat{s})^2)^{1/2}}^{\infty} \mathbb{E}_{\mathcal{H}} \left[ e^{-\rho(\hat{s}_2-\hat{s})} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}_2}^{\hat{s}_2,v_l} \cup E_{\hat{h}}^{\hat{s},v_l} \right) \right] dh_1 dt_1, \end{aligned} \quad (8.47) $$
where the point $(t_1, h_1)$ with $t_1 < 0$ and of type 2, must lie outside the region $E_{\hat{h}}^{\hat{s},v_2}$. So we have separated the integral in two parts, $(-\infty, \hat{s} - \hat{h}/v_2] \times \mathbb{R}^+$ and $[\hat{s} - \hat{h}/v_2, 0] \times \mathbb{R}^+ \setminus E_{\hat{h}}^{\hat{s},v_2}$.

The third term in (8.45) equals

$$ \zeta\left[ \begin{smallmatrix} 1, & 1 \\ 1, & 1, & 2 \end{smallmatrix} \right] = 2\lambda_2 v_2 \int_0^{\hat{s}+\hat{h}/v_2} \int_{(\hat{h}^2-v_2^2(t_1-\hat{s})^2)^{1/2}}^{\infty} \mathbb{E}_{\mathcal{H}} \left[ e^{-\rho(\hat{s}_2-\hat{s})} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}_2}^{\hat{s}_2,v_l \cup E_{\hat{h}}^{\hat{s},v_l}} \right) \right] \mathrm{d}h_1 \mathrm{d}t_1 \\ + 2\lambda_2 v_2 \int_{\hat{s}+\hat{h}/v_2}^{\infty} \int_0^{\infty} \mathbb{E}_{\mathcal{H}} \left[ e^{-\rho(\hat{s}_2-\hat{s})} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}_2}^{\hat{s}_2,v_l \cup E_{\hat{h}}^{\hat{s},v_l}} \right) \right] \mathrm{d}h_1 \mathrm{d}t_1, \quad (8.48) $$

since the point $(t_1, h_1)$ with $t_1 \ge 0$ and of type 2, must lie outside the region $E_{\hat{h}}^{\hat{s},v_2}$. So we have separated the integral in two parts, $[0, \hat{s} + \hat{h}/v_2] \times \mathbb{R}^+ \setminus E_{\hat{h}}^{\hat{s},v_2}$ and $(\hat{s} + \hat{h}/v_2, \infty) \times \mathbb{R}^+$.

Computing the void probabilities in the terms from (8.46)-(8.48) and substituting the result to (8.45), we obtain the inner expectation in (8.42) as a sum

$$ \begin{aligned} & \zeta\left[ \begin{smallmatrix} 1, & 1 \\ 1, & 1, & 2 \end{smallmatrix} \right] + \zeta\left[ \begin{smallmatrix} 1, & 2 \\ 1, & 1, & 2 \end{smallmatrix} \right] + \zeta\left[ \begin{smallmatrix} 1, & 1 \\ 1, & 1, & 2 \end{smallmatrix} \right] \\ &= 2\lambda_1 v_1 \int_0^{\hat{s}+\hat{h}/v_1} \int_{(\hat{h}^2-v_1^2(t_1-\hat{s})^2)^{1/2}}^{\infty} e^{-\rho(\hat{s}'-\hat{s})} e^{-\sum_{l=1,2} 2\lambda_l v_l} |E_{\hat{h}'}^{\hat{s}',v_l \cup E_{\hat{h}}^{\hat{s},v_l}}| \mathrm{d}h_1 \mathrm{d}t_1 \\ &\quad + 2\lambda_1 v_1 \int_{\hat{s}+\hat{h}/v_1}^{\infty} \int_0^{\infty} e^{-\rho(\hat{s}'-\hat{s})} e^{-\sum_{l=1,2} 2\lambda_l v_l} |E_{\hat{h}'}^{\hat{s}',v_l \cup E_{\hat{h}}^{\hat{s},v_l}}| \mathrm{d}h_1 \mathrm{d}t_1 \\ &\quad + 2\lambda_2 v_2 \int_{-\infty}^{\hat{s}-\hat{h}/v_2} \int_0^{\infty} e^{-\rho(\hat{s}_2-\hat{s})} e^{-\sum_{l=1,2} 2\lambda_l v_l} |E_{\hat{h}_2}^{\hat{s}_2,v_l \cup E_{\hat{h}}^{\hat{s},v_l}}| \mathrm{d}h_1 \mathrm{d}t_1 \\ &\quad + 2\lambda_2 v_2 \int_{\hat{s}-\hat{h}/v_2}^{\hat{s}+\hat{h}/v_2} \int_{(\hat{h}^2-v_2^2(t_1-\hat{s})^2)^{1/2}}^{\infty} e^{-\rho(\hat{s}_2-\hat{s})} e^{-\sum_{l=1,2} 2\lambda_l v_l} |E_{\hat{h}_2}^{\hat{s}_2,v_l \cup E_{\hat{h}}^{\hat{s},v_l}}| \mathrm{d}h_1 \mathrm{d}t_1 \\ &\quad + 2\lambda_2 v_2 \int_{\hat{s}+\hat{h}/v_2}^{\infty} \int_0^{\infty} e^{-\rho(\hat{s}_2-\hat{s})} e^{-\sum_{l=1,2} 2\lambda_l v_l} |E_{\hat{h}_2}^{\hat{s}_2,v_l \cup E_{\hat{h}}^{\hat{s},v_l}}| \mathrm{d}h_1 \mathrm{d}t_1 \\ &:= \zeta_{1,1}(h, t', h'). \end{aligned} \quad (8.49) $$

Using the expression from (8.49) in (8.42) yields

$$ \mathbb{E}_{W_1}^0 [e^{-\rho T}] = \frac{4\lambda_1^2 v_1^2}{L_1} \int_{(\mathbb{R}^+)^3} \zeta_{1,1}(h, t', h') dt' dh' dh. \quad (8.50) $$

This completes the proof for the case $\tau_n = 1 = \tau_p$. $\square$

### 8.3.6. Proof of Lemma 8.19, type $(\frac{1}{2,2})$ or equivalently $[\frac{1}{2,2}]$.

*Proof.* Using Lemma 8.3 for the second term in (8.38), with $\tau_n = 2 = \tau_p$ and $q = 1$, we have

$$ \mathbb{E}_{W_2}^0 [e^{-\rho T}] = \frac{4\lambda_2^2 v_2^2}{L_2} \int_{(\mathbb{R}^+)^3} \mathbb{E}_{\mathcal{H}} \left[ e^{-\rho(T_1 \circ \theta_{\hat{s}})} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} (E_{\hat{h}_l}^{\hat{s},v_l}) \right] \mathrm{d}t' \mathrm{d}h' \mathrm{d}h, \quad (8.51) $$

where we consider two radial birds of type $\tau_n = 2 = \tau_p$ at $(t_n, h_n) = (0, h)$ and $(t_p, h_p) = (-t', h)$ and where, in short, we wrote $(\hat{s}(0, h, -t', h'), \hat{h}(0, h, -t', h')) = (\hat{s}, \hat{h})$. Note that, since $\tau_n = 2$ and $q = 1$, the station corresponding to the head point $(0, h)$ is the serving station after time $\hat{s}$. Since $(0, h)$ is of type 2, we define $\hat{h}_2(t) := (v_2^2 t^2 + h^2)^{1/2}$, which we write in short as $\hat{h}_t$ within this proof. Notice that

$$ T \circ \theta_{\hat{s}} = T_1(\hat{s}) - \hat{s}, $$
where $T_1(\hat{s})$ is the time to the handover after time $\hat{s}$. By Corollary 8.14, the inner expectation in (8.51) can be determined by exploring the next handover as an intersection of the radial bird at $(0, h)$ and a third radial bird with its head at, say $(T_j, H_j)$, that lies outside of $E_{\hat{h}}^{\hat{s},v_1}$ or $E_{\hat{h}}^{\hat{s},v_2}$, depending on its type. We have the following cases:

**Case 1.** ($T_j, H_j$) is of type 1. In this case, we must have $(T_j, H_j) \notin E_{\hat{h}}^{\hat{s},v_1}$. Along with this, we must assume that $T_j \ge \hat{s}$. In case $T_j < \hat{s}$, the intersection created by the bird with head at $(T_j, H_j)$ either contributes to a handover that happens before time $\hat{s}$, or is such that the intersection lies in $\mathcal{L}_e^+(v_1, v_2)$, see Figure 8.4. In this case $T_j \ge \hat{s}$ and the unexplored region is $Q_{\hat{s}} \setminus E_{\hat{h}}^{\hat{s},v_1}$. There are two intersection points $(\hat{s}_1(0, h, T_j, H_j), \hat{h}_1(0, h, T_j, H_j))$ and $(\hat{s}_2(0, h, T_j, H_j), \hat{h}_2(0, h, T_j, H_j))$. For any of these points to represent a handover we need the extra regions $(\bigcup_{t \in [\hat{s}, \hat{s}_1(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}}^{\hat{s}, v_l})^\circ$ and $(\bigcup_{t \in [\hat{s}, \hat{s}_2(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}}^{\hat{s}, v_l})^\circ$ to be empty of points from $\mathcal{H}^l$, for $l = 1, 2$, where $\hat{h}_t = (v_2^2 t^2 + h^2)^{\frac{1}{2}}$. The reason for the union is the same as above. Since $\hat{s}_1(0, h, T_j, H_j) \le \hat{s}_2(0, h, T_j, H_j)$, we have the inclusion

$$ \bigcup_{t \in [\hat{s}, \hat{s}_1(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}}^{\hat{s}, v_l} \subset \bigcup_{t \in [\hat{s}, \hat{s}_2(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}}^{\hat{s}, v_l}. $$

Hence, it cannot happen that $\hat{s}_2(0, h, T_j, H_j)$ corresponds to a handover, while $\hat{s}_1(0, h, T_j, H_j)$ does not. We must take the first intersection point as a handover, and thus $T_1(\hat{s}) := \hat{s}_1(0, h, T_j, H_j)$.

Additionally, the point $(T_j, H_j)$ must be chosen in such a way that $T_j \in D_\ell(0, h, H_j)$ or $T_j \in D_r(0, h, H_j)$, depending on whether $T_j < 0$ or $T_j \ge 0$, respectively, given $h, H_j$, see Figure 8.5. The type of the pair of consecutive handovers is $[\begin{smallmatrix} 1, 2 \\ 2, 2, 1 \end{smallmatrix}]$ or $[\begin{smallmatrix} 1, 1 \\ 2, 2, 1 \end{smallmatrix}]$, depending on whether $T_j < 0$ or $T_j \ge 0$, respectively.

<figure><img src="image_42.png" /></figure>

(1) In this case, the intersection of $(0, h)$ and $(t_1, h_1)$, lies in $\mathcal{L}_e^+(v_1, v_2)$.

<figure><img src="image_43.png" /></figure>

(2) In this case, the intersection of $(0, h)$ and $(t_1, h_1)$ gives a handover before time $\hat{s}$.

<figure><img src="image_44.png" /><figcaption>FIGURE 8.4. The next head point $(T_j, H_j) = (t_1, h_1) \notin E_{\hat{h}}^{\hat{s},v_1}$, but it cannot be such that $t_1 < \hat{s}$.</figcaption></figure>

(1) For the intersection $(\hat{s}, \hat{h})$ and $(\hat{s}', \hat{h}')$ to represent two consecutive handovers, the regions $S_1 \cup E_{\hat{h}}^{\hat{s},v_1} \cup E_{\hat{h}'}^{\hat{s}',v_1}$ and $E_{\hat{h}}^{\hat{s},v_2} \cup E_{\hat{h}'}^{\hat{s}',v_2}$ must have no points from $\mathcal{H}^1$ and $\mathcal{H}^2$, respectively. Here $(\hat{s}', \hat{h}')$ is a realization of $(\hat{s}_1(0, h, T_j, H_j), \hat{h}_1(0, h, T_j, H_j))$.

<figure><img src="image_45.png" /></figure>

(2) For the intersections $(\hat{s}, \hat{h})$ and $(\hat{s}', \hat{h}')$ to represent two consecutive handovers, the regions $S_1 \cup E_{\hat{h}}^{\hat{s},v_1} \cup E_{\hat{h}'}^{\hat{s}',v_1}$ and $E_{\hat{h}}^{\hat{s},v_2} \cup E_{\hat{h}'}^{\hat{s}',v_2}$ must have no points from $\mathcal{H}^1$ and $\mathcal{H}^2$, respectively. Here $(\hat{s}', \hat{h}')$ is a realization of $(\hat{s}_1(0, h, T_j, H_j), \hat{h}_1(0, h, T_j, H_j))$.

<caption>FIGURE 8.5. The case where $(T_j, H_j) = (t_1, h_1)$ is of type 1. The two cases considered are, $t_1 < 0$ and $t_1 \ge 0$, as in the first two terms in (8.52).</caption>
**Case 2.** ($T_j, H_j$) is of type 2. The intersection of the radial birds of type 2 at $(0, h)$ and $(T_j, H_j) \notin E_{\hat{h}}^{\hat{s},v_2}$, gives rise to a handover if $T_j \ge 0$, see Figure 8.6. Otherwise, if $T_j < 0$, then the intersection given by the birds at $(T_j, H_j)$ and $(0, h)$ is either contained in $\mathcal{L}_e^+(v_1, v_2)$ or it gives rise to a handover before time $\hat{s}$. So we must have $T_j \ge 0$ and the unexplored region is $Q_0 \setminus E_{\hat{h}}^{\hat{s},v_2}$. The type of the pair of handovers is $[\begin{smallmatrix} 1, & 1 \\ 2, & 2 \\ 2, & 2 \end{smallmatrix}]$. Let $(\hat{s}(0, h, T_j, H_j), \hat{h}(0, h, T_j, H_j))$ be their intersection point. Here we can consider that $T_1(\hat{s}) := \hat{s}(0, h, T_j, H_j)$. For this intersection to give that the next handover takes place after time $\hat{s}$, the extra region $(\bigcup_{t \in [\hat{s}, \hat{s}(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}}^{\hat{s}, v_l})^\circ$, beyond $E_{\hat{h}}^{\hat{s}, v_l}$, must have no point of $\mathcal{H}^l$ for $l = 1, 2$, where $\hat{h}_t = (v_2^2 t^2 + h^2)^{\frac{1}{2}}$.

Based on the last discussions with Case 1 and Case 2, we can decompose the inner expectation in (8.51) as

$$
\begin{align}
& \mathbb{E}_{\mathcal{H}} \left[ e^{-\rho(T_1(\hat{s})-\hat{s})} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l}(E_{\hat{h}}^{\hat{s},v_l})=0 \right] \nonumber \\
& = \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\exists(T_j,H_j) \in \mathcal{H}^1: \hat{s} \le T_j \le 0, T_j \in D_\ell(0,h,H_j)} \cap_{l=1,2} \left\{ \mathcal{H}^l \left( \bigcup_{t \in [\hat{s}, \hat{s}_1(0,h,T_j,H_j)]} E_{\hat{h}_t}^{t,v_l} \setminus E_{\hat{h}}^{\hat{s},v_l} \right)^\circ =0 \right\} \right. \right. \nonumber \\
& \qquad \left. \left. \times e^{-\rho(\hat{s}_1(0,h,T_j,H_j)-\hat{s})} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l}(E_{\hat{h}}^{\hat{s},v_l})=0 \right] \right. \nonumber \\
& + \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\exists(T_j,H_j) \in \mathcal{H}^1: T_j \in D_r(0,h,H_j), \cap_{l=1,2} \left\{ \mathcal{H}^l \left( \bigcup_{t \in [\hat{s}, \hat{s}_1(0,h,T_j,H_j)]} E_{\hat{h}_t}^{t,v_l} \setminus E_{\hat{h}}^{\hat{s},v_l} \right)^\circ =0 \right\} e^{-\rho(\hat{s}_1(0,h,T_j,H_j)-\hat{s})} \right. \right. \nonumber \\
& \qquad \left. \left. \times \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l}(E_{\hat{h}}^{\hat{s},v_l})=0 \right] \right. \nonumber \\
& + \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\exists(T_j,H_j) \in \mathcal{H}^2: T_j \ge 0, \cap_{l=1,2} \left\{ \mathcal{H}^l \left( \bigcup_{t \in [\hat{s}, \hat{s}(0,h,T_j,H_j)]} E_{\hat{h}_t}^{t,v_l} \setminus E_{\hat{h}}^{\hat{s},v_l} \right)^\circ =0 \right\} e^{-\rho(\hat{s}(0,h,T_j,H_j)-\hat{s})} \right. \right. \nonumber \\
& \qquad \left. \left. \times \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l}(E_{\hat{h}}^{\hat{s},v_l})=0 \right] . \tag{8.52}
\end{align}
$$

Similarly to the final step in (8.44), there exists a unique head point satisfying the condition for

<figure><img src="image_46.png" /><figcaption>FIGURE 8.6. The case where $(t_1, h_1)$ is of type 2, with $t_1 \ge 0$, as in the last term in (8.52). For the intersections $(\hat{s}, \hat{h})$ and $(\hat{s}', \hat{h}')$ to represent two consecutive handovers, the regions $S_2 \cup E_{\hat{h}}^{\hat{s},v_1} \cup E_{\hat{h}'}^{\hat{s}',v_1}$ and $E_{\hat{h}}^{\hat{s},v_2} \cup E_{\hat{h}'}^{\hat{s}',v_2}$ must have no points from $\mathcal{H}^1$ and $\mathcal{H}^2$, respectively. Here $(\hat{s}', \hat{h}')$ is a realization of $(\hat{s}(0, h, T_j, H_j), \hat{h}(0, h, T_j, H_j))$.</figcaption></figure>
individual term in (8.52), the last expression equals

$$
\mathbb{E}_{\mathcal{H}} \left[ \sum_{(T_j, H_j) \in \mathcal{H}^1 : \hat{s} \le T_j \le 0, T_j \in D_\ell(0, h, H_j)} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( \bigcup_{t \in [\hat{s}, \hat{s}_1(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \cup E_{\hat{h}}^{\hat{s}, v_l} \right) \right] = 0 \\
+ \mathbb{E}_{\mathcal{H}} \left[ \sum_{(T_j, H_j) \in \mathcal{H}^1 : T_j \in D_r(0, h, H_j)} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( \bigcup_{t \in [\hat{s}, \hat{s}_1(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \cup E_{\hat{h}}^{\hat{s}, v_l} \right) \right] = 0 \\
+ \mathbb{E}_{\mathcal{H}} \left[ \sum_{(T_j, H_j) \in \mathcal{H}^2 : T_j \ge 0} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( \bigcup_{t \in [\hat{s}, \hat{s}(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \cup E_{\hat{h}}^{\hat{s}, v_l} \right) \right] = 0
\quad (8.53)
$$

We write the sum of three terms in (8.53) as $\zeta[\begin{smallmatrix} 1,2 \\ 2,2,1 \end{smallmatrix}] + \zeta[\begin{smallmatrix} 1,1 \\ 2,2,1 \end{smallmatrix}] + \zeta[\begin{smallmatrix} 1,1 \\ 2,2,2 \end{smallmatrix}]$. Using Remark 7.10 for $l=1$, the region in the first two terms in (8.53) satisfies

$$
\bigcup_{t \in [\hat{s}, \hat{s}_1(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_1} \cup E_{\hat{h}}^{\hat{s}, v_1} = E_{\hat{h}}^{\hat{s}, v_1} \cup S_1 \cup E_{\hat{h}_1(0, h, T_j, H_j)}^{\hat{s}_1(0, h, T_j, H_j), v_1}.
$$

In the last union, $S_1$ is the region outside $E_{\hat{h}}^{\hat{s},v_1} \cup E_{\hat{h}_1(0,h,T_j,H_j)}^{\hat{s}_1(0,h,T_j,H_j),v_1}$, but below the hyperbola with equation
$u^2 - \frac{v_1^2 v_2^2}{v_1^2 - v_2^2} t^2 = h^2$, in the $(t, u)$-coordinate system similar to (7.17), and in the time interval

$$
\left[ \left(1 - \frac{v_2^2}{v_1^2}\right) \hat{s}, \left(1 - \frac{v_2^2}{v_1^2}\right) \hat{s}_1(0, h, T_j, H_j) \right].
$$

For $l=2$, using Part (ii) of Proposition 8.16, the increasing property of $\{E_{\hat{h}_t}^{t,v_2} \setminus E_{\hat{h}}^{\hat{s},v_2}\}_{t \ge \hat{s}}$ in the first and second term in (8.53), we get that

$$
\bigcup_{t \in [\hat{s}, \hat{s}_1(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_2} \cup E_{\hat{h}}^{\hat{s}, v_2} = E_{\hat{h}_1(0, h, T_j, H_j)}^{\hat{s}_1(0, h, T_j, H_j), v_2} \cup E_{\hat{h}}^{\hat{s}, v_2}.
$$

The next head point $(T_j, H_j)$ must be in the region $R_1(h, \hat{s}, \hat{h})$, where

$$
R_1(h, \hat{s}, \hat{h}) := \left\{ (t, u) : u \ge 0, t \ge \hat{s}, v_1^2(t - \hat{s})^2 + u^2 \ge \hat{h}^2 \text{ and } u^2 - \frac{v_1^2 v_2^2}{v_1^2 - v_2^2} t^2 \le h^2 \right\}, \quad (8.54)
$$

as indicated in the first picture of Figure 8.5. In (8.54), the first inequality $t \ge \hat{s}$ is due to the discussion
in Case 1 of this proof. The second inequality $v_1^2(t - \hat{s})^2 + u^2 \ge \hat{h}^2$, signifies that the next head point
must be outside $E_{\hat{h}}^{\hat{s}, v_l}$. The last inequality $u^2 - \frac{v_1^2 v_2^2}{v_1^2 - v_2^2} t^2 \le h^2$, signifies that the next head point must be
below the hyperbola (7.17) corresponding to the head point $(0, h)$, which is more geometric version of
the intersection criteria. Then, implementing the last discussions, the first two terms in (8.53) together
are equal to

$$
\mathbb{E}_{\mathcal{H}} \left[ \sum_{(T_j, H_j) \in \mathcal{H}^1 : \hat{s} \le T_j \le 0, T_j \in D_\ell(0, h, H_j)} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}}^{\hat{s}, v_1} \cup S_1 \cup E_{\hat{h}_1(0, h, T_j, H_j)}^{\hat{s}_1(0, h, T_j, H_j), v_1} \right) \right] = 0 \\
\times e^{-\rho(\hat{s}_1(0, h, T_j, H_j) - \hat{s})} \\
+ \mathbb{E}_{\mathcal{H}} \left[ \sum_{(T_j, H_j) \in \mathcal{H}^1 : T_j \in D_r(0, h, H_j)} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}}^{\hat{s}, v_1} \cup S_1 \cup E_{\hat{h}_1(0, h, T_j, H_j)}^{\hat{s}_1(0, h, T_j, H_j), v_1} \right) \right] = 0 \\
\times \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}}^{\hat{s}, v_2} \cup E_{\hat{h}_1(0, h, T_j, H_j)}^{\hat{s}_1(0, h, T_j, H_j), v_2} \right) = 0 \\
\times e^{-\rho(\hat{s}_1(0, h, T_j, H_j) - \hat{s})} \right]. \quad (8.55)
$$
Applying the Campbell-Mecke formula in (8.55) together and using the region $R_1(h, \hat{s}, \hat{h})$ defined in (8.54), we get

$$ \zeta\begin{bmatrix} 1, & 2 \\ 2, & 2, 1 \end{bmatrix} + \zeta\begin{bmatrix} 1, & 1 \\ 2, & 2, 1 \end{bmatrix} = 2\lambda_1 v_1 \int_{R_1(h, \hat{s}, \hat{h})} e^{-\rho(\hat{s}_1 - \hat{s})} \mathbb{E}_{\mathcal{H}}\left[ \mathbb{1}_{\mathcal{H}^l}\left(E_{\hat{h}}^{\hat{s}, v_1} \cup S_1 \cup E_{\hat{h}_1}^{\hat{s}_1, v_1}\right) = 0 \mathbb{1}_{\mathcal{H}^l}\left(E_{\hat{h}}^{\hat{s}, v_2} \cup E_{\hat{h}_1}^{\hat{s}_1, v_2}\right) = 0 \right] dt_1 dh_1, \quad (8.56) $$

where in short, we wrote $(\hat{s}_1, \hat{h}_1)$ for $(\hat{s}_1(0, h, t_1, h_1), \hat{h}_1(0, h, t_1, h_1))$. Observe that the curves $v_1^2(t-\hat{s})^2 + u^2 = \hat{h}^2$ and $u^2 - \frac{v_1^2 v_2^2}{v_1^2 - v_2^2} t^2 = h^2$ intersect exactly at one point, whose abscissa is $\frac{v_1^2 - v_2^2}{v_1^2} \hat{s}$. So we divide the domain of integration in two parts and use the void probabilities to get that the last expression is equal to

$$ \begin{aligned} & 2\lambda_1 v_1 \int_{\frac{v_1^2 - v_2^2}{v_1^2} \hat{s}} \int_{(\hat{h}^2 - v_1^2(t_1 - \hat{s})^2)^{1/2}}^{\left(\frac{v_1^2 v_2^2}{v_1^2 - v_2^2} t_1^2 + h^2\right)^{1/2}} e^{-\rho(\hat{s}_1 - \hat{s})} e^{-2\lambda_1 v_1} \left| E_{\hat{h}}^{\hat{s}, v_1} \cup S_1 \cup E_{\hat{h}_1}^{\hat{s}_1, v_1} \right| e^{-2\lambda_2 v_2} \left| E_{\hat{h}}^{\hat{s}, v_2} \cup E_{\hat{h}_1}^{\hat{s}_1, v_2} \right| dh_1 dt_1 \\ & + 2\lambda_1 v_1 \int_{\hat{s} + \hat{h}/v_1}^{\infty} \int_{0}^{\left(\frac{v_1^2 v_2^2}{v_1^2 - v_2^2} t_1^2 + h^2\right)^{1/2}} e^{-\rho(\hat{s}_1 - \hat{s})} e^{-2\lambda_1 v_1} \left| E_{\hat{h}}^{\hat{s}, v_1} \cup S_1 \cup E_{\hat{h}_1}^{\hat{s}_1, v_1} \right| e^{-2\lambda_2 v_2} \left| E_{\hat{h}}^{\hat{s}, v_2} \cup E_{\hat{h}_1}^{\hat{s}_1, v_2} \right| dh_1 dt_1. \end{aligned} \quad (8.57) $$

For the third term in (8.53), using Remark 7.10 for $l=1$,

$$ \bigcup_{t \in [\hat{s}, \hat{s}(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_1} \cup E_{\hat{h}}^{\hat{s}, v_1} = E_{\hat{h}}^{\hat{s}, v_1} \cup S_2 \cup E_{\hat{h}(0, h, T_j, H_j)}^{\hat{s}(0, h, T_j, H_j), v_1}, $$

where $S_2$ is the region outside $E_{\hat{h}}^{\hat{s}, v_1} \cup E_{\hat{h}(0, h, T_j, H_j)}^{\hat{s}(0, h, T_j, H_j), v_1}$, but below the hyperbola given by the equation $u^2 - \frac{v_1^2 v_2^2}{v_1^2 - v_2^2} t^2 = h^2$ in the $(t, u)$-coordinate system similar to (7.17) and in the time interval

$$ \left[ \left(1 - \frac{v_2^2}{v_1^2}\right) \hat{s}, \left(1 - \frac{v_2^2}{v_1^2}\right) \hat{s}(0, h, T_j, H_j) \right]. $$

For $l=2$, using Part (i) of Proposition 8.16, the increasing property of $\{E_{\hat{h}_t}^{t, v_2} \setminus E_{\hat{h}}^{\hat{s}, v_2}\}_{t \ge \hat{s}}$, we have

$$ \bigcup_{t \in [\hat{s}, \hat{s}(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_2} \cup E_{\hat{h}}^{\hat{s}, v_2} = E_{\hat{h}(0, h, T_j, H_j)}^{\hat{s}(0, h, T_j, H_j), v_2} \cup E_{\hat{h}}^{\hat{s}, v_2}. $$

The next responsible head point $(T_j, H_j)$ must belong to the region $R_2(h, \hat{s}, \hat{h})$, where

$$ R_2(h, \hat{s}, \hat{h}) := \{(t, u) : v_2^2(t - \hat{s})^2 + u^2 \ge \hat{h}^2, \ u \ge 0 \text{ and } t \ge 0\}. \quad (8.58) $$

The region $R_2(h, \hat{s}, \hat{h})$ in (8.58) is defined in such a way that, the next head point must lie outside $E_{\hat{h}}^{\hat{s}, v_2}$ and on the right of $(0, h)$ as discussed in Case 2 above. Applying the Campbell-Mecke formula, the third term in (8.53), after invoking the last discussions, equals,

$$ \begin{aligned} \zeta\begin{bmatrix} 1, & 1 \\ 2, & 2, 2 \end{bmatrix} & = \mathbb{E}_{\mathcal{H}}\left[ \sum_{(T_j, H_j) \in \mathcal{H}^2 : T_j \ge 0} \mathbb{1}_{\mathcal{H}^l}\left(E_{\hat{h}}^{\hat{s}(0, h, T_j, H_j), v_1} \cup S_2 \cup E_{\hat{h}}^{\hat{s}, v_1}\right) = 0 \mathbb{1}_{\mathcal{H}^l}\left(E_{\hat{h}}^{\hat{s}(0, h, T_j, H_j), v_2} \cup E_{\hat{h}}^{\hat{s}, v_2}\right) = 0 \right] e^{-\rho(\hat{s}(0, h, T_j, H_j) - \hat{s})} \\ & = 2\lambda_2 v_2 \int_{R_2(h, \hat{s}, \hat{h})} e^{-\rho(\hat{s}' - \hat{s})} \mathbb{E}_{\mathcal{H}}\left[ \mathbb{1}_{\mathcal{H}^1}\left(E_{\hat{h}'}^{\hat{s}', v_1} \cup S_2 \cup E_{\hat{h}}^{\hat{s}, v_1}\right) = 0 \mathbb{1}_{\mathcal{H}^2}\left(E_{\hat{h}'}^{\hat{s}', v_2} \cup E_{\hat{h}}^{\hat{s}, v_2}\right) = 0 \right] dt_1 dh_1 \\ & = 2\lambda_2 v_2 \int_{0}^{\infty} \int_{(\hat{h}^2 - v_2^2(t_1 - \hat{s})^2)^{1/2}} e^{-\rho(\hat{s}' - \hat{s})} e^{-2\lambda_1 v_1} \left| E_{\hat{h}'}^{\hat{s}', v_1} \cup S_2 \cup E_{\hat{h}}^{\hat{s}, v_1} \right| e^{-2\lambda_2 v_2} \left| E_{\hat{h}'}^{\hat{s}', v_2} \cup E_{\hat{h}}^{\hat{s}, v_2} \right| dh_1 dt_1 \\ & + 2\lambda_2 v_2 \int_{\hat{s} + \hat{h}/v_2}^{\infty} \int_{0}^{\infty} e^{-\rho(\hat{s}' - \hat{s})} e^{-2\lambda_1 v_1} \left| E_{\hat{h}'}^{\hat{s}', v_1} \cup S_2 \cup E_{\hat{h}}^{\hat{s}, v_1} \right| e^{-2\lambda_2 v_2} \left| E_{\hat{h}'}^{\hat{s}', v_2} \cup E_{\hat{h}}^{\hat{s}, v_2} \right| dh_1 dt_1. \end{aligned} \quad (8.59) $$
in short, we wrote $(\hat{s}', \hat{h}')$ for $(\hat{s}(0, h, t_1, h_1), \hat{h}(0, h, t_1, h_1))$. We denote the sum of all the terms in (8.57) and (8.59) as

$$ \zeta_{2,2}(h, t', h') := \zeta\left[\begin{smallmatrix} 1, & 2 \\ 2, & 2, 1 \end{smallmatrix}\right] + \zeta\left[\begin{smallmatrix} 1, & 1 \\ 2, & 2, 1 \end{smallmatrix}\right] + \zeta\left[\begin{smallmatrix} 1, & 1 \\ 2, & 2, 2 \end{smallmatrix}\right]. \quad (8.60) $$

Finally, we obtain the result by substituting the expression of $\zeta_{2,2}(h, t', h')$ to (8.51). $\square$

### 8.3.7. Proof of Lemma 8.20, type $(\begin{smallmatrix} 1 \\ 2, 1 \end{smallmatrix})$ or equivalently $\begin{bmatrix} 1 \\ 2, 1 \end{bmatrix}$.

*Proof.* In this case we have $\tau_n = 1$, $\tau_p = 2$, $q = 1$. Using Lemma 8.4, for the mixed Palm expectations in (8.38) with these values of $\tau_n, \tau_p, q$ and $f(\mathcal{H}) = e^{-\rho T}$, we get

$$ \frac{L_{2,1}^{(1)}}{4\lambda_1\lambda_2 v_1 v_2} \mathbb{E}_{W_{2,1}^{(1)}}[e^{-\rho T}] = \int_0^\infty \int_0^h \int_{t^*}^\infty \mathbb{E}_{\mathcal{H}} \left[ e^{-\rho(T \circ \theta_{\hat{s}_1})} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}_1}^{\hat{s}_1, v_l} \right) = 0 \right] dt' dh' dh \\ + \int_0^\infty \int_h^\infty \int_0^\infty \mathbb{E}_{\mathcal{H}} \left[ e^{-\rho(T \circ \theta_{\hat{s}_1})} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}_1}^{\hat{s}_1, v_l} \right) = 0 \right] dt' dh' dh, \quad (8.61) $$

where $(\hat{s}_1, \hat{h}_1) = (\hat{s}_1(0, h, -t', h'), \hat{h}_1(0, h, -t', h'))$ denotes the coordinates of the first intersection between the radial birds at $(t_n, h_n) = (0, h)$ of type $\tau_n = 1$ and $(t_p, h_p) = (-t', h')$ type $\tau_p = 2$, which are responsible for a handover. The coordinates $(\hat{s}_2, \hat{h}_2) = (\hat{s}_2(0, h, -t', h'), \hat{h}_2(0, h, -t', h'))$ denote the second intersection. Since $\tau_n = 1$ and $q = 1$, the station corresponding to the head point $(0, h)$ becomes the serving station after time $\hat{s}_1$. The head point $(0, h)$ being of type 1, we define $\hat{h}_1(t) := (v_1^2 t^2 + h^2)^{\frac{1}{2}}$, which we write in short as $\hat{h}_t$ locally. Observe that

$$ T \circ \theta_{\hat{s}_1} = T_1(\hat{s}_1) - \hat{s}_1, $$

where $T_1(\hat{s}_1)$ is the time of the next handover after time $\hat{s}_1$. For the next handover, the extra region $(\bigcup_{t \in [\hat{s}_1, T_1(\hat{s}_1)]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_l})^\circ$ beyond $E_{\hat{h}_1}^{\hat{s}_1, v_l}$ must have no points of $\mathcal{H}^l$, for $l = 1, 2$. Here we have for any $\hat{s} \ge \hat{s}_2$

$$ \bigcup_{t \in [\hat{s}_1, \hat{s}_2]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_l} \subset \bigcup_{t \in [\hat{s}_1, \hat{s}]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_l}. $$

Hence, the next handover can happen before or at $\hat{s}_2$, i.e., $T_1(\hat{s}_1) < \hat{s}_2$ or $T_1(\hat{s}_1) = \hat{s}_2$, which leads to another layer of decomposition.

**Case 1.** $T_1(\hat{s}_1) < \hat{s}_2$. By Corollary 8.14, the inner expectation in both terms in (8.61) can be evaluated by looking for the next handover created by a radial bird with head at, say $(T_j, H_j)$, which is of type 1 or type 2. The head point $(T_j, H_j)$ must be located outside $E_{\hat{h}_1}^{\hat{s}_1, v_1}$ and $E_{\hat{h}_1}^{\hat{s}_1, v_2}$, depending on its type 1 or 2 (see Figure 8.7), as follows:

**Subcase 1.1.** $(T_j, H_j)$ is of type 1. Here, it must be the case that $T_j \ge 0$ and $(T_j, H_j) \notin E_{\hat{h}_1}^{\hat{s}_1, v_1}$. Indeed, since, for any radial bird with the head point at $(T_j, H_j) \notin E_{\hat{h}_1}^{\hat{s}_1, v_1}$, such that $T_j < 0$, the intersection to the radial bird with head at $(0, h)$, does not contribute to a future handover as it intersects the one with head at $(0, h)$, before time $\hat{s}_1$. Thus the unexplored region must be $Q_0 \setminus E_{\hat{h}_1}^{\hat{s}_1, v_1}$ and the type of the pair of handovers is $\begin{bmatrix} 1, & 1 \\ 2, & 2, 1 \end{bmatrix}$. In this case, the handover is given by the intersection $(\hat{s}(0, h, T_j, H_j), \hat{h}(0, h, T_j, H_j))$, as both $(0, h)$ and $(T_j, H_j)$ are of the same type, see Picture (1) of Figure 8.7. Thus we have $T_1(\hat{s}_1) := \hat{s}(0, h, T_j, H_j)$. For this intersection to give the next handover immediately after time $\hat{s}_1$, the extra region $(\bigcup_{t \in [\hat{s}_1, \hat{s}(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_l})^\circ$, beyond $E_{\hat{h}_1}^{\hat{s}_1, v_l}$, must have no point of $\mathcal{H}^l$ for $l = 1, 2$, where $\hat{h}_t = (v_1^2 t^2 + h^2)^{\frac{1}{2}}$. The union is due to Definition 8.13 and to the fact that the sets in the union potentially lack monotonicity.

**Subcase 1.2.** $(T_j, H_j)$ is of type 2. Since $(T_j, H_j) \notin E_{\hat{h}_1}^{\hat{s}_1, v_2}$, the pair of points $(0, h), (T_j, H_j)$ naturally satisfies the intersection criterion, by part (i) of Lemma 7.12. For $(T_j, H_j)$ to be an eligible
candidate, we must have $T_j \ge -t'$. Otherwise if $T_j < -t'$, then the radial bird with head point at $(T_j, H_j)$ intersects the one with head point at $(0, h)$, at a point contained in the open region above the lower envelope, as a result this does not contribute to any handover. Hence we assume that $T_j \ge -t'$ and the unexplored region is $Q_{-t'} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_2}$. Depending on whether $-t' \le T_j < 0$ or $T_j \ge 0$, the type of the pair of consecutive handovers is $[1, 2]$ or $[1, 1]$ respectively.

In both cases, $-t' \le T_j < 0$ or $T_j \ge 0$, there are two intersections $(\hat{s}_1(0, h, T_j, H_j), \hat{h}_2(0, h, T_j, H_j))$ and $(\hat{s}_2(0, h, T_j, H_j), \hat{h}_2(0, h, T_j, H_j))$ between the radial birds at $(0, h)$ and $(T_j, H_j)$. The first intersection $(\hat{s}_1(0, h, T_j, H_j), \hat{h}_2(0, h, T_j, H_j))$ can never correspond to a handover, because it is contained in $\mathcal{L}_e^+(v_1, v_2)$. So the only intersection that can possibly be responsible for a handover is $(\hat{s}_2(0, h, T_j, H_j), \hat{h}_2(0, h, T_j, H_j))$, as described in Figure 8.8. Hence, we have, $T_1(\hat{s}_1) := \hat{s}_2(0, h, T_j, H_j)$. For this intersection to give the next handover after time $\hat{s}_1$, the region $(\bigcup_{t \in [\hat{s}_1, \hat{s}_2(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_l})^o$, beyond $E_{\hat{h}_1}^{\hat{s}_1, v_l}$, must have no point of $\mathcal{H}^l$ for $l=1,2$, where $\hat{h}_t = (v_1^2 t^2 + h^2)^{1/2}$. The union is due to Definition 8.13 and the individual sets in the union possibly lack monotonicity.

**Case 2.** $T_1(\hat{s}_1) = \hat{s}_2$. Then the next handover is given by the second intersection $(\hat{s}_2, \hat{h}_2)$ of the radial birds with head points $(0, h)$, $(-t', h')$, provided the region $(\bigcup_{t \in [\hat{s}_1, \hat{s}_2]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_l})^o$ has no points of $\mathcal{H}^l$, for $l=1,2$, see Figure 8.8, where the union is due to Definition 8.13 and to the fact that the individual sets therein may not be monotonic. The type of the pair of consecutive handover is $[1, 2]$.

<figure><img src="image_47.png" /></figure>

(1) For the intersections $(\hat{s}_1, \hat{h}_1)$ and $(\tilde{s}, \tilde{h})$ to represent two consecutive handovers, the regions $E_{\hat{h}_1}^{\hat{s}_1, v_1} \cup E_{\tilde{h}}^{\tilde{s}, v_1}$ and $E_{\hat{h}_1}^{\hat{s}_1, v_2} \cup E_{\tilde{h}}^{\tilde{s}, v_2}$ must have no points from $\mathcal{H}^1$ and $\mathcal{H}^2$, respectively. Here $(\tilde{s}, \tilde{h})$ is a realization of $(\hat{s}(0, h, T_j, H_j), \hat{h}(0, h, T_j, H_j))$.

(2) For the intersections $(\hat{s}_1, \hat{h}_1)$ and $(\tilde{s}, \tilde{h})$ to represent two consecutive handovers, the regions $E_{\hat{h}_1}^{\hat{s}_1, v_1} \cup E_{\tilde{h}}^{\tilde{s}, v_1}$ and $E_{\hat{h}_1}^{\hat{s}_1, v_2} \cup E_{\tilde{h}}^{\tilde{s}, v_2}$ must have no points from $\mathcal{H}^1$ and $\mathcal{H}^2$, respectively. Here $(\tilde{s}, \tilde{h})$ is a realization of $(\hat{s}_2(0, h, T_j, H_j), \hat{h}_2(0, h, T_j, H_j))$.

<figure><img src="image_48.png" /><figcaption>FIGURE 8.8. The case corresponding to the last term in (8.63). For the intersections $(\hat{s}_1, \hat{h}_1)$ and $(\hat{s}_2, \hat{h}_2)$ to represent two consecutive handovers, the regions $E_{\hat{h}_1}^{\hat{s}_1, v_1} \cup E_{\hat{h}_2}^{\hat{s}_2, v_1}$ and $E_{\hat{h}_1}^{\hat{s}_1, v_2} \cup E_{\hat{h}_2}^{\hat{s}_2, v_2}$ must have no points from $\mathcal{H}^1$ and $\mathcal{H}^2$, respectively.</figcaption></figure>

Thus, based on the last arguments, we write the inner expectation in (8.61), as the following sum of three terms:
$$
\begin{align}
& \mathbb{E}_{\mathcal{H}} \left[ e^{-\rho(T_1(\hat{s}_1) - \hat{s}_1)} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} (E_{\hat{h}_1}^{\hat{s}_1, v_l}) = 0 \right] \nonumber \\
&= \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\exists(T_j, H_j) \in \mathcal{H}^1 : T_j \ge 0, \cap_{l=1,2} \left\{ \mathcal{H}^l \left( \bigcup_{t \in [\hat{s}_1, \hat{s}(0,h,T_j,H_j)]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_l} \right)^{\circ} = 0 \right\}} \mathbb{1}_{\hat{s}_1 \le \hat{s}(0,h,T_j,H_j) \le \hat{s}_2} \right. \nonumber \\
& \qquad \left. \times e^{-\rho(\hat{s}(0,h,T_j,H_j) - \hat{s}_1)} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} (E_{\hat{h}_1}^{\hat{s}_1, v_l}) = 0 \right] \nonumber \\
&\quad + \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\exists(T_j, H_j) \in \mathcal{H}^2 : T_j \ge -t', \cap_{l=1,2} \left\{ \mathcal{H}^l \left( \bigcup_{t \in [\hat{s}_1, \hat{s}_2(0,h,T_j,H_j)]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_l} \right)^{\circ} = 0 \right\}} \mathbb{1}_{\hat{s}_1 \le \hat{s}_2(0,h,T_j,H_j) \le \hat{s}_2} \right. \nonumber \\
& \qquad \left. \times e^{-\rho(\hat{s}(0,h,T_j,H_j) - \hat{s}_1)} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} (E_{\hat{h}_1}^{\hat{s}_1, v_l}) = 0 \right] \nonumber \\
&\quad + \mathbb{E}_{\mathcal{H}} \left[ e^{-\rho(\hat{s}_2 - \hat{s}_1)} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( \bigcup_{t \in [\hat{s}_1, \hat{s}_2]} E_{\hat{h}_t}^{t, v_l} \cup E_{\hat{h}_1}^{\hat{s}_1, v_l} \right) = 0 \right]. \tag{8.62}
\end{align}
$$

Similarly to (8.44), there exists a unique head point that satisfies the condition in each term of the sum in (8.62), the last expression can be written as

$$
\begin{align}
& \mathbb{E}_{\mathcal{H}} \left[ \sum_{(T_j, H_j) \in \mathcal{H}^1 : T_j \ge 0} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( \bigcup_{t \in [\hat{s}_1, \hat{s}(0,h,T_j,H_j)]} E_{\hat{h}_t}^{t, v_l} \cup E_{\hat{h}_1}^{\hat{s}_1, v_l} \right) = 0 \right] \mathbb{1}_{\hat{s}_1 \le \hat{s}(0,h,T_j,H_j) \le \hat{s}_2} e^{-\rho(\hat{s}(0,h,T_j,H_j) - \hat{s}_1)} \nonumber \\
& + \mathbb{E}_{\mathcal{H}} \left[ \sum_{(T_j, H_j) \in \mathcal{H}^2 : T_j \ge -t'} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( \bigcup_{t \in [\hat{s}_1, \hat{s}_2(0,h,T_j,H_j)]} E_{\hat{h}_t}^{t, v_l} \cup E_{\hat{h}_1}^{\hat{s}_1, v_l} \right) = 0 \right] \mathbb{1}_{\hat{s}_1 \le \hat{s}_2(0,h,T_j,H_j) \le \hat{s}_2} e^{-\rho(\hat{s}(0,h,T_j,H_j) - \hat{s}_1)} \nonumber \\
& + \mathbb{E}_{\mathcal{H}} \left[ e^{-\rho(\hat{s}_2 - \hat{s}_1)} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( \bigcup_{t \in [\hat{s}_1, \hat{s}_2]} E_{\hat{h}_t}^{t, v_l} \cup E_{\hat{h}_1}^{\hat{s}_1, v_l} \right) = 0 \right]. \tag{8.63}
\end{align}
$$

Using the increasing property of the sequence of regions $\{E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_l}\}_{t \ge \hat{s}}$ from part (i) of Proposition 8.16, in all the terms in (8.63), we get

$$
\bigcup_{t \in [\hat{s}_1, \hat{s}(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \cup E_{\hat{h}_1}^{\hat{s}_1, v_l} = E_{\hat{h}(0, h, T_j, H_j)}^{\hat{s}(0, h, T_j, H_j), v_l} \cup E_{\hat{h}_1}^{\hat{s}_1, v_l},
$$

$$
\bigcup_{t \in [\hat{s}_1, \hat{s}_2(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \cup E_{\hat{h}_1}^{\hat{s}_1, v_l} = E_{\hat{h}_2(0, h, T_j, H_j)}^{\hat{s}_2(0, h, T_j, H_j), v_l} \cup E_{\hat{h}_1}^{\hat{s}_1, v_l} \quad \text{and} \quad \bigcup_{t \in [\hat{s}_1, \hat{s}_2]} E_{\hat{h}_t}^{t, v_l} \cup E_{\hat{h}_1}^{\hat{s}_1, v_l} = E_{\hat{h}_2}^{\hat{s}_2, v_l} \cup E_{\hat{h}_1}^{\hat{s}_1, v_l},
$$

for $l = 1, 2$, where the first two equalities depend on whether $(T_j, H_j)$ is of type 1 or 2 and the last one is for $\hat{s}_1$ and $\hat{s}_2$ to be consecutive handovers. Thus the three terms in (8.63) can be written as

$$
\begin{align}
& \mathbb{E}_{\mathcal{H}} \left[ \sum_{(T_j, H_j) \in \mathcal{H}^1 : T_j \ge 0} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}(0,h,T_j,H_j)}^{\hat{s}(0,h,T_j,H_j),v_l} \cup E_{\hat{h}_1}^{\hat{s}_1,v_l} \right) = 0 \right] \mathbb{1}_{\hat{s}_1 \le \hat{s}(0,h,T_j,H_j) \le \hat{s}_2} e^{-\rho(\hat{s}(0,h,T_j,H_j) - \hat{s}_1)} \nonumber \\
& + \mathbb{E}_{\mathcal{H}} \left[ \sum_{(T_j, H_j) \in \mathcal{H}^2 : T_j \ge -t'} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}_2(0,h,T_j,H_j)}^{\hat{s}_2(0,h,T_j,H_j),v_l} \cup E_{\hat{h}_1}^{\hat{s}_1,v_l} \right) = 0 \right] \mathbb{1}_{\hat{s}_1 \le \hat{s}_2(0,h,T_j,H_j) \le \hat{s}_2} e^{-\rho(\hat{s}(0,h,T_j,H_j) - \hat{s}_1)} \nonumber \\
& + \mathbb{E}_{\mathcal{H}} \left[ e^{-\rho(\hat{s}_2 - \hat{s}_1)} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}_2}^{\hat{s}_2,v_l} \cup E_{\hat{h}_1}^{\hat{s}_1,v_l} \right) = 0 \right] := \zeta \begin{bmatrix} 1, 1 \\ 2, 1, 1 \end{bmatrix} + \zeta \begin{bmatrix} 1, 1 \\ 2, 1, 2 \end{bmatrix} + \zeta \begin{bmatrix} 1, 2 \\ 2, 1, 2^* \end{bmatrix}, \tag{8.64}
\end{align}
$$
where the type of the pair of handovers marked with `*` means that we have reappearance of the previous station as an upcoming one, as seen in the last column of Table 8.2.

Let $R_1(h, t', h') \subset (\mathbb{R}^+)^2 \setminus E_{\hat{h}}^{\hat{s}, v_1}$ be the region defined as

$$R_1(h, t', h') := \{(t, u) : \hat{s}_1 \le \hat{s}(0, h, t, u) \le \hat{s}_2\},$$

which contains all possible head points that produce such a handover as discussed in Subcase 1.1. It can be proved, using Lemma 7.13, that $R_1(h, t', h') = E_{\hat{h}_2}^{\hat{s}_2, v_1} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_1}$. Applying the Campbell-Mecke formula, the first term in (8.64) equals

$$\begin{aligned}
\zeta\left[\begin{smallmatrix} 1, & 1 \\ 2, & 1, & 1 \end{smallmatrix}\right] &= 2\lambda_1 v_1 \int_{E_{\hat{h}_2}^{\hat{s}_2, v_1} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_1}} e^{-\rho(\hat{s}(0, h, t_1, h_1) - \hat{s}_1)} \mathbb{E}_{\mathcal{H}}\left[\prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l}\left(E_{\hat{h}(0, h, t_1, t_1)}^{\hat{s}(0, h, t_1, h_1), v_l \cup E_{\hat{h}_1}^{\hat{s}_1, v_l}}\right) = 0\right] dt_1 dh_1 \\
&= 2\lambda_1 v_1 \int_{E_{\hat{h}_2}^{\hat{s}_2, v_1} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_1}} e^{-\rho(\hat{s}(0, h, t_1, h_1) - \hat{s}_1)} e^{-\sum_{l=1,2} 2\lambda_l v_l |E_{\hat{h}(0, h, t_1, t_1)}^{\hat{s}(0, h, t_1, h_1), v_l \cup E_{\hat{h}_1}^{\hat{s}_1, v_l}|} dt_1 dh_1. \quad (8.65)
\end{aligned}$$

Let $R_2(h, t', h') \subset Q_{-t'} \setminus E_{\hat{h}}^{\hat{s}, v_2}$ be the region defined as $R_2(h, t', h') := \{(t, u) : \hat{s}_1 \le \hat{s}_2(0, h, t, u) \le \hat{s}_2\}$, containing the head point that gives the required handover, as seen in Subcase 1.2. It can be proved using Lemma 7.13, that the region $R_2(h, t', h') = E_{\hat{h}_2}^{\hat{s}_2, v_2} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_2}$. Applying the Campbell-Mecke formula, the second term in (8.64) equals

$$\begin{aligned}
\zeta\left[\begin{smallmatrix} 1, & 1 \\ 2, & 1, & 2 \end{smallmatrix}\right] &= 2\lambda_2 v_2 \int_{E_{\hat{h}_2}^{\hat{s}_2, v_2} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_2}} e^{-\rho(\hat{s}_2(0, h, t_1, h_1) - \hat{s}_1)} \mathbb{E}_{\mathcal{H}}\left[\prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l}\left(E_{\hat{h}_2(0, h, t_1, t_1)}^{\hat{s}_2(0, h, t_1, h_1), v_l \cup E_{\hat{h}_1}^{\hat{s}_1, v_l}}\right) = 0\right] dt_1 dh_1 \\
&= 2\lambda_2 v_2 \int_{E_{\hat{h}_2}^{\hat{s}_2, v_2} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_2}} e^{-\rho(\hat{s}_2(0, h, t_1, h_1) - \hat{s}_1)} e^{-\sum_{l=1,2} 2\lambda_l v_l |E_{\hat{h}_2(0, h, t_1, t_1)}^{\hat{s}_2(0, h, t_1, h_1), v_l \cup E_{\hat{h}_1}^{\hat{s}_1, v_l}|} dt_1 dh_1. \quad (8.66)
\end{aligned}$$

The third term in (8.64) equals

$$\zeta\left[\begin{smallmatrix} 1, & 2 \\ 2, & 1, & 2^* \end{smallmatrix}\right] = e^{-\rho(\hat{s}_2 - \hat{s}_1)} e^{-\sum_{l=1,2} 2\lambda_l v_l |E_{\hat{h}_2}^{\hat{s}_2, v_l} \cup E_{\hat{h}_1}^{\hat{s}_1, v_l}|}. \quad (8.67)$$

Define

$$\zeta_{2,1}^{(1)}(\rho, h, t', h') := \zeta\left[\begin{smallmatrix} 1, & 1 \\ 2, & 1, & 1 \end{smallmatrix}\right] + \zeta\left[\begin{smallmatrix} 1, & 1 \\ 2, & 1, & 2 \end{smallmatrix}\right] + \zeta\left[\begin{smallmatrix} 1, & 2 \\ 2, & 1, & 2^* \end{smallmatrix}\right], \quad (8.68)$$

using (8.65)-(8.67). We obtain the result by substituting the expression of $\zeta_{2,1}^{(2)}(\rho, h, t', h')$ from (8.68) back to both terms in (8.61), as

$$\begin{aligned}
\frac{L_{2,1}^{(1)}}{4\lambda_1\lambda_2 v_1 v_2} \mathbb{E}_{W_{2,1}^{(1)}}^0 [e^{-\rho T}] &= \int_0^\infty \int_0^h \int_{t^*}^\infty \zeta_{2,1}^{(1)}(\rho, t', h', h) dt' dh' dh + \int_0^\infty \int_h^\infty \int_0^\infty \zeta_{2,1}^{(1)}(\rho, t', h', h) dt' dh' dh \\
&:= \xi_{2,1}^{(1)}(\rho, v_2, v_1), \quad (8.69)
\end{aligned}$$

which is a function of $\rho, v_2, v_1$ only. $\square$

### 8.3.8. Proof of Lemma 8.20, type $(2, 1)$ or equivalently $[\begin{smallmatrix} 2 \\ 1, 2 \end{smallmatrix}]$.

*Proof.* Using Lemma 8.4, the mixed Palm expectations in (8.38) with $\tau_n = 2, \tau_p = 1, q = 2$, and $f(\mathcal{H}) = e^{-\rho T}$, we have

$$\begin{aligned}
\frac{L_{1,2}^{(2)}}{4\lambda_1\lambda_2 v_1 v_2} \mathbb{E}_{W_{1,2}^{(2)}}^0 [e^{-\rho T}] &= \int_0^\infty \int_0^h \int_{t^*}^\infty \mathbb{E}_{\mathcal{H}}\left[e^{-\rho(T \circ \theta_{\hat{s}_2})} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l}\left(E_{\hat{h}_2}^{\hat{s}_2, v_l}\right) = 0\right] dt' dh' dh \\
&\quad + \int_0^\infty \int_h^\infty \int_0^\infty \mathbb{E}_{\mathcal{H}}\left[e^{-\rho(T \circ \theta_{\hat{s}_2})} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l}\left(E_{\hat{h}_2}^{\hat{s}_2, v_l}\right) = 0\right] dt' dh' dh, \quad (8.70)
\end{aligned}$$

where there are two radial birds of type $\tau_n = 2$ and $\tau_p = 1$, respectively, with head at $(t_p, h_p) = (-t', h')$ and $(t_n, h_n) = (0, h)$ and we write in short $(\hat{s}_2, \hat{h}_2)$ for $(\hat{s}_2(0, h, -t', h'), \hat{h}_2(0, h, -t', h'))$ to denote the
second intersection, see Figure 8.9. Note that, since $\tau_n = 2$ and $q = 2$, the station corresponding to the head point $(-t', h')$ is the serving station after time $\hat{s}_2$. Since $(-t', h')$ is of type 2, we define $\hat{h}_t := (v_2^2(t+t')^2 + h'^2)^{\frac{1}{2}}$. Also note that

$$T \circ \theta_{\hat{s}_2} := T_1(\hat{s}_2) - \hat{s}_2,$$

where $T_1(\hat{s}_2)$ is the time point of the next handover after time $\hat{s}_2$. The inner expectation in both terms in (8.70) can be evaluated by looking for the next handover created by a radial bird with its head at $(T_j, H_j)$ (say), such that it must lie outside $E_{\hat{h}_2}^{\hat{s}_2, v_1}$ or $E_{\hat{h}_2}^{\hat{s}_2, v_2}$, depending on its type.

**Case 1.** $(T_j, H_j)$ is of type 1. We must have $T_j \ge \hat{s}_2$. Otherwise, if $T_j < \hat{s}_2$, then the intersection produced by the radial bird with head at $(T_j, H_j)$, with the one at $(0, h)$, is either contained in $\mathcal{L}_e^+(v_1, v_2)$ or it is responsible for a handover before the time $\hat{s}_2$. Thus $T_j \ge \hat{s}_2$ and the unexplored region is $Q_{\hat{s}_2} \setminus E_{\hat{h}_2}^{\hat{s}_2, v_1}$. The type of the pair of consecutive handover is $[\begin{smallmatrix} 2, 1 \\ 1, 2, 1 \end{smallmatrix}]$. Also $(T_j, H_j)$ must be such that, $T_j \in D_r(-t', h', H_j)$, given $t', h', H_j$, for $(T_j, H_j)$ to be in this case. Suppose $(\hat{s}_1(-t', h', T_j, H_j), \hat{h}_1(-t', h', T_j, H_j))$ and $(\hat{s}_2(-t', h', T_j, H_j), \hat{h}_2(-t', h', T_j, H_j))$ be the two intersection point of the radial birds with head at $(-t', h')$, $(T_j, H_j)$, such that $\hat{s}_1(-t', h', T_j, H_j) \le \hat{s}_2(-t', h', T_j, H_j)$. Observe that, the point $(T_j, H_j)$ being outside $E_{\hat{h}_2}^{\hat{s}_2, v_1}$, we have $\hat{s}_2 \le \hat{s}_1(-t', h', T_j, H_j)$, see Picture (1) of Figure 8.9. For the two intersection points to represent handovers, we must have no point of $\mathcal{H}^l$, for $l = 1, 2$, in the extra regions $(\bigcup_{t \in [\hat{s}_2, \hat{s}_1(-t', h', T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_2}^{\hat{s}_2, v_l})^o$ and $(\bigcup_{t \in [\hat{s}_2, \hat{s}_2(-t', h', T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_2}^{\hat{s}_2, v_l})^o$, respectively, beyond $E_{\hat{h}_2}^{\hat{s}_2, v_l}$, where $\hat{h}_t = (v_2^2(t+t')^2 + h'^2)^{\frac{1}{2}}$. Since $\hat{s}_1(-t', h', T_j, H_j) \le \hat{s}_2(-t', h', T_j, H_j)$, we have

$$\bigcup_{t \in [\hat{s}_2, \hat{s}_1(-t', h', T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_2}^{\hat{s}_2, v_l} \subset \bigcup_{t \in [\hat{s}_2, \hat{s}_2(-t', h', T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_2}^{\hat{s}_2, v_l}.$$

Hence, the first intersection $(\hat{s}_1(-t', h', T_j, H_j), \hat{h}_1(-t', h', T_j, H_j))$ gives the next handover. Thus we have $T_1(\hat{s}_1) := \hat{s}_1(0, h, T_j, H_j)$.

**Case 2.** $(T_j, H_j)$ is of type 2. In this case, the point must be such that $T_j \ge -t'$. Otherwise, the birds with head at $(T_j, H_j)$ and $(-t', h')$, intersects before time $\hat{s}_2$, which is not of interest here, since we are looking a future handover after $\hat{s}_2$. Thus the unexplored region is $Q_{-t'} \setminus E_{\hat{h}_2}^{\hat{s}_2, v_2}$ and the type of the pair of consecutive handover is $[\begin{smallmatrix} 2, 1 \\ 1, 2, 2 \end{smallmatrix}]$. The handover must be given by $(\hat{s}(-t', h', T_j, H_j), \hat{h}(-t', h', T_j, H_j))$, the only intersection between the birds at $(-t', h')$ and $(T_j, H_j)$, see Picture (2) of Figure 8.9. Additionally, we have $T_1(\hat{s}_1) := \hat{s}_2(0, h, T_j, H_j)$. The union is due to the Definition 8.13 and the individual sets in the union may not be monotonic.

<figure><img src="image_49.png" /></figure>

(1) For the intersection $(\hat{s}_2, \hat{h}_2)$ and $(\tilde{s}, \tilde{h})$ to represent two consecutive handovers, the regions $S_1 \cup E_{\hat{h}_2}^{\hat{s}_2, v_1} \cup E_{\tilde{h}}^{\tilde{s}, v_1}$ and $E_{\hat{h}_2}^{\hat{s}_2, v_2} \cup E_{\tilde{h}}^{\tilde{s}, v_2}$ must have no points from $\mathcal{H}^1$ and $\mathcal{H}^2$, respectively. Here $(\tilde{s}, \tilde{h})$ is a realization of $(\hat{s}_1(-t', h', T_j, H_j), \hat{h}_1(-t', h', T_j, H_j))$.

(2) For the intersection $(\hat{s}_2, \hat{h}_2)$ and $(\tilde{s}, \tilde{h})$ to represent two consecutive handovers, the regions $S_2 \cup E_{\hat{h}_2}^{\hat{s}_2, v_1} \cup E_{\tilde{h}}^{\tilde{s}, v_1}$ and $E_{\hat{h}_2}^{\hat{s}_2, v_2} \cup E_{\tilde{h}}^{\tilde{s}, v_2}$ must have no points from $\mathcal{H}^1$ and $\mathcal{H}^2$, respectively. Here $(\tilde{s}, \tilde{h})$ is a realization of $(\hat{s}(-t', h', T_j, H_j), \hat{h}(-t', h', T_j, H_j))$.

<caption>FIGURE 8.9. The two pictures correspond to the two cases in (8.71), where the head point $(T_j, H_j) = (t_1, h_1)$ is of type 1 or 2.</caption>
The last discussion leads to a decomposition of the inner expectation in (8.70) as

$$
\begin{align}
& \mathbb{E}_{\mathcal{H}} \left[ e^{-\rho(T_1(\hat{s}_2) - \hat{s}_2)} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}_2}^{\hat{s}_2, v_l} \right) = 0 \right] \nonumber \\
&= \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\exists(T_j, H_j) \in \mathcal{H}^1 : T_j \ge \hat{s}_2, T_j \in D_r(-t', h', H_j), \cap \left( \bigcup_{l=1,2} \left\{ \mathcal{H}^l \left( \bigcup_{t \in [\hat{s}_2, \hat{s}_1(-t', h', T_j, H_j)] E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_2}^{\hat{s}_2, v_l} \right)^{\circ} = 0 \right\} \times e^{-\rho(\hat{s}(-t', h', T_j, H_j) - \hat{s}_2)} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}_2}^{\hat{s}_2, v_l} \right) = 0 \right)} \right] \nonumber \\
&\quad + \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\exists(T_j, H_j) \in \mathcal{H}^2 : \cap \left( \bigcup_{l=1,2} \left\{ \mathcal{H}^l \left( \bigcup_{t \in [\hat{s}_2, \hat{s}(-t', h', T_j, H_j)] E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_2}^{\hat{s}_2, v_l} \right)^{\circ} = 0 \right\} e^{-\rho(\hat{s}(-t', h', T_j, H_j) - \hat{s}_2)} \times \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}_2}^{\hat{s}_2, v_l} \right) = 0 \right)} \right] \tag{8.71}
\end{align}
$$

Since there exists a unique head point satisfying the condition in the last sum, it reduces to

$$
\begin{align}
& \mathbb{E}_{\mathcal{H}} \left[ \sum_{(T_j, H_j) \in \mathcal{H}^1 : T_j \ge \hat{s}_2, T_j \in D_r(-t', h', H_j)} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( \bigcup_{t \in [\hat{s}_2, \hat{s}_1(-t', h', T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \cup E_{\hat{h}_2}^{\hat{s}_2, v_l} \right) e^{-\rho(\hat{s}_1(-t', h', T_j, H_j) - \hat{s}_2)} \right] \nonumber \\
&\quad + \mathbb{E}_{\mathcal{H}} \left[ \sum_{(T_j, H_j) \in \mathcal{H}^2} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( \bigcup_{t \in [\hat{s}_2, \hat{s}(-t', h', T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \cup E_{\hat{h}_2}^{\hat{s}_2, v_l} \right) e^{-\rho(\hat{s}(-t', h', T_j, H_j) - \hat{s}_2)} \right] \nonumber \\
&:= \zeta \left[ \begin{smallmatrix} 2, & 1 \\ 1, & 2, 1 \end{smallmatrix} \right] + \zeta \left[ \begin{smallmatrix} 2, & 1 \\ 1, & 2, 2 \end{smallmatrix} \right]. \tag{8.72}
\end{align}
$$

For the first term in (8.72), using Remark 7.10 for $l=1$, we have

$$
\bigcup_{t \in [\hat{s}_2, \hat{s}_1(-t', h', T_j, H_j)]} E_{\hat{h}_t}^{t, v_1} \cup E_{\hat{h}_2}^{\hat{s}_2, v_1} = E_{\hat{h}_2}^{\hat{s}_2, v_1} \cup S_1 \cup E_{\hat{h}_1(-t', h', T_j, H_j)}^{\hat{s}_1(-t', h', T_j, H_j), v_1},
$$

where $\hat{h}_t = (v_2^2(t+t')^2 + h'^2)^{1/2}$ and $S_1$ is the region outside $E_{\hat{h}_2}^{\hat{s}_2, v_1} \cup E_{\hat{h}_1(-t', h', T_j, H_j)}^{\hat{s}_1(-t', h', T_j, H_j), v_1}$ but below the
hyperbola given by the equation $u^2 - \frac{v_1^2 v_2^2}{v_1^2 - v_2^2}(t+t')^2 = h'^2$, in the $(t, u)$-coordinate system similar to
(7.17) and within the time interval

$$
\left[ \frac{1}{v_1^2} \left( (v_1^2 - v_2^2)\hat{s}_2 - v_2^2 t' \right), \frac{1}{v_1^2} \left( (v_1^2 - v_2^2)\hat{s}_1(-t', h', T_j, H_j) - v_2^2 t' \right) \right].
$$

For $l=2$, using the increasing property of $\{E_{\hat{h}_t}^{t,v_2} \setminus E_{\hat{h}}^{\hat{s},v_2}\}_{t \ge \hat{s}_2}$, from part (ii) of Proposition 8.16, we
have

$$
\bigcup_{t \in [\hat{s}_2, \hat{s}_1(-t', h', T_j, H_j)]} E_{\hat{h}_t}^{t, v_2} \cup E_{\hat{h}}^{\hat{s}, v_2} = E_{\hat{h}_1(-t', h', T_j, H_j)}^{\hat{s}_1(-t', h', T_j, H_j), v_2} \cup E_{\hat{h}}^{\hat{s}, v_2}.
$$

Then the first term in (8.72) equals

$$
\mathbb{E}_{\mathcal{H}} \left[ \sum_{(T_j, H_j) \in \mathcal{H}^1 : T_j \ge \hat{s}_2, T_j \in D_r(-t', h', H_j)} \mathbb{1}_{\mathcal{H}^1} \left( E_{\hat{h}_1(-t', h', T_j, H_j)}^{\hat{s}_1(-t', h', T_j, H_j), v_1} \cup S_1 \cup E_{\hat{h}_2}^{\hat{s}_2, v_1} \right) = 0 \right. \\
\left. \times \mathbb{1}_{\mathcal{H}^2} \left( E_{\hat{h}_1(-t', h', T_j, H_j)}^{\hat{s}_1(-t', h', T_j, H_j), v_2} \cup E_{\hat{h}_2}^{\hat{s}_2, v_2} \right) = 0 \right] e^{-\rho(\hat{s}_1(-t', h', T_j, H_j) - \hat{s}_2)}. \quad (8.73)
$$
The point $(T_j, H_j)$ must lie in the region $R_1(h, t', h')$, where

$$R_1(h, t', h') := \left\{ (t, u) : t \ge \hat{s}_2, v_1^2(t - \hat{s}_2)^2 + u^2 \ge \hat{h}_2^2 \text{ and } u^2 - \frac{v_1^2 v_2^2}{v_1^2 - v_2^2} (t + t')^2 \le h'^2 \right\}. \quad (8.74)$$

For the region $R_1(h, t', h')$ in (8.74), the first inequality $t \ge \hat{s}_2$ is due to the fact that the head point should lie on the right of $(\hat{s}_2, \hat{h}_2)$, as discussed in Case 1 of this proof. The second and the third condition are because the next head point must lie outside $E_{\hat{h}_2}^{\hat{s}_2, v_1}$ and below the hyperbola $u^2 - \frac{v_1^2 v_2^2}{v_1^2 - v_2^2} (t + t')^2 = h'^2$, by Property II. The latter is equivalent to the intersection criteria of two birds.

Applying the Campbell-Mecke formula, the first term in (8.72) equals

$$\begin{align}
\zeta \begin{bmatrix} 2, 1 \\ 1, 2, 1 \end{bmatrix} & = 2\lambda_1 v_1 \int_{R_1(h, t', h')} e^{-\rho(\hat{s}_1(-t', h', t_1, h_1) - \hat{s}_2)} \mathbb{E}_H \left[ \mathbb{1}_{\mathcal{H}^1} \left( E_{\hat{h}_1(-t', h', t_1, h_1)}^{\hat{s}_1(-t', h', t_1, h_1), v_1} \cup S_1 \cup E_{\hat{h}_2}^{\hat{s}_2, v_1} \right) = 0 \right. \nonumber \\
& \qquad \left. \times \mathbb{1}_{\mathcal{H}^2} \left( E_{\hat{h}_1(-t', h', t_1, h_1)}^{\hat{s}_1(-t', h', t_1, h_1), v_2} \cup E_{\hat{h}_2}^{\hat{s}_2, v_2} \right) = 0 \right] dt_1 dh_1 \nonumber \\
& = 2\lambda_1 v_1 \int_{\hat{s}_2}^{\hat{s}_2 + \hat{h}_2 / v_1} \int_{(\hat{h}_2^2 - v_1^2)(t_1 - \hat{s}_2)^2)^{1/2}} \left( \frac{v_1^2 v_2^2}{v_1^2 - v_2^2} (t_1 + t')^2 + h'^2 \right)^{1/2} e^{-\rho(\hat{s}_1(-t', h', t_1, h_1) - \hat{s}_2)} \mathbb{E}_H \left[ \mathbb{1}_{\mathcal{H}^1} \left( E_{\hat{h}_1(-t', h', t_1, h_1)}^{\hat{s}_1(-t', h', t_1, h_1), v_1} \cup S_1 \cup E_{\hat{h}_2}^{\hat{s}_2, v_1} \right) = 0 \right. \nonumber \\
& \qquad \left. \times \mathbb{1}_{\mathcal{H}^2} \left( E_{\hat{h}_1(-t', h', t_1, h_1)}^{\hat{s}_1(-t', h', t_1, h_1), v_2} \cup E_{\hat{h}_2}^{\hat{s}_2, v_2} \right) = 0 \right] dh_1 dt_1 \nonumber \\
& + 2\lambda_1 v_1 \int_{\hat{s}_2 + \hat{h}_2 / v_1}^{\infty} \int_0^{\left( \frac{v_1^2 v_2^2}{v_1^2 - v_2^2} (t_1 + t')^2 + h'^2 \right)^{1/2}} e^{-\rho(\hat{s}_1(-t', h', t_1, h_1) - \hat{s}_2)} \mathbb{E}_H \left[ \mathbb{1}_{\mathcal{H}^1} \left( S_1 \cup E_{\hat{h}_1(-t', h', t_1, h_1)}^{\hat{s}_1(-t', h', t_1, h_1), v_1} \cup E_{\hat{h}_2}^{\hat{s}_2, v_1} \right) = 0 \right. \nonumber \\
& \qquad \left. \times \mathbb{1}_{\mathcal{H}^2} \left( E_{\hat{h}_1(-t', h', t_1, h_1)}^{\hat{s}_1(-t', h', t_1, h_1), v_2} \cup E_{\hat{h}_2}^{\hat{s}_2, v_2} \right) = 0 \right] dh_1 dt_1. \tag{8.75}
\end{align}$$

For the second term in (8.72), using Remark 7.10 for $l=1$,

$$\bigcup_{t \in [\hat{s}_2, \hat{s}(-t', h', T_j, H_j)]} E_{\hat{h}_t}^{t, v_1} \cup E_{\hat{h}_2}^{\hat{s}_2, v_1} = E_{\hat{h}_2}^{\hat{s}_2, v_1} \cup S_2 \cup E_{\hat{h}}^{\hat{s}(-t', h', T_j, H_j), v_1},$$

where $\hat{h}_t = (v_2^2(t+t')^2 + h'^2)^{1/2}$ and $S_2$ is the region outside $E_{\hat{h}_2}^{\hat{s}_2, v_1} \cup E_{\hat{h}}^{\hat{s}(-t', h', T_j, H_j), v_1}$ but below the hyperbola given by the equation $u^2 - \frac{v_1^2 v_2^2}{v_1^2 - v_2^2} (t+t')^2 = h'^2$, in the $(t, u)$-coordinate system similar to (7.17) and within the time interval

$$\left[ \frac{1}{v_1^2} \left( (v_1^2 - v_2^2)\hat{s}_2 - v_2^2 t' \right), \frac{1}{v_1^2} \left( (v_1^2 - v_2^2)\hat{s}(-t', h', T_j, H_j) - v_2^2 t' \right) \right].$$

For $l=2$, using part (ii) of Proposition 8.16, the increasing property of $\{E_{\hat{h}_t}^{t, v_2} \setminus E_{\hat{h}}^{\hat{s}, v_2}\}_{t \ge \hat{s}_2}$, we have

$$\bigcup_{t \in [\hat{s}_2, \hat{s}(-t', h', T_j, H_j)]} E_{\hat{h}_t}^{t, v_2} \cup E_{\hat{h}}^{\hat{s}, v_2} = E_{\hat{h}}^{\hat{s}(-t', h', T_j, H_j), v_2} \cup E_{\hat{h}}^{\hat{s}, v_2}.$$

The second term in (8.72) equals

$$\mathbb{E}_{\mathcal{H}} \left[ \sum_{(T_j, H_j) \in \mathcal{H}^2} \mathbb{1}_{\mathcal{H}^1} \left( E_{\hat{h}_2}^{\hat{s}_2, v_1} \cup S_2 \cup E_{\hat{h}}^{\hat{s}(-t', h', T_j, H_j), v_1} \right) = 0 \right] \mathbb{1}_{\mathcal{H}^1} \left( E_{\hat{h}_2}^{\hat{s}_2, v_2} \cup E_{\hat{h}}^{\hat{s}(-t', h', T_j, H_j), v_2} \right) = 0 e^{-\rho(\hat{s}(-t', h', T_j, H_j) - \hat{s}_2)}.$$

The point $(T_j, H_j)$ must lie in the region $R_2(h, t', h')$, where

$$R_2(h, t', h') := \left\{ (t, u) : t \ge -t', u \ge 0, \text{ and } v_2^2(t - \hat{s}_2)^2 + u^2 \ge \hat{h}_2^2 \right\}. \quad (8.76)$$
For the region $R_2(h, t', h')$ in (8.76), the first condition $t \ge -t'$, is as discussed in Case 2 of the proof. The last condition is due to the fact that the next head point must lie outside $E_{\hat{h}_2}^{\hat{s}_2, v_2}$.

Applying the Campbell-Mecke formula and later evaluating void probabilities, the second term in (8.72) equals

$$
\begin{align*}
\zeta\left[\begin{smallmatrix} 2,1 \\ 1,2,2 \end{smallmatrix}\right] &= 2\lambda_2 v_2 \int_{R_2(h,t',h')} e^{-\rho(\hat{s}(-t',h',t_1,h_1)-\hat{s}_2)} \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\mathcal{H}^1}\left(E_{\hat{h}(-t',h',t_1,h_1)}^{\hat{s}(-t',h',t_1,h_1),v_1} \cup S_2 \cup E_{\hat{h}_2}^{\hat{s}_2,v_1}\right) = 0 \right. \\
&\qquad \left. \times \mathbb{1}_{\mathcal{H}^2}\left(E_{\hat{h}(-t',h',t_1,h_1)}^{\hat{s}(-t',h',t_1,h_1),v_2} \cup E_{\hat{h}_2}^{\hat{s}_2,v_2}\right) = 0 \right] \mathrm{d}t_1 \mathrm{d}h_1 \\
&= 2\lambda_2 v_2 \int_{-t'}^{\hat{s}_2} \int_{(\hat{h}_2^2 - v_2^2(t_1 - \hat{s}_2)^2)^{\frac{1}{2}}}^{\infty} e^{-\rho(\hat{s}(-t',h',t_1,h_1)-\hat{s}_2)} e^{-2\lambda_1 v_1} \left| E_{\hat{h}(-t',h',t_1,h_1)}^{\hat{s}(-t',h',t_1,h_1),v_1} \cup S_2 \cup E_{\hat{h}_2}^{\hat{s}_2,v_1} \right| \\
&\qquad \times e^{-2\lambda_2 v_2} \left| E_{\hat{h}(-t',h',t_1,h_1)}^{\hat{s}(-t',h',t_1,h_1),v_2} \cup E_{\hat{h}_2}^{\hat{s}_2,v_2} \right| \mathrm{d}t_1 \mathrm{d}h_1 \\
&\quad + 2\lambda_2 v_2 \int_{\hat{s}_2}^{\hat{s}_2+\hat{h}_2/v_2} \int_{(\hat{h}_2^2 - v_2^2(t_1 - \hat{s}_2)^2)^{\frac{1}{2}}}^{\infty} e^{-\rho(\hat{s}(-t',h',t_1,h_1)-\hat{s}_2)} e^{-2\lambda_1 v_1} \left| E_{\hat{h}(-t',h',t_1,h_1)}^{\hat{s}(-t',h',t_1,h_1),v_1} \cup S_2 \cup E_{\hat{h}_2}^{\hat{s}_2,v_1} \right| \\
&\qquad \times e^{-2\lambda_2 v_2} \left| E_{\hat{h}(-t',h',t_1,h_1)}^{\hat{s}(-t',h',t_1,h_1),v_2} \cup E_{\hat{h}_2}^{\hat{s}_2,v_2} \right| \mathrm{d}t_1 \mathrm{d}h_1 \\
&\quad + 2\lambda_2 v_2 \int_{\hat{s}_2+\hat{h}_2/v_2}^{\infty} \int_{0}^{\infty} e^{-\rho(\hat{s}(-t',h',t_1,h_1)-\hat{s}_2)} e^{-2\lambda_1 v_1} \left| E_{\hat{h}(-t',h',t_1,h_1)}^{\hat{s}(-t',h',t_1,h_1),v_1} \cup S_2 \cup E_{\hat{h}_2}^{\hat{s}_2,v_1} \right| \\
&\qquad \times e^{-2\lambda_2 v_2} \left| E_{\hat{h}(-t',h',t_1,h_1)}^{\hat{s}(-t',h',t_1,h_1),v_2} \cup E_{\hat{h}_2}^{\hat{s}_2,v_2} \right| \mathrm{d}t_1 \mathrm{d}h_1. \tag{8.77}
\end{align*}
$$

Let us denote the sum of all the terms in (8.75) and (8.77) as $\zeta_{1,2}^{(2)}(\rho, t', h', h) = \zeta\left[\begin{smallmatrix} 2,1 \\ 1,2,1 \end{smallmatrix}\right] + \zeta\left[\begin{smallmatrix} 2,1 \\ 1,2,2 \end{smallmatrix}\right]$.
Using the expression for $\zeta_{1,2}^{(2)}(\rho, t', h', h)$ back in (8.70) yields

$$
\begin{align}
\frac{L_{1,2}^{(2)}}{4\lambda_1\lambda_2 v_1 v_2} \mathbb{E}_{W_{1,2}^{(2)}}[e^{-\rho T}] &= \int_0^\infty \int_0^h \int_{t^*}^\infty \zeta_{1,2}^{(2)}(\rho, t', h', h) \mathrm{d}t' \mathrm{d}h' \mathrm{d}h + \int_0^\infty \int_h^\infty \int_0^\infty \zeta_{1,2}^{(2)}(\rho, t', h', h) \mathrm{d}t' \mathrm{d}h' \mathrm{d}h \nonumber \\
&:= \xi_{1,2}^{(2)}(\rho, v_2, v_1), \tag{8.78}
\end{align}
$$

which is a function of $\rho, v_2, v_1$ only. $\square$

8.3.9. *Proof of Lemma 8.20, type (1) or equivalently $[\begin{smallmatrix} 2 \\ 2,1 \end{smallmatrix}]$.*

*Proof.* Using Lemma 8.4, for the mixed Palm expectations in (8.38) with $\tau_n = 2, \tau_p = 1, q = 2$ and $f(\mathcal{H}) = e^{-\rho T}$, we have

$$
\begin{align}
\frac{L_{2,1}^{(2)}}{4\lambda_1\lambda_2 v_1 v_2} \mathbb{E}_{W_{2,1}^{(2)}}[e^{-\rho T}] &= \int_0^\infty \int_0^h \int_{t^*}^\infty \mathbb{E}_{\mathcal{H}} \left[ e^{-\rho(T \circ \theta_{\hat{s}_1})} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}_1}^{\hat{s}_1, v_l} \right) = 0 \right] \mathrm{d}t' \mathrm{d}h' \mathrm{d}h \nonumber \\
&\quad + \int_0^\infty \int_h^\infty \int_0^\infty \mathbb{E}_{\mathcal{H}} \left[ e^{-\rho(T \circ \theta_{\hat{s}_1})} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}_1}^{\hat{s}_1, v_l} \right) = 0 \right] \mathrm{d}t' \mathrm{d}h' \mathrm{d}h. \tag{8.79}
\end{align}
$$

where $(\hat{s}_1, \hat{h}_1) = (\hat{s}_1(0, h, -t', h'), \hat{h}_1(0, h, -t', h'))$ is the first intersection of the birds at $(t_p, h_p) = (0, h)$ and $(t_n, h_n) = (-t', h')$, which are of type $\tau_p = 2$ and $\tau_n = 1$, respectively.

The station corresponding to the head point $(-t', h')$ is the serving station after time $\hat{s}_1$. Since $(-t', h')$ is of type 1, we define $\hat{h}_1(t) := (v_1^2(t+t')^2 + h'^2)^{\frac{1}{2}}$, which write in short as $\hat{h}_t$. Observe that

$$T \circ \theta_{\hat{s}_1} = T_1(\hat{s}_1) - \hat{s}_1,$$
where $T_1(\hat{s}_1)$ is the time point of the first handover after time $\hat{s}_1$. The intersection point $(\hat{s}_1, \hat{h}_1)$ corresponding to the typical handover in this case, the second intersection $(\hat{s}_2, \hat{h}_2)$ will correspond to the next handover, if there is no radial bird that intersects the bird at $(0, h)$ before time $\hat{s}_2$, i.e., $T_1(\hat{s}_1) < \hat{s}_2$ or the next handover happens at time $\hat{s}_2$, in which case $T_1(\hat{s}_1) = \hat{s}_2$. The reason for the next handover to happen at most before $\hat{s}_2$, will be clear later in the following cases.

**Case 1.** $T_1(\hat{s}_1) < \hat{s}_2$. The next handover can be given by a bird with head point, say $(T_j, H_j)$, outside the region $E_{\hat{h}_1}^{\hat{s}_1, v_1}$ or $E_{\hat{h}_1}^{\hat{s}_1, v_2}$, depending on its type. Thus we have two sub-cases:

**Subcase 1.1.** $(T_j, H_j)$ is of type 1. In this case, $(T_j, H_j)$ must be at least on the right of $(-t', h')$, i.e., $T_j \ge -t'$. Otherwise, the next radial bird with head point $(T_j, H_j)$ being outside $E_{\hat{h}_1}^{\hat{s}_1, v_1}$ and $T_j < -t'$, intersects the one with head at $(-t', h')$, at a point that is either contained in $\mathcal{L}_e^+(v_1, v_2)$ or contributes to a handover before time $\hat{s}_1$, which is not of interest. Then, we must have $T_j \ge -t'$ and the unexplored region is $Q_{-t'} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_1}$. The type of the pair of consecutive handovers is $[2, 1]$. There is just one intersection, say $(\hat{s}(-t', h', T_j, H_j), \hat{h}(-t', h', T_j, H_j))$ between the radial birds at $(-t', h')$ and $(T_j, H_j)$ and handover is given by that, see Picture (1) of Figure 8.10. Thus we have $T_1(\hat{s}_1) := \hat{s}(-t', h', T_j, H_j)$. In this case, the extra region $(\bigcup_{t \in [\hat{s}_1, \hat{s}(-t', h', T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}}^{\hat{s}_1, v_l})^\circ$ must have no points of $\mathcal{H}^l$ for $l=1,2$. The union is due to Definition 8.13 and to the fact that the individual sets in the union may lack monotonicity.

**Subcase 1.2.** $(T_j, H_j)$ is of type 2. Here it must be the case that $T_j \ge 0$. Otherwise, if $T_j < 0$, then the two birds with head point $(T_j, H_j)$ and $(0, h)$ intersect at a point before time $\hat{s}_1$, which is not relevant in this case. Then, we must have $T_j \ge 0 \ge -t'$ and the unexplored region is $Q_0 \setminus E_{\hat{h}_1}^{\hat{s}_1, v_2}$. The type of the pair of consecutive handovers is $[2, 1, 2]$. By Lemma 7.12, there are two intersections $(\hat{s}_1(-t', h', T_j, H_j), \hat{h}_1(-t', h', T_j, H_j))$ and $(\hat{s}_2(-t', h', T_j, H_j), \hat{h}_2(-t', h', T_j, H_j))$ between the birds at $(-t', h')$, $(T_j, H_j)$. The first intersection cannot give rise to a handover, as it is contained in $\mathcal{L}_e^+(v_1, v_2)$. So the next handover must be given by the second intersection, see Picture (2) of Figure 8.10. Thus we have $T_1(\hat{s}_1) := \hat{s}_2(-t', h', T_j, H_j)$. The last intersection to give a handover, the extra region $(\bigcup_{t \in [\hat{s}_1, \hat{s}_2(-t', h', T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_l})^\circ$ must have no points of $\mathcal{H}^l$ for $l=1,2$. The union is due to Definition 8.13 and the fact that individual sets in the union potentially lack monotonicity.

<figure><img src="image_50.png" /></figure>

(1) For the intersections $(\hat{s}_1, \hat{h}_1)$ and $(\tilde{s}, \tilde{h})$ to represent two consecutive handovers, the regions $E_{\hat{h}_1}^{\hat{s}_1, v_1} \cup E_{\hat{h}}^{\hat{s}_1, v_1}$ and $E_{\hat{h}_1}^{\hat{s}_1, v_2} \cup E_{\hat{h}}^{\hat{s}_1, v_2}$ must have no points from $\mathcal{H}^1$ and $\mathcal{H}^2$, respectively. Here $(\tilde{s}, \tilde{h})$ is a realization of $(\hat{s}(-t', h', T_j, H_j), \hat{h}(-t', h', T_j, H_j))$.

(2) For the intersections $(\hat{s}_1, \hat{h}_1)$ and $(\tilde{s}, \tilde{h})$ to represent two consecutive handovers, the regions $E_{\hat{h}_1}^{\hat{s}_1, v_1} \cup E_{\hat{h}}^{\hat{s}_1, v_1}$ and $E_{\hat{h}_1}^{\hat{s}_1, v_2} \cup E_{\hat{h}}^{\hat{s}_1, v_2}$ must have no points from $\mathcal{H}^1$ and $\mathcal{H}^2$, respectively. Here $(\tilde{s}, \tilde{h})$ is a realization of $(\hat{s}_2(-t', h', T_j, H_j), \hat{h}_2(-t', h', T_j, H_j))$.

<caption>FIGURE 8.10. The cases corresponding to the first and second term in (8.80), when the head point $(T_j, H_j) = (t_1, h_1)$ is of type 1 and type 2, respectively.</caption>

**Case 2.** $T_1(\hat{s}_1) = \hat{s}_2$. In this case, the next handover is given by the second intersection $(\hat{s}_2, \hat{h}_2)$ of the radial birds with head points $(0, h), (-t', h')$ and the extra region $(\bigcup_{t \in [\hat{s}_1, \hat{s}_2]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_l})^\circ$ has no point of $\mathcal{H}^l$, for $l=1,2$, see Figure 8.11. The next handover in question happens as late as $\hat{s}_2$, since
for any $\hat{s} > \hat{s}_2$

$$ \bigcup_{t \in [\hat{s}_1, \hat{s}_2]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_l} \subset \bigcup_{t \in [\hat{s}_1, \hat{s}]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_l}. $$

Thus the type of the pair of consecutive handovers is $[ \begin{smallmatrix} 2, 1 \\ 2, 1, 2^* \end{smallmatrix} ]$.

Based on last discussions we have the following decomposition of the inner expectation in (8.79),

$$
\begin{align}
& \mathbb{E}_{\mathcal{H}} \left[ e^{-\rho(T_1(\hat{s}_1) - \hat{s}_1)} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} (E_{\hat{h}_1}^{\hat{s}_1, v_l}) = 0 \right] \nonumber \\
&= \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\exists(T_j, H_j) \in \mathcal{H}^1: T_j \ge -t', \cap_{l=1,2} \{\mathcal{H}^l \left( \bigcup_{t \in [\hat{s}_1, \hat{s}(-t',h',T_j,H_j)] E_{\hat{h}_t}^{t,v_l} \setminus E_{\hat{h}}^{\hat{s},v_l} \right)^{\circ} = 0 \}} \mathbb{1}_{\hat{s}_1 \le \hat{s}(-t',h',T_j,H_j) \le \hat{s}_2} \right. \nonumber \\
& \qquad \left. \times e^{-\rho(\hat{s}(-t',h',T_j,H_j) - \hat{s}_1)} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} (E_{\hat{h}_1}^{\hat{s}_1, v_l}) = 0 \right] \nonumber \\
& \quad + \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\exists(T_j, H_j) \in \mathcal{H}^2: T_j \ge 0, \cap_{l=1,2} \{\mathcal{H}^l \left( \bigcup_{t \in [\hat{s}_1, \hat{s}_2(-t',h',T_j,H_j)] E_{\hat{h}_t}^{t,v_l} \setminus E_{\hat{h}}^{\hat{s},v_l} \right)^{\circ} = 0 \}} \mathbb{1}_{\hat{s}_1 \le \hat{s}_2(-t',h',T_j,H_j) \le \hat{s}_2} \right. \nonumber \\
& \qquad \left. \times e^{-\rho(\hat{s}_2(-t',h',T_j,H_j) - \hat{s}_1)} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} (E_{\hat{h}_1}^{\hat{s}_1, v_l}) = 0 \right] \nonumber \\
& \quad + \mathbb{E}_{\mathcal{H}} \left[ e^{-\rho(\hat{s}_2 - \hat{s}_1)} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( \bigcup_{t \in [\hat{s}_1, \hat{s}_2]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_l} \right)^{\circ} = 0 \right. \nonumber \\
& \qquad \left. \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} (E_{\hat{h}_1}^{\hat{s}_1, v_l}) = 0 \right] \tag{8.80}
\end{align}
$$

where the pair type marked with * means that we have recurrence of the previous station as an upcoming station, seen in the last column of Table 8.2. Similarly to (8.63), there exists a unique head point satisfying the condition in each term in (8.80), and hence the sum can be written as

$$
\begin{align}
& \mathbb{E}_{\mathcal{H}} \left[ \sum_{(T_j, H_j) \in \mathcal{H}^1: T_j \ge -t'} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l \left( \bigcup_{t \in [\hat{s}_1, \hat{s}(-t',h',T_j,H_j)] E_{\hat{h}_t}^{t,v_l} \setminus E_{\hat{h}}^{\hat{s},v_l} \right)^{\circ} = 0} \mathbb{1}_{\hat{s}_1 \le \hat{s}(-t',h',T_j,H_j) \le \hat{s}_2} \right. \nonumber \\
& \qquad \left. \times e^{-\rho(\hat{s}(-t',h',T_j,H_j) - \hat{s}_1)} \right] \nonumber \\
& \quad + \mathbb{E}_{\mathcal{H}} \left[ \sum_{(T_j, H_j) \in \mathcal{H}^2: T_j \ge 0} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l \left( \bigcup_{t \in [\hat{s}_1, \hat{s}_2(-t',h',T_j,H_j)] E_{\hat{h}_t}^{t,v_l} \setminus E_{\hat{h}}^{\hat{s},v_l} \right)^{\circ} = 0} \mathbb{1}_{\hat{s}_1 \le \hat{s}_2(0,h,T_j,H_j) \le \hat{s}_2} \right. \nonumber \\
& \qquad \left. \times e^{-\rho(\hat{s}_1 \le \hat{s}(0,h,T_j,H_j) - \hat{s}_1)} \right] \nonumber \\
& \quad + \mathbb{E}_{\mathcal{H}} \left[ e^{-\rho(\hat{s}_2 - \hat{s}_1)} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l \left( \bigcup_{t \in [\hat{s}_1, \hat{s}_2]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_l} \right)^{\circ} = 0} \right]. \tag{8.81}
\end{align}
$$

For the first term in (8.81), by part (i) of Proposition 8.16, the increasing property of $\{E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_l}\}_{t \ge \hat{s}_1}$, we have

$$ \bigcup_{t \in [\hat{s}_1, \hat{s}(-t',h',T_j,H_j)]} E_{\hat{h}_t}^{t, v_l} \cup E_{\hat{h}_1}^{\hat{s}_1, v_l} = E_{\hat{h}(-t',h',T_j,H_j)}^{\hat{s}(-t',h',T_j,H_j),v_l} \cup E_{\hat{h}_1}^{\hat{s}_1, v_l}. $$
<figure><img src="image_51.png" /><figcaption>FIGURE 8.11. The case corresponding to the last term in (8.80). For the intersections $(\hat{s}_1, \hat{h}_1)$ and $(\hat{s}_2, \hat{h}_2)$ to represent two consecutive handovers, the regions $E_{\hat{h}_1}^{\hat{s}_1, v_1} \cup E_{\hat{h}_2}^{\hat{s}_2, v_1}$ and $E_{\hat{h}_1}^{\hat{s}_1, v_2} \cup E_{\hat{h}_2}^{\hat{s}_2, v_2}$ must have no points from $\mathcal{H}^1$ and $\mathcal{H}^2$, respectively.</figcaption></figure>

The point $(T_j, H_j)$ must lie in the region $R_1(h, t', h') \subset [-t', \infty) \times \mathbb{R}^+ \setminus E_{\hat{h}_1}^{\hat{s}_1, v_1}$ be the region defined as

$$R_1(h, t', h') := \{(t, u) : \hat{s}_1 \le \hat{s}(-t', h', t, u) \le \hat{s}_2\},$$

which contains such a possible head point, as seen in Subcase 1.1. Using Lemma 7.13 it can be proved that the region $R_1(h, t', h') = E_{\hat{h}_2}^{\hat{s}_2, v_1} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_1}$. By the Campbell-Mecke formula, after implementing the last discussions, the first term in (8.81) equals

$$ \begin{aligned} \zeta\left[\begin{smallmatrix} 2, 1 \\ 2, 1, 1 \end{smallmatrix}\right] &= 2\lambda_1 v_1 \int_{E_{\hat{h}_2}^{\hat{s}_2, v_1} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_1}} e^{-\rho(\hat{s}(-t', h', t_1, h_1)-\hat{s}_1)} \mathbb{E}_{\mathcal{H}}\left[\prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l\left(E_{\hat{h}(-t', h', t_1, h_1), v_l \cup E_{\hat{h}_1}^{\hat{s}_1, v_l}}\right)}=0\right] dt_1 dh_1 \\ &= 2\lambda_1 v_1 \int_{E_{\hat{h}_2}^{\hat{s}_2, v_1} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_1}} e^{-\rho(\hat{s}(-t', h', t_1, h_1)-\hat{s}_1)} e^{-\sum_{l=1,2} 2\lambda_l v_l |E_{\hat{h}(-t', h', t_1, h_1), v_l \cup E_{\hat{h}_1}^{\hat{s}_1, v_l}}|} dt_1 dh_1. \end{aligned} \quad (8.82) $$

For the second term in (8.81), by the increasing property of $\{E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_l}\}_{t \ge \hat{s}_1}$ from part (i) of Proposition 8.16, we have

$$ \bigcup_{t \in [\hat{s}_1, \hat{s}_2(-t', h', T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \cup E_{\hat{h}_1}^{\hat{s}_1, v_l} = E_{\hat{h}_2(-t', h', T_j, H_j)}^{\hat{s}_2(-t', h', T_j, H_j), v_l} \cup E_{\hat{h}_1}^{\hat{s}_1, v_l}. $$

The point $(T_j, H_j)$ must lie in the region $R_2(h, t', h') \subset (\mathbb{R}^+)^2 \setminus E_{\hat{h}_1}^{\hat{s}_1, v_2}$ be the region defined as

$$R_2(h, t', h') := \{(t, u) : \hat{s}_1 \le \hat{s}_2(-t', h', t, u) \le \hat{s}_2\}.$$

that contains the required head points, as discussed in Subcase 1.2. It can be proved using Lemma 7.13, that the region $R_2(h, t', h') = E_{\hat{h}_2}^{\hat{s}_2, v_2} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_2}$. Using this along with the application of the Campbell-Mecke formula, we get that the second term in (8.81) equals

$$ \zeta\left[\begin{smallmatrix} 2, 1 \\ 2, 1, 2 \end{smallmatrix}\right] = 2\lambda_2 v_2 \int_{E_{\hat{h}_2}^{\hat{s}_2, v_2} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_2}} e^{-\rho(\hat{s}_2(-t', h', t_1, h_1)-\hat{s}_1)} \mathbb{E}_{\mathcal{H}}\left[\prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l\left(E_{\hat{h}_2(-t', h', t_1, h_1), v_l \cup E_{\hat{h}_1}^{\hat{s}_1, v_l}}\right)}=0\right] dt_1 dh_1. \quad (8.83) $$

Using void probabilities in (8.83) we obtain that

$$ \zeta\left[\begin{smallmatrix} 2, 1 \\ 2, 1, 2 \end{smallmatrix}\right] = 2\lambda_2 v_2 \int_{E_{\hat{h}_2}^{\hat{s}_2, v_2} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_2}} e^{-\rho(\hat{s}_2(-t', h', t_1, h_1)-\hat{s}_1)} e^{-\sum_{l=1,2} 2\lambda_l v_l |E_{\hat{h}_2(-t', h', t_1, h_1), v_l \cup E_{\hat{h}_1}^{\hat{s}_1, v_l}}|} dt_1 dh_1. \quad (8.84) $$

For the third term in (8.81), we have

$$ \bigcup_{t \in [\hat{s}_1, \hat{s}_2]} E_{\hat{h}_t}^{t, v_l} \cup E_{\hat{h}_1}^{\hat{s}_1, v_l} = E_{\hat{h}_2(-t', h', T_j, H_j)}^{\hat{s}_2, v_l} \cup E_{\hat{h}_1}^{\hat{s}_1, v_l}, $$
and hence this third term equals

$$ \zeta\left[\begin{smallmatrix} 2, 1 \\ 2, 1, 2^* \end{smallmatrix}\right] = e^{-\rho(\hat{s}_2 - \hat{s}_1)} e^{-\sum_{l=1,2} 2\lambda_l v_l \left|E_{\hat{h}_2}^{\hat{s}_2, v_l} \cup E_{\hat{h}_1}^{\hat{s}_1, v_l}\right|} \quad (8.85) $$

Let us denote the sum of all the terms in (8.82)-(8.85) as $\zeta_{2,1}^{(2)}(\rho, t', h', h) = \zeta\left[\begin{smallmatrix} 2, 1 \\ 2, 1, 1 \end{smallmatrix}\right] + \zeta\left[\begin{smallmatrix} 2, 1 \\ 2, 1, 2 \end{smallmatrix}\right] + \zeta\left[\begin{smallmatrix} 2, 1 \\ 2, 1, 2^* \end{smallmatrix}\right]$. Using the expression for $\zeta_{2,1}^{(2)}(\rho, t', h', h)$ in (8.79) yields

$$ \frac{L_{2,1}^{(2)}}{4\lambda_1\lambda_2 v_1 v_2} \mathbb{E}_{W_{2,1}^{(2)}}[e^{-\rho T}] = \int_0^\infty \int_0^h \int_{t^*}^\infty \zeta_{2,1}^{(2)}(\rho, t', h', h) dt' dh' dh + \int_0^\infty \int_h^\infty \int_0^\infty \zeta_{2,1}^{(2)}(\rho, t', h', h) dt' dh' dh \\ := \xi_{2,1}^{(2)}(\rho, v_1, v_2), \quad (8.86) $$

which is a function of $\rho, v_1, v_2$ only. $\square$

8.3.10. *Proof of Lemma 8.20, type (2) or equivalently $\begin{bmatrix} 1 \\ 1, 2 \end{bmatrix}$.*

*Proof.* Using Lemma 8.4, the mixed Palm expectations in (8.38) with $\tau_n = 2, \tau_p = 1, q = 1$, and $f(\mathcal{H}) = e^{-\rho T}$, we have

$$ \frac{L_{1,2}^{(1)}}{4\lambda_1\lambda_2 v_1 v_2} \mathbb{E}_{W_{1,2}^{(1)}}[e^{-\rho T}] = \int_0^\infty \int_0^h \int_{t^*}^\infty \mathbb{E}_{\mathcal{H}} \left[ e^{-\rho(T \circ \theta_{\hat{s}_2})} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}_2}^{\hat{s}_2, v_l} \right) =0 \right] dt' dh' dh \\ + \int_0^\infty \int_h^\infty \int_0^\infty \mathbb{E}_{\mathcal{H}} \left[ e^{-\rho(T \circ \theta_{\hat{s}_2})} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}_2}^{\hat{s}_2, v_l} \right) =0 \right] dt' dh' dh. \quad (8.87) $$

where $(\hat{s}_2, \hat{h}_2) = (\hat{s}_2(0, h, -t', h'), \hat{h}_2(0, h, -t', h'))$ is the second intersection of the birds at $(t_n, h_n) = (0, h)$, $(t_p, h_p) = (-t', h')$, which are of type $\tau_n = 2$ and $\tau_p = 1$, respectively. We write this intersection $(\hat{s}_2, \hat{h}_2)$ in short. Note that the station corresponding to the head point $(0, h)$ is the serving station after time $\hat{s}_2$. Since $(0, h)$ is of type 2, we define $\hat{h}_2(t) := (v_2^2 t^2 + h^2)^{\frac{1}{2}}$, which we write in short as $\hat{h}_t$. Notice that

$$ T \circ \theta_{\hat{s}_2} := T_1(\hat{s}_2) - \hat{s}_2, $$

where $T_1(\hat{s}_2)$ is the handover that happens for the first time after time $\hat{s}_2$. The next handover point can be found by looking for the first head point, say $(T_j, H_j)$, the radial bird at which, intersects the one with head point $(0, h)$. Depending on the type of $(T_j, H_j)$ there are two cases as follows:

**Case 1.** $(T_j, H_j)$ is of type 1. For the point $(T_j, H_j)$ to be eligible for a handover, it must be outside $E_{\hat{h}_2}^{\hat{s}_2, v_1}$ and satisfy $T_j \ge \hat{s}_2$. Otherwise, if $T_j < \hat{s}_2$, then the intersection between the birds with head at $(T_j, H_j)$ and $(0, h)$, is either contained in $\mathcal{L}_e^+(v_1, v_2)$ or it produces a handover before time $\hat{s}_2$. So it must be the case that $T_j \ge \hat{s}_2$ and the unexplored region is $Q_{\hat{s}_2} \setminus E_{\hat{h}_2}^{\hat{s}_2, v_1}$.

In addition to that, if $\hat{s}_2 < 0$, we must have $T_j \in D_\ell(0, h, H_j)$ or $T_j \in D_r(0, h, H_j)$, when $\hat{s}_2 \le T_j \le 0$ and the type of the pair of consecutive handovers can be $\begin{bmatrix} 1, 2 \\ 1, 2, 1 \end{bmatrix}$ or $\begin{bmatrix} 1, 1 \\ 1, 2, 1 \end{bmatrix}$, depending on whether $\hat{s}_2 \le T_j < 0$ or $T_j \ge 0$. If $\hat{s}_2 \ge 0$, we must have $T_j \in D_r(0, h, H_j)$ and the type of the pair of consecutive handovers can be $\begin{bmatrix} 1, 1 \\ 1, 2, 1 \end{bmatrix}$ only. Suppose the two intersection points are $(s_1(0, h, T_j, H_j), \hat{h}_1(0, h, T_j, H_j))$ and $(s_2(0, h, T_j, H_j), \hat{h}_2(0, h, T_j, H_j))$, both of which can be eligible for handovers. For the two intersection points to represent handovers we must have no point of $\mathcal{H}^l$, for $l = 1, 2$, in the extra regions $(\bigcup_{t \in [\hat{s}_2, \hat{s}_1(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_2}^{\hat{s}_2, v_l})^\circ$ and $(\bigcup_{t \in [\hat{s}_2, \hat{s}_2(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_2}^{\hat{s}_2, v_l})^\circ$, respectively, beyond $E_{\hat{h}_2}^{\hat{s}_2, v_l}$. Since $\hat{s}_1(0, h, T_j, H_j) \le \hat{s}_2(0, h, T_j, H_j)$, we have

$$ \bigcup_{t \in [\hat{s}_2, \hat{s}_1(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_2}^{\hat{s}_2, v_l} \subset \bigcup_{t \in [\hat{s}_2, \hat{s}_2(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_2}^{\hat{s}_2, v_l}, $$
Hence, we choose the first intersection to be responsible for the next handover. Thus we have $T_1(\hat{s}_2) := \hat{s}_1(0, h, T_j, H_j)$ and the extra region $(\bigcup_{t \in [\hat{s}_2, \hat{s}_1(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_2}^{\hat{s}_2, v_l})^\circ$ has no points of $\mathcal{H}^l$, for $l = 1, 2$, where $\hat{h}_t = (v_2^2 t^2 + h^2)^{\frac{1}{2}}$, see Picture (1) of Figure 8.12. This union is due to Definition 8.13 and to the fact that the individual sets in the union may not be monotonic.

**Case 2.** ($T_j, H_j$) is of type 2. For the point $(T_j, H_j)$ to be eligible, it must be from outside of $E_{\hat{h}_2}^{\hat{s}_2, v_2}$ and satisfy $T_j \ge 0$. Otherwise if $T_j < 0$, the two birds with heads at $(T_j, H_j)$ and $(0, h)$ intersect at a point which is contained in $\mathcal{L}_e^+(v_1, v_2)$ or produce a handover before time $\hat{s}_2$, which is not interesting in this case. Thus we must have $T_j \ge 0$ and the unexplored region is $Q_0 \setminus E_{\hat{h}_2}^{\hat{s}_2, v_2}$. The type of the pair of consecutive handovers is $[1, 1, 2, 2]$. Hence we have $T_1(\hat{s}_2) := \hat{s}(0, h, T_j, H_j)$ and the region $(\bigcup_{t \in [\hat{s}_2, \hat{s}(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_2}^{\hat{s}_2, v_l})^\circ$ has no points of $\mathcal{H}^l$, for $l = 1, 2$, where $\hat{h}_t = (v_2^2 t^2 + h^2)^{\frac{1}{2}}$, see Picture (2) of Figure 8.12.

Based on the reasoning in Case 1 and Case 2 we can decompose the inner expectation in (8.87) as:

$$
\begin{align}
& \mathbb{E}_{\mathcal{H}} \left[ e^{-\rho(T_1(\hat{s}_2) - \hat{s}_2)} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}_2}^{\hat{s}_2, v_l} \right) = 0 \right] \nonumber \\
&= \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\exists(T_j, H_j) \in \mathcal{H}^1: \hat{s}_2 \le T_j \le 0, T_j \in D_\ell(0, h, H_j), \cap_{l=1,2} \left\{ \mathcal{H}^l \left( \bigcup_{t \in [\hat{s}_2, \hat{s}_1(0, h, T_j, H_j)] E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_2}^{\hat{s}_2, v_l} \right)^\circ = 0 \right\}} e^{-\rho(\hat{s}_1(-t', h', T_j, H_j) - \hat{s}_2)} \right. \nonumber \\
& \qquad \left. \times \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}_2}^{\hat{s}_2, v_l} \right) = 0 \right] \nonumber \\
&\quad + \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\exists(T_j, H_j) \in \mathcal{H}^1: T_j \in D_r(0, h, H_j), \cap_{l=1,2} \left\{ \mathcal{H}^l \left( \bigcup_{t \in [\hat{s}_2, \hat{s}_1(0, h, T_j, H_j)] E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_2}^{\hat{s}_2, v_l} \right)^\circ = 0 \right\}} e^{-\rho(\hat{s}_1(-t', h', T_j, H_j) - \hat{s}_2)} \right. \nonumber \\
& \qquad \left. \times \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}_2}^{\hat{s}_2, v_l} \right) = 0 \right] \nonumber \\
&\quad + \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\exists(T_j, H_j) \in \mathcal{H}^2: T_j \ge 0, \cap_{l=1,2} \left\{ \mathcal{H}^l \left( \bigcup_{t \in [\hat{s}_2, \hat{s}(0, h, T_j, H_j)] E_{\hat{h}_t}^{t, v_l} \setminus E_{\hat{h}_2}^{\hat{s}_2, v_l} \right)^\circ = 0 \right\}} e^{-\rho(\hat{s}(0, h, T_j, H_j) - \hat{s}_2)} \right. \nonumber \\
& \qquad \left. \times \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( E_{\hat{h}_2}^{\hat{s}_2, v_l} \right) = 0 \right] . \tag{8.88}
\end{align}
$$

There exists a unique head point satisfying the condition in each term of last sum in (8.88), and hence it can be written as

$$
\begin{align}
& \mathbb{E}_{\mathcal{H}} \left[ \sum_{(T_j, H_j) \in \mathcal{H}^1: \hat{s}_2 \le T_j \le 0} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( \bigcup_{t \in [\hat{s}_2, \hat{s}_1(0, h, T_j, H_j)] E_{\hat{h}_t}^{t, v_l} \cup E_{\hat{h}_2}^{\hat{s}_2, v_l}} e^{-\rho(\hat{s}_1(0, h, T_j, H_j) - \hat{s}_2)} \right) \right] \nonumber \\
& + \mathbb{E}_{\mathcal{H}} \left[ \sum_{(T_j, H_j) \in \mathcal{H}^1: T_j \in D_r(0, h, H_j)} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( \bigcup_{t \in [\hat{s}_2, \hat{s}_1(0, h, T_j, H_j)] E_{\hat{h}_t}^{t, v_l} \cup E_{\hat{h}_2}^{\hat{s}_2, v_l}} e^{-\rho(\hat{s}_1(0, h, T_j, H_j) - \hat{s}_2)} \right) \right] \nonumber \\
& + \mathbb{E}_{\mathcal{H}} \left[ \sum_{(T_j, H_j) \in \mathcal{H}^2: T_j \ge 0} \prod_{l=1,2} \mathbb{1}_{\mathcal{H}^l} \left( \bigcup_{t \in [\hat{s}_2, \hat{s}(0, h, T_j, H_j)] E_{\hat{h}_t}^{t, v_l} \cup E_{\hat{h}_2}^{\hat{s}_2, v_l}} e^{-\rho(\hat{s}(0, h, T_j, H_j) - \hat{s}_2)} \right) \right] \tag{8.89} \\
&:= \zeta \left[ \begin{smallmatrix} 1, 2 \\ 1, 2, 1 \end{smallmatrix} \right] + \zeta \left[ \begin{smallmatrix} 1, 1 \\ 1, 2, 1 \end{smallmatrix} \right] + \zeta \left[ \begin{smallmatrix} 1, 1 \\ 1, 2, 2 \end{smallmatrix} \right]. \tag{8.90}
\end{align}
$$
<figure><img src="image_52.png" /></figure>

(1) For the intersections $(\hat{s}_2, \hat{h}_2)$ and $(\tilde{s}_1, \tilde{h}_1)$ to represent two consecutive handovers, the regions $S_1 \cup E_{\hat{h}_2}^{\hat{s}_2, v_1} \cup E_{\tilde{h}_1}^{\tilde{s}, v_1}$ and $E_{\hat{h}_2}^{\hat{s}_2, v_2} \cup E_{\tilde{h}_1}^{\tilde{s}, v_2}$ must have no points from $\mathcal{H}^1$ and $\mathcal{H}^2$, respectively. Here $(\tilde{s}, \tilde{h})$ is a realization of $(\hat{s}_1(0, h, T_j, H_j), \hat{h}_1(0, h, T_j, H_j))$.

<figure><img src="image_53.png" /></figure>

(2) For the intersections $(\hat{s}_2, \hat{h}_2)$ and $(\tilde{s}_1, \tilde{h}_1)$ to represent two consecutive handovers, the regions $S_2 \cup E_{\hat{h}_2}^{\hat{s}_2, v_1} \cup E_{\tilde{h}_1}^{\tilde{s}_1, v_1}$ and $E_{\hat{h}_2}^{\hat{s}_2, v_2} \cup E_{\tilde{h}_1}^{\tilde{s}_1, v_2}$ must have no points from $\mathcal{H}^1$ and $\mathcal{H}^2$, respectively. Here $(\tilde{s}_1, \tilde{h}_1)$ is a realization of $(\hat{s}(0, h, T_j, H_j), \hat{h}(0, h, T_j, H_j))$.

FIGURE 8.12. Case 1 and Case 2, when the head point $(T_j, H_j) = (t_1, h_1)$ is of type 1 or 2, respectively.

Using Remark 7.10 for the first two terms in (8.89) for $l=1$, we have

$$ \bigcup_{t \in [\hat{s}_2, \hat{s}_1(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_1} \cup E_{\hat{h}_2}^{\hat{s}_2, v_1} = E_{\hat{h}_2}^{\hat{s}_2, v_1} \cup S_1 \cup E_{\hat{h}_1(0, h, T_j, H_j)}^{\hat{s}_1(0, h, T_j, H_j), v_1}, $$

where $\hat{h}_t = (v_2^2 t^2 + h^2)^{1/2}$ and $S_1$ is the region outside $E_{\hat{h}_2}^{\hat{s}_2, v_1} \cup E_{\hat{h}_1(0, h, T_j, H_j)}^{\hat{s}_1(0, h, T_j, H_j), v_1}$, but below the hyperbola given by the equation $u^2 - \frac{v_1^2 v_2^2}{v_1^2 - v_2^2} t^2 = h^2$, in the $(t, u)$-coordinate system similar to (7.17) and within the time interval

$$ \left[ \frac{v_1^2 - v_2^2}{v_1^2} \hat{s}_2, \frac{v_1^2 - v_2^2}{v_1^2} \hat{s}_1(0, h, T_j, H_j) \right]. $$

For $l=2$, using part (ii) of Proposition 8.16, the increasing property of $\{E_{\hat{h}_t}^{t, v_2} \setminus E_{\hat{h}}^{\hat{s}, v_2}\}_{t \ge \hat{s}_2}$, we have

$$ \bigcup_{t \in [\hat{s}_2, \hat{s}_1(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_2} \cup E_{\hat{h}}^{\hat{s}, v_2} = E_{\hat{h}_1(0, h, T_j, H_j)}^{\hat{s}_1(0, h, T_j, H_j), v_2} \cup E_{\hat{h}}^{\hat{s}, v_2}. $$

The first two terms in (8.90) become

$$ \begin{aligned} & \zeta \begin{bmatrix} 1, 2 \\ 1, 2, 1 \end{bmatrix} + \zeta \begin{bmatrix} 1, 1 \\ 1, 2, 1 \end{bmatrix} \\ &= \mathbb{E}_{\mathcal{H}} \left[ \sum_{(T_j, H_j) \in \mathcal{H}^1 : \hat{s}_2 \le T_j \le 0, T_j \in D_\ell(0, h, H_j)} \mathbb{1}_{\mathcal{H}^1} \left( E_{\hat{h}_2}^{\hat{s}_2, v_1} \cup S_1 \cup E_{\hat{h}_1(0, h, T_j, H_j)}^{\hat{s}_1(0, h, T_j, H_j), v_1} \right) = 0 \right] \mathbb{1}_{\mathcal{H}^2} \left( E_{\hat{h}_2}^{\hat{s}_2, v_2} \cup E_{\hat{h}_1(0, h, T_j, H_j)}^{\hat{s}_1(0, h, T_j, H_j), v_2} \right) = 0 \\ & \qquad \times e^{-\rho(\hat{s}_1(0, h, T_j, H_j) - \hat{s}_2)} \right] \\ & + \mathbb{E}_{\mathcal{H}} \left[ \sum_{(T_j, H_j) \in \mathcal{H}^1 : T_j \in D_r(0, h, H_j)} \mathbb{1}_{\mathcal{H}^1} \left( E_{\hat{h}_2}^{\hat{s}_2, v_1} \cup S_1 \cup E_{\hat{h}_1(0, h, T_j, H_j)}^{\hat{s}_1(0, h, T_j, H_j), v_1} \right) = 0 \right] \mathbb{1}_{\mathcal{H}^2} \left( E_{\hat{h}_2}^{\hat{s}_2, v_2} \cup E_{\hat{h}_1(0, h, T_j, H_j)}^{\hat{s}_1(0, h, T_j, H_j), v_2} \right) = 0 \\ & \qquad \times e^{-\rho(\hat{s}_1(0, h, T_j, H_j) - \hat{s}_2)} \right]. \end{aligned} \quad (8.91) $$

The point $(T_j, H_j)$ must belong to the region $R_1(h, t', h') \subset [\hat{s}_2, \infty) \times \mathbb{R}^+ \setminus E_{\hat{h}_2}^{\hat{s}_2, v_1}$, where

$$ R_1(h, t', h') := \left\{ (t, u) : t > \hat{s}_2, u \ge 0, v_1^2(t - \hat{s}_2)^2 + u^2 \ge \hat{h}_2^2, u^2 - \frac{v_1^2 v_2^2}{v_1^2 - v_2^2} t^2 \le h^2 \right\}. \quad (8.92) $$
For the region $R_1(h, t', h')$ in (8.92), the first condition $t > \hat{s}_2$ is due to the discussion in Case 1. The last two conditions is due to the fact that the next head point must lie outside $E_{\hat{h}_2}^{\hat{s}_2, v_1}$ and below the curve of the hyperbola $u^2 - \frac{v_1^2 v_2^2}{v_1^2 - v_2^2} t^2 = h^2$.

Applying the Campbell-Mecke formula, the first two terms in (8.90) become

$$
\begin{align}
\zeta\begin{bmatrix} 1, 2 \\ 1, 2, 1 \end{bmatrix} + \zeta\begin{bmatrix} 1, 1 \\ 1, 2, 1 \end{bmatrix} & = 2\lambda_1 v_1 \int_{R_1(h, t', h')} e^{-\rho(\hat{s}_1(0, h, t_1, h_1) - \hat{s}_2)} e^{-2\lambda_1 v_1} \left| E_{\hat{h}_1(0, h, t_1, h_1)}^{\hat{s}_1(0, h, t_1, h_1), v_1} \cup S_1 \cup E_{\hat{h}_2}^{\hat{s}_2, v_1} \right| e^{-2\lambda_2 v_2} \left| E_{\hat{h}_1(0, h, t_1, h_1)}^{\hat{s}_1(0, h, t_1, h_1), v_2} \cup E_{\hat{h}_2}^{\hat{s}_2, v_2} \right| dt_1 \, dh_1 \nonumber \\
& = 2\lambda_1 v_1 \int_{\frac{v_1^2 - v_2^2}{v_1^2 + v_2^2} \hat{s}_2} \int_{(\hat{h}_2^2 - v_1^2(t_1 - \hat{s}_2)^2)^{1/2}} \frac{1}{2} e^{-\rho(\hat{s}_1(0, h, t_1, h_1) - \hat{s}_2)} e^{-2\lambda_1 v_1} \left| E_{\hat{h}_1(0, h, t_1, h_1)}^{\hat{s}_1(0, h, t_1, h_1), v_1} \cup S_1 \cup E_{\hat{h}_2}^{\hat{s}_2, v_1} \right| \nonumber \\
& \qquad \times e^{-2\lambda_2 v_2} \left| E_{\hat{h}_1(0, h, t_1, h_1)}^{\hat{s}_1(0, h, t_1, h_1), v_2} \cup E_{\hat{h}_2}^{\hat{s}_2, v_2} \right| dt_1 \, dh_1 \nonumber \\
& + 2\lambda_1 v_1 \int_{\hat{s}_2 + \hat{h}_2 / v_1}^{\infty} \int_{0}^{\frac{1}{2}} \left( \frac{v_1^2 v_2^2}{v_1^2 - v_2^2} t_1^2 + h^2 \right)^{1/2} e^{-\rho(\hat{s}_1(0, h, t_1, h_1) - \hat{s}_2)} e^{-2\lambda_1 v_1} \left| E_{\hat{h}_1(0, h, t_1, h_1)}^{\hat{s}_1(0, h, t_1, h_1), v_1} \cup S_1 \cup E_{\hat{h}_2}^{\hat{s}_2, v_1} \right| \nonumber \\
& \qquad \times e^{-2\lambda_2 v_2} \left| E_{\hat{h}_1(0, h, t_1, h_1)}^{\hat{s}_1(0, h, t_1, h_1), v_2} \cup E_{\hat{h}_2}^{\hat{s}_2, v_2} \right| dt_1 \, dh_1. \tag{8.93}
\end{align}
$$

For the third term in (8.90), using Remark 7.10 for $l=2$, the region satisfies

$$
\bigcup_{t \in [\hat{s}_2, \hat{s}(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_1} \cup E_{\hat{h}_2}^{\hat{s}_2, v_1} = E_{\hat{h}_2}^{\hat{s}_2, v_1} \cup S_2 \cup E_{\hat{h}(0, h, T_j, H_j)}^{\hat{s}(0, h, T_j, H_j), v_1},
$$

where $\hat{h}_t = (v_2^2 t^2 + h^2)^{1/2}$ and $S_2$ is the region outside $E_{\hat{h}_2}^{\hat{s}_2, v_1} \cup E_{\hat{h}(0,h,T_j,H_j)}^{\hat{s}(0,h,T_j,H_j),v_1}$, but below the hyperbola given by the equation $u^2 - \frac{v_1^2 v_2^2}{v_1^2 - v_2^2} t^2 = h^2$, in the $(t, u)$-coordinate system similar to (7.17) and within the time interval

$$
\left[ \frac{v_1^2 - v_2^2}{v_1^2} \hat{s}_2, \frac{v_1^2 - v_2^2}{v_1^2} \hat{s}(0, h, T_j, H_j) \right].
$$

For $l=2$, using the increasing property of $\{E_{\hat{h}_t}^{t, v_2} \setminus E_{\hat{h}}^{\hat{s}, v_2}\}_{t \ge \hat{s}_2}$ from Part (ii) of Proposition 8.16, we have that

$$
\bigcup_{t \in [\hat{s}_2, \hat{s}(0, h, T_j, H_j)]} E_{\hat{h}_t}^{t, v_2} \cup E_{\hat{h}}^{\hat{s}, v_2} = E_{\hat{h}(0, h, T_j, H_j)}^{\hat{s}(0, h, T_j, H_j), v_2} \cup E_{\hat{h}}^{\hat{s}, v_2}.
$$

Then the third term in (8.90) equals,

$$
\zeta\begin{bmatrix} 1, 1 \\ 1, 2, 2 \end{bmatrix} = \mathbb{E}_{\mathcal{H}}\left[ \sum_{(T_j, H_j) \in \mathcal{H}^2 : T_j \ge 0} \mathbb{1}_{\mathcal{H}^1}\left(E_{\hat{h}(0,h,T_j,H_j)}^{\hat{s}(0,h,T_j,H_j),v_1} \cup S_2 \cup E_{\hat{h}_2}^{\hat{s}_2,v_l}\right) = 0 \mathbb{1}_{\mathcal{H}^2}\left(E_{\hat{h}(0,h,T_j,H_j)}^{\hat{s}(0,h,T_j,H_j),v_2} \cup E_{\hat{h}_2}^{\hat{s}_2,v_2}\right) = 0 \times e^{-\rho(\hat{s}(0,h,T_j,H_j)-\hat{s}_2)} \right]. \quad (8.94)
$$

The point $(T_j, H_j)$ must lie in the region $R_2(h, t', h') \subset (\mathbb{R}^+)^2 \setminus E_{\hat{h}_2}^{\hat{s}_2, v_2}$, defined as

$$
R_2(h, t', h') := \{(t, u) : t \ge 0, u \ge 0, v_2^2(t - \hat{s}_2)^2 + u^2 \ge \hat{h}_2^2\}. \quad (8.95)
$$

For the region $R_2(h, t', h')$ in (8.95), the first condition $t \ge 0$ is due to the discussion in Case 2. The last two conditions is due to the fact that the next head point must lie outside $E_{\hat{h}_2}^{\hat{s}_2, v_2}$. By the Campbell-Mecke
The state at the k-th step implicitly depends on the previous step as the head point $(t_k^p, h_k^p)$ is involved in previous handover(s). The following theorem establishes the Markovianity of the sequence $\{(H_k^l, T_k^r, H_k^r), type(k)\}_{k \in \mathbb{Z}}$, following the foot steps of Theorem 6.1.

**Theorem 9.1.** The sequence $\{(H_k^l, T_k^r, H_k^r), type(k)\}_{k \in \mathbb{Z}}$ forms a homogeneous Markov chain on the state space $(\mathbb{R}^+)^3 \times \mathcal{T}$.

*Proof.* Fix $k \in \mathbb{Z}$. Let $type(k) = [\begin{smallmatrix} q \\ i,j \end{smallmatrix}]$ be the type of the k-th handover, for some $i, j \in \{1, 2\}$. Given $(H_k^l, T_k^r, H_k^r) = (h_k^l, t_k^r, h_k^r)$, suppose the locations of the responsible head points are $(0, h_k^l)$ and $(t_k^r, h_k^r)$, respectively and the intersection point of the corresponding radial birds for the handover in question is $(\hat{s}_k, \hat{h}_k)$. Define $\tau(t, h)$ to be the type of a head point $(t, h) \in \mathbb{H}^+$. For any $s \in \mathbb{R}$, define $Q_s := \{(t, h) \in \mathbb{H}^+ : t \ge s\}$, to be the quadrant on $\mathbb{H}^+$ and on the right of the vertical line $t = s$. Depending on the type $[\begin{smallmatrix} q \\ i,j \end{smallmatrix}]$, we can assign their corresponding type to the head points and carry out our analysis in the following two cases:

**Case 1.** $q = 1$. In this case, the head point corresponding to new station is on the right and so we have $\tau(t_k^r, h_k^r) = j$ and $\tau(0, h_k^l) = i$. Here, we have four possibilities for the handover types: $[\begin{smallmatrix} q \\ i,j \end{smallmatrix}] = [\begin{smallmatrix} 1 \\ 1,1 \end{smallmatrix}, [\begin{smallmatrix} 1 \\ 2,1 \end{smallmatrix}, [\begin{smallmatrix} 1 \\ 1,2 \end{smallmatrix}]$ and $[\begin{smallmatrix} 1 \\ 2,2 \end{smallmatrix}]$. Let $m \ne j \in \{1, 2\}$. The precise unexplored region for different cases plays an essential role in establishing the Markovianity of the process in question. To determine the $(k+1)$-st step of the process, one first needs to find the unexplored regions for potential location of the upcoming head point. Observe that there are two unexplored regions, as the upcoming head point can be of two different types. For any given type $[\begin{smallmatrix} 1 \\ i,j \end{smallmatrix}]$ for the k-th handover, let $\mathcal{U}_j^1$ and $\mathcal{U}_m^1 \subset \mathbb{H}^+$ be the unexplored region of the upcoming head point of type $j$ and $m$, respectively. Table 9.1 lists out the unexplored regions $\mathcal{U}_j^1$ and $\mathcal{U}_m^1$ for the case $q = 1$, by gathering the details in various parts of the proofs of Lemma 8.19 and Lemma 8.20 corresponding to different handover types. These unexplored regions only depend on the state $((h_k^l, t_k^r, h_k^r), type(k))$ at step $k$.

<table><caption>**TABLE 9.1.** Unexplored regions $U_j^1, U_m^1$, for $m \ne j$, when the k-th handover type is $[\begin{smallmatrix} 1 \\ i,j \end{smallmatrix}]$. The reference to figures are mentioned next to the handover types, resembles the corresponding scenario and they can be used to determine the unexplored regions.</caption><thead><tr><th>Types: <span style="font-size: 1.5em; vertical-align: middle;">[</span><span style="font-size: 0.8em; vertical-align: middle;">1</span><span style="font-size: 1.5em; vertical-align: middle;">]</span><br>U<sub>j</sub><sup>1</sup>, U<sub>m</sub><sup>1</sup>, m ≠ j</th><th>[<span style="font-size: 0.8em; vertical-align: middle;">1</span><span style="font-size: 1.5em; vertical-align: middle;">]</span>, Fig. 8.3</th><th>[<span style="font-size: 0.8em; vertical-align: middle;">1</span><span style="font-size: 1.5em; vertical-align: middle;">]</span>, Fig. 8.7 & 8.8</th><th>[<span style="font-size: 0.8em; vertical-align: middle;">1</span><span style="font-size: 1.5em; vertical-align: middle;">]</span>, Fig. 8.12</th><th>[<span style="font-size: 0.8em; vertical-align: middle;">1</span><span style="font-size: 1.5em; vertical-align: middle;">]</span>, Fig. 8.5 & 8.6</th></tr></thead><tbody><tr><th>U<sub>j</sub><sup>1</sup></th><td>Q<sub>t<sub>k</sub><sup>r</sup></sub> \ E<sub>\hat{h}_k</sub><sup>s\hat{k}, v<sub>1</sub></sup></td><td>Q<sub>t<sub>k</sub><sup>r</sup></sub> \ E<sub>\hat{h}_k</sub><sup>s\hat{k}, v<sub>1</sub></sup></td><td>Q<sub>t<sub>k</sub><sup>r</sup></sub> \ E<sub>\hat{h}_k</sub><sup>s\hat{k}, v<sub>2</sub></sup></td><td>Q<sub>t<sub>k</sub><sup>r</sup></sub> \ E<sub>\hat{h}_k</sub><sup>s\hat{k}, v<sub>2</sub></sup></td></tr><tr><th>U<sub>m</sub><sup>1</sup></th><td>\mathbb{H}<sup>+</sup> \ E<sub>\hat{h}_k</sub><sup>s\hat{k}, v<sub>2</sub></sup></td><td>Q<sub>0</sub> \ E<sub>\hat{h}_k</sub><sup>s\hat{k}, v<sub>2</sub></sup></td><td>Q<sub>s\hat{k}</sub> \ E<sub>\hat{h}_k</sub><sup>s\hat{k}, v<sub>1</sub></sup></td><td>Q<sub>s\hat{k}</sub> \ E<sub>\hat{h}_k</sub><sup>s\hat{k}, v<sub>1</sub></sup></td></tr></tbody></table>

As explained in (8.36) and Lemma 8.12, one can construct a sequence of non-decreasing open sets $\{S_t^j\}_{t \ge \hat{s}_k}$ and $\{S_t^m\}_{t \ge \hat{s}_k}$ in the unexplored region $U_j^1$ and $U_m^1$, using the sequence $\{h_j(t)\}_{t \ge \hat{s}_k}$ where $h_j(t) := (v_j^2(t - t_k^r)^2 + (h_k^r)^2)^{\frac{1}{2}}$. By Corollary 8.14, there exists a head point say $(T_u, H_u)$ of type $\tau_u$ in the unexplored region $U_j^1$ or $U_m^1$ such that $(k+1)$-st handover is given by the intersection point, say $(\hat{S}_{k+1}, \hat{H}_{k+1})$, of the radial birds at $(t_k^r, h_k^r)$ and $(T_u, H_u)$. Here, the letter $u$ stands for *upcoming*. The pair of stopping set is $(S_{\hat{S}_{k+1}}^j, S_{\hat{S}_{k+1}}^m)$, and the upcoming head point $(T_u, H_u)$ lie on the boundary $\partial S_{\hat{S}_{k+1}}^j$ or $\partial S_{\hat{S}_{k+1}}^m$ of the respective stopping sets, depending on whether $\tau_u = j$ or $m$. Then the type of this handover is $[\begin{smallmatrix} q' \\ j, \tau_u \end{smallmatrix}]$, where

$$q' := \begin{cases} 1 & \text{if } T_u \ge t_k^r, \\ 2, & \text{if } T_u < t_k^r. \end{cases} \qquad (9.2)$$
Subsequently we define

$$ (H_{k+1}^l, T_{k+1}^r, H_{k+1}^r) := \begin{cases} (h_k^r, T_u - t_k^r, H_u) & \text{if } q' = 1, \\ (h_k^r, t_k^r - T_u, H_u), & \text{if } q' = 2. \end{cases} \quad (9.3) $$

This is the 'spatial' part of the state at step $k+1$.

**Case 2.** $q = 2$. In this case, the head point corresponding to the new station is on the left and so we have $\tau(0, h_k^l) = j$ and $\tau(t_k^r, h_k^r) = i$. The possibilities of the handover types in this case are $[q] = [2, 1]$ and $[1, 2]$. Let $m \neq j \in \{1, 2\}$. The head point of the upcoming station can be found in the unexplored region, say $\mathcal{U}_j^2$ and $\mathcal{U}_m^2$, listed in the Table 9.2 in the case $q = 2$, obtained using the same recipe as for Table 9.1 in the case $q = 1$, as follows:

<table><caption>TABLE 9.2. Unexplored regions $\mathcal{U}_j^2, \mathcal{U}_m^2$, for $m \neq j$, when the $k$-th handover type is $[\begin{smallmatrix} 2 \\ i,j \end{smallmatrix}]$. The figures mentioned next to the handover types, are used to determine the unexplored regions.</caption><thead><tr><th>Types: <sup>2</sup><sub>i,j</sub></th><th><sup>2</sup><sub>2,1</sub>, Fig. 8.10</th><th><sup>2</sup><sub>1,2</sub>, Fig. 8.9</th></tr></thead><tbody><tr><td>$\mathcal{U}_j^2, \mathcal{U}_m^2, m \neq j$</td><td>$Q_0 \setminus E_{\hat{h}_k}^{\hat{s}_k, v_1}$</td><td>$Q_0 \setminus E_{\hat{h}_k}^{\hat{s}_k, v_2}$</td></tr><tr><td>$\mathcal{U}_j^2$</td><td>$Q_{t_k^r} \setminus E_{\hat{h}_k}^{\hat{s}_k, v_2}$</td><td>$Q_{\hat{s}_k} \setminus E_{\hat{h}_k}^{\hat{s}_k, v_1}$</td></tr><tr><td>$\mathcal{U}_m^2$</td><td></td><td></td></tr></tbody></table>

These unexplored regions only depend on the state $((h_k^l, t_k^r, h_k^r), type(k))$ at step $k$. Similarly to the case $q = 1$, we construct a sequence of non-decreasing open sets $\{S_t^j\}_{t \ge \hat{s}_k}$ and $\{S_t^m\}_{t \ge \hat{s}_k}$, representing explored regions in the unexplored region $\mathcal{U}_j^2$ and $\mathcal{U}_m^2$, using the sequence $\{h_j(t)\}_{t \ge \hat{s}_k}$, where $h_j(t) := (v_j^2 t^2 + (h_k^l)^2)^{\frac{1}{2}}$. By Corollary 8.14, there exists a head point, say $(T_u, H_u)$, of type $\tau_u$ in the unexplored region $\mathcal{U}_j^2$ or $\mathcal{U}_m^2$ such that $(k+1)$-st handover is given by the intersection point, say $(\hat{S}_{k+1}, \hat{H}_{k+1})$, of the radial birds at $(0, h_k^l)$ and $(T_u, H_u)$. The pair of stopping sets is $(S_{\hat{S}_{k+1}}^j, S_{\hat{S}_{k+1}}^m)$, and the upcoming head point $(T_u, H_u)$ lie on the boundary of the respective stopping sets, depending on whether $\tau_u = j$ or $m$. From Table 9.2, it is clear that $T_u \ge 0$. Then, the type of the upcoming handover is $[\begin{smallmatrix} q' \\ j, \tau_u \end{smallmatrix}]$, with $q'$ equal to 1. Also note that $[\begin{smallmatrix} q' \\ j, \tau_u \end{smallmatrix}] = [\begin{smallmatrix} 2 \\ j, j \end{smallmatrix}]$ is not possible for obvious reasons. Subsequently, we define the 'spatial' part of the state at $(k+1)$-st step as

$$ (H_{k+1}^l, T_{k+1}^r, H_{k+1}^r) := (H_k^r, T_u - t_k^r, H_u). \quad (9.4) $$

The Markov property is satisfied by the sequence $\{(H_k^l, T_k^r, H_k^r), type(k)\}_{k \in \mathbb{Z}}$, since the state at the $(k+1)$-st step, $((H_{k+1}^l, T_{k+1}^r, H_{k+1}^r), type(k+1))$, is determined using the unexplored regions, $\mathcal{U}_j^1$ and $\mathcal{U}_m^1$ in Case 1 for $q = 1$ and $\mathcal{U}_j^2$ and $\mathcal{U}_m^2$ in Case 2 for $q = 2$, where one finds a Poisson point process of heads which is conditionally independent of the past, given the state $((h_k^l, t_k^r, h_k^r), type(k))$ at step $k$. This completes the proof of Theorem 9.1. $\square$

**Remark 9.2.** Our technique allows one to predict the upcoming handover type $[\begin{smallmatrix} q' \\ j, \tau_u \end{smallmatrix}]$ and the location of the responsible head point, say $(T_u, H_u)$ of type $\tau_u$, given the state at step $k$. In addition, it enables us to derive a representation of the transition kernel of the chain, for a transition to $((H_{k+1}^l, T_{k+1}^r, H_{k+1}^r), [\begin{smallmatrix} q' \\ j, \tau_u \end{smallmatrix}])$, given the $k$-th state is $((H_k^l, T_k^r, H_k^r), [\begin{smallmatrix} q \\ i,j \end{smallmatrix}])$ along the line of the single-speed case.
## 10. HANDOVER PALM DISTRIBUTION AND APPLICATIONS: MULTI-SPEED CASE

In this section we first determine the handover Palm probability distribution with the goal to determine the Palm probability distribution of inter-handover times and typical handover distance. An identical analysis follows to come up with a result similar to the mass transport principle (Lemma 5.1 for the single-speed and Lemma 8.1 for the two-speed case) for the multi-speed case.

**10.1. Palm probability with respect to handovers.** Without loss of generality, here also we assume that $v_1 > v_2 > \cdots > v_n$. Identically to the two-speed case, here also the reference probability space is considered as $(\Omega, \mathcal{F}, \mathbb{P})$, with $(\Omega, \mathcal{F})$ the canonical space of head point process $\mathcal{H} = \sum_{l \in [n]} \mathcal{H}^l$ on $\mathbb{H}^+$, where $\mathcal{H}^l$ is the head point process corresponding to speed $v_l$. A point $\omega \in \Omega$ representing a realization of the point process $\mathcal{H}$. The measure space is equipped with the shift $\{\theta_t\}_{t \in \mathbb{R}}$ along the time axis, defined as

$$ \theta_t(\mathcal{H}) = \sum_{l \in [n]} \theta_t(\mathcal{H}^l) = \sum_{l \in [n]} \sum_{i \in \mathbb{N}} \delta_{(T_i^l - t, H_i^l)}, \quad (10.1) $$

which is ergodic for $\mathbb{P}$. As in the two-speed case, the law $\mathbb{P}$ of the head point process $\mathcal{H}$ is left invariant by this shift.

We have constructed the handover point process $\mathcal{V}$ in Subsection 7.2.3, using the pure handover point processes $\mathcal{V}_l$ for $l \in [n]$, and the mixed handover point process $\mathcal{V}_{l,r}^{(k)}$ for $l \neq r \in [n], k \in \{1, 2\}$, depending on all possible type of intersections. The handover point process $\mathcal{V}$ is stationary from Lemma 7.17 and has a positive and finite intensity from Theorem 7.20. This allows us to define the Palm probability w.r.t. $\mathcal{V}$ on the probability space $(\Omega, \mathcal{F})$, which will be denoted as $\mathbb{P}_{\mathcal{V}}^0$.

Recall the point processes $\mathcal{R}_l$ for $l \in [n]$ and $\mathcal{R}_{l,r}^{(k)}$ for $l \neq r \in [n], k \in \{1, 2\}$, corresponding to the right most head points. The same mass transport principle, Lemma 8.1, under the bijections defined in Lemma 7.27, applies to all the individual Palm probability measures, stated in the following.

**Lemma 10.1.** For all non-negative measurable functions $f$ on $(\Omega, \mathcal{F})$ we have:

$$ (i). \text{ For all } l \in [n], \qquad \mathbb{E}_{\mathcal{V}_l}^0[f(\mathcal{H})] = \mathbb{E}_{\mathcal{R}_l}^0[f(\theta_{\beta_l(0)}(\mathcal{H}))], \quad (10.2) $$
where $\beta_l$ is the bijection between the point processes $\mathcal{V}_l$ and $\mathcal{R}_l$.

$$ (ii). \text{ For all } l \neq r \in [n], k = 1, 2, \qquad \mathbb{E}_{\mathcal{V}_{l,r}^{(k)}}^0[f(\mathcal{H})] = \mathbb{E}_{\mathcal{R}_{l,r}^{(k)}}^0[f(\theta_{\beta_{l,r}^{(k)}(0)}(\mathcal{H}))], \quad (10.3) $$
where $\beta_{l,r}^{(k)}$ is the bijection between the point processes $\mathcal{V}_{l,r}^{(k)}$ and $\mathcal{R}_{l,r}^{(k)}$, of type $\binom{k}{l,r}$.

Hence, we have the following decomposition of $\mathbb{E}_{\mathcal{V}}^0[f(\mathcal{H})]$, similar to Theorem 8.2, based on different types of handovers, using the Palm probability measure for superposition of multi-type independent point processes (cf. [4, Subsection 1.6, p. 36]).

**Theorem 10.2.** For all non-negative measurable functions $f$ on $(\Omega, \mathcal{F})$,

$$ \mathbb{E}_{\mathcal{V}}^0[f(\mathcal{H})] = \sum_{l \in [n]} \frac{\Lambda_l}{\lambda_{\mathcal{V}}} \mathbb{E}_{\mathcal{V}_l}^0[f(\mathcal{H})] + \sum_{l \neq r \in [n], k=1,2} \frac{\Lambda_{l,r}^{(k)}}{\lambda_{\mathcal{V}}} \mathbb{E}_{\mathcal{V}_{l,r}^{(k)}}^0[f(\mathcal{H})]. \quad (10.4) $$

Similar to the supporting results, Lemma 8.3, Lemma 8.4, individual Palm probability measures are expressed as follows:
**Lemma 10.3.** For $l \in [n]$ and for all non-negative measurable functions $f$ on $(\Omega, \mathcal{F})$,

$$ \mathbb{E}_{\mathcal{V}_l}^0[f(\mathcal{H})] = \frac{4\lambda_l^2 v_l^2}{\Lambda_l} \int_{(\mathbb{R}^+)^3} \mathbb{E}_{\mathcal{H}} \left[ f\left(\theta_{\hat{s}(0,h,-t',h')}(\mathcal{H}^l + \delta_{(0,h)} + \delta_{(-t',h')} + \sum_{m \neq l} \mathcal{H}^m)\right) \times \prod_i \mathbb{1}_{\mathcal{H}^i} \left(E_{\hat{h}(0,h,-t',h')}^{\hat{s}(0,h,-t',h')}, v_i\right) \right] dt' dh' dh, $$

where $(\hat{s}(t, h, t', h'), \hat{h}(t, h, t', h'))$ are the coordinates of the intersection of the two radial birds with heads $(t, h)$ and $(t', h')$.

The proof Lemma 10.3 follows the steps of Lemma 5.3 and Lemma 8.3, except for the extra condition, for the void probabilities, the multiple half-ellipses to be empty of head points corresponding to the different speeds, as in Theorem 7.29 and Figure 7.10. Having said this, we move towards the interesting case of mixed handovers similar to Lemma 8.4, for which we provide more details in the following.

**Lemma 10.4.** For $r \neq l \in [n]$, $k = 1, 2$ and for all non-negative measurable functions $f$ on $(\Omega, \mathcal{F})$,

$$ \frac{\Lambda_{l,r}^{(k)}}{4\lambda_l \lambda_r v_l v_r} \mathbb{E}_{\mathcal{V}_{l,r}^{(k)}}[f(\mathcal{H})] = \int_0^\infty \int_0^h \int_{t^*}^\infty \mathbb{E}_{\mathcal{H}} \left[ f\left(\theta_{\hat{s}(0,h,-t',h')} (\mathcal{H}^r + \delta_{(0,h)} + \mathcal{H}^l + \delta_{(-t',h')} + \sum_{m \neq l,r} \mathcal{H}^m) \times \prod_i \mathbb{1}_{\mathcal{H}^i} \left(E_{\hat{h}_k(0,h,-t',h')}^{\hat{s}_k(0,h,-t',h')}, v_i\right) \right) \right] dt' dh' dh \\ + \int_0^\infty \int_h^\infty \int_0^\infty \mathbb{E}_{\mathcal{H}} \left[ f\left(\theta_{\hat{s}(0,h,-t',h')} (\mathcal{H}^r + \delta_{(0,h)} + \mathcal{H}^l + \delta_{(-t',h')} + \sum_{m \neq l,r} \mathcal{H}^m) \times \prod_i \mathbb{1}_{\mathcal{H}^i} \left(E_{\hat{h}_k(0,h,-t',h')}^{\hat{s}_k(0,h,-t',h')}, v_i\right) \right) \right] dt' dh' dh, $$

where, $(\hat{s}_k(t, h, t', h'), \hat{h}_k(t, h, t', h'))$ are the coordinates of the k-th intersection of the two radial birds with heads at $(t, h)$ and $(t', h')$, $t^* = \frac{1}{v_l v_r} (h^2 - h'^2)^{\frac{1}{2}} (v_r^2 - v_l^2)^{\frac{1}{2}}$.

**Remark 10.5.** The symmetry envisioned earlier in Lemma 8.7 can also be shown to hold, using the same time-reversal argument, under the very specific assumption that the function $f$ is symmetric, i.e., $f(\mathcal{H}) = f(-\mathcal{H})$, defined in Subsection 8.1.1. In other words,

$$ \mathbb{E}_{\mathcal{V}_{i,j}^{(1)}}[f(\mathcal{H})] = \mathbb{E}_{\mathcal{V}_{j,i}^{(2)}}[f(\mathcal{H})] \text{ and } \mathbb{E}_{\mathcal{V}_{i,j}^{(2)}}[f(\mathcal{H})] = \mathbb{E}_{\mathcal{V}_{j,i}^{(1)}}[f(\mathcal{H})], $$

for all $i \neq j \in [n]$. Under the symmetric assumption on the function $f$, we have the decomposition of $\mathbb{E}_{\mathcal{V}}^0[f(\mathcal{H})]$ from Theorem 10.2 that

$$ \mathbb{E}_{\mathcal{V}}^0[f(\mathcal{H})] = \sum_{i \in [n]} \frac{\Lambda_i}{\lambda_{\mathcal{V}}} \mathbb{E}_{\mathcal{V}_i}^0[f(\mathcal{H})] + 2 \sum_{j<i \in [n], k=1,2} \frac{\Lambda_{i,j}^{(k)}}{\lambda_{\mathcal{V}}} \mathbb{E}_{\mathcal{V}_{i,j}^{(k)}}^0[f(\mathcal{H})]. \quad (10.5) $$

**10.2. Typical handover distance: multi-speed case.** Let $\hat{H}$ be the random variable for handover distance and $\hat{H}_0$ be the distance of the typical handover. Based on the different types of intersections that happen at time 0 between the corresponding radial birds, we just state the following result for the Laplace transform of $\hat{H}^2$ under the Palm probability measure $\mathbb{P}_{\mathcal{V}}^0$, in the multi-speed setting. This can be proved along the same lines of Lemma 8.9, by applying the same ideas along with the formula (10.5) and the symmetry, adapted to the multi-speed setting.

**Lemma 10.6.** For any $\gamma \ge 0$, the Laplace transform of $\hat{H}^2$ under the Palm probability measure $\mathbb{P}_{\mathcal{V}}^0$ of handover is

$$ \mathcal{L}_{\hat{H}^2}^0(\gamma) = \frac{1}{\lambda_{\mathcal{V}}} \frac{4\sqrt{\lambda}}{\pi} \left(1 + \frac{\gamma}{\lambda\pi}\right)^{-3/2} \sum_{i \in [n]} v_i \left(\frac{\lambda_i}{\lambda}\right)^2 + \frac{8}{\lambda_{\mathcal{V}}} \sum_{j<i \in [n], k \in \{1,2\}} \lambda_i \lambda_j v_i v_j \eta_{i,j}^{(k)}(\gamma), \quad (10.6) $$
where $\eta_{i,j}^{(k)}(\gamma)$ can be determined as in (8.30) and (8.31) for $j < i \in [n]$ and $k \in \{1, 2\}$ and $\Lambda_{i,j}^{(k)} = 4\lambda_i \lambda_j v_i v_j \eta_{i,j}^{(k)}(0)$, for $k = 1, 2$.

**10.3. Distribution of inter-handover time: multi-speed case.** Let us first convert the handover types bijectively to the new set of notations

$$ \mathcal{T}_n := \left\{ \begin{bmatrix} q \\ l,m \end{bmatrix} : q = 1 \text{ if } l = m, \text{ and } q = 1,2 \text{ for } l \neq m, l,m \in [n] \right\}, $$

using a table similar to Table 8.1 for the $n$ types of speeds. As a result for any $l \in [n]$, there exists a handover type $l$ or equivalently $\begin{bmatrix} 1 \\ l,l \end{bmatrix}$, and pure handover point process $\mathcal{V}_l \stackrel{d}{=} \mathcal{W}_l$. Also, for any $i \neq j \in [n]$ and $k \in \{1, 2\}$, there exists $l \neq m \in [n]$ and $q \in \{1, 2\}$ such that, $\mathcal{V}_{i,j}^{(k)} \stackrel{d}{=} \mathcal{W}_{l,m}^{(q)}$, where the left hand side correspond to handover type $\begin{pmatrix} k \\ i,j \end{pmatrix}$ (in the old notations) and right hand side correspond to handover type $\begin{bmatrix} q \\ l,m \end{bmatrix}$. With this in hand, we write the handover point process as

$$ \mathcal{V} \equiv \mathcal{W} := \sum_{i \in [n]} \mathcal{W}_i + \sum_{l \neq m \in [n], q \in \{1,2\}} \mathcal{W}_{l,m}^{(q)}. $$

Let $L_l$ be the intensity of the pure handover process $\mathcal{W}_l$, for $l \in [n]$ and $L_{l,m}^{(q)}$ be the intensity of the mixed handover process $\mathcal{W}_{l,m}^{(q)}$ of type $\begin{bmatrix} q \\ l,m \end{bmatrix}$, for $l \neq m \in [n]$, $q \in \{1, 2\}$. Then we have the decomposition of the Laplace transform of the inter-handover time $T$, identical to Theorem 8.18, under the Palm probability measure $\mathbb{P}_{\mathcal{W}}^0 \equiv \mathbb{P}_{\mathcal{V}}^0$.

**Theorem 10.7.** Let $\rho \ge 0$. In the multi-speed case, the Laplace transform of $T$ under the Palm probability measure $\mathbb{P}_{\mathcal{V}}^0$ is given by:

$$ \mathbb{E}_{\mathcal{W}}^{0}[e^{-\rho T}] = \sum_{l \in [n]} \frac{L_l}{\lambda_{\mathcal{W}}} \mathbb{E}_{\mathcal{W}_l}^{0}[e^{-\rho T}] + \sum_{l \neq m \in [n], k \in \{1,2\}} \frac{L_{l,m}^{(q)}}{\lambda_{\mathcal{W}}} \mathbb{E}_{\mathcal{W}_{l,m}^{(q)}}^{0}[e^{-\rho T}]. \quad (10.7) $$

*Proof of Theorem 10.7.* The result can be proved similarly to Theorem 8.18. We derive the expressions for $\mathbb{E}_{\mathcal{W}_l}^0[e^{-\rho T}]$, for $l \in [n]$ and $\mathbb{E}_{\mathcal{W}_{l,m}^{(q)}}^0[e^{-\rho T}]$, for $l \neq m \in [n]$, $q \in \{1, 2\}$, with the help of Lemma 10.3 and Lemma 10.4, following the same steps as those of Lemma 8.19 and Lemma 8.20 in the two-speed case, adapted to the multi-speed setting, along with the help of Remark 8.23. $\square$

## 11. FUTURE WORK

**Related problems.** Our model and the corresponding results are the first steps towards understanding the handover phenomenon in a model with mobile stations and static user, through the dynamical Poisson-Voronoi tessellation in the simplest case. One can gradually relax the simplifications along the following line of research on far richer class of dynamical tessellations.

1. **Random environment:** Exploring the motion of the stations on a random environment, for example on the Poisson line process, in the single and the multi-speed settings.

2. **Higher dimensions:** Generalizing all these models to 3 and higher dimensions in both single and multiple speed scenarios.

3. **Finite visibility:** Understanding all these models in 2 and higher dimension, where the user or the stations or both have finite visibility. This restriction to the user and stations is not so unrealistic, as it naturally appears in the spherical case.

4. **Spherical case:** It would be interesting to analyze handovers in the spherical case, where stations are moving along randomly oriented orbits, as in the communication model with LEO and MEO constellations. We believe that the tools of spherical stochastic geometry and the knowledge acquired in all the scenarios defined in the present paper, will certainly help us understand this spherical case in the future.