# On A Class Of Dynamical Poisson-Voronoi Tessellations

François Baccelli, Sanjoy Kumar Jhawar

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François Baccelli, Sanjoy Kumar Jhawar. On A Class Of Dynamical Poisson-Voronoi Tessellations.
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# ON A CLASS OF DYNAMICAL POISSON-VORONOI TESSELLATIONS

FRANÇOIS BACCELLI¹,² AND SANJOY KUMAR JHAWAR²,¹

**ABSTRACT.** Consider a dynamical network model featuring mobile stations on the Euclidean plane. The initial locations of the stations are given by a homogeneous Poisson point process. The stations are all moving at a constant speed and in a random direction. Consider fixed users located in the Euclidean plane, which are served by the mobile stations. Each user stays connected to the nearest station at any given point of time. Since the stations are moving, an user disconnects and connects with different stations over time, by always selecting which ever station is the closest. This gives rise to a dynamical version of the Poisson-Voronoi tessellation. The focus of this paper is on the sequence of “handover” events of a typical user, which are the epochs when its association changes. This defines a point process on the time-axis, the “handover point process”. We show that this point process is stationary and we determine its main properties, in particular its intensity and the joint distribution of its inter-event times. We also analyze the handover Palm distributions of several variables of practical interest. This includes the distance to the closest mobile stations and the point process of all other mobile stations at handover epochs. The analysis is conducted both in the single-speed and in the multi-speed scenarios. It leads to the identification of the three dimensional state variables that “Markovize” the association dynamics. The analysis is based on a specific system of non-compact particles. The motivations are in the modeling of low or medium orbit satellite wireless communication networks. The model studied here is a planar “caricature” of this problem, which is initially defined on the sphere.

**Keywords:** Palm calculus; stochastic geometry; Poisson point process; dynamic Voronoi tessellation; handover; terrestrial and non-terrestrial networks; dynamic wireless communication; satellite communication.

**Mathematics Subject Classification (2020):** Primary 60D05; 60G55; 82C22; Secondary 60K37.

## CONTENTS

<table><tr><td>1. Introduction</td><td>2</td></tr><tr><td>2. Setting and main objects</td><td>3</td></tr><tr><td>3. Questions, strategy and structure of the paper</td><td>4</td></tr><tr><td>4. Handover frequency: single-speed case</td><td>7</td></tr><tr><td>5. Palm probability with respect to handovers and applications</td><td>21</td></tr><tr><td>6. Handover process as a factor of a Markov chain: single-speed case</td><td>32</td></tr><tr><td>7. Handover frequency: two and multi-speed cases</td><td>34</td></tr><tr><td>8. Handover Palm distribution and applications: two-speed case</td><td>53</td></tr><tr><td>9. Handover process as a factor of a Markov chain: two-speed case</td><td>90</td></tr><tr><td>10. Handover Palm distribution and applications: multi-speed case</td><td>93</td></tr><tr><td>11. Future work</td><td>95</td></tr><tr><td>12. Table of Notation</td><td>96</td></tr><tr><td>Appendix A.</td><td>98</td></tr><tr><td>References</td><td>109</td></tr></table>

Date: November 21, 2025.

¹INRIA Paris, ²Télécom Paris.
# 1. INTRODUCTION

Consider stations moving at constant speed and in random directions on the Euclidean plane. A typical initial condition for the stations is the set of atoms of a homogeneous Poisson point process $\Phi$ on $\mathbb{R}^2$. Consider a user located at the origin (a user located at any other locus would see the same statistics as those described below). This user is bound to keep connected to the closest station at any time.

Here is a set of practical questions within this setting. At what frequency does the user have to swap from a station to another one? These events are called *handovers* and the corresponding time points are called *handover epochs*. More generally, what is the full structure of the handover (time) point process, for instance the distribution of the duration between two handovers or more precisely, the joint distribution of the inter-handover times?

Here are further questions, which assume that one can define the state of the planar dynamical system of interest at a typical handover epoch, or more precisely the Palm distribution of the planar dynamical system with respect to the time handover point process: What is the distribution of the distance from the user to the two stations involved in the handover? What is the distribution of the point process made of the stations not involved in the handover?

For the first set of questions, it is interesting to consider the following dual dynamical system where the stations are not mobile and the user is mobile in place, with a constant speed and a random direction. Here is a first teaser for the reader: for some motion speed, are there more handovers in the mobile station, fixed user model or in the dual fixed stations, mobile user case?

For the second set of questions, in the PPP model, the distribution, at a typical time of the dynamical system, of the distance $R$ to the closest station is well known to be Rayleigh, and the other stations are known to form a Poisson point process of the same intensity outside the ball of radius $R$. Here is a second teaser for the reader: under the Palm distribution of the handover point process, is the distribution of the distance to the stations involved in the handover still Rayleigh? What is the law of the point process made of the other stations?

The aim of the machinery introduced in the present paper is to answer these questions and many others of the same kind. These questions are of independent mathematical interest as the simplest and most natural questions about the dynamics of Poisson-Voronoi tessellations. They are also of central importance in the wireless communication setting alluded to above. The frequency of handovers is important as any such event means an interruption of the communication stream and a signaling overhead from the serving station to the next. The distribution of the distance to the closest station at handover times determines the signal power at these epochs. That of the other stations determines the interference power. Altogether, this allows one to determine the distribution of the signal to interference and noise ratio (SINR) and signal to noise ratio (SNR), which in turn determine the distribution of the Shannon rate of the link to the user at these critical epochs. These quantities are some of the key performance metrics in the context of satellite communication network [5].

Communication networks based on Low Earth Orbit (LEO) or Medium Earth Orbit (MEO) satellite constellations feature moving stations, more precisely moving on orbits. One can for instance consider the random locations of the satellites to be a Cox point process on certain Earth orbits, at the same or different altitudes with respect to the surface of the Earth. The ever changing locations of the satellites and the rotation of the Earth lead to networks with stations having a dynamics of the type described in the abstract when devices on the Earth's surface connect to the closest visible satellite at any given time. The central object in this regard is the spherical dynamical tessellations formed by the projection of the satellite point process on the surface of the Earth. One of the important questions is to mathematically characterize the structure and the properties of the handover point process under this dynamical setting. Spherical tessellations under various types of LEO constellation of satellites, was first considered in [23]. We refer the reader to [24] and the references therein for a study of communication networks based on LEO and MEO satellite constellations, from a stochastic geometry perspective.

The objective of this work is to investigate this class of problems in a simplified setting. The first simplification is that where the spherical geometry is replaced by a planar one. The second one is
that where the Cox motion alluded to above is simplified to a simple random way-point model. These simplifications, which may look extreme, are in fact natural, or even necessary to define first steps that are tractable in this line of thoughts. As we shall see, even the planar caricature of the problem is already challenging. The steps towards relaxing these simplifications are discussed in the paper in terms of future research in Section 11.

Consider the static scenario where all the stations are fixed and a mobile user is moving along a straight line at speed $v$, without changing its direction. In this case the user performs handover whenever it crosses the boundary of the static Voronoi tessellation. It has been shown in [9] using the linear contact length distribution of Voronoi tessellation for stationary point processes in [13], that the handover frequency is $\frac{4v\sqrt{\lambda}}{\pi}$, which is expressed in terms of the intensity $\lambda$ of stations and the speed $v$ of the user. This result is also discussed in [7], [6], [8], [16]. Further statistics of the 2 and higher dimensional Poisson-Voronoi tessellations have also been studied in [11], [15], [17], [20], [16], where as generalization to the case of stationary point processes have been studied in [13].

In a computational geometry perspective, dynamic Voronoi tessellations were first considered in [12] in a finite setting for the study of geometric data structure using the underlying Delaunay triangulation of the Voronoi tessellation evolving over time and further explored in [19], [2], [1], [18]. To the best of our knowledge, the current work about the dynamic Poisson-Voronoi tessellations with applications to wireless communication in a stochastic geometry setting, appears to be considered for the first time.

## 2. SETTING AND MAIN OBJECTS

Consider a grand probability space $(\Omega, \mathcal{F}, \mathbb{P})$ on which all our measures and random variables are defined. To start with a simple model on $\mathbb{R}^2$, consider a static user $u$ located at the origin $o = (0,0)$. At time $0$, consider stations located at all atoms of a homogeneous Poisson point process $\Phi_\lambda$ on $\mathbb{R}^2$ with intensity $\lambda$. For simplicity, we write $\Phi$ for $\Phi_\lambda$, as long as the intensity $\lambda$ is fixed. We denote the intensity measure by $\mu$ and its density by $d\mu = \lambda dx$ and denote the probability distribution of $\Phi$ as $\mathbb{P}_\Phi$. We refer to the locations of the mobile stations at time $0$ as their original locations $(\Phi)$. At time $0$, the configuration of the atoms $\{X_i\}_{i \in \mathbb{N}} \subset \mathbb{R}^2$ of the point process $\Phi$ are such that $X_0 := \arg \min\{|X_i|: i \in \mathbb{N}\}$, where $|x|$ denotes the norm $||x||_2$. This is the location of the nearest station to the user at time $0$.

Let us now define how stations move. At time $0$, they are located at the atoms of $\Phi$. We suppose that each station is moving at a constant speed $v$, but in a random direction, chosen uniformly and independently in $(-\pi, \pi]$, with respect to the $x$-axis. The stations move without any obstacle, collision or change of direction. In the simplest model, the velocities of the mobile stations located at the atoms $\{X_i\}_{i \in \mathbb{N}}$ of $\Phi$, are defined as an i.i.d. collection of vectors $\{V_n\}_{n \in \mathbb{N}}$, where $V_n := (v \cos \Theta_n, v \sin \Theta_n)$ and $\Theta_n \stackrel{D}{\sim} U[-\pi, \pi)$. One can think of the point process $\Phi$ as being equipped with marks $\{\Theta_i\}_{i \in \mathbb{N}_0}$ representing the i.i.d. directions of motion of the stations. Incorporating the direction of motion with the initial locations, we construct a marked Poisson point process $\tilde{\Phi} := \sum_{i \in \mathbb{N}} \delta_{(X_i, \Theta_i)}$ with intensity measure $m_{\tilde{\Phi}}$ having density

$$m_{\tilde{\Phi}}(dx, d\theta) := \lambda dx \otimes \frac{1}{2\pi} d\theta.$$

As we will see later, our analysis remains the same in case the law for the direction of motion is arbitrary, for instance $\delta_\theta$ for $\theta \in [-\pi, \pi)$ fixed or $\sum_{i=1}^n p_i \delta_{\theta_i}$, for some $\theta_i \in [-\pi, \pi)$ fixed, $n \in \mathbb{N}$ and $p_i \in [0, 1]$ such that $\sum_{i=1}^n p_i = 1$. Nevertheless we proceed with $U[-\pi, \pi)$ distributed random variables for the directions in several parts of this work.

At time $t$, the location of a mobile station, which was initially located at $X_i^0 = X_i$, is $X_i^t := X_i + V_i t$. We denote the set of locations of all the mobile stations at time $t$ by $\tilde{\Phi}^t$. For brevity, we shall write $\tilde{\Phi}^0 = \Phi$. By the displacement theorem [14, Section 5.5], [3, Theorem 2.2.17], the point process $\Phi$ corresponding to the initial locations of the stations and the new locations, given by $\tilde{\Phi}^t$, have the same distribution for all times $t$. The displacement theorem also guarantees that there is no collision almost surely under the dynamics. For completeness, we shall detail the result in Appendix A (Proposition A.1). We refer the reader to [10], for an interactive simulation of the joint mobility of the stations in the single-speed case.
The collective motion of the points on the plane gives rise to a dynamical Voronoi tessellation on the plane. Due to the motion of the stations, the corresponding Voronoi cells change their shapes continuously. The user being fixed at the origin, the nearest station to the user gets changed to a new one whichever becomes the closest. In other words, the Voronoi cell that covers the user is replaced by a new cell corresponding to the station nearest to the user after some time. The old cell spends a random amount of time covering the user, before another one replaces it. We call the event of change of cell a handover and the change of cell events create a point process on the time axis, called as the handover point process. Indeed, as we will show, under this dynamics, we have a countable collection of handover epochs without accumulation on the time axis, for each realization of the marked Poisson point process $\tilde{\Phi}$. The handover point process is denoted by $V_\lambda := \sum_{t \in S} \delta_t$, where informally,

$$S := \{ t \in \mathbb{R} : \exists (X_i, \Theta_i), (X_j, \Theta_j) \text{ atoms of } \tilde{\Phi}, \text{ such that } |X_i^t| = |X_j^t| \text{ and } \tilde{\Phi}^t(B_{|X_i^t|}(o)) = 0 \}.$$

We write $B_r(x)$ for an open ball of radius $r$, centered at $x \in \mathbb{R}^2$. For simplicity we shall write $V$ for $V_\lambda$. We are interested in the statistical properties of the point process $V$, for example its stationarity and its intensity, denoted by $\lambda_V$.

Another important quantity we are interested in this context is the duration between two successive handovers, namely the inter-handover time. In terms of the dynamic Voronoi tessellation, the inter-handover time is nothing but the time spent by a station while serving the user. The structure of the “history” of the inter-handover times is important as well.

It is equally important to consider the case when the stations are moving at different speeds and the directions are sampled uniformly at random as above. Suppose the set of speed is $v_{[n]} := \{v_i\}_{i \in [n]}$ for some $n \in \mathbb{N}$. For each $i \in [n]$, a station of type $i$ moves with speed $v_i$. The initial locations of the stations given by the point process $\Phi$ can be thought of as a superposition $\sum_{i \in [n]} \Phi^i$ of the collection of Poisson point processes $\Phi^i$ with intensity $\lambda_i$, corresponding to the stations of type $i$. The total intensity of $\Phi$ is $\lambda = \sum_{i \in [n]} \lambda_i$. The directions of motion are again chosen uniformly at random as above. Although, the directions having arbitrary distribution give us the same result. The displacement theorem (similar to the one in single-speed case in Proposition A.1) can also be shown to hold, even in the multi-speed setting.

Like in the single-speed setting, we will suppose that the system obeys the same nearest neighbor association rule. The corresponding dynamical Voronoi tessellation is of slightly different nature. The Voronoi cells corresponding to the faster stations pierce through the tessellation while respecting the space required for other cells. In short, the handover point process consists of points representing handover between stations of the same type and handover between stations of different types. The inter-handover time is again defined as the duration between two consecutive handovers, which is essentially the time spent by a station, of certain type, while serving a user.

The multi-speed scenario serves as a natural generalization of the model with single-speed. Apart from this, the motivation for considering this type of multi-speed mobility model comes from the non-terrestrial networks. Consider a communication network consisting of LEO satellite constellation where satellites are moving in different altitude orbits and are serving the users on the Earth directly. The speeds of satellites are different at different orbital altitudes, like for instance in satellite constellation consisting of multi-altitude LEO satellites. In this type of communication networks, the user on Earth performs a “handover” whenever it receives a better signal than the previous one. As the power of signal is a function of distance, the best signal can be considered to be from the nearest visible satellite and in this case, the two satellite types have different velocities.

### 3. QUESTIONS, STRATEGY AND STRUCTURE OF THE PAPER

One of the main goals of this article is to formally define the handover point process, as well as to determine its geometric and statistical properties aligned with the motivations provided in the preceding sections. We give a formal construction of the handover point process for the single-speed case in Subsection 4.5. This construction allows us to prove the stationarity of the handover point process based on a specific non-compact particle system, described in Subsection 4.3. Using this construction we derive our main result about the intensity of the handover point process in Theorem 4.26.
Informally, this is based on the distance function (defined in (4.3)) for each of the mobile stations, measuring the distance from the user. For each station, the distance function gives rise to an infinite closed subset of the Euclidean plane. Each of these closed sets, which we call radial birds are non-compact particles, which are branches of hyperbolas described in Section 4. Our analysis is based on the Poisson point process on $H^+$, called as the head point process and denoted as $\mathcal{H}$, given by the collection of all hyperbola vertices or equivalently radial bird heads, see Subsection 4.1. The intersections of certain pairs of radial birds satisfying an “empty half-ball” condition (or “empty half-ellipse” condition for speed $v \neq 1$) allows us to define the handover point process $\mathcal{V}$, see Lemma 4.18.

We will need to properly define the Palm measure with respect to the handover point process, informally defined as, $P_V^0(\cdot) := P_V(\cdot|\text{ a handover at time } 0)$. The formal construction of this handover Palm probability measure is the main objective of Section 5. An important mathematical tool of the paper lies in derivation of this Palm measure in terms of the mass transport principle in Theorem 5.4. This is instrumental in characterizing the handover Palm distribution of several geometric features of interest, as described in Section 4 and Section 5. The first application of this is the handover Palm distribution of inter-handover time, denoted by $T$, in Theorem 5.17 in Subsection 5.4. This also includes in particular the handover Palm distribution of the distance to the two stations involved in a handover (Subsection 5.2), and that of the distance point process of other stations at that time (Subsection 4.7).

The tools developed in this work also allow us to devise a Markov chain in $(\mathbb{R}^+)^3$ w.r.t. which the handover local characteristics are factors and the state of which is sufficient to predict essential information about the next handover. The reader should refer to Section 6 for more details. This opens up a natural way to prove further structural properties of the handover process including a central limit theorem, the law of large numbers or even large deviations as properties of the Markov chain mentioned above.

Our result about the handover frequency in the two-speed case is presented in Theorem 7.20 in Section 7. This can be considered as the preparation for further properties of the multi-speed case in Theorem 7.29. The distribution of inter-handover time under the Palm probability of handover is evaluated in Theorem 8.18 and Theorem 10.7, for the two-speed and the multi-speed setting. These results are inspired from the same result in the single-speed case. Because of the presence of slower and faster stations in the system, it happens that the faster stations serve the user for a very short period of time. On the other hand, the faster ones do not let the slower ones stay as the serving station for very long. For these reasons, it will be essential to augment our analysis with the extra information about the type of handover which is seen. Our main result in the two-speed case (Theorem 8.18) about the handover Palm probability distribution of inter-handover time captures all possible intricacies of the multi-type settings.

The intricacies envisioned here due to the multiple types of stations, still allow for a Markov chain representation of the dynamics. This not only allows one to predict the information about the next handover given the Markov state like in the single-speed case, but also to predict the type of next serving base stations, for example: handover from slow to slow, slow to fast, fast to fast stations and vice-versa. More details can be found in Section 9.

In the following we briefly provide a structure and summary of the article by considering Figure 3.1 and Figure 3.2 as guiding line, while also indicating the key geometric objects of interest in the single-speed and the two-speed case, respectively.

**3.1. Summary for the single-speed case.** Figure 3.1 gives a summary of the main contributions in the single-speed case. First of all, the radial bird corresponding to each station are depicted by the curves of the form $\mathcal{U}$ in Figure 3.1. The first contribution of the paper concern the handover point process, depicted by the symbols like $\times$ in Figure 3.1, whose intensity is determined in Theorem 4.26. We construct it in a rigorous way in Subsection 4.5, from the “lower envelope” of the radial birds, which can be found in Definition 4.14, depicted by thick blue curve ($\sim$) in Figure 3.1. The inter-handover time, for which we derive the handover Palm distribution in Subsection 5.4, is represented by the symbol $\leftrightarrow$ on the time-axis in Figure 3.1. The “handover distance”, whose handover Palm distribution is determined in Subsection 5.2, is represented by a vertical dashed line $\langle \ldots \rangle$ aligned with the vertical line depicted by $\text{---}$ at typical handover time, in Figure 3.1.
<figure><img src="image_2.png" /></figure>

<figure><img src="image_3.png" /><figcaption>FIGURE 3.1. Summary for the handover process in the single-speed case.</figcaption></figure>

We also study the point process of visible heads (this correspond to the 'best' location of the serving station), depicted by ● in Figure 3.1, as well as the Palm distribution of geometric objects of interest with respect to this visible head point process (Subsection 5.3.1). The distance of the serving station at its best location from the user, is depicted by the vertical dashed line ←---→ aligned with the vertical line represented by the symbol ┄ at that typical time, in Figure 3.1. The distance to the serving station at any typical time is represented by a vertical dashed line ←---→, measured on the vertical line depicted by the line ┄ at a typical time, in Figure 3.1. We represent the points of the point process of distances to all other stations by ● in Figure 3.1 on the vertical line at different typical times, see Subsection 4.7.

Section 6 concerns the construction of the handover process as a factor of a discrete time Markov chain, essentially derived from the head point process. We represent the state of the Markov chain by a structure ┄ resembling a fork, and the steps of the Markov chain by arrows of the form ↕. The inter-step time of the Markov chain is depicted by the symbol ←→ in Figure 3.1.

## 3.2. Summary for the two-speed case.
The diagram (Figure 3.2) for the handover process for the two-speed case is organized similarly. This contains two types of radial birds corresponding to the two speeds (depicted by curves of the form ↕ and ┄ in Figure 3.2). The radial bird particle process and the underlying head point process are defined in Subsection 7.1.1. Under the “empty half-ellipse” condition for multiple ellipses (Lemma 7.4), the intersections of certain pairs of radial birds (of the same or different types) give rise to the multi-type handover point process V, (depicted by × on the time-axis in Figure 3.2). See Subsection 7.1.7 for its construction. In this case, we depict the “lower envelope” (defined in Subsection 7.1.2) using red-blue thick curve of the form ~. We denote the handover types using a combinatorial like notation $[i,j]$, for some $i,j,q \in \{1,2\}$, which encodes the types of the birds involved and their relative order, is formally defined in Subsection 8.3.1. We derive the main result about the handover frequency for the two-speed case in Theorem 7.20. Section 8 gives the Palm probability distribution with respect to handovers and its application, in particular, the handover Palm distribution of the distance of the two stations involved in the handover (Lemma 8.9, Subsection 8.2) and that of inter-handover time (Theorem 8.18, Subsection 8.3). We use symbols similar to those in
Figure 3.1 to depict the geometric objects of interest in Figure 3.2, except for the visible heads of type 1 and type 2, which are depicted using symbols ● and ●, respectively, in Figure 3.2.

The final result is about the construction of the handover process as a factor of a discrete time Markov chain (Section 9). The Markov chain has a spatial part associated to head point process and a discrete part associated with the handover types of the form $[\begin{smallmatrix} q \\ i,j \end{smallmatrix}]$. We represent the spatial part of the chain with a structure of the form □ and the steps of the Markov chain using the arrows of the form ↕. The inter-step times of the Markov chain are again depicted by symbol ↔ in Figure 3.2.

<figure><img src="image_4.png" /><figcaption>FIGURE 3.2. Summary for the handover process in the two-speed case.</figcaption></figure>

### 3.3. Sequential organization of the article.
The organization of the rest of the article is as follows. In Section 4, we first set up the ground work for defining the radial bird particle process and it’s fundamental properties that will be used in deriving our result about handover frequency in Theorem 4.26. The heart of the problem lies in determining the Palm probability distribution with respect to handovers, which is the main content of the Section 5. This also contains the probability distribution of the inter-handover time under the Palm probability measure $P_V^0$ of handovers. This is done in Theorem 5.17, which is derived as an application of Theorem 5.4, along with many other applications. The construction of the handover process as a factor of a discrete time Markov chain is presented in Section 6 for the single-speed case. Section 7 contains the handover frequency analysis for the two and multi-speed scenarios, whereas Section 8 and Section 10 are dedicated for determining the distribution of handover time for the two and multi-speed settings. Section 9 discuss the fact that the handover process is a factor of a discrete time Markov chain, for the two-speed setting, as an adaptation of Section 6. For the purpose of completeness and ease of reading, we have provided a table of notation in Section 12. Section 11 lists a series of questions that need to be addressed in the future when moving towards more practical settings.

## 4. HANDOVER FREQUENCY: SINGLE-SPEED CASE

In what follows, we devise a method allowing us to define the handover point process in a rigorous way. For this, suppose that at time instant 0, there is a mobile station at $X$, an atom of $\Phi$ with polar coordinates $(R = |X|, \Psi)$, and with velocity vector $V := (\cos \Theta, \sin \Theta)$. For the sake of simplicity, we
assume that the stations are moving at unit speed. The results for general speed parameter $v$ hold up to a scaling that will be discussed below. Let $\alpha \equiv (\Psi - \Theta)$ (mod $2\pi$) be the relative angle between the location vector $X$ and the velocity vector $V$. Our method relies on the use of a distance function (a function of the time variable) defined for each of the mobile stations, and which helps to simplify our analysis by keeping a minimum number of variables.

For a station starting at $X$, the new location at time $t$ is $X^t = X + Vt$. The distance at time $t$ from the mobile station to the user $u$ at the origin $o$ is hence

$$|X^t| = (|X|^2 + 2t|X| \cos \alpha + t^2)^{\frac{1}{2}}, \qquad (4.1)$$

where $\alpha \equiv (\Psi - \Theta)$ (mod $2\pi$) is the relative angle. So the distance at time $t$ only depends on the initial distance $|X|$, the initial relative angle $\alpha$, and the time $t$. It will be convenient to use the point process $\tilde{\Phi}$ capturing the initial distances of mobile stations along with their relative motion directions. It is easy to see that the marked point process $\tilde{\Phi} := \sum_{i \in \mathbb{N}} \delta(R_i, \alpha_i)$, on the space $\mathbb{R}^+ \times [-\pi, \pi)$, is Poisson with intensity measure $\mu$ of density

$$d\mu := 2\pi r \lambda dr \otimes \frac{1}{2\pi} d\alpha. \qquad (4.2)$$

The variable $\alpha$ captures the randomness coming from the relative angle of motion of the station with respect to the user, which is uniform on the interval $[-\pi, \pi)$.

**Remark 4.1.** One can also consider the direction of motion $\Theta$ having any law, for instance $\delta_{\theta}$ for fixed $\theta \in [-\pi, \pi)$ or $\sum_{i=1}^n p_i \delta_{\theta_i}$, for fixed $\theta_i \in [-\pi, \pi)$, some $n \in \mathbb{N}$ and $p_i \in [0, 1]$ such that $\sum_{i=1}^n p_i = 1$. In both cases, the relative angle $\alpha \equiv (\Psi - \Theta)$ (mod $2\pi$), can be proved to be a uniformly distributed random variable on $[-\pi, \pi)$.

**Distance function:** For an atom $(R, \alpha)$ of $\tilde{\Phi}$, corresponding to a station starting at $X$, the distance at time $t$ from the user is

$$|X^t| = (R^2 + 2tR \cos \alpha + t^2)^{\frac{1}{2}} := f((R, \alpha), t). \qquad (4.3)$$

We call $f((R, \alpha), \cdot)$ the distance function. As a function of the time variable $t$, it possesses a unique minimum. Heuristically the collection of all unique minimum time-space locations corresponding to all stations gives rise to a point process on the upper half plane, which we analyze in the following subsection.

**4.1. Head point process.** Let $\mathbb{H}^+ := \mathbb{R} \times \mathbb{R}^+$, where the first component represents the time coordinate and the second one the height coordinate. The unique minimum of the function $f((R, \alpha), \cdot)$, defined in (4.3), with respect to the time variable, gives the distance of the mobile station to the user along its trajectory. Thus the nearest time and nearest location of the mobile station starting at $X$ are

$$(T, H) := \left( \arg \inf_{t \in \mathbb{R}} \{ f((R, \alpha), t) \}, f\left(\arg \inf_{t \in \mathbb{R}} \{ f((R, \alpha), t) \}\right) \right) = (-R \cos \alpha, R |\sin \alpha|). \qquad (4.4)$$

The locations $(T, H)$ are intrinsic and do not change with time, once the initial distances and the relative angles are fixed.

**Definition 4.2 (Head point process).** The point process

$$\mathcal{H} := \sum_{i \in \mathbb{N}} \delta_{(T_i, H_i)}, \qquad (4.5)$$

on $\mathbb{H}^+$, formed by the minimum time-space points $(T_i, H_i)$, as defined in (4.4), associated with the atoms $(R_i, \alpha_i)$ of $\tilde{\Phi}$, is called the head point process.

**Lemma 4.3 (Mapping Lemma).** The point process $\mathcal{H} = \sum_{i \in \mathbb{N}} \delta_{(T_i, H_i)}$ is a homogeneous Poisson point process on $\mathbb{H}^+$ with intensity measure $\nu$, where $d\nu := dt \otimes 2\lambda dh$.

*Proof Lemma 4.3.* Consider two Poisson point processes $\tilde{\Phi}^1$ on $\mathbb{R}^+ \times [0, \pi)$ and $\tilde{\Phi}^2$ on $\mathbb{R}^+ \times [-\pi, 0)$, of intensity measures $\mu_1, \mu_2$ respectively, where $d\mu_1 := \pi r \lambda dr \otimes \frac{1}{\pi} d\alpha$ and $d\mu_2 := \pi r \lambda dr \otimes \frac{1}{\pi} d\alpha$. The reason we need the intensity measures to be $\mu_1, \mu_2$ in the above form is that the location of mobile station at distance $r$ from the user can be uniformly distributed on the semi-circle of radius $r$ and centered at $o$. The point processes $\tilde{\Phi}^1, \tilde{\Phi}^2$ are two sub point processes of $\tilde{\Phi}$, and we can write $\tilde{\Phi} := \tilde{\Phi}^1 + \tilde{\Phi}^2$. Define
two bijective maps $g_1 : \mathbb{R}^+ \times [0, \pi) \to \mathbb{H}^+$, $g_2 : \mathbb{R}^+ \times [-\pi, 0) \to \mathbb{H}^+$ as follows:

$$g_1(r, \alpha) := (-r \cos \alpha, r \sin \alpha), \quad g_2(r, \alpha) := (-r \cos \alpha, -r \sin \alpha). \qquad (4.6)$$

By the mapping theorem, the Poisson point processes $\tilde{\Phi}^1$ under $g_1$ and $\tilde{\Phi}^2$ under $g_2$ map to two independent point processes $\tilde{\Phi}^1 \circ g_1^{-1}$ and $\tilde{\Phi}^2 \circ g_2^{-1}$, with intensity measures $\nu_1 := \mu_1 \circ g_1^{-1}$ and $\nu_2 := \mu_2 \circ g_2^{-1}$ on $\mathbb{H}^+$, respectively. Thus $\mathcal{H} := (\tilde{\Phi}^1 \circ g_1^{-1}) + (\tilde{\Phi}^2 \circ g_2^{-1})$ is also a Poisson point process with intensity measures $\nu := \mu_1 \circ g_1^{-1} \oplus \mu_2 \circ g_2^{-1}$ on $\mathbb{H}^+$ and we are interested in determining an exact expression for its intensity measure $\nu$ or its density.

For any measurable function $f : \mathbb{H}^+ \to \mathbb{R}$ we have, by the change of variable formula,

$$\begin{aligned}
& \int_0^\infty \int_{-\infty}^\infty f(t, h)dt \otimes 2\lambda dh \\
&= 2\lambda \int_0^\infty \int_0^\pi f(-r \cos \alpha, r \sin \alpha)d\alpha r dr \\
&= \lambda \int_0^\infty \int_0^\pi f(-r \cos \alpha, r \sin \alpha)d\alpha r dr + \lambda \int_0^\infty \int_{-\pi}^0 f(-r \cos \alpha, -r \sin \alpha)d\alpha r dr \\
&= \int_0^\infty \int_0^\pi f(-r \cos \alpha, r \sin \alpha)\frac{1}{\pi}d\alpha \otimes \pi r \lambda dr + \int_0^\infty \int_{-\pi}^0 f(-r \cos \alpha, -r \sin \alpha)\frac{1}{\pi}d\alpha \otimes \pi r \lambda dr. \quad (4.7)
\end{aligned}$$

Using the maps $g_1, g_2$ from (4.6) in (4.7) we obtain that

$$\begin{aligned}
& \int_0^\infty \int_{-\infty}^\infty f(t, h)dt \otimes 2\lambda dh \\
&= \int_0^\infty \int_0^\pi f \circ g_1(r, \alpha) \frac{1}{\pi} d\alpha \otimes \pi r \lambda dr + \int_0^\infty \int_{-\pi}^0 f \circ g_2(r, \alpha) \frac{1}{\pi} d\alpha \otimes \pi r \lambda dr. \quad (4.8)
\end{aligned}$$

Above, for both maps $g_1(r, \alpha) = (-r \cos \alpha, r \sin \alpha)$ and $g_2(r, \alpha) = (-r \cos \alpha, -r \sin \alpha)$, the Jacobian is $r$. In particular for any measurable set $A \subset \mathbb{H}^+$,

$$\mathbb{E}[\mathcal{H}(A)] = \mathbb{E}[\tilde{\Phi}^1 \circ g_1^{-1}(A)] + \mathbb{E}[\tilde{\Phi}^2 \circ g_2^{-1}(A)] = \mu_1 \circ g_1^{-1}(A) + \mu_2 \circ g_2^{-1}(A) = \nu(A).$$

The above equivalences establish the fact that the Poisson point processes $\tilde{\Phi}^1, \tilde{\Phi}^2$ on $\mathbb{R}^+ \times [0, \pi)$ and $\mathbb{R}^+ \times [-\pi, 0)$, with intensity measures $\mu_1, \mu_2$, respectively, can be mapped to another Poisson point process on $\mathbb{H}^+$ of intensity measure $\nu$, with density $d\nu := dt \otimes 2\lambda dh$, bijectively, through the maps $g_1, g_2$ together. On the other hand, we get a 2 in the density $dt \otimes 2\lambda dh$ of the intensity measure of the heads because there exist two points, one on each of the spaces $\mathbb{R}^+ \times [0, \pi)$ and $\mathbb{R}^+ \times [-\pi, 0)$, that have the same image in $\mathbb{H}^+$ under the maps $g_1, g_2$. $\square$

**Remark 4.4.** We write the probability measure $\mathbb{P}_{\mathcal{H}}$ as the law of the head point process $\mathcal{H}$ and $\mathbb{E}_{\mathcal{H}}$ denotes the expectation under $\mathbb{P}_{\mathcal{H}}$. From now onward, our analysis will essentially rely on the head point process $\mathcal{H}$ instead the station point process $\tilde{\Phi}$, exploiting the power of the mapping lemma (Lemma 4.3).

**Remark 4.5.** (Speed $v \neq 1$) Suppose the stations are moving at a speed $v \neq 1$. As before, we consider $\tilde{\Phi}$ as a marked Poisson point process on $\mathbb{R}^+ \times [-\pi, \pi]$. For any $(R, \alpha) \in \mathbb{R}^+ \times [-\pi, \pi]$, corresponding to any station, it's nearest location to the user is $(T, H) = (-\frac{R}{v} \cos \alpha, R |\sin \alpha|)$. Indeed, as it is the time-space minimum of the distance function

$$f_v((R, \alpha), t) = (R^2 + 2tvR \cos \alpha + v^2t^2)^{\frac{1}{2}}. \qquad (4.9)$$

In this case, the intensity of the head point process $\mathcal{H}$ is given by

$$d\nu = vdt \otimes 2\lambda dh.$$

Indeed, the maps required in Lemma 4.3 are

$$g_1(R, \alpha) = \left(-\frac{R}{v} \cos \alpha, R \sin \alpha\right), \quad g_2(R, \alpha) = \left(-\frac{R}{v} \cos \alpha, -R \sin \alpha\right),$$

and, in (4.8), the corresponding Jacobian for both maps $g_1, g_2$ turns out to be $R/v$. This remark will be very useful in determining the handover frequency in the single-speed ($v \neq 1$) and multi-speed cases.
**Observation 4.6.** In order to determine which station is the nearest at any given time, it will be important to keep track of the time and distance coordinate data $\{(t, f((R, \alpha), t)) : t \in \mathbb{R}\} \subset \mathbb{H}^{+}$, for each station corresponding to $(R, \alpha)$ in the point process $\tilde{\Phi}$.

**Remark 4.7** (Distance function as a branch of a hyperbola on $\mathbb{H}^{+}$). Consider a station initially at distance $R$ and moving at a speed $v$ with a relative angle $\alpha$. The equation of the distance function in (4.9) can be rephrased as the equation of the hyperbola:

$$h^2 - v^2 \left( t + \frac{R}{v} \cos \alpha \right)^2 = R^2 \sin^2 \alpha, \qquad (4.10)$$

in $\mathbb{H}^{+}$ in $(t, h)$-coordinate as where $h$ denotes the distance of the station at time $t$. The center of the hyperbola is $(-\frac{R}{v} \cos \alpha, 0)$ and its vertex is $(-\frac{R}{v} \cos \alpha, R|\sin \alpha|)$.

**Definition 4.8** (Radial bird closed set). For each $(R, \alpha)$ atom of $\tilde{\Phi}$, let

$$C_{(R,\alpha)} := \{(t, f((R, \alpha), t)) : t \in \mathbb{R}\}. \qquad (4.11)$$

We call the infinite closed set $C_{(R,\alpha)} \subset \mathbb{H}^{+}$ a radial bird closed set or a radial bird particle or simply a radial bird.

**Remark 4.9.** For each $(R, \alpha)$ atom of the point process $\tilde{\Phi}$, the closed set $C_{(R,\alpha)}$ defined by the radial distance of the mobile station to the user at different times and the set looks like a bird, hence the name.

In the following, we showcase some properties of the distance function $f$ and of the corresponding radial bird. This is a preparation to the “radial bird particle process” in Subsection 4.3.

## 4.2. Properties of the distance function.
Consider an atom $(R, \alpha)$ of $\tilde{\Phi}$. The corresponding distance function and radial bird, $f((R, \alpha), \cdot)$ and $C_{(R,\alpha)}$, respectively, possess the following features:

1. **Head:** the minimum time-space location $(T, H)$ is the head of the radial bird.

2. **Distance to the head:** the distance function $f((R, \alpha), \cdot)$, which gives the distance of the station to the user at time $t$, satisfies the relation,

$$f((R, \alpha), t) = (R^2 + 2tR \cos \alpha + t^2)^{\frac{1}{2}} = ((t + R \cos \alpha)^2 + (R|\sin \alpha|)^2)^{\frac{1}{2}} = |(t, 0) - (T, H)|, \quad (4.12)$$

for any $t \in \mathbb{R}$. A key observation in (4.12) is that the distance at time $t$ to the mobile station starting at $X$ is also $|(t, 0) - (T, H)|$, see Figure 4.1 (Picture 1). This essentially means that the radial bird closed sets are purely determined by their heads.

<figure><img src="image_5.png" /><figcaption>(1) The distance of the station at time $t$ is the distance between $(t, 0)$ and the head point at $(T, H)$.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(2) A family of curves with their heads located at a vertical line $t = s$ and the straight lines $h = s + t$, $h = s - t$ are their asymptotes.</figcaption></figure>

<caption>FIGURE 4.1. Properties of the distance function.</caption>

3. **Family of curves:** Fix some $s \in \mathbb{R}$. For $u \in \mathbb{R}^{+}$, $g_{s,u}(t) := |(t, 0) - (s, u)|$ is the height at time $t$ of the bird with head at $(s, u)$. Then $\{g_{s,u}(\cdot)\}_{u \in \mathbb{R}^{+}}$ defines the set of all radial birds with head at $(s, u)$ when varying $u \in \mathbb{R}^{+}$. Each element in this family has the lines $h = s + t$ and $h = s - t$, for $h \in \mathbb{R}^{+}$, as its asymptotes (see Picture 2 of Figure 4.1).

4. **Translation:** Observe that, for fixed $u \in \mathbb{R}^{+}$ and $s_1 < s_2 \in \mathbb{R}$, the two subsets of $\mathbb{H}^{+}$, $A_1 := \{(t, g_{s_1,u}(t)) : t \in \mathbb{R}\}$ and $A_2 := \{(t, g_{s_2,u}(t)) : t \in \mathbb{R}\}$ are just translated versions of each other. For instance, $A_2 = (s_2 - s_1, 0) + A_1$. This is the pivotal reason for the time-stationarity of the spatially marked point process described later.
5. **Unique intersection:** For $(s_1, u_1)$, $(s_2, u_2) \in \mathbb{H}^+$ with $s_1 \neq s_2$, the intersection of the two sets, $A_1 := \{(t, g_{s_1, u_1}(t)) : t \in \mathbb{R}\}$ and $A_2 := \{(t, g_{s_2, u_2}(t)) : t \in \mathbb{R}\}$, contains exactly one point.

6. **Non-unit speed:** If $v \neq 1$, the head points are given by $(T, H) = \left(-\frac{R}{v} \cos \alpha, R|\sin \alpha|\right)$, for some $(R, \alpha)$ from $\tilde{\Phi}$. Then similarly to (4.12) the distance function $f_v((R, \alpha), \cdot)$ is given by

$$f_v((R, \alpha), t) = \left( v^2 \left( t + \frac{R}{v} \cos \alpha \right)^2 + (R|\sin \alpha|)^2 \right)^{\frac{1}{2}} = (v^2(t-T)^2 + H^2)^{\frac{1}{2}}, \quad (4.13)$$

where we only scale the time coordinate. For fixed $v \neq 1$, let $g_{s,u}^{(v)}(t) := (v^2(t-s)^2 + u^2)^{\frac{1}{2}}$. Then each element of the family of curves $\{g_{s,u}^{(v)}(\cdot)\}_{u \in \mathbb{R}^+}$, has the lines $h = s + vt$ and $h = s - vt$, for $h \in \mathbb{R}^+$, as its asymptotes. All the above properties, (1), (3), (4), (5) are also true for any $v \neq 1$.

### 4.3. Radial bird particle process.

**Definition 4.10 (Radial bird Particle Process (RBPP)).** The radial bird particle process is the point process

$$P_c := \sum_{i \in \mathbb{N}} \delta_{C_i},$$

on $C(\mathbb{H}^+)$, where $C_i$ is the particle corresponding to the atom $(R_i, \alpha_i)$ of $\tilde{\Phi}$.

We have the following result about $P_c$ as a Poisson set/particle process. See [3, Definition 9.2.17], for the reference of Poisson set processes.

**Lemma 4.11.** The following holds true for the radial bird particle process $P_c$:

(i) $P_c$ is a Poisson particle process on the space $(C(\mathbb{H}^+), \mathcal{B}(C(\mathbb{H}^+)))$, with support on the set of closed sets of the form (4.11).

(ii) Both $P_c$ and $\cup_i C_i$ are stationary along the time coordinate.

<figure><img src="image_7.png" /><figcaption>FIGURE 4.2. The birds together with their heads form the radial bird particle process $P_c \equiv \mathcal{H}_c$.</figcaption></figure>

*Proof of Lemma 4.11 (i).* There is a bijection between the atom $(R_i, \alpha_i)$, the corresponding radial bird $C_i \equiv C_{(R_i, \alpha_i)}$ and its head $(T_i, H_i)$. The proof then follows from the inverse construction of set valued marked Poisson point process from the set process described in [3, Proposition 10.2.10], using the Poisson point process $\mathcal{H} = \sum_{i \in \mathbb{N}} \delta_{(T_i, H_i)}$ of the heads as the background process, see Figure 4.2. $\square$

*Proof of Lemma 4.11 (ii).* Let $t \in \mathbb{R}$ and $S_t : C(\mathbb{H}^+) \to C(\mathbb{H}^+)$ be the map defined as $S_t(C) := C - (t, 0)$. The particle process $S_t(P_c)$ is the particle process $P_c$ shifted by $(t, 0)$, namely

$$S_t(P_c) := \sum_{i \in \mathbb{N}} \delta_{S_t(C_i)} = \sum_{i \in \mathbb{N}} \delta_{C_i - (t, 0)}.$$
Under the shift $S_t$, each of the radial bird particles is shifted towards the left along the time axis. Using Property 4, $S_t(P_c)$ has the same distribution as $P_c$. Hence we have time-stationarity of $P_c$. Alternatively, the shape of the set valued mark depends only on the height coordinate of the head, not on the time coordinate, see Properties 2 and 4. So the time-stationarity of the radial bird particle process $P_c$ follows from the time stationarity of the background point process $\mathcal{H}$ corresponding to the heads of the radial bird particles. $\square$

**Remark 4.12.** Rather than the particle process $P_c$, one can consider the marked point process

$$ \mathcal{H}_c := \sum_{i \in \mathbb{N}} \delta_{(T_i, H_i, C_i)}, \qquad (4.14) $$

on $\mathbb{H}^+$, with marks on the space of closed sets of $\mathbb{H}^+$. It is clear that this Poisson point process is stationary along the time axis as well.

**Discussion 4.13 (RBPP: stochastic process on $(\mathbb{R}^+)^N$).** For each $t \in \mathbb{R}$, we can construct a sequence of positive random variables that represent the distances of the stations at time $t$. For each atom $(R_i, \alpha_i)$ of $\tilde{\Phi} = \sum_{i \in \mathbb{N}} \delta_{(R_i, \alpha_i)}$, let $\xi_i^t := f((R_i, \alpha_i), t)$. We shall re-order the elements of the sequence $\{\xi_i^t\}_{i \in \mathbb{N}}$ by defining $\xi_{(i)}^t$ to be the distance of the $i$-th closest station at time $t$. Formally, for each $t$, we construct the sequence $\xi(t) := \{\xi_{(k)}^t\}_{k \in \mathbb{N}}$ as

$$ \xi_{(1)}^t := \inf\{\xi_i^t\}_{i \in \mathbb{N}} \text{ and, for } k > 1, \xi_{(k)}^t := \inf\{\{\xi_i^t\}_{i \in \mathbb{N}} \setminus \{\xi_{(i)}^t\}_{1 \le i < k}\}, $$

which defines the ordered sequence of distances of the stations. So the radial bird particle process $P_c$ allows one to build a stochastic process $\Xi := \{\xi(t)\}_{t \in \mathbb{R}}$ on the space $(\mathbb{R}^+)^N$. For any $t_0 \in \mathbb{R}$, the intersection of the vertical line $t = t_0$ with the union of the radial bird closed sets, $\cup_{i \in \mathbb{N}} C_i$, is given by the set $\{(t_0, \xi_{(i)}^{t_0})\}_{i \in \mathbb{N}}$, as seen in Figure 4.3, where $C_i \equiv C_{R_i, \alpha_i}$ with $(R_i, \alpha_i)$ the points of $\tilde{\Phi}$. What is even more useful is the data $\xi(t) = \{\xi_{(i)}^t\}_{i \in \mathbb{N}}$, at time $t$, which correspond to the distances to the stations. In the words of wireless communication, this enable us to compute the signal and the interference power experienced by the user at any given time $t$. In Subsection 4.7, we determine the distribution of $\xi(t)$ as a point process on $\mathbb{R}^+$, at different time epochs of interest, in particular at handover epochs and beyond.

<figure><img src="image_8.png" /><figcaption>FIGURE 4.3. The blue $\times$'s forms the handover point process on the time-axis. The blue nodes ($\bullet$), block nodes ($\bullet$) and red nodes ($\bullet$), respectively, on individual vertical blue, black and red lines, respectively, forms the point process corresponding to the distance of all the mobile base stations at handover times, at a time corresponding to a head and at a typical time, respectively.</figcaption></figure>

Having defined the radial bird particle process, we are now interested in understanding its properties, in particular the lower envelope process, which plays a fundamental role for defining the handover epochs.
**Definition 4.14 (Lower envelope process).** Associated with the marked Poisson point process of the stations $\tilde{\Phi} = \sum_{i \in \mathbb{N}} \delta_{(R_i, \alpha_i)}$, define the function

$$t \mapsto L(t) := \inf_{i \in \mathbb{N}} f((R_i, \alpha_i), t).$$

The stochastic process $\{\mathcal{L}(t) = (t, L(t))\}_{t \in \mathbb{R}}$ is called the lower envelope process, where the random variable $L(t)$ represents the distance of the closest station at time $t$. The **lower envelope** is the random closed set on $\mathbb{H}^+$ defined as

$$\mathcal{L}_e := \{(t, L(t)), t \in \mathbb{R}\} \subset \mathbb{H}^+.$$

**Lemma 4.15.** The lower envelope process $\mathcal{L}(\cdot)$ is stationary with respect to time.

*Proof of Lemma 4.15.* The time stationarity of $\mathcal{L}(\cdot)$ follows from the time stationarity of the head point process $\mathcal{H}$. The lower envelope process $\mathcal{L}(\cdot)$ is a factor of the head point process $\mathcal{H}$, from Lemma 4.11. □

**Remark 4.16.** Let us denote the open region above the lower envelope, as $\mathcal{L}_e^+$. Suppose $(s, u) \in \mathbb{H}^+$ is an intersection of two radial birds.

**4.4. Characterization of the lower envelope $\mathcal{L}_e$.** In this subsection, we give a characterization of the lower envelope $\mathcal{L}_e$, namely a criterion for a point $(t, h) \in \mathbb{H}^+$ to be a handover point, which will be instrumental in what follows. We first consider the unit speed case. For $h \in \mathbb{R}^+$, let $U_h := B_h(o) \cap \mathbb{H}^+$, be the upper open half-ball of radius $h$ and centered at $(0, 0)$. We shall write $U_h^s := (s, 0) + U_h$, so that $U_h^0 := U_h$.

**Observation 4.17.** Let $(s, h), (s', h') \in \mathbb{H}^+$. Let $C$ be the radial bird with its head at $(s, h)$. Let $\hat{h}$ be the height at which the radial bird $C$ intersects the vertical line $t = s'$. Then $(s, h) \in U_{h'}^{s'}$ if and only if $\hat{h} < h'$, as depicted in Figure 4.4. This follows from the alternative description of the distance function in (4.12).

<figure><img src="image_9.png" /><figcaption>FIGURE 4.4. Scenario in Observation 4.17. All birds with head point inside and outside, respectively, of the half-ball of height $h'$, intersect the vertical line below and above the level $h'$, respectively.</figcaption></figure>

**Lemma 4.18.** For any point $(t, h) \in \cup_{i \in \mathbb{N}} C_i$, the following holds:

(i). The point $(t, h)$ belongs to $\mathcal{L}_e$ if and only if $\mathcal{H}(U_h^t) = 0$, where $\mathcal{H}$ is the head point process.

(ii). The probability of such an event is

$$\mathbb{P}((t, h) \in \mathcal{L}_e) = e^{-\lambda \pi h^2}.$$

*Proof of Lemma 4.18 (i).* Without loss of generality, we shall prove the result for $t = 0$. Suppose $(0, h) \in \mathcal{L}_e$. Then there exists a radial bird $C' \equiv C_{(R', \alpha')}$, for some atom $(R', \alpha')$ of $\tilde{\Phi}$, such that $(0, h) \in C'$. We have two cases, depending on whether $(0, h)$ is a head of the radial bird $C'$ or not. Let $(\mathbb{H}^+)^*$ denote the open upper half plane $\mathbb{R} \times (0, \infty)$ and, for any open set $A \subset \mathbb{R}^2$, let $\partial^* A := \partial A \cap (\mathbb{H}^+)^*$, denote the boundary of an open set $A$ contained inside $(\mathbb{H}^+)^*$. We refer to the horizontal axis as the $t$-axis and to the vertical axis as the $h$-axis.

**Case 1:** $(0, h)$ is a head point. Suppose $(0, h)$ is the head of $C'$, as in Figure (4.5) (Picture 1). We shall first prove that any radial bird with its head on $\partial^* U_h$, passes through $(0, h)$. Suppose $(s, u)$ is an
atom of $\mathcal{H}$ such that $(s, u) \in \partial U_h$, then $s^2 + u^2 = h^2$. On the other hand, for the same $(s, u)$, there exists $(R, \alpha)$, an atom of $\tilde{\Phi}$ such that $(s, u) = (-R \cos \alpha, R|\sin \alpha|)$ and $s^2 + u^2 = R^2$. The corresponding radial bird is given by

$$f((R, \alpha), t) = (R^2 + 2tR \cos \alpha + t^2)^{1/2}. \qquad (4.15)$$

The curve (4.15) passes through $(0, h)$, since $f((R, \alpha), 0) = R = (s^2 + u^2)^{1/2} = h$. The result follows from Observation 4.17, that any radial bird with its head inside $U_h$ intersects the $h$-axis at a height lower than $h$. So, if there is any atom of $\mathcal{H}|_{U_h}$, then the corresponding radial bird prevents $(0, h)$ from being on the lower envelope $\mathcal{L}_e$.

<figure><img src="image_10.png" /></figure>

(1) The point $(0, h)$ is the head of a radial bird (in red). A radial bird (in blue) with its head on $\partial^*U_h$ passes through $(0, h)$ and any other radial bird (in black) with its head inside $U_h$ intersects the $h$-axis below the level $h$.

<figure><img src="image_11.png" /></figure>

(2) The point $(0, h)$ is on a radial bird (in red). A radial bird (in blue) on $\partial^*U_h$ passes through $(0, h)$ and any other radial bird (in black) with its head inside $U_h$ intersects the $h$-axis below level $h$.

<caption>FIGURE 4.5. The two scenarios of $(0, h)$ being or not being a head point.</caption>

**Case 2:** $(0, h)$ is not a head point. Suppose $(0, h)$ is any other point of $C'$ than its head, as in Figure (4.5) (Picture 2). Let $(s, u)$ be a point on $\partial^*U_h$. Using the same argument as in case (5), one can prove that the radial bird with its head at $(s, u)$ passes through $(0, h)$. Due to Observation 4.17, any other radial bird with its heads inside $U_h$ intersects the $h$-axis below level $h$. So it must be the case that $\mathcal{H}(U_h) = 0$; otherwise $(0, h) \notin \mathcal{L}_e$. Hence we have the result. $\square$

*Proof of Lemma 4.18 (ii).* Using part (i), the event a point $(t, h)$ lies on the lower envelope we have,

$$\{(t, h) \in \mathcal{L}_e\} = \{\mathcal{H}(U_h^s) = 0\}.$$

The probability of such an event equals $e^{-2\lambda|U_h^s|} = e^{-2\lambda\pi h^2/2} = e^{-\lambda\pi h^2}$. $\square$

**Remark 4.19.** Let $(0, h)$ be the intersection point of two radial birds with their heads at $(t_1, h_1)$ and $(t_2, h_2)$, respectively. Then the semi-circle $\partial^*U_h$ passes through both heads at $(t_1, h_1)$ and $(t_2, h_2)$ (see Figure 4.6 (Picture 1)). This implies that any other radial bird with its head inside $U_h$ intersects the height-axis below the level $h$, and this prevents $(0, h)$ from being a handover point. As a consequence, we have that $\mathcal{H}(U_h) = 0$.

<figure><img src="image_12.png" /></figure>

(1) The semicircle of radius $h$, passing through the intersection of two birds, passes through their heads, for any pair of birds. For $(0, h)$ to be a handover point, the regions $U_h$ must be empty of heads.

<figure><img src="image_13.png" /></figure>

(2) The half-ellipse of semi-major axis of length $h$, passing through the intersection of two birds, passes through their heads, for any pair of birds. For $(0, h)$ to be a handover point, the regions $E_h^v$ must be empty of heads.

<caption>FIGURE 4.6. Half-ball condition and half-ellipse condition.</caption>

**Remark 4.20.** We often refer to the key geometric result, Lemma 4.18, as the **half-ball condition** in case of $v = 1$.
**Discussion 4.21 (Speed $v \neq 1$).** Consider the general setup where the stations are moving at speed $v \neq 1$. Then the region that needs to be empty of head points, for a point $(0, h)$ to be on the lower envelope process, is the region enclosed by the upper half ellipse with equation $v^2x^2 + y^2 = h^2$, see Figure 4.6 (Picture 2). For any $s \in \mathbb{R}$, we denote by $E_h^{s,v}$ the open set enclosed by the upper half-ellipse given by the equation $v^2(x-s)^2 + y^2 = h^2$ and $E_h^v := E_h^{0,v}$. The probability that $E_h^{s,v}$ is empty of head points from $\mathcal{H}$ is

$$ \mathbb{P}(\mathcal{H}(E_h^{s,v}) = 0) = e^{-2\lambda v|E_h^{s,v}|} = e^{-2\lambda v \times \pi \frac{h^2}{2v}} = e^{-\lambda \pi h^2}, $$

using Remark 4.5. We refer to Lemma 4.18 as the **half-ellipse condition** in the case of $v \neq 1$. Similarly, we can generalize this to the multi-speed case, which will be the content of Section 7.

**Remark 4.22.** The void probability remains the same as in the unit speed case. In the fast movement case ($v > 1$), the increase in the intensity of heads is compensated by the region of the upper half ellipse having smaller area. In the slow case ($v < 1$), the decrease in the intensity of heads is balanced by the area of a larger upper half ellipse. In this case, we denote the lower envelope as $\mathcal{L}_e(v)$.

**Remark 4.23.** We say that an intersection point $(s, u)$ of two radial birds, does not correspond to a handover if $(s, u) \in \mathcal{L}_e^+$. For speed $v \neq 1$, we denote the open region above the lower envelope as $\mathcal{L}_e^+(v)$.

4.5. **Construction of the handover point process.** This subsection showcases how the handover point process can be derived from the lower envelope process, more precisely using the marked head point process $\mathcal{H}_c$ with set valued marks defined in (4.14). The construction of the handover point process $\mathcal{V}$ from $\mathcal{H}_c$ is based on finding pairs of head points corresponding to a handover epoch, along with a non-negative mark attached to them. The former represents the time-coordinate at which the two radial birds intersect and the mark represents the height at which the corresponding birds intersect. A thinning is used to retain only those pairs of heads that gives rise to handovers.

By Remark 4.12, $\mathcal{H}_c$ is defined using the head point process $\mathcal{H}$ along with the corresponding radial bird attached to each head as a mark. By Property (2), each of the closed set valued mark is determined by the coordinate of head point. Also the intersection point of two birds is a function of the pair of heads of the corresponding birds. For this reason, we can detect the handovers just using the head point process. We adopt the convention to denote the radial bird with its head at $(t, h)$ by $C_{(t,h)}$, in view of Property 2.

<figure><img src="image_14.png" /><figcaption>FIGURE 4.7. Construction of the handover point process $\mathcal{V}$ and its marked version $\mathcal{V}^*$.</figcaption></figure>

Let $(t, h) \neq (t', h') \in \mathbb{H}^+$ be such that $t' < t$ and the corresponding radial birds $C_{(t,h)}$, $C_{(t',h')}$ intersect at $(\hat{s}, \hat{h})$. The quantities $\hat{s}$ and $\hat{h}$ are derived later in (4.22) and (4.23), respectively, in terms of $(t, h)$, $(t', h')$. Let $A(\hat{s}, \hat{h})$ be the event that the intersection point $(\hat{s}, \hat{h})$ represents a handover. By Lemma 4.18, the
event $A(\hat{s}, \hat{h})$ is equivalent to having no points of $\mathcal{H}$ in the upper half-ellipse $U_{\hat{h}}^{\hat{s}}$, i.e., $\{\mathcal{H}(U_{\hat{h}}^{\hat{s}}) = 0\}$. In this sense the point process given by

$$ \hat{\mathcal{V}} := \sum_{(T_i, H_i) \in \mathcal{H}} \sum_{(T_j, H_j) \in \mathcal{H}: T_j < T_i} \delta_{(\hat{s}, \hat{H})} \mathbb{1}_{A(\hat{s}, \hat{H})}, \qquad (4.16) $$

represents the collection of intersection points responsible for handovers, where $(\hat{s}, \hat{H})$ is the intersection point of the radial birds at $(T_i, H_i)$ and $(T_j, H_j)$. We have considered the pair of head points $(T_i, H_i)$ and $(T_j, H_j)$ such that $T_j < T_i$, just to avoid any double counting. In Figure 4.7, the points $\{\times, \times\}$ are the projections of intersections of the birds, on the time axis, representing the corresponding pair of head points. We shall discard the pair of heads corresponding to the black ones ($\times$) and retain only the pair for blue ones ($\times$) to form the handover point process. The blue nodes ($\bullet$) correspond to the handover epochs and the distances at which the handovers happen. Finally, the handover point process is given by

$$ \mathcal{V} := \sum_{(T_i, H_i) \in \mathcal{H}} \sum_{(T_j, H_j) \in \mathcal{H}: T_j < T_i} \delta_{\hat{S}} \mathbb{1}_{A(\hat{s}, \hat{H})}. \qquad (4.17) $$

We call the point process $\hat{\mathcal{V}}$ the marked version of the handover point process $\mathcal{V}$, where the handover points are associated to non-negative marks that represents the height at which the corresponding radial birds intersect. In another words, the mark denotes the handover distance. The mark $\hat{H}$ is also used in for the retention of the pair of head points corresponding to the intersection at $(\hat{s}, \hat{H})$.

**Remark 4.24.** Alternatively, in view of (4.17) one can define the handover point process $\mathcal{V}$ as

$$ \mathcal{V} := \sum_{(T_i, H_i) \in \mathcal{H}} \sum_{(T_j, H_j) \in \mathcal{H}: T_j > T_i} \delta_{\hat{S}} \mathbb{1}_{A(\hat{s}, \hat{H})}, \qquad (4.18) $$

by counting pair of head points $(T_i, H_i)$, $(T_j, H_j) \in \mathcal{H}$ such that $T_j > T_i$.

In the following result, the time-stationarity of the handover point process $\mathcal{V}$, is inherited from the same property of the head point process $\mathcal{H}$.

**Lemma 4.25** (Time-stationarity of $\mathcal{V}$). The handover point process $\mathcal{V}$ is stationary with respect to time.

## 4.6. Handover frequency for the single-speed.

**Theorem 4.26** (Handover frequency for the single-speed). The handover frequency or intensity of the handover point process $\mathcal{V}$ is $\lambda_{\mathcal{V}} = \frac{4\nu\sqrt{\lambda}}{\pi}$, where $\lambda$ is the intensity of mobile stations and $\nu$ is their speed.

The proof relies on the representation of the two-point Palm probability distribution of $\mathcal{H}$, which we define in the following before going into the proof of the last theorem.

4.6.1. Two point Palm probability distribution of $\mathcal{H}$. It is clear from (4.16) that each handover event is characterized as an intersection of two radial bird particles, that correspond to two atoms of the head point process $\mathcal{H}$ satisfying certain properties. This requires us to define the two-point Palm probability measure for $\mathcal{H}$. Recall that, from Lemma 4.3, the head point process $\mathcal{H}$ is Poisson with intensity measure $\nu$ having for density $d\nu = dt \otimes 2\lambda dh$.

Let $M(\mathbb{H}^+)$ be the space of all locally finite measures on $\mathbb{H}^+$. We define $M(\mathbb{H}^+)$ as the $\sigma$-algebra on $M(\mathbb{H}^+)$, generated by maps of the form $\mu \mapsto \mu(A)$, for any $A \in \mathcal{B}(\mathbb{H}^+)$ and $\mu \in M(\mathbb{H}^+)$, where $\mathcal{B}(\mathbb{H}^+)$ is the Borel $\sigma$-algebra on $\mathbb{H}^+$, see [3] for more details. The following result is classical for Poisson point processes:

**Lemma 4.27** (Two-point Palm probability distribution of $\mathcal{H}$). The two point Palm probability distribution of the head point process is

$$ \mathbb{P}_{\mathcal{H}}^{(t,h),(t',h')}(B) := \mathbb{P}(\mathcal{H}^{(t,h),(t',h')} \in B), $$

for any $B \in \mathcal{M}(\mathbb{H}^+)$, with $\mathcal{H}^{(t,h),(t',h')} := \mathcal{H} + \delta_{(t,h)} + \delta_{(t',h')}$ and $\mathcal{H}$ the stationary version of the head point process.
This result extends naturally to the marked version of the head point process $\mathcal{H}_c$ defined in (4.14).

**Lemma 4.28** (Two-point Palm probability distribution of $\mathcal{H}_c$). The two point Palm probability distribution of $\mathcal{H}_c$ is

$$ \mathbb{P}_{\mathcal{H}_c}^{(t,h,C),(t',h',C')}(\cdot) := \mathbb{P}\left(\mathcal{H}_c^{(t,h,C),(t',h',C')} \in \cdot\right), $$

with $\mathcal{H}_c^{(t,h,C),(t',h',C')} := \mathcal{H}_c + \delta_{(t,h,C)} + \delta_{(t',h',C')}$ and $\mathcal{H}_c$ the stationary version of the head point process. Here $C$ and $C'$ are the radial birds with head point at $(t, h)$ and $(t', h')$, respectively.

Suppose there are two radial bird particles $C, C'$ with their heads at $(t, h), (t', h')$, respectively. Let $(\hat{s}, \hat{h})$ be the intersection point of $C, C'$, where $\hat{s}$ is the time when the two radial birds intersect and $\hat{h}$ is the distance at which they intersect. Let $U_{\hat{h}}^{\hat{s}}$ be the upper half-ball centered at $(\hat{s}, 0)$ and of radius $\hat{h}$. It follows from Lemma 4.18 and Remark 4.19 that $\partial^* U_{\hat{h}}^{\hat{s}}$ also passes through both heads $(t, h)$ and $(t', h')$, see Figure 4.8. It also follows from Lemma 4.18 that a handover happens at $(\hat{s}, \hat{h})$ if and only if $\mathcal{H}(U_{\hat{h}}^{\hat{s}}) = 0$.

<figure><img src="image_15.png" /><figcaption>FIGURE 4.8. Given $(t, h)$ and $(t', h')$, the intersection $(\hat{s}, \hat{h})$ is a handover point if and only if $\mathcal{H}(U_{\hat{h}}^{\hat{s}}) = 0$.</figcaption></figure>

We have the following result about the independence of the head point processes restricted to $U_{\hat{h}}^{\hat{s}}$ and $(U_{\hat{h}}^{\hat{s}})^c$, under the two point Palm probability measure $\mathbb{P}_{\mathcal{H}_c}^{(t,h,C),(t',h',C')}$. We state it as a corollary without proof, as it directly follows from the fact that Poisson point process on disjoint sets are independent, together with Lemma 4.18.

**Corollary 4.29.** Under the two-point Palm probability measure $\mathbb{P}_{\mathcal{H}}^{(t,h),(t',h')}$,

(i) The head point processes $\mathcal{H}|_{U_{\hat{h}}^{\hat{s}}}$ and $\mathcal{H}|_{(U_{\hat{h}}^{\hat{s}})^c}$ are independent.

(ii) Additionally, under the event $\{\mathcal{H}(U_{\hat{h}}^{\hat{s}}) = 0\}$, the point process $\mathcal{H}|_{(U_{\hat{h}}^{\hat{s}})^c}$ is Poisson with intensity of density $dt \otimes 2\lambda dh$.

*Proof of Theorem 4.26 (Handover frequency).* The proof mainly relies on the characterization for a point to be on the lower envelope $\mathcal{L}_e$ in terms of void probability of half-balls (Lemma 4.18). Let $X_i, X_j$ be the initial locations of two stations and $(R_i, \alpha_i), (R_j, \alpha_j)$ are corresponding to $X_i, X_j$. Let $f((R_i, \alpha_i), \cdot)$ and $f((R_j, \alpha_j), \cdot)$ be the curves of two radial birds $C_i, C_j$, with their heads at $(T_i, H_i)$ and $(T_j, H_j)$, respectively. For $(T_i, H_i)$ and $(T_j, H_j)$ with $T_j < T_i$, let $(S_{i,j}, H_{i,j})$ be the intersection point of the associated radial birds. We will see later that $(S_{i,j}, H_{i,j})$ can be expressed in terms of $(T_i, H_i)$ and $(T_j, H_j)$. For any $(s, h) \in \mathbb{H}^+$, define

$$ A(s, h) := \{(s, h) \text{ is a handover point}\}. \quad (4.19) $$

We can write for the event $A(S_{i,j}, H_{i,j}) (\equiv A_{i,j})$ that, $1_{A_{i,j}} \equiv 1_{A_{i,j}}(\mathcal{H})$. Then the handover frequency can be written as

$$ \lambda_{\mathcal{H}} = m_{\mathcal{H}}[0, 1] = \mathbb{E} \left[ \sum_{(T_i, H_i) \in \mathcal{H}: T_i \in [0, 1]} \sum_{(T_j, H_j) \in \mathcal{H}: T_j < T_i} 1_{A_{i,j}} \right], \quad (4.20) $$
based on the construction of the handover point process in Subsection 4.5, in particular at Equation (4.16). The last expression clearly suggests using now the factorial power of order 2 of $\mathcal{H}$. The point process $\mathcal{H}$ being Poisson, any factorial power can be expressed in terms of a product measure. More precisely, by applying the multivariate Campbell-Mecke formula for the factorial power of order 2 of $\mathcal{H}$, i.e., $\mathcal{H}^{2,\neq}$ (Proposition 2.3.24 and Proposition 2.3.25, in [3]), we get that

$$ 
\begin{aligned}
\lambda_{\mathcal{H}} &= \mathbb{E} \left[ \sum_{(T_i, H_i) \in \mathcal{H}: T_i \in [0,1]} \sum_{(T_j, H_j) \in \mathcal{H}: T_j < T_i} \mathbb{1}_{A_{i,j}} \right] \\
&= (2\lambda)^2 \int_0^1 \int_0^\infty \int_{-\infty}^{t_1} \int_0^\infty \mathbb{E}_{\mathcal{H}}^{(t_1,h_1),(t_2,h_2)} [\mathbb{1}_{A(s,h)}(\mathcal{H})] \, dh_2 \, dt_2 \, dh_1 \, dt_1 \\
&= 4\lambda^2 \int_0^1 \int_0^\infty \int_{-\infty}^{t_1} \int_0^\infty \mathbb{E}_{\mathcal{H}} [\mathbb{1}_{A(s,h)}(\mathcal{H} + \delta_{(t_1,h_1)} + \delta_{(t_2,h_2)})] \, dh_2 \, dt_2 \, dh_1 \, dt_1, \quad (4.21)
\end{aligned}
$$

where $(s, h)$ is the point that corresponds to the intersection of the radial birds associated with the heads located at $(t_1, h_1)$ and $(t_2, h_2)$. Here $\mathcal{H}^{(t_1,h_1),(t_2,h_2)} = \mathcal{H} + \delta_{(t_1,h_1)} + \delta_{(t_2,h_2)}$, is the two-point Palm version of $\mathcal{H}$ and it's two-point Palm expectation $\mathbb{E}_{\mathcal{H}}^{(t_1,h_1),(t_2,h_2)}$ is as described in Lemma 4.27.

The handover point $(s, h)$ can be expressed in terms of $(t_1, h_1)$, $(t_2, h_2)$, as follows. There exists a pair of points $(r_1, \alpha_1)$, $(r_2, \alpha_2) \in \mathbb{R}^+ \times (-\pi, \pi]$ such that $(t_1, h_1) = (-r_1 \cos \alpha_1, r_1|\sin \alpha_1|)$ and $(t_2, h_2) = (-r_2 \cos \alpha_2, r_2|\sin \alpha_2|)$. Let $f((r_1, \alpha_1), \cdot)$ and $f((r_2, \alpha_2), \cdot)$ be the curves of the radial birds $C_1, C_2$, as defined in (4.3). The time coordinate of handover $s$ satisfies $f((r_1, \alpha_1), s) = f((r_2, \alpha_2), s)$, i.e.,

$$ s^2 + 2sr_1 \cos \alpha_1 + r_1^2 = s^2 + 2sr_2 \cos \alpha_2 + r_2^2. $$

This implies,

$$ s = \frac{r_1^2 - r_2^2}{2(r_2 \cos \alpha_2 - r_1 \cos \alpha_1)} = \frac{t_1^2 + h_1^2 - t_2^2 - h_2^2}{2(t_1 - t_2)}. \quad (4.22) $$

The time coordinate $s$ is unique, which comes from the fact that radial birds $C_1, C_2$ with different $t_1, t_2$ must intersect exactly once, see Property 5. Substituting the value of $s$ from (4.22), we get the height $h$ of the intersection point from:

$$ 
\begin{aligned}
h^2 &= s^2 - 2st_1 + r_1^2 \\
&= \frac{1}{4(t_1 - t_2)^2} [(t_1^2 + h_1^2 - t_2^2 - h_2^2)^2 - 4t_1(t_1 - t_2)(t_1^2 + h_1^2 - t_2^2 - h_2^2) + 4(t_1^2 + h_1^2)(t_1 - t_2)^2] \\
&= \frac{1}{4(t_1 - t_2)^2} [(t_2 - t_1)^4 + 2(h_1^2 + h_2^2)(t_1 - t_2)^2 + (h_2^2 - h_1^2)^2] \\
&= \frac{1}{4} \left[ (t_1 - t_2)^2 + 2(h_1^2 + h_2^2) + \frac{(h_2^2 - h_1^2)^2}{(t_1 - t_2)^2} \right]. \quad (4.23)
\end{aligned}
$$

Recall that $A(s, h)$ is the event that $C_1$ and $C_2$ intersect at $(s, h)$, which is a handover point. A handover happens at $(s, h)$ if and only if $\mathcal{H}(U_h^s) = 0$, by Lemma 4.18. Since $\mathcal{H}$ is a Poisson point process, by the Slivnyak-Mecke theorem, the probability of the event $A(s, h)$ in (4.21), is

$$ \mathbb{E}_{\mathcal{H}} [\mathbb{1}_{A(s,h)} (\mathcal{H} + \delta_{(t_1,h_1)} + \delta_{(t_2,h_2)})] = \mathbb{P}_{\mathcal{H}} (A(s,h)) = \mathbb{P}_{\mathcal{H}} (\mathcal{H}(U_h^s) = 0) = e^{-\lambda \pi h^2}, \quad (4.24) $$

which does not depend on the time coordinate $s$. Using the probability from (4.24) in (4.21), we obtain that

$$ \lambda_{\mathcal{H}} = 4\lambda^2 \int_0^1 \int_0^\infty \int_0^\infty \int_{-\infty}^{t_1} e^{-\lambda \pi h^2} dt_2 \, dh_2 \, dt_1, \quad (4.25) $$

as $t_2$ lies on the left of $t_1$. Since the height $h$ is a function of the difference $t_1 - t_2$ as seen in (4.23), the integral in (4.25) can be converted into an integral in three variables after changing the variable $t_1 - t_2$ to $t$, as follows

$$ \lambda_{\mathcal{H}} = 4\lambda^2 \int_0^\infty \int_0^\infty \int_0^\infty e^{-\lambda \pi h^2} dt \, dh_1 \, dh_2. \quad (4.26) $$
Using the expression for $h^2$ from (4.23) and keeping aside the term $\frac{1}{2}(h_1^2+h_2^2)$, we evaluate the inner-most integral in (4.26) only with respect to $t$ as

$$ \int_0^\infty e^{-\frac{\lambda\pi}{4}\left[t^2+\frac{(h_2^2-h_1^2)^2}{t^2}\right]} dt = e^{-\frac{\lambda\pi}{2}|h_2^2-h_1^2|} \frac{\sqrt{\pi}}{2\sqrt{\frac{\lambda\pi}{4}}} = e^{-\frac{\lambda\pi}{2}|h_2^2-h_1^2|} \frac{1}{\sqrt{\lambda}}, \quad (4.27) $$

where we used the following relation:

$$ \int_0^\infty e^{-ax^2 - \frac{b}{x^2}} dx = \frac{\sqrt{\pi}}{2\sqrt{a}} e^{-2\sqrt{ab}}, \quad (4.28) $$

for $a = \frac{\lambda\pi}{4} > 0$ and $b = \frac{\lambda\pi}{4}(h_2^2 - h_1^2)^2 > 0$. The exact value of the integral in (4.28) can be evaluated from the relation

$$ \int e^{-ax^2 - \frac{b}{x^2}} dx = -\frac{\sqrt{\pi}e^{-2\sqrt{ab}}}{4a} \left[ e^{4\sqrt{ab}} \operatorname{erfc}(\sqrt{a}x + \sqrt{b}/x) + \operatorname{erfc}(\sqrt{a}x - \sqrt{b}/x) \right] + c, \quad (4.29) $$

for any $a, b > 0$ and a constant $c$, where $\operatorname{erfc}(x) := \frac{2}{\sqrt{\pi}} \int_x^\infty e^{-t^2} dt$, $\operatorname{erfc}(\infty) = 0$ and $\operatorname{erfc}(-\infty) = 2$.

Substituting the resultant from (4.27) to the integral (4.26), along with an extra factor of $e^{-\frac{\lambda\pi}{2}(h_1^2+h_2^2)}$, the right hand side in (4.26) turns out to be

$$ \begin{aligned} \lambda_V &= \frac{4\lambda^2}{\sqrt{\lambda}} \int_0^\infty \int_0^\infty e^{-\frac{\lambda\pi}{2}(h_1^2+h_2^2)} e^{-\frac{\lambda\pi}{2}|h_2^2-h_1^2|} dh_1 dh_2 \\ &= 4\lambda^{\frac{3}{2}} \int_0^\infty \int_0^\infty e^{-\frac{\lambda\pi}{2}(h_1^2+h_2^2+|h_2^2-h_1^2|)} dh_1 dh_2 \\ &= 4\lambda^{\frac{3}{2}} \int_0^\infty \int_0^\infty e^{-\lambda\pi(h_1^2+h_2^2)} dh_1 dh_2 \\ &= 4\lambda^{\frac{3}{2}} \frac{1}{\lambda\pi} \int_0^\infty \int_0^\infty e^{-(x^2+y^2)} dx dy = \frac{4\sqrt{\lambda}}{\pi}, \end{aligned} \quad (4.30) $$

since it can be shown that $\int_0^\infty \int_0^\infty e^{-(x^2+y^2)} dx dy = 1$. This completes the proof of Theorem 4.26. $\square$

**Remark 4.30.** In view of (4.18), we can give an alternative definition of the handover frequency, instead of the one in (4.20), as follows:

$$ \lambda_V = m_V[0, 1] = \mathbb{E} \left[ \sum_{(T_i, H_i) \in \mathcal{H}: T_i \in [0, 1]} \sum_{(T_j, H_j) \in \mathcal{H}: T_j > T_i} \mathbb{1}_{A_{i,j}} \right]. $$

Under this definition we can carry out the entire computation of the handover frequency $\lambda_V$.

<figure><img src="image_16.png" /><figcaption>FIGURE 4.9. In the dual system the handover points are represented by the crossings of the Voronoi boundaries by the trajectory of an user moving along a horizontal line.</figcaption></figure>
**Remark 4.31** (Speed $v \neq 1$). If the mobile stations move at a general speed $v$, the handover frequency can be computed to be $\lambda_V = \frac{4v\sqrt{\lambda}}{\pi}$. This is justified since, in view of Lemma 4.3 and Remark 4.5, the speed $v$ scales the time axis proportionally.

**Remark 4.32.** We would like to mention that the proof technique developed above, naturally lifts to the multi-speed case, where the stations are moving at different speeds, which is the main content of Section 7. In the present section, we have first analyzed the case of single-speed as a preparation for the multi-speed case.

**Remark 4.33** (Comparison with the static scenario). Consider the same model, but with static stations and a mobile user moving along a straight line with speed $v$, see Figure 4.9. The static model has been studied in various papers of the literature, for example [6], [8], [7], [9]. The handover frequency is computed as the intensity of the point process corresponding to the intersection of an infinite line with the boundaries of cells of the corresponding Poisson-Voronoi tessellation. This uses the contact and cord length distribution of the Poisson-Voronoi tessellation [21]. The frequency of handovers in the simple dual static model is also $\lambda_V = \frac{4\sqrt{\lambda}}{\pi}$, as analytically proved in [9], [17, 7.22] and [15]. This fact will be discussed further in the following, as an alternative proof of Theorem 4.26.

### 4.6.2. Another proof of Theorem 4.26.

*Proof.* We give here a simpler proof of Theorem 4.26. Unfortunately, this simpler approach does not extend to the multi-speed case, which will centrally rely on the methodology developed for the first proof.

Let us first consider the simpler model with the direction of motion fixed and same for all stations, let us say $\Theta = \theta$. Then at time $t$, the distance of the mobile station, starting at $X = (|X|, \Psi)$, from the user is given by,

$$|X^t| := (|X|^2 + 2t|X| \cos \alpha + t^2)^{\frac{1}{2}} = (R'^2 + 2tR' \cos \alpha' + t^2)^{\frac{1}{2}} := f((R', \alpha'), t),$$

where $|X| := R'$ and $\alpha' \equiv (\Psi - \theta)$ (mod $2\pi$). Collectively the pairs of the form $(R', \alpha')$ give rise to a marked Poisson point process $\tilde{\Phi}' := \sum_{i \in \mathbb{N}} \delta(R'_i, \alpha'_i)$ on $\mathbb{R}^+ \times [-\pi, \pi]$. Subsequently, one can construct the head point process $\mathcal{H}' := \sum_{i \in \mathbb{N}} \delta(T'_i, H'_i)$, where $(T'_i, H'_i) := (-R'_i \cos \alpha'_i, R'_i |\sin \alpha'_i|)$ and the marked head point process $\mathcal{H}'_c := \sum_{i \in \mathbb{N}} \delta(T'_i, H'_i, C'_i)$, where $C'_i := \{(t, f((R'_i, \alpha'_i), t)) : t \in \mathbb{R}\}$ are the radial birds. The lower envelope process in this case $\mathcal{L}'(\cdot)$ is defined as

$$\mathcal{L}'(t) := \{(t, L'(t)) : t \in \mathbb{R}\}, \text{ where, } L'(t) := \inf_{i \in \mathbb{N}} f((R'_i, \alpha'_i), t).$$

The key observation is that $\alpha'$ has the same distribution as $\alpha$. Hence the marked head point processes $\mathcal{H}'_c$ and $\mathcal{H}_c$, the one in the random direction case (see the Definition 4.10), have the same distribution. As a result, the lower envelope process $\mathcal{L}'(\cdot)$ has the same distribution as the one in the random direction case defined in Definition 4.14, see Figure 4.2.

As we have seen in Subsection 4.5, the handover point process is totally determined by the lower envelope process $\mathcal{L}'(\cdot)$ of the marked head point process $\mathcal{H}'_c$. Hence, the handover point process has the same intensity in both the random and non-random direction cases. Having just a single direction of motion $\theta$ for all mobile stations is equivalent to the situation where the user is moving in the $\pi + \theta$ direction, while all the mobile stations are static, as considered in [8]. There the handover frequency is $\frac{4\sqrt{\lambda}}{\pi}$ in the unit speed case. Since the radial bird particle process is oblivious of the direction of motion of the stations, the handover frequency coincides with the one in the static case. This completes the second proof of Theorem 4.26. $\square$

**Remark 4.34.** Moving beyond handover frequency, our analysis successfully captures the statistical properties of geometric object of interest, studied in the rest of the article. Note that our results hold for both the dynamic and the dual static model, which are captured together through Figure 3.1.

### 4.7. State at typical times and handover times.
Let $L_s$ be the vertical line $t = s$, for some $s \in \mathbb{R}$. Observe that all the radial birds intersect the vertical line $L_s$ at certain heights. The point process of interest here, representing the ordered distances of the base stations, is given by the intersection of the birds with this vertical line, see Figure 4.10. The further away a head point is from $L_s$, on the right or
left, the higher its intersection with the corresponding radial bird. This gives rise to a point process on $L_s$, which can be seen as a point process on $\mathbb{R}^+$. Recall that $\pi_2(A) = \{\pi_2(x, y) : \text{ for all } (x, y) \in A\}$, for any $A \subset \mathbb{R}^2$, where $\pi_2 : \mathbb{H}^+ \to \mathbb{R}^+$ is the projection map to the h-axis such that $\pi_2(x, y) = y$ for $(x, y) \in \mathbb{H}^+$.

Let $\eta_s$ be the point process on $\mathbb{R}^+$ corresponding to the intersection points of all the birds with the vertical line $L_s$, i.e.,

$$ \eta_s := \sum_{i \in \mathbb{N} : (T_i, H_i) \in \mathcal{H}} \delta_{\pi_2(C_i \cap L_s)}, $$

where $C_i$ is the radial bird corresponding to $(T_i, H_i) \in \mathcal{H}$. The point process $\eta_s$ gives the distances between the user and the mobile stations at time $s$. We would like to characterize the point processes $\{\eta_s\}_{s \in \mathbb{R}}$ on $\mathbb{R}^+$, in particular at any typical time and also at handover times. Let $\hat{h}_s := \inf(\text{Supp}(\eta_s))$.

<figure><img src="image_17.png" /><figcaption>FIGURE 4.10. All the radial birds intersects the line $L_s$ at different heights and creates a point process $\eta_s$.</figcaption></figure>

Then $\hat{h}_s$ is the distance to the serving station at time $s$, which characterizes the power of the signal received by the user. On the other hand the point process $\eta_s - \delta_{\hat{h}_s}$ enables us to encode the distance of all other non-serving stations, contributing to the power of the interference experienced by the user.

Suppose there is a handover at time 0, namely consider the system under the Palm probability measure $\mathbb{P}_V^0$ of handovers (to be determined rigorously in Theorem 5.4 in Section 5). Consider the point process $\eta_0$ for which $\inf(\text{Supp}(\eta_0)) = \hat{h}$, say. At a typical handover, there are two stations at equal distance $\hat{h}$. We call $\hat{h}$ the typical handover distance and we derive its Palm probability distribution in Subsection 5.2. In the following, we give a result about the distribution of point process $\eta_0 - \delta_{\hat{h}}$.

**Lemma 4.35.** Conditioned on $\hat{h}$, under the Palm probability measure $\mathbb{P}_V^0$, the point process $\eta_0$ is a Poisson point process on $[\hat{h}, \infty)$ with intensity measure $\hat{\mu}$ with density $d\hat{\mu} := 2\pi\lambda r dr$.

*Proof.* Suppose that, under the Palm probability measure $\mathbb{P}_V^0$ of handovers, there is a handover at time 0 at a distance $\hat{h}$. Then there exists $(t, h), (t', h') \in \mathcal{H}$ such that, the radial birds $C_{(t,h)}$, $C_{(t',h')}$ intersects at $(0, \hat{h})$ and $\mathcal{H}(U_{\hat{h}}^0) = 0$. Since $\mathcal{H}$ itself is a Poisson point process, we have from Corollary 4.29, sub point processes $\mathcal{H}|_{\overline{U_{\hat{h}}^0}}$ and $\mathcal{H}|_{(\overline{U_{\hat{h}}^0})^c}$ which are independent and such that $\mathcal{H}|_{(\overline{U_{\hat{h}}^0})^c}$ is a Poisson point process. Note that the point process $\eta_0 - \delta_{\hat{h}}$ of the distances of the mobile stations at time 0 is created by all the radial birds with the head point process $\mathcal{H}|_{(\overline{U_{\hat{h}}^0})^c}$ and $\text{Supp}(\eta_0) \subset [\hat{h}, \infty)$, conditioned on $\hat{h}$. Since $\mathcal{H}|_{(\overline{U_{\hat{h}}^0})^c}$ is an independent Poisson point process, the point process $\eta_0 - \delta_{\hat{h}}$ is a Poisson point process with intensity measure $\hat{\mu}$ having density $d\hat{\mu} := 2\pi\lambda r dr$ on $[\hat{h}, \infty)$, as seen in the radial part of the one at (4.2). This completes the proof. $\square$

## 5. PALM PROBABILITY WITH RESPECT TO HANDOVERS AND APPLICATIONS

The goal of this section is to determine the Palm probability distribution w.r.t. the handover epochs. This is done in Subsection 5.1, by extending the method used in the proof of Theorem 4.26. This is then leveraged to determine the Palm distribution of the distance to the stations involved in the handover and the Palm distribution of the time to the next handover.
5.1. **Palm probability with respect to handovers.** In this section, the reference probability space is $(\Omega, \mathcal{F}, \mathbb{P})$, with $(\Omega, \mathcal{F})$ the canonical space of point processes on $\mathbb{H}^+$. So a point $\omega$ is a realization of the point process $\mathcal{H} = \sum_{i \in \mathbb{N}} \delta_{(T_i, H_i)}$. One can equip this measure space with the shift $\{\theta_t\}_{t \in \mathbb{R}}$ along the time axis. Its action on $\mathcal{H}$ is defined by

$$ \theta_t(\mathcal{H}) = \sum_{i \in \mathbb{N}} \delta_{(T_i - t, H_i)}. \qquad (5.1) $$

The law $\mathbb{P}$ of the Poisson point process $\mathcal{H}$ is left invariant by this shift. This shift is ergodic for $\mathbb{P}$.

Consider the handover point process $\mathcal{V}$, as constructed in Subsection 4.5. This point process is stationary ($\theta_t$-compatible) from Lemma 4.25 and has a positive and finite intensity from Theorem 4.26. So one can define the Palm probability w.r.t. $\mathcal{V}$ on the probability space $(\Omega, \mathcal{F})$, which will be denoted as $\mathbb{P}_{\mathcal{V}}^0$.

Consider the pairs of points $((t, h), (t', h'))$ of $\mathbb{H}^+$, such that $t' \le t$ and the condition that $\mathcal{H}(U_{\hat{h}}^{\hat{s}}) = 0$ holds, where $(\hat{s}, \hat{h})$ are the coordinates of the intersection of the two radial birds with heads at $(t, h)$ and $(t', h')$, respectively. The sequence of rightmost points of such pairs that are in the support of $\mathcal{H}$ forms a stationary ($\theta_t$-compatible) point process $\mathcal{R}$ on $\mathbb{R}$. This point process has a positive and finite intensity. So one can define the Palm probability w.r.t. $\mathcal{R}$ (on the probability space $(\Omega, \mathcal{F}))$, which will be denoted as $\mathbb{P}_{\mathcal{R}}^0$. As already explained, there is a bijection between the points of $\mathcal{R}$ and the points of

<figure><img src="image_18.png" /><figcaption>FIGURE 5.1. The bijection between point process $\mathcal{R}$ and the handover point process $\mathcal{V}$ via the head point process $\mathcal{H}$ and the intersection points of the radial birds.</figcaption></figure>

$\mathcal{V}$, that will be denoted by $\beta$, (see Figure 5.1). Note that if $((t, h), (t', h'))$ is as above, then $\beta(t) = \hat{s}$, the abscissa of the intersection of the two radial birds with these heads. The following lemma is obtained from Theorem 6.1.35 in [3] (which is itself a consequence of the mass transport principle for stationary point processes):

**Lemma 5.1.** For all non-negative measurable functions $f$ on $(\Omega, \mathcal{F})$,

$$ \mathbb{E}_{\mathcal{V}}^{0}[f(\mathcal{H})] = \mathbb{E}_{\mathcal{R}}^{0}[f(\theta_{\beta(0)}(\mathcal{H}))]. \qquad (5.2) $$

*Proof of Lemma 5.1.* The formula (5.2) is a special case of that given in [3, Theorem 6.1.35] when taking $\Phi = \mathcal{V}$, $\Phi' = \mathcal{R}$ and $g(y, \omega) = f(\omega)1_{\{y=\beta^{-1}(0)\}}$. $\square$

Let us denote the intensity of the point process $\mathcal{R}$, as $\lambda_{\mathcal{R}}$. Due to the above bijection $\beta$ between $\mathcal{V}$ and $\mathcal{R}$ in the mass transport principle under the Palm probability measures, we get the equality of intensities:

**Corollary 5.2.** The intensity of the point processes $\mathcal{V}$ and $\mathcal{R}$ are equal, $\lambda_{\mathcal{R}} = \lambda_{\mathcal{V}}$.

We now use the Poisson structure of $\mathcal{H}$ to get:

**Lemma 5.3.** For all non-negative measurable functions $f$ on $(\Omega, \mathcal{F})$,

$$ \mathbb{E}_{\mathcal{R}}^{0}[f(\theta_{\beta(0)}(\mathcal{H}))] = \frac{4\lambda^2}{\lambda_{\mathcal{R}}} \int_{(\mathbb{R}^+)^3} \mathbb{E}_{\mathcal{H}} \left[ 1_{\mathcal{H}}(U_{\hat{h}(0,h,-t',h')}) = 0 \right] f(\theta_{\hat{s}(0,h,-t',h')} (\mathcal{H} + \delta_{(0,h)} + \delta_{(-t',h')})) \, dh' \, dh \, dt', $$

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 $(t', h')$ and $\lambda_{\mathcal{R}} = \frac{4\sqrt{\lambda}}{\pi}$.
Proof of Lemma 5.3. For all $(T_i, H_i)$ and $(T_j, H_j)$ in the support of $\mathcal{H}$, let $(\hat{S}_{i,j}, \hat{H}_{i,j})$ denote the coordinates of the intersection of the two radial birds with these heads. By the very definition of $\mathbb{P}_{\mathcal{R}}^0$,

$$ \mathbb{E}_{\mathcal{R}}^{0}[f(\theta_{\beta(0)}(\mathcal{H}))] = \frac{1}{\lambda_{\mathcal{R}}} \mathbb{E} \left[ \sum_{T_i \in \mathcal{R} : 0 \le T_i \le 1} f(\theta_{\beta(0)} \circ \theta_{T_i}(\mathcal{H})) \right]. \quad (5.3) $$

Using the property of the shift $\{\theta_t\}_{t \in \mathbb{R}}$, we have $\theta_{\beta(0)} \circ \theta_{T_i} = \theta_{T_i + \beta(T_i)}$ and hence

$$ \begin{aligned} & \mathbb{E}_{\mathcal{R}}^{0}[f(\theta_{\beta(0)}(\mathcal{H}))] \\ &= \frac{1}{\lambda_{\mathcal{R}}} \mathbb{E} \left[ \sum_{T_i \in \mathcal{R}, 0 \le T_i \le 1} f(\theta_{T_i + \beta(T_i)}(\mathcal{H})) \right] \\ &= \frac{1}{\lambda_{\mathcal{R}}} \mathbb{E} \left[ \sum_{(T_i, H_i) \in \mathcal{H}, 0 \le T_i \le 1} \mathbb{1}_{\exists (T_j, H_j) \in \mathcal{H}: T_j < T_i} \text{ and } \mathcal{H}(U_{\hat{H}_{i,j}}^{\hat{s}_{i,j}}) = 0} f(\theta_{T_i + \beta(T_i)}(\mathcal{H})) \right] \\ &= \frac{1}{\lambda_{\mathcal{R}}} \mathbb{E} \left[ \sum_{(T_i, H_i) \in \mathcal{H}, 0 \le T_i \le 1} \sum_{(T_j, H_j) \in \mathcal{H}: T_j < T_i} \mathbb{1}_{\mathcal{H}(U_{\hat{H}_{i,j}}^{\hat{s}_{i,j}}) = 0} f(\theta_{T_i + \beta(T_i)}(\mathcal{H})) \right] \\ &= \frac{1}{\lambda_{\mathcal{R}}} \mathbb{E} \left[ \sum_{(T_i, H_i) \in \mathcal{H}, 0 \le T_i \le 1} \sum_{(T_j, H_j) \in \mathcal{H}: T_j < T_i} \mathbb{1}_{\mathcal{H}(U_{\hat{H}_{i,j}}^{\hat{s}_{i,j}}) = 0} f(\theta_{\hat{s}_{i,j}}(\mathcal{H})) \right]. \end{aligned} \quad (5.4) $$

Applying the Campbell-Mecke formula for the factorial power of order 2 of $\mathcal{H}$, we obtain from (5.4) that

$$ \begin{aligned} & \mathbb{E}_{\mathcal{R}}^{0}[f(\theta_{\beta(0)}(\mathcal{H}))] \\ &= \frac{(2\lambda)^2}{\lambda_{\mathcal{R}}} \int_{0}^{1} \int_{0}^{\infty} \int_{0}^{\infty} \int_{-\infty}^{t} \mathbb{E}_{\mathcal{H}}^{(t,h),(t',h')} \left[ \mathbb{1}_{\mathcal{H}(U_{\hat{h}(t,h,t',h')})} = 0} f(\theta_{\hat{s}(t,h,t',h')}(\mathcal{H})) \right] dt'dh'dhdt \quad (5.5) \\ &= \frac{4\lambda^2}{\lambda_{\mathcal{R}}} \int_{0}^{1} \int_{0}^{\infty} \int_{0}^{\infty} \int_{-\infty}^{t} \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\mathcal{H}(U_{\hat{h}(t,h,t',h')})} = 0} f(\theta_{\hat{s}(t,h,t',h')}(\mathcal{H} + \delta_{(t,h)} + \delta_{(t',h')})) \right] dt'dh'dhdt \\ &= \frac{4\lambda^2}{\lambda_{\mathcal{R}}} \int_{0}^{1} \int_{0}^{\infty} \int_{0}^{\infty} \int_{t-t'=\infty}^{0} \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\mathcal{H}(U_{\hat{h}(0,h,t'-t,h')})} = 0} f(\theta_{\hat{s}(0,h,t'-t,h')}(\mathcal{H} + \delta_{(0,h)} + \delta_{(t'-t,h')})) \right] dt'dh'dhdt, \end{aligned} \quad (5.6) $$

where we used the Slivnyak-Mecke formula for the last but one equation. We obtain the final result by the change of variable $t - t'$ to $t'$ in (5.6), similar to the one in (4.26). $\square$

We hence get the following result using $\lambda_{\mathcal{R}} = \lambda_{\mathcal{V}} = \frac{4\sqrt{\lambda}}{\pi}$:

**Theorem 5.4.** For all measurable and non-negative $f$

$$ \mathbb{E}_{\mathcal{V}}^{0}[f(\mathcal{H})] = \pi \lambda^{\frac{3}{2}} \int_{(\mathbb{R}^{+})^3} \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\mathcal{H}(U_{\hat{h}(0,h,-t',h')})} = 0} f(\theta_{\hat{s}(0,h,-t',h')}(\mathcal{H} + \delta_{(0,h)} + \delta_{(-t',h')})) \right] dt'dh'dh. \quad (5.7) $$

**Remark 5.5.** Observe that the steps until (5.5) are true for any stationary point process on $\mathbb{H}^+$. In that sense Lemma 5.3 is far more general. The same holds true for Lemma 5.1.

5.2. Typical handover distance. Suppose $\hat{H}$ is the random for distance of a typical handover. The following result evaluates the Laplace transform of $\hat{H}^2$ under the Palm distribution $\mathbb{P}_{\mathcal{V}}^0$ of $\mathcal{V}$ based on Theorem 5.4. This enables us to determine the distribution of $\hat{H}$.

**Lemma 5.6.** The Laplace transform of $\hat{H}^2$ under the Palm probability measure $\mathbb{P}_{\mathcal{V}}^0$ is

$$ \mathcal{L}_{\hat{H}^2}^{0}(\gamma) = \mathbb{E}_{\mathcal{V}}^{0} \left[ e^{-\gamma \hat{H}^2} \right] = \left(1 + \frac{\gamma}{\lambda \pi}\right)^{-\frac{3}{2}}, \quad (5.8) $$
for any $\gamma > 0$. The squared handover-distance $\hat{H}^2$ follows a Gamma distribution with parameters $(\frac{3}{2}, \frac{1}{\lambda\pi})$. Moreover, the handover distance $\hat{H}$ follows a Nakagami distribution with parameters $(\frac{3}{2}, \frac{3}{2\lambda\pi})$.

**Proof of Lemma 5.6.** In order to evaluate the Laplace transform of $\hat{H}^2$ under the Palm probability measure $\mathbb{P}_V^0$, we directly apply Theorem 5.4, by taking $f(\mathcal{H}) = e^{-\gamma\hat{H}^2}$ as follows:

$$ 
\begin{aligned}
\mathcal{L}_{\hat{H}^2}^0(\gamma) &= \mathbb{E}_V^0 \left[ e^{-\gamma \hat{H}^2} \right] \\
&= \pi \lambda^{3/2} \int_0^\infty \int_0^\infty \int_0^\infty \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\mathcal{H}(U_{\hat{h}}^{\hat{s}})=0} e^{-\gamma (\hat{H}^2 \circ \theta_{\hat{s}})} \right] dt' dh' dh \\
&= \pi \lambda^{3/2} \int_0^\infty \int_0^\infty \int_0^\infty e^{-(\lambda\pi+\gamma)\hat{h}^2} dt' dh' dh,
\end{aligned} 
\quad (5.9) $$

where $\hat{s}(0, h, -t', h') \equiv \hat{s}$ and $\hat{h}(0, h, -t', h') \equiv \hat{h}$. The last step (5.9) is due to the fact that

$$ \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\mathcal{H}(U_{\hat{h}}^{\hat{s}})=0} e^{-\gamma (\hat{H}^2 \circ \theta_{\hat{s}})} \right] = e^{-\gamma \hat{h}^2} \mathbb{P}_{\mathcal{H}} \left( \mathcal{H} \left( U_{\hat{h}}^{\hat{s}} \right) = 0 \right) = e^{-\lambda \pi \hat{h}^2} e^{-\gamma \hat{h}^2}, $$

as the handover distance $\hat{h}$ does not change due to the shift $\theta_{\hat{s}}$. Recall from (4.23) that

$$ \hat{h}^2 = \frac{1}{4} \left[ t'^2 + \frac{(h^2 - h'^2)^2}{t'^2} + 2(h^2 + h'^2) \right]. \quad (5.10) $$

Using the expression for $\hat{h}^2$, keeping aside the term $\frac{1}{2}(h^2 + h'^2)$, we evaluate the innermost integral in (5.9), with respect to $t$, to be

$$ \int_0^\infty e^{-\frac{\lambda\pi+\gamma}{4} \left[ t'^2 + \frac{(h^2-h'^2)^2}{t'^2} \right]} dt' = \left( \frac{\pi}{\lambda\pi+\gamma} \right)^{\frac{1}{2}} e^{-\frac{\lambda\pi+\gamma}{2} |h^2-h'^2|}, \quad (5.11) $$

evaluated using the definite integral $\int_0^\infty e^{-ax^2 - \frac{b}{x^2}} dx = \frac{\sqrt{\pi}}{2\sqrt{a}} e^{-2\sqrt{ab}}$ from (4.28) with $a = \frac{\lambda\pi+\gamma}{4} > 0$ and $b = \frac{\lambda\pi+\gamma}{4}(h^2 - h'^2)^2 > 0$. Substituting the resultant from (5.11) to (5.9) along with an extra factor of $e^{-\frac{\lambda\pi+\gamma}{2}(h^2+h'^2)}$, the right hand side in (5.9) turns out to be

$$ 
\begin{aligned}
\mathcal{L}_{\hat{H}^2}^0(\gamma) &= \pi\lambda^{3/2} \left(\frac{\pi}{\lambda\pi + \gamma}\right)^{\frac{1}{2}} \int_0^\infty \int_0^\infty e^{-\frac{\lambda\pi+\gamma}{2}(h^2+h'^2)} e^{-\frac{\lambda\pi+\gamma}{2}|h^2-h'^2|} dh' dh \\
&= \left(\frac{\pi^3\lambda^3}{\lambda\pi + \gamma}\right)^{\frac{1}{2}} \int_0^\infty \int_0^\infty e^{-\frac{\lambda\pi+\gamma}{2}(h^2+h'^2+|h^2-h'^2|)} dh' dh \\
&= \left(\frac{\pi^3\lambda^3}{\lambda\pi + \gamma}\right)^{\frac{1}{2}} \int_0^\infty \int_0^\infty e^{-(\lambda\pi+\gamma)(h^2\vee h'^2)} dh' dh \\
&= \left(\frac{\pi^3\lambda^3}{\lambda\pi + \gamma}\right)^{\frac{1}{2}} \frac{1}{\lambda\pi + \gamma} \int_0^\infty \int_0^\infty e^{-(x^2\vee y^2)} dx dy = \left(1 + \frac{\gamma}{\lambda\pi}\right)^{-\frac{3}{2}},
\end{aligned} 
\quad (5.12) $$

as it can be shown that $\int_0^\infty \int_0^\infty e^{-(x^2\vee y^2)} dx dy = 1$. Hence it is clear that $\hat{H}^2$ is distributed as a $\Gamma(\frac{3}{2}, \frac{1}{\lambda\pi})$ random variable and, as a result, the typical handover distance $\hat{H}$ follows a Nakagami distribution with parameters $(\frac{3}{2}, \frac{3}{2\lambda\pi})$. The probability density for $\hat{H}$ is

$$ f_{\hat{H}}(h) = 4\pi\lambda^{3/2}h^2e^{-\lambda\pi h^2}, \quad \text{for } h \ge 0, \quad (5.13) $$

and its mean is $\mathbb{E}[\hat{H}] = \frac{2}{\pi\sqrt{\lambda}}$. $\square$

**Remark 5.7** (Speed $v \neq 1$). In the single-speed case with any non-unit station speed $v$, the typical handover distance $\hat{H}$ also follows a Nakagami distribution with parameters $(\frac{3}{2}, \frac{3}{2\lambda\pi})$, as the Laplace transform of $\hat{H}^2$, under the Palm probability measure $\mathbb{P}_V^0$, remains the same, namely $\mathcal{L}_{\hat{H}^2}^0(\gamma) = (1 + \frac{\gamma}{\lambda\pi})^{-\frac{3}{2}}$.
Indeed,

$$ \mathcal{L}_{\hat{H}^2}^0(\gamma) = \frac{4\lambda^2 v^2}{\lambda_V} \int_0^\infty \int_0^\infty \int_0^\infty e^{-(\lambda\pi+\gamma)\hat{h}^2} dt' dh' dh, \quad (5.14) $$

since the head point process $\mathcal{H}$ has intensity $2\lambda v$, where $\hat{h}^2 = \frac{1}{4} \left[ v^2 t'^2 + \frac{(h^2-h'^2)^2}{v^2 t'^2} + 2(h^2+h'^2) \right]$ and $\lambda_V = \frac{4v\sqrt{\lambda}}{\pi}$. Using this in (5.14) yields

$$ \begin{aligned} \mathcal{L}_{\hat{H}^2}^0(\gamma) &= \pi\lambda^{3/2}v \int_0^\infty \int_0^\infty e^{-\frac{\lambda\pi+\gamma}{2}(h^2+h'^2)} \int_0^\infty e^{-\frac{\lambda\pi+\gamma}{4}\left[v^2t'^2+\frac{(h^2-h'^2)^2}{v^2t'^2}\right]} dt' dh' dh \\ &= \pi\lambda^{3/2} \int_0^\infty \int_0^\infty e^{-\frac{\lambda\pi+\gamma}{2}(h^2+h'^2)} \int_0^\infty e^{-\frac{\lambda\pi+\gamma}{4}\left[t^2+\frac{(h^2-h'^2)^2}{t^2}\right]} dt dh' dh \\ &= \pi\lambda^{3/2} \left(\frac{\pi}{\lambda\pi+\gamma}\right)^{\frac{1}{2}} \int_0^\infty \int_0^\infty e^{-\frac{\lambda\pi+\gamma}{2}\left((h^2+h'^2)+|h^2-h'^2|\right)} dh' dh \\ &= \left(\frac{\pi^3\lambda^3}{\lambda\pi+\gamma}\right)^{\frac{1}{2}} \frac{1}{\lambda\pi+\gamma} = \left(1+\frac{\gamma}{\lambda\pi}\right)^{-\frac{3}{2}}, \end{aligned} $$

where the last two steps are as in (5.12).

**5.3. Comparison with other typical distances.** In this section, we compare the statistical properties of the distance of the two stations involved in a typical handover already determined in Lemma 5.6, the distance to a typical head (Lemma 5.11) and the distance of the nearest station at a typical time (5.20).

**5.3.1. Distance to a typical head on $\mathcal{L}_e$.** Recall that, $\mathcal{H} := \sum_{i \in \mathbb{N}} \delta_{(T_i, H_i)}$ be the head point process and suppose $\mathcal{H}_V \le \mathcal{H}$, i.e., $\text{Supp}(\mathcal{H}_V) \subset \text{Supp}(\mathcal{H})$, contains only those heads that lies on the lower envelope $\mathcal{L}_e$. In other words, we say that $\mathcal{H}_V$ is nothing but $\mathcal{H}$ restricted to the lower envelope $\mathcal{L}_e$, i.e., $\mathcal{H}_V := \mathcal{H}|_{\mathcal{L}_e}$. We use the subscript $v$ for the points being “visible from the time axis”, or equivalently being on the lower envelope $\mathcal{L}_e$. Let us write the point process $\mathcal{H}_V$ as

$$ \mathcal{H}_V := \sum_{j \in \mathbb{N}} \delta_{(T_j^V, H_j^V)} = \sum_{i \in \mathbb{N}} \delta_{(T_i, H_i)} \mathbb{1}_{\{(T_i, H_i) \in \mathcal{L}_e\}}, \quad (5.15) $$

where we use the notation $(T_j^V, H_j^V)$ for the visible head points. The sequence of points in $\text{Supp}(\mathcal{H}_V)$ gives rise to a stationary point process, say $\mathcal{A}_V$ on $\mathbb{R}$, of the abscissas of the points in $\text{Supp}(\mathcal{H}_V)$. In other words,

$$ \mathcal{A}_V := \sum_{j \in \mathbb{N}: (T_j^V, H_j^V) \in \mathcal{H}_V} \delta_{T_j^V}, \quad (5.16) $$

is the point process of the projections of the atoms of $\mathcal{H}_V$ on the time axis. The point processes $\mathcal{H}_V$ and $\mathcal{A}_V$ time-stationarity from that of $\mathcal{H}$.

**Lemma 5.8.** *The intensity of the point process $\mathcal{A}_V$ is $\sqrt{\lambda}$.*

*Proof of Lemma 5.8.* For $(s, h) \in \mathbb{H}^+$, the event that $\{(s, h) \text{ is an atom of } \mathcal{H}_V\}$ is equivalent to the event that $\mathcal{H}(U_h^s) = 0$. Thus the intensity of the point process $\mathcal{A}_V$, i.e., the expected number of points of $\mathcal{A}_V$ in the unit interval $[0, 1]$ of the time axis is

$$ \lambda_V = \mathbb{E} \left[ \sum_{T_j^V \in \mathcal{A}_V : 0 \le T_j^V \le 1} 1 \right] = \mathbb{E} \left[ \sum_{(T_i, H_i) \in \mathcal{H} : 0 \le T_i \le 1} \mathbb{1}_{\mathcal{H}(U_{H_i}^{T_i})=0} \right], $$

using (5.15). By applying the Campbell-Mecke formula for $\mathcal{H}$, we have

$$ \lambda_V = 2\lambda \int_0^1 \int_0^\infty e^{-2\lambda|U_h^t|} dh dt = 2\lambda \int_0^\infty \int_0^1 e^{-\lambda\pi h^2} dt dh = 2\lambda \int_0^\infty e^{-\lambda\pi h^2} dh = \sqrt{\lambda}. \quad \square $$
**Remark 5.9.** In the non-unit speed case ($v \neq 1$), the intensity of $A_V$ is $v\sqrt{\lambda}$. This result also recovers the assertion made in [7, Theorem 3].

<figure><img src="image_19.png" /><figcaption>FIGURE 5.2. The mobile station, corresponding to the radial bird in red, partly serves the user, but not during the time when it is at its closest distance.</figcaption></figure>

**Remark 5.10.** The point process $H_V$ consists of only those atoms of $H$ that lie on the lower envelope. There can be some atoms of $H$ for which a part of the corresponding radial bird contributes to the lower envelope, but are such that the head point itself does not lie on $L_e$, see Figure 5.2. In other words, the mobile stations corresponding to those points serve as the closest mobile station but not during the time when they are at their shortest distance to the user. Clearly, the point process $H_V$ is not sufficient to provide the true handover frequency, since $\lambda_V = \frac{4\sqrt{\lambda}}{\pi} > \sqrt{\lambda} = \lambda_V$. As seen earlier, we certainly need more points to obtain the complete picture of the handover point process $V$.

One can define the Palm probability, denoted by $P_{A_V}^0$, with respect to $A_V$. Let $H_V$ be the distance of the typical head point corresponding to $A_V$. This distance determines the best signal power received from the serving station, which is of practical interest.

**Lemma 5.11.** For any $\gamma \ge 0$, the Laplace transform of $H_V^2$, under the Palm probability distribution $P_{A_V}^0$ of $A_V$, is given by

$$ \mathcal{L}_{H_V^2}^0(\gamma) = \left(1 + \frac{\gamma}{\lambda\pi}\right)^{-1/2}, \qquad (5.17) $$

that is, $H_V$ follows a Nakagami distribution with parameter $(\frac{1}{2}, \frac{1}{2\lambda\pi})$.

*Proof.* Let $\gamma \ge 0$. The Laplace transform of the random variable $H_V^2$, under the Palm probability distribution $P_{A_V}^0$ of the $A_V$, is

$$ \begin{aligned} \mathcal{L}_{H_V^2}^0(\gamma) &:= \mathbb{E}_{A_V}^0 [e^{-\gamma H_V^2}] = \frac{1}{\lambda_V} \mathbb{E} \left[ \sum_{(T_j^V, H_j^V) \in \mathcal{H}_V : T_j^V \in [0,1]} e^{-\gamma (H_j^V)^2} \right] \\ &= \frac{1}{\lambda_V} \mathbb{E} \left[ \sum_{(T_i, H_i) \in \mathcal{H} : T_i \in [0,1]} e^{-\gamma H_i^2} \mathbb{1}_{\{(T_i, H_i) \in \mathcal{L}_e\}} \right] \\ &= \frac{1}{\lambda_V} \mathbb{E} \left[ \sum_{(T_i, H_i) \in \mathcal{H} : T_i \in [0,1]} e^{-\gamma H_i^2} \mathbb{1}_{\{\mathcal{H}(U_{H_i}^{T_i})=0\}} \right], \end{aligned} \qquad (5.18) $$

where we have used the fact that for any $(s, h) \in \mathbb{H}^+$, $\{(s, h) \in \mathcal{L}_e\} = \{\mathcal{H}(U_h^s) = 0\}$, by part (i) of Lemma 4.18. Applying the Campbell-Mecke formula for the point process $H$ in (5.18), we get

$$ \begin{aligned} \mathcal{L}_{H_V^2}^0(\gamma) &= \frac{2\lambda}{\sqrt{\lambda}} \int_0^1 dt \int_0^\infty e^{-\gamma h^2} e^{-\pi\lambda h^2} dh = 2\sqrt{\lambda} \int_0^\infty e^{-(\pi\lambda+\gamma)h^2} dh \\ &= 2\sqrt{\lambda} \int_0^\infty e^{-x} \frac{1}{2\sqrt{x(\pi\lambda+\gamma)}} dx \\ &= \left(\frac{\lambda}{\pi\lambda+\gamma}\right)^{\frac{1}{2}} \int_0^\infty e^{-x} x^{-1/2} dx = \left(1+\frac{\gamma}{\lambda\pi}\right)^{-\frac{1}{2}}, \end{aligned} \qquad (5.19) $$

since $\int_0^\infty e^{-x} x^{-\frac{1}{2}} dx = \pi^{\frac{1}{2}}$. Thus $H_V^2$ is distributed as a $\Gamma(\frac{1}{2}, \frac{1}{\lambda\pi})$ random variable and consequently $H_V$ follows a Nakagami distribution with parameter $(\frac{1}{2}, \frac{1}{2\lambda\pi})$. $\square$
5.3.2. Distance of nearest station at a typical time. For any typical time, let $\tilde{H}$ be the distance to the nearest station to user. Suppose the typical time is $t$, then we write $\tilde{H} \equiv \tilde{H}_t := \inf_i ||X_i^t||$, where $X_i \in \Phi$ is the initial location of a station and $X_i^t$ at time $t$. Then

$$ \mathbb{P}(\tilde{H}_t > h) = \mathbb{P}(\inf_i ||X_i^t|| > h) = \mathbb{P}(\tilde{\Phi}^t(B_h(o)) = 0) = \mathbb{P}(\Phi(B_h(o)) = 0) = e^{-\lambda\pi h^2}, \quad (5.20) $$

since $\tilde{\Phi}^t$ and $\Phi$ has the same distribution. So, $\tilde{H}$ is distributed as Rayleigh with parameter $\frac{1}{\sqrt{2\lambda\pi}}$.

5.3.3. Laplace transform order. We say that two random variables $X, Y$ are ordered for the Laplace transform order, i.e., $X \le_{Lt} Y$, if the corresponding Laplace transforms are ordered, i.e., $\mathcal{L}_X(\gamma) \le \mathcal{L}_Y(\gamma)$, for all $\gamma \ge 0$. The theory of Laplace transform order is studied in [26].

**Lemma 5.12.** The typical distances $\hat{H}, \tilde{H}$ and $H_V$ satisfy

$$ \hat{H} \le_{Lt} \tilde{H} \le_{Lt} H_V. \quad (5.21) $$

*Proof.* Recall the probability densities of the random variables $\hat{H}, \tilde{H}$ and $H_V$

$$ f_{\hat{H}}(h) = 4\pi\lambda^{3/2}h^2e^{-\lambda\pi h^2}, f_{\tilde{H}}(h) := 2\lambda\pi h e^{-\lambda\pi h^2} \text{ and } f_{H_V}(h) = 2\lambda^{1/2}e^{-\lambda\pi h^2}, \text{ for all } h \ge 0, $$

respectively. Observe that $f_{\tilde{H}}(h) \le f_{H_V}(h)$ on $[0, 1/\pi\sqrt{\lambda}]$ and $f_{\tilde{H}}(h) > f_{H_V}(h)$ on $(1/\pi\sqrt{\lambda}, \infty)$. In terms of Laplace transforms for all $\gamma \ge 0$, we have

$$ \begin{aligned} \mathcal{L}_{H_V}(\gamma) - \mathcal{L}_{\tilde{H}}(\gamma) &= \int_0^\infty e^{-\gamma h} (f_{H_V}(h) - f_{\tilde{H}}(h)) \, dh \\ &= \int_0^{1/\pi\sqrt{\lambda}} e^{-\gamma h} (f_{H_V}(h) - f_{\tilde{H}}(h)) \, dh - \int_{1/\pi\sqrt{\lambda}}^\infty e^{-\gamma h} (f_{\tilde{H}}(h) - f_{H_V}(h)) \, dh. \end{aligned} $$

Using the value $h = \frac{1}{\pi\sqrt{\lambda}}$ gives a lower bound, i.e.,

$$ \begin{aligned} \mathcal{L}_{H_V}(\gamma) - \mathcal{L}_{\tilde{H}}(\gamma) &\ge e^{-\frac{\gamma}{\pi\sqrt{\lambda}}} \left[ \int_0^{1/\pi\sqrt{\lambda}} (f_{H_V}(h) - f_{\tilde{H}}(h)) \, dh - \int_{1/\pi\sqrt{\lambda}}^{\infty} (f_{\tilde{H}}(h) - f_{H_V}(h)) \, dh \right] \\ &= e^{-\frac{\gamma}{\pi\sqrt{\lambda}}} \int_0^{\infty} (f_{H_V}(h) - f_{\hat{H}}(h)) \, dh = 0. \end{aligned} $$

Thus $\mathcal{L}_{H_V}(\gamma) \ge \mathcal{L}_{\tilde{H}}(\gamma)$ for all $\gamma \ge 0$, which implies $\tilde{H} \le_{Lt} H_V$. On the other hand, observe that $f_{\hat{H}}(h) \le f_{\tilde{H}}(h)$ on $[0, 1/2\sqrt{\lambda}]$ and $f_{\hat{H}}(h) > f_{\tilde{H}}(h)$ on $(1/2\sqrt{\lambda}, \infty)$. In terms of Laplace transforms, we have

$$ \begin{aligned} \mathcal{L}_{\tilde{H}}(\gamma) - \mathcal{L}_{\hat{H}}(\gamma) &= \int_0^\infty e^{-\gamma h} (f_{\tilde{H}}(h) - f_{\hat{H}}(h)) \, dh \\ &= \int_0^{1/2\sqrt{\lambda}} e^{-\gamma h} (f_{\tilde{H}}(h) - f_{\hat{H}}(h)) \, dh - \int_{1/2\sqrt{\lambda}}^\infty e^{-\gamma h} (f_{\hat{H}}(h) - f_{\tilde{H}}(h)) \, dh. \end{aligned} $$

Using the value $h = \frac{1}{2\sqrt{\lambda}}$ again gives the lower bound

$$ \begin{aligned} \mathcal{L}_{\tilde{H}}(\gamma) - \mathcal{L}_{\hat{H}}(\gamma) &\ge e^{-\frac{\gamma}{2\sqrt{\lambda}}} \left[ \int_0^{1/2\sqrt{\lambda}} (f_{\tilde{H}}(h) - f_{\hat{H}}(h)) \, dh - \int_{1/2\sqrt{\lambda}}^{\infty} (f_{\hat{H}}(h) - f_{\tilde{H}}(h)) \, dh \right] \\ &= e^{-\frac{\gamma}{2\sqrt{\lambda}}} \int_0^{\infty} (f_{\tilde{H}}(h) - f_{\hat{H}}(h)) \, dh = 0. \end{aligned} $$

Thus $\mathcal{L}_{\tilde{H}}(\gamma) \ge \mathcal{L}_{\hat{H}}(\gamma)$ for all $\gamma \ge 0$, which implies $\hat{H} \le_{Lt} \tilde{H}$. The completes the proof of the Laplace transform ordering (5.21) among the typical distances. $\square$

5.4. Distribution of inter-handover time. Let us formally define the inter-handover time under Palm in terms of the head point process $\mathcal{H}$. For any $s \in \mathbb{R}$, we denote the first time a handovers happens after time $s$ as $T_1(s)$. Suppose there is a handover at time 0 given by the intersection of two
radial birds $C_{(t_1,h_1)}$ and $C_{(t_2,h_2)}$, for some $(t_1, h_1)$ and $(t_2, h_2)$ such that $t_2 \le t_1$. Let $T$ denotes the inter-handover time, which coincides with $T_1(0)$. Then the next handover will be given by an intersection that happens on the part of the radial bird $C_{(t_1,h_1)}$ lying on the right of the vertical line $t = 0$.

Consider the sequence $\{U_{\hat{h}_t}^t\}_{t \ge 0}$ of open half-balls of heights $\{\hat{h}_t\}_{t \ge 0}$, where

$$ \hat{h}_t := ((t - t_1)^2 + h_1^2)^{\frac{1}{2}}. $$

Let us also consider the collection $\{S_t\}_{t \ge 0}$, where

$$ S_t := \left( \bigcup_{t' \in [0, t]} U_{\hat{h}_{t'}}^{t'} \setminus U_{\hat{h}_0}^0 \right)^{\circ}. \qquad (5.22) $$

We claim that the collection $\{S_t\}_{t \ge 0}$ of open sets is increasing in $t$, using the following Lemma 5.13, for which we provide a geometric proof in Appendix A.2.

**Lemma 5.13.** The collection of sets $\{U_{\hat{h}_t}^t \setminus U_{\hat{h}_0}^0\}_{t \ge 0}$ is increasing in $t$.

**Remark 5.14 (Increasing wing property).** In general, we call the decreasing and the increasing branches of the curve of a radial bird as its decreasing wing and increasing wing, respectively. Suppose $(t_1, h_1) \in \mathbb{H}^+$ with $t_1 \ge 0$, such that there is a radial bird $C_{(t_1,h_1)}$ with head at $(t_1, h_1)$ and $(0, \hat{h}_0) \in C_{(t_1,h_1)}$. Then the collection $\{U_{\hat{h}_t}^t \setminus U_{\hat{h}_0}^0\}_{t \ge t_1}$ is non-decreasing, where $\hat{h}_t = ((t - t_1)^2 + h_1^2)^{\frac{1}{2}}$, since the point $(t, \hat{h}_t)$ lies on the increasing wing of the radial bird at $(t_1, h_1)$. We often refer to this property as the increasing wing property for the single speed case.

Using the increasing property of the collection $\{U_{\hat{h}_t}^t \setminus U_{\hat{h}_0}^0\}_{t \ge 0}$, as in Lemma 5.13, we have

$$ S_t = \left( \bigcup_{t' \in [0, t]} U_{\hat{h}_{t'}}^{t'} \setminus U_{\hat{h}_0}^0 \right)^{\circ} = U_{\hat{h}_t}^t \setminus \overline{U_{\hat{h}_0}^0}. $$

Also observe that the collection of sets $\{S_t\}_{t \ge 0}$ lies entirely in the region $Q_0 \setminus \overline{U_{\hat{h}_0}^0}$, where $Q_s := \{(t, h) \in \mathbb{H}^+ : t \ge s\}$ is the right quadrant with respect to the vertical line $t = s$.

**Definition 5.15 (Inter-handover time).** Given that there is a handover at time 0, the time taken for the next handover is defined as

$$ T := \inf \{s > 0 : \forall t \in (0, s], \mathcal{H}(S_t) = 0, \text{ but } \mathcal{H}(\overline{S_s}) = 2\}, \qquad (5.23) $$

which is equivalent to $T_1(0)$ under the handover Palm probability distribution.

As a consequence of the increasing property of the sets $\{S_t\}_{t \ge 0}$, we can conclude that $\mathcal{H}(S_{T_1(0)}) = 0$, implies $\mathcal{H}(S_t) = 0$, for all $t \in [0, T_1(0)]$. Additionally $\mathcal{H}(\overline{S_{T_1(0)}}) = 2$. This essentially implies there exists a third head point $(t_3, h_3)$ such that $C_{(t_1,h_1)}$ and $C_{(t_3,h_3)}$ intersect at $(T_1(0), \hat{h}_{T_1(0)})$, $\mathcal{H}(S_{T_1(0)}) = 0$ and $\mathcal{H}(\overline{S_{T_1(0)}}) = 2$, see Figure 5.3. Hence we have the following corollary of Lemma 5.13:

<figure><img src="image_20.png" /><figcaption>FIGURE 5.3. The sequence of increasing sets of the form $S_t = (U_{\hat{h}_t}^t \setminus U_{\hat{h}_0}^0)^{\circ}$ has no points of $\mathcal{H}$, for all $t \in [0, T_1(0)]$, we just draw a two of them in red and blue. Here $\mathcal{H}(\overline{S_{T_1(0)}}) = 2$.</figcaption></figure>
<figure><img src="image_21.png" /><figcaption>FIGURE 5.4. The sequence of increasing sets of the form $S_t = (U_{\hat{h}_t}^t \setminus U_{\hat{h}_0}^0)^\circ$ has no points of $\mathcal{H}$, for all $t \in [0, T_1(0)]$, we just draw a two of them in red and blue. Here $\mathcal{H}(\overline{S_{T_1(0)}}) = 2$.</figcaption></figure>

**Corollary 5.16.** For the time point s meeting the stopping condition in the definition of T in (5.23), there exists a unique point, say $(t_3, h_3) \in \mathbb{H}^+$ with $t_3 \ge t_1$, such that the intersection of the radial bird with that point as head and the radial bird $C_{(t_1,h_1)}$, produces a handover at time T.

In the following, we determine the Laplace transform of the inter-handover time T under the Palm probability of handovers.

**Theorem 5.17.** Consider the motion model with speed $v = 1$. Under the Palm probability measure $\mathbb{P}_{\mathcal{V}}^0$, the Laplace transform of the inter-handover time T is given by,

$$ \mathbb{E}_{\mathcal{V}}^{0}[e^{-\rho T}] = 2\pi \lambda^{\frac{5}{2}} \int_{(\mathbb{R}^{+})^{5}} e^{-\rho(\hat{s}_{1}-\hat{s})} e^{-2\lambda|U_{\hat{h}_{1}}^{\hat{s}_{1}} \cup U_{\hat{h}}^{\hat{s}}|} dt_{3} dh_{3} dt_{2} dh_{2} dh_{1}, \quad (5.24) $$

for any $\rho \ge 0$, where $\hat{s} = \frac{h_1^2-t_2^2-h_2^2}{2t_2}$, $\hat{h}^2 = \frac{1}{4}[t_2^2+2(h_1^2+h_2^2)+\frac{(h_2^2-h_1^2)^2}{t_2^2}]$, $\hat{s}_1 = \frac{t_3^2+h_2^2-h_1^2}{2t_3}$ and $\hat{h}_1^2 = \frac{1}{4}[t_3^2+2(h_1^2+h_3^2)+\frac{(h_3^2-h_1^2)^2}{t_3^2}]$ and $|U_{\hat{h}}^{\hat{s}} \cup U_{\hat{h}_1}^{\hat{s}_1}|$ is evaluated in Lemma 5.18.

*Proof.* Taking $f(\mathcal{H}) = e^{-\rho T}$ in Formula (5.7) from Theorem 5.4 with $(0, h) = (0, h_1)$ and $(-t', h') = (-t_2, h_2)$, we obtain

$$ \mathbb{E}_{\mathcal{V}}^{0}[e^{-\rho T}] = \pi \lambda^{\frac{3}{2}} \int_{(\mathbb{R}^{+})^{3}} \mathbb{E}_{\mathcal{H}}\left[\mathbb{1}_{\mathcal{H}(U_{\hat{h}}^{\hat{s}})=0}e^{-\rho(T \circ \hat{s})}\right] dt_2 dh_2 dh_1, \quad (5.25) $$

where $(\hat{s}, \hat{h}) = (\hat{s}(0, h_1, -t_2, h_2), \hat{h}(0, h_1, -t_2, h_2))$ and from, (4.22) and (4.23),

$$ \hat{s}(0, h_1, -t_2, h_2) = \frac{h_1^2 - t_2^2 - h_2^2}{2t_2} \quad \text{and} \quad \hat{h}(0, h_1, -t_2, h_2) = \frac{1}{4} \left[ t_2^2 + 2(h_1^2 + h_2^2) + \frac{(h_2^2 - h_1^2)^2}{t_2^2} \right]. \quad (5.26) $$

In (5.25) we have

$$ T \circ \theta_{\hat{s}} = T_1(\hat{s}) - \hat{s}, \quad (5.27) $$

where $T_1(\hat{s})$ is the time of the first handover after time $\hat{s}$. The next handover can be determined by exploring the head points outside the region $U_{\hat{h}}^{\hat{s}}$, the radial bird of which intersects the radial bird $C_{(0,h_1)}$ at a point with abscissa more than $\hat{s}$. Then there is a unique head point with the property of Corollary 5.16. Moreover, that particular head point must lie one the right of $(0, h_1)$. This is true, since, any other radial bird with head point being on the left of $(0, h_1)$ and outside $U_{\hat{h}}^{\hat{s}}$ intersects the radial bird $C_{(0,h_1)}$ at a point, which is either contained in the open region above the lower envelope, i.e., $\mathcal{L}_e^+$ or produces a handover before time $\hat{s}$. So the next eligible head point must be on the right of $(0, h_1)$ and outside $\overline{U_{\hat{h}}^{\hat{s}}}$. So the unexplored region is $Q_0 \setminus \overline{U_{\hat{h}}^{\hat{s}}}$. Suppose the next handover is given by a random head point $(T_j, H_j)$, then the intersection of $C_{(0,h_1)}$ and $C_{(T_j, H_j)}$ is $(\hat{s}(0, h_1, T_j, H_j), \hat{h}(0, h_1, T_j, H_j))$ and we can define $T_1(\hat{s}) := \hat{s}(0, h_1, T_j, H_j)$. Based on the last discussion, we get that the inner expectation
<figure><img src="image_22.png" /><figcaption>FIGURE 5.5. There exists a head point $(t_3, h_3)$ which is responsible for the first handover at time $\hat{s}_1$ after time $\hat{s}$. There should not be any points of $\mathcal{H}$ in the region $U_{\hat{h}_1}^{\hat{s}_1} \setminus \overline{U_{\hat{h}}^{\hat{s}}}.$</figcaption></figure>

in (5.25) is equal to

$$
\begin{align}
& \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\mathcal{H}}(U_{\hat{h}}^{\hat{s}}) = 0 e^{-\rho(T_1(\hat{s}) - \hat{s})} \right] \nonumber \\
&= \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\mathcal{H}}(U_{\hat{h}}^{\hat{s}}) = 0 \mathbb{1}_{\exists (T_j, H_j) \in \mathcal{H}: T_j \ge 0} \left\{ \mathcal{H} \left( \bigcup_{t \in [0, \hat{s}(0, h_1, T_j, H_j)]} U_{\hat{h}_t}^t \setminus U_{\hat{h}}^{\hat{s}} \right)^{\circ} = 0 \right\} e^{-\rho(\hat{s}(0, h_1, T_j, H_j) - \hat{s})} \right] \nonumber \\
&= \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\mathcal{H}}(U_{\hat{h}}^{\hat{s}}) = 0 \sum_{(T_j, H_j) \in \mathcal{H}: T_j \ge 0} \mathbb{1}_{\mathcal{H}} \left( \bigcup_{t \in [0, \hat{s}(0, h_1, T_j, H_j)]} U_{\hat{h}_t}^t \setminus U_{\hat{h}}^{\hat{s}} \right)^{\circ} e^{-\rho(\hat{s}(0, h_1, T_j, H_j) - \hat{s})} \right]. \tag{5.28}
\end{align}
$$

In the random sum in (5.28), there exists a unique such point $(T_j, H_j)$, as shown in Corollary 5.16. Using the increasing property of Lemma 5.13 in (5.28) we have

$$
\mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\mathcal{H}}(U_{\hat{h}}^{\hat{s}}) = 0 e^{-\rho(T_1(\hat{s}) - \hat{s})} \right] = \mathbb{E}_{\mathcal{H}} \left[ \sum_{(T_j, H_j) \in \mathcal{H}: T_j \ge 0} \mathbb{1}_{\mathcal{H}}(U_{\hat{h}(0, h_1, T_j, H_j)}^{\hat{s}(0, h_1, T_j, H_j)} \cup U_{\hat{h}}^{\hat{s}}) = 0} e^{-\rho(\hat{s}(0, h_1, T_j, H_j) - \hat{s})} \right]. \quad (5.29)
$$

Using the Campbell-Mecke formula in (5.29), we have

$$
\begin{align}
& 2\lambda \int_0^\hat{h} \int_{(\hat{h}^2 - (t_3 - \hat{s})^2)^{1/2}}^\infty \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\mathcal{H}(U_{\hat{h}(0,h_1,t_3,h_3)}^{\hat{s}(0,h_1,t_3,h_3)} \cup U_{\hat{h}}^{\hat{s}}) = 0} e^{-\rho(\hat{s}(0,h_1,t_3,h_3)-\hat{s})} \right] \mathrm{d}h_3 \mathrm{d}t_3 \nonumber \\
& \quad + 2\lambda \int_{\hat{h}}^\infty \int_0^\infty \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\mathcal{H}(U_{\hat{h}(0,h_1,t_3,h_3)}^{\hat{s}(0,h_1,t_3,h_3)} \cup U_{\hat{h}}^{\hat{s}}) = 0} e^{-\rho(\hat{s}(0,h_1,t_3,h_3)-\hat{s})} \right] \mathrm{d}h_3 \mathrm{d}t_3 \nonumber \\
& = 2\lambda \int_0^\hat{h} \int_{(\hat{h}^2 - (t_3 - \hat{s})^2)^{1/2}}^\infty e^{-\rho(\hat{s}_1 - \hat{s})} \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\mathcal{H}(U_{\hat{h}_1}^{\hat{s}_1} \cup U_{\hat{h}}^{\hat{s}}) = 0} \right] \mathrm{d}h_3 \mathrm{d}t_3 \nonumber \\
& \quad + 2\lambda \int_{\hat{h}}^\infty \int_0^\infty e^{-\rho(\hat{s}_1 - \hat{s})} \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{\mathcal{H}(U_{\hat{h}_1}^{\hat{s}_1} \cup U_{\hat{h}}^{\hat{s}}) = 0} \right] \mathrm{d}h_3 \mathrm{d}t_3. \tag{5.30}
\end{align}
$$

Using the void probability, the last sum in (5.30) equals

$$
\begin{align}
& 2\lambda \int_0^\hat{h} \int_{(\hat{h}^2 - (t_3 - \hat{s})^2)^{1/2}}^\infty e^{-2\lambda |U_{\hat{h}_1}^{\hat{s}_1} \cup U_{\hat{h}}^{\hat{s}}|} \mathrm{d}h_3 \mathrm{d}t_3 + 2\lambda \int_{\hat{h}}^\infty \int_0^\infty e^{-\rho(\hat{s}_1 - \hat{s})} e^{-2\lambda |U_{\hat{h}_1}^{\hat{s}_1} \cup U_{\hat{h}}^{\hat{s}}|} \mathrm{d}h_3 \mathrm{d}t_3 \nonumber \\
& \quad := \zeta(\rho, h_1, t_2, h_2), \tag{5.31}
\end{align}
$$

where $(\hat{s}_1, \hat{h}_1) \equiv (\hat{s}(0, h_1, t_3, h_3), \hat{h}(0, h_1, t_3, h_3))$ and from (4.22) and (4.23)

$$
\hat{s}(0, h_1, t_3, h_3) := \frac{t_3^2 + h_3^2 - h_1^2}{2t_3} \quad \text{and} \quad \hat{h}^2(0, h_1, t_3, h_3) := \frac{1}{4} \left[ t_3^2 + 2(h_1^2 + h_3^2) + \frac{(h_3^2 - h_1^2)^2}{t_3^2} \right]. \quad (5.32)
$$

In (5.31), the function $\zeta(\rho, h_1, t_2, h_2)$ denotes the Laplace transform of the inter-handover time $T$, given that there is a handover produced by $(0, h_1)$ and $(-t_2, h_2)$. Figure 5.5 depicts the existence of the void region $U_{\hat{h}_1}^{\hat{s}_1} \cup U_{\hat{h}}^{\hat{s}}$. The handover distances $\hat{h}$ and $\hat{h}_1$ do not change due to the shift $\{\theta_t\}_t$. By replacing
the resultant from (5.31) in (5.25) yields

$$
\begin{align}
\mathbb{E}_{\mathcal{V}}^{0}[e^{-\rho T}] &= \pi \lambda^{\frac{3}{2}} \int_{(\mathbb{R}^{+})^{3}} \zeta(\rho, h_{1}, t_{2}, h_{2}) \, dt_{2} \, dh_{2} \, dh_{1} \nonumber \\
&= 2\pi \lambda^{\frac{5}{2}} \int_{(\mathbb{R}^{+})^{3}} \int_{0}^{\hat{h}} \int_{(\hat{h}^{2} - (t_{3}-\hat{s})^{2})^{\frac{1}{2}}}^{\infty} e^{-\rho(\hat{s}_{1}-\hat{s})} e^{-2\lambda|U_{\hat{h}_{1}}^{\hat{s}_{1}} \cup U_{\hat{h}}^{\hat{s}}|} \, dh_{3} \, dt_{3} \, dt_{2} \, dh_{2} \, dh_{1} \nonumber \\
&\quad + 2\pi \lambda^{\frac{5}{2}} \int_{(\mathbb{R}^{+})^{3}} \int_{\hat{h}}^{\infty} \int_{0}^{\infty} e^{-\rho(\hat{s}_{1}-\hat{s})} e^{-2\lambda|U_{\hat{h}_{1}}^{\hat{s}_{1}} \cup U_{\hat{h}}^{\hat{s}}|} \, dh_{3} \, dt_{3} \, dt_{2} \, dh_{2} \, dh_{1}. \tag{5.33}
\end{align}
$$

The area of the region $U_{\hat{h}}^{\hat{s}} \cup U_{\hat{h}_{1}}^{\hat{s}_{1}}$ is evaluated below in Lemma 5.18. Using the expressions of $\hat{s}, \hat{h}, \hat{s}_{1}$ and $\hat{h}_{1}$ from (5.26) and (5.32), one can evaluate the final expression for $\mathbb{E}_{\mathcal{V}}^{0}[e^{-\rho T}]$ from (5.33). $\square$

We now evaluate the area of the region $U_{\hat{h}}^{\hat{s}_{1}} \cup U_{\hat{h}}^{\hat{s}}$ in terms of the original variables $h_1, t_2, h_2, t_3, h_3$, using
the following geometric lemma, see Figure 5.5 for illustration. The is presented in Appendix A.

**Lemma 5.18.** Suppose $s, s' \in \mathbb{R}$ with $s < s'$ and $h, h' \in \mathbb{R}^{+}$ for which $U_{\hat{h}}^{s} \cap U_{\hat{h}'}^{s'} \neq \emptyset$ and $\exists h_1 \ge 0$ such that $\partial^{*}U_{\hat{h}}^{s} \cap \partial^{*}U_{\hat{h}'}^{s'} = \{(0, h_1)\}$. Then we have

$$
\left| U_h^s \cup U_{h'}^{s'} \right| = \begin{cases} \frac{\pi}{2} h^2 + \frac{1}{2} h_1 (s' - s) + \frac{1}{2} F_1(h_1, h, h'), & \text{if } s < s' < 0, \\ \frac{\pi}{2} (h^2 + h'^2) + \frac{1}{2} h_1 (s' - s) - \frac{1}{2} F_2(h_1, h, h'), & \text{if } s < 0 < s', \\ \frac{\pi}{2} h'^2 + \frac{1}{2} h_1 (s' - s) - \frac{1}{2} F_1(h_1, h, h'), & \text{if } 0 < s < s', \end{cases} \quad (5.34)
$$

where

$$
F_1(h_1, h, h') = h'^2 \arcsin(h_1/h') - h^2 \arcsin(h_1/h), \quad F_2(h_1, h, h') = h'^2 \arcsin(h_1/h') + h^2 \arcsin(h_1/h).
$$

**Remark 5.19.** From (5.33), for $\rho = 0$, we have

$$
\int_{(\mathbb{R}^+)^3} \left[ \int_0^{\hat{h}} \int_{(\hat{h}^2 - (t_3-\hat{s})^2)^{\frac{1}{2}}}^{\infty} e^{-2\lambda |U_{\hat{h}_1}^{\hat{s}_1} \cup U_{\hat{h}}^{\hat{s}}|} dh_3 \, dt_3 + \int_{\hat{h}}^{\infty} \int_0^{\infty} e^{-2\lambda |U_{\hat{h}_1}^{\hat{s}_1} \cup U_{\hat{h}}^{\hat{s}}|} dh_3 \, dt_3 \right] dt_2 \, dh_2 \, dh_1 = \frac{1}{2\pi \lambda^{\frac{5}{2}}}. \quad (5.35)
$$

Also, for small values of $\rho$, we have $\frac{1}{\rho}(1 - E_{\mathcal{V}}^0[e^{-\rho T}]) \approx E_{\mathcal{V}}^0[T] = \frac{1}{\lambda_{\mathcal{V}}} = \frac{\pi}{4\sqrt{\lambda}}$ and hence from (5.33) we have

$$
\int_{(\mathbb{R}^+)^3} \int_0^{\hat{h}} \int_{(\hat{h}^2 - (t_3 - \hat{s})^2)^{\frac{1}{2}}}^{\infty} e^{-\rho(\hat{s}_1 - \hat{s})} e^{-2\lambda |U_{\hat{h}_1}^{\hat{s}_1} \cup U_{\hat{h}}^{\hat{s}}|} dh_3 \, dt_3 \, dt_2 \, dh_2 \, dh_1 \\
+ \int_{(\mathbb{R}^+)^3} \int_{\hat{h}}^{\infty} \int_0^{\infty} e^{-\rho(\hat{s}_1 - \hat{s})} e^{-2\lambda |U_{\hat{h}_1}^{\hat{s}_1} \cup U_{\hat{h}}^{\hat{s}}|} dh_3 \, dt_3 \, dt_2 \, dh_2 \, dh_1 \approx \frac{1 - \rho E_{\mathcal{V}}^0[T]}{2\pi \lambda^{\frac{5}{2}}} = \frac{1 - \frac{\pi \rho}{4\sqrt{\lambda}}}{2\pi \lambda^{\frac{5}{2}}}. \quad (5.36)
$$

In (5.35) and (5.36), we have evaluated these multiple integrals without directly computing them, for
any $\rho \in [0, \varepsilon)$, for some small $\varepsilon > 0$. They can independently be proved to achieve these values, by
analytical and numerical approaches.

**Remark 5.20.** For $v \neq 1$ we have

$$
\begin{align}
\mathbb{E}_{\mathcal{V}}^{0}[e^{-\rho T}] &= 2v^2\pi\lambda^{\frac{5}{2}} \int_{(\mathbb{R}^+)^3} \int_0^{\hat{h}/v} \int_{(\hat{h}^2-v^2(t_3-\hat{s})^2)^{\frac{1}{2}}}^{\infty} e^{-\rho(\hat{s}_1-\hat{s})} e^{-2v\lambda|E_{\hat{h}_1}^{\hat{s}_1,v} \cup E_{\hat{h}}^{\hat{s},v}|} dh_3 dt_3 dt_2 dh_2 dh_1 \nonumber \\
&\quad + 2v^2\pi\lambda^{\frac{5}{2}} \int_{(\mathbb{R}^+)^3} \int_{\hat{h}/v}^{\infty} \int_0^{\infty} e^{-\rho(\hat{s}_1-\hat{s})} e^{-2v\lambda|E_{\hat{h}_1}^{\hat{s}_1,v} \cup E_{\hat{h}}^{\hat{s},v}|} dh_3 dt_3 dt_2 dh_2 dh_1, \tag{5.37}
\end{align}
$$

where the area of the union of half-ellipses $E_{\hat{h}_1}^{\hat{s}_1,v}$, $E_{\hat{h}}^{\hat{s},v}$ is evaluated in Lemma A.4 in Appendix A, after
expressing $\hat{s}, \hat{h}, \hat{s}_1$ and $\hat{h}_1$ in terms of $h_1, t_2, h_2, t_3, h_3$ and $v$, similarly to the one in Theorem 5.17.
## 6. HANDOVER PROCESS AS A FACTOR OF A MARKOV CHAIN: SINGLE-SPEED CASE

Without loss of generality, let us first assume that the speed $v = 1$. The analysis for the case $v \neq 1$ is similar, except for the fact that we use the empty half-ellipse condition, instead of the empty half-ball condition. The preceding analysis of the handover frequency and the Palm distribution of inter-handover time is governed only by the head point process $\mathcal{H}$. Indeed, the radial bird particle process $\mathcal{H}_c$, the collection of their intersection points and the handover point process $\mathcal{V}$ are factors of $\mathcal{H}$. The focus of this section is the joint distribution of inter-handover times and distances, as a factor of a Markov chain. Consider the sequence $\{(H_n^l, T_n^r, H_n^r)\}_{n \in \mathbb{Z}}$ of triples of $(\mathbb{R}^+)^3$, where $H_n^l$ is the distance of the left head point of the $n$-th handover, $T_n^r$ is the time of the right head point of the $n$-th handover, where $T_n^r$ is measured from the time of the left head point, and $H_n^r$ is the distance of the right head point of the $n$-th handover.

**Theorem 6.1.** The sequence $\{(H_n^l, T_n^r, H_n^r)\}_{n \in \mathbb{Z}}$ forms a homogeneous Markov chain on the state space $(\mathbb{R}^+)^3$. A representation of the transition kernel is given in (6.7).

*Proof.* Fix $n \in \mathbb{Z}$. Suppose the $n$-th handover is given by the intersection of the radial birds with head points $(0, H_n^l), (T_n^r, H_n^r)$, with $T_n^r \ge 0$. Then the intersection point $(\hat{S}_n, \hat{H}_n)$ of the radial birds at the two head points $(0, H_n^l), (T_n^r, H_n^r)$, is such that $\mathcal{H}(U_{\hat{H}_n}^{\hat{S}_n}) = 0$. The possible head point responsible for the next handover must be on the right of $(T_n^r, H_n^r)$ and outside $\overline{U_{\hat{H}_n}^{\hat{S}_n}}$. Indeed, any radial bird with head point lying on the left of $(T_n^r, H_n^r)$ and out side $\overline{U_{\hat{H}_n}^{\hat{S}_n}}$, intersects $C_{(T_n^r, H_n^r)}$ before $\hat{S}_n$. The intersection point is not eligible to be considered as a future handover. We define the region $Q_{T_n^r} \setminus \overline{U_{\hat{H}_n}^{\hat{S}_n}}$ to be the unexplored region, where, $Q_s := \{(t, h) \in \mathbb{H}^+ : t \ge s\}$, is the quadrant on the right of the vertical line $t = s$ on $\mathbb{H}^+$.

As explained in (5.22), one can construct a sequence of increasing open sets $\{S_t\}_{t \ge \hat{S}_n}$ in the unexplored region $Q_{T_n^r} \setminus \overline{U_{\hat{H}_n}^{\hat{S}_n}}$, where $S_t := U_{h(t)}^t \setminus \overline{U_{\hat{H}_n}^{\hat{S}_n}}$ and $h(t) := ((t - T_n^r)^2 + (H_n^r)^2)^{\frac{1}{2}}$. See Figure 5.3 and Figure 5.4 for two different cases. Subsequently, by Corollary 5.16, there exists almost surely a unique head point, say $(T_u, H_u)$ with $T_u \ge T_n^r$, in the region $Q_{T_n^r} \setminus \overline{U_{\hat{H}_n}^{\hat{S}_n}}$, such that the next handover is given by the intersection $(\hat{S}_{n+1}, \hat{H}_{n+1})$ of the radial birds at $(T_n^r, H_n^r)$ and $(T_u, H_u)$, where $\hat{H}_{n+1} = h(\hat{S}_{n+1})$. Thus $S_{\hat{S}_{n+1}} := U_{\hat{H}_{n+1}}^{\hat{S}_{n+1}} \setminus \overline{U_{\hat{H}_n}^{\hat{S}_n}} \subset Q_{T_n^r} \setminus \overline{U_{\hat{H}_n}^{\hat{S}_n}}$, is the stopping set until one finds the next head point $(T_u, H_u)$ at its boundary. We define the next triple of the chain as

$$ (H_{n+1}^l, T_{n+1}^r, H_{n+1}^r) := (H_n^r, T_u - T_n^r, H_u). \qquad (6.1) $$

The Markov property of the sequence $\{(H_n^l, T_n^r, H_n^r)\}_{n \in \mathbb{Z}}$ follows from the fact that, given the state $(H_n^l, T_n^r, H_n^r)$ at step $n$, state of the process as defined in (6.1), in this unexplored region $Q_{T_n^r} \setminus \overline{U_{\hat{H}_n}^{\hat{S}_n}}$, the head point is conditionally independent of the past and Poisson.

For the sake of completeness we give below a representation of the transition kernel of this Markov chain. Given $(H_n^l, T_n^r, H_n^r) = (h_n^l, t_n^r, h_n^r)$, let $C_{(0,h_n^l)}$ and $C_{(t_n^r, h_n^r)}$ be the radial birds with head points at $(0, h_n^l)$ and $(t_n^r, h_n^r)$. Also suppose the intersection point is $(\hat{s}_n, \hat{h}_n)$. For any $t \ge \hat{s}_n + \hat{h}_n$, there exists a unique upper half circle that has its top most point lying on $C_{(t_n^r, h_n^r)}$, its right most point at $(t, 0)$, and is such that the half-circle passes through $(t_n^r, h_n^r)$. Consider the family of upper half-circles $\{D_t\}_{t \ge \hat{s}_n + \hat{h}_n}$ satisfying the last properties and defined by the equation

$$ (u - \hat{s}(t))^2 + h^2 = \hat{h}(t)^2, $$

in the $(u, h)$-coordinate system, where $(\hat{s}(t), \hat{h}(t))$ is the intersection point of the radial bird at $(t, 0)$, with the radial bird $C_{(t_n^r, h_n^r)}$, see Figure 6.1.
<figure><img src="image_23.png" /><figcaption>FIGURE 6.1. The sequence of half-circles {$D_t$}$_{t \ge \hat{s}_n + \hat{h}_n}$. We just draw a few of them, for example in red, blue and violet, respectively, and each of which passes through ($t_n^r, h_n^r$).</figcaption></figure>

Let $U_t$ be the open upper half-ball contained below the half-circle $D_t$. Then we have

$$ \mathbb{P} \left( (T_u, H_u) \notin U_t \mid (H_n^l, T_n^r, H_n^r) = (h_n^l, t_n^r, h_n^r) \right) = e^{-2\lambda |\overline{U_t} \setminus \overline{U_{\hat{h}_n}^{\hat{s}_n}}|} \quad (6.2) $$

Given ($h_n^l, t_n^r, h_n^r$), define the stopping time,

$$ \tau(h_n^l, t_n^r, h_n^r) := \inf \left\{ t \ge \hat{s}_n + \hat{h}_n : \mathcal{H} \left( \overline{U_t} \setminus \overline{U_{\hat{h}_n}^{\hat{s}_n}} \right) > 0 \right\}. $$

In short we write $\tau(h_n^l, t_n^r, h_n^r) = \tau$. By Corollary 5.16, $\tau < \infty$, $\mathbb{P}$-almost surely. Hence there exists a unique head point at $(T_u, H_u)$ distributed on $D_\tau \setminus \overline{U_{\hat{h}_n}^{\hat{s}_n}}$ with a certain law which we compute below. Before that, observe that, given ($h_n^l, t_n^r, h_n^r$), the conditional probability density $f_\tau$ of $\tau$ is

$$ f_{\tau}(t) = -\frac{\partial}{\partial t} \left[ e^{-2\lambda |\overline{U_t} \setminus \overline{U_{\hat{h}_n}^{\hat{s}_n}}|} \right] = 2\lambda e^{-2\lambda |\overline{U_t} \setminus \overline{U_{\hat{h}_n}^{\hat{s}_n}}|} \frac{\partial}{\partial t} |\overline{U_t} \setminus \overline{U_{\hat{h}_n}^{\hat{s}_n}}|. \quad (6.3) $$

The quantity $\frac{\partial}{\partial t} |\overline{U_t} \setminus \overline{U_{\hat{h}_n}^{\hat{s}_n}}|$ is the rate at which the exploration region grows as a function of $t \ge \hat{s}_n + \hat{h}_n$. By the application of the increasing wing property, Remark 5.14, we have that $\frac{\partial}{\partial t} |\overline{U_t} \setminus \overline{U_{\hat{h}_n}^{\hat{s}_n}}| \ne 0$.

<figure><img src="image_24.png" /><figcaption>FIGURE 6.2. If $\tau \in [t, t + \Delta t]$, the next head point $(u, h)$ lies in the region $\overline{U_{t+\Delta t}} \setminus \overline{U_t}$.</figcaption></figure>

Note that the point $(T_u, H_u)$ is non-uniformly distributed on $D_\tau \setminus \overline{U_{\hat{h}_n}^{\hat{s}_n}}$. Indeed, if $\tau \in [t, t + \Delta t]$, for $t \ge \hat{s}_n + \hat{h}_n$ and an infinitesimal element $\Delta t$, the head point, say $(u, h)$, is uniformly distributed on the region $(\overline{U_{t+\Delta t}} \setminus \overline{U_{\hat{h}_n}^{\hat{s}_n}}) \setminus (\overline{U_t} \setminus \overline{U_{\hat{h}_n}^{\hat{s}_n}})$ with the shape of a crescent. The region is nothing but $\overline{U_{t+\Delta t}} \setminus \overline{U_t}$, see Figure 6.2. It is more likely for $(u, h)$ to lie on the thicker side of $\overline{U_{t+\Delta t}} \setminus \overline{U_t}$. Let $(\hat{s}, \hat{h})$ and $(\hat{s}', \hat{h}')$ be the intersection points of $C_{(t_n^r, h_n^r)}$ with the radial birds at the right most points $(t, 0)$ and $(t + \Delta t, 0)$, respectively. These intersection points can be computed using (4.22) and (4.23). The equation of the two half-circles $D_t$ and $D_{t+\Delta t}$ are given by

$$ (u - \hat{s})^2 + h^2 = \hat{h}^2 \quad \text{and} \quad (u - \hat{s}')^2 + h^2 = (\hat{h}')^2, \quad (6.4) $$

in $(u, h)$-coordinate system, where $(t, 0)$ and $(t + \Delta t, 0)$ are the right most point of the half-circles $D_t$ and $D_{t+\Delta t}$.
Let $X_t = (X_t^1, X_t^2)$ be a random variable uniformly distributed on the region $\overline{U_{t+\Delta t}} \setminus \overline{U_t}$, as a representative of the next head point $(T_{n+1}^r + t_n^r, H_{n+1}^r) = (T_u, H_u)$, using (6.1). For $x \in [t_n^r, t + \Delta t]$, let $D_t(x)$ and $D_{t+\Delta t}(x)$ be the heights derived from the equation (6.4) of the half circles $D_t$ and $D_{t+\Delta t}$, respectively, expressed as

$$D_t(x) = (\hat{h}^2 - (x - \hat{s})^2)^{\frac{1}{2}} \text{ and } D_{t+\Delta t}(x) = ((\hat{h}')^2 - (x - (\hat{s}'))^2)^{\frac{1}{2}}.$$

Let $s, s' \in [t_n^r, t + \Delta t]$ be such that $s < s'$. Then

$$\mathbb{P}(X_t^1 \in [s, s']) = \int_s^{s'} \frac{1}{|\overline{U_{t+\Delta t}} \setminus \overline{U_t}|} (D_{t+\Delta t}(x) - D_t(x)) \, dx \approx \frac{\Delta t}{|\overline{U_{t+\Delta t}} \setminus \overline{U_t}|} \int_s^{s'} \frac{\partial}{\partial t} D_t(x) \, dx. \quad (6.5)$$

Taking the limit $\Delta t \to 0$ in (6.5), we have $\lim_{\Delta t \to 0} \frac{|\overline{U_{t+\Delta t}} \setminus \overline{U_t}|}{\Delta t} = \frac{\partial}{\partial t} |\overline{U_t} \setminus \overline{U_{\hat{h}_n}^{\hat{s}_n}}|$. Since $|\overline{U_t} \setminus \overline{U_{\hat{h}_n}^{\hat{s}_n}}| \neq 0$, we get that the conditional probability density of $T_{n+1}^r + t_n^r$ at $x \in [t_n^r, t]$ is equal to $\left(\frac{\partial}{\partial t} |\overline{U_t} \setminus \overline{U_{\hat{h}_n}^{\hat{s}_n}}|\right)^{-1} \frac{\partial}{\partial t} D_t(x)$.

For all $t \ge t_n^r$, $A \subset [0, t - t_n^r]$ and $B \subset \mathbb{R}^+$, using (6.5), (6.1) and the underlying time-stationarity, we hence have

$$\begin{aligned} \mathbb{P} (T_{n+1}^r \in A, H_{n+1}^r \in B | (H_n^l, T_n^r, H_n^r) = (h_n^l, t_n^r, h_n^r), \tau = t) \\ = \left( \frac{\partial}{\partial t} |\overline{U_t} \setminus \overline{U_{\hat{h}_n}^{\hat{s}_n}}| \right)^{-1} \int_{A+t_n^r} \frac{\partial D_t(x)}{\partial t} \mathbb{1}_B(D_t(x)) \, dx. \end{aligned} \quad (6.6)$$

Here is then the announced representation of the transition kernel. For all $t \ge t_n^r$, $A \subset [0, t - t_n^r]$ and $B, C \subset \mathbb{R}^+$,

$$\begin{aligned} \mathbb{P} (T_{n+1}^r \in A, H_{n+1}^r \in B, H_{n+1}^l \in C | (H_n^l, T_n^r, H_n^r) = (h_n^l, t_n^r, h_n^r)) \\ = \delta_{h_n^r}(C) \times \int_{\hat{s}_n + \hat{h}_n}^{\infty} \left( \frac{\partial}{\partial t} |\overline{U_t} \setminus \overline{U_{\hat{h}_n}^{\hat{s}_n}}| \right)^{-1} \left( \int_{A+t_n^r} \frac{\partial D_t(x)}{\partial t} \mathbb{1}_B(D_t(x)) \, dx \right) f_{\tau}(t) \, dt \\ = \delta_{h_n^r}(C) \times \int_{\hat{s}_n + \hat{h}_n}^{\infty} \left( \int_{A+t_n^r} \frac{\partial D_t(x)}{\partial t} \mathbb{1}_B(D_t(x)) \, dx \right) 2\lambda e^{-2\lambda|\overline{U_t} \setminus \overline{U_{\hat{h}_n}^{\hat{s}_n}}|} \, dt, \end{aligned} \quad (6.7)$$

where $f_{\tau}$ is the conditional probability density of $\tau$, given $(h_n^l, t_n^r, h_n^r)$, as derived in (6.3). This completes the proof of Theorem 6.1. $\square$

**Remark 6.2.** The sojourn time of the Markov chain $\{(H_n^l, T_n^r, H_n^r)\}_{n \in \mathbb{Z}}$ is nothing but the inter-step time of the Markov chain, as depicted in Figure 3.1. The Markov process enables us to explore the structural features of the handover process, which we leave for future research.

## 7. HANDOVER FREQUENCY: TWO AND MULTI-SPEED CASES

### 7.1. Two-speed case.
As already explained, this part is motivated by communication networks comprising of, for example, low earth orbit (LEO) satellites along with medium earth orbit (MEO) satellites. The LEO and MEO satellites move at different speeds depending on their altitude. Naturally, as a first step towards understanding the handover phenomenon, we would like to consider a model on Euclidean plane with two different types of mobile stations, moving at speed $v_1$ and $v_2$, respectively, as described in Section 2.

Suppose the initial locations are independent homogeneous Poisson point processes $\tilde{\Phi}^1$ and $\tilde{\Phi}^2$ in polar coordinates on $\mathbb{R}^+ \times (-\pi, \pi]$, with $\tilde{\Phi}^l = \sum_{i \in \mathbb{N}} \delta_{(R_i^{(l)}, \alpha_i^{(l)})}$, for two different speeds $v_l$, with $l = 1, 2$. The intensity measures $\mu_1, \mu_2$ have for density

$$d\mu_1 := 2\pi r \lambda_1 dr \otimes \frac{1}{2\pi} d\alpha \text{ and } d\mu_2 := 2\pi r \lambda_2 dr \otimes \frac{1}{2\pi} d\alpha, \quad (7.1)$$

respectively, where $\lambda_1$ and $\lambda_2$ are the intensities. In short we shall write $\tilde{\Phi} := \tilde{\Phi}^1 + \tilde{\Phi}^2$ and $\lambda := \lambda_1 + \lambda_2$. Essentially the model inherits all the properties of the single-speed model, except that the radial bird particle process consists of two types of particles. Let us denote the two types of radial bird particles by
type 1 and type 2, corresponding to the two speeds $v_1, v_2$, respectively. A key new fact is that the birds of different types can intersect each other twice or never almost surely, unlike the unique intersection property 5 of the radial birds in the single-speed case. This fact will be more clear later in Lemma 7.5, Lemma 7.8 and Remark 7.6.

### 7.1.1. Head point process and radial bird particle process.
For any speed $v \in \{v_1, v_2\}$, the radial bird particle corresponding to an atom $(R, \alpha)$ of $\tilde{\Phi} = \tilde{\Phi}^1 + \tilde{\Phi}^2$ is given by the distance function

$$f_v((R, \alpha), t) := (R^2 + 2vtR \cos \alpha + v^2t^2)^{\frac{1}{2}} = \left( v^2 \left( t + \frac{R}{v} \cos \alpha \right)^2 + (R|\sin \alpha|)^2 \right)^{\frac{1}{2}}, \quad (7.2)$$

whereas the head of the radial bird is given by $(-\frac{R}{v} \cos \alpha, R|\sin \alpha|)$, see Remark 4.5. For an atom $(R_i^{(l)}, \alpha_i^{(l)})$ of $\tilde{\Phi}^l$, the head point is at

$$\left(-\frac{R_i^{(l)}}{v_l} \cos \alpha_i^{(l)}, R_i^{(l)} |\sin \alpha_i^{(l)}|\right) := \left(T_i^{(l)}, H_i^{(l)}\right).$$

Then the head point processes on $\mathbb{H}^+$ are $\mathcal{H}^l := \sum_{i \in \mathbb{N}} \delta(T_i^{(l)}, H_i^{(l)})$, for $l = 1, 2$. We have a result similar to Lemma 4.3:

**Lemma 7.1.** For each $l = 1, 2$, $\mathcal{H}^l$ is a Poisson point process on $\mathbb{H}^+$ with intensity measure $\nu_l$, where $d\nu_l := v_l dt \otimes 2\lambda_l dh$, and $\mathcal{H}^1$ and $\mathcal{H}^2$ are independent.

*Proof Lemma 7.1.* The proof follows from constructing $\mathcal{H}^1$ and $\mathcal{H}^2$ separately from the Poisson point processes $\tilde{\Phi}^1$ and $\tilde{\Phi}^2$, respectively, in the same way as we have constructed $\mathcal{H}$ from $\tilde{\Phi}$ in Lemma 4.3 for the single-speed case. The point processes $\mathcal{H}^1$ and $\mathcal{H}^2$ are independent because $\tilde{\Phi}^1$ and $\tilde{\Phi}^2$ are independent. $\square$

We write the law of the head point process $\mathcal{H} := \mathcal{H}^1 + \mathcal{H}^2$, as $\mathbb{P}_{\mathcal{H}} := \mathbb{P}_{\mathcal{H}^1} \otimes \mathbb{P}_{\mathcal{H}^2}$, which is the product law of $\mathbb{P}_{\mathcal{H}^1}$ and $\mathbb{P}_{\mathcal{H}^2}$, since $\mathcal{H}^1$ and $\mathcal{H}^2$ are independent. Observe that, in (7.2), the speed factor only scales the time coordinate of the space $\mathbb{H}^+$, reflected only on the intensity measure $\nu_l$, for $l = 1, 2$. In this sense the speed $v \in \{v_1, v_2\}$ can be considered as the shape parameter of the radial birds, which are defined as

$$C^l := \left\{ \left( t, f_{v_l} \left( (R^{(l)}, \alpha^{(l)}), t \right) \right) : t \in \mathbb{R} \right\}, \quad (7.3)$$

where $C^l$ corresponds to the atom $(R^{(l)}, \alpha^{(l)})$ of $\tilde{\Phi}^l$, for $l = 1, 2$. The radial bird particle process is defined as

$$\mathcal{P}_c(v_1, v_2) := \sum_{l=1,2} \sum_{i \in \mathbb{N}} \delta_{C_i^l}.$$

We also denote by $C_i^l$ the radial bird with head point $(T_i^{(l)}, H_i^{(l)}) \in \mathcal{H}^l$. In the following, we have the same properties for $\mathcal{P}_c(v_1, v_2)$ as those explored in Lemma 4.11 for the single-speed case.

**Lemma 7.2.** The following hold true the radial bird particle process $\mathcal{P}_c(v_1, v_2)$:

(i) $\mathcal{P}_c(v_1, v_2)$ is a Poisson particle process on the space $(C(\mathbb{H}^+), \mathcal{B}(C(\mathbb{H}^+)))$, with support on the set of closed sets of the form (7.3).

(ii) Both $\mathcal{P}_c(v_1, v_2)$ and $\cup_{l=1,2} \cup_i C_i^l$ are stationary along the time coordinate.

In view of Remark 4.12, we can equivalently write the radial bird particle process $\mathcal{P}_c(v_1, v_2)$ in terms of marked head point process $\mathcal{H}_c(v_1, v_2) := \mathcal{H}_c(v_1) + \mathcal{H}_c(v_2)$, where

$$\mathcal{H}_c(v_l) := \sum_{i \in \mathbb{N}} \delta((T_i^{(l)}, H_i^{(l)}), C_i^l),$$

for $l = 1, 2$. The time-stationarity of $\mathcal{H}_c(v_1, v_2)$ is inherited from the same property of the head point process $\mathcal{H} = \mathcal{H}^1 + \mathcal{H}^2$, as seen in part (ii) of Lemma 7.2, for the two-speed case.
### 7.1.2. Joint lower envelope process.
The joint lower envelope process is a stochastic process defined as $\{L_{v_1,v_2}(t)\}_{t \in \mathbb{R}}$, where

$$L_{v_1,v_2}(t) := \min_{l=1,2} \inf_{i \in \mathbb{N}} f_{v_l} \left( (R_i^{(l)}, \alpha_i^{(l)}), t \right).$$

The joint lower envelope of all the radial birds of the two types together, is a random closed subset of $\mathbb{H}^+$ depicted in Figure 3.2, defined as

$$\mathcal{L}_e(v_1, v_2) := \{(t, L_{v_1, v_2}(t)) : t \in \mathbb{R}\}. \qquad (7.4)$$

Let us denote the open region above the lower envelope $\mathcal{L}_e(v_1, v_2)$ as

$$\mathcal{L}_e^+(v_1, v_2) := \{(t, h) : h > L_{v_1, v_2}(t)\}. \qquad (7.5)$$

The following result about the time-stationarity of the joint lower envelope process can be proved in the same way as in Lemma 4.15.

**Lemma 7.3.** *The joint lower envelope process $\{(t, L_{v_1,v_2}(t))\}_{t \in \mathbb{R}}$ is stationary with respect to time.*

In the following, we describe the characterization for any point of a radial bird to be on the lower envelope, which is analogous to Lemma 4.18. For any point $(s, u) \in \mathbb{H}^+$ and for any speed $v \in \{v_1, v_2\}$, the curve with respect to the $(t, h)$-coordinates associated with the equation

$$v^2(t-s) + h^2 = u^2 \qquad (7.6)$$

is an ellipse containing $(s, u)$. Let $E_u^{s,v}$ be the open region enclosed by the upper half-ellipse given by (7.6) and above the time axis.

**Lemma 7.4 (Half-ellipse condition).** For any point $(s, u) \in \cup_{l=1,2} \cup_{i \in \mathbb{N}} C_i^l$, the following holds true:

(i) The point $(s, u) \in \mathcal{L}_e(v_1, v_2)$ if and only if $\mathcal{H}^l(E_u^{s,v_l}) = 0$, for $l = 1, 2$.

(ii) The probability of such an event is,

$$\mathbb{P}((s, u) \in \mathcal{L}_e(v_1, v_2)) = e^{-\lambda \pi u^2}, \qquad (7.7)$$

where $\lambda = \lambda_1 + \lambda_2$.

*Proof of Lemma 7.4.* Part (i) can be proved using Lemma 4.18 and Discussion 4.21 for the point processes $\mathcal{H}^1$ and $\mathcal{H}^2$ separately. Given two speeds $v_1, v_2$, there are two upper half ellipses $E_u^{s,v_1}$ and $E_u^{s,v_2}$ which should be empty of points from $\mathcal{H}^1$ and $\mathcal{H}^2$, respectively.

Part (ii) can be proved using the independence of $\mathcal{H}^1$ and $\mathcal{H}^2$ as follows:

$$\mathbb{P}((s, u) \in \mathcal{L}_e(v_1, v_2)) = \mathbb{P}_{\mathcal{H}^1}(\mathcal{H}^1(E_u^{s,v_1}) = 0) \times \mathbb{P}_{\mathcal{H}^2}(\mathcal{H}^2(E_u^{s,v_2}) = 0) = e^{-2\lambda_1 v_1 \frac{\pi u^2}{2v_1}} e^{-2\lambda_2 v_2 \frac{\pi u^2}{2v_2}} = e^{-\lambda \pi u^2}, \\ \text{where } \lambda = \lambda_1 + \lambda_2. \quad \square$$

Before moving to the construction of the handover point process and deriving its intensity for the two-speed case, we present some geometric results, determining the condition on the location of heads for which there are two intersections between the two birds. For a pair of birds of different types, we will refer to the following results, Lemma 7.5 and Lemma 7.8, as the intersection criterion. For this we suppose, without loss of generality, that $v_1 > v_2$.

### 7.1.3. Intersection criterion.

**Lemma 7.5 (Intersection criterion).** Let $t_1 \in \mathbb{R}$ and $h_1, h_2 \in \mathbb{R}^+$ be such that there is a bird at $(t_1, h_1)$ of speed $v_1$. Then there exist two sets $D_\ell \equiv D_\ell(t_1, h_1, h_2)$, $D_r \equiv D_r(t_1, h_1, h_2) \subsetneq \mathbb{R}$, such that, for all $t_2 \in (D_\ell \cup D_r)^\circ$ or $\partial(D_\ell \cup D_r)$ or $(D_\ell \cup D_r)^c$, the radial bird of speed $v_2$ and head at $(t_2, h_2)$ intersects the first bird, twice, once or never, respectively. The sets $D_\ell, D_r$ are given by,

$$D_\ell = \begin{cases} (-\infty, t_1], & \text{if } h_2 \ge h_1, \\ (-\infty, t_1 - t^*], & \text{if } h_2 < h_1, \end{cases} \qquad D_r = \begin{cases} [t_1, \infty), & \text{if } h_2 \ge h_1, \\ [t_1 + t^*, \infty), & \text{if } h_2 < h_1, \end{cases} \qquad (7.8)$$

where $t^* = \frac{1}{v_1 v_2} (h_1^2 - h_2^2)^{\frac{1}{2}} (v_1^2 - v_2^2)^{\frac{1}{2}}$.
**Remark 7.6.** Given $t_1 \in \mathbb{R}$ and $h_1, h_2 \in \mathbb{R}^+$, suppose there is a bird at $(t_1, h_1)$ of speed $v_1$. In the case $h_1 \le h_2$, the bird at $(t_2, h_2)$ with speed $v_2$ intersects the first one exactly once if $t_2 = t_1$ and $h_2 = h_1$. In the other case, namely $h_1 > h_2$, the bird at $(t_2, h_2)$ with speed $v_2$ intersects the first one exactly once if $t_2 = t_1 - t^*$ or $t_1 + t^*$. Hence there exists almost surely two or no intersection, between the birds of different types.

**Remark 7.7 (Symmetry).** Suppose two radial birds with heads located at $(t_1, h_1)$ and $(t_2, h_2)$, of two different types intersect. Then, $t_2 \in D_\ell(t_1, h_1, h_2)$ if and only if $-t_2 \in D_r(-t_1, h_1, h_2)$.

We also have an intersection criterion for the dual scenario as follows, which can be proved similarly.

**Lemma 7.8 (Intersection criterion).** Let $t_2 \in \mathbb{R}$ and $h_1, h_2 \in \mathbb{R}^+$ be such that there is a bird at $(t_2, h_2)$ of speed $v_2$. Then there exist two sets $D'_\ell \equiv D'_\ell(t_1, h_1, h_2)$, $D'_r \equiv D'_r(t_1, h_1, h_2) \subsetneq \mathbb{R}$, such that, for all $t_1 \in (D'_\ell \cup D'_r)^\circ$ or $\partial(D'_\ell \cup D'_r)$ or $(D'_\ell \cup D'_r)^c$, the radial bird of speed $v_1$ and head at $(t_1, h_1)$ intersects the first bird, twice, once or never, respectively. The sets $D'_\ell, D'_r$ are given by,

$$D'_{\ell} = \begin{cases} (-\infty, t_2], & \text{if } h_1 \le h_2, \\ (-\infty, t_2 - t^*], & \text{if } h_1 > h_2, \end{cases} \qquad D'_{r} = \begin{cases} [t_2, \infty), & \text{if } h_1 \le h_2, \\ [t_2 + t^*, \infty), & \text{if } h_1 > h_2, \end{cases} \qquad (7.9)$$

$$\text{where } t^* = \frac{1}{v_1 v_2} (h_1^2 - h_2^2)^{\frac{1}{2}} (v_1^2 - v_2^2)^{\frac{1}{2}}.$$

*Proof of Lemma 7.5.* Unlike in the single-speed case, each pair of radial birds with different speeds intersect each other twice or never almost surely. Let $(s, h) \in \mathcal{L}_e(v_1, v_2)$ be an intersection point of the two radial birds $C_1, C_2$ with heads at $(t_1, h_1)$ and $(t_2, h_2)$ and with speeds $v_1, v_2$, respectively. We prove the result only for the case with $t_2 \le t_1$. The other case $t_2 \ge t_1$ can be proved similarly. There exists a pair $(r_1, \alpha_1)$ and $(r_2, \alpha_2)$ of points of the initial marked Poisson point process $\tilde{\Phi}$ of the stations, such that

$$(t_1, h_1) = \left( -\frac{r_1}{v_1} \cos \alpha_1, r_1 |\sin \alpha_1| \right) \quad \text{and} \quad (t_2, h_2) = \left( -\frac{r_2}{v_2} \cos \alpha_2, r_2 |\sin \alpha_2| \right). \qquad (7.10)$$

For finding the time-coordinate of the intersections, we solve the quadratic equation

$$v_1^2 s^2 + 2sv_1 r_1 \cos \alpha_1 + r_1^2 = v_2^2 s^2 + 2sv_2 r_2 \cos \alpha_2 + r_2^2,$$

that is,

$$(v_1^2 - v_2^2)s^2 - 2s(v_1^2 t_1 - v_2^2 t_2) + r_1^2 - r_2^2 = 0. \qquad (7.11)$$

The time coordinates of intersections are given by the solutions to the quadratic equation (7.11), which are

$$\begin{align}
\hat{s}_1 &\equiv \hat{s}_1(t_1, t_2, h_1, h_2) := \frac{1}{v_1^2 - v_2^2} \left[ v_1^2 t_1 - v_2^2 t_2 - \sqrt{\Delta} \right], \tag{7.12} \\
\hat{s}_2 &\equiv \hat{s}_2(t_1, t_2, h_1, h_2) := \frac{1}{v_1^2 - v_2^2} \left[ v_1^2 t_1 - v_2^2 t_2 + \sqrt{\Delta} \right], \nonumber
\end{align}$$

where

$$\Delta \equiv \Delta(t_1, t_2, h_1, h_2) := (v_1^2 t_1 - v_2^2 t_2)^2 - (r_1^2 - r_2^2)(v_1^2 - v_2^2) = v_1^2 v_2^2 (t_1 - t_2)^2 - (h_1^2 - h_2^2)(v_1^2 - v_2^2). \quad (7.13)$$

The last simplification is due to the fact that, the head point $(t_i, h_i)$ corresponding to speed $v_i$ satisfies relation $r_i^2 = v_i^2 t_i^2 + h_i^2$, for $i = 1, 2$, which can be derived from (7.10).

If $h_2 \ge h_1$, the quantity $\Delta$ is larger than or equal to 0. On the other hand, $\Delta$ can become negative for $h_2 < h_1$, but depending on $t_1, t_2$. So one can say that the radial birds with different $t_1, t_2$ and different $h_1, h_2$, intersect each other twice, once or never, depending on whether $\Delta > 0$, $\Delta = 0$ or $\Delta < 0$, respectively. For $s_1, s_2$ to be real, we need $\Delta$ to be non-negative, that is $(t_1 - t_2)^2 \ge (h_1^2 - h_2^2) \frac{(v_1^2 - v_2^2)}{v_1^2 v_2^2}$. Given the two speeds $v_1, v_2$, the head point $(t_1, h_1)$ of the first radial bird, and $h_2$, the height of the head point of the second radial bird, let

$$D_{\ell} \equiv D_{\ell}(t_1, h_1, h_2) := \left\{ t_2 \le t_1 : (t_1 - t_2)^2 \ge (h_1^2 - h_2^2) \frac{(v_1^2 - v_2^2)}{v_1^2 v_2^2}, \text{ i.e., } \Delta \ge 0 \right\}, \quad (7.14)$$

be the set of all possible values of $t_2$, the first coordinate of the head of the second radial bird, for which $\Delta$ is non-negative. In case $h_1 < h_2$, we have $D_\ell = (-\infty, t_1]$, but in case $h_1 \ge h_2$, we have
$D_\ell \subsetneq (-\infty, t_1]$, depending on the specific choice of $t_1$. The main goal of the rest of the proof is to derive the region $D_\ell$, when $h_1 \ge h_2$.

The situations where the two radial birds do not intersect or do intersect, are depicted on the first picture in Figure 7.1. Given the speeds $v_1 > v_2$, the heights of the heads $h_1 \ge h_2$, and the first coordinate $t_1$ of the first radial bird, there exist $t_2$, far enough from $t_1$, and $t'_2$, close enough to $t_1$, such that $(t_1 - t_2)^2 \ge (h_1^2 - h_2^2) \frac{(v_1^2 - v_2^2)}{v_1^2 v_2^2}$ and $(t_1 - t'_2)^2 < (h_1^2 - h_2^2) \frac{(v_1^2 - v_2^2)}{v_1^2 v_2^2}$. In other words, $\Delta \ge 0$ for $t_2$ and $\Delta < 0$ for $t'_2$. As a consequence, $t_2 \in D_\ell$, but $t'_2 \notin D_\ell$. Figure 7.1 illustrates the set $D_\ell$.

<figure><img src="image_25.png" /></figure>

(1) There exists $t_2, t'_2 < t_1$, such that the radial bird at $(t_2, h_2)$ intersects the radial bird at $(t_1, h_1)$, but the other one at $(t'_2, h_2)$ does not.

(2) There exists $t^*$ such that, two radial birds at $(t_1, h_1)$, $(t_2, h_2)$ intersects each other twice for all $t_2 < t_1 - t^*$ and $t_2 > t_1 + t^*$.

<caption>FIGURE 7.1. Criterion for intersections with $v_1 > v_2$, $h_1 > h_2$, fixed $t_1$ but variable $t_2$.</caption>

In view of the last observation, given $t_1$ and $h_1 \ge h_2$, there exists a maximum value of $t_2$ such that there is a unique intersection point of the two radial birds $C_1, C_2$, see Picture 7.12 of Figure 7.1. In other words, it corresponds to the value of $t_2$ such that $\Delta = 0$, which is $t_1 - t^*$, given $t_1$ and $h_1 \ge h_2$, where,

$$t^* := \frac{1}{v_1 v_2} (h_1^2 - h_2^2)^{\frac{1}{2}} (v_1^2 - v_2^2)^{\frac{1}{2}}. \qquad (7.15)$$

Hence the set $D_\ell$ turns out to be $(-\infty, t_1 - t^*]$, given $t_1$ and $h_1 \ge h_2$. Similarly we can determine the other part in the case $t_2 \ge t_1$, where $D_r = D_r(t_1, h_1, h_2)$ and $D_r(t_1, h_1, h_2) = [t_1 + t^*, \infty)$, for $h_1 \ge h_2$ and $D_r(t_1, h_1, h_2) = [t_1, \infty)$, for $h_1 < h_2$. $\square$

### 7.1.4. Geometric version of the intersection criteria.
Suppose there are two radial birds of type 1 and 2, with their heads at $(t_1, h_1)$ and $(t_2, h_2)$, respectively. Keeping the head of one of them fixed, we can find the trajectory of the other one, such that the two radial birds just touch each other. This leads to a geometric version of the intersection criterion that we describe now together with some applications.

**I. Fixed** $(t_1, h_1)$. For a given radial bird at $(t_1, h_1)$ of type 1, consider the collection of points

$$\tilde{C}_{(t_1,h_1)} := \{(t, h) \in \mathbb{H}^+ : t = t_1 - t^* \text{ or } t = t_1 + t^*\},$$

where $t^* = \frac{1}{v_1 v_2} (h_1^2 - h^2)^{\frac{1}{2}} (v_1^2 - v_2^2)^{\frac{1}{2}}$, given $(t_1, h_1)$, as defined in (7.15). Any radial bird with head point on $\tilde{C}_{(t_1,h_1)}$ and of type 2, intersects $C_{(t_1,h_1)}^1$ exactly once; in other words, they just touch each other. Geometrically, $\tilde{C}_{(t_1,h_1)}$ is given by the curve governed by the equation $t - t_1 = \pm t^*$, i.e.,

$$(t - t_1)^2 = \frac{1}{v_1^2 v_2^2} (h_1^2 - h^2)(v_1^2 - v_2^2), \quad \text{i.e.,} \quad \frac{v_1^2 v_2^2}{v_1^2 - v_2^2} (t - t_1)^2 + h^2 = h_1^2, \qquad (7.16)$$

which is an ellipse in $\mathbb{H}^+$, see Picture 1 of Figure 7.2. In short, the head point, say $(t_2, h_2)$, of the radial bird of type 2 must lie outside the open region in $\mathbb{H}^+$ enclosed by the ellipse given by (7.16), to have guaranteed intersection with $C_{(t_1,h_1)}^1$, as seen in Lemma 7.5, the intersection criterion.

**II. Fixed** $(t_2, h_2)$. For a fixed radial bird of type 2 at $(t_2, h_2)$, similarly define

$$\tilde{C}_{(t_2,h_2)} := \{(t, h) \in \mathbb{H}^+ : t = t_2 - t^* \text{ or } t = t_2 + t^*\},$$
<figure><img src="image_26.png" /><figcaption>(1) For fixed $(t_1, h_1)$ of type 1, the trajectory of the head points of the radial birds that touch $C_{(t_1, h_1)}^1$ forms an ellipse (in color teal) given by (7.16).</figcaption></figure>

<caption>(2) Given $(t_2, h_2)$ of type 2, the trajectory of the head of the radial birds of type 1, that touch $C_{(t_2, h_2)}^2$ forms a hyperbola (in color teal) given by (7.17).</caption>

FIGURE 7.2. Geometric interpretation of intersection criterion.

where $t^* = \frac{1}{v_1 v_2} (h^2 - h_2^2)^{\frac{1}{2}} (v_1^2 - v_2^2)^{\frac{1}{2}}$, as defined in (7.15). Any radial bird with head point on $\tilde{C}_{(t_2, h_2)}$ and of type 1, intersects $C_{(t_2, h_2)}^2$ exactly once, or equivalently, these two birds just touch each other. Here also, the set $\tilde{C}_{(t_2, h_2)}$ is characterised by the curve given by $t - t_2 = \pm t^*$, i.e.,

$$ (t - t_2)^2 = \frac{1}{v_1^2 v_2^2} (h^2 - h_2^2)(v_1^2 - v_2^2), \quad \text{i.e.,} \quad h^2 - \frac{v_1^2 v_2^2}{v_1^2 - v_2^2} (t - t_2)^2 = h_2^2, \qquad (7.17) $$

which is a branch of a hyperbola on $\mathbb{H}^+$, see Picture 2 of Figure 7.2. This curve can also be considered as a radial bird with head at $(t_2, h_2)$, with a different speed $v_1 v_2 / (v_1^2 - v_2^2)^{\frac{1}{2}}$. There is guaranteed intersection between $C_{(t_2, h_2)}^2$ and the radial bird of type 1 with head at, say $(t_1, h_1)$, if $(t_1, h_1)$ lies below the curve of the hyperbola given by (7.17), as seen in the intersection criterion of Lemma 7.8.

**Remark 7.9 (Bijection between $C_{(t_2, h_2)}^2$ and $\tilde{C}_{(t_2, h_2)}$).** Given a radial bird of type 2 at $(t_2, h_2)$, the set $\tilde{C}_{(t_2, h_2)}$ is constructed in such a way that for each point $(\hat{s}, \hat{h}) \in C_{(t_2, h_2)}^2$, there exists a unique point, say $(t_1, h_1) \in \tilde{C}_{(t_2, h_2)}$ such that there is a radial bird of type 1 with head point at $(t_1, h_1)$, which touches the bird $C_{(t_2, h_2)}^2$ at $(\hat{s}, \hat{h})$. In this sense, there exists a natural bijection, say $\gamma_{(t_2, h_2)}: C_{(t_2, h_2)}^2 \to \tilde{C}_{(t_2, h_2)}$. Indeed, given $(t_2, h_2)$, if we take $(\hat{s}, \hat{h}) \in C_{(t_2, h_2)}^2$, then there exists a unique point $(t_1, h_1)$, such that $\hat{s}$ satisfies (7.12) and the pair $(\hat{s}, \hat{h})$ satisfies (7.17), i.e.,

$$ \hat{s} = \frac{1}{v_1^2 - v_2^2} (v_1^2 t_1 - v_2^2 t_2) \quad \text{and} \quad \hat{h}^2 = \frac{v_1^2 v_2^4}{(v_1^2 - v_2^2)^2} (t_1 - t_2)^2 + h_1^2, \qquad (7.18) $$

see Figure 7.3. Then $(t_1, h_1)$ is given by

$$ t_1 = \frac{1}{v_1^2} [\hat{s}(v_1^2 - v_2^2) + v_2^2 t_2], \qquad (7.19) $$

and

$$ h_1^2 = \hat{h}^2 - \frac{v_1^2 v_2^4}{(v_1^2 - v_2^2)^2} \left( \frac{1}{v_1^2} [\hat{s}(v_1^2 - v_2^2) + v_2^2 t_2] - t_2 \right)^2 = \hat{h}^2 - \frac{v_2^4}{v_1^2} (\hat{s} - t_2)^2. \qquad (7.20) $$

Given $(t_2, h_2) \in \mathbb{H}^+$, $\gamma_{(t_2, h_2)}((\hat{s}, \hat{h})) = (t_1, h_1)$, which is determined by (7.19) and (7.20).

**Remark 7.10 (Region below $\tilde{C}_{(t_2, h_2)}$).** For any $(\hat{s}, \hat{h}) \in C_{(t_2, h_2)}^2$, the upper half ellipse enclosed by the curve $v_1^2(t - \hat{s})^2 + h^2 = \hat{h}^2$ lies completely below the hyperbola branch given by (7.17). Define a sequence $\{\hat{h}_t\}_{t \in \mathbb{R}}$, where $\hat{h}_t := (v_2^2(t - t_2)^2 + h_2^2)^{\frac{1}{2}}$, corresponding to the height of the radial bird $C_{(t_2, h_2)}^2$ at different time $t$. Consider $(\hat{s}_1, \hat{h}_1) \in C_{(t_2, h_2)}^2$ such that $\hat{s}_1 \ge \hat{s}$. Then, using the bijection $\gamma_{(t_2, h_2)}$, we have the image of the points $(\hat{s}, \hat{h})$, $(\hat{s}_1, \hat{h}_1)$, expressed as in (7.19), (7.20), lying on the curve $\tilde{C}_{(t_2, h_2)}$.
<figure><img src="image_27.png" /><figcaption>FIGURE 7.3. Under the bijection $\gamma_{(t_2,h_2)}$, the point $(\hat{s}, \hat{h})$ is mapped to a point $(t_1, h_1)$, such that the radial birds $C^1_{(t_1,h_1)}$, if it exists, touches $C^2_{(t_2,h_2)}$ at $(\hat{s}, \hat{h})$.</figcaption></figure>

Note that for $(\hat{s}, \hat{h})$ and $(\hat{s}_1, \hat{h}_1)$ to represent two consecutive handovers, it is necessary that along with $E^{\hat{s},v_1}_{\hat{h}} \cup E^{\hat{s}_1,v_1}_{\hat{h}_1}$, the extra region $S$, outside $E^{\hat{s},v_1}_{\hat{h}} \cup E^{\hat{s}_1,v_1}_{\hat{h}_1}$, but below the hyperbola $\tilde{C}_{(t_2,h_2)}$ between the two points $\gamma_{(t_2,h_2)}(\hat{s}, \hat{h})$ and $\gamma_{(t_2,h_2)}(\hat{s}_1, \hat{h}_1)$, be empty of head points of type 1.

Then we have

$$ \bigcup_{t \in [\hat{s}, \hat{s}_1]} E^{t, v_1}_{\hat{h}_t} = E^{\hat{s}, v_1}_{\hat{h}} \cup S \cup E^{\hat{s}_1, v_1}_{\hat{h}_1}, $$

where $S$ is the region outside $E^{\hat{s},v_1}_{\hat{h}} \cup E^{\hat{s}_1,v_1}_{\hat{h}_1}$ but below the hyperbola (7.17) within the time interval

$$ \left[ \frac{1}{v_1^2} (\hat{s}(v_1^2 - v_2^2) + v_2^2 t_2), \frac{1}{v_1^2} (\hat{s}_1(v_1^2 - v_2^2) + v_2^2 t_2) \right], $$

see Figure 7.3.

### 7.1.5. Some more geometric results about mixed intersections.
Here we present some more geometric properties intersection of radial birds of different types (called as mixed intersections) and the corresponding handovers.

**Observation 7.11.** Suppose there are two radial birds with their heads at $(t_1, h_1)$, $(t_2, h_2)$, with speed $v_1$ and $v_2$, respectively, such that $v_1 > v_2$. Also suppose that the pair $(t_1, h_1)$, $(t_2, h_2)$ satisfies one of the intersection criteria and they intersect each other at $(\hat{s}_1, \hat{h}_1)$ and $(\hat{s}_2, \hat{h}_2)$, such that $\hat{s}_1 \le \hat{s}_2$. Then the following holds:

(i). $\hat{s}_1 \le t_1 \le \hat{s}_2$;

(ii). $t_2 \le \hat{s}_1 \le t_1$, if $t_2 \le t_1$ and $t_1 \le \hat{s}_2 \le t_2$, if $t_1 \le t_2$.

In the following, we present a property that guarantees intersection between different types of birds.

**Lemma 7.12.** Let $(s, u) \in \mathbb{H}^+$ and let $(t_1, h_1)$, $(t_2, h_2) \in \mathbb{H}^+$ be such that the radial bird at $(t_1, h_1)$ is of type 1 and satisfies $(t_1, h_1) \in \partial E^{s,v_1}_u$, but $(t_2, h_2) \notin E^{s,v_2}_u$, see Figure 7.4. Then the following holds:

(i). The radial bird of type 2 intersects that of type 1 twice almost surely.

(ii). For $(\hat{s}_1, \hat{h}_1)$ and $(\hat{s}_2, \hat{h}_2)$ being the first and second intersections of the two radial birds, respectively, we have $\hat{s}_1 \le s \le \hat{s}_2$.

*Proof of part (i).* By the intersection criterion in Lemma 7.5 and Lemma 7.8, the two radial birds $C^1_{(t_1,h_1)}$ and $C^2_{(t_2,h_2)}$ must intersect each other if $h_1 \le h_2$, for any $t_1, t_2 \in \mathbb{R}$. We now prove the result for the case $h_1 > h_2$. Without loss of generality, we prove the result for $s = 0$ and $t_2 > t_1 > 0$; one of the other case, $t_2 < t_1 < 0$, can be proved similarly. The two more cases $t_2 < 0 < t_1$ and $t_1 < 0 < t_2$ can be proved as a corollary of the cases $t_2 > t_1 > 0$ and $t_2 < t_1 < 0$, respectively.
Suppose $t_2 > t_1 > 0$. Let us denote the vertical line $t = 0$ as $L_0$. Since $(t_2, h_2) \notin E_u^{0,v_2}$, by Observation 4.17 (for the single-speed case), the radial bird of type 2 with head at $(t_2, h_2)$ must intersect the vertical line $L_0$ at a level higher than $u$. On the other hand, the radial bird of type 1 at $(t_1, h_1)$ intersects the line $L_0$ exactly at level $u$, see Figure 7.4. This implies that the two radial birds at $(t_1, h_1)$ and $(t_2, h_2)$ intersect each other twice almost surely. This completes the proof of this part. $\square$

<figure><img src="image_28.png" /><figcaption>FIGURE 7.4. The bird at $(t_1, h_1)$ intersects the line $L_0$ at level $u$, but the bird at $(t_2, h_2)$ intersects the line $L_0$ at level higher than $u$.</figcaption></figure>

*Proof of part (ii).* Without loss of generality, we prove the result for $s = 0$ also. We also assume $0 < t_1 < t_2$, whereas the other case $t_2 < t_1 < 0$, can be proved similarly. The two more cases $t_2 < 0 < t_1$ and $t_1 < 0 < t_2$ can be derived from the cases $0 < t_1 < t_2$ and $t_2 < t_1 < 0$, respectively. Observe that each side of the curve of the radial birds is monotonic. Since the bird at $(t_2, h_2)$ intersects the vertical line $L_0$ at a level higher than $u$, then using monotonicity and the fact that $0 \le t_1$, the first intersection must be on the left of $(0, u)$ and the second intersection on the right of $(0, u)$, i.e., $\hat{s}_1 \le 0 \le \hat{s}_2$. This completes the proof. $\square$

We present another property of mixed intersections, which will be useful when deriving the Palm probability distribution with respect to handovers of different types.

**Lemma 7.13.** Suppose the radial birds at $(t_1, h_1)$, $(t_2, h_2) \in \mathbb{H}^+$, of type 1 and 2, respectively, intersect at $(\hat{s}_1, \hat{h}_1)$ and $(\hat{s}_2, \hat{h}_2)$, where $\hat{s}_1 \le \hat{s}_2$ and $\mathcal{H}^l(E_{\hat{h}_1}^{\hat{s}_1, v_1}) = 0$, for $l = 1, 2$, so that $(\hat{s}_1, \hat{h}_1)$ is a handover. Also suppose there exists a radial bird at $(t_0, h_0)$ of type $l$, that intersects the bird of type 1 at $(\tilde{s}, \tilde{h})$, if $l = 1$, and at $(\tilde{s}_1, \tilde{h}_1)$, $(\tilde{s}_2, \tilde{h}_2)$, if $l = 2$. Then the following holds:

(i). For $l = 1$, $\hat{s}_1 \le \tilde{s} \le \hat{s}_2$ if and only if $(t_0, h_0) \in E_{\hat{h}_2}^{\hat{s}_2, v_1} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_1}$.

(ii). For $l = 2$, $\hat{s}_1 \le \tilde{s}_2 \le \hat{s}_2$ if and only if $(t_0, h_0) \in E_{\hat{h}_2}^{\hat{s}_2, v_2} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_2}$.

*Proof.* Assume that $(t_0, h_0)$ is of type 1 and $\hat{s}_1 \le \tilde{s} \le \hat{s}_2$. Let $L_{\hat{s}_1}$ and $L_{\hat{s}_2}$ be the vertical lines $t = \hat{s}_1$ and $t = \hat{s}_2$, respectively. Using the monotonicity of the two sides of the radial bird $C_{(t_1, h_1)}^1$, the condition $\hat{s}_1 \le \tilde{s} \le \hat{s}_2$ is equivalent to the fact that the radial bird at $(t_0, h_0)$ intersects the line $L_{\hat{s}_1}$ at a height above $\hat{h}_1$ and the line $L_{\hat{s}_2}$ at a height below $\hat{h}_2$, see Observation 4.17 for the single speed case. This is possible if and only is $(t_0, h_0)$ lies inside $E_{\hat{h}_2}^{\hat{s}_2, v_1}$ and outside $E_{\hat{h}_1}^{\hat{s}_1, v_2}$, see Picture (1) of Figure 7.5. In fact, $(t_0, h_0)$ can not lie inside $E_{\hat{h}_1}^{\hat{s}_1, v_1}$, by assumption.

Similarly, for $(t_0, h_0)$ of type 2, $\hat{s}_1 \le \tilde{s}_2 \le \hat{s}_2$ if and only if $(t_0, h_0) \in E_{\hat{h}_2}^{\hat{s}_2, v_2} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_2}$, see Picture (2) of Figure 7.5. This is true because the radial bird of type 2 at $(t_0, h_0) \in E_{\hat{h}_2}^{\hat{s}_2, v_2} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_2}$ intersects the line $L_{\hat{s}_1}$ at a height above $\hat{h}_1$ and the line $L_{\hat{s}_2}$ at a height below $\hat{h}_2$. $\square$

7.1.6. *Distance of intersections to the time axis.* Given $t_1, t_2, h_1, h_2$, suppose $\hat{s}_1, \hat{s}_2$ in (7.12) exist and are solutions to (7.11). Then the distances $\hat{h}_1 \equiv \hat{h}_1(t_1, t_2, h_1, h_2)$ and $\hat{h}_2 \equiv \hat{h}_2(t_1, t_2, h_1, h_2)$, are given
<figure><img src="image_29.png" /></figure>

(1) Any radial bird of type 1, whose head point is in $E_{\hat{h}_2}^{\hat{s}_2, v_1} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_1}$, intersects the vertical line $L_{\hat{s}_1}$ above $\hat{h}_1$ and the vertical line $L_{\hat{s}_2}$ below $\hat{h}_2$.

<figure><img src="image_30.png" /></figure>

(2) Any radial bird of type 2, whose head point is in $E_{\hat{h}_2}^{\hat{s}_2, v_2} \setminus E_{\hat{h}_1}^{\hat{s}_1, v_2}$, intersects the vertical line $L_{\hat{s}_1}$ above $\hat{h}_1$ and the vertical line $L_{\hat{s}_2}$ below $\hat{h}_2$.

<caption>FIGURE 7.5. Scenario described in Lemma 7.13.</caption>

by

$$ \begin{aligned} \hat{h}_1^2 &:= v_1^2 s_1^2 - 2v_1^2 s_1 t_1 + r_1^2 = v_2^2 s_1^2 - 2v_2^2 s_1 t_2 + r_2^2, \\ \hat{h}_2^2 &:= v_1^2 s_2^2 - 2v_1^2 s_2 t_1 + r_1^2 = v_2^2 s_2^2 - 2v_2^2 s_2 t_2 + r_2^2. \end{aligned} \quad (7.21) $$

Using the expressions of $\hat{s}_1, \hat{s}_2$ from (7.12) yields

$$ \hat{h}_1^2 = v_1^2 \left( \frac{v_2^2(t_1 - t_2) - \sqrt{\Delta}}{v_1^2 - v_2^2} \right)^2 + h_1^2, \quad \hat{h}_2^2 = v_1^2 \left( \frac{v_2^2(t_1 - t_2) + \sqrt{\Delta}}{v_1^2 - v_2^2} \right)^2 + h_1^2. \quad (7.22) $$

The distances $\hat{h}_1, \hat{h}_2$ are indeed functions of the difference $t_1 - t_2$ and $h_1, h_2$, so we write $\hat{h}_k \equiv \hat{h}_k(t_1 - t_2, h_1, h_2)$, for $k = 1, 2$.

**Observation 7.14.** A key observation is that, for fixed $v_1 > v_2$ and $h_2 > h_1$, the intersection distances computed in (7.22) are symmetric functions of $t_1, t_2$, i.e., they remain invariant if we take $t_1 > t_2$ or $t_1 \le t_2$, see Figure 7.6. In the case $t_1 \le t_2$, we have a different solution $\tilde{s}_1, \tilde{s}_2$ to (7.11) with $\tilde{s}_2 \le t_2 \le \tilde{s}_1$,

$$ \tilde{s}_1 := \frac{1}{v_1^2 - v_2^2} \left[ v_2^2 t_2 - v_1^2 t_1 - \sqrt{\Delta} \right], \quad \tilde{s}_2 := \frac{1}{v_1^2 - v_2^2} \left[ v_2^2 t_2 - v_1^2 t_1 + \sqrt{\Delta} \right]. \quad (7.23) $$

It can be verified that the distances of intersections are $\tilde{h}_1(t_2 - t_1, h_1, h_2) \equiv \tilde{h}_1 = \ell_1$ and $\tilde{h}_1(t_2 - t_1, h_1, h_2) \equiv \tilde{h}_2 = \ell_2$, expressed differently as

$$ \begin{aligned} \tilde{h}_1^2 &:= v_1^2 \tilde{s}_1^2 - 2v_1^2 \tilde{s}_1 t_1 + r_1^2 = v_2^2 \tilde{s}_1^2 - 2v_2^2 \tilde{s}_1 t_2 + r_2^2 = \hat{h}_1^2, \\ \tilde{h}_2^2 &:= v_1^2 \tilde{s}_2^2 - 2v_1^2 \tilde{s}_2 t_1 + r_1^2 = v_2^2 \tilde{s}_2^2 - 2v_2^2 \tilde{s}_2 t_2 + r_2^2 = \hat{h}_2^2. \end{aligned} \quad (7.24) $$

So the key idea is that, in the two scenarios $t_1 > t_2$ and $t_1 \le t_2$, the intersection times are different

<figure><img src="image_31.png" /><figcaption>FIGURE 7.6. The handover distances $\ell_1, \ell_2$ are symmetric functions of $t_1 - t_2$, when $v_1 > v_2$.</figcaption></figure>

but the intersection distances remain the same.

7.1.7. Construction of the handover point process. In the following, we formally describe the construction of the handover point process $\mathcal{V}$ in the two-speed case. As seen before in the single-speed case,
handovers are given by intersections of two radial birds under a certain empty half-ball or half-ellipse condition. By Lemma 7.4, for the two-speed case, an intersection point $(\hat{s}, \hat{h})$ between two radial birds at $(t, h)$, $(t', h')$, gives rise to a handover if and only if the region $E_{\hat{h}}^{\hat{s}, v_l}$ has no point of $\mathcal{H}^l$, for $l = 1, 2$.

In order to avoid over counting, we introduce some ordering, for the pair of head points. With this ordering in mind we can decompose the handover point process depending on the type of intersection. There are in total six types of intersection happens between all possible radial birds, as we will see later. Two of them are of pure type and four of them are of mixed type, defined below.

**Definition 7.15** (Pure handovers). For $l = 1, 2$, an intersection is of type $l$ if both radial birds are of type $l$. We call these intersections pure intersections and handovers given by them as pure handovers.

We denote the corresponding handover point process as $\mathcal{V}_l$. Formally, as in (4.17) for the single-speed case, the pure handover point process is defined as

$$ \mathcal{V}_l := \sum_{(T_i^l, H_i^l) \in \mathcal{H}^l} \sum_{(T_j^l, H_j^l) \in \mathcal{H}^l : T_j^l < T_i^l} \delta_{\hat{S}} \mathbb{1}_{A(\hat{S}, \hat{H})}, \quad (7.25) $$

for $l = 1, 2$, where $(\hat{S}, \hat{H})$ is the intersection point between the radial birds with their heads at $(T_i^l, H_i^l)$ and $(T_j^l, H_j^l) \in \mathcal{H}^l$ such that $T_j^l < T_i^l$ and $A(\hat{S}, \hat{H})$ is the event that the intersection point $(\hat{S}, \hat{H})$ exists and is responsible for a handover. In other words, $\mathcal{V}_l$ is the point process of abscissas of the intersection points of any two radial bird of type $l$, which give rise to a handover.

Suppose $l \neq r \in \{1, 2\}$. As seen in the intersection criterion, described in Lemma 7.5 and Lemma 7.8, there are either 2 or 0 intersection almost surely, between any two radial birds of type $l$ and $r$.

**Definition 7.16** (Mixed handovers). For $l, r, k \in \{1, 2\}$ with $l \neq r$, we say that an intersection is of type $(\binom{k}{l,r})$, if it is the $k$-th intersection between the radial birds of type $l$ and $r$. In the combinatorial like notation $(\binom{k}{l,r})$, the fact that the position of $l$ is on the left means that the head of the bird of type $l$ is on the left of the head of the bird of type $r$, which is on the right. The two values of $k$ refer to the two intersections, as in (7.12), of radial birds of two different types, where $k = 1$ means the left intersection and $k = 2$ means the right intersection. We call intersections of these type mixed intersections and the handovers given by these intersections as mixed handovers.

We denote the point processes corresponding to the mixed handovers as $\mathcal{V}_{l,r}^{(k)}$. Formally, for $k \in \{1, 2\}$,

$$ \mathcal{V}_{2,1}^{(k)} := \sum_{(T_i^1, H_i^1) \in \mathcal{H}^1} \sum_{(T_j^2, H_j^2) \in \mathcal{H}^2 : T_j^2 \in D_\ell(T_i^1, H_i^1, H_j^2)} \delta_{\hat{S}^k} \mathbb{1}_{A(\hat{S}^k, \hat{H}^k)}, \quad (7.26) $$

and

$$ \mathcal{V}_{1,2}^{(k)} := \sum_{(T_i^2, H_i^2) \in \mathcal{H}^2} \sum_{(T_j^1, H_j^1) \in \mathcal{H}^1 : T_j^1 \in D_\ell(T_i^2, H_i^2, H_j^1)} \delta_{\hat{S}^k} \mathbb{1}_{A(\hat{S}^k, \hat{H}^k)}, \quad (7.27) $$

where $(\hat{S}^k, \hat{H}^k)$ is the $k$-th intersection of the radial birds with their heads at $(T_i^1, H_i^1) \in \mathcal{H}^1$ and $(T_j^2, H_j^2) \in \mathcal{H}^2$ such that $T_j^2 \in D_\ell(T_i^1, H_i^1, H_j^2)$ in the first case and heads at $(T_i^2, H_i^2) \in \mathcal{H}^2$ and $(T_j^1, H_j^1) \in \mathcal{H}^1$ such that $T_j^1 \in D_\ell(T_i^2, H_i^2, H_j^1)$ in the second case, respectively. The set $D_\ell(T_i^1, H_i^1, H_j^2)$ given $T_i^1, H_i^1, H_j^2$ and $D_\ell(T_i^2, H_i^2, H_j^1)$ given $T_i^2, H_i^2, H_j^1$ is defined in Lemma 7.5 and Lemma 7.8, respectively, for the intersection criterion. The event $A(\hat{S}^k, \hat{H}^k)$ is for the intersection point $(\hat{S}^k, \hat{H}^k)$ is responsible for a handover.

Equivalently, $\mathcal{V}_{2,1}^{(k)}$ and $\mathcal{V}_{2,1}^{(k)}$ are the point processes of abscissas of the intersections, which give rise to handovers of types $(\binom{k}{2,1})$ and $(\binom{k}{1,2})$, respectively, for $k = 1, 2$, as shown in Figure 7.7. All the point processes $\mathcal{V}_l$ for $l \in \{1, 2\}$ and $\mathcal{V}_{l,r}^{(k)}$ for $l, r, k \in \{1, 2\}$ with $l \neq r$, inherit the time-stationarity property of the head point process $\mathcal{H}$. In total the handover point process in the two-speed case is

$$ \mathcal{V} := \sum_{l=1,2} \mathcal{V}_l + \sum_{l,r,k=1,2 : l \neq r} \mathcal{V}_{l,r}^{(k)}. \quad (7.28) $$
<figure><img src="image_32.png" /><figcaption>FIGURE 7.7. The labels at the intersections are the corresponding types, where the red and blue birds are of type 1 and 2 with speeds $v_1 > v_2$.</figcaption></figure>

In the following, we 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 7.17 (Time-stationarity of $\mathcal{V}$).** In the two-speed case, the handover point process $\mathcal{V}$ is stationary with respect to time.

Let $\mathcal{R}_l$ be the point processes on $\mathbb{R}$ made of the abscissas of the right most head points corresponding to pure handovers, for $l \in \{1, 2\}$. Let $\mathcal{R}_{l,r}^{(k)}$ be the point process made of the abscissas of the right most point of a pair of heads corresponding to mixed handovers, for $l \neq r, k \in \{1, 2\}$ and the right-most among the two heads, is of type $r$ and left-most is of type $l$. The time-stationarity of the point processes $\mathcal{R}_l$, for $l \in \{1, 2\}$, and $\mathcal{R}_{l,r}^{(k)}$, for $l \neq r, k \in \{1, 2\}$, is also inherited from that of $\mathcal{H}$.

Based on the bijection for the single-speed case in Lemma 5.1, in the following, we also have the same bijection among the handover point processes and the point processes corresponding to the right-most head points. This is crucial for deriving mass transport principle extending Lemma 5.1.

**Lemma 7.18.** (i). For all $l \in \{1, 2\}$, there exists a bijection $\beta_l$ between $\mathcal{V}_l$ and $\mathcal{R}_l$.

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

Let $\Lambda_l$ denote the intensity of $\mathcal{V}_l$ for $l \in \{1, 2\}$. We also denote the intensities of $\mathcal{V}_{l,r}^{(k)}$ by $\Lambda_{l,r}^{(k)}$ for $l \neq r, k \in \{1, 2\}$. We obtain the equality of intensities as a corollary:

**Corollary 7.19.** (i). For $l = 1, 2$, the intensities 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 $l \neq r$ and $k \in \{1, 2\}$, the intensities of the point processes $\mathcal{V}_{l,r}^{(k)}$ and $\mathcal{R}_{l,r}^{(k)}$ are equal, $\lambda_{\mathcal{R}_{l,r}^{(k)}} = \lambda_{\mathcal{V}_{l,r}^{(k)}} := \Lambda_{l,r}^{(k)}$.

7.1.8. *Handover frequency.* Suppose there are two types of stations with speed $v_1$ and $v_2$, respectively. Without loss of generality, assume that $v_1 > v_2$. The total handover frequency is

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

We now present the result about the handover frequencies in the two-speed case. We use the same technique as in the proof of Theorem 4.26 to obtain the pure handover frequency, whereas the mixed handover frequency requires a slightly different approach. Nevertheless, one can analyze both pure and mixed handovers, only using the head point processes $\mathcal{H}^1, \mathcal{H}^2$.

**Theorem 7.20 (Handover frequency: two-speed case).** Suppose there are two types of stations with intensities $\lambda_1, \lambda_2$ and speeds $v_1 > v_2$. Let $\lambda_1 + \lambda_2 = \lambda$. Then we have:

(i). The pure handover frequencies are

$$ \Lambda_l = \frac{4v_l\sqrt{\lambda}}{\pi} \left(\frac{\lambda_l}{\lambda}\right)^2, \quad \text{for } l = 1, 2. \qquad (7.29) $$
(ii). The mixed handover frequencies satisfies the relations $\Lambda_{1,2}^{(1)} = \Lambda_{2,1}^{(2)}$ and $\Lambda_{1,2}^{(2)} = \Lambda_{2,1}^{(1)}$.

(iii). For $k = 1, 2$,

$$ \Lambda_{2,1}^{(k)} = 4\lambda_1\lambda_2 v_1 v_2 \int_0^\infty \left[ \int_0^{h_1} \int_{t^*}^\infty e^{-\lambda\pi\hat{h}_k^2(t,h_1,h_2)} dt \, dh_2 + \int_{h_1}^\infty \int_0^\infty e^{-\lambda\pi\hat{h}_k^2(t,h_1,h_2)} dt \, dh_2 \right] dh_1, \quad (7.30) $$

where

$$ \hat{h}_k^2 = v_1^2 \left( \frac{v_2^2 t + (-1)^k \sqrt{\Delta}}{v_1^2 - v_2^2} \right)^2 + h_1^2 \text{ and } \Delta = v_1^2 v_2^2 t^2 - (h_1^2 - h_2^2)(v_1^2 - v_2^2). $$

The last two expressions are functions of $t, h_1, h_2$, determined in (7.22) and (7.13), respectively. The quantity

$$ t^* = \frac{1}{v_1 v_2} (h_1^2 - h_2^2)^{\frac{1}{2}} (v_1^2 - v_2^2)^{\frac{1}{2}}, $$

is the non-negative solution of $\Delta = 0$.

*Proof of Theorem 7.20 part (i).* For handovers of the pure type, choose a speed, say $v_1$ in $\{v_1, v_2\}$. From Equation (4.20), we have a contribution from the pure handovers which is similar to (4.21). By applying the multivariate Campbell Mecke formula for the factorial power of order 2 of $\mathcal{H}^1$, we obtain

$$ \Lambda_1 = (2v_1\lambda_1)^2 \int_0^1 \int_0^\infty \int_{-\infty}^{t_1} \int_0^\infty \mathbb{E}_{\mathcal{H}}^{1,(t_1,h_1),(t_2,h_2)} [\mathbb{1}_{A(s,h)} (\mathcal{H}^1 + \mathcal{H}^2)] dh_2 dt_2 dh_1 dt_1 \\ = 4v_1^2 \lambda_1^2 \int_0^1 \int_0^\infty \int_{-\infty}^{t_1} \int_0^\infty \mathbb{E}_{\mathcal{H}}^{1} [\mathbb{1}_{A(s,h)} (\mathcal{H}^1 + \delta_{(t_1,h_1)} + \delta_{(t_2,h_2)} + \mathcal{H}^2)] dh_2 dt_2 dh_1 dt_1, \quad (7.31) $$

where $A(s, h)$ is the event that there is a handover, at time $s$ and at a distance $h$, between the stations corresponding to the radial birds with heads located at $(t_1, h_1)$ and $(t_2, h_2)$.

Throughout this article, we make the following notational convention, which simply means that a point $(t, h)$ will naturally inherit the information about the type of the head point, whenever the point mass $\delta_{(t,h)}$ is written on the right of $\mathcal{H}^1$ or $\mathcal{H}^2$. This convention applies to multi-speed case as well. For $i \neq j \in \{1, 2\}$,

(1). $\mathcal{H}^i + \delta_{(t_1,h_1)} + \delta_{(t_2,h_2)} + \mathcal{H}^j$ is the two-point Palm version of $\mathcal{H}^i + \mathcal{H}^j$, with $(t_1, h_1), (t_2, h_2)$ of type i.

(2). $\mathcal{H}^i + \delta_{(t_1,h_1)} + \mathcal{H}^j + \delta_{(t_2,h_2)}$ is the two-point Palm version of $\mathcal{H}^i + \mathcal{H}^j$, with $(t_1, h_1), (t_2, h_2)$ of type i and j, respectively.

(3). $\mathcal{H}^i + \delta_{(t_1,h_1)} + \delta_{(t_2,h_2)} + \mathcal{H}^j = \mathcal{H}^j + \mathcal{H}^i + \delta_{(t_1,h_1)} + \delta_{(t_2,h_2)}$ and $\mathcal{H}^i + \delta_{(t_1,h_1)} + \mathcal{H}^j + \delta_{(t_2,h_2)} = \mathcal{H}^j + \delta_{(t_2,h_2)} + \mathcal{H}^i + \delta_{(t_1,h_1)}$.

In (7.31), the expectation $\mathbb{E}_{\mathcal{H}}^1$ is under the probability measure

$$ \mathbb{P}_{\mathcal{H}}^{1,(t_1,h_1),(t_2,h_2)} := \mathbb{P}_{\mathcal{H}^1}^{(t_1,h_1),(t_2,h_2)} \otimes \mathbb{P}_{\mathcal{H}^2}, \quad (7.32) $$

which is the two-point Palm probability measure on $\mathcal{H}$, conditioned on the two points $(t_1, h_1), (t_2, h_2)$ in $\mathcal{H}^1$. We can find the coordinate $(s, h)$ of the intersection point of two radial birds in terms of their heads at $(t_1, h_1)$ and $(t_2, h_2)$, similarly to (4.22) and (4.23).

By Lemma 7.4, a handover happens at $(s, h)$ if and only if the regions $E_h^{s,v_1}$, $E_h^{s,v_2}$ have no point of $\mathcal{H}^1$, $\mathcal{H}^2$ respectively. Since $\mathcal{H}^1$ is a Poisson point process, by the Slivnyak-Mecke theorem, the probability of such an event $A(s, h)$, in (7.31), is

$$ \begin{aligned} \mathbb{E}_{\mathcal{H}}^{1} [\mathbb{1}_{A(s,h)} (\mathcal{H}^1 + \delta_{(t_1,h_1)} + \delta_{(t_2,h_2)} + \mathcal{H}^2)] &= \mathbb{P}_{\mathcal{H}^1} (\mathcal{H}^1 (E_h^{s,v_1}) = 0) \mathbb{P}_{\mathcal{H}^2} (\mathcal{H}^2 (E_h^{s,v_2}) = 0) \\ &= e^{-2\lambda_1 v_1 \pi h^2 / 2v_1} e^{-2\lambda_2 v_2 \pi h^2 / 2v_2} = e^{-\lambda\pi h^2}, \end{aligned} \quad (7.33) $$

where $\lambda = \lambda_1 + \lambda_2$. Even though the intersection is between radial birds of the same type, the open regions $E_h^{s,v_1}$ and $E_h^{s,v_2}$ should have no head points of their respective types, for the point $(s, h)$ to be a handover point.
Then the frequency of the handovers is obtained from the proof of Theorem 4.26 as

$$
\begin{align}
\Lambda_1 &= 4\lambda_1^2 v_1^2 \int_0^1 \int_0^\infty \int_0^\infty \int_{-\infty}^{t_1} e^{-\lambda\pi h^2} dt_2 dh_2 dh_1 dt_1 \nonumber \\
&= \left(\frac{\lambda_1}{\lambda}\right)^2 \times 4\lambda^2 v_1^2 \int_0^1 \int_0^\infty \int_0^\infty \int_{-\infty}^{t_1} e^{-\lambda\pi h^2} dt_2 dh_2 dh_1 dt_1 = \left(\frac{\lambda_1}{\lambda}\right)^2 \times \frac{4v_1\sqrt{\lambda}}{\pi}, \tag{7.34}
\end{align}
$$

since the multiple integral in (7.34) along with the factor $4v_1^2\lambda^2$, equals $\frac{4v_1\sqrt{\lambda}}{\pi}$, which is the handover frequency in the single-speed case with speed $v_1$, see Remark 4.31. Similarly we can prove that

$$
\Lambda_2 = \frac{4v_2 \sqrt{\lambda}}{\pi} \left(\frac{\lambda_2}{\lambda}\right)^2, \qquad (7.35)
$$

by performing the same computations as above for $\Lambda_1$, using the two-point Palm probability measure $\mathbb{P}_{\mathcal{H}^2, (t_1, h_1), (t_2, h_2)} := \mathbb{P}_{\mathcal{H}^1} \otimes \mathbb{P}_{\mathcal{H}^2, (t_1, h_1), (t_2, h_2)}$. $\square$

*Proof of Theorem 7.20 part (ii).* Let us first prove that $\Lambda_{1,2}^{(1)} = \Lambda_{2,1}^{(2)}$. The radial birds corresponding to speeds $v_1$ and $v_2$, with the assumption $v_1 > v_2$, are referred to as *fast radial birds* and *slow radial birds*, respectively.

Let $(T_i^1, H_i^1)$, $(T_j^2, H_j^2)$ be the locations of the heads of the radial birds $C_1, C_2$ corresponding to two stations moving at speeds $v_1, v_2$, respectively. Just to make sure intersection happens between $C_1$ and $C_2$, we assume the following. Given the triple $(T_j^2, H_j^2, H_i^1)$, $T_i^1 \in D'_\ell(T_j^2, H_j^2, H_i^1)$, when $T_i^1 < T_j^2$ or $T_i^1 \in D'_r(T_j^2, H_j^2, H_i^1)$, when $T_i^1 > T_j^2$. On the other hand, given the triple $(T_i^1, H_i^1, H_j^2)$, $T_j^2 \in D_\ell(T_i^1, H_i^1, H_j^2)$ when $T_j^2 < T_i^1$, or $T_j^2 \in D_r(T_i^1, H_i^1, H_j^2)$ when $T_j^2 > T_i^1$. The pairs of sets $(D_\ell, D_r)$ and $(D'_\ell, D'_r)$ are defined as in (7.8) and (7.9), respectively. We say that $C_1$ lies on the left (resp. right) of $C_2$ if $T_i^1 \le T_j^2$ (resp. $T_j^2 < T_i^1$). Given the pair $(T_i^1, H_i^1)$, $(T_j^2, H_j^2)$ satisfying the intersection criteria (Lemma 7.5), consider the events $A_{1,2}^{(k)}$ and $A_{2,1}^{(k)}$, for $k \in \{1, 2\}$ defined by

$$
A_{1,2}^{(k)} := \{ \text{handover at } (S_{i,j}^k, H_{i,j}^k), \text{ with } T_i^1 \le T_j^2 \},
$$

$$
A_{2,1}^{(k)} := \{ \text{handover at } (S_{j,i}^k, H_{j,i}^k), \text{ with } T_j^2 \le T_i^1 \},
$$

where for $k \in \{1, 2\}$, $(S_{i,j}^k, H_{i,j}^k)$, and $(S_{j,i}^k, H_{j,i}^k)$, are the pairs of intersection points of the birds $C_1, C_2$, when $T_i^1 < T_j^2$ and $T_i^1 \ge T_j^2$, respectively, which are given by (7.12). In words, $A_{1,2}^{(k)}$ is the event that there is a handover of type $\binom{k}{1,2}$, given by the $k$-th intersection of $C_1, C_2$, with the fast radial bird on the left of the slow one, and $A_{2,1}^{(k)}$ is the event that there is a handover of type $\binom{k}{2,1}$, i.e., a handover due to the $k$-th intersection of $C_1, C_2$, with the slow radial bird being on the left of the fast one. By definition, the mixed handover frequencies are

$$
\Lambda_{1,2}^{(1)} = \mathbb{E} \left[ \sum_{(T_j^2, H_j^2) \in \mathcal{H}^2 : T_j^2 \in [0,1]} \sum_{(T_i^1, H_i^1) \in \mathcal{H}^1 : T_i^1 \in D'_\ell(T_j^2, H_j^2, H_i^1)} \mathbb{1}_{A_{1,2}^{(1)}} \right], \quad (7.36)
$$

$$
\Lambda_{2,1}^{(2)} = \mathbb{E} \left[ \sum_{(T_i^1, H_i^1) \in \mathcal{H}^1 : T_i^1 \in [0,1]} \sum_{(T_j^2, H_j^2) \in \mathcal{H}^2 : T_j^2 \in D_\ell(T_i^1, H_i^1, H_j^2)} \mathbb{1}_{A_{2,1}^{(2)}} \right]. \quad (7.37)
$$

where the sets $D_\ell$ and $D'_\ell$ are as defined in (7.8) and (7.9), respectively. Alternatively, when counting backward in time, due to the symmetry seen in Figure 7.8, it is also true that

$$
\Lambda_{1,2}^{(1)} = \mathbb{E} \left[ \sum_{(T_j^2, H_j^2) \in \mathcal{H}^2 : T_j^2 \in [0,1]} \sum_{(T_i^1, H_i^1) \in \mathcal{H}^1 : T_i^1 \in D'_r(T_j^2, H_j^2, H_i^1)} \mathbb{1}_{A_{2,1}^{(2)}} \right], \quad (7.38)
$$

$$
\Lambda_{2,1}^{(2)} = \mathbb{E} \left[ \sum_{(T_i^1, H_i^1) \in \mathcal{H}^1 : T_i^1 \in [0,1]} \sum_{(T_j^2, H_j^2) \in \mathcal{H}^2 : T_j^2 \in D_r(T_i^1, H_i^1, H_j^2)} \mathbb{1}_{A_{1,2}^{(1)}} \right], \quad (7.39)
$$

where the sets $D_r$ and $D'_r$ are as defined in (7.8) and (7.9), respectively.
The two definitions (7.36) and (7.38) are equivalent since, counting the number of handovers given by the first intersection between the pair of radial birds located at ($T_i^1, H_i^1$) and ($T_j^2, H_j^2$), with $T_i^1 < T_j^2$, is equivalent to counting backward in time the number of handovers given by the second intersection between the pair of radial birds located at ($T_i^1, H_i^1$) and ($T_j^2, H_j^2$), with $T_j^2 < T_i^1$. By the same reasoning, the two definitions (7.37) and (7.39) are also equivalent.

<figure><img src="image_33.png" /><figcaption>FIGURE 7.8. The mixed handovers seen in reverse time.</figcaption></figure>

We now show that $\Lambda_{1,2}^{(1)} = \Lambda_{2,1}^{(2)}$. Due to the bijection $\beta_{1,2}^{(1)}$ in part (ii) of Lemma 7.18, one can associate each mixed intersection to the right-most head points or equivalently to the left-most head point. Let $\mathcal{L}_{1,2}^{(1)}$ be the point process on $\mathbb{R}$ made of the abscissas of the left-most head point corresponding to the handovers with the faster bird being on the left of the slower one. Both the point processes $\mathcal{R}_{1,2}^{(1)}$ and $\mathcal{L}_{1,2}^{(1)}$ are stationary with respect to the time axis, which follows from the stationarity of $\mathcal{V}_{1,2}^{(1)}$. Thus we can redefine the mixed handover frequency $\Lambda_{1,2}^{(1)}$ using the point process $\mathcal{L}_{1,2}^{(1)}$, as

$$ \mathbb{E} \left[ \sum_{T_j^2 \in \mathcal{R}_{1,2}^{(1)} : T_j^2 \in [0,1]} 1 \right] = \Lambda_{1,2}^{(1)} = \mathbb{E} \left[ \sum_{T_i^1 \in \mathcal{L}_{1,2}^{(1)} : T_i^1 \in [0,1]} 1 \right] \quad (7.40) $$

and hence, from (7.40), we have

$$ \begin{aligned} \Lambda_{1,2}^{(1)} &= \mathbb{E} \left[ \sum_{T_i^1 \in \mathcal{L}_{1,2}^{(1)} : T_i^1 \in [0,1]} 1 \right] \\ &= \mathbb{E} \left[ \sum_{(T_i^1, H_i^1) \in \mathcal{H}^1 : T_i^1 \in [0,1]} \mathbb{1}_{\exists(T_j^2, H_j^2) \in \mathcal{H}^2 : T_j^2 \in D_r(T_i^1, H_i^1, H_j^2)} \mathbb{1}_{A_{1,2}^{(1)}} \right] \\ &= \mathbb{E} \left[ \sum_{(T_i^1, H_i^1) \in \mathcal{H}^1 : T_i^1 \in [0,1]} \sum_{(T_j^2, H_j^2) \in \mathcal{H}^2 : T_j^2 \in D_r(T_i^1, H_i^1, H_j^2)} \mathbb{1}_{A_{1,2}^{(1)}} \right] \\ &\stackrel{(7.37),(7.39)}{=} \mathbb{E} \left[ \sum_{(T_i^1, H_i^1) \in \mathcal{H}^1 : T_i^1 \in [0,1]} \sum_{(T_j^2, H_j^2) \in \mathcal{H}^2 : T_j^2 \in D_\ell(T_i^1, H_i^1, H_j^2)} \mathbb{1}_{A_{2,1}^{(2)}} \right] = \Lambda_{2,1}^{(2)}. \end{aligned} \quad (7.41) $$

We can prove the other equality $\Lambda_{1,2}^{(2)} = \Lambda_{2,1}^{(1)}$ similarly. $\square$

*Proof of Theorem 7.20 part (iii).* Let us now compute the contribution of mixed handovers by evaluating $\Lambda_{2,1}^{(1)}$ and $\Lambda_{2,1}^{(2)}$. By applying the multivariate Campbell Mecke formula to the factorial power of order 2 of $\mathcal{H}$ in (7.36), similarly to what was done for Equation (4.21) in Theorem 4.26 for the single-speed case, we obtain that, for $k = 1, 2$,

$$ \begin{aligned} \Lambda_{2,1}^{(k)} &= 4\lambda_1\lambda_2 v_1 v_2 \int_0^1 \int_0^\infty \int_0^\infty \int_{D_\ell(t_1,h_1,h_2)} \mathbb{E}_{\mathcal{H}}^{(t_1,h_1),(t_2,h_2)} \left[ \mathbb{1}_{A_{2,1}^{(k)}} (\mathcal{H}^1 + \mathcal{H}^2) \right] dt_2 dh_2 dh_1 dt_1 \\ &= 4\lambda_1\lambda_2 v_1 v_2 \int_0^1 \int_0^\infty \int_0^\infty \int_{D_\ell(t_1,h_1,h_2)} \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{A_{2,1}^{(k)}} (\mathcal{H}^1 + \delta_{(t_1,h_1)} + \mathcal{H}^2 + \delta_{(t_2,h_2)}) \right] dt_2 dh_2 dh_1 dt_1, \end{aligned} \quad (7.42) $$
where the probability measure is

$$ \mathbb{P}_{\mathcal{H}}^{(t_1,h_1),(t_2,h_2)} := \mathbb{P}_{\mathcal{H}^1}^{(t_1,h_1)} \otimes \mathbb{P}_{\mathcal{H}^2}^{(t_2,h_2)}, \qquad (7.43) $$

namely the two-point Palm probability measure on $\mathcal{H}$ with the first point belonging to $\mathcal{H}^1$ and the second point belonging to $\mathcal{H}^2$. Separating the integral (7.42) into the cases, $h_2 \in [0, h_1]$, for which $D_\ell(t_1, h_1, h_2) = (-\infty, t_1 - t^*]$, and $h_2 \in (h_1, \infty)$, for which $D_\ell = (-\infty, t_1]$, from Lemma 7.5, we obtain

$$ \Lambda_{2,1}^{(k)} = 4\lambda_1\lambda_2 v_1 v_2 \int_0^1 \int_0^\infty \int_0^{h_1} \int_{-\infty}^{t_1-t^*} \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{A_{2,1}^{(k)}} (\mathcal{H}^1 + \delta_{(t_1,h_1)} + \mathcal{H}^2 + \delta_{(t_2,h_2)}) \right] dt_2 dh_2 dh_1 dt_1 \\ + 4\lambda_1\lambda_2 v_1 v_2 \int_0^1 \int_0^\infty \int_{h_1}^\infty \int_{-\infty}^{t_1} \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{A_{2,1}^{(k)}} (\mathcal{H}^1 + \delta_{(t_1,h_1)} + \mathcal{H}^2 + \delta_{(t_2,h_2)}) \right] dt_2 dh_2 dh_1 dt_1, \quad (7.44) $$

where $t^* = \frac{1}{v_1 v_2}(h_1^2 - h_2^2)^{\frac{1}{2}}(v_1^2 - v_2^2)^{\frac{1}{2}}$ as in (7.15). Using the intersection points $(\hat{s}_1, \hat{h}_1)$, $(\hat{s}_2, \hat{h}_2)$ of the radial birds, determined in (7.12) and (7.22), we can redefine the event $A_{2,1}^{(k)}$, from (7.36), for $k = 1, 2$, as

$$ A_{2,1}^{(k)} := \{(\hat{s}_k, \hat{h}_k) \text{ is a handover point}\}. \qquad (7.45) $$

Under the two-point Palm probability measure $\mathbb{P}_{\mathcal{H}}^{(t_1,h_1),(t_2,h_2)}$, the required probability of the events $A_{2,1}^{(k)}$, for $k = 1, 2$, turns out to be a function of $t_1 - t_2$, given $h_1, h_2$. This will be clear later in (7.51) in the proof of the following lemma.

**Lemma 7.21.** Suppose $h_1, h_2 \in \mathbb{R}^+$ and $t_1, t_2 \in \mathbb{R}$ such that $t_2 \le t_1$, for $h_1 < h_2$ and $t_2 \le t_1 - t^*$, for $h_1 > h_2$, where $t^*$ is defined in (7.15). Under the two-point Palm probability measure $\mathbb{P}_{\mathcal{H}}^{(t_1,h_1),(t_2,h_2)}$, the probability of the event $A_{2,1}^{(k)}$, for $k = 1, 2$, is

$$ \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{A_{2,1}^{(k)}} (\mathcal{H}^1 + \delta_{(t_1,h_1)} + \mathcal{H}^2 + \delta_{(t_2,h_2)}) \right] = e^{-\lambda \pi \hat{h}_k^2}, $$

where $\hat{h}_k$ is determined in (7.22) as a function of $t_1 - t_2, h_1, h_2$.

Using Lemma 7.21, we can simplify (7.44) to get

$$ \Lambda_{2,1}^{(k)} = 4\lambda_1\lambda_2 v_1 v_2 \int_0^1 \int_0^\infty \int_0^{h_1} \int_{-\infty}^{t_1-t^*} e^{-\lambda\pi\hat{h}_k^2(t_1-t_2,h_1,h_2)} dt_2 dh_2 dh_1 dt_1 \\ + 4\lambda_1\lambda_2 v_1 v_2 \int_0^1 \int_0^\infty \int_{h_1}^\infty \int_{-\infty}^{t_1} e^{-\lambda\pi\hat{h}_k^2(t_1-t_2,h_1,h_2)} dt_2 dh_2 dh_1 dt_1. \qquad (7.46) $$

Making the change of variable $t_1 - t_2 = t$ in the expression of $\hat{h}_k(t_1 - t_2, h_1, h_2)$ in (7.46) yields

$$ \Lambda_{2,1}^{(k)} = 4\lambda_1\lambda_2 v_1 v_2 \int_0^\infty \int_0^{h_1} \int_{t^*}^\infty e^{-\lambda\pi\hat{h}_k^2(t,h_1,h_2)} dt dh_2 dh_1 \\ + 4\lambda_1\lambda_2 v_1 v_2 \int_0^\infty \int_{h_1}^\infty \int_0^\infty e^{-\lambda\pi\hat{h}_k^2(t,h_1,h_2)} dt dh_2 dh_1. \qquad (7.47) $$

The total the handover frequency is

$$ \lambda_V = \sum_{l \in \{1,2\}} \Lambda_l + 2 \sum_{l<r, k \in \{1,2\}} \Lambda_{l,r}^{(k)}. \qquad (7.48) $$

This completes the proof of Theorem 7.20, provided Lemma 7.21 holds. $\square$

**Proof of Lemma 7.21.** For the two intersection points $(\hat{s}_1, \hat{h}_1)$, $(\hat{s}_2, \hat{h}_2)$, given by (7.12) and (7.22), we have, for the event $A_{2,1}^{(k)}$ with the new definition in (7.45),

$$ \mathbb{E}_{\mathcal{H}} \left[ \mathbb{1}_{A_{2,1}^{(k)}} (\mathcal{H}^1 + \delta_{(t_1,h_1)} + \mathcal{H}^2 + \delta_{(t_2,h_2)}) \right] = \mathbb{P}_{\mathcal{H}}^{(t_1,h_1),(t_2,h_2)} \left( (\hat{s}_k, \hat{h}_k) \text{ is a handover point} \right). \qquad (7.49) $$

In the rest of the proof, we evaluate the probability in (7.49), for $k = 1, 2$. Let us now assume that $t_1 > t_2$ and $v_1 > v_2$ and consider that $(\hat{s}_k, \hat{h}_k)$ is a handover point. The upper half ellipses corresponding
<figure><img src="image_34.png" /><figcaption>(1) The regions $E_{\hat{s}_1}^{s_1, v_1}$ and $E_{\hat{s}_1}^{s_1, v_2}$ must be empty of heads from fast and slow stations respectively, in case the mixed intersection $(\hat{s}_1, \hat{h}_1)$ represents an handover.</figcaption></figure>

<figure><img src="image_35.png" /><figcaption>(2) The regions $E_{\hat{s}_2}^{s_2, v_1}$ and $E_{\hat{s}_2}^{s_2, v_2}$ must be empty of heads from fast and slow stations respectively, in case the mixed intersection $(\hat{s}_2, \hat{h}_2)$ represents an handover.</figcaption></figure>

<caption>FIGURE 7.9. Mixed handovers due to first and second intersection of birds of two types.</caption>

to $(\hat{s}_k, \hat{h}_k)$ are given by,

$$v_1^2(t - \hat{s}_k)^2 + h^2 = \hat{h}_k^2 \quad \text{and} \quad v_2^2(t - \hat{s}_k)^2 + h^2 = \hat{h}_k^2 \qquad (7.50)$$

and the corresponding regions are $E_{\hat{h}_k}^{\hat{s}_k, v_1}$ and $E_{\hat{h}_k}^{\hat{s}_k, v_2}$, respectively. For a point $(\hat{s}_k, \hat{h}_k)$ to be a handover point, we need that the regions $E_{\hat{h}_k}^{\hat{s}_k, v_1}$ and $E_{\hat{h}_k}^{\hat{s}_k, v_2}$ be empty of points from $\mathcal{H}^1$ and $\mathcal{H}^2$, respectively (see Figure 7.9). Since $\mathcal{H}^1, \mathcal{H}^2$ are Poisson point processes, the corresponding probability is

$$\begin{aligned} \mathbb{P}_{\mathcal{H}}^{(t_1, h_1), (t_2, h_2)} \left( (\hat{s}_k, \hat{h}_k) \text{ is a handover point} \right) &= \prod_{l=1,2} \mathbb{P}_{\mathcal{H}^l} \left( \mathcal{H}^l (E_{\hat{h}_k}^{\hat{s}_k, v_l}) = 0 \right) \\ &= \prod_{l=1,2} e^{-2\lambda_l v_l} \frac{\pi \hat{h}_k^2}{2v_l} = e^{-\pi \hat{h}_k^2 (\lambda_1 + \lambda_2)} = e^{-\pi \lambda \hat{h}_k^2}, \end{aligned} \qquad (7.51)$$

since $\lambda_1 + \lambda_2 = \lambda$.

**Remark 7.22.** We cannot directly recover the handover frequency for the single-speed case from the two-speed case. However we can use part of the above proof to recover the result. For this assume that, $v_1 = v_2 = v$, $\lambda_1 = p\lambda$, $\lambda_2 = (1-p)\lambda$, for some $p \in [0, 1]$, then the sum of the first two term in (7.47) is $\Lambda_1 + \Lambda_2 = \frac{4v\sqrt{\lambda}}{\pi} (p^2 + (1-p)^2)$. The rest of the contribution concerns the mixed handover frequency as a sum $\sum_{l \neq r, k \in \{1,2\}} \Lambda_{l,r}^{(k)} = \Lambda_{1,2} + \Lambda_{2,1}$, where $\Lambda_{1,2}^{(2)} = \Lambda_{1,2}$, $\Lambda_{2,1}^{(1)} = \Lambda_{2,1}$, $\Lambda_{1,2}^{(1)} = 0$ and $\Lambda_{2,1}^{(2)} = 0$, since both birds are of the same type and hence intersect exactly once. Using the equality $\Lambda_{1,2} = \Lambda_{2,1}$, we have in total $\sum_{l \neq r, k \in \{1,2\}} \Lambda_{l,r}^{(k)} = 2\Lambda_{2,1}$, where

$$\Lambda_{2,1} = 4p(1-p)\lambda^2 v^2 \int_0^\infty \int_0^\infty \int_0^\infty e^{-\lambda\pi\hat{h}^2} dt \, dh_2 \, dh_1,$$

where, $\hat{h}^2 = \frac{1}{4} [v^2 t^2 + 2(h_1^2 + h_2^2) + \frac{(h_2^2 - h_1^2)^2}{v^2 t^2}]$, given $t_1 - t_2 = t$ and $h_1, h_2$. We employ the usual analysis as earlier in (7.34) to get

$$\Lambda_{2,1} = p(1-p) \times 4\lambda^2 v^2 \int_{(\mathbb{R}^+)^3} e^{-\lambda\pi\hat{h}^2} dt \, dh_2 \, dh_1 = p(1-p) \frac{4v\sqrt{\lambda}}{\pi}.$$

In total we have

$$\lambda_v = \Lambda_1 + \Lambda_2 + 2\Lambda_{2,1} = \frac{4v\sqrt{\lambda}}{\pi} (p^2 + (1-p)^2) + 2p(1-p) \frac{4v\sqrt{\lambda}}{\pi} = \frac{4v\sqrt{\lambda}}{\pi},$$

which is indeed the handover frequency in the single-speed case.

**Remark 7.23** ($\lambda_1 = \lambda/2 = \lambda_2$). Consider the case of equal proportions of stations of different speeds, i.e., $\lambda_1 = \lambda/2 = \lambda_2$. Then, from (7.47), we can simplify the term $\Lambda_{2,1}^{(k)}$ for $k = 1, 2$ as

$$\Lambda_{2,1}^{(k)} = \lambda^2 v_1 v_2 \int_0^\infty \left[ \int_0^{h_1} \int_{t^*}^\infty e^{-\pi \lambda \hat{h}_k^2} dt \, dh_2 + \int_{h_1}^\infty \int_0^\infty e^{-\pi \lambda \hat{h}_k^2} dt \, dh_2 \right] dh_1, \qquad (7.52)$$