## 6.1 First order heuristic and Poisson estimate for $\lambda_c$

The goal of this first chapter is to develop two heuristics to estimate the spatial densities of users in the stationary regime. The first heuristic we develop is a mean-field heuristic: in this heuristic, we assume that all the subprocess of users of class $C \in \mathcal{P}(K)$ are homogenous PPP, and we use this assumption to derive an equation giving the first-order heuristic. Such Poisson heuristics have already been studied in different setups, like in replica mean-field models: in [19], the authors prove that as the number of replicas goes to infinity, they behave like independently distributed Poisson processes. In another setup, [5] and [45] provide a quantification of the error of the mean-field approximation. We then define a second heuristic using a cavity approximation (see [80]) in order two refine the approximation. We say that this heuristic is of the second order because we rely on the cross correlation function $\rho^{(2)}$. We then discuss the relative performance of these two heuristics in a numerical setup.

Let $\mu_C = \frac{1}{|\mathcal{D}|} \mathbb{E}[\Phi_{C,0}(\mathcal{D})]$ denote the user density of the point process $\Phi_{C,0}$ of users of class $C \in \mathcal{P}(K)$ in the stationary regime.

A first heuristic to estimate $\mu_C$ is the Poisson heuristic: we assume that, in the stationary regime, all the point processes $\Phi_{C,0}$ for $C \in \mathcal{P}(K)$ are independent Poisson point processes with intensity $\mu_C^f$. As mentioned earlier, Poisson approximations appear in to replica mean-field methods, where we consider multiple realizations of our stochastic process and interactions between users are picked uniformly at random among replicas of the dynamics. We know that (see [19]) as the number of replicas go to infinity, the processes behave like independent Poisson point processes, which justifies the Poisson approximation. The Poisson heuristic presented here falls in the same line of thought. We define the following heuristic:

**Heuristic 1.** We define the Poisson heuristic $\mu_C^f$ for $\mu_C$ as the smallest solution of the following equation:

$$ \mu_C^f \int_0^\infty e^{-z\mathcal{N}_0} \exp \left[ -\sum_{U \in \mathcal{P}(K)} \mu_U^f \mathcal{J}(z, |C \cap U|) \right] dz = \frac{\lambda p_C L_C}{|C|\ell(r)}. \quad (6.1) $$

$$ \text{where } \mathcal{J}(z, k) = \int_{\mathcal{D}} (1 - e^{-zk\ell(\|x\|)}) dx. $$

We numerically observe that the values of the intensities $\mu_C^f$ obtained by solving (6.1) do not depend on the value of $\mathcal{N}_0 > 0$. This observation is consistent with the fluid system and the stability condition in the network being independent of the value of $\mathcal{N}_0$.

*Proof.* Let us apply the RCP to the process $\Phi_{0,C}$ of users of class $C \in \mathcal{P}(K)$ during a time interval $dt$:
6.1. FIRST ORDER HEURISTIC AND POISSON ESTIMATE FOR $\lambda_c$ 79

$$ \lambda p_C L_C |\mathcal{D}| = \mathbb{E} \left[ \sum_{x \in \Phi_{C,0}} R(x, \Phi_0) \right]. $$

We use the definition of the Palm probability measure to rewrite this as:

$$ \lambda p_C L_C |\mathcal{D}| = \mathbb{E}_{\Phi_{C,0}}^0 [R(0^C, \Phi_0)] \mathbb{E}[\Phi_0^C(\mathcal{D})]. \quad (6.2) $$

Using this, we get that, in the stationary regime:

$$ \lambda p_C L_C = \mu_C \mathbb{E}_{\Phi_{C,0}}^0 [R(0^C, \Phi_0)] = |C| \mu_C \ell(r) \mathbb{E}_{\Phi_{C,0}}^0 \left[ \frac{1}{N_0 + I_C} \right]. $$

This is equivalent to:

$$ \mu_C = \frac{\lambda p_C L_C}{|C| \ell(r)} \left( \mathbb{E}_{\Phi_{C,0}}^0 \left[ \frac{1}{N_0 + I_C} \right] \right)^{-1}. $$

Let us state use this Lemma (see Lemma 1 from [80]):

**Lemma 6.1.** Let $Y$ be a positive random variable with finite expectation and $c > 0$ be a real number. Then:

$$ \mathbb{E} \left[ \frac{1}{c + Y} \right] = \int_{0}^{\infty} e^{-zc} \mathbb{E}[e^{-zY}] dz. $$

Using Lemma 6.1 with $Y \equiv I(0^C, \Phi_0)$, which has a finite first moment and with $c \equiv \mathcal{N}_0$, we get:

$$ \mathbb{E}_{\Phi_{C,0}}^{0} \left[ \frac{1}{\mathcal{N}_{0} + I_{C}} \right] = \int_{0}^{\infty} e^{-z\mathcal{N}_{0}} \mathbb{E}_{\Phi_{C,0}}^{0} [e^{-zI_{C}}] dz. $$

Let us assume that the processes $\Phi_{0,U}$ are independent Poisson point processes denoted by $\Psi^U$. Using this, combined with Slivnyak's theorem, we get:

$$ \mathbb{E}_{\Phi_{C,0}}^{0} \left[ \frac{1}{N_0 + I_C} \right] \approx \int_{0}^{\infty} e^{-z\mathcal{N}_0} \prod_{U \in \mathcal{P}(K)} \mathbb{E} \left[ e^{-z \sum_{x \in \Phi_0^U \setminus \{0\} \cup \{C\}} P|C \cap U| \ell(\|x\|)} \right] dz. $$

Using the formula for the Laplace transform of a Poisson point process, we get:

$$ \mathbb{E} \left[ e^{-z \sum_{x \in \Phi_0^U \setminus \{0\} \cup \{C\}} |C \cap U| \ell(\|x\|)} \right] = \exp \left[ - \mu_U^f \int_{\mathcal{D}} (1 - e^{-z|C \cap U| \ell(\|x\|)}) dx \right]. $$

To conclude the proof, we combine these equations:

$$ \mu_C^f \int_0^\infty e^{-z\mathcal{N}_0} \exp \left[ - \sum_{U \in \mathcal{P}(K)} \mu_U^f \mathcal{J}(z, |C \cap U|) \right] dz = \frac{\lambda p_C L_C}{|C| \ell(r)}, $$

which is the announced result. $\square$
An observation we can make comes from the symmetry hypothesis. We can conjecture that under this assumption, the stationary user densities follow the same property, i.e., are such that whenever $|C| = |D|$, $\mu_C = \mu_D$. The proof for this result is not presented in this work, but numerical simulations we performed suggest that it holds.

We can use this observation to obtain another version of the system of equations in (6.1): if we define $\mu_j = \frac{1}{|\mathcal{D}|} \mathbb{E} \left[ \sum_{U:|U|=j} \Phi_{U,0}(\mathcal{D}) \right] = \sum_{U:|U|=j} \mu_U$ as the intensity of users communicating on $j$ channels, for $1 \le j \le K$, with $L_j = L_C$ and $p_j = \binom{K}{j} p_C$ for all $C$ such that $|C| = j$, we obtain the following system of equations from Heuristic 1:

$$ \mu_j^f \int_0^\infty e^{-z\mathcal{N}_0} \exp \left[ -\sum_{l=1}^K \mu_l^f \sum_{m=1}^{\min(j,l)} \alpha_{m,j,l} \mathcal{J}(z,m) \right] dz = \frac{\lambda p_j L_j}{j\ell(r)}, \quad (6.3) $$

where $\alpha_{m,j,l} = \frac{\binom{j}{m}\binom{j-l}{l-m}}{\binom{K}{j}}$. This result if a consequence of Lemma 3.5 applied to the vector of user densities.

This result can be interesting computationally, in the case where we want to quickly estimate the density of users transmitting on a given class in the symmetric system: in contrast to the system proposed in Heuristic 1, which needs to solve $2^K - 1$ equations, the symmetric system from (6.3) possesses $K$ equations. As $K$ grows larger, the gain in performance becomes non-negligible.

## A heuristic for $\lambda_c$

An interesting result arising from Heuristic 1 is an estimate for the value of $\lambda_c$: a necessary condition for the fixed point equation (6.1) to admit solutions is that the system is in its stationary regime, i.e., if $\lambda < \lambda_c$. We define the following estimate for $\lambda_c$:

**Heuristic 2.** Let $\lambda_p$ be the largest value of $\lambda$ such that Equation (6.1) admits a solution (or Equation (6.3) in the symmetric network). Then, $\lambda_p$ is an estimate for $\lambda_c$, which we call the Poisson heuristic estimate.

This estimate is interesting because it comes from a completely different approach and is only relying on the study of the stationary regime of Poisson approximates of the network. We will discuss in Section 6.3 the performance of this heuristic.

The reason why this heuristic captures the stability region of our dynamics is not yet understood. We do not know if the value of $\lambda_p$ is equal to the value of $\lambda_c$ presented in Theorem 2.2 or if this value is a precise numerical approximation of the critical arrival rate.
## 6.2 Second order heuristic

In an effort to refine the first order heuristic for user densities from Heuristic 1, we introduce a second order heuristic, which uses an approximation for the pair-wise correlation function in the stationary regime.

**Heuristic 3.** The second-order heuristic for the intensity of $\Phi_{0,C}$ is $\mu_C^s$ defined as:

$$ \mu_C^s = \frac{\lambda p_C L_C}{|C| \ell(R)} (\mathcal{N}_0 + I_C), \qquad (6.4) $$

where $I_C \triangleq \mathbb{E}_{\Phi_{C,0}}^0 [I(0^C, \Phi_0)]$ is the interference experienced by the typical user of class C in the stationary regime. The vector $\mathbf{I} = (I_C)_{C \in \mathcal{P}(K)}$ is the solution to the following equation:

$$ I_C = \sum_{U \in \mathcal{P}(K)} \frac{|C \cap U|}{\mu_C^s} \int_{x \in \mathcal{D}} \ell(\|x\|) \rho_{C,U}^{(2)}(x, 0) dx \quad \forall C \in \mathcal{P}(K), \qquad (6.5) $$

with the second order moment measure $\rho_{C,U}^{(2)}$ is defined as a function of $\mathbf{I}$:

$$ \rho_{C,U}^{(2)}(x, y) = \frac{\lambda(\mu_U^s p_C + \mu_C^s p_U)}{\frac{1}{L_C} \frac{|C|\ell(R)}{\mathcal{N}_0 + |C \cap U|\ell(\|x-y\|) + I_C} + \frac{1}{L_U} \frac{|D|\ell(R)}{\mathcal{N}_0 + |C \cap D|\ell(\|x-y\|) + I_U}}. \qquad (6.6) $$

*Proof.* To obtain this heuristic, we make the following cavity approximation in the system: let us consider a pair of points $(x, y)$ in $\mathcal{D}$, where $x$ is of class C and $y$ is of class D.

The arrival rate of the pair is $\lambda p_C \mu_D + \lambda p_D \mu_C$, taking into account the contributions of the two processes. To obtain the departure rate, let us fix the locations of the two points. The interference experienced by a point located at $x$ of class C in the stationary regime conditioned on a user of class D being present at $y$ is equal to:

$$ I(x) = |C \cap D|\ell(\|x-y\|) + \sum_{U \in \mathcal{P}(K)} \sum_{z \in \Phi_{0,U} \setminus \{y\}} |C \cap U|\ell(\|z-x\|) \\ = |C \cap D|\ell(\|x-y\|) + I_C, \qquad (6.7) $$

where $I_C$ is the interference experienced by the typical user of class C in the network, i.e. $I_C = \mathbb{E}_{\Phi_0}^0 [I(0^C, \Phi_0)]$. Using the definition of the second order moment measure relative to processes $\Phi_{C,0}$ and $\Phi_{U,0}$, $\rho_{C,U}^{(2)}$, we have:

$$ I_C = \mathbb{E}_{\Phi_0}^0 [I(0^C, \Phi_0)] = \sum_U \mathbb{E}_{\Phi_0}^0 [I(0^C, \Phi_0^U)] \\ = \frac{1}{\mu_C} \sum_U |C \cap U| \int_{x \in \mathcal{D}} \ell(\|x\|) \rho_{C,U}^{(2)}(x, 0) dx. $$
We obtain the departure rate for the pair $(x, y)$ with $x$ being of type $C$ and $y$ being of type $D$:

$$d_{C,D}(x, y) = \frac{1}{L_C \mathcal{N}_0 + |C \cap D| \ell(\|x - y\|) + I_C} + \frac{1}{L_D \mathcal{N}_0 + |C \cap D| \ell(\|x - y\|) + I_D} \frac{|C| \ell(R)}{|D| \ell(R)}$$

In the stationary regime, on average, the umber of arrivals compensate the departures, which gives us:

$$\lambda(\mu_U p_C + \mu_C p_U) = \rho_{C,U}^{(2)}(x, y)d_{C,D}(x, y).$$

Finally, to obtain the second-order heuristic for $\mu_C$, we use Miyazawa's Rate Conservation principle for the process $\Phi_{C,0}(\mathcal{D})$, assuming that all the quantities are distributed according to their stationary distribution:

$$\lambda p_C |\mathcal{D}| = \mathbb{E} [\Phi_{C,0}(\mathcal{D})] \frac{|C| \ell(r)}{\mathcal{N}_0 + I_C},$$

which gives the intended result.

<figure><img src="image_31.png" /></figure>

## 6.3 Numerical simulations and performance of the heuristics

We can compare the heuristics from Heuristic 1 and 2 to the value we obtain through simulation. To compute this intensity, we use an ergodic approximation of the form:

$$\mu_C \approx \frac{1.28}{t} \int_u^{u+t} \frac{\Phi_{w,C}(\mathcal{D})}{|\mathcal{D}|} dw, \qquad (6.8)$$

where $u$ is taken sufficiently large to ensure we are in the stationary regime of the system and $t$ is large enough so that the approximation is correct. The 1.28 factor arising in (6.8) comes from a Palm bias: in the simulation, when we compute the average number of users in $\mathcal{D}$, we introduce an observation bias when estimating the stationary user density, which we correct by using this multiplicative factor.

Figure 6.1 displays the results of both the Poisson (in red, dashed) against the numerical estimation of the user density (in blue, plain) for the process $\Phi_{\{1\}}$ using (6.8).

As we expected from previous results (see Section IV from [80]), the Poisson heuristic for spatial wireless networks does not estimate precisely the stationary user densities. This difference in predicted values can be easily explained by the fact that the stationary point process are not Poisson: making a Poisson approximation amounts to removing the spatial interactions between point processes in the stationary regime, which is likely to
<figure><img src="image_32.png" /><figcaption>Figure 6.1: Value of $\mu_{\{1\}}$ as a function of $\lambda/\lambda_c$ in a symmetric configuration.</figcaption></figure>

not be true in our systems. From [80], we know that the single-class system shows clustering in the stationary regime, which renders the Poisson approximation less accurate when predicting the steady-state user densities. A sensible conjecture to make is that our spatial multiclass wireless network displays the same behavior in the stationary regime, which would provide an explanation for this observation.

On the other hand, we can see that the second-order heuristic is more performant when it comes to predicting the user densities, although the heuristic is more computationally heavy as we have to evaluate double integrals. One can further refine the approximation by developing higher order heuristics. In such a setup, to develop a heuristic of order $n$, we would have to assume the locations of $n$ points in the system, which we can approximate using the heuristic of order $n-1$. This would lead us to building recursively a sequence of heuristics for the $n$-th order of which we can study the convergence (see [21]) in order to predict the spatial user densities in the stationary regime.

### Poisson heuristic

The Poisson heuristic for $\lambda_c$ gives us an interesting result: Figure 6.2 shows the respective values of $\lambda_c$ and $\lambda_p$ both in the symmetric and non-symmetric case for different values of $p_{\{1,2\}}$. In the symmetric case, we use $p_{\{1\}} = p_{\{2\}} = \frac{1-p_{\{1,2\}}}{2}$ and we use the value of $\lambda_c$ from Theorem 2.2. For the non-symmetric case, we add a dysymmetry factor $\alpha$, and we set $p_{\{1\}} = \alpha x$, $p_{\{2\}} = (1-\alpha)x$ and $p_{\{1,2\}} = \frac{1-x}{2}$. For the value of $\lambda_c$, we solve the system defined in Theorem 4.5.

We can see that the Poisson heuristic for $\lambda_c$ is very performant in both cases. To explain this result, we can make the following conjecture: the first order heuristic fails to predict correctly the spatial user density, but from Figure 6.1, the difference in the predicted value tends to become smaller as $\lambda/\lambda_c$ tends to 1. We can thus conjecture that in the heavy-traffic regime, when $\lambda$ becomes arbitrarily close to $\lambda_c$, the subprocesses $\Phi_{0,C}$ become Poisson as the spatial interactions average out. Thus, the value of $\mu_C$ predicted
<figure><img src="image_33.png" /><figcaption>Figure 6.2: Comparison between $\lambda_c$ and its Poisson heuristic as a function of $p_{\{1,2\}}$ in the network in the symmetric and the non-symmetric case.</figcaption></figure>

through Heuristic 1 becomes exact, and as a consequence, the value of $\lambda_p$ becomes equal to $\lambda_c$.
CLUSTERING

We remind here the definition of clustering, taken from [80]: a point process is said to be clustered if the dynamics show attraction in the stationary regime, i.e.:

**Definition 7.1.** A PP $\Phi_0$ will be said to be clustered if and only if we have, for all non-negative, non-increasing bounded function f:

$$ \mathbb{E}[F(0, \Phi_0)] \leq \mathbb{E}_{\Phi_0}^0 [F(0, \Phi_0)]. \quad (7.1) $$

where $F(x, \Phi_0) = \sum_{y \in \Phi_0 \setminus \{x\}} f(\|x - y\|)$ for $x \in \Phi_0$.

The goal in this chapter is to study this property in the stationary regime, and we will establish that the dynamics in the steady-state are clustered. This question is of interest to allow for a better description of the stationary regime of the dynamics, and to explain the precision of the Poisson heuristic defined in the previous section.

A useful tool we will use to display clustering in the system is the Ripley-K function, defined as:

$$ K_{\Phi_0}(r) = \frac{1}{\lambda} \mathbb{E}_{\Phi_0}^0 [\phi(B(0, r) - 1)], \quad (7.2) $$

where $\lambda$ is the intensity of $\Phi_0$. Using $f(x) = 1\{x \leq r\}$ in (7.1), we obtain a necessary condition for clustered point processes: let $\Phi$ be a clustered point process with intensity $\lambda > 0$ and $\Psi$ be a homogenous PPP with intensity $\lambda$. Then, we have, for all $r > 0$:

$$ K_{\Phi}(r) \geq K_{\Psi}(r) = \pi r^2. $$

This results gives the classical interpretation for clustering, where users in the stationary regime attract each other. We know that clustering was proved in the single-class case (see [80]), and we would like to see how this translates in the multiclass setting. First, let us state the following lemma:
**Lemma 7.1.** If $\Phi_t$ is stable, we have, for all $C \in \mathcal{P}(K)$:

$$p_C \mathbb{E}[F(0_C, \Phi_0)] + \sum_{U \in \mathcal{P}(K)} p_U \mathbb{E}[F(0_U, \Phi_{0,C})] \\ \leq \mathbb{E}_{\Phi_{0,C}}^0 [F(0_C, \Phi_0)] + \sum_{U \in \mathcal{P}(K)} \mathbb{E}_{\Phi_{0,U}}^0 [F(0_U, \Phi_{0,C})]. \quad (7.3)$$

*Proof.* Let $C \in \mathcal{P}(K)$ and let us denote $F_{t,C} = \sum_{x \in \Phi_{t,C}} F(x, \Phi_t)$, where $F$ is a shot-noise following the conditions stated in Definition 7.1.

Let us assume that a user arrives at $a \in \mathcal{D}$ between times $t$ and $t+dt$ so that $\Phi_{t+dt} = \Phi_t \cup \{a\}$. By making a simple case study, if $a$ is of class $C$, we have:

$$F_{t+dt,C} = \sum_{x \in \Phi_{t+dt,C}} F(x, \Phi_{t+dt}) = \sum_{x \in \Phi_{t,C}} F(x, \Phi_{t+dt}) + F(a, \Phi_{t+dt}) \\ = \sum_{x \in \Phi_{t+dt,C}} \left( \sum_{y \in \Phi_{t+dt} \setminus \{x\}} f(\|x - y\|) \right) + \sum_{y \in \Phi_{t+dt} \setminus \{a\}} f(\|a - y\|) \\ = \sum_{x \in \Phi_{t,C}} \left( \sum_{y \in \Phi_t \setminus \{x\}} f(\|x - y\|) \right) + \sum_{y \in \Phi_{t,C} \setminus \{a\}} f(\|a - y\|) + F(a, \Phi_t) \\ = F_{t,C} + F(a, \Phi_{t,C}) + F(a, \Phi_t).$$

If $a$ is of class $U \neq C$, we make a similar calculation to get:

$$F_{t+dt,C} = F_{t,C} + F(a, \Phi_{t,C}).$$

Let $F_{0,C}$ denote the stationary regime of the stochastic process $F_{t,C}$, and let $\mathbf{A}F_C = F_{0+,C} - F_{0,C}$ be the increase in $F_{t,C}$ in the stationary regime after one arrival.

An arriving user in the network chooses class $C$ with probability $p_C$. Let $\mathbb{E}^\uparrow$ be the Palm probability associated with the PP of arrival instants in $\mathbb{R}$. We use the PASTA property i.e., we can assume that an arriving user is the typical user using this class. This means that the increase in $F_{0,C}$ induced by an arrival of a user of class $C \in \mathcal{P}(K)$ is equal to $\mathbb{E}[F(0_C, \Phi_{0,C} + F(0_C, \Phi_0))]$, and to $\mathbb{E}[F(0_D, \Phi_{0,C})]$ if the arriving user is of class $D \neq C$. We have:

$$\begin{aligned} \mathbb{E}^{\uparrow}[\mathbf{A}F_C] &= p_C \mathbb{E}[F(0_C, \Phi_{0,C}) + F(0_C, \Phi_0)] + \sum_{U \neq C} p_U \mathbb{E}[F(0_U, \Phi_{0,C})] \\ &= p_C \mathbb{E}[F(0_C, \Phi_0)] + \sum_{U \in \mathcal{P}(K)} p_U \mathbb{E}[F(0_U, \Phi_{0,C})]. \end{aligned} \quad (7.4)$$

We consider the point process corresponding to deaths instants of users of class $C$. It admits as stochastic intensity $\mathbf{R}_{t,C} = \frac{1}{L_C} \sum_{x \in \Phi_{t,C}} R(x, \Phi_t)$ with respect to the filtration $\mathcal{F}_{t,C} = \sigma\{\Phi_{C,s}, s < t\}$. Using Papangelou's theorem
(see Theorem 1.9.2 in [14]), the associated Palm probability measure admits a Radon-Nikodym derivative equal to:

$$ \frac{d\mathbb{P}^\downarrow}{d\mathbb{P}_C} = \frac{\mathbf{R}_{0,C}}{\mathbb{E}[\mathbf{R}_{0,C}]} \qquad (7.5) $$

A user located at $b \in \Phi_0$ with class $D \in \mathcal{P}(K)$ leaves the system with probability $\frac{R(x, \Phi_0)}{L_D \mathbb{E}[\mathbf{R}_{0,D}]}$. Let $\mathbf{DF}_C = F_{0,C} - F_{0-,C}$ denote the decrement process of $F_{0,C}$. We can write the expected decrease in $F_{t,C}$ caused by a departure from the system:

$$ \mathbb{E}^\downarrow[\mathbf{DF}_C] = \mathbb{E} \left[ \frac{\mathbf{R}_{0,C}}{\mathbb{E}[\mathbf{R}_{0,C}]} \sum_{x \in \Phi_{0,C}} \frac{R(x, \Phi_0)}{L_C \mathbf{R}_{0,C}} (F(x, \Phi_0) + F(x, \Phi_{0,C})) \right] \\ + \mathbb{E} \left[ \sum_{D \neq C} \frac{\mathbf{R}_{0,D}}{\mathbb{E}[\mathbf{R}_{0,D}]} \sum_{x \in \Phi_{0,D}} \frac{R(x, \Phi_{0,D})}{L_D \mathbf{R}_{0,D}} F(x, \Phi_{0,C}) \right]. $$

We can rearrange the previous equation by using the definition of the Palm expectation to get:

$$ \mathbb{E}^\downarrow[\mathbf{DF}_C] = \frac{\mathbb{E}[\Phi_{0,C}(\mathcal{D})]}{L_C \mathbb{E}[\mathbf{R}_{0,C}]} \mathbb{E}_{\Phi_{0,C}}^0 [R(0_C, \Phi_0) F(0_C, \Phi_0)] + \\ \sum_{U \in \mathcal{P}(K)} \frac{\mathbb{E}[\Phi_{0,U}(\mathcal{D})]}{L_U \mathbb{E}[\mathbf{R}_{0,U}]} \mathbb{E}_{\Phi_{0,U}}^0 [R(0_U, \Phi_{0,C}) F(0_U, \Phi_{0,C})] . \quad (7.6) $$

Let us apply the RCP to the quantity of information present in the network. We get:

$$ \begin{aligned} \mathbb{E}[\mathbf{R}_{0,C}] &= \lambda p_C |\mathcal{D}| \\ \mathbb{E}_{\Phi_{0,C}}^0 [R(0_C, \Phi_0)] \mathbb{E}[\Phi_{0,C}(\mathcal{D})] &= \lambda p_C L_C |\mathcal{D}|. \end{aligned} \qquad (7.7) $$

Let $U, C \in \mathcal{P}(K)$. Then, $R(0_U, \Phi_{0,C})$ and $F(0_U, \Phi_{0,C})$ are negatively correlated: if $C \cap U \neq \emptyset$, an arrival of a user of class $U$ increases the shot-noise on users of class $C$, and slow down the rate-of-transfer for these users. If $C \cap U = \emptyset$, there exists $D$ such that $C \cap D \neq \emptyset$ and $C \cap U \neq \emptyset$ (for example, $D = C \cup U$). An arrival of a user of class $U$ increases the shot-noise for users of class $D$ and decrease the rate-of-transfer for these users. Consequently, the rate-of-transfer for users of class $U$ slows down as well.

Moreover, $R(0_C, \Phi_0)$ and $F(0_C, \Phi_0)$ are also negatively correlated. Using the FKG inequality in (7.6) and the two relations from (7.7) yields:

$$ \mathbb{E}^\downarrow[\mathbf{DF}_C] \leq \mathbb{E}_{\Phi_{0,C}}^0 [F(0_C, \Phi_0)] + \sum_{U \in \mathcal{P}(K)} \mathbb{E}_{\Phi_{0,U}}^0 [F(0_U, \Phi_{0,C})]. \quad (7.8) $$

A last application of the RCP to the number of users of class $C$ in the system in the stationary regime gives:
$$ \mathbb{E}^{\uparrow}[\mathbf{A}\mathbf{F}_C] = \mathbb{E}^{\downarrow}[\mathbf{D}\mathbf{F}_C], \qquad (7.9) $$

which gives the following inequality, for all $C \in \mathcal{P}(K)$:

$$ \begin{aligned} & p_C \mathbb{E}[F(0_C, \Phi_0)] + \sum_{U \in \mathcal{P}(K)} p_U \mathbb{E}[F(0_U, \Phi_{0,C})] \\ & \leq \mathbb{E}_{\Phi_{0,C}}^0 [F(0_C, \Phi_0)] + \sum_{U \in \mathcal{P}(K)} \mathbb{E}_{\Phi_{0,U}}^0 [F(0_U, \Phi_{0,C})], \quad (7.10) \end{aligned} $$

which is the intended result. □

We can see that Lemma 7.1 is a consequence of the clustering of the subprocesses $\Phi_{0,C}$: if each of these processes was clustered, then we would have:

$$ \mathbb{E}[F(0_C, \Phi_0)] \leq \mathbb{E}_{\Phi_{0,C}}^0 [F(0_C, \Phi_0)], $$

which immediately yields the same result.

Let us sum (7.3) over $C \in \mathcal{P}(K)$. Using the law of total expectations, and reminding that the typical user is of type C with probability $p_C$ and that $\Phi_0$ is the superposition of the processes $\Phi_{0,C}$, we have:

$$ \begin{aligned} & \sum_C \left[ p_C \mathbb{E}[F(0_C, \Phi_0)] + \sum_{U \in \mathcal{P}(K)} p_U \mathbb{E}[F(0_U, \Phi_{0,C})] \right] \\ &= \sum_C p_C \mathbb{E}[F(0, \Phi_{0,C} | 0^C)] + \sum_C \sum_{U \in \mathcal{P}(K)} p_U \mathbb{E}[F(0_U, \Phi_{0,C})] \\ &= \mathbb{E}[F(0, \Phi_0)] + \sum_U \sum_C \mathbb{E}[F(0, \Phi_{0,C} | 0^U)] \\ &= 2\mathbb{E}[F(0, \Phi_0)], \end{aligned} $$

where (a) uses the fact that $\sum_C \mathbb{E}[F(0, \Phi_{0,C} | 0^U)] = \mathbb{E}[F(0, \Phi_0 | 0^U)]$. This yields:

$$ 2\mathbb{E}[F(0, \Phi_0)] \leq 2 \sum_{C \in \mathcal{P}(K)} \mathbb{E}_{\Phi_{0,C}}^0 [F(0_C, \Phi_0)]. \qquad (7.11) $$

We thus obtain the following result:

**Theorem 7.2.** In the stationary regime, we have:

$$ \mathbb{E}[F(0, \Phi_0)] \leq \sum_{C \in \mathcal{P}(K)} \mathbb{E}_{\Phi_{0,C}}^0 [F(0_C, \Phi_0)]. \qquad (7.12) $$

The condition established in (7.2) is weaker than the clustering condition established in Definition 7.1. To see why, we can iterate the result from Lemma A.4 over all $C \in \mathcal{P}(K)$ to obtain:
$$ \sum_{C \in \mathcal{P}(K)} \frac{\mu_C}{\mu_0} \mathbb{E}_{\Phi_{0,C}}^0 [F(0_C, \Phi_0)] = \mathbb{E}_{\Phi_0}^0 [F(0, \Phi_0)]. $$

In the general case, we have no way to directly compare $\sum_C \frac{\mu_C}{\mu_0} \mathbb{E}_{\Phi_{0,C}}^0 [F(0_C, \Phi_0)]$ and $\sum_C \mathbb{E}_{\Phi_{0,C}}^0 [F(0_C, \Phi_0)]$, so we cannot establish clustering in the network as it is defined in [80].

Though, the result from Theorem 7.2 can be interpreted. Firstly, we can remark that in case $K = 1$, we obtain the statistical clustering observed in [80], which gives attraction between the points in the single-tier case. In the case $K \ge 2$, we use $f(x) = 1\{x \ge r\}$ and Slyvniak's theorem to obtain the following condition:

$$ \pi r^2 \le \sum_{C \in \mathcal{P}(K)} \frac{\mu_C}{\mu_0} K_{\Phi_{0,C}}(r). \qquad (7.13) $$

Equation (7.13) states that the average number of points we see around around a given user is larger than its Poisson equivalent, i.e. that the system shows some kind of attraction in the stationary regime. The reason why we cannot reach per-class attraction in the stationary regime comes from the multiclass setup of the network: due to the difference in relative interference power in the users, some users attract other users more than others. With this observation, users of class C will be attracted by other users of class C, but also by users of class D such that $D \cap C \ne \emptyset$. The condition from Theorem 7.2 averages the observation over all classes to remove this effect.
<figure><img src="image_34.png" /></figure>
# HIGHER ORDER MOMENT MEASURES

In the stationary regime, the definition of stability implies that the first order moment measure is finite. Higher order moment measures are an interesting object of study. In [22], Banerjee and Sankararaman study interference queuing networks on the lattice $\mathbb{Z}^d$ and obtain the existence and finiteness of exponential moment measures. The architecture of this section is based on these results.

From Theorem 2.1, we know that the SBD process $\Phi_0(\mathcal{D})$ can be bounded from above and below by two interference queuing networks (as defined in [81]). The goal of this chapter is to study how the results obtained on this category of queuing network translates to our dynamics, and if we can obtain the existence of exponential moments for the jump process $\Phi_0(\mathcal{D})$. This question is of interest as it allows to obtain information about the stationary distribution of the queuing process $\Phi_0(\mathcal{D})$, which represent the population of users in the network, and allows to bound the tails of the distribution using the probability generating function.

The following Theorem is the main result in this section:

**Theorem 8.1.** In the symmetric case, for all $\lambda \le \lambda_c$, there exists $c_0 > 0$ such that for all $c \in [0, c_0)$, we have:

$$ \mathbb{E}[e^{c\Phi_0(\mathcal{D})}] < \infty. $$

*Proof.* We will start by proving that the queuing system $\bar{X}_\epsilon$ has exponential moments. We will then conclude with the existence of exponential moments for $\Phi_0(\mathcal{D})$ using the stochastic domination from Theorem 1.2.

Let us start the following Lemma:

**Lemma 8.2.** Let $y \in \mathbb{R}^{N_e \times 2^K}$. Then, we have:
$$ \bar{\lambda}_c \sum_{i,C} y_{i,C}^m \le \sum_{i,C} y_{i,C}^m \frac{1}{L_C p_C \varepsilon^2} R_{i,C}(\mathbf{y}), \quad (8.1) $$

*Proof.* Using the Cauchy-Schwartz inequality, we have:

$$ \left(\sum_{i,C} y_{i,C}^m\right)^2 \le \left(\sum_{i,C} y_{i,C}^m \frac{1}{L_C p_C \varepsilon^2} R_{i,C}(\mathbf{y})\right) \left(\sum_{i,C} \frac{y_{i,C}^m}{L_C p_C \varepsilon^2} R_{i,C}(\mathbf{y})\right). \quad (8.2) $$

Let us focus on the second term of the right-hand side of the inequality. We know, from the definition of $\bar{\lambda}_c$, that it is the largest $\nu > 0$ such that there exists $\mathbf{z} \in \mathbb{R}_+^{N_e \times 2^K-1}$ such that for all $i, C$, $\nu p_C \varepsilon^2 \le \frac{1}{L_C} \sum_{k,U} \ell^\varepsilon(a_k, a_i) |C \cap U| z_{k,U}$.

As a consequence, for all $\mathbf{y} \in \mathbb{R}_+^{N_e \times 2^K-1}$, we have:

$$ \frac{1}{L_C} R_{i,C}(\mathbf{y}) \ge \bar{\lambda}_c p_C \varepsilon^2. $$

This yields:

$$ \sum_{i,C} \frac{y_{i,C}^m}{L_C p_C \varepsilon^2} R_{i,C}(\mathbf{y}) \le \sum_{i,C} \frac{1}{\bar{\lambda}_c} y_{i,C}^m. $$

Injecting this result into (8.2) gives:

$$ \bar{\lambda}_c \sum_{i,C} y_{i,C}^m \le \sum_{i,C} y_{i,C}^m \frac{1}{L_C p_C \varepsilon^2} R_{i,C}(\mathbf{y}), \quad (8.3) $$

which concludes the proof.

<figure><img src="image_35.png" /></figure>

We can now prove this next lemma:

**Lemma 8.3.** Let $\lambda < \bar{\lambda}_c$ and $D_\varepsilon = \frac{\bar{\lambda}_c - \lambda}{\lambda + 1 / \bar{L} p \varepsilon^2}$, where $\bar{L} \bar{p} = \min_C L_C p_C$. Then, we have:

$$ D \sum_{i,C} \mathbb{E} [X_{i,C}^m] \le \sum_{i,C} \binom{m+1}{k} E [X_{i,C}^k]. $$

*Proof.* Let us consider the function $V(\mathbf{y}) = \sum_{i,C} \frac{1}{p_C \varepsilon^2} y_{i,C}^{m+1}$, and let $\mathcal{L}$ be the infinitesimal generator associated with the Markov chain $\bar{\mathbf{X}}$. Since the system is stationary, $\mathbb{E}[\mathcal{L}V(\bar{\mathbf{X}})] = 0$, which translates as:

$$ \sum_{i,C} \lambda \mathbb{E} \left[ (\bar{X}_{i,C} + 1)^{m+1} - \bar{X}_{i,C}^{m+1} \right] + \sum_{i,C} \mathbb{E} \left[ \frac{1}{L_C p_C \varepsilon^2} R_{i,C}(\bar{\mathbf{X}}) \left( (\bar{X}_{i,C} - 1)^{m+1} - \bar{X}_{i,C}^{m+1} \right) \right] = 0, $$
After separating the terms of index $m$ on the left side of the equations, and the others in the right side, we get:

$$ \sum_{i,C} \mathbb{E} \left[ \left( -\lambda + \frac{1}{L_C p_C \varepsilon^2} R_{i,C}(\bar{\mathbf{X}}) \right) \bar{X}_{i,C}^m \right] = \sum_{i,C} \sum_{k=0}^{m-1} \binom{m+1}{k} \mathbb{E} \left[ \left( \lambda + \frac{1}{L_C p_C \varepsilon^2} R_{i,C}(\bar{\mathbf{X}}) \right) (-1)^{k+1-l} \bar{X}_{i,C}^k \right]. $$

We then remind that we have, for all $i, C$, $\frac{1}{L_C p_C \varepsilon^2} R_{i,C}(\bar{\mathbf{X}}) \le \frac{1}{L_C p_C \varepsilon^2}$. We apply this to the two sides of the equation, and we use Lemma 8.2 to the left-hand side to get:

$$ \sum_{i,C} (\bar{\lambda}_c - \lambda) \mathbb{E} [\bar{X}_{i,C}^m] \le \sum_{i,C} \sum_{k=0}^{m-1} (\lambda + \frac{1}{L_C p_C \varepsilon^2}) \binom{m+1}{k} \mathbb{E} [\bar{X}_{i,C}^k] $$

Let $\bar{L}\bar{p} = \min_C p_C L_C$. We get:

$$ (\bar{\lambda}_c - \lambda) \sum_{i,C} \mathbb{E} [\bar{X}_{i,C}^m] \le \left( \lambda + \frac{1}{\bar{L}\bar{p}\varepsilon^2} \right) \sum_{i,C} \sum_{k=0}^{m-1} \binom{m+1}{k} \mathbb{E} [\bar{X}_{i,C}^k] $$

Let us set $D_\varepsilon = \frac{\bar{\lambda}_c - \lambda}{\lambda + \frac{1}{\bar{L}\bar{p}\varepsilon^2}}$, which concludes the proof. $\square$

We can move to proving the main result of the theorem. We multiply each side by $\frac{c^m}{m!}$ sum the inequalities for $1 \le m \le N$:

$$ \begin{aligned} D_\varepsilon \sum_{m=1}^N \mathbb{E} \left[ \sum_{i,C} \bar{X}_{i,C}^m \right] \frac{c^m}{m!} &\le \sum_{m=1}^N \sum_{k=0}^{m-1} \frac{c^m}{k!(m+1-k)!} \mathbb{E} \left[ \sum_{i,C} \bar{X}_{i,C}^k \right] \\ &\le \sum_{i,C} \sum_{k=0}^{N-1} \sum_{m=k+1}^N \frac{\mathbb{E} \left[ \bar{X}_{i,C}^k \right]}{k!} \sum_{m=k+1}^N \frac{c^k}{(m+1-k)!} \\ &= \sum_{i,C} \sum_{k=0}^{N-1} \frac{\mathbb{E} \left[ \bar{X}_{i,C}^k \right]}{k!} \sum_{u=0}^{N-k-1} \frac{c^{k+u+1}}{(u+2)!} \\ &\le \sum_{i,C} \sum_{k=0}^{N-1} \mathbb{E} \left[ \bar{X}_{i,C}^k \right] \frac{c^{k+1}}{k!} \sum_{u=0}^{\infty} \frac{c^u}{(u+2)!} \\ &\le c e^c \sum_{i,C} \sum_{k=0}^N \mathbb{E} \left[ \bar{X}_{i,C}^k \right] \frac{c^k}{k!}. \end{aligned} $$

Let $S_N = \sum_{i,C} \sum_{m=0}^N \mathbb{E} [\bar{X}_{i,C}^m] \frac{c^m}{m!}$. The last inequality can be rearranged as:
$$D_{\varepsilon}(S_N - 1) \leq c e^c S_N \Leftrightarrow S_N \leq \frac{D_{\varepsilon}}{D_{\varepsilon} - c e^c}. \qquad (8.4)$$

We thus obtain that for each $i, C$, $\mathbb{E}[\bar{X}_{i,C}^m] \leq S_N \leq \frac{D_{\varepsilon}}{D_{\varepsilon}-c e^c}$. Since the sequence is increasing, and bounded from above, it converges, and we can conclude that $\mathbb{E}[e^{c\bar{X}_{i,C}}]$ is finite as long as $c e^c \leq D_{\varepsilon}$. We set $\bar{c}_0 = W(D_{\varepsilon})$, where $W$ is the Lambert W function, i.e. for all $x$, $W(x)$ is the unique value such that $x = W(x)e^{W(x)}$.

We have proven that any queue taken at random in $\bar{X}$ has finite exponential moments, whenever $\lambda < \bar{\lambda}_c$. We can easily observe that $\lim_{\varepsilon \to 0^+} D_{\varepsilon} = 0$, which gives as a consequence, that $\mathbb{E}[e^{c\Phi_0(\mathcal{D})}]$ is finite if and only if $c = 0$ (due to the fact that $W(0) = 0$).

We remind that $\Phi_0(\mathcal{D}) \leq \sum_{i,C} \bar{X}_{i,C}$, for all $\varepsilon > 0$. Instead of letting $\varepsilon$ go to 0, we can take its largest possible value, i.e. $\varepsilon^2 = |\mathcal{D}|$, which corresponds to the setup where there is only one queue in the region. Let $\bar{\lambda}_0$ be the critical arrival rate for this system, and we define:

$$D_0 = \frac{\bar{\lambda}_0 - \lambda}{\lambda + \frac{1}{L\bar{p}|\mathcal{D}|}}.$$

We can use stochastic domination to conclude that for all $c < c_0$, $\mathbb{E}[e^{c\Phi_0(\mathcal{D})}] < \infty$, with $c_0 = W(D_0)$, whenever $\lambda < \bar{\lambda}_c$. $\square$

Among the notable consequences of Theorem 8.1, we can obtain interesting bounds for the number of users in the stationary regime:

**Theorem 8.4.** There exists $c_1 > 0$ and $x_0$ such that for all $x > x_0$, we have:

$$\mathbb{P}[\Phi_0(\mathcal{D}) > x] \leq e^{-c_1 x}.$$

*Proof.* We apply Markov's inequality to the variable $e^{c\Phi_0(\mathcal{D})}$, with $0 < c < c_0$:

$$\mathbb{P}[e^{c\Phi_0(\mathcal{D})} > e^{cx}] \leq \frac{\mathbb{E}[e^{c\Phi_0(\mathcal{D})}]}{e^{cx}}. \qquad (8.5)$$

Let $K > 0$ that $\mathbb{E}[e^{c\Phi_0(\mathcal{D})}] < K < \infty$. Because the function $x \mapsto e^{cx}$ is positive and increasing, we obtain:

$$\mathbb{P}[\Phi_0(\mathcal{D}) > x] \leq K e^{-cx}. \qquad (8.6)$$

Through calculation, we can obtain that there exists $x_0 > 0$ and $c_1 > 0$ such that for $x \geq x_0$, $Ke^{-cx} \leq e^{-c_1 x}$, which concludes the proof. $\square$

The result from Theorem 8.4 can provide for bounds for the staying time for users in the system, using Little's law (see [64]), which provides an interesting result about transmitting time in the network.

Figure 8.1 displays the population in the system for two values of $\lambda$, one in the stable regime, the other in the unstable regime. The form of the histograms tend to show that in the stable regime, the population in
<figure><img src="image_36.png" /><figcaption>Figure 8.1: Population distribution for two realizations of the network, in the stable regime (left, $\lambda = 0.9\lambda_C$) and unstable regime (right, $\lambda = 1.1\lambda_C$). In the stable regime, we see that the stationary distribution has bounded tails whereas in the unstable regime, the stationary distribution is unbounded.</figcaption></figure>

the network has bounded tails, whereas the unstable system has unbounded tails. This result is displayed in Figure 8.2 shows the distribution tails in the system in the two regimes, and outlines the result of Theorem 8.4.

<figure><img src="image_37.png" /><figcaption>Figure 8.2: Population tails in the symmetric system, with $\lambda = 0.95\lambda_C$</figcaption></figure>

The result of Theorem 8.4 allows to obtain an interesting bound for the population of users in the stationary regime. Using this result with Little's law (see [64]), we can also obtain that the distribution of service times in the system has exponential tails, which is a very interesting property to have in communication networks.
<figure><img src="image_38.png" /></figure>
# Part III

## Velocity-based association policies in multitier wireless networks
<figure><img src="image_39.png" /></figure>
# TWO-TIER 5G WIRELESS NETWORK

## 9.1 Network setup

The classical setup for multi-tier networks relies on maximum power association policies [39], where MUs always connect to the tier and the BS that offers the best received power. These policies only rely on the geometry of the network and do not take into account MU mobility. In [73], the authors studied a class of association policies purely based on user velocity, consisting in setting a velocity threshold deciding which tier an MU should be associated with. The main result of [73] lies in finding an optimal threshold, that maximizes the expected mean effective Shannon rate the typical user in the network receives. These threshold association policies are able to outperform the classical max power association policies for densely populated networks with high mobility MUs, but offer poor performance in other mobility or user-density patterns.

The goal of this part and the following ones is to define a class of association policies incorporating both the ideas of max power and velocity-aware association policies, that rely on MU velocity, using only the network parameters and easily obtainable data, such as velocity and received power. This line of thought leads us to introduce *velocity-biased max power (VBMP)* association policies, where users are associated with the network tier that offers the best received power, up to a *bias factor* dictated by the velocity of the MU.

For our study, we propose a simple two-tier network architecture using two types of base stations (BSs) with the same frequency bands and differences in spatial intensity and transmit power: smaller *micro* cells overlay larger *macro* cells. We assume that base stations (BSs) are located according to two independent homogeneous Poisson point processes (PPP): the macro BSs form a PPP $\Phi_M \subset \mathbb{R}^2$ of intensity $\lambda_M$ and the micro BSs a PPP $\Phi_\mu$ of
intensity $\lambda_\mu$, with $\lambda_\mu > \lambda_M$.

We assume that mobile users (MUs) are located according to a third stationary PPP $\Phi_u$ of intensity $\lambda_u$. Each MU moves on a straight line with an orientation chosen at random and independent of other MUs. Each MU travels at a constant velocity $v$, sampled from a given distribution $f$, independently for each user. Without loss of generality, using the homogeneity and isotropy of the PPP, we can assume that the typical MU is located at the origin and that its motion is along the x-axis.

We assume that all BSs are always active. Some of the results presented here can be extended to the case where BSs with no users are shut off, but this does not fall in the scope of this work.

Base stations use directional beamforming with a sector approximation to communicate with the typical MU (see Section B.3). Under this assumption, the probability that the typical MU lies in the main (resp. side) lobe of its serving BS in the macro tier is thus equal to $p_{M,m} = \frac{1}{n_M}$ (resp. $1 - p_{M,m}$). The antenna gain of the typical MU is thus equal to:

$$g_{M,x} = \begin{cases} G_m^M & \text{w.p. } p_{M,m} \\ G_s^M & \text{w.p. } 1 - p_{M,m}. \end{cases} \qquad (9.1)$$

The micro and macro tiers share the same spectrum, so they interfere with one another. We call the tier with which the typical MU is associated the association tier, and the other tier, the interferer tier. An MU in the network always connects to the closest BS in its association tier. This BS association results in cells of BSs forming a Poisson-Voronoi (PV) tessellation (see Section A.3).

Let $\ell(x) \triangleq Kx^{-\alpha}$, with $\alpha > 2$ and $K = (\frac{c}{4\pi f_c})^2$, be the path-loss function for the system, where $f_c$ is the carrier frequency and $c$ is the speed of light. We assume Rayleigh fading with mean 1 between the BS located at $x$ and the typical MU at the origin, denoted by $h_x$. Let $P_M$ and $P_\mu$ denote the transmit powers for each tier, with $P_\mu < P_M$, and $G_x^M$ and $G_y^\mu$ the antenna gains for macro and micro BSs located at $x$ and $y$, respectively. We define the interference experienced by the typical MU from macro tier BSs as:

$$I_M(\Phi_M) = \sum_{x \in \Phi_M} h_x g_{M,x} P_M \ell(\|x\|).$$

Let $X_{M,0}$ be the closest macro BS to the typical MU, and $X_{\mu,0}$ be its micro counterpart. Conditioned on that the closest macro BS to which the typical MU connects is at distance $r > 0$, the SINR at the typical MU, denoted as $\text{SINR}_M$ is expressed as

$$\text{SINR}_M = \frac{h_0 P_M g_{M,0} Kr^{-\alpha}}{\sigma^2 + I_M(\Phi_M \setminus \{X_{M,0}\}) + I_\mu(\Phi_\mu)}, \qquad (9.2)$$

where $g_{M,0}$ (given by (B.8)) is the gain of the serving BS of the typical MU and $\sigma^2$ is the thermal noise density.
## Parameters

We consider that both tiers use a carrier frequency $f_c$ in the 3.5 GHz range with a bandwidth $W = 100$ MHz. The transmit powers of macro and micro BSs are assumed to be 20 W and 4 W, respectively. The antenna gain profile we consider for numerical results is that the main lobe gain is $G_m^M = n_M$ and the side lobe gain is $G_s^M = 1/n_M$, but other gain profiles can be used. The thermal noise $\sigma^2$ is $-174$ dBm.Hz$^{-1}$. Finally, MU velocities are sampled from an exponential distribution with scale parameter $v_u$. Table 9.1 gives the parameters used in the numerical results.

<table><caption>Table 9.1: Values of network parameters in Part III</caption><thead><tr><th>Parameter</th><th>Micro tier</th><th>Macro tier</th></tr></thead><tbody><tr><td>Carrier frequency ($f_c$)</td><td colspan="2">3.5 GHz</td></tr><tr><td>Bandwidth ($W$)</td><td colspan="2">100 MHz</td></tr><tr><td>Thermal noise density ($\sigma^2$)</td><td colspan="2">$-174$ dBm.Hz$^{-1}$</td></tr><tr><td>Transmit powers ($P_\mu, P_M$)</td><td>36 dBm</td><td>43 dBm</td></tr><tr><td>Beam reselection time ($T_{\mu,b}, T_{M,b}$)</td><td colspan="2">23 ms</td></tr><tr><td>BS handover time ($T_{\mu,c}, T_{M,c}$)</td><td colspan="2">43 ms</td></tr><tr><td>SSB burst periodicity ($\tau$)</td><td colspan="2">20 ms</td></tr><tr><td>Path-loss exponent ($\alpha$)</td><td colspan="2">4</td></tr><tr><td>Number of beams ($n_\mu, n_M$)</td><td colspan="2">8</td></tr><tr><td>MU intensity ($\lambda_u$)</td><td colspan="2">$1$ m$^{-2}$</td></tr><tr><td>BS intensities ($\lambda_\mu, \lambda_M$)</td><td>0.1 m$^{-2}$</td><td>0.02 m$^{-2}$</td></tr><tr><td>Velocity scale parameter ($v_u$)</td><td colspan="2">$[1, 10, 35]$ m.s$^{-1}$</td></tr><tr><td>Maximum SINR ($Q_{\max}$)</td><td colspan="2">30 dB</td></tr></tbody></table>

## 9.2 Shannon rate

We denote the Shannon rate per Hertz in each tier by $\mathcal{R}_M = \mathbb{E}[\log(1 + \text{SINR}_M)]$ and $\mathcal{R}_\mu = \mathbb{E}[\log(1 + \text{SINR}_\mu)]$. Due to RF imperfections and modulation schemes, we set $Q_{\max}$ to be the maximum achievable SINR. Thus, we have

$$ \mathcal{R}_M = \int_0^{Q_{\max}} \frac{p_M(T)}{1+T} dT \qquad (9.3) $$

in the macro tier. The same applies in the micro tier by using the appropriate values. Using Theorem 10.1, the Shannon rate achieved by the typical MU can be expressed as

$$ \mathcal{R}_M(v) = (1 - p_{\text{bm}}^M(v))\mathcal{R}_{M,m} + p_{\text{bm}}^M(v)\mathcal{R}_{M,s}, \qquad (9.4) $$

where $\mathcal{R}_{M,m}$ is the Shannon rate achieved with no beam misalignment, while $\mathcal{R}_{M,s}$ is its equivalent with beam misalignment (as discussed in Section B.3).
Equation (9.3) gives the Shannon rate for the typical MU without considering the overheads associated with beam reselections during intra-cell mobility and BS handovers during inter-cell mobility. To take these overheads into account, we now define the effective Shannon rate.

## 9.3 Mean Effective Shannon rate under an association policy

The metric we use to compare association policies is the ESR received by the typical MU as given by (B.6). In this chapter and the following, we define an association policy $\mathcal{P}$ as a property for MUs $\mathcal{P}$ such that if for a MU located at $x$, if $\mathcal{P}(x)$ holds, the MU is associated with the micro tier of the network, and if $\mathcal{P}$ does not hold, it is associated with the macro tier.

In the case of velocity-based association policies, the property $\mathcal{P}$ depends only on the velocity $v$ of the MU, and we denote as $\mathcal{P}(v)$ the truth value of the property. $\bar{\mathcal{P}}$ represents the negation of the proposition $\mathcal{P}$. We have:

**Lemma 9.1.** For a given velocity-based association policy $\mathcal{P}$, the average ESR received by the typical MU is given by:

$$ \mathcal{R}(\mathcal{P}) = \int_{0}^{\infty} \left( \mathcal{R}_{\mu, \text{eff}}(v) \mathbb{1}_{\mathcal{P}(v)} + \mathcal{R}_{M, \text{eff}}(v) \mathbb{1}_{\bar{\mathcal{P}}(v)} \right) f(v) dv, \quad (9.5) $$

where $\mathbb{1}_{(\cdot)}$ is the indicator function and $\bar{\mathcal{P}}$ the negation of $\mathcal{P}$.

This result uses and ergodic interpretation: to obtain the average MESR the typical user experiences over its trajectory, we compute the spatial average of the ESR over a large ball of radius $R > 0$ centered at the origin, with MUs having velocities sampled i.i.d. from the distribution $f$. Under a velocity-based association policy, the spatial average of the MESR of MUs in this ball becomes an average over the velocities of the MUs. Using Birkhoff's ergodic theorem [27], we can prove that this spatial average converges to a constant, $\mathcal{R}(\mathcal{P})$, which is also equal to the ESR experienced by the typical MU.

This metric will be used throughout this part to compare the performance of the different policies we use. Due to the isotropy and stationarity of the MU and BS processes, finding bounds and computing results for the average MESR ensures the performance of the network, which are then verified experimentally by simulating the dynamics described here. In the next chapter, we will use the result of Lemma 9.1 to a first array of association policies to obtain a maximal MESR for the network.
# VELOCITY-BASED ASSOCIATION POLICIES

Among the variety of velocity-based association policies for multi-tier networks, a simple family to study is the family of threshold policies.

**Definition 10.1.** Let $v_T > 0$. The threshold policy associated with $v_T$ is the association policy such that, for a MU moving at velocity $v$:

- if $v \le v_T$, the MU is associated with the micro tier,

- if $v > v_T$, the MU is associated with the macro tier.

This family of velocity-based association policies have the advantage of being simple to implement, as they act as a binary decision problem, depending only on one parameter which we can estimate with a certain precision for mobile phones (using for instance GPS location data). In the chapter, we propose an in-depth study of threshold policies and how they affect network performance.

## 10.1 Coverage probability

The first quantity we need to evaluate is the coverage probability under a given threshold association policy

**Theorem 10.1 (Coverage probability).** The coverage probability with the macro tier association is

$$p_M(v, T) = (1 - p_{bm}^M(v))q_M(G_m^M, T) + p_{bm}^M(v)q_M(G_s^M, T), \quad (10.1)$$
where

$$ q_M(G, T) = \pi \lambda_M \int_{r \ge 0} e^{-\pi \lambda_M r} \exp \left( -\frac{T \sigma^2}{P_M K G} r^{1/\delta} \right) \\ \exp \left( -\pi r \left( \frac{T}{P_M G} \right)^{\delta} \left( \lambda_M P_M^{\delta} \rho_M(G, T) + \lambda_{\mu} P_{\mu}^{\delta} \kappa_{\mu} \right) \right) dr. $$

Here $\delta \triangleq 2/\alpha$, and $\rho_M$ and $\kappa_\mu$ are given by

$$ \rho_M(G, T) = p_{M,m}(G_m^M)^\delta \int_{\left(\frac{T G_m^M}{G}\right)^{-\delta}}^\infty \frac{du}{1+u^{1/\delta}} + (1-p_{M,m})(G_s^M)^\delta \int_{\left(\frac{T G_s^M}{G}\right)^{-\delta}}^\infty \frac{du}{1+u^{1/\delta}}, $$

$$ \kappa_{\mu} = \left( p_{\mu,m} (G_{m}^{\mu})^{\delta} + (1 - p_{\mu,m}) (G_{s}^{\mu})^{\delta} \right) \int_{0}^{\infty} \frac{du}{1 + u^{1/\delta}}. $$

The coverage probability in the micro tier is obtained by swapping the role of both tiers.

*Proof.* The proof for this result is given in Section 12.1 of Chapter 12. □

We can verify the result given by Theorem 10.1 numerically. We start by sampling two homogenous PPP for the base stations. We then sample a large number of velocities for the typical MU, and we compute the coverage probability. Figure 10.1 displays the coverage probability, $p_c(\beta) = \mathbb{P}[\text{SINR}_0 > \beta]$ for the macro and micro tier for values of $\beta$ ranging from 0.01 to 100 dB.

<figure><img src="image_40.png" /><figcaption>Figure 10.1: Coverage probability in each tier (blue: in the macro tier, red: in the micro tier). Here, the typical MU moves at a constant velocity $v = 10 \text{m.s}^{-1}$.</figcaption></figure>

An interesting case to look at is the interference-limited setup, where $\sigma^2$ can be considered to be equal to 0. In this case, the functions $q_M$ and $q_\mu$ simplify to:

$$ q_M(G, T) = \frac{1}{1 + \left(\frac{T}{G}\right)^{\delta} \left(\rho_M(G, T) + \Omega \kappa_{\mu}\right)} \quad (10.2) $$

$$ q_{\mu}(G, T) = \frac{1}{1 + \left(\frac{T}{G}\right)^{\delta} \left(\rho_{\mu}(G, T) + \frac{1}{\Omega} \kappa_{M}\right)}, \quad (10.3) $$
where we set $\Omega = \frac{\lambda_{\mu} P_{\mu}^{\delta}}{\lambda_{M} P_{M}^{\delta}}$. When setting $\Omega = 0$, which corresponds to setting either $\lambda_{\mu} = 0$ or $P_{\mu} = 0$, i.e., removing the micro tier, we obtain the same formula as the one from [7].

## 10.2 Optimal threshold association policy

A question arising when studying association policies is that of the optimal policy, i.e., the policy that maximizes a certain network metric. In the case we are studying, the metric of interest we consider is the average MESR. Our question then becomes: what is the threshold policy that maximizes the average MESR in the system?

Under a threshold policy, the average ESR is given by:

$$ \mathcal{R}(v_T) = \int_{0}^{v_T} \mathcal{R}_{\mu, \text{eff}}(v) f(v) dv + \int_{v_T}^{\infty} \mathcal{R}_{M, \text{eff}}(v) f(v) dv. \quad (10.4) $$

We have the following Theorem when considering one MU per antenna:

**Theorem 10.2 (Optimal threshold policy).** Under the foregoing assumptions, there exists a unique threshold maximizing the average ESR per user. This optimal threshold $v_T^*$ does not depend on the velocity distribution $f$ of MUs, and is the unique solution to the following equation:

$$ \mathcal{R}_{\mu, \text{eff}}(v_T^*) = \mathcal{R}_{M, \text{eff}}(v_T^*). \quad (10.5) $$

*Proof.* The proof for this Theorem is available in Section 12.2 in Chapter 12. $\square$

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

Figure 10.2 shows the normalized ESR, i.e. the average MESR as a percentage of its maximum value for three exponential distributions for MUs. We can see that the maximum value of the MESR is attained for the same velocity threshold in all cases.

This network setup mimics an ideal situation, in which the antenna have infinite radio resources. In such a setup, we can consider that the system behaves as if only one MU is connected to each antenna, as allocating radio resources to one MU does not affect transmission for the other transmitting MUs. Moreover, the rate received by an MU in the setup corresponds to the peak rate that it receives in the network. A more realistic setup with multiple MUs is studied in the next section.

## 10.3 Load-dependent threshold policies

If we consider that radio resources for each antenna are not infinite anymore, we have to consider the number of MUs connected to a given antenna. The radio resources are now equally shared among these users. This sharing can either be done temporally, where transmission are scheduled sequentially
<figure><img src="image_42.png" /><figcaption>Figure 10.2: Normalized ESR in the network as a function of the velocity threshold $v_T$. The optimal threshold does not depend on the scale parameter $v_u$.</figcaption></figure>

(corresponding to a TDMA setup) or spatially, where transmission are encoded in the same signal (corresponding to a CSMA setup). In our study, both cases translate with the same modification in the network metrics, which we detail here.

Let $\mathcal{P}$ be a given velocity-based association policy. Under the association policy $\mathcal{P}$, let $Z_{\mu}^{0}(\mathcal{P})$ denote the number of users associated with the micro BS corresponding to the 0-cell, i.e., the Voronoi cell containing the typical MU located at the origin 0, and $Z_{M}^{0}(\bar{\mathcal{P}})$ be its macro counterpart. Note that $Z_{\mu}^{0}(\mathcal{P})$ and $Z_{M}^{0}(\bar{\mathcal{P}})$ are random variables due to the random network geometry and velocity. For this setup, we can modify (9.5) to rewrite the average ESR $\mathcal{R}_{load}$ as follows:

$$ \mathcal{R}_{load}(\mathcal{P}) = \mathbb{E} \left[ \frac{\mathcal{R}_{\mu, \text{eff}}(v)}{Z_{\mu}^{0}(\mathcal{P})} 1_{\mathcal{P}(v)} + \frac{\mathcal{R}_{M, \text{eff}}(v)}{Z_{M}^{0}(\bar{\mathcal{P}})} 1_{\bar{\mathcal{P}}(v)} \right]. \quad (10.6) $$

In (10.6), the number of MUs associated to the 0-cell and the ESR both depend on the geometry of each PV tessellation. We make the following assumption: let us assume that the number of users in the 0-cell of the PV tessellation in each tier is independent from the SINR. Under this assumption, we can define the following heuristic to approximate $\mathcal{R}_{load}$:

$$ \hat{\mathcal{R}}_{load}(\mathcal{P}) = \mathbb{E} \left[ \frac{1}{Z_{\mu}^{0}(\mathcal{P})} \right] \mathbb{E} \left[ \mathcal{R}_{\mu, \text{eff}}(v) 1_{\mathcal{P}(v)} \right] + \mathbb{E} \left[ \frac{1}{Z_{M}^{0}(\bar{\mathcal{P}})} \right] \mathbb{E} \left[ \mathcal{R}_{M, \text{eff}}(v) 1_{\bar{\mathcal{P}}(v)} \right], \quad (10.7) $$

where $\mathbb{E}[\frac{1}{Z_{\mu}^{0}(\mathcal{P})}]$ and $\mathbb{E}[\frac{1}{Z_{M}^{0}(\bar{\mathcal{P}})}]$ are the average inverse load in the 0-cell of micro and macro tiers, respectively.

Like in Section 10.2, threshold policies remain good candidates to maximize the average ESR experienced by the typical MU. Let $v_T$ be a velocity
threshold. Under the threshold policy with threshold $v_T$, the intensity of the PPP of users associated with the micro tier is equal to $\lambda_u F(v_T)$, with $F$ being the CDF of the velocity distribution, and the intensity of the PPP of users associated with the macro tier is equal to $\lambda_u(1 - F(v_T))$. Although we do not have an analytical formula for the average inverse load in the network, we can use the heuristic developed in Lemma A.8.

We set the following notation:

$$ 
\begin{aligned}
\mathbb{E}\left[\frac{1}{Z_{\mu}^{0}(v_{T})}\right] &= L\left(\frac{\lambda_{\mu}}{\lambda_{u}F(v_{T})}\right) \equiv L_{\mu}(v_{T}) \\
\mathbb{E}\left[\frac{1}{Z_{M}^{0}(v_{T})}\right] &= L\left(\frac{\lambda_{M}}{\lambda_{u}(1-F(v_{T}))}\right) \equiv L_{M}(v_{T}).
\end{aligned}
$$

Under a velocity-based threshold policy, the average ESR in the TDMA setup becomes

$$ \hat{\mathcal{R}}_{\text{load}}(v_T) = L_{\mu}(v_T) \int_{0}^{v_T} \mathcal{R}_{\mu, \text{eff}}(v) f(v) dv + L_{M}(v_T) \int_{v_T}^{\infty} \mathcal{R}_{M, \text{eff}}(v) f(v) dv. \qquad (10.8) $$

We define the load-threshold heuristic for the optimal threshold policy in the TDMA setup as follows:

**Definition 10.2 (Load-threshold heuristic).** The load-threshold heuristic is the threshold $v_{LT}$ that maximizes $\hat{\mathcal{R}}_{\text{load}}$, i.e., such that:

$$ v_{LT} \triangleq \arg\max \hat{\mathcal{R}}_{\text{load}}(v_T). \qquad (10.9) $$

The optimal load-threshold policy is the threshold policy associated with $v_{LT}$.

As shown in Fig. 10.3, we numerically observe that $v_{LT}$ is uniquely defined although we cannot prove it in the general case that the function $\hat{\mathcal{R}}_{\text{load}}(v_T)$ goes through a unique maximum. Specifically, Fig. 10.3 compares the simulations under TDMA and the heuristic from (10.9). The difference between the simulation and the predicted value of the heuristic comes from the fact that the inverse of the load and the ESR are not independent.

Further simulation can give us an order of magnitude for the difference between the optimal value of the MESR computed and the MESR obtained when selecting the threshold obtained in the load-threshold heuristic from Definition 10.2. Table 10.3 displays the difference in value between these two values.
<figure><img src="image_43.png" /><figcaption>Figure 10.3: Comparison of the heuristic in (10.8) (in plain line) and simulations (circles)</figcaption></figure>

<table><caption>Table 10.1: Numerical illustration of the load-threshold heuristic</caption><tr><td>v<sub>u</sub> (m.s<sup>-1</sup>)</td><td>1</td><td>10</td><td>35</td></tr><tr><td>v<sub>LT</sub> (m.s<sup>-1</sup>)</td><td>1.30</td><td>4.90</td><td>9.41</td></tr><tr><td>R<sub>load</sub>(v<sub>LT</sub>)</td><td>0.294</td><td>0.242</td><td>0.201</td></tr><tr><td>Optimal MESR</td><td>0.289</td><td>0.252</td><td>0.195</td></tr></table>

## 10.4 Discussion and comparison with a classical association policies

When interpreting these results, we can observe that the load-threshold heuristic is rather performant to estimate the maximal MESR in the network with a low computational cost, as the method is based on solving an only fixed point equation. One can try to refine this heuristic by using more performant methods, but this search will come at the cost of an increase in computational costs.

Another interesting question is to investigate the dependence of $v_{LT}$ on the MU parameters. The density $\lambda_u$ of MUs allows us to understand how well the network performs under different load conditions. Specifically, as $\lambda_u \to 0$, the ESR in the network becomes equal to the Shannon rate without TDMA, which is its peak rate. This translates into $v_{LT}$ becoming equal to $v_T^*$ from Theorem 10.2. Conversely, as $\lambda_u \to \infty$, the number of MUs associated with the micro tier becomes arbitrarily large. Because its contribution towards the ESR is larger than that from the macro tier, to maximize the average ESR, the optimal velocity threshold approaches zero.

The behavior of the optimal threshold for intermediate values of $\lambda_u$ depends on the velocity distribution. Figure 10.4 gives the optimal velocity threshold for the heuristic given in (10.9). The MU velocity is a limiting factor in the network: if the velocity of MUs is higher, the average overhead per unit of time in the macro tier increases, leading to a decrease in the contribution of the macro tier to the average ESR. To balance this effect,
<figure><img src="image_44.png" /><figcaption>Figure 10.4: Optimal velocity threshold obtained for the heuristic from (10.9) (in plain line) for three velocity distributions against the density of users. In black, dashed, the equivalent network without load.</figcaption></figure>

more MUs are associated with the micro tier, resulting in an increase in the
optimal velocity threshold.

We now compare the optimal average ESR using the optimal load-threshold heuristic based on 10.2 to a classical association policy based on the *maximium received power* (MRP), where MUs connect to the BS with the highest average received power. Because MUs move fast in the network, we do not take into account fading in the signal: the typical MU located at the origin connects to the macro tier BS located at $X_{0,M}$ if the received power in the macro tier $RP_M = P_M G_{0,M} \ell(|X_{0,M}|)$ is higher than the received power from the micro BS located at $X_{0,\mu}$, $RP_\mu = P_\mu G_{0,\mu} \ell(|X_{0,\mu}|)$. It is to note that the MARP policy is primarily based on the received *power*, and thus, the network geometry, and the threshold policy on *velocity*. Hence, this comparison allows us a fine-grained view at the ESR, which has two main components as seen from (10.8): $R_M$, the Shannon rate as a function of the SINR and hence the received power, and $T_o^M(v)$, the overhead as a function of the velocity $v$ of an MU.

As shown in Fig. 10.5, for a given value of MU density, the average ESR decreases faster with velocity under the MRP policy than under the load-threshold heuristic: while for low values of $v_u$, the MRP policy outperforms the load-threshold heuristic, the threshold policy provides MUs with a better average ESR for velocities $v_u$ higher than 11 m.s$^{-1}$, up to a gain of 45%. The fact that these results are obtained for parameters representing accurately the sub-6 GHz setting of 5G new radio (NR) supports the importance of the study of velocity-based threshold policies.
<figure><img src="image_45.png" /><figcaption>Figure 10.5: Comparison between the MARP policy (square markers, dashed line) and the threshold policy under $v_{LT}$ (circles, plain line) with 95% confidence intervals for three different velocity distributions.</figcaption></figure>
# VBMP ASSOCIATION POLICIES

In the last section of the previous chapter, we compared the optimal load-threshold heuristic with a max-power association policy, and observed through simulation that the former policy works better for densely populated network with fast-moving MUs. On the other hand, this policy is not interesting when it comes to sparsely populated networks or with slowly-moving MUs, as it performs worse than the *maximum received power* (MRP) policy, i.e. the policy such that a user associates to the tier that provides the maximal received power at its location.

The goal of this chapter is to look for policies that outperform both threshold and MRP policies. To do so, we define a class of association policies incorporating both the ideas of max power and velocity-aware association policies. This line of thought leads us to introduce *velocity-biased max power* (VBMP) association policies, where users are associated with the network tier that offers the best received power, up to a *bias factor* dictated by the velocity of the MU.

## 11.1 VBMP policies

We keep the same network setup as we did in the previous chapter, but for simplicity purposes, we decide to omit beam misalignment. This feature can easily be applied to all the results presented in this section by taking similar steps as the ones we used for threshold policies and adding the adequate terms when considering the main or the side lobes of the antenna.

We define the velocity-biased max power policies as follows:

**Definition 11.1.** Let $K$ be a positive real function. The velocity-biased max power (VBMP) policy associated with $K$ is the association policy such that:

- If $R_M \ge K(v)R_\mu$, we associate the MU with the macro tier
- If $R_M < K(v)R_\mu$, we associate the MU with the micro tier

The function $K$ acts as a bias term in the classical MRP policy, in order to take into account MU velocity in the association between the two tiers. It is to note that threshold policies and MRP policies are both types of VBMP policies. The MRP policy is the VBMP policy with the function $K = 1$, and the threshold policy with threshold $v_T$ is associated with the function $K$ such that $K(v) = 0$ if $v < v_T$, and $K(v) = +\infty$ if $v > v_T$.

In this chapter, we will look for the function $K$ that maximizes the MESR in the network. In order to do so, we will use variational calculus, so we will focus only on differentiable functions.

## Coverage probability

Let us take a VBMP policy with an associated function $K$. A first metric of interest is the coverage probability. We have the following theorem:

**Theorem 11.1 (Coverage Probability).** Let $\theta > 0$. Under a VBMP policy, the coverage probability in the macro tier is equal to:

$$ \mathbb{P}[\text{SINR}_{0,M} > \theta] = \delta C \int_{z=0}^{1/K(v)} \frac{z^{-\delta-1} e^{-\theta z}}{(1 + C z^{-\delta} + \theta^{\delta} F(\theta, \delta) + C \theta^{\delta} F(zT, \delta))^{2}} dz. $$

where $\delta = 2/\alpha$, $\Omega = \frac{\lambda_\mu}{\lambda_M} \left(\frac{P_\mu}{P_M}\right)^\delta$ and $F(\theta, \delta) = \int_{\theta-\delta}^{\infty} \frac{du}{1+u^{1/\delta}}$. The coverage probability in the micro tier, defined as the probability that $\text{SINR}_{0,\mu} > \theta$ and the typical MU is associated with the micro tier - denoted as Micro - is obtained by swapping the roles of the two tiers:

$$ \mathbb{P}[\text{SINR}_{0,\mu} > \theta] = \delta \Omega \int_{z=0}^{1/K(v)} \frac{z^{-\delta-1} e^{-\theta/z}}{\left(1 + \Omega z^{-\delta} + \theta^{\delta} F\left(\frac{\theta}{z}, \delta\right) + \Omega \theta^{\delta} F\left(\frac{T}{z}, \delta\right)\right)^2} dz. $$

The proof for this result is presented in Appendix 12.3

## MESR under a VBMP policy

We can compute the MESR under a VBMP policy using the coverage probability from Theorem 11.1 and using an integration by parts (see (9.3)):

$$ \mathcal{R}_{M, \text{eff}}(v) = (1 - T_{o,M}(v))^{+} \int_{0}^{Q_{m}} \mathbb{P}[\text{SINR}_{0,M} > \theta, \text{Macro}] \frac{d\theta}{\theta + 1}. $$

We can define the average MESR for the typical user located at the origin in a similar way as in Lemma 9.1:
$$
\begin{align}
\mathcal{R}(K) &= \int_{0}^{v_{\max}} \left( (1 - T_{o,M}(v))^{+} \mathbb{E} \left[ \log(1 + \mathrm{SINR}_{0,M}(v, K)) \right] \right) \tag{11.1} \\
&\quad + (1 - T_{o,\mu}(v))^{+} \mathbb{E} \left[ \log(1 + \mathrm{SINR}_{0,\mu}(v, K)) \right] f(v) \mathrm{d}v \nonumber \\
&= \int_{0}^{v_{\max}} \left( \mathcal{R}_{M,\mathrm{eff}}(K, v) + \mathcal{R}_{\mu,\mathrm{eff}}(K, v) \right) f(v) \mathrm{d}v. \tag{11.2}
\end{align}
$$

This quantity can be expressed in terms of network parameters by using Theorem 11.1. In the general case, there is no closed-form formula for $\mathcal{R}(K)$ depending on $K$ and $v$, but we will see in the next section that this result is sufficient to obtain an equation to define the function $K$ that maximizes the average MESR in the network.

11.2 Optimal VBMP policy

We want to find the VBMP policy that maximizes the average MESR in the
system. This amounts to finding the function *K* that maximizes *R*(*K*). We
define an optimal VBMP policy:

**Definition 11.2.** The optimal VBMP policy is the VBMP policy with a function *K*\* that maximizes the average MESR in the system.

From (11.2), the function $\mathcal{R}$ has the following form:

$$
\mathcal{R}(K) = \int_{v=0}^{v_{\max}} J(v, K) f(v) dv, \quad (11.3)
$$

with $J(v, K) = \mathcal{R}_{M,\text{eff}}(K, v) + \mathcal{R}_{\mu,\text{eff}}(K, v)$. We use the Euler-Lagrange cri-
terion to obtain:

$$
\frac{\partial}{\partial K} \left( \mathcal{R}_{M,\text{eff}}(K^*, v) + \mathcal{R}_{\mu,\text{eff}}(K^*, v) \right) = 0. \quad (11.4)
$$

Using the definition for the MESR and the result from Theorem 11.1, we
can rewrite equation (11.4) in terms of network parameters as follows:

$$
\begin{align*}
\frac{\partial}{\partial K} \mathcal{R}_{M,\text{eff}}(K^*, v) &= -(1 - T_{o,M}(v))^{+} \int_{0}^{Q_m} \frac{\delta \Omega K^*(v)^{\delta-1}}{\theta + 1} e^{-\frac{\theta}{K^*(v)}} \mathrm{d}\theta \\
&\quad \left( (1 + \Omega K^*(v))^{\delta} + \theta^{\delta} F(\theta, \delta) + \Omega \theta^{\delta} F\left(\frac{\theta}{K^*(v)}, \delta\right) \right)^2 \mathrm{d}\theta \\
\frac{\partial}{\partial K} \mathcal{R}_{\mu,\text{eff}}(K, v) &= (1 - T_{o,\mu}(v))^{+} \int_{0}^{Q_m} \frac{\delta \Omega K^*(v)^{\delta-1}}{\theta + 1} e^{-\theta K^*(v)} \mathrm{d}\theta \\
&\quad \left( (1 + \Omega K^*(v))^{\delta} + \Omega (\theta K^*(v))^{\delta} F(\theta, \delta) + (\theta K^*(v))^{\delta} F(\theta K^*(v), \delta) \right)^2 \mathrm{d}\theta.
\end{align*}
$$

We can state the following theorem:
<figure><img src="image_46.png" /><figcaption>Figure 11.1: Plot of the function $K^*$ as a function of $v$ against the MRP and the velocity-threshold functions.</figcaption></figure>

**Theorem 11.2.** For each $v > 0$, we define $K^*(v)$ as the smallest positive solution to (11.4). The optimal VBMP policy is the policy associated with $K^*$.

We observe numerically, in the network setups we investigated, that (11.4) admits a unique solution, but we did not manage to prove it in the general case.

An important observation to make is that the optimal VBMP policy does not depend on the distribution of velocities in the network: when applying the Euler-Lagrange's criterion to our dynamics, all the terms depending on $f(v)$ disappear in the equation. Thus, this VBMP policy is optimal for any velocity distribution profile.

Numerically, in our network setup, we can observe that (11.4) has a unique solution. Although this result is not proved, we can conjecture that this solution is actually unique. Figure 11.1 shows the solution of (11.4) as a function of $v$ (in green). We can see that for immobile users, the optimal VBMP policy has the same criterion as for the MRP policy, i.e., gives a max-power association. As the MUs move faster, the bias term for the micro tier becomes lower, i.e., we associate more users with the macro tier. This can be explained by the fact that the micro tier gives the best performance, and we prefer to prevent as much as possible fast moving users to connect to it.

Figure 11.2 shows a comparison between the optimal VBP policy with the MRP and the load-threshold policies. As expected, it performs better than the load-threshold in the one MU per cell case, as it showed poor performance, and the gain in performance compared to the MRP policy is negligible. This result could be predicted from the fact that, without tak-
<figure><img src="image_47.png" /><figcaption>Figure 11.2: Comparison between the MRP policy (red), the load-threshold policy (blue) and the optimal VBMP policy (green) in the one MU per cell case.</figcaption></figure>

ing into account the load in the network, max-power association performs extremely well.

## 11.3 Load-dependent VBMP policy

As the VBMP policies do not seem to provide a significative improvement to the MESR in the network, we can look into the load-dependent case. We saw in the previous chapter that this network setup changes drastically compared to the one MU per cell. We use the same expression for the MESR as in (10.6).

$$ \mathcal{R}_{\text{load}}(K) = \int_{0}^{v_{\text{max}}} \mathbb{E} \left[ \frac{\log(1 + \text{SINR}(v, K))}{Z_{0}(K)} (1 - T_{o}(v))^{+} \right] f(v) dv, $$

Let $Z_{0,M}(K)$ and $Z_{0,\mu}(K)$ respectively denote the random variables associated with the number of users in the 0-cell in the macro and the micro tier under the VBMP policy associated with $K$. We use the following heuristic:

$$ \hat{\mathcal{R}}(K) = \int_{0}^{v_{\text{max}}} \mathbb{E} \left[ \frac{1}{Z_{0,M}(K)} \right] \mathcal{R}_{M,\text{eff}}(v, K) f(v) dv + \int_{0}^{v_{\text{max}}} \mathbb{E} \left[ \frac{1}{Z_{0,\mu}(K)} \right] \mathcal{R}_{\mu,\text{eff}}(v, K) f(v) dv. $$
We can use Lemma A.8 to obtain the values of the average inverse load in the network. To do so, we remind that the point processes of MUs associated with each network tier are independent thinnings of the PPP of transmitting MUs $\Phi_u$. We thus need to find the probability that a MU is associated with one tier or the other given a VBMP policy $K$.

**Lemma 11.3.** The probability for an MU moving at velocity $v$ under a given VBMP policy to be associated with the macro tier is equal to:

$$G_M(K) = \mathbb{P}[\text{Macro}] = \int_{v=0}^{v_{\max}} \frac{f(v)}{1 + \Omega K(v)^{\delta}} \mathrm{d}v. \quad (11.5)$$

The probability of being associated with the micro tier is $\mathbb{P}[\text{Micro}] = 1 - G_M(K)$.

The detail for this result is given in Appendix 12.4. We thus obtain the following average inverse load in the network:

$$\begin{align}
\mathbb{E}\left[\frac{1}{Z_{0,M}(K)}\right] &= L\left(\frac{\lambda_M}{\lambda_u G_M(K)}\right) \triangleq L_M(K) \nonumber \\
\mathbb{E}\left[\frac{1}{Z_{0,\mu}(K)}\right] &= L\left(\frac{\lambda_\mu}{\lambda_u(1-G_M(K))}\right) \triangleq L_\mu(K), \tag{11.6}
\end{align}$$

where $L(x) = x\left(1 - \left(\frac{1}{1+\frac{2}{7x}}\right)^{7/2}\right)$. We get this formula for $\hat{\mathcal{R}}(K)$:

$$\hat{\mathcal{R}}(K) = \int_{0}^{v_{\max}} \left( L_{M}(K) \mathcal{R}_{M,\text{eff}}(v, K) + L_{\mu}(K) \mathcal{R}_{\mu,\text{eff}}(v, K) \right) f(v) \mathrm{d}v.$$

We can thus define the optimal load-dependent VBMP policy as:

**Definition 11.3.** The optimal load-aware VBMP policy is the function $K^\dagger$ that corresponds to a maximum of $\mathcal{R}_{\text{load}}$. The load-K heuristic is the function $K_{\text{load}}^*$ that maximizes $\hat{\mathcal{R}}$.

Unlike the one MU per BS case, we cannot use the Euler-Lagrange equation to find the function $K^\dagger$. Here, the terms under the integral are not depending on $K$ and its derivatives at the same point, which are local conditions and allow us to obtain the Euler-Lagrange equation. Instead, we have a global condition, in the form of $G_M(K)$, which prevents us from using the classical results in calculus of variations. To obtain an equation defining the optimal load-dependent VBMP policy, let us define the function $F$ as:

$$F(v, K, G(K)) = (\mathcal{R}_{M,\text{eff}}(K, v)L_M(G(K)) + \mathcal{R}_{\mu,\text{eff}}(K, v)L_\mu(G(K))),$$

so that $\hat{\mathcal{R}}(K) = \int_0^\infty F(v, K, G(K)) \mathrm{d}v$. Let us take $\varepsilon > 0$ and $h$ an integrable function such that $h(0) = 0$ and $\lim_{x \to \infty} h(x) = 0$. We define $K_\varepsilon = K + \varepsilon h$
and $F_{\epsilon} = F(v, K_{\epsilon}, G(K_{\epsilon}))$. Using Euler's condition, we know that $\hat{\mathcal{R}}(K)$ is maximal whenever:

$$ \frac{d}{d\epsilon} \hat{\mathcal{R}}(K_{\epsilon}) = \int_{0}^{v_{\max}} \frac{d}{d\epsilon} F_{\epsilon} dv = 0. $$

We take the total derivative of $F_{\epsilon}$ with respect to $\epsilon$:

$$ \frac{d}{d\epsilon} F_{\epsilon} = \frac{dK_{\epsilon}}{d\epsilon} \frac{\partial F_{\epsilon}}{\partial K_{\epsilon}} + \frac{dG(K_{\epsilon})}{d\epsilon} \frac{\partial F_{\epsilon}}{\partial G(K_{\epsilon})} $$

We know that:

$$ \frac{dK_{\epsilon}}{d\epsilon} = h(v) $$

$$ \frac{\partial F_{\epsilon}}{\partial K_{\epsilon}} = L_{M}(G(K_{\epsilon})) \frac{\partial}{\partial K} \mathcal{R}_{M, \text{eff}}(K_{\epsilon}, v) + L_{\mu}(G(K_{\epsilon})) \frac{\partial}{\partial K} \mathcal{R}_{\mu, \text{eff}}(K_{\epsilon}, v). $$

After some calculation, we can also obtain:

$$ \frac{dG(K_{\epsilon})}{d\epsilon} = C\delta \int_{0}^{\infty} \frac{h(u)K(u)^{\delta-1}}{(1+CK(u)^{\delta})} f(u)du, $$

and:

$$ \frac{\partial F_{\epsilon}}{\partial G(K_{\epsilon})} = \left( - \frac{\lambda_{M}}{\lambda_{u}G(K_{\epsilon})^2} L' \left( \frac{\lambda_{M}}{\lambda_{u}G(K_{\epsilon})} \right) \mathcal{R}_{M, \text{eff}}(v, K) + \frac{\lambda_{\mu}}{\lambda_{u}(1-G(K_{\epsilon}))^2} L' \left( \frac{\lambda_{\mu}}{\lambda_{u}(1-G(K_{\epsilon}))} \right) \mathcal{R}_{\mu, \text{eff}}(v, K) \right) f(v). $$

Unfortunately, due to the nature of the dependency of $\frac{dG(K_{\epsilon})}{d\epsilon}$ on $h$, we cannot use the fundamental lemma of calculus of variations (see Lemma 1.1.1 in [52]) to obtain an equation defining $K^{\dagger}$. A way to obtain a numerical approximation of the function $K^{\dagger}$ is to introduce a discretization for the problem.

Let us assume that velocities can only take a finite number of values $(v_i)_{0 \le i \le N-1}$, ranging from 0 to a velocity $v_M$, separated by a step $\epsilon = \frac{v_M}{N-1}$, so that $v_i = v_0 + i\epsilon$. Let us denote $p_i = \mathbb{P}[v_i \le v < v_{i+1}]$, $K_i = K(v_i)$ and $\mathbf{K} = (K_i)_{0 \le i \le N-1}$.

This discretization allows us to rewrite the quantities defined earlier as sums:

$$ G(\mathbf{K}) = \sum_{i=0}^{N-1} \frac{p_i}{1 + CK_i^{\delta}}, $$

$$ L_M(\mathbf{K}) = L\left(\frac{\lambda_M}{\lambda_u G(\mathbf{K})}\right), $$

$$ L_{\mu}(\mathbf{K}) = L\left(\frac{\lambda_{\mu}}{\lambda_{u}G(\mathbf{K})}\right). $$
<figure><img src="image_48.png" /><figcaption>Figure 11.3: Plot of the estimate $K^\dagger$ (green) of the optimal VBMP policy for an exponential distribution with parameter $v_u = 10$ against the MARP policy (red) and the optimal threshold policy (blue).</figcaption></figure>

We can define a newer version of the target $\hat{\mathcal{R}}$:

$$ \mathcal{R}(\mathbf{K}) = \sum_{i=0}^{N-1} (\mathcal{R}_{M,\text{eff}}(K_i)L_M(\mathbf{K}) + \mathcal{R}_{\mu,\text{eff}}(K_i)L_\mu(\mathbf{K})) p_i. $$

Let $K^\dagger \in \mathbb{R}_+^N$ that maximizes the function $\mathcal{R}$. Using the dominated convergence theorem as well as Riemann sums, we know that as $\epsilon$ goes to 0, the vector $K^\dagger$ approaches the function $K^\dagger$.

## Numerical resolution

We can now move on to solving the system using known algorithmic method. The results presented here are obtained using Python's `scipy.optimization.minimize` function, which implements multiple gradient descent algorithms to solve for $K^\dagger$.

Using a gradient descent algorithm leads to numerous evaluations of $\mathcal{R}_{M,\text{eff}}(K_i)$ and $\mathcal{R}_{\mu,\text{eff}}(K_i)$, which can become computationally heavy when looking for a small value of $\epsilon$. The regular integration packages in Python to evaluate double integrals use the library QUADPACK ([71]), which achieves slow convergence in the cases we consider here. Among the other methods existing to evaluate multiple integrals, we decide to use the Genz-Malik scheme (as presented in [46]) to evaluate the integrals, which allows to obtain a value of $K^\dagger$ in reasonable time.
<figure><img src="image_49.png" /><figcaption>Figure 11.4: Comparison of the three association policies: the optimal VBMP policy (green), the MARP policy (red) and the optimal threshold policy (blue) as a function of $v_u$.</figcaption></figure>

## 11.4 Discussion and comparison

We can compare the optimal load-aware VBMP policy defined in Definition 11.3 to the two other association policies defined earlier: the optimal load-threshold policy. Figure 11.4 displays the comparison between the three policies for different values of $v_u$, where the MU velocity is exponentially distributed with mean $v_u$.

We can see that the proposed association policy performs better than the classical MARP policy at all velocities, but still is outperformed by the load-threshold policy for high MU velocities, although we originally designed this VBMP policy to outperform it. This observation can be explained by two factors.

The first factor is the precision of the numerical integration methods we use to evaluate $\mathcal{R}_{M,\text{eff}}(K_i)$ and $\mathcal{R}_{\mu,\text{eff}}(K_i)$ at each iteration. In the numerical simulations, we use an adaptive Genz-Malik scheme over the classical numerical integration method to increase the speed of computation. Another candidate to evaluate quickly multiple integrals is Monte-Carlo integration, which is also known to be performant, but can have high variance. We can compare the three methods of integration to compute $\mathcal{R}_M(K_i)$ for values of $K_i$ ranging from 0 to 1 to see the differences in the three methods.

Figure 11.5 shows the three methods of integration, and we can see that the Genz-Malik scheme and the quadpack integration both give similar results for the value of $\mathcal{R}_M(K_i)$, while the Monte-Carlo integration presents high variance, and sometimes can prove to have a very large interval confidence. In the simulation ran here, we can see that it can also present outliers and miss the exact value of the integral. On the other hand, we can see that
<figure><img src="image_50.png" /><figcaption>Figure 11.5: Comparison of the three QUADPACK integration (in red), the Genz-Malik scheme (in blue) and the Monte-Carlo integration (in green) with 200 000 samples.</figcaption></figure>

the Genz-Malik integration scheme predicts relatively well that value of the integral, which asserts its utility here.

The second factor that can impact the precision of the load-dependent VBMP policy we obtain is the minimization method we use to obtain the vector $K^\dagger$. The library `scipy.optimization` implement by default the L-BFGS-D method ([32]), but other methods can be investigated to see if they yield better results in the cases we are considering, such as Powell's method ([74]), which achieves better results with noisy inputs. Investigating this point to find - or design - a better method to maximize the MESR under a load-dependent VBMP policy needs to be further investigated and may be the focus of a future work.

Although the results in this part did not lead to the original goal we set - obtaining an association policy able to outperform both the load-threshold policy defined in Definition 10.2, we still managed to obtain an association policy that outperforms the classical MARP policy when the load of the antennas is taken into account: Figure 11.4 shows that the proposed load-dependent VBMP policy performs better at all velocities, with the simple addition of a bias term in the original MARP policy.

The work proposed in this part opens a new line of though about incorporating velocity in the design of association policies in multitier wireless networks. We proposed two user-centric, simple associations policies centered around MU velocity, and we showed that they can both outperform a classical max-power association policy when we consider multiple MU per cell. The work presented here can be seen as a first step into designing more comprehensive association policies incorporating MU velocity, in order to achieve better network performance.
# PROOF OF THEOREMS OF CHAPTERS 10 AND 11

## 12.1 Proof of Theorem 10.1

Base stations in the macro tier are distributed according a homogenous PPP with intensity $λ_M$. The probability that the BS X is located at a distance $r > 0$ from the origin and the associated distribution function:

$$ 
\begin{aligned}
F_M(r) &= \mathbb{P}[\|X\| \le r] = 1 - e^{-\lambda_M \pi r^2} \\
f_M(r) &= \frac{d}{dr} F_x(r) = 2\pi r e^{-\lambda_M \pi r^2}.
\end{aligned}
$$

We assume that the typical MU is located in the main lobe of its association beam, i.e. $g_0^M = G_m^M$. We get:

$$ 
\begin{aligned}
\mathbb{P}[\text{SINR}_{M,0} > T | r, g_0^M = G_m^M] &= \mathbb{P}\left[\frac{P_M G_k^M h_0 r^{-\alpha}}{\sigma^2 + I_{M,a} + I_{\mu,i}} > T | r\right] \\
&= \mathbb{P}\left[h_0 > \frac{Tr^{\alpha}}{P_M G_k^M}(\sigma^2 + I_{M,a} + I_{\mu,i}) | r\right] \\
&= e^{-\frac{\mu T r^{\alpha}}{P_M G_k^M}} \mathbb{E}\left[\exp\left(-\frac{\mu T}{P_M G_k^M} I_{M,a} - \frac{\mu T}{P_M G_k^M} I_{\mu,i}\right)\right].
\end{aligned}
$$

Focusing on the first term in the expectation, we get:
$$
\begin{align*}
\mathbb{E} \left[ \exp \left( -\frac{\mu T}{P_M g_{M,0}} I_{M,a} \right) \right] &= \mathbb{E} \left[ \prod_{x \in \Phi^M \setminus \{X\}} \mathbb{E} \left[ \exp \left( -\frac{\mu T G_x}{g_0^M} r^\alpha h_x \ell(|x|) \right) \right] \right] \\
&= \mathbb{E} \left[ \prod_{x \in \Phi^M \setminus \{X\}} \mathbb{E} \left[ \frac{1}{1 + \frac{T G_x}{g_0^M} r^\alpha \ell(|x|)} \right] \right] \\
&= \mathbb{E} \left[ \prod_{x \in \Phi^M \setminus \{X\}} \frac{p_{M,m}}{1 + \frac{T G_m^M}{g_0^M} r^\alpha \ell(|x|)} + \frac{1 - p_{M,m}}{1 + \frac{T G_s^M}{g_0^M} r^\alpha \ell(|x|)} \right],
\end{align*}
$$

Using the formula for the Laplace transform of a PPP, we have:

$$
\begin{align*}
& \mathbb{E} \left[ \exp \left( -\frac{\mu T}{P_M g_{M,0}} I_{M,a} \right) \right] \\
&= \exp \left( -2\pi \lambda_M \int_r^\infty \left( 1 - \frac{p_{M,m}}{1 + \frac{T G_m^M}{g_{M,0}} r^\alpha \ell(|x|)} - \frac{1 - p_{M,m}}{1 + \frac{T G_s^M}{g_{M,0}} r^\alpha \ell(|x|)} \right) zdz \right) \\
&\stackrel{(a)}{=} \exp \left( -\pi \lambda_M r^2 T^\delta \left( p_{M,m} \left( \frac{G_m^M}{g_{M,0}} \right)^\delta \int_{\left( \frac{T G_j^M}{g_{M,0}} \right)^{-\delta}}^\infty \frac{du}{1 + u^{1/\delta}} \right. \right. \\
&\qquad \left. \left. + (1 - p_{M,m}) \left( \frac{G_s^M}{g_{M,0}} \right)^\delta \int_{\left( \frac{T G_j^M}{g_{M,0}} \right)^{-\delta}}^\infty \frac{du}{1 + u^{1/\delta}} \right) \right).
\end{align*}
$$

Where (a) uses the change of variables $u = \left( \frac{G_m^M}{T g_{M,0}} \right)^\delta \frac{z^2}{r^2}$ for the first integral and $u = \left( \frac{G_s^M}{T g_{M,0}} \right)^\delta \frac{z^2}{r^2}$ for the second one. Using the same steps and some results developed in Appendix A, the second expectation yields:

$$
\begin{align*}
& \mathbb{E} \left[ \exp \left( -\frac{\mu T}{P_M g_{M,0}} I_{\mu,i} \right) \right] \\
&= \exp \left( -\pi \lambda_{\mu} r^2 T^{\delta} \left( \frac{P_{\mu}}{P_M g_{M,0}} \right)^{\delta} \left( p_{\mu,m} (G_M^{\nu})^{\delta} + (1-p_{\mu,m}) (G_M^{\nu})^{\delta} \right) C(\delta) \right).
\end{align*}
$$

Where $C(\delta) = \int_{0}^{\infty} \frac{du}{1+u^{1/\delta}} = \frac{\delta\pi}{\sin(\pi\delta)}$. Reminding that the typical MU is aligned, we have $g_{M,0} = G_{m}^{M}$, which leads to:

$$
\mathbb{E} \left[ \mathbb{P} \left[ \text{SINR}_{M,0} > T | r, g_0^M = G_m^M \right] \right] = \pi \lambda_M \int_{r \ge 0} e^{-\pi \lambda_M r} e^{-\frac{\mu T r^{1/\delta}}{P_M G_m^M}} \exp \left( \left( \frac{T}{P_M G_m^M} \right)^{\delta} \left( \lambda_M P_M^{\delta} \rho^M(G_m^M, T) + \lambda_{\mu} P_{\mu}^{\delta} \kappa_{\mu}(\delta) \right) \right) dr.
$$

Finally, we obtain:
$$q_m^M(T) = \int_{r \ge 0} \mathbb{E} [\mathbb{P}[\text{SINR}_{M,0} > T | r, g_0^M = G_m^M]] 2\pi \lambda_M r e^{-\pi \lambda_M r^2} dr. \quad (12.1)$$

Making the change of variables $v = r^2$ gives the intended result for $q_m^M(T)$. To obtain the result for $q_s^M$, $q_M^\nu$ and $q_s^\mu$, we simply swap the roles of the two network tiers and use the adequate values.

## 12.2 Proof of Theorem 10.2

Differenciating (10.4) w.r.t. the velocity threshold $v_T$ gives:

$$\frac{\mathcal{D}}{\mathcal{D}v_T} \mathcal{R}(v) = \mathcal{R}_{\mu, \text{eff}}(v) f(v) - \mathcal{R}_{M, \text{eff}}(v) f(v). \quad (12.2)$$

Granted that $f > 0$, the points where the derivative is equal to zero are solutions to:

$$\mathcal{R}_{\mu, \text{eff}}(v) = \mathcal{R}_{M, \text{eff}}(v). \quad (12.3)$$

Using definitions from Section 9.3, we know that:

$$\mathcal{R}_{M, \text{eff}}(v) = (1 - T_o^M(v))^+ \mathcal{R}_M$$

and that $T_o^M(v) = \nu_{M,c}(v)T_{M,c} + \nu_{M,e}(v)T_{M,b}$. We also know that $\nu_{M,e}(v)$ is equal to $\nu_{M,b}(v)$ for $v < v_{M,\tau} = \frac{\pi}{n_M \tau \sqrt{\lambda_M}}$, and to $\frac{1}{\tau}$ else. We can solve for $T_o^M(v) = 1$ depending on the value of $v$. We obtain two solutions:

$$\begin{cases} w_{M,1} = \frac{\pi}{\sqrt{\lambda_M}(4T_{c,M} + n_M T_{b,M})} & \text{if } v < v_{M,\tau} \\ w_{M,2} = \frac{\pi}{\sqrt{\lambda_M} 4T_{c,M}} \left( \frac{T_{M,b}}{\tau} - 1 \right) & \text{if } v \ge v_{M,\tau}. \end{cases}$$

Using the numerical parameters from Table 9.1, we obtain $v_{M,\tau} = 138.84$ m.s$^{-1}$, $w_{M,1} = 62.40$ m.s$^{-1}$ and $w_{M,2} = 19.37$ m.s$^{-1}$. Because $w_{M,2} < v_{M,\tau}$, we reject this second solution, which gives us the following formula for $\mathcal{R}_{M,\text{eff}}(v)$:

$$\mathcal{R}_{M, \text{eff}}(v) = \begin{cases} \mathcal{R}_M \left( 1 - v \frac{\sqrt{\lambda_M}}{\pi} (4T_{c,M} + n_M T_{b,M}) \right) & \text{for } v \le w_{M,1} \\ 0 & \text{else} \end{cases}$$

Conducting the same analysis for the micro network, we define $v_{\mu,\tau}$, $w_{\mu,1}$ and $w_{\mu,2}$ similarly. When evaluating the numerical values, we reject the solution $w_{\mu,2}$, because $w_{\mu,2} < v_{\mu,\tau}$, and we obtain the following expression for $\mathcal{R}_{\mu,\text{eff}}$:
$$ \mathcal{R}_{\mu, \text{eff}}(v) = \begin{cases} \mathcal{R}_{\mu} \left( 1 - v \frac{\sqrt{\lambda_{\mu}}}{\pi} (4T_{c,\mu} + n_{\mu}T_{b,\mu}) \right) & \text{for } v \le w_{\mu,1} \\ 0 & \text{else} \end{cases} $$

$\mathcal{R}_{M,\text{eff}}(v)$ and $\mathcal{R}_{\mu,\text{eff}}(v)$ are two linear functions of $v$, with negative slopes respectively equal to $-\mathcal{R}_{M} \frac{\sqrt{\lambda_{M}}}{\pi} (4T_{c,M} + n_{M}T_{b,M})$ and $-\mathcal{R}_{\mu} \frac{\sqrt{\lambda_{\mu}}}{\pi} (4T_{c,\mu} + n_{\mu}T_{b,\mu})$. Because $\mathcal{R}_{\mu} > \mathcal{R}_{M}$ and $\lambda_{\mu} > \lambda_{M}$, the lines intersect at a unique point, and we can conclude that (12.3) admits a unique positive solution, denoted as $v_{T}^{*}$.

Finally, because $\mathcal{R}_{\mu,\text{eff}} \ge \mathcal{R}_{M,\text{eff}}$ for $v < v_{T}^{*}$ and $\mathcal{R}_{\mu,\text{eff}} \le \mathcal{R}_{M,\text{eff}}$ for $v > v_{T}^{*}$, the threshold $v_{T}^{*}$ corresponds to a maximum of the function $\mathcal{R}$.

## 12.3 Proof of Theorem 11.1

Let $\theta > 0$. The coverage probability in the macro tier as the probability that $\text{SINR}_{0,M}$ is greater than $\theta > 0$ and the typical MU is associated with the macro tier. To evaluate this probability, we condition the event $\text{SINR}_{0,M} > \theta$ and that the typical MU is associated with the macro tier on the closest BS in the macro tier $X_{0,M}$ being at a distance $r$ of the origin and its micro counterpart, $X_{0,\mu}$, being at a distance $s$. We have:

$$ \begin{aligned} & \mathbb{P}[\text{SINR}_{0,M} > \theta, \text{Macro} \quad ||X_{0,\mu}| = s, |X_{0,M}| = r] \\ &= \mathbb{P}\left[h_0 > \frac{\theta r^{\alpha}}{P_M \kappa} \left(\sigma^2 + I_M(\Phi_M \setminus \{X_{0,M}\}) + I_{\mu}(\Phi_{\mu})\right)\right] \\ &= \mathbb{E}\left[\exp\left(-\frac{\theta r^{\alpha}}{\kappa P_M} I_{\mu}(\Phi_{\mu})\right), \text{Macro} \quad |X_{0,\mu}| = s\right] \\ &= \mathbb{E}\left[\exp\left(-\frac{\theta r^{\alpha}}{\kappa P_M} I_M(\Phi_M \setminus \{X_{0,M}\})\right), \text{Macro} \quad |X_{0,M}| = r\right] \end{aligned} $$

The first expectation gives:

$$ \begin{aligned} & \mathbb{E}\left[\exp\left(-\frac{\theta r^{\alpha}}{\kappa P_M} I_M(\Phi_M \setminus \{X_{0,M}\})\right), \text{Macro} \quad |X_{0,M}| = r\right] \\ &= \mathbb{E}\left[\prod_{x \in \Phi_M \setminus \{X_{0,M}\}} \mathbb{E}\left[\exp\left(-\theta h_x \left(\frac{r}{x}\right)^{\alpha}\right)\right]\right] \\ &\stackrel{(a)}{=} \mathbb{E}\left[\prod_{x \in \Phi_M \setminus \{X_{0,M}\}} \frac{1}{1 + \theta \left(\frac{r}{x}\right)^{\alpha}}\right] \\ &\stackrel{(b)}{=} \exp\left(-2\pi \lambda_M \int_r^{\infty} \frac{dx}{1 + \theta \left(\frac{r}{x}\right)^{\alpha}}\right), \end{aligned} $$

where (a) uses the Laplace transform of an exponential variable with parameter 1, and (b) uses the Laplace transform of a Poisson process. With the change of variables $u = \theta^{-\delta} \left(\frac{x}{z}\right)^2$, we get:
$$
\begin{align*}
& \mathbb{E} \left[ \exp \left( -\frac{\theta r^\alpha}{\kappa P_M} I_M (\Phi_M \setminus \{X_{0,M}\}) \right), \text{Macro} \middle| |X_{0,M}| = r \right] \\
&= \exp \left( -2\pi \lambda_M r^2 \theta^\delta \int_{\theta-\delta}^\infty \frac{du}{1+u^{1/\delta}} \right).
\end{align*}
$$

For the second expectation, we remind that the closest BS $X_{0,\mu}$ is at a
distance $s$ of the origin. We have:

$$
\begin{align*}
& \mathbb{E} \left[ \exp \left( -\frac{\theta r^\alpha}{\kappa P_M} I_\mu (\Phi_\mu) \right), \text{Macro} \middle| |X_{0,\mu}| = s \right] \\
&= \mathbb{E} \left[ \exp \left( -\frac{\theta P_\mu}{P_M} \left(\frac{r}{s}\right)^\alpha - \frac{\theta r^\alpha}{\kappa P_M} I_\mu (\Phi_\mu \setminus \{X_{0,\mu}\}) \right) \right] \\
&= e^{-\frac{\theta P_\mu}{P_M} \left(\frac{r}{s}\right)^\alpha} \mathbb{E} \left[ \prod_{x \in \Phi_\mu \setminus \{X_{0,\mu}\}} \frac{1}{1 + \frac{\theta P_\mu}{P_M} \left(\frac{r}{x}\right)^\alpha} \right] \\
&= e^{-\frac{\theta P_\mu}{P_M} \left(\frac{r}{s}\right)^\alpha} \exp \left( -2\pi \lambda_\mu \int_s^\infty \frac{dx}{1 + \frac{\theta P_\mu}{P_M} \left(\frac{s}{x}\right)^\alpha} \right).
\end{align*}
$$

We introduce the change of variables $u = \theta^\delta \left(\frac{P_\mu}{P_M}\right)^\delta \frac{r^2}{y^2}$ in the inside integral to get:

$$
\mathbb{E} \left[ \exp \left( -\frac{\theta r^\alpha}{\kappa P_M} I_\mu (\Phi_\mu) \right), \text{Macro} \middle| |X_{0,\mu}| = s \right] \\
= e^{-\frac{\theta P_\mu}{P_M} \left(\frac{r}{s}\right)^\alpha} \exp \left( -\pi \lambda_\mu \int_{u=\left(\frac{P_M}{P_\mu}\right)^{\frac{\delta}{2}} \frac{s^2}{r^2}}^{\infty} \frac{dx}{1+u^{1/\delta}} \right).
$$

The probability that a BS $X$ is located at a distance $r > 0$ from the origin
and the associated distribution function in the macro tier are equal to:

$$
\begin{align*}
F_M(r) &= \mathbb{P} [|X_{0,M} \le r] = 1 - \exp(-\pi \lambda_M r^2) \\
f_M(r) &= \frac{d}{dr} F_M(r) = 2\pi \lambda_M r e^{-\pi \lambda_M r^2},
\end{align*}
$$

with a similar expression for the micro tier by using the adequate values.
To obtain the coverage probability, we integrate over all the realisations
of the PPP $\Phi_M$ and $\Phi_\mu$. A user is associated with the macro tier if $s >$
$(\frac{P_\mu K(v)}{P_M})^\alpha r$, and to the micro tier if $0 \le s \le (\frac{P_\mu K(v)}{P_M})^\alpha r$. With these
bounds of integration, we get:

$$
\mathbb{P}[\text{SINR}_{0,M} > \theta] = \int_{r=0}^{\infty} \int_{s=\left(\frac{P_{\mu}K(v)}{P_{M}}\right)^{1/\alpha} r}^{\infty} e^{-\pi(\lambda_{M}r^2+\lambda_{\mu}s^2)} \\
\times \mathbb{P}[\text{SINR}_{0,M} > \theta | |X_{0,\mu}| = s, |X_{0,M}| = r] 4\pi^2\lambda_{\mu}r\lambda_{M}sdsdr.
$$
We introduce the change of variables $y = \pi\lambda_{\mu}s^2$ and $x = \pi\lambda_{M}r^2$ to obtain:

$$ \mathbb{P}[\text{SINR}_{0,M} > \theta] = \int_{x=0}^{\infty} \int_{y=CK(v)^{\delta}x}^{\infty} e^{-\theta\left(\frac{Cx}{y}\right)^{1/\delta}} e^{-x-y} e^{-x-y} \left(-x\left(\theta^{\delta}\int_{\theta-\delta}^{\infty}\frac{du}{1+u^{1/\delta}}+C\theta^{\delta}\int_{\frac{y}{x}}^{\infty}\frac{du}{C\theta^{\delta}}\frac{1}{1+u^{1/\delta}}\right)\right) dy dx. $$

We introduce a second change of variables for the inside integral, $z = \left(\frac{Cx}{y}\right)^{1/\delta}$, with $dy = -\delta \frac{Cx}{z^{1+\delta}} dz$. We have

$$ \mathbb{P}[\text{SINR}_{0,M} > \theta] = \int_{x=0}^{\infty} \int_{z=\frac{1}{K(v)}}^{0} e^{-\theta z} \left(-\delta \frac{Cx}{z^{\delta+1}}\right) e^{-x\left(\theta^{\delta}\int_{\theta-\delta}^{\infty}\frac{du}{1+u^{1/\delta}}+C\theta^{\delta}\int_{\frac{1}{(zT)^{\delta}}}^{\infty}\frac{du}{1+u^{1/\delta}}\right)} e^{-x\left(1+\frac{C}{z^{\delta}}\right)} dx dz. $$

After rearranging the terms and using Fubini's Theorem to swap the two integrals and introducing $F(\theta, \delta) = \int_{\theta-\delta}^{\infty} \frac{du}{1+u^{1/\delta}}$, we get

$$ \mathbb{P}[\text{SINR}_{0,M} > \theta] = \delta C \int_{z=0}^{1/K(v)} \frac{z^{-\delta-1} e^{-\theta z}}{\left(1 + Cz^{-\delta} + \theta^{\delta}F(\theta, \delta) + C\theta^{\delta}F(zT, \delta)\right)^2} dz. $$

For the formula in the micro tier, we swap the role of the two network tiers and use the appropriate values to get the intended result, which concludes the proof.

## 12.4 Proof of Lemma 11.3

Let us consider an MU located at the origin and moving at a given velocity $v > 0$. It is associated with the macro tier if and only if $R_M > K(v)R_{\mu}$. Let us assume that the closest macro BS to the origin, $X_{0,M}$ is located at distance $r > 0$ and its micro counterpart, $X_{0,\mu}$, is at distance $s > 0$.

The MU is thus associated with the macro tier if and only if $s > \left(\frac{P_{\mu}K(v)}{P_M}\right)^{1/\alpha} r$. This leads to:

$$ \mathbb{P}[\text{Macro} | v] = 4\pi^2 \int_{r=0}^{\infty} \int_{s=\left(\frac{P_{\mu}K(v)}{P_M}\right)^{1/\alpha} r}^{\infty} e^{-\pi\lambda_M r^2 - \pi\lambda_{\mu}s^2} \lambda_M \lambda_{\mu} ds dr. $$

We use the new variables $x = \pi\lambda r^2$ and $y = \pi\lambda_{\mu}s^2$, and introduce $C = \frac{\lambda_{\mu}}{\lambda_M} \left(\frac{P_M}{P_{\mu}}\right)^{\delta}$, which yields:

$$ \mathbb{P}[\text{Macro} | v] = \int_{x=0}^{\infty} \int_{y=CK(v)^{\delta}x}^{\infty} e^{-x-y} dx dy = \frac{1}{1 + CK(v)^{\delta}}. $$

To obtain the probability that the typical MU is associated with the macro tier, we integrate with respect to the MU velocity distribution $f$, which gives the intended result.
# CONCLUSION

In this thesis, we developed frameworks using stochastic geometry and elements of queuing theory to study some aspects of the dynamics of spatial queuing systems, aimed at modelling wireless networks. In this chapter, we will summarize the results obtained, focus on some questions of interest that arose during our study, and propose some prospects for future work.

## Summary

The first system we studied is a multiclass spatial birth-and-death process. Spatial queuing systems and birth-and-death processes have been studied in various setups ([15], [80]), and some results already exist. In our work, we introduced a form of service differentiation to modify the classical wireless network dynamics, under which users are allocated a given number of frequency bands to transmit on, and interfere only with users that have overlapping frequency bands. This differentiation is motivated by the introduction of bandwidth partitioning ([68]) algorithms in the latest generation of wireless networks as a means to increase network performance.

The first step of our study was to develop a broader framework than the one proposed in [80] to capture the multiclass nature of the interactions we consider here, and to establish several useful properties (monotonicity, irreducibility). We then moved on to studying the stability of the dynamics under a symmetry assumption. To study the stability of this system, we first introduced two queuing systems that bound the dynamics of the original system from above and below, and relied on fluid limits ([78]), which give helpful relations linking the behavior of a system of differential equations and the recurrence of these queuing systems. Once stability for these two systems was reached, we could reach the stability condition for the symmetric spatial birth-and-death process. After this, we relaxed the symmetry assumption and showed that we can reach a more general stability condition for non-symmetric networks using similar fluid limit argument, using a reciprocal to
a stability condition obtained in [84].

In a second part, we looked into the stationary regime of the dynamics introduced in Part I. The first quantities of interest we looked into are spatial user densities, which characterize the steady-state dynamics of this network. We derived two heuristics: a first one relying on a Poisson approximation, similar in a way to replica mean-field methods, and a second one leveraging a cavity approximation and using a second-order approximation. A most interesting result in this section is that the Poisson approximation allowed us to obtain a precise heuristic to estimate the critical arrival rate of the system. The second interesting property of the stationary regime is clustering. We know that such a property arises in the single-class system ([80]). In the multiclass system though, we reached a weaker condition, which can be explained by the fact that not all users interact with each other with the same power, weakening the attraction-inducing properties of wireless interactions. Finally, we completed the study of the stationary regime by showing that the queuing process admits exponential tails, which is of great interest for communication networks as it help to bound precisely latency and service times - two of the requirements of URLLC networks.

The third part of the manuscript focused on a different perspective: instead of considering time dynamics, where immobile users appear and disappear in the network, we consider spatial dynamics, where users move on a straight line inside a fixed network consisting in two types of base stations: macro and micro base stations. Macro BSs provide more power but have a larger coverage than micro BSs, which are more directional but less powerful. This model is motivated by multitier networks using beam-based communications ([39], [40]), which can be found in the newly deployed 5G wireless networks ([49]). The goal of this study of such a network is to design user-centric association policies taking into account user mobility, which are easily implementable and have a low computational cost. This lead us to define two families of association policies: one only considering user velocity, and the second one, offering a trade-off between network geometry and user mobility. Using relevant numerical computations and heuristics, we could prove that these family of policies can be tuned to maximize a certain network performance metric, which can be tied to spectral efficiency. Under this framework, we could compare these two association policies to a classical max-power association policies to find out that the two proposed association policies outperform this max-power association in several cases, notably, in densely populated networks with fast-moving users.

Although our study is separated in two parts leveraging two different frameworks and mathematical tools, the study we present here is a first step towards building a comprehensive framework to model time-spatial interactions in a multitier 5G network. In the next two sections, we will focus on some questions of interest that appeared during this work, and some prospects and natural extensions that can be investigated using the tools we developed here.
Discussion and questions of interest

The systems we considered in this thesis are idealized version of real-world
systems, yet the framework we used encapsulates the main features of wireless
communications (beam-based communication, bandwidth partitioning, user
mobility, communication overheads, etc.). The work we presented here can
be seen as a first step on building a more comprehensive framework to model
future wireless networks.

The results proved in Part I can be extended to other queuing networks.
Stability conditions for spatial queuing systems have already been studied
([15], [28], [58]) using various methods. The framework we present here has
the advantage of being robust, as it can be applied to other types of dynamics.
We can adapt Theorem 1.2 and the discretization to other types of departing
processes, and reach the stability condition using the method developed in
the proof of Theorem 2.3. This would allow us to reach stability conditions
for any spatial queuing systems with monotonicity and where the arrival rate
is constant and the departure rates is inversely proportional to a shot-noise
of users in the system.

In Part II, we developed heuristics to estimate the spatial user densities
in the network, and we derived from the Poisson heuristic an estimate for
the value of $\lambda_c$, which turns to be an accurate estimate for the critical arrival
rate in the system. The main question arising from this is the relevance of
the Poisson heuristic and how can such a result be explained. We know that
in replica mean-field systems and under some hypothesis, the replicas behave
as independent Poisson processes (see [19]). One can wonder if such a result
can be extended to our dynamics, by designing a replica mean-field version
of our system and studying the behavior of such a system as the number of
copies go to infinity.

When comparing association policies in Part III, we saw that numerically,
the optimal load-dependent VBMP policy outperformed the classical max-
power association policy for all user velocities, but that it failed to do better
than the load-threshold policy for high velocities. The question that arises
from this is whether it is possible to obtain an association policy that outper-
forms both. We saw that the optimization problem for which the theoretical
best VBMP policy was not solvable using the classical techniques of calcu-
lus of variations, and that we had to rely on an estimate by discretizing that
velocity space. This method of estimating the optimal VBMP policy faces
two obstacles: the computational cost of decreasing the grid size of velocities
is high, and the optimization methods used may face convergence issues to
find the global maximum of the MESR. Another explanation why the op-
timal VBMP policy does not outperform the load-threshold policy is that
the VBMP policies use bias factors that are differentiable (even, $C^1$) while
the bias factor in the load-threshold policy is a Heaviside function. These
questions need to be investigated and may be the subject of future work.
Prospects and future work

**Networks with heterogenous interactions** An interesting question coming from this line of thought would be spatial dynamical systems where the interactions between users are of different nature. An instance of such a heterogenous network is the following: let us imagine a network mixing users that transmit by peer-to-peer connection (as presented in [15]) and users that transmit using wireless transmissions. Under this setup, we would have a first class of users for which the intensity of the departure process is proportional to a shot-noise of the point process of users of the class, and a second class of users for which the departure process will have a stochastic intensity inversely proportional to a shot-noise of users of both classes.

Another interesting thing to note is that the framework we developed linking stability with fluid limits is an equivalence between Markov chains and Markov jump processes with systems of differential equations and their stability conditions. In the case we studied, stability regions are intervals of $\mathbb{R}_+$ containing 0 due to the monotonicity of the network, but one can wonder whether other network setups could give different behaviors for the fluid limit and how it could transpose for the original Markov chain.

**Load balancing and allocation policy** In Part I, the user type distribution $(p_C)_{C \in \mathcal{P}(K)}$ is determined by the user needs in the system, and can be empirically obtained by studying wireless traffic. From Theorem 2.2 and Theorem 4.4, we know that the critical arrival rate $\lambda_c$ is defined as the maximum rate of transmission of the class that is able to transmit the less (see Theorem 4.4). A first question arising is the question of the maximization of the stability region, although an immediate analysis of the result from Theorem 4.4 gives that the maximum value of $\lambda_c$ is reached when we set $p_{C^*} = 1$, where $C^*$ is the class with the lowest $L_C$. Another problem we can look into is load balancing in the network: how can we select the value of $(p_C)$ to limit the queue size of certain classes? Theorem 8.1 gives bounds for the queue length of each process linked with the value of $\lambda_c$, which is linked as well to the service times in the system, through Little's law ([64]).

We can also consider the case of adaptive allocation policies, which are critical in 5G NR ([63]). Among the possible allocation policies, a possibility is to let each $p_C$ be inversely proportional to its queue length. In such a setup, the larger a queue grows, the smaller the arrival rate in such queue is and we obtain a sort of dampening effect that would prevent queue length from becoming too large. Studying such a network would require to change the network setup a reestablish the fluid limits from Theorem 2.3 in order to account for these changes, and study the new stability region.

**Poisson heuristic and critical arrival rate** One of the most interesting question arisen in this work comes from the Poisson heuristic for $\lambda_c$, which proves to be very efficient to determine an approximate value of $\lambda_c$. We can wonder whether this result holds in more general cases. We know from [19]
that replica mean-field models have arrival processes that converge to Poisson
point processes as the number of replica goes to infinity.

An idea to prove this result would be to consider *n* parallel instances
of our network, and for each user, to sample users among all the replicas
uniformly at random, and to study the evolution of such a system as *n* goes
to infinity using a similar methodology. Such a result would be interesting
for two reasons. The first is that it would allow for an easy way to reach
a stability condition for spatial birth-and-death processes, as stochastic ge-
ometry tools and rate conservation arguments allow to easily reach simple
results when working with Poisson point processes. The second one would
be more theoretical, as it could be seen as a commutative diagram property,
where we could obtain an equality between a certain temporal average in the
system (the fluid limit) and a spatial average (the Poisson approximation in
the stationary regime).

**Spatial birth-and-death processes with mobility** As we saw in Part III of the thesis, mobility is a key factor when taking into account wireless communication, which can be easily included in our framework: assume that users move on a straight line with an angle $\theta$ with the x-axis i.i.d sampled from a distribution $g$ at a velocity $v$ sampled from distribution $f$, which forms a spatial birth-and-death move process ([60]). Due to the exponential distribution for file sizes in the system, the system is still a Markov jump process, and still possesses the domination property that allows us to bound the dynamics from above and below by two interference queuing networks. To obtain the stability condition, we have to add two terms, one to represent the users leaving the cell, the second for the users arriving in the cell; these two terms will depend on the distributions $f$ and $g$. Once the queueing equations for the two systems is established, we could obtain fluid limits and investigate the stability condition of such a network.

An situation where such processes would arise can be found from vehicular
networks. A first instance comes from wireless sensor networks, which have
been studied for some time ([61]). In the framework of 5G URLLC networks,
using service differentiation would allow for more flexibility and possibly low
latency in vehicular networks. In such a setup, sensors are represented by
transmitters and the onboard computer of car is the receiver, which are
located at a fixed distance from each other, which would form a SBD move
process. Providing a study for such networks could allow for more efficient
wireless sensor networks, and provide useful results about latency. A second
instance featuring the dynamics we studied in this thesis is the modelling a
fleet of autonomous vehicles. In this framework, moving vehicles transmit in
a cellular network and transmit to base stations placed at given locations in
space. Such systems have already been deployed ([50]), and leverage features
from both the systems we studied in this work.
<figure><img src="image_51.png" /></figure>
# RÉSUMÉ DU MANUSCRIT EN FRANÇAIS

Cette thèse s'intéresse à différents aspects de l'application de la théorie des processus ponctuels pour modéliser les nouvelles générations de réseaux de communication sans fil. Ce manuscrit s'articule en trois parties. Dans une première partie, nous nous intéressons à une classe de processus de saut markoviens qui implémentent une forme de différenciation de service, le partitionement de bande passante, similaire aux échnologies introduites dans les réseaux 5G. Dans une seconde partie, nous étudions les propriétés du régime stationnaire des dynamiques introduites dans la partie I, en étudiant les mesures moments et leur estimation, la distribution stationnaire du processus de saut ainsi que la présence d'attraction statistique. Pour finir, en partie III, nous changeons d'objet d'étude : considérons un réseau sans fil de type 5G avec plusieurs niveaux (dans notre exemple, 2 : un niveau *micro* et un niveau *macro*). Une question importante est la politique d'association dans un tel réseau. Dans les réseaux 5G, la mobilité des utilisateurs est une dimension importante à prendre en compte, car elle affecte de manière non-négligeable les performances du réseau. La question que nous traitons dans cette dernière partie est la performance de politiques d'associations dépendant de la vitesse des utilisateurs.

## Partie I - Stabilité d'un processus de vie et de mort spatial mutliclasse avec des interactions sans fil

Dans cette première partie, nous nous intéressons à un modèle similaire à celui présenté dans [80]. Considérons une région fermée et bornée $\mathcal{D}$ du plan euclidien. Par simplicité, supposons que $\mathcal{D}$ est un tore carré plat.
Des récepteurs arrivent dans $\mathcal{D}$ selon une pluie de Poisson d'intensité $\lambda > 0$. Lors de leur arrivée dans le système, chaque récepteur arrive avec un transmetteur situé à une distance $r > 0$ dans une direction aléatoire. Pour transmettre, les paires de récepteurs/transmetteurs (ou utilisateurs) peuvent utiliser $K > 1$ canaux de communication orthogonaux, dépendant de leurs besoins (par exemple, des utilisateurs devant transmettre plus d'informations pourraient demander plus de canaux de communication). Pour $C \subset [1, \dots, K]$, un utilisateur recevra l'ensemble de configuration (ou classe) $C$ sur lesquels transmettre avec une probabilité $p_C > 0$, et reçoit un fichier distribué indépendamment selon une loi exponentielle de moyenne $L_C > 0$. Comme les canaux sont orthogonaux, un utilisateur de classe $C$ recevra une interférence d'un utilisateur de classe $D$ avec une puissance proportionnelle à la taille de l'intersection de leurs classes, $|D \cap C|$.

Le système ainsi définit forme un processus de vie et de mort spatial, et est une extension du système introduit par Sanakararaman et Baccelli ([80]) dans le cas où plusieurs canaux de communications sont disponibles. Notons $\Phi^T$ le processus ponctuel des transmetteurs, $\Phi^R$, celui des récepteurs, $\Phi$ le processus des paires récepteurs-transmetteurs. Supposons que l'atténuation du signal dans le système est modélisée par une fonction $\ell$ de path-loss positive, non-croissante et bornée, et nous avons la présence d'un bruit extérieur de densité $\mathcal{N}_0$.

Pour définir le taux de départ des utilisateurs, nous pouvons exprimer l'interférence subie par un point situé en $x$ avec un récepteur situé en $y$ comme un bruit de grenaille (shot-noise) du processus des utilisateurs:

$$I(x, \Phi_t) = \sum_{z \in \Phi_t^T \setminus \{y\}} |C_x \cap C_z| \ell(\|x - z\|).$$

En considérant l'interférence dans le système comme du bruit et en supposant que le signal est faible devant l'interférence et le bruit, nous pouvons définir un taux de transfert de fichier comme suit:

$$R(x, \Phi_t) = \frac{|C|}{\mathcal{N}_0 + I(x, \Phi_t)}$$

En l'absence d'abandon de communication, le taux de départ des utilisateurs de classe $C$ est égal à $\frac{1}{L_C} \sum_{x \in \Phi_{t,C}} R(x, \Phi_t)$.

Le processus $\Phi$ est un processus de saut markovien, que nous appellerons *processus BWP*. Nous disons que ce processus est stable si et seulement s'il est positif récurrent. La question principale de la partie I est la définition de la région de stabilité de tels processus en fonction du paramètre d'arrivée $\lambda$.

L'étude de ce genre de système dynamique peut s'avérer complexe sans information supplémentaire sur les propriétés du système. Parmi les propriétés qui s'avèrent utiles, nous pouvons compter la *monotonie stochastique*. Cette propriété utilise un ordre partiel sur les processus stochastiques, la domination stochastique, pour lier la stabilité ou l'instabilité d'un processus de Markov à celle d'un autre en utilisant des majoration ou des minorations adéquates.
Au vu de la définition de la dynamique, nous pouvons identifier quatre quantités qui ont une influence sur le système : le taux d'arrivée $\lambda$, la condition initiale $\Phi_0$, le vecteur des tailles moyennes de fichiers $L$ et la fonction de transfert de fichier $R$ (ou, de manière équivalente, la path-loss $\ell$).

En utilisant des arguments de couplages, nous pouvons prouver que les quatre conditions suivantes sont suffisantes pour qu'un processus BWP $\Psi'$ domine un processus $\Psi$ (avec les trois autres paramètres pris égaux - les quantités marquées d'un prime se rapportent à $\Psi'$ et les autres, à $\Psi$):

- $\lambda < \lambda'$,

- $\Psi_0 \subset \Psi'_0$,

- $L \le L'$ (par coordonnée),

- Pour tout $\Psi_1 \subset \Psi_2$ et $x \in \mathcal{D}$, $R'(x, \Psi_1) \le R(x, \Psi_2)$, ou, de manière équivalente, pour tout $s \ge 0$, $\ell(s) \le \ell'(s)$.

En utilisant les propriétés de monotonie stochastique, nous pouvons déduire que si $\Psi'$ domine $\Psi$, si $\Psi'$ est stable, $\Psi$ est stable et si $\Psi$ est instable, $\Psi'$ est instable.

Pour obtenir la stabilité du processus $\Phi$, nous allons définir deux systèmes en utilisant une discrétisation de la région $\mathcal{D}$ en cellules de taille $\varepsilon$, puis définir deux nouvelles path-loss $\ell^\varepsilon$ et $\ell_\varepsilon$ dans le système, de sorte que tous les utilisateurs dans la même cellule voient la même atténuation du signal des autres cellules, et telles que :

$$ \ell_{\varepsilon}(x, y) \le \ell(\|x - y\|) \le \ell^{\varepsilon}(x, y). $$

En utilisant les deux nouvelles fonction de path-loss, nous pouvons définir deux interference queueing networks (voir [81]), qui sont des réseaux de file d'attente avec un espace d'état fini. Un premier, $\bar{X}_\varepsilon$ qui domine $\Phi$ et un second, $\underline{X}_\varepsilon$, qui est dominé par $\Phi$. Nous pouvons ensuite obtenir la stabilité de $\bar{X}_\varepsilon$ en utilisant des limites fluides ([84]) et l'instabilité de $\underline{X}_\varepsilon$ en utilisant de la domination stochastique avec une fille M/M/1 adéquate.

Nous obtenons deux conditions de stabilité:

- $\bar{X}_\varepsilon$ est stable si $\lambda < \bar{\lambda}_\varepsilon$,

- $\underline{X}_\varepsilon$ est instable si $\lambda > \underline{\lambda}_\varepsilon$.

En faisant tendre $\varepsilon$ vers 0, nous obtenons que $\bar{\lambda}_\varepsilon$ et $\underline{\lambda}_\varepsilon$ ont une limite commune, $\lambda^*$, ce qui nous permet d'obtenir le résultat principal de la partie I:

**Theorem.** Le modèle BWP est stable si $\lambda < \lambda^*$ est instable si $\lambda > \lambda^*$, où:

$$ \lambda^* = \frac{1}{p_C L_C} \frac{|C| z_C^*}{\langle \ell_{\mathcal{D}} \rangle \sum_U |C \cap U| z_U^*}, $$

et où $z^*$ est solution du système:
$$ \forall C, D \in \mathcal{P}(K) \quad \frac{1}{p_C L_C} \frac{|C| z_C^*}{\sum_U |C \cap U| z_U^*} = \frac{1}{p_D L_D} \frac{|D| z_D^*}{\sum_U |D \cap U| z_D^*}. $$

## Partie II - étude de processus ponctuels stationnaire en interaction

La partie II de ce manuscrit se concentre sur l'étude du système défini en partie I dans son régime stationnaire. L'étude de ce régime stationnaire permet d'obtenir des informations intéressantes sur la distribution spatiale des utilisateurs soumis à ce genre de dynamiques. Dans cette partie, nous étudions trois propriétés de ce régime stationnaire : l'estimation des mesures moment, l'existence de moments exponentiels et l'attraction statistique.

Les mesures moments, et plus précisément, l'estimation des densités spatiales d'utilisateurs dans le réseau en régime stationnaire. Pour estimer cette quantité, nous ne pouvons pas obtenir de forme close, mais nous définissons deux heuristiques.

La première heuristique s'appuie sur une approximation de champ moyen : nous supposons que dans le régime stationnaire, les processus $\Phi_{0,C}$ des utilisateurs de classe C sont des processus de Poisson d'intensité $\mu_C$. En utilisant des lois de conservation dans le système, nous piouvons obtenir sous cette hypothèse, un système d'équations dont les $\mu_C$ sont solutions. Cette heuristique poissonienne fournit une première approximation des densités spatiales. Un résultat intéressant, conséquence de ces calcules, est le suivant: si le système a un régime stationnaire, donc la dynamique est stable, nous pouvons calculer les $\mu_C$ par cette méthode. Par contraposée, si nous ne pouvons pas les calculer, le système est instable. Si $\lambda_p$ est la plus grande valeur de $\lambda$ telle que nous pouvons calculer les $\mu_C$ par cette méthode, nous obtenons une estimation de la valeur de $\lambda^*$, taux critique d'arrivée dans le système. En comparant cette valeur, obtenue numériquement, et la valeur obtenue en partie I, $\lambda_p$ est une estimation assez précise de $\lambda^*$. La raison de ce résultat n'est pas étudiée dans ce manuscrit.

La seconde heuristique que nous développons est une approximation par cavité, provenant de la physique statistique. En supposant la présence de deux utilisateurs, un en $x$ de classe C et l'autre en $y$ de classe D, nous pouvons estimer la fonction de corrélation croisée $\rho_{C,D}^{(2)}$ et appliquer des lois de conservation dans le système pour obtenir une seconde estimation, meilleure que la première, pour les densités spatiales d'utilisateurs $\mu_C$, au prix d'un calcul plus coûteux.

Pour ce qui est des mesures moments d'ordre supérieur, nous avons obtenu l'existence de moments exponentiels dans le système, i.e. l'existence de $c_0 > 0$ tel que pour tout $0 \le c < c_0$, $E[e^{c\Phi_{0,c(D)}}] < \infty$. L'existence de moments exponentiels dans le système permet de borner les queues de la distribution stationnaire et d'obtenir des garanties de performance dans le réseau.
La dernière propriété intéressante sur laquelle nous nous sommes penchés est l'attraction statistique en régime stationnaire. Nous savons que le cas $K = 1$ crée de l'attraction en régime stationnaire, i.e. que la dynamique fait que les utilisateurs se regroupent en grappes où les communications sont plus lentes. Dans le cas $K \ge 2$, nous avons prouvé une version relaxée de cette attraction, que nous pensons être due aux non-linéarités introduites par la différenciation de service dans notre modèle.

# Partie III - Politique d'association basée sur la mobilité dans des réseaux sans fil 5G

Dans la troisième partie de ce manuscrit, nous nous intéressons à un réseau sans fil 5G possédant deux niveaux d'antennes : un premier niveau, appelé *macro*, avec des antennes puissantes à large couverture et un second niveau, *micro*, d'antennes plus faibles avec une plus faible couverture.

Dans les réseaux 5G, les antennes utilisent des faisceaux étroits et directifs pour communiquer avec les appareils connectés. Lorsqu'un appareil sort du faisceau de la station à laquelle il est connecté, il doit resélectionner un nouveau faisceau. Cette opération prend un certain temps pendant lequel l'appareil ne peut pas transmettre avec le réseau. De la même façon, si l'appeil connecté sort de la couverture de la station à laquelle il est connecté, il doit sélectionner une nouvelle station, opération pendant laquelle il ne pourra pas communiquer avec le réseau. Suivant cette obseruation, la prise en compte de la mobilité des appareils connectés dans les réseaux 5G sans fil devient une composante importante de la mesure des performances du réseau. Plus particulièrement, dans les réseaux étagés, nous allons nous demander comment la prendre en compte pour concevoir des politiques d'association intéressantes.

Nous considérons un réseau sans fil à deux niveaux, macro et micro. Les stations de base (SBs) de chacun des deux niveaux sont distribuées selon deux processus ponctuels de Poisson indépendants, $\Phi_M$ et $\Phi_\mu$, d'intensités respectives $\lambda_M$ et $\lambda_\mu$, possédant $n_M$ et $n_\mu$ faisceaux et de puissances $P_M$ et $P_\mu$. Les utilisateurs mobiles (UMs) dans le réseau sont distribués selon un troisième PPP indépendant $\Phi_u$ d'intensité $\lambda_u$. Ces UMs se déplacent en ligne droite à une vitesse distribuée indépendamment selon une distribution f.

On suppose que les communications sont descendantes dans le réseau, et les UMs se connectent à la SB la plus proche d'eux dans leur niveau d'association, ce qui forme deux tessellations de Poisson-Voronoï - une par niveau du réseau.

En utilisant l'isotropie et l'invariance du réseau, nous pouvons résumer l'étude des performances dans le réseau à l'étude de l'utilisateur typique situé à l'origine du repère. Nous pouvons également supposer qu'il se déplace sur une ligne droite selon l'axe des abscisses.

Nous pouvons calculer les intensités temporelles de réspections de fais-
<figure><img src="image_52.png" /><figcaption>Figure 14.1: Illustration de l’overhead par unité de temps dans un réseau mobile. Les resélections de faisceaux sont en jaune, les changements de SB, en orange</figcaption></figure>

ceaux et de changement de cellule (voir [53]) en fonction des paramètres du réseau et en fonction de la vitesse v d'un UM:

$$ \nu_{c,M} = \frac{4\sqrt{\lambda_M}}{\pi}v \qquad \nu_{b,M} = \frac{n_M\sqrt{\lambda_M}}{\pi}v $$

Nous définissons alors l'overhead par unité de temps comme la proportion du temps pendant laquelle l'UM ne transmet pas avec le réseau:

$$ T_o^M = \nu_{c,M}T_{c,M} + \nu_{b,M}T_{b,M}. $$

Grâce à une interprétation ergodique, nous pouvons définir un taux de Shannon effectif dans un niveau du réseau comme étant le taux de Shannon global pondéré par la proportion du temps pendant laquelle l'UM transmet, en prenant en compte l'overhead, ce qui mène à la définition suivante:

$$ \mathcal{R}_{M,\text{eff}}(v) = \mathcal{R}_M (1 - T_o^M(v))^+, $$

Nous allons donc définir des politiques d'association ayant pour but de maximiser le taux moyen effectif de Shannon (MESR) dans le réseau.

La première famille de politiques que nous étudions sont les politiques de seuil: en fixant un seuil $v_T > 0$, tous les UM de vitesse inférieure à $v_T$ sont associés avec le niveau micro, et tous ceux ayant une vitesse supérieure à $v_T$ sont associés avec le niveau macro. Nous pouvons calculer le MESR de l'utilisateur typique dans le réseau sous une politique de seuil donnée:

$$ \mathcal{R}(v_T) = \int_0^{v_T} \mathcal{R}_{\mu,\text{eff}}(v) f(v) dv + \int_{v_T}^{\infty} \mathcal{R}_{M,\text{eff}}(v) f(v) dv. $$

Dans un premier temps, nous supposons qu'il n'y a qu'un seul UM par SB dans le réseau. Nous obtenons le théorème suivant:
**Theorem (Politique de seuil optimale).** Il existe un unique seuil $v_T$ qui maximise le MESR dans le réseau. Ce seuil optimal $v_T^*$ ne dépend pas de la distribution de vitesse $f$, et est solution de l'équation suivante:

$$ \mathcal{R}_{\mu, \text{eff}}(v_T^*) = \mathcal{R}_{M, \text{eff}}(v_T^*). $$

Si nous relaxons la supposition selon laquelle nous avons un seul UM par SB, nous introduisons $Z_M^0(v_T)$ la variable aléatoire dénotant le nombre d'UM partageant la SB de l'utilisateur typique sous une politique de seuil dans le niveau macro. Nous pouvons redéfinir le MESR dans le réseau :

$$ \hat{\mathcal{R}}_{\text{load}}(v_T) = \mathbb{E} \left[ \frac{\mathcal{R}_{\mu, \text{eff}}(v)}{Z_{\mu}^0(v_T)} 1_{v \le v_T} + \frac{\mathcal{R}_{M, \text{eff}}(v) 1_{v > v_T}}{Z_M^0(\bar{v}_T)} \right], $$

Dans le cas général, nous ne pouvons pas calculer facilement le maximum de cette fonction. Nous devons utiliser deux heuristiques pour simplifier cette expression:

- Nous supposons que le SINR et les $Z^0$ sont indépendantes,

- Nous utilisons une heuristique pour la taille des cellules d'une tessel-lation de Poisson-Voronoi pour approcher la moyenne de $1/Z_M^0(v_T)$ et $1/Z_\mu^0(v_T)$ par deux fonctions $L_M(v_T)$ et $L_\mu(v_T)$.

Le MESR moyen dans le réseau devient alors, sous ceux deux heuristiques:

$$ \hat{\mathcal{R}}_{\text{load}}(v_T) = L_{\mu}(v_T) \int_{0}^{v_T} \mathcal{R}_{\mu, \text{eff}}(v) f(v) dv + L_{M}(v_T) \int_{v_T}^{\infty} \mathcal{R}_{M, \text{eff}}(v) f(v) dv. $$

Nous pouvons définir une nouvelle politique de seuil optimale en prenant en compte la charge dans le système:

**Theorem.** *L'heuristique de charge-seuil est le seuil $v_{LT}$ qui maximise $\hat{\mathcal{R}}_{\text{load}}$, i.e., tel que :*

$$ v_{LT} \triangleq \arg \max \hat{\mathcal{R}}_{\text{load}}(v_T). $$

*La politique de charge-seuil optimale est la politique de seuil associée avec $v_{LT}$.*

Nous pouvons comparer cette politique de charge-seuil optimale à une politique d'association classique : la politique max-power (MARP), où les UMs sont associés au niveau du réseau qui leur fournit le meilleur SINR dans le cas où la vitesse de UM est distribuée selon une distribuion exponentielle.

Nous pouvons voir sur la Figure 14.2 que la politique de charge-seuil optimale offre des performances supérieures à celles de la politique MARP dès que la vitesse moyenne de UM devient grande, ce qui donne un exemple simple de cas de figure où les politiques définies dans ce travail surpassent une politique déjà établie.
<figure><img src="image_53.png" /><figcaption>Figure 14.2: Comparaison entre la politique max-power (marquers carrés, en pointillés) et la politique de charge-seuil optimale (cercles, ligne pleine).</figcaption></figure>

La seconde catégorie de politiques d'association que nous explorons sont des politiques de biabs dépendant de la vitesse. Les politique de seuil dépendent uniquement de la vitesse des UMs, et ignorent la géométrie du réseau. Le but de cette étude est de concevoir une famille de politiques d'association combinant les idées de la politique MARP et des politiques de seuil.

Soit $K$ une fonction continue, positive. Nous définission la politique VBMP (max-power biaisée par la vitesse) de la façon suivante : si $R_M$ et $R_\mu$ sont les puissances reçues par l'UM typique située à l'origine, alors :

- si $R_M > K(v)R_\mu$, on associe l'UM avec le niveau macro,

- sinon, on l'associe avec le niveau micro.

Nous pouvons reformuler les résultats obtenus avec les politiques de seuil pour obtenir un MESR dépendant de la charge sous une politique VBMP:

$$ \mathcal{R}_{\text{load}}(K) = \int_{0}^{v_{\text{max}}} \mathbb{E} \left[ \frac{\log(1 + \text{SINR}(v, K))}{Z_{0}(K)} (1 - T_{o}(v))^{+} \right] f(v) dv. $$

Nous allons utiliser les mêmes heuristiques que celles utilisées précédemment pour déifnir la politique VBMP qui maximise le MESR sujet à la charge dans le réseau:

$$ \hat{\mathcal{R}}(K) = \int_{0}^{v_{\text{max}}} \left( L_{M}(K) \mathcal{R}_{M,\text{eff}}(v, K) + L_{\mu}(K) \mathcal{R}_{\mu,\text{eff}}(v, K) \right) f(v) dv, $$

où:

- $L_M(K) = L\left(\frac{\lambda_M}{\lambda_u G_M(K)}\right)$,

- $G_M(K) = \int_{v=0}^{v_{\text{max}}} \frac{f(v)}{1+\Omega K(v)^\delta} dv$ est la probabilité que l'UM typique est associée avec le niveau macro.
<figure><img src="image_54.png" /><figcaption>Figure 14.3: Comparaison des trois politiques d'associations : MARP (en rouge), charge-seuil optimale (en bleu) et VBMP optimale (en vert)</figcaption></figure>

La forme du MESR $\hat{\mathcal{R}}(K)$ que nous cherchons à maximiser est particulière, car elle ne nous permet pas d'utiliser des résultats de calculs de variations pour obtenir une condition sur la fonction K qui maximise le MESR.

Pour approcher le maximum, nous utilisons une discrétisation de l'espace des vitesses et de la fonction K, et nous utilisons des méthodes d'optimisation numériques pour chercher le maximum du MESR. Malheureusement, les méthodes utilisées ne nous ont pas permis d'obtenir le résultat escompté : la politique que nous obtenons est meilleure que la politique MARP, mais offre des performances inférieures à la politique de charge-seuil. La raison de ce résultat reste une question ouverte dans ce manuscrit (voir Figure 14.3).

## Conclusion

Dans ce manuscrit, nous avons exploré deux modèles mathématiques faisant intervenir des technologies mises en oeuvre dans les dernières générations de réseaux de communication sans fil : la différenciation de service, le beamforming directionnel et les réseaux étagés.

Dans les deux premières parties, nous avons étudié la différenciation de service et son impact sur un réseau de files d'attente spatial en proposant un cadre d'étude qui peut être généralisé à un ensemble de dynamiques. Nous avons également obtenu des résultats sur les mesures moments, la distribution stationnaire des dynamiques et un résultat sur l'attraction statistique que les dynamiques sans fil engendrent.

La troisième partie se concentre sur l'étude d'un réseau 5G à deux niveaux, où la mobilité joue un rôle important. Nous avons proposé un cadre simple présentant des particularités des réseaux 5G de nouvelle génération, défini une métrique et étudié deux familles de politiques d'associations : une pre-
mière basée uniquement sur la vitesse des utilisateurs dans le réseau et une seconde, mélangeant mobilité et géométrie. Nous les avons ensuite comparées à une politique d'association classique pour estimer leurs performances.

Le travail présenté dans ce manuscrit peut être vu comme une première étape vers la définition d'un cadre d'étude plus large pour modéliser des réseaux plus complexes. Les résultats présentés dans ce manuscrit peuvent être étendus à des cadres d'étude voisins possédant des propriétés similaires. Les travaux présentés dans cette thèse ouvrent également de nouvelles perspectives, avec plusieurs questions ouvertes sur certains résultats présentés, qui pourraient faire l'objet de futures recherches.
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<figure><img src="image_55.png" /></figure>
<figure><img src="image_56.png" /></figure>
# RÉSUMÉ

Grâce aux nouveaux paradigmes introduits dans la dernière génération de réseaux sans fil, les attentes concernant le temps de service, la latence et la performance du réseau ont augmenté. Pour modéliser ces réseaux, la théorie des processus ponctuels et la géométrie stochastique se sont avérées utiles car elles fournissent un cadre polyvalent et robuste pour obtenir des résultats lorsque l'on travaille avec ces réseaux sans fil. L'ajout d'une dynamique markovienne pour modéliser les connexions et les temps de service complète le cadre d'analyse de ces réseaux sans fil.

La première contribution du travail présenté dans cette thèse réside dans l'analyse de la différenciation des services : les réseaux 5G NR ont introduit le partitionnement de la bande passante comme outil pour augmenter la performance du réseau. Dans cette configuration de réseau, tous les utilisateurs n'interfèrent pas les uns avec les autres avec la même puissance : les utilisateurs qui émettent avec un spectre de fréquence d'émission plus large auront une plus grande bande passante, mais ils rencontreront également plus d'interférences de la part des autres utilisateurs dans le réseau. En revanche, les utilisateurs dont le spectre est plus étroit subiront moins d'interférences. Nous définissons un cadre markovien pour étudier un tel processus spatial multiclasse de naissance et de mort, et nous décrivons sa région de stabilité. Pour de tels systèmes, les propriétés du régime stationnaire sont analysées, telles que les mesures moment ou l'attraction, ce qui permet de mieux comprendre cette dynamique.

Le deuxième problème que nous examinons est celui de la mobilité, qui est devenue une caractéristique importante des réseaux sans fil en raison de l'utilisation d'antennes hautement directionnelles. En utilisant une architecture simple pour un réseau cellulaire à deux niveaux, nous étudions deux familles de politiques d'association : une première famille qui s'appuie uniquement sur la mobilité de l'utilisateur, et une seconde qui offre un compromis entre la géométrie du réseau et la mobilité de l'utilisateur afin d'augmenter les performances du réseau. Ces politiques sont ensuite comparées à une politique d'association classique de puissance maximale afin d'évaluer leurs performances.

# MOTS CLÉS

Géométrie aléatoire, Chaînes de Markov, Processus ponctuels, Réseaux de communication sans fil, Limites fluides, Calcul de Palm

# ABSTRACT

Thanks to the new paradigms introduced in the latest generation of wireless networks, expectations concerning service time, latency and network performance have increased. To model such networks, point process theory and stochastic geometry have proven to be useful as they provide a versatile and robust framework to obtain results when working with such wireless networks. Adding to this Markov dynamics to model connections and service times completes this framework to analyze such wireless networks.

The first contribution of the work presented in this thesis lies in the analysis of service differentiation: 5G NR networks have introduced bandwidth partitioning as a tool to increase network performance. Under this network setup, not all users interfere with each other with the same power: users transmitting with a broader transmitting frequency spectrum will have a larger bandwidth, but they will also encounter more interference from the other users in the network. In contrast, users with a narrower spectrum will experience less interference. We define a Markovian framework to study such a multiclass spatial birth-and-death process, and we describe its stability region. For such systems, properties of the stationary regime are analyzed, such as moment measures or statistical clustering, leading to a better understanding of these dynamics.

The second problem we look into is mobility, which has become an important feature in wireless networks due to the use of highly directional antennas. Using a simple architecture for a two-tier cellular network, we study two families of association policies: a first family which only relies on user mobility, and the second offers a trade-off between network geometry and user mobility to increase network performance. These policies are then compared to a classical max-power association policy to assert their performance.

# KEYWORDS

Stochastic geometry, Markov chains, Point processes, Wireless communication networks, Fluid limits, Palm calculus