# Apprentissage Statistique Cellule par Cellule de la Charge à Partir de la Demande de Trafic dans les Réseaux Cellulaires Sans Fil

Lucas Darlavoix, Bartlomiej Blaszczyszyn, Mohamed Kadhem Karray, Zeinab Nehme

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Lucas Darlavoix, Bartlomiej Blaszczyszyn, Mohamed Kadhem Karray, Zeinab Nehme. Apprentissage Statistique Cellule par Cellule de la Charge à Partir de la Demande de Trafic dans les Réseaux Cellulaires Sans Fil. CORES 2025 - 10èmes Rencontres Francophones sur la Conception de Protocoles, l'Evaluation de Performances et l'Expérimentation des Réseaux de Communication, Jun 2025, Saint Valery-sur-Somme, France. hal-05030767

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# *Apprentissage Statistique Cellule par Cellule
de la Charge à Partir de la Demande de Trafic
dans les Réseaux Cellulaires Sans Fil*

L. Darlavoix<fnref n="1" />,<fnref n="2" />, B. Błaszczyszyn<fnref n="2" />, M. K. Karray<fnref n="1" /> et Z. Nehme<fnref n="1" />

Nous proposons une approche statistique pour prédire la charge spécifique à chaque cellule dans les réseaux sans fil opérationnels, en utilisant la demande de trafic comme variable principale. Contrairement à un modèle linéaire global, qui ajuste les données de toutes les cellules de manière uniforme sans tenir compte des spécificités de chaque cellule, notre approche linéaire par cellule se distingue par deux éléments clés : (i) un modèle unidimensionnel, qui prédit la charge de chaque cellule uniquement en fonction de sa propre demande de trafic, et (ii) un modèle multidimensionnel, qui intègre la demande de trafic de l'ensemble des cellules. À l'aide de données réelles massives, nous montrons que le modèle unidimensionnel surpasse les approches globales en termes de précision. Bien que le modèle multidimensionnel offre une flexibilité supplémentaire et des améliorations sur certains aspects spécifiques, notamment en cas de trafic élevé, ses coefficients manquent souvent d'interprétabilité physique, même avec une régularisation de crête (ridge regularization). Cela souligne la nécessité de développer des techniques de régularisation avancées, prenant en compte la géométrie des réseaux sans fil.

**Mots-clefs** : Évaluation de performances, Qualité de Service / Qualité d'Usage, Réseaux sans fil

## 1 Introduction and state of the art

With the annual rise in network traffic, telecommunication companies must ensure their infrastructure meets customer demands. While real-time metrics offer performance insights, predicting the relationship between cell loads and traffic demands in multi-cellular networks is critical for assessing service quality. Today, physics-based mathematical models [BK19] provide global predictions but lack accuracy at the cell level without detailed propagation data. In contrast, AI-driven statistical models leverage massive datasets to capture these relationships. Prior work [BBDHK23] has also focused on global predictions, while this study aims to enable accurate, cell-level forecasts. These state-of-the-art global approaches are described below.

**Mathematical model** A global approach, based on the model in [BK19], links the load of an "extra" base station—placed at the origin and superimposed on a Poisson point process $\Psi = \{X\}$ representing the network—to its traffic demand. This demand corresponds to the traffic generated within its Voronoi cell, competing with all other stations in $\Psi$. The influence of these stations is captured by $R(SINR)$, which models the rate-SINR relationship for a given technology while accounting for network interference. On average, i.e., in terms of mathematical expectations, by *Palm calculus*, this *mean model* yields the following relationship between the traffic $\bar{\rho}$ per unit area and the corresponding load $\bar{\theta}$ of this abstract cell:

$$ \bar{\theta} = \frac{\bar{\rho}}{\lambda_{BS}} \mathbb{E} \left[ 1/R \left( \frac{P_{X_*}}{N + \bar{\theta} \sum_{X \in \Psi \setminus \{X_*\}} P_{X_*}} \right) \right], \quad (1) $$

where $\lambda_{BS}$ is the base station density, (i.e., the number of stations per unit area in $\Psi$), $X_*$ represents the station of $\Psi$ closest to the origin, and $P_{X_*}, P_{X_*}$ represents the received power at the origin from station $X, X_* \in \Psi$, $N$ is the noise power, and the expectation $\mathbb{E}$ accounts for statistical averaging over the network geometry $\Psi$ (and possibly randomized received power). This method is referred to as the "mean model" in future sections.

<footnote n="1">¹Orange Innovation - Campus Gardens, 92320 Châtillon, France</footnote>

<footnote n="2">²INRIA / ENS Paris, 75013 Paris 13, France</footnote>
<figure><img src="image_2.png" /><figcaption>FIGURE 1: Predicted load versus measured load for state-of-the-art methods using global approaches, via a mathematical model (A) and via a statistical model (B). More information on the dataset used is available in Section 3.</figcaption></figure>

**Statistical approach** Several researchers have explored load prediction using linear fitting methods and this work is particularly inspired by [BBDHK23]. Their algorithm is trained on traffic-load pairs collected over a month to estimate the slope $\beta_1$ and intercept $\beta_0$, leading to the linear relation:

$$ \theta = \beta_0 + \beta_1 \rho. \qquad (2) $$

This method is referred to as the "BBDHK" approach in future sections.

Figure 1 compares the predictions of these global approaches against real data from an operational network. The results suggest that these models are complementary: the mathematical model performs better in low and high traffic scenarios, while the statistical model is more accurate in moderate traffic conditions, but neither approach is fully adequate for cellwise predictions across the entire traffic range.

## 2 Cellwise linear statistical model

In contrast to the global approach, the cellwise approach builds a separate statistical model for each cell $C_i$ in the network $\{C_i\}_{i=1}^N$. Specifically, for each $i = 1, \dots, N$, this model provides a function $\vartheta_i(\rho)$ designed to predict the charge $\vartheta_i$ of cell $C_i$ based on the traffic demand. The input $\rho \in \mathbb{R}^N$ of this function represents the traffic demand, which may include the demand from all cells, i.e., $\rho := (\rho_j)_{j=1}^N$, and in particular, the traffic demand $\rho_i$ of the specific cell $C_i$.

**Class of multilinear functions.** A function of the form

$$ \vartheta_i(\rho) = \hat{\beta}_i \cdot (1, \rho)^\top := \hat{\beta}_{i0} + \sum_{j=1}^{N} \hat{\beta}_{ij} \rho_j, \qquad (3) $$

is sought, where the coefficients $\hat{\beta}_{ij} \in \mathbb{R}$ are to be determined for $i = 1, \dots, N$ and $j = 1, \dots, N$. In this notation: $\hat{\beta}_i$. and $\rho$ are row vectors, $(1, \rho) := (1, \rho_1, \dots, \rho_N)$ is an extended input vector including a bias term, $(\cdot)^\top$ denotes the transpose, making $(1, \rho)^\top$ a column vector. The coefficients $\hat{\beta}_i$ are estimated by fitting the model to a set of training data collected from the network.

**Training data set.** Specifically, the dataset consists of observations of the traffic demand $\rho_{ki}$ and the corresponding load $\theta_{ki}$ for each cell $i = 1, \dots, N$ at distinct instants $k = 1, \dots, M$ (e.g., different hours and dates when data was collected in a commercial network). This dataset is represented as: $\{(\rho_{ki}, \theta_{ki}) : k = 1, \dots, M, i = 1, \dots, N\}$. It is customary to introduce standard matrix notation for these training data: Let $\Phi \in \mathbb{R}^{M \times (1+N)}$ be the design matrix, where the k-th row corresponds to $(1, \rho_{ki})$ for $k = 1, \dots, M$ and $\theta \in \mathbb{R}^{M \times N}$ be the response matrix, where each column corresponds to $\theta_{,i}$ for $i = 1, \dots, N$.

**Fitting (loss models).** The function $\vartheta_i(\rho)$ is determined by minimizing the least square error, yielding a multilinear cellwise statistical model. While $M \ge N + 1$ is sufficient in principle, only for $M \gg N$ can the model reliably capture the true relationship between traffic demand and cell load. However, if $N$ and thus the number of parameters $\hat{\beta}$, become too large, making $M \gg N$ computationally infeasible, the model risks overfitting, that is, memorizing training data instead of learning a generalizable relationship. This consequently degrades predictive performance on unseen data. To mitigate overfitting, two common
strategies are ridge regularization and dimensionality reduction of the model, the latter incorporating prior physical knowledge to constrain the model. These two methods, in addition to the ordinary multidimensional model, are discussed below (see Appendix A for more details).

**Multidimensional model.** In this model, the function $\vartheta_i(\rho)$ is determined by minimizing the least-squares error with respect to the training data $\Phi, \theta$. Specifically, the coefficients are given by

$$ \hat{\beta}_i^T = (\Phi^T \Phi)^{-1} \Phi^T \theta_i. \quad (4) $$

**Ridge-regularized model.** In this approach, the function $\vartheta_i(\rho)$ is again determined by minimizing the least-squares error with respect to the training data $\Phi, \theta$, while also incorporating an additional term, involving a regularization parameter $\alpha > 0$, discouraging large values of $\beta_i$. This gives

$$ \hat{\beta}_i^T = (\Phi^T \Phi + \alpha I)^{-1} \Phi^T \theta_i. \quad (5) $$

**Cellwise unidimensional model.** In this simplified framework, each cell's model is trained solely on its own traffic. Specifically, the function $\vartheta_i(\rho)$ is determined by minimizing the least-squares error with respect to the training data, while setting the parameters $\hat{\beta}_{ij} = 0$ for $j = 2, \dots, N$, giving

$$ \hat{\beta}_{i1} = \frac{(\rho_i - \bar{\rho}_i)^T (\theta_i - \bar{\theta}_i)}{\|\rho_i - \bar{\rho}_i\|^2}, \quad \hat{\beta}_{i0} = \bar{\theta}_i - \hat{\beta}_{i1} \bar{\rho}_i, \quad (6) $$

where $\bar{\theta}_i = \frac{1}{M} \sum_{k=1}^M \theta_{ki}$ and $\bar{\rho}_i = \frac{1}{M} \sum_{k=1}^M \rho_{ki}$.

**Remark.** The global statistical model BBDHK, mentioned in the Introduction, mathematically simplifies to the unidimensional model by considering a single cell $C_1$ (i.e., setting $N = 1$) and 'flattening' the structure of the training data. This involves combining the traffic and load data into one vector each: the traffic vector $(\rho_{11}, \dots, \rho_{(NM)1})$ and the load vector $(\theta_{11}, \dots, \theta_{(NM)1})$.

## 3 Numerical results

In this section, several experiments are conducted using a dataset (confidential due to company restrictions), made of 4G 2600 MHz downlink traffic and load values for $N = 7979$ cells, stemming from a large area representative of some European country and comprising a mix of urban, suburban and rural zones.

**Cellwise methods** To compare all approaches, a preliminary investigation was conducted under the same learning and testing conditions as the BBDHK study. Cellwise unidimensional and multidimensional linear models were trained, using the scikit-learn *LinearRegression* function, on traffic-load pairs for each station over a month (30 days), excluding the midday hour (12 PM) of the 30th day, which was used for testing. This resulted in $M = 24 \times 30 - 1 = 719$ training instants. The same dataset was used for the BBDHK model (Figure 1B), the mean model (Figure 1A), and the two cellwise methods (Figure 2).

**Unidimensional approach** Along with the comparison between Figure 2A and Figure 1, Table 1 provides metrics for the unidimensional cellwise approach and the two state-of-the-art methods. The absolute error is calculated using RMSE = $\sqrt{m[(\hat{\theta} - \theta)^2]}$, with $\hat{\theta}$ and $\theta$ the predicted and measured load for the testing data respectively and $m$ the empirical mean. The relative error is given by $RMSE/\sqrt{m[\theta^2]}$.

The results show that the fine-grained unidimensional cellwise model outperforms state-of-the-art methods. This illustrates that using a cell-specific prediction approach improves results over global models.

**Multidimensional approach** The cellwise multidimensional model, accounting for inter-cell interactions, outperforms its unidimensional counterpart, as shown in Figure 2B and the "Multidim." column of Table 2, operating fifty times faster due to matrix formulation. It enhances predictions for high loads—critical for telecommunication companies assessing QoS. However, these gains must be viewed in context, as the condition $M \gg N$ is not satisfied because of computational limitations, inducing a memorization effect due to overfitting. Additionally, since the test set is temporally close to the training set, both come from the same distribution, raising concerns about the model's ability to truly capture the underlying physics of the network.
<figure><img src="image_3.png" /><figcaption>**FIGURE 2:** Cellwise linear fitting in the unidimensional case (A) and multidimensional case (B).</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**FIGURE 3:** Impact of $\alpha$ on error metrics (A) and on $\hat{\beta}$ coefficients (versus inter-cell distances) for $\alpha = 10^4$, $\alpha = 10^{11}$ and $\alpha = 10^{15}$ (B).</figcaption></figure>

<table><caption>**TABLE 1:** Error table comparing cellwise unidimensional linear fitting and state-of-the-art global methods, via a mean model approach (1) and via the BBDHK statistical approach (2).</caption><thead><tr><th colspan="2">Error type</th><th>Mean model</th><th>BBDHK</th><th>Unidim.</th></tr></thead><tbody><tr><td rowspan="2">Absolute error</td><td>Median</td><td>5.5 × 10<sup>-2</sup></td><td>5.8 × 10<sup>-2</sup></td><td>3.8 × 10<sup>-2</sup></td></tr><tr><td>80th-percentile</td><td>1.2 × 10<sup>-1</sup></td><td>1.3 × 10<sup>-1</sup></td><td>9.4 × 10<sup>-2</sup></td></tr><tr><td rowspan="2">Relative error</td><td>Median</td><td>0.13</td><td>0.14</td><td>0.10</td></tr><tr><td>80th-percentile</td><td>0.28</td><td>0.33</td><td>0.23</td></tr></tbody></table>

To improve robustness, a new experiment was conducted using proper dataset division (3 weeks training, 1 week testing) and limiting training and testing to daytime hours (8 AM to 8 PM). While these adjustments lead to some performance degradation, as shown in the "Robust Multidim." column of Table 2, they help the model better account for network complexity and enhance overall robustness.

<table><caption>**TABLE 2:** Error table comparing performances of cellwise unidimensional linear fitting (learning on all hours for 29 days, testing on the midday hour of one day), cellwise multidimensional linear fitting (also learning on all hours for 29 days, testing on the midday hour of one day) and a more robust implementation of the multidimensional method (learning in daytime on 23 days, testing on daytime on 7 days).</caption><thead><tr><th colspan="2">Error type</th><th>Unidim.</th><th>Multidim.</th><th>Robust Multidim.</th></tr></thead><tbody><tr><td rowspan="2">Absolute error</td><td>Median</td><td>3.8 × 10<sup>-2</sup></td><td>2.3 × 10<sup>-2</sup></td><td>4.2 × 10<sup>-2</sup></td></tr><tr><td>80th-percentile</td><td>9.4 × 10<sup>-2</sup></td><td>5.6 × 10<sup>-2</sup></td><td>8.5 × 10<sup>-2</sup></td></tr><tr><td rowspan="2">Relative error</td><td>Median</td><td>0.10</td><td>0.06</td><td>0.11</td></tr><tr><td>80th-percentile</td><td>0.23</td><td>0.14</td><td>0.22</td></tr></tbody></table>

**Ridge regularization** In order to mitigate overfitting, which causes the memorization effect, ridge regularization was applied. As depicted in [Bac24, Fig 3.3, p. 58] and shown in Figure 3A, selecting an optimal regularization parameter $\alpha$ is crucial: low values have little effect, while high values cause error metrics to explode $^\dagger$. Empirical analysis finds an optimal $\alpha^* = 9 \times 10^{10} \approx 10^{11}$, but with only marginal performance improvements. Furthermore, Figure 3B shows that in this case, ridge regularization does not enhance the physical interpretability of model coefficients: the balance of positive and negative values, as well as the long-range dependencies between cells, are difficult to justify both theoretically and in engineering practice.

## 4 Conclusion

Cellwise load prediction with linear fitting models outperforms state-of-the-art approaches but suffers from overfitting when capturing cell interactions in a multidimensional model, especially in large networks, despite ridge regularization. Thus, future work will incorporate further physical insights, such as the immediate neighborhood of a cell, as leveraging geometry is key to making these methods robust, scalable and adaptable.

<footnote>$^\dagger$. Also, as observed on Figure 3B for $\alpha = 10^{15}$, high values of $\alpha$ overly shrink coefficients, causing the loss of valuable information.</footnote>
# References

[Bac24] Francis Bach. *Learning Theory from First Principles*. MIT Press, 2024.

[BBDHK23] Lynda Berrah, Adrien Bufort, Maroua Drissi, and Imed Hadj Kacem. Orange internal report, 2023. Private communication.

[BK19] Bartlomiej Blaszczyszyn and Mohamed Kadhem Karray. Performance analysis of cellular networks with opportunistic scheduling using queueing theory and stochastic geometry. *IEEE Transactions on Wireless Communications*, 18(12):5952–5966, 2019.

# A Appendix: Statistical background

## A.1 Multidimensional model

In this model, the function $\vartheta_i(\rho)$ is determined by minimizing the least-squares error with respect to the training data $\Phi, \theta$. Specifically, for each cell $C_i$, $i = 1, \dots, N$, the coefficients of the multilinear function are obtained by solving the optimization problem:

$$ \hat{\beta}_i = (\hat{\beta}_{i0}, \dots, \hat{\beta}_{iN}) := \arg \min_{\beta_i = (\beta_{i0}, \dots, \beta_{iN})} \sum_{k=1}^{M} |\theta_{ki} - \beta_i \cdot (1, \rho_k)^\top|^2. \quad (7) $$

Using the design and response matrices for training data, the objective function in (7) can be rewritten in matrix form as $\|\theta_{.i} - \Phi\beta_{.i}\|_2^2$, where $\| \cdot \|_2$ denotes the Euclidean norm in dimension $M$. If the matrix $\Phi$ has full column rank (i.e., rank $N+1$), equivalently if $\Phi^\top\Phi$ is invertible, then the unique minimizer of this function is given by (7); see [Bac24, Proposition 3.1].

## A.2 Ridge-regularized model

In this model, the function $\vartheta_i(\rho)$ is determined by minimizing the least-squares error with respect to the training data $\Phi, \theta$, while incorporating an additional regularization term. This approach helps mitigate potential overfitting of the multilinear statistical model, particularly when $M$ is not sufficiently large relative to $N$, as discussed earlier.

Specifically, for each cell $C_i$, $i = 1, \dots, N$, the coefficients of the multilinear function are obtained by solving:

$$ \hat{\beta}_i = (\hat{\beta}_{i0}, \dots, \hat{\beta}_{iN}) := \arg \min_{\beta_i = (\beta_{i0}, \dots, \beta_{iN})} \sum_{k=1}^{M} |\theta_{ki} - \beta_i \cdot (1, \rho_k)^\top|^2 + \alpha \sum_{j=0}^{N} |\beta_{ij}|^2, \quad (8) $$

where $\alpha > 0$ is the regularization parameter. The second term introduces an additional penalty in the minimization process, discouraging large values of $\beta_i$. This not only helps prevent overfitting but fundamentally ensures that the optimization problem is well-posed even for $N+1 < M$. Indeed, the objective function in (8) can be expressed in matrix form as $\|\theta_{.i} - \Phi\beta_{.i}\|_2^2 + \alpha\|\beta_{.i}\|_2^2$, where $\| \cdot \|_2$ denotes the Euclidean norm, applied in dimensions $M$ and $1+N$ for the first and second terms, respectively. For $\alpha > 0$, this function has a unique minimizer given by (5); see [Bac24, Proposition 3.6 and Exercise 3.5].

## A.3 Cellwise unidimensional model

In this model, the function $\vartheta_i(\rho)$ is determined by minimizing the least-squares error with respect to the training data, while setting the parameters $\hat{\beta}_{ij} = 0$ for $j = 2, \dots, N$. For each cell $C_i$, $i = 1, \dots, N$, the coefficients $\hat{\beta}_{i0}$ and $\hat{\beta}_{i1}$ are obtained by solving the optimization problem (7) under this additional condition.

This optimization is unique (provided $M \ge 2$ and the traffic vector $\rho_{.i}$ for cell $i$ is not constant), yielding (6) as its solution.