Anthony Bardou

Thomas Begin

Analysis of a Decentralized Bayesian Optimization Algorithm for Improving Spatial Reuse in Dense WLANs

Keywords: Machine Learning, Bayesian Optimization, Spatial Reuse, IEEE 802.11, Power Control

Despite representing the prominent means of accessing the Internet, WLANs remain subject to performance issues, which may be mitigated through more efficient spatial reuse of radio channels. In this perspective, the IEEE 802.11ax amendment enables the dynamic update of two key parameters in wireless transmission: the transmission power (TX POWER) and the sensitivity threshold (OBSS PD). In this paper, we present INSPIRE, a distributed online learning solution performing local Bayesian optimizations based on Gaussian processes to improve spatial reuse in WLANs. INSPIRE makes no explicit assumptions on the WLANs' topology and favors altruistic behaviors of the access points in their search for adequate configurations of their TX POWER and OBSS PD parameters. INSPIRE can easily be extended to work with a limited number of observations to throttle its computational complexity. We demonstrate the superiority of INSPIRE over other state-of-the-art strategies using the ns-3 simulator and two examples inspired by real-life deployments of dense WLANs. Our results show that, in only a few seconds, INSPIRE is able to drastically increase the quality of service of operational WLANs by improving their fairness and throughput. Finally, we discuss the configurations recommended by INSPIRE. We show that they comply with an 802.11ax empirical recommendation, and we correlate their values with some graph-based metrics of the WLAN topologies.

Introduction

Wireless networks have become ubiquitous and part of our everyday life in modern societies. According to Cisco's forecasts [START_REF]Cisco visual networking index: Forecast and trends, 2017-2022 white paper[END_REF], Wireless Local Area Networks (WLANs) accounted for 57% of the Internet traffic in 2022 when mobile and wired networks represented 22% and 21%, respectively. WLANs typically follow the guidelines of the IEEE standard 802.11 (commercially known as Wi-Fi). They comprise one or multiple Access Points (APs) that relay back-and-forth frames to a set of associated wireless stations (STAs). In addition to their wireless interface, APs also embed a wired interface to provide access to the Internet.

The spatial reuse of radio frequency resources, together with improvements in physical layers and allocation of new radio frequency bands, has been and remains a prominent leverage to increase the efficiency of wireless networks. This problem of assigning radio resources is eased by the breaking of radio bands into separate channels. Indeed, APs assigned to different orthogonal channels can transmit simultaneously without interfering with each other. Furthermore, APs assigned to the same channel can transmit concurrently and successfully if their distance makes the power attenuation on their signal strong enough.

However, from a higher perspective, the spatial reuse of WLANs is somewhat less advanced than that of mobile networks. In the latter networks, this concept, often coined as "small cell" or "cell densification", has led, over the years, the cell radius to decrease from 35 km for early 2G systems to 5 km for 3G systems, 100 m for 4G systems, and about 25 m in 5G [START_REF] Haas | Lifi is a paradigm-shifting 5g technology[END_REF]. Conversely, the radio range of APs in WLANs has remained constant since the inception of 802.11 in the 1990s with the transmission power of APs set to 100 mW (20 dBm). But this has changed with the recent release of the IEEE 802.11ax amendment [START_REF]Ieee standard for information technology-telecommunications and information exchange between systems local and metropolitan area networks-specific[END_REF], opening the path to major gains through more effective spatial reuse in WLANs.

The 802.11ax amendment, released by IEEE in 2021 and commercially branded by the Wi-Fi Alliance as Wi-Fi 6, enables APs to dynamically adjust/tune two key parameters at each AP: TX PWR and OBSS PD. The former parameter specifies the power level (in dBm) at which the AP transmits its data. The latter parameter defines the sensitivity threshold (in dBm). If the energy received is below this level, this indicates to the AP that the radio channel is clear and thus available for transmission. Otherwise, the AP must defer its transmissions. While prior amendments to 802.11 held these TX PWR (a) With the default configuration of TX PWR and OBSS PD, the two APs are within each other's detection range so that they cannot transmit simultaneously.

(b) The value of TX PWR is reduced on both AP so that they do not belong to each other's detection range. Under this configuration, concurrent transmissions from the two AP may occur at the same time. and OBSS PD parameters constant (typically 20 dBm and -82 dBm respectively), 802.11ax has made them dynamic with their values spanning from 1 to 21 dBm for the former and from -82 to -62 dBm for the latter. Adjusting the configurations of TX PWR and OBSS PD can help overcome the limitations of spatial reuse in dense environments by allowing APs that are close to each other to transmit on the same channel.

Figure 1 depicts a simple example of two APs operating on the same radio channel and illustrates how different configurations of the TX PWR parameter can lead to different performance. Note that in this simple example, concurrent transmissions of the two APs could also be attained by increasing OBSS PD at each AP (and keeping TX PWR at their default value). Although the two options produce similar effects here, in general, reducing TX PWR and increasing OBSS PD may affect the WLANs' performance differently (see Table 2 of [START_REF] Wilhelmi | Potential and pitfalls of multiarmed bandits for decentralized spatial reuse in wlans[END_REF] for more details).

Despite the potential of 802.11ax to improve the spatial reuse of radio channels, finding an adequate configuration of TX PWR and OBSS PD for the APs in a WLAN is a complex problem. First, an adequate configuration is very topology-specific. In other words, knowing an efficient configuration for a given scenario is of (almost) no value for another scenario. Second, a distributed solution would be more appreciated than a centralized solution. Not only does this avoid the search in an otherwise very high dimensional space but this avoids the assumption of having a centralized entity (e.g., a controller) deciding the configurations of all APs. Additionally, a centralized entity can be an acceptable assumption if all the interfering APs belong to the same WLAN but unrealistic if they belong to concurrent WLANs. Third, forecasting the performance of WLANs with an analytical model that can subsequently help establish optimal configurations is difficult. The degree of details in the models will be either too coarse and thus inapplicable, or adequate but unscalable when scenarios involve multiple APs and STAs. An alternative to this centralized, analytical strategy is a decentralized, learning strategy where APs can apply new configurations of their parameters, measure the effect of these changes on their performance, and exchange their experience with surrounding APs. This paves the way for the use of online and reinforcement learning techniques in a distributed manner.

In this paper, we present a decentralized online learning strategy, based on Bayesian optimization and Gaussian Processes (GPs) to efficiently address the issue of improving the spatial reuse of a radio channel in WLANs. This leads us to define a distributed algorithm, known as INSPIRE, which makes no explicit assumptions about the topology of WLANs or the radio environments and thus can apply to any WLANs (including when APs belong to different WLANs). The current paper extends our previous work presented in [START_REF] Bardou | Inspire: Distributed bayesian optimization for improving spatial reuse in dense wlans[END_REF]. Overall, the contributions of the paper are as follows:

• We demonstrate the ability of GPs at approximating the reward function that reflects the performance of WLANs, and at exploring efficient AP configurations;

• We establish the superiority of a divide-and-conquer approach to handle the complex problem of setting the TX PWR and OBSS PD parameters at each AP;

• We introduce INSPIRE a distributed solution that lets the APs of concurrent WLANs automatically adapt their internal parameters' setting in their own interest as well as in the interest of obtaining more efficient spatial reuse of radio channels;

• We propose a lightweight version of INSPIRE, referred to as INSPIRE LIM, which can be of practical interest when limited computing resources are available to conduct the search for an adequate setting of TX PWR and OBSS PD parameters.

• We evaluate the efficiency of INSPIRE on real-life inspired case studies using a detailed network discrete-event simulator and compare its performance with several state-of-the-art solutions (centralized or distributed).

• We correlate the values found for TX PWR and OBSS PD by INSPIRE with metrics related to the network topology under study.

The rest of the paper is organized as follows. In Section 2, we discuss the related work. We use Section 3 to present our proposed strategy and the main theoretical results to handle the issue of spatial reuse of a radio channel in WLANs. The performance evaluation of our strategy is reported in Section 4. We use Section 5 to deepen the understanding of the numerical results as well as the parameter settings made by INSPIRE. Section 6 concludes this paper.

Background

Spatial Reuse with TX PWR and OBSS PD

The release of the IEEE 802.11ax amendment [START_REF]IEEE standard for information technology-telecommunications and information exchange between systems -local and metropolitan area networks-specific requirements -part 11[END_REF] in late 2021 marks a new era for the spatial reuse of radio channels of WLANs: Nodes can dynamically adjust their transmission power (TX PWR) and sensitivity threshold (OBSS PD) parameters. For a detailed explanation of how this new feature is implemented, we refer the interested reader to [START_REF] Wilhelmi | Spatial reuse in ieee 802.11ax wlans[END_REF], which also provides simple scenarios to illustrate its potential benefits.

Years before IEEE released the 802.11ax amendment, the idea of dynamically updating TX PWR and OBSS PD has been explored by some researchers. The pioneering work of [START_REF] Zhu | Adapting physical carrier sensing to maximize spatial reuse in 802.11 mesh networks[END_REF] presents an analytical model that, based on the current radio channel conditions, dynamically configures OBSS PD on each node of a Wi-Fi-based mesh network. Concurrently, [START_REF] Kim | SP-TPC: a selfprotective energy efficient communication strategy for[END_REF] established that adapting TX PWR can lead to increased throughput and reduced energy consumption. More recently, in 2020, [START_REF] Qiu | Joint access point placement and power-channel-resource-unit assignment for 802.11ax-based dense wifi with qos requirements[END_REF] casts the issues of positioning the APs of a WLAN and choosing their TX PWR as an optimization problem. The authors provides a solution to this problem that delivers a static configuration of TX PWR for a WLAN. But their solution does no account for the number of STAs nor the type of traffic in the WLAN.

The difficulty of accurately modeling the dependency between the configuration parameters of a large WLAN and its performance is a strong hurdle to the development of spatial reuse strategies based on analytical models. As a result, most of the proposed strategies are data-driven. Adaptive by construction, they constitute promising candidates in the search for configurations that improve the spatial reuse of a radio channel and thus, the performance of WLANs.

Machine learning (ML) techniques are natural candidates for addressing problems requiring a data-driven approach, and the spatial reuse problem is no exception to that rule. [START_REF] Ak | Fsc: Two-scale ai-driven fair sensitivity control for 802.11ax networks[END_REF] addresses the problem of configuring TX PWR and OBSS PD with a two-scale solution using artificial neural networks (ANN). In their strategy, STAs and APs first adjust their value of OBSS PD to minimize interference. Then, an ANN, which was trained offline through simulation, is used to increase the fairness between STAs in terms of attained throughput. However, given the vast diversity of WLAN topologies, the offline learning of the ANN appears as a clear limitation to the generalization of this strategy. An online learning procedure is proposed by [START_REF] Bardou | Improving the spatial reuse in ieee 802.11ax wlans: A multi-armed bandit approach[END_REF], which uses reinforcement learning and more precisely the Multi-Armed Bandit (MAB) framework to find the optimal configuration of TX PWR and OBSS PD in a WLAN. The approach comprises two agents with one sampling promising configurations through a multivariate normal distribution, and the other identifying the best configuration among those already sampled with Thompson sampling and Normal-Gamma priors. A similar strategy is proposed by [START_REF] Bardou | Mitigating starvation in dense wlans: A multi-armed bandit solution[END_REF], which recommends the use of a mixture of hyperspheres to subsample the configuration space. These three ML solutions [START_REF] Ak | Fsc: Two-scale ai-driven fair sensitivity control for 802.11ax networks[END_REF][START_REF] Bardou | Improving the spatial reuse in ieee 802.11ax wlans: A multi-armed bandit approach[END_REF][START_REF] Bardou | Mitigating starvation in dense wlans: A multi-armed bandit solution[END_REF] are tested on the network simulator ns-3 and lead to significant WLAN improvements. However, in order to perform their optimization, they all assume the presence of a central controller that has access and control over all the APs in the WLANs. By construction, these approaches are centralized, and hence cannot be applied to cases where concurrent WLANs managed by different owners interfere with others.

Distributed approaches are undisputedly better fit than centralized approaches to handle cases with a set of concurrent WLANs. [START_REF] Shahwaiz Afaqui | Evaluation of dynamic sensitivity control algorithm for ieee 802.11ax[END_REF] introduces a distributed algorithm named Dynamic Sensitive Control which is run on every STAs of a WLAN. In short, each STA tries to dynamically reduce its value of OBSS PD to favor concurrent transmissions while keeping it high enough to ensure a high quality signal reception. Similarly, [START_REF] Lee | Lsr: Link-aware spatial reuse in ieee 802.11ax wlans[END_REF] proposes Link-aware Spatial Reuse (LSR), a distributed algorithm designed for the APs. In LSR, each AP chooses another AP, which is allowed to transmit concurrently, and then prescribes a value of TX PWR for the selected AP. These two algorithms rely on a single measurement metric reflecting the quality of the received signal, namely the Received Signal Strength, to choose the nodes configura-tion. More recently, strategies using distributed MAB approaches have been proposed [START_REF] Wilhelmi | Collaborative spatial reuse in wireless networks via selfish multi-armed bandits[END_REF][START_REF] Wilhelmi | Potential and pitfalls of multiarmed bandits for decentralized spatial reuse in wlans[END_REF]. They both use Thompson sampling with Gaussian priors to find the best couple of TX PWR and OBSS PD at each AP. In [START_REF] Wilhelmi | Collaborative spatial reuse in wireless networks via selfish multi-armed bandits[END_REF], each AP seeks to maximize the throughput of its associated STAs. On the other hand, in [START_REF] Wilhelmi | Potential and pitfalls of multiarmed bandits for decentralized spatial reuse in wlans[END_REF], the authors assume that every AP has access to the performance of all other APs in the WLAN; then each AP attempts to maximize a global reward that takes into account the performance of all the other nodes. Both strategies [START_REF] Wilhelmi | Collaborative spatial reuse in wireless networks via selfish multi-armed bandits[END_REF][START_REF] Wilhelmi | Potential and pitfalls of multiarmed bandits for decentralized spatial reuse in wlans[END_REF] solutions were evaluated in a self-made simulator with simple random scenarios.

Table 1 summarizes the main characteristics of the data-driven strategies discussed above. It shows that, out of the eight considered strategies, two (namely [START_REF] Shahwaiz Afaqui | Evaluation of dynamic sensitivity control algorithm for ieee 802.11ax[END_REF][START_REF] Ak | Fsc: Two-scale ai-driven fair sensitivity control for 802.11ax networks[END_REF]) only focus on the configuration of the OBSS PD parameter (keeping the TX PWR parameter fixed). To help in the comparison of the different strategies, we introduce two concepts: "pull area" and "push area". The pull area indicates the area on which each node is assumed to obtain information (this typically includes parameter configurations and performance measurements). Depending on the strategy being considered, the pull area can include just the node itself, the surrounding nodes, or the whole set of nodes in the WLANs. The push area designates the area which each AP can influence typically through the prescription of parameter configurations. In the case of centralized strategies (e.g., [START_REF] Ak | Fsc: Two-scale ai-driven fair sensitivity control for 802.11ax networks[END_REF][START_REF] Bardou | Improving the spatial reuse in ieee 802.11ax wlans: A multi-armed bandit approach[END_REF][START_REF] Bardou | Mitigating starvation in dense wlans: A multi-armed bandit solution[END_REF]), the pull and push areas naturally cover the whole set of APs. We distinguish partially distributed strategies (e.g., [START_REF] Wilhelmi | Potential and pitfalls of multiarmed bandits for decentralized spatial reuse in wlans[END_REF]) wherein either the pull or push area includes the whole set of APs with fully distributed strategies (e.g., [START_REF] Shahwaiz Afaqui | Evaluation of dynamic sensitivity control algorithm for ieee 802.11ax[END_REF][START_REF] Lee | Lsr: Link-aware spatial reuse in ieee 802.11ax wlans[END_REF][START_REF] Wilhelmi | Collaborative spatial reuse in wireless networks via selfish multi-armed bandits[END_REF]) in which both the pull and push areas differ from the whole set of APs. We observe in Table 1 that only three out of the eight state-of-the-art strategies can be considered as fully distributed. The last four columns of Table 1 pertain to the performance evaluation used to validate each of these strategies. It appears that most strategies were evaluated without considering the dynamical selection of the Modulation Coding Scheme (MCS) for the speed of the wireless links, nor bidirectional (with upstream and downstream) traffic. This can be seen as a strong limitation since this overlooks some associated trade-offs. For instance, increasing the value of TX PWR certainly enables the communication to operate with a faster data rate (larger MCS), but at the same time, it increases the level of interference with surrounding APs. Additionally, most strategies were evaluated on relatively simple scenarios (with a few APs and a limited number of radio channels), often using a self-made network simulator. This paper is an extension of the previous work [START_REF] Bardou | Inspire: Distributed bayesian optimization for improving spatial reuse in dense wlans[END_REF] that proposes a fully distributed strategy to address the problem of the spatial reuse of radio channels in WLANs. The proposed strategy, named INSPIRE, can be applied to any arrangement of WLANs and its novelties are mostly twofold. First, to the best of our knowledge, it is the first strategy making use of Gaussian Processes to explore promising WLAN configurations in the quest of discovering the optimal one. Gaussian processes are recognized tools to deal with the exploration vs. exploitation dilemma (see [START_REF] Srinivas | Gaussian process optimization in the bandit setting: No regret and experimental design[END_REF][START_REF] Ray | On kernelized multi-armed bandits[END_REF]) which is at the center of the spatial reuse problem. Second, unlike the existing fully distributed strategies, INSPIRE allows each AP to account for its surroundings thanks to pull and push areas broader than a single node. Through the use of a simple consensus method, APs of the WLANs achieve to behave altruistically selecting configurations for the "greater good" of the WLANs. We also introduce realistic scenarios, inspired by real-life WLANs, with dynamic MCS and bidirectionnal traffic, to evaluate and compare the efficiency of all the considered strategies.

Bayesian Optimization with Gaussian Processes

Before describing our solution, let us introduce Bayesian Optimization (BO) and Gaussian Processes (GPs) in this section, as they play a key role in our proposed solution to the spatial reuse problem in WLANs.

Gaussian Process. A GP is a stochastic process, that is, a collection of random variables {Y (x)} x∈C indexed by a set C. The specificity that makes a Gaussian process is that any finite set of random variables {Y (x 1 ), • • • , Y (x n )} has a joint multivariate Gaussian distribution. It can be fully specified by its mean function µ and its covariance function

Σ µ(x) = E [Y (x)] (1) Σ (x, x ′ ) = E [(Y (x) -µ(x)) (Y (x ′ ) -µ(x ′ ))] . (2)
In most cases, a BO algorithm rely on the pioneering work [START_REF] Christopher | Gaussian processes for regression[END_REF], that proposes a unified framework for regression tasks using GPs. We describe this approach in a few lines and we illustrate it with Figure 2. A GP is considered as a surrogate model for a black-box objective function f . It requires a prior GP, describing the random variables {f (x)} x∈C , by a prior mean function µ 0 (x) and a covariance function k(x, x ′ ). The vast majority of BO algorithms consider, without loss of generality, that µ 0 (x) = 0. For any point x ∈ C, it is known by construction that f (x) ∼ N (µ 0 (x), σ 2 0 (x)), with σ 2 0 (x) = Σ(x, x). Given t observations, described by the t × d matrix of datapoints X = (x i ) ⊤ i∈ [1,t] and by the corresponding t-dimensional vector of outputs y = (y i ) ⊤ i∈ [1,t] , the posterior distribution of f (x) is obtained from the t + 1-dimensional joint Gaussian distribution of the t observations along with the queried f (x), by conditioning on the observed data. Then, a posteriori, we have

f (x) ∼ N (µ t (x), σ 2 t (x)) with µ t (x) = Σ(x, X) ⊤ K -1 y
(3)

σ 2 t (x) = σ 2 0 (x) -Σ(x, X) ⊤ K -1 Σ(x, X) (4)
with Σ(x, X) = (Σ (x, x i )) i∈[1,t] the t-dimensional vector built by applying the covariance function to x and each datapoint in X and K = (Σ (x i , x j )) i,j∈ [1,t] the t × t matrix built by applying the covariance function to each possible couple of datapoints in X. A BO algorithm needs to provide, at each time step t, a query x t that achieves a trade-off between exploration and exploitation. The BO algorithm typically bases its querying policy on the maximization of an acquisition function that quantifies the benefits of observing f (x). Common acquisition functions include Probability of Improvement [START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF], Expected Improvement [START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF] and GP-UCB [START_REF] Srinivas | Information-theoretic regret bounds for gaussian process optimization in the bandit setting[END_REF]. Further details are provided about the choice of the acquisition function in our case and its maximization in Section 3.3. The use of an efficient acquisition function allows us to sequentially extend our observed data set with promising points that improve the quality of the GP regression and ultimately get closer and closer to the optimal configuration.

Under some assumptions, BO offers optimality guarantees regarding the optimization of an arbitrary objective function f , and it has obtained excellent empirical performance in a variety of black-box optimization tasks [START_REF] Bergstra | Making a science of model search: Hyperparameter optimization in hundreds of dimensions for vision architectures[END_REF][START_REF] Daniel Lizotte | Automatic gait optimization with gaussian process regression[END_REF][START_REF] González | Bayesian optimization for synthetic gene design[END_REF]. Since the spatial reuse of a WLAN can be seen as a black-box objective function, we propose to use BO to optimize the performance of WLANs. We are now ready to introduce our proposed solution, called INSPIRE and described in the next section.

Positioning

The current paper extends our previous work presented in [START_REF] Bardou | Inspire: Distributed bayesian optimization for improving spatial reuse in dense wlans[END_REF]

Proposed solution

Assumptions on the WLANs under study

Let W denote the set of concurrent WLANs under study, each of which being comprised of one or more APs. We let V be the set of APs in W that operate on the radio channel of interest. We denote by N the number of APs, by s i the set of STAs associated with AP i and by S the total number of STAs in the considered radio channel of W. Thus, we have: S = N i=1 |s i |. Finally, we use N i to designate the set of APs that are within the communication range of AP i (when every AP is under the default configuration of the TX PWR and OBSS PD parameters). Note that AP i itself belongs to N i . We refer to the APs in N i as the surroundings of AP i.

We make no assumptions on W, including on the specific arrangement of its APs and STAs, other than the three detailed below.

First, we assume that every AP i is able to exchange control frames (possibly through its beacon frames) with its surrounding APs (i.e., the ones in N i ). By the same token, we suppose that at least one AP i has another AP in its communication range (i.e., ∃i ∈ 1, N , N i ̸ = {i}), otherwise the spatial reuse of the radio channel would already be at its apex.

Second, we assume that the N APs have their TX PWR and OBSS PD parameters configurable (as defined by the 802.11ax amendment). We use x t i to denote the configuration of AP i with regards to its two TX PWR and OBSS PD parameters at time t. Analogously, x t represents the configuration of the N APs from W at time t. Thus, we have:

x t i ∈ C = -82, -62 × 1, 21 dBm and x t ∈ C N .
Lastly, we assume that each AP in W can periodically run performance tests and obtain, in return, the mean throughput attained by each of its STAs over a short time interval ∆t. More formally, we use the vector T t ∈ R +S to denote the throughput attained by the S STAs of W given the WLAN configuration x t at time t. Throughout this paper, we sometimes refer to T t as T (x t ) to explicitly exhibit the dependency between the STAs' throughputs and APs' configurations.

In this work, we seek to discover an adequate configuration x * of the N APs composing W that improves the collective experience of the S STAs through a better reuse of their radio channel. We address this problem as a reinforcement learning task in which, at regular time intervals t, the APs collect measurements T t associated to their current configuration x t , and need to decide their next configuration x t+1 . The obstacles towards that objective are mostly threefold. (i) We need to define a meaningful objective function that APs will attempt to optimize collectively; (ii) We are facing the well-known exploration vs. exploitation dilemma since the search for an adequate configuration of the WLANs should be as seamless as possible (without disrupting the STAs). This leads us to cast the problem as a MAB problem where the arms are the WLANs' configurations. Following the MAB terminology, we refer to the objective function as the reward function; (iii) We are looking for a strategy that can be applied in a distributed way since it would be in general unrealistic to assume that (concurrent) APs have a fine knowledge beyond their surroundings.

Reward function

We need to define a reward function R that appraises the "goodness" of a configuration x with regards to the WLANs performance. Because multiple criteria may be considered in the definition of R, there is no universal definition. However, assessing the quality of a configuration x can be derived from the STAs throughputs T (x) obtained with APs configured with x. Among all easily computable reward functions, R(x) = T i ∈T (x) T i is called the proportional fairness (PF) and provides a convenient trade-off between fairness and cumulated throughput. However, PF is often criticized for its high variability, since ∂R ∂T i = T j ∈T (x),j̸ =i T j . To overcome this drawback, we consider the logarithm of PF. This lowers its variability, which becomes: ∂R ∂T i = 1 T i (note that T i is typically much larger than 1). This also emphasizes the contribution of STAs with low throughputs in the computation of R and provides a pleasant closed-form to optimize. For an arbitrary set of APs X, we can define R X = log i∈X j∈s i T j (x). Then, our global reward function R is:

R(x) = R V (x) = log i∈V j∈s i T j (x) = i∈V j∈s i log T j (x) (5)
However, to compute Equation 5, an AP must have a complete knowledge of the performance attained by the STAs of all APs or, at least, be able to communicate with all the APs in W. This is in contradiction with our assumption that APs only have a partial knowledge of W, limited to their surrounding APs. To design a reward function compatible with the distributed case, we proceed as follows. Each AP i applies Equation 5 but restricted to the set of its associated STAs and obtains in return a "selfish" reward denoted by R {i} . Previous work [START_REF] Wilhelmi | Collaborative spatial reuse in wireless networks via selfish multi-armed bandits[END_REF] have showed that considering such selfish rewards may have a positive but limited impact on the WLANs performance. Therefore, we introduce a more altruistic reward, denoted by R i that accounts not only for the "selfish" reward of AP i (i.e., R {i} ) but also for the rewards of the surrounding APs (i.e., the ones in N i ). The "altruistic" local reward of AP i is computed as:

R i (x) = j∈N i R {j} (x) |N j | (6)
Note that R i defined with Equation 6 ensures that a configuration x maximizing all the local rewards is also a maximum for the global reward R since we have:

N i=1 R i (x) = R(x).
Proof. This is a straightforward property since

N i=1 R i (x) = N i=1 j∈N i R {j} (x) |N j | (7)
By noticing that i ∈ N j ⇐⇒ j ∈ N i , we can permute the indices and use the definition of R {j} :

N j=1 R {j} (x) |N j | i∈N j 1 = N j=1 k∈s j log T k (x) = R(x) (8)

Local reward maximization

For the sake of clarity and since all variables in this section are relative to an AP i, we often omit the subscript i in the notations. Now that each AP i has its own local reward function, we need a model of the knowledge of AP i about R i in order to find its argmax. We represent the beliefs of AP i about R i by defining a prior distribution on the reward function space with a GP.

In our case, a GP can be defined as a collection of random variables indexed by configurations of APs in

N i : {Y c ; c ∈ C |N i | } such that every finite collection (Y c 1 , • • • , Y cn ) ∼ N (µ, Σ).
Without loss of generality, we assume the GP to have zero mean so that it is entirely determined by its covariance function Σ :

C |N k | × C |N k | → R + . We use X t to denote the t × 2|N k | features matrix gathering the tested configurations (x 1 , • • • , x t )
T and Y t to denote the t × 1 label vector gathering the corresponding local reward values

(R i (x 1 ), • • • , R i (x t ))
T . Given X t and Y t , we can use Equations 3 and 4 to infer the distribution of the reward value for an arbitrary configuration x, R i (x). Since GPs can be used as a prior on a function space, they are useful to solve regression problems as well as maximization tasks. In our case, the AP i uses a GP to model R i and to assist the exploration of promising configurations of the APs in N i , maximizing R i in a Bayesian way.

Choosing the covariance function Σ is a critical step when designing a GP as it determines some key features such as its isotropy and smoothness. Since the reward function, which quantifies the quality of spatial reuse in N i , is likely to exhibit threshold effects, we choose a covariance function that decreases rapidly as the distance between two considered configurations increases. This allows the GP to model the true function instead of considering a smooth approximation of it. Because we have no incentive to prefer any particular direction over another, we let the covariance function Σ(x, x ′ ) depend only on ||x -x ′ || to ensure the isotropy of the GP. This leads us to use a Matérn kernel [START_REF] Marc | Classes of kernels for machine learning: a statistics perspective[END_REF] with parameter ν = 3 2 , which is defined as

Σ(x, x ′ ) = s 2 1 + √ 3||x -x ′ || ρ e - √ 3||x-x ′ || ρ (9)
where s 2 and ρ are two hyperparameters whose values are approximated by maximizing the likelihood of Y t (which is Gaussian) during the learning process.

As discussed before, each AP i faces the exploitation vs exploration dilemma in its attempt to find the optimal configuration. A common way in the MAB framework to appraise a given strategy π is then to consider the cumulative regret Γ (π). In our problem, Γ (π) is expressed with Equation 10for an episode of D steps, since it is expressed as the cumulative sum of the differences between the best reward that the AP can get and R i (π(t)), which is the actual reward obtained at time t for the strategy π.

Γ (π) = D t=1 max x∈C |N i | R i (x) -R i (π(t)) (10)
Minimizing the cumulative regret with GP models is usually performed by defining a strategy π that derives from the maximization of an acquisition function A: π(t) = arg max x∈C |N i | A t (x). However, this assumes that our search space C |N i | is continuous. Since each AP i deals with discrete configurations of APs in N i , we systematically round the recommendation of the GP to the nearest valid WLAN configuration. Many acquisition functions exist, such as Knowledge Gradient (KG) [START_REF] Gupta | Bayesian look ahead one-stage sampling allocations for selection of the best population[END_REF], GP-UCB [START_REF] Srinivas | Information-theoretic regret bounds for gaussian process optimization in the bandit setting[END_REF] or the Expected Improvement (EI) [START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF]. We choose EI over KG (whose computational cost can rapidly become prohibitive) and GP-UCB (which was found to be less efficient in our experiments). The EI acquisition function is expressed

as A t (x) = E [(µ t+1 (x) -max 1≤k≤t R i (x k )) + ] given that X t+1 = (X t , x), Y t+1 = (Y t , R i (x)). Since R i (x) ∼ N (µ(x), σ 2 (x))
, we can derive a convenient closed-form for EI, as shown in Equation 11.

EI(x) = (µ(x) -R * i,t )Φ(Z) + σ(x)ϕ(Z) (11)
with

R * i,t = max 1≤k≤t R i (x k ), Z = µ(x)-R * i,t
σ(x) , Φ and ϕ being respectively the CDF and the PDF of a standard Gaussian distribution.

Then, the AP i can try to maximize Equation 11 by differentiating it and performing a gradient ascent. By applying its strategy π i (t) = arg max x∈C |N i | EI(x) and classical gradient ascent techniques on Equation 11, AP i provides promising configurations for its surrounding APs in N i .

Aggregation of local prescriptions

In the previous sections, we have described how each AP i computes its local reward and relies on its model GP i to explore promising configurations for the APs in N i .

However, more coordination between APs is required. By construction, the collection F = (N k ) 1≤k≤N is a cover of the set of APs in W but not a partition. In fact, if F had only null intersections (i.e., ∀j, k, N j ∩ N k = ∅), then the spatial reuse of the radio channel would already be at its apex and there is no need for improvement. Figure 3 illustrates an example with 5 APs in which the collection F = (N 1 , • • • , N 5 ) exhibits multiple non-null intersections. As a result, most APs will receive a set of different prescriptions for the configuration of their TX PWR and OBSS PD parameters at their next iteration. For instance, AP 1 will receive prescriptions from APs 2 and 4 in addition to its own prescription. Since APs can only test one configuration at a time, one of those prescriptions must be chosen, or preferably, a consensus between them must be reached.

In general, independently maximizing each local reward function is very likely to lead to a sub-optimal situation since, for non-linear optimization problems, individual interests are often not aligned with the global interest (e.g., the famous Tragedy of the Commons [START_REF] Hardin | The tragedy of the commons[END_REF]). Without more information on the relation between the configuration of the APs and the measured throughputs of STAs, it seems difficult to provide an expression for the maximal argument of the global reward function R given the maximal argument of the local reward functions R i . However, recall that N i=1 R i (x) = R(x). We can leverage this property to provide guarantees. Theorem 1. Let {R i } 1≤i≤N be a set of Lipschitzian functions, w ∈ [0, 1] N , ||w|| 1 = 1 be a weight vector and {x i j } i∈N j be the prescriptions received by the AP j, with x i = arg max x∈C |N i | R i (x), then the weighted marginal median of the received sets of prescriptions is a maximin optimum x of N i=1 w i R i (x N i ):

xj = median { x i j , w i } i∈N j (12)
Proof. Let x i be the prescription of AP i, with x i j the prescription of AP i for AP j and w ∈ [0, 1] N a normalized weight vector (||w|| 1 = 1). Consider the quantities

R * = N i=1 w i R i (x i ) and R = N i=1 w i R i (x N i ) for a given consensus x ∈ C N . If x i = arg max x∈C |N i | R i (x), then it follows that ∀x ∈ C N , R * -R > 0.
We want to minimize this difference. If all the functions R i are λ-Lipschitzian, we have:

R * -R = N i=1 w i R i (x i ) - N i=1 w i R i (x N i ) = N i=1 w i (R i (x i ) -R i (x N i )) ≤ λ N i=1 w i ||x i -xN i || 1 ≤ λ N i=1 w i j∈N i D d=1 |x i j,d -xj,d | = Ψ (x) (13)
by expliciting the L 1 norm over the D dimensions of C. By rearranging the indices and splitting the absolute values, we have:

Ψ (x) = λ D d=1 N j=1 i∈N j w i |x i j,d -xj,d | = λ D d=1 N j=1      i∈N j x i j,d <x j,d w i (x j,d -x i j,d ) + i∈N j x i j,d ≥x j,d w i (x i j,d -xj,d )      (14)
We want to minimize Ψ (x). This is equivalent to finding x so that ∇Ψ is 0, which can be written as, ∀(j, d) ∈ 1, N × 1, D :

∂Ψ ∂ xj,d = λ      i∈N j x i j,d <x j,d w i - i∈N j x i j,d ≥x j,d w i      = 0 (15)
The only value of xj,d which ensures this relation is the weighted median of the sample {x i j,d } i∈N j , provided that the weights cancel out. It remains to verify that this critical point is a minimum by considering the coefficients of the Hessian matrix. To this aim, we can rewrite the partial derivative of Ψ :

∂Ψ ∂ xj,d = -λ i∈N j w i sign x i j,d -xj,d = -λ i∈N j w i 2H 1 2 x i j,d -xj,d -1
with H1

2 the Heaviside function. Then, we obtain the Hessian matrix coefficients:

∂Ψ ∂ xj,d ∂ xj ′ ,d ′ = 0 if (j, d) ̸ = (j ′ , d ′ ), 2λ i∈N j w i δ x i j,d -xj,d otherwise. ( 16
)
with δ the Dirac impulsion. These coefficients define a positive definite matrix if xj,d ∈ {x i j } i∈N j , which is necessarily the case if xj,d is a weighted median of the sample. Note that we must not create new data: if two elements are eligible to be the weighted median, we must select one of them and not taking the average between the two.

Since the weighted marginal median of the prescriptions minimizes the upper bound of R * -R, it is a maximin optimum for R = N i=1 w i R i (x N i ).

Since, by definition of the local reward functions, N i=1 R i (x) = R(x) (see Eq. 8), we can apply Theorem 1 with, ∀i ∈ 1, N , w i = 1 N and state that taking the marginal median of the prescriptions is a good way to reach high values of the global reward R.

Algorithm and complexity

Algorithm 1 summarizes the main steps of our proposed strategy INSPIRE run on each AP of the WLANs.

Contrary to what one might think, the most resource intensive operation in Algorithm 1 is not the inversion of the Σ(X t , X t ) matrix. Since, at step t, we already know the Cholesky decomposition of Σ(X t-1 , X t-1 ) = LL T , the Cholesky decomposition of Σ(X t , X t ) is easily obtained. The most resourceintensive operation is the maximization of the acquisition function through gradient ascent. Since this requires computing many matrix-vector multiplications for at most m steps, the computational complexity of Algorithm 1 at time t is O (mt 2 ). Broadcast x i to APs in N i

5:

Receive the prescriptions x j i from AP j, j ̸ = i, j ∈ N i 6:

Compute the consensus xt+1 i with Equation 127:

Test x t+1 i for ∆t seconds and compute its selfish reward R {i} with Equation 5 applied only to AP i

Add the pattern xt+1

N i , R i to GP i 12: end while It is worth noting that the dimensionality of the problem (i.e., dim(C

|N i | ) = |N i | dim C)
does not appear in the expression of the asymptotic computational complexity of INSPIRE. This interesting property results from the use of a kernel function by GPs to compare WLANs configurations. This gives INSPIRE the ability to handle arbitrarily dense WLANs, or to optimize more parameters than just TX PWR and OBSS PD, without taking a hefty toll on its execution time. In fact, the real burden to the execution time of INSPIRE is the optimization step t. This compels us to bound the size of X t and to find a balance between the amount of collected data on the WLANs' performance and configuration and a quick execution time.

In practice, to limit the computational burden, we can apply a windowing method (e.g. sliding window) to bound the size of X t and so the computational complexity of INSPIRE. We denote this new version by INSPIRE LIM and we evaluate the impact of limiting the size of the data set on the empirical performance of the proposed solution in the following section.

Performance Evaluation

Experimental settings

To evaluate the ability of INSPIRE at improving the spatial reuse of a radio channel through the configuration of the TX PWR and OBSS PD parameters, we consider two distinct scenarios.

The first scenario is inspired by the WLAN deployment of Cisco in their offices in San Francisco. In [START_REF]Cisco. High density wi-fi deployments[END_REF], Cisco provides the location of 60 APs that together deliver wireless connectivity to their employees on a floor. To account for the WLANs' activity from other floors, we consider a three-floor building and we replicate on each floor the same arrangement of APs as in Cisco's offices. This leads us to a total number of 180 APs spanned over three floors. Assuming 18 independent radio channels, we run a radio channel allocation algorithm to determine the radio channel used by each AP. For our first scenario, we consider the subgraph resulting from the channel allocation with the highest density. We use T1 to refer to this topology (i.e., arrangements of APs and STAs), which is illustrated in Figure 4a. T1 exhibits a total of 10 APs and we associate a number of 5 STAs per AP.

The second scenario addresses the case of many single-AP WLANs deployed and operated independently in a relatively limited area. This is typically the case in housing units where each apartment is equipped with its own AP so that the APs are often only a few meters away from a number of others. More specifically, we consider a nine-story building with 216 apartments of 25 m² each. We randomly position an AP within each apartment as well as 4 STAs per AP. Then, similarly to the first scenario, we apply a radio channel allocation algorithm given a total of 18 radio channels, to obtain the topology of interest denoted by T2. Note that T2 consists of 14 APs and 56 STAs. Figure 4b depicts the topology T2.

For each scenario, we consider heavily loaded conditions. APs attempt to transmit frames to each of their associated STAs at a rate of 50 Mbps while the latter attempt to send their frames to the AP at a lower rate of 3.33 Mbps. These assumptions are in line with the downstream traffic largely exceeding the upstream traffic in WLANs. Given the speed of wireless links in 802.11ax, the buffers of the APs will always be full of frames waiting to be sent. More generally, considering APs in saturation represents undoubtedly the most difficult case when dealing with the spatial reuse of a radio channel. Therefore, if INSPIRE manages to significantly improve the WLANs' performance under these circumstances, then it can only do better under normal

conditions.

To better appraise the quality of INSPIRE, we also consider a "control strategy", several state-of-the-art solutions which were discussed in Section 2 as well as a version of INSPIRE that caps the amount of collected data with a windowing method. All the considered approaches are briefly summarized below:

• DEFAULT: Every AP keeps its default configuration for the TX PWR and OBSS PD parameters (i.e., (-82, 20) dBm);

• WCNC'15: Each AP implements a simple distributed algorithm to dynamically update its OBSS PD parameter [START_REF] Shahwaiz Afaqui | Evaluation of dynamic sensitivity control algorithm for ieee 802.11ax[END_REF];

• JNCA'19: Each AP solves a MAB problem using Thompson sampling to dynamically update their TX PWR and OBSS PD parameters [START_REF] Wilhelmi | Potential and pitfalls of multiarmed bandits for decentralized spatial reuse in wlans[END_REF];

• MSWiM'21: Similar to JNCA'19, except that the sampling of new configurations is performed through a multivariate Gaussian mixture, and that the solution is centralized [START_REF] Bardou | Improving the spatial reuse in ieee 802.11ax wlans: A multi-armed bandit approach[END_REF];

• ADHOC'23: Similar to MSWiM'21, except that the sampling of new configurations is performed through a mixture of hyperspheres [START_REF] Bardou | Improving the spatial reuse in ieee 802.11ax wlans: A multi-armed bandit approach[END_REF];

• INSPIRE LIM: Similar to INSPIRE, except that only the last 50 observations are considered to make predictions.

We implemented INSPIRE (based on the open-source Gaussian process library LibGP [START_REF] Blum | Libgp. GitHub[END_REF]) as well as the six strategies described above in the opensource network simulator ns-3 [START_REF]The Network Simulator ns-3[END_REF]. ns-3 is a well-established realistic discreteevent simulator that implements most of the network protocols involved in the WLANs communication from the Physical up to the Application layer. We report in Table 2 the simulation parameters used in the rest of this section. Unlike previous works (e.g., [START_REF] Bardou | Improving the spatial reuse in ieee 802.11ax wlans: A multi-armed bandit approach[END_REF][START_REF] Shahwaiz Afaqui | Evaluation of dynamic sensitivity control algorithm for ieee 802.11ax[END_REF][START_REF] Ak | Fsc: Two-scale ai-driven fair sensitivity control for 802.11ax networks[END_REF][START_REF] Wilhelmi | Collaborative spatial reuse in wireless networks via selfish multi-armed bandits[END_REF][START_REF] Wilhelmi | Potential and pitfalls of multiarmed bandits for decentralized spatial reuse in wlans[END_REF])) with the exception of [START_REF] Lee | Lsr: Link-aware spatial reuse in ieee 802.11ax wlans[END_REF], our simulations incorporate the mechanism of rate adaptation that let APs and STAs dynamically vary the speed of their wireless links (through the use of different Modulation Coding Scheme (MCS)) in response to the quality of the received signal. This is particularly important for the sake of our study since changing the value of TX PWR necessarily affects the quality of the received signal and thus the MCS. Since our simulated WLANs take place in buildings, we choose an appropriate path loss by combining the models ItuR1238 and InternalWallsLoss, both implemented by ns-3. With these propagation models, the signal is decreased by an additional attenuation coefficient each time it goes through a floor or a wall. The attenuation coefficients are respectively -4 dBm (which is the default value in ItuR1238) and -8 dBm.

We instrumented ns-3 to collect and compute a number of performance metrics. At the end of each iteration, the quality of the spatial reuse is assessed with Equation 5, although distributed strategies internally use the local reward defined in Equation 6. Then, we compute the classical metric to analyze the efficiency of a strategy at dealing with a MAB problem: (i) The cumulative regret (with Equation 10 using a normalized version of the global reward in Equation 5). We also collect the following performance metrics to reflect the effect of each strategy on the behavior of the WLANs and of their STAs: (ii) The execution time of the solution in seconds, on a logarithmic scale, (iii) The number of starving STAs, which we define as STAs experiencing a very low throughput (namely, less than 10% of their attainable throughput) and (iv) The cumulated throughput, which simply sums all STAs' throughput.

Each simulation lasts 30 seconds and we replicated them independently 22 times to obtain and visualize their first, second, and third quartiles. When the quartiles of a performance metric vary too much within a single simu-

Wi-Fi Manager

IdealWifiManager lation, we apply an exponential moving average (with α = 0.04) to extract the underlying trends of the quartiles sequences. The metrics are collected throughout the whole duration of the simulation. At the end of each iteration, we compute all the performance metrics and then we refer to the current strategy to decide what will be the next configuration of the WLANs. Since an iteration lasts ∆t = 75 ms and a simulation lasts 30 seconds, the quality of each solution is assessed over 400 iterations.

Numerical results

Figure 5 illustrates the performance metrics delivered by the ns-3 simulator for each strategy in the case of topology T1. The cumulative regret, represented in Figure 5a, indicates which strategy has performed the best at any time of the simulation. INSPIRE is found to be the most efficient strategy, reducing the cumulative regret by 70% more than DEFAULT and by over 45% than ADHOC'23, which happens to be the most efficient state-of-the-art strategy. We now look at the other performance metrics to better understand how much INSPIRE is able to improve the behavior of the WLAN and of its STAs. Taking DEFAULT as a baseline, Figure 5b shows that INSPIRE reduces the number of STAs in starvation by 80% while Figure 5c demonstrates that our proposed solution manages to increase the cumulated throughput (+400%). Considering Figure 5d, which reports the response time of the We now turn to the case of topology T2. First, we observe in Figure 6a that among the six considered strategies, INSPIRE is the one that manages to decrease the most the cumulative regret with a decline of about 36% compared to the DEFAULT configuration at the end of the simulation. The proposed solution also outperforms WCNC'15, which is found to be the best state-of-the-art strategy on this topology, by a margin of 14%. Looking at the performance of WLANs and of their STAs, Figure 6b shows that INSPIRE is able to limit the number of STAs starving from throughput by a degree of 36% when compared to the DEFAULT configuration. Similarly, the cumulated throughput of STAs have their value increased by 28% and nearly doubled with INSPIRE (see Figure 6c). Although the dimensionality of a configuration is larger for T2 than for T1 (28 against 20), the response time of INSPIRE is virtually equivalent for the two topologies and remains around a tenth of a second. This is because the additional time required to compare higher-dimensional configurations is marginal compared to the additional time required to make inferences on a larger dataset (see Section 3.5).

Eventually, note that INSPIRE LIM is slightly outperformed by INSPIRE, but remains roughly equivalent to the best considered state-of-the-art strategy.

Overall, through the study of topologies T1 and T2, INSPIRE demonstrates its superiority over the other state-of-the-art strategies. Its caped version, INSPIRE LIM, also demonstrates its empirical competitiveness. The significant improvements brought by our proposed solution on all performance metrics are permanently obtained after 100 iterations only (corresponding to 7.5 seconds of simulated time). In other words, in less than 10 seconds, INSPIRE manages to significantly improve the behavior of the WLANs and of the associated STAs thanks to a better spatial reuse of the radio channel. This efficiency in searching and finding an adequate configuration of the TX PWR and OBSS PD parameters at each AP of the WLANs mostly results from the distributed, altruistic use of Gaussian processes which we further discuss in the next section.

Discussion

Seemingly similar problems with vastly different complexity

The topologies T1 and T2 may look similar, but INSPIRE performed differently on each of them. By the end of the optimization process (i.e., 400 steps), the performance metrics for T1 were improved by at least 70% from their initial values under the DEFAULT configuration, and only 7 STAs (representing 14% of the STAs) were still starving from throughput. In case of T2, the progress was less with 14 STAs (representing 25% of the STAs) remaining in starvation. This difference results from the location of STAs relatively to the APs. Looking at Figure 4, it appears that STAs in T2 are further from their associated AP than the ones in T1. As a consequence, STAs are also closer to a concurrent AP in T2 than in T1. Indeed, while STAs in T1 are on average 10 times closer to their associated AP than to a concurrent AP, this ratio drops to an average value of 4 for STAs on T2. With STAs closer to concurrent APs, the spatial reuse problem becomes more difficult. As a matter of fact, to reach its associated STA, the AP must transmit at a greater power, increasing its chance to cause interference to the surrounding APs. Similarly, STAs that are far away from their AP are significantly affected by the transmissions of concurrent APs.

To verify that T2 constitutes a more complex example than T1, we examine the shape of the reward function in both cases. Because of the high dimensionality the reward function and the lack of a closed-form expression, we resort to a slicing technique to provide a visualization of the reward function in Equation 5. We postpone to Appendix A the details of this slicing. Figure 7 illustrates the obtained random slices in case of T1 and T2. Interestingly, Figure 7a shows that INSPIRE succeeded to find a configuration that is maximal in this (random) slice for the case of T1. We also notice that many configurations of equivalent efficiency exist, which also tends to ease the search for an adequate configuration. Conversely, in the case of T2, INSPIRE does not find the best configuration since the slice of Figure 7b shows that, at least, a 6% better configuration exist. Nonetheless, INSPIRE was able to find an efficient configuration.

Benefits of distributed prescriptions

Eventually, to justify our choice of letting APs exploit only local information and prescribe network configurations to their surrounding APs, we consider alternative versions of INSPIRE: (i) GPs w/o agg. where each AP keeps using the local, altruistic reward of Equation 6 but does not aggregate local prescriptions, and prescribes only for its own configuration and (ii) Single GP, a centralized version of INSPIRE where a single GP has a complete knowledge of the WLANs and decides on the configuration of every AP. We compare these alternative strategies with INSPIRE and the DEFAULT strategy by considering their cumulative regret on topologies T1 and T2 in Figure 8. Figure 8a shows that, on T1, the alternative strategies have a cumulative regret 46% lower than DEFAULT. We start by providing a visualization for the configurations found by INSPIRE for each AP in T1 and T2 in Figure 9. We observe that, for a large majority of APs, the larger OBSS PD, the lower TX PWR (with a correlation coefficient of -0.47 and -0.86 for T1 and T2, respectively). Interestingly, this is somehow in agreement with a constraint in the 802.11ax amendment [START_REF]Ieee standard for information technology-telecommunications and information exchange between systems local and metropolitan area networks-specific[END_REF] that enforces the following relation.

OBSS PD ≤ max(-82, min(-62, -82 + (20 -TX PWR))).

(

) 17
The boundary of the constraint described in Equation 17 is depicted with a red dashed line on Figure 9. Interestingly, although Equation 17was not included as a constraint in our optimization task, the vast majority of APs configured with the configurations recommended by INSPIRE satisfy it.

We now study the relationships between the centrality of an AP in WLAN topology and its recommended configuration for TX PWR and OBSS PD. We consider the eigenvector centrality [START_REF] Mohammed J Zaki | Data mining and analysis: fundamental concepts and algorithms[END_REF], defined as the normalized eigenvector associated with the largest eigenvalue of the adjacency matrix of the conflict graph of the considered topology (displayed in Figure 4). An AP with an eigenvector centrality close to 1 is in conflict with a lot of APs, which are themselves in conflict with many APs. Conversely, an AP with an eigenvector centrality close to 0 means little to no conflicts with APs which are themselves isolated.

Figure 10 reports the results. We start by considering the TX PWR of the APs with respect to their eigenvector centrality for T1 and T2, in Figures 10a and10b respectively. It appears that the more an AP is central, the lower its TX PWR. However, the low magnitude of the correlation coefficients (-0.03 and -0.32 for T1 and T2, respectively) indicates that there is maybe more to understand about the found values for TX PWR of the APs. Conversely, regarding the OBSS PD reported in Figures 10c and10d, a stronger relationship can be observed with the eigenvector centrality of an AP. There is a clear positive correlation between the two quantities (0.51 and 0.56 for T1 and T2, respectively), meaning that the more an AP is central, the larger its OBSS PD.

Overall, these results confirm that the optimization of the AP configuration is not an easy problem since no systematic, strong (i.e. with a magnitude close to 1) correlations between the optimal values of TX PWR, OBSS PD and the considered graph properties of the WLAN topology seem to exist. Note that we considered other definitions of centrality, which ended up exhibiting lower correlations with the recommended configurations. To conclude, as suggested by Figure 10, our results indicate that, in general, the more central an AP, the higher its OBSS PD and, to a lesser extent, the lower its TX PWR.

Conclusions

In this work, we have presented INSPIRE, a bayesian optimization method to improve the spatial reuse of radio channels in WLANs by configuring two parameters of APs: the transmission power (TX PWR) and the sensitivity threshold (OBSS PD), which can be dynamically configured with the latest Wi-Fi amendments. To address the difficult problem of sharing efficiently and fairly the resource of a radio channel, INSPIRE works as a distributed solution where each AP solves a local Multi-Armed Bandit problem with the help of information and actions limited to its surrounding APs (i.e. within its communication range). The development of the solution includes (i) an intuitive quantification (based on STAs throughputs) of the "goodness" of a configuration of TX PWR and OBSS PD for concurrent APs in WLANs, both at local and global scales, (ii) the use of an acquisition function and Gaussian processes to find local configurations that maximize approximations of local reward functions, and (iii) an altruistic behavior facilitated by prescriptions to surrounding APs along with a consensus method which aggregates the prescriptions of surrounding APs for the "greater good" of the WLANs.

We show that INSPIRE can be easily extended to a version working with a limited number of observations (through the use of a sliding window) to limit its computational complexity while only slightly affecting its ability to find an efficient AP configuration.

INSPIRE has been evaluated and compared with other state-of-the-art strategies addressing the same problem, using the open-source network simulator ns-3 that implements all the layers of the network stack. The different strategies were compared on two examples inspired by real-life deployments of dense WLANs in both professional and domestic environments, representing two optimization tasks of different complexities. INSPIRE was found to outperform other state-of-the-art strategies by significantly reducing the number of STAs in starvation and increasing the cumulated throughput of the WLANs in only a few seconds. Interestingly, it has also demonstrated significantly better empirical performance than its centralized counterpart, which brings another practical example of the superiority of a distributed approach over a centralized approach when addressing a complex issue.

As future work, we plan to assess the quality of INSPIRE for a specific class of WLANs where STAs are mobile (e.g., customers in a shopping mall). Another natural follow-up would be to experiment INSPIRE with real material on a testbed.

Figure 1 :

 1 Figure 1: Adequately configuring the TX PWR parameter of APs can significantly improve the spatial reuse of radio channels in WLANs.

Figure 2 :

 2 Figure 2: Gaussian Process Regression. The ground truth f (x) is depicted with a grey dashed line and the noisy observations (X, y) with black dots. The posterior mean µ t (x) is shown with a solid orange line and the posterior standard deviation σ t (x) as an orange interval.

  in several ways: (i) It studies the response time of the INSPIRE algorithm; (ii) It introduces a lightweight version of INSPIRE where only the latest observations (50 in our case) of the WLAN performance are considered; (iii) It sheds light on why seemingly similar network topologies may actually result into optimization problems of different complexity; (iv) It correlates the best values found for TX PWR and OBSS PD at each AP with metrics related to the network topology.

Figure 3 :

 3 Figure 3: A WLAN represented by a graph with APs depicted as labelled triangles and STAs as black dots. An edge exists between two APs when they are in the communication range of each other. We use different colors to illustrate the surroundings of each AP in F.

Algorithm 1 1 : 3 :

 113 INSPIRE run at each AP i Input: subset N i of APs Initialize the Gaussian Process GP i 2: while true do Find a prescription x i = arg max x∈C |N i | EI t i (x) by gradient ascent 4:

8 : 9 : 10 :

 8910 Broadcast R {i} , |N i | and xt+1 i to APs in N i Receive R {j} , |N j | and xt+1 j from AP j, j ̸ = i, j ∈ N i Compute the local reward R i with Equation 6 and the local configuration xt+1 N i 11:

  (a) Topology T1. (b) Topology T2.

Figure 4 :

 4 Figure 4: The two considered topologies. APs are shown as red triangles, connected by a arrow if they lie in each other's communication range. Associated STAs are shown as dots colored according to their throughputs: warm, yellowish colors indicate that the STA has enough throughput most of the time while, on the contrary, cool, blueish colors indicate that the STA has mostly not enough throughput under the default configuration of 802.11: 20 dBm for TX PWR and -82 dBm for OBSS PD.

Figure 5 :

 5 Figure 5: Performance metrics delivered on topology T1 by each strategy.

Figure 6 :

 6 Figure 6: Performance metrics delivered on topology T2 by each strategy.

  Figure 7a suggests a relatively smooth reward function in T1. On the other hand, the reward function in the case of T2 is much more erratic, featuring a lot of local maxima as shown by Figure 7b.

  (a) Slice for T1, with the best solution found by INSPIRE at (0, 0) and two scaled random unit vectors. The maximum of the slice is shown with a red circle. (b) Slice for T2, with the best solution found by INSPIRE at (0, 0) and two scaled random unit vectors. The maximum of the slice is shown with a red circle.

Figure 7 :

 7 Figure 7: Random slices of the global reward function for the topologies T1 and T2.

  However, INSPIRE manages to reduce its cumulative regret by an extra 25%. Despite the greater complexity of the function in T2, this extra reduction factor persists but at a value of 13% as shown in Figure 8b. Given the significant gap between INSPIRE and GPs w/o agg., it is clear that prescribing for surrounding APs and aggregating those prescriptions leads to a more altruistic behaviour, which in turn brings additional benefits at the scale of the WLANs. More surprisingly, INSPIRE outperforms its centralized counterpart Single GP. At first glance, this is counter-intuitive since Single GP has a complete knowledge and control over the APs of the WLANs. However, Single GP involves a single agent to optimize a function of high complexity, which has to deal with K|C| variables. Decentralization breaks down this task into simpler local optimization problems of lower dimension. With INSPIRE, each AP k only deals with |N k ||C| variables, which is significantly less than K|C| for large WLANs. Overall, APs run by INSPIRE solve simpler optimization problems and behave altruistically by ensuring a consensus with surrounding APs. By doing so, they manage to improve the spatial reuse of the radio channel at the scale of the WLANs.

  (a) Cumulative regret on topology T1.(b) Cumulative regret on topology T2.

Figure 8 :

 8 Figure 8: Comparison of the cumulative regrets of the different versions of INSPIRE on T1 and T2.

  (a) Recommended configuration for T1. (b) Recommended configuration for T2.

Figure 9 :

 9 Figure 9: Recommended configurations according to INSPIRE for each AP in topologies T1 and T2. Each dot represents an AP. An AP violates the constraint described in Equation 17 when it is located above the dashed red line.

  (a) TX PWR for each AP w.r.t. its eigenvector centrality in T1. (b) TX PWR for each AP w.r.t. its eigenvector centrality in T2. (c) OBSS PD for each AP w.r.t. its eigenvector centrality in T1. (d) OBSS PD for each AP w.r.t. its eigenvector centrality in T2.

Figure 10 :

 10 Figure 10: Visualization of the recommended configurations w.r.t. the eigenvector centrality of the APs in topologies T1 and T2. Each dot represents an AP.

Table 1 :

 1 Comparison of the state-of-the-art data-driven strategies. The last column refers to the size of the scenarios involved in the performance evaluation of the strategy. For instance, 216/18 means the evaluation comprises 216 APs distributed over 18 radio channels.

	Traffic Simulator APs /	Up/Down channels	Up Self-made 100/3	Down ns-3 6/1	Up/Down ns-3 3/1	Down Self-made 8/1	Down Self-made 8/1	Down ns-3 10/1	Up/Down ns-3 216/18	Up/Down ns-3 216/18
	Dynamic	MCS
	Degree of centralization	Pull area Push area	Associated STAs Associated STAs	AP itself AP itself	All APs All APs	AP itself AP itself	All APs AP itself	All APs All APs	All APs All APs	Surrounding APs Surrounding APs
	Tuning of	TX PWR
	Proposed	solution	WCNC'15 [14]	WCNC'21 [15]	Globecom'20 [11]	ADHOC'19 [16]	JNCA'19 [4]	MSWiM'21 [12]	ADHOC'23 [13]	INSPIRE

Table 2 :

 2 ns-3 parameters.

	Parameter	Value
	ns-3 version	3.31
	Number of repetitions	22
	Simulation duration	30 s
	Duration of an iteration (∆t) 75 ms
	Packet size	1,464 bytes
	Downlink traffic	50.0 Mbps
	Uplink traffic	3.33 Mbps
	Channel size	20 MHz
	Frequency band	5 GHz
	A-MDPU Aggregation	4
	Path loss	HybridBuildings (ItuR1238 +
		InternalWallsLoss)

Acknowledgements

This work was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR).

Appendix A. Building of the reward slices

We detail in this appendix how we build the random slices of the reward function showed in Figure 7. For the sake of completeness, we also provide the numerical values of the unit vectors of our (randomly) selected bases.

For visualization purposes, we build a random vector basis in two dimensions. Considering the best configuration c * found by INSPIRE on a given topology, we uniformly draw two random configurations û, v ∈ C K and use (c * , û, v) as a vector basis. By plotting the reward of each configuration attainable through this basis (i.e., any valid configuration c that can be expressed as c = c * + xû + yv, (x, y) ∈ R 2 ), we obtain a 3d plot representing a random slice of the considered reward function, with the best configuration found at the origin of the plot.

Details of the vector basis for topology T1 (Figure 7a) are as follows:


Jin Yan
email: jin.yan@eurecom.fr

Jérôme Härri
email: jerome.haerri@eurecom.fr

MCS Analysis for 5G-NR V2X Sidelink Broadcast Communication

Leveraging Modulation and Coding Schemes (MCS) in 5G New Radio (NR) Sidelink represents one key strategy to provide sufficient capacity required by future 5G for Vehicle-to-Everything (V2X) services for intelligent vehicles. Early studies either directly adopt the previously optimised QPSK 1/2 by 802.11p/C-V2X or suggest an optimal MCS value under a particular context. In this paper, we identify a MCS value optimal under any context, by evaluating the impact of MCS on V2X broadcast communication considering multiple varying parameters (e.g. variable packet size, transmit rate or density) representative of different 5G V2X services.

I. INTRODUCTION

Vehicle-to-everything (V2X) communication is a key paradigm for advanced driver-assistance systems, providing intelligent vehicles with complete awareness of their environments, while enabling interactions with other vehicles, pedestrians, roadside units or any connected devices. Introduced in 3GPP 5G NR rel. 16, V2X 5G NR Sidelink enables 5G low latency, high capacity and reliability V2X communication.

As described by Garcia et al. [1], most V2X services and applications for intelligent vehicles are based on a broadcast communication paradigm considering ad hoc resource allocations. Accordingly, performance of 5G NR V2X communications strongly depends on predefined transmit parameters. While most of them are set in V2X profiles either for WiFi or Cellular V2X (e.g. ETSI EN 302 571 [2]) which is popularly adopted by recent studies (e.g. [3]), these parameters are not yet formally defined for 5G NR V2X. Modulation and Coding Schemes (MCS) are of particular interest, as transposing default values from WiFi-or LTE-V2X directly into the innovative 5G NR physical layer will lead to inefficient V2X channel usage.

The seminal work from Jiang et al. [START_REF] Jiang | Communication Density: A Channel Load Metric for Vehicular Communications Research[END_REF] identified QPSK (1/2) as an optimal MCS value for WiFi V2X, a MCS value widely adopted in all standards and subsequent researches. Quite interestingly, LTE V2X adopted an equivalent MCS value without investigation. Ali et al. [START_REF] Ali | 3GPP NR V2X Mode 2: Overview, Models and System-level Evaluation[END_REF] performed the first system-level evaluation of an optimal MCS value for 5G NR V2X broadcast communication, leading to 16 QAM 1/2 (MCS-14). However, their study was limited to a particular cooperative awareness context considering constant and rather low transmit rate and packet sizes. Future 5G V2X services are expected to operate under more stringent conditions, with various and deeply intervened V2X messages and services offering V2X communication conditions with highly dynamic packet sizes and transmit patterns, leading to the impossibility of identifying a particular context.

In this paper, we investigate the optimal 5G NR Sidelink V2X MCS value under a more stringent set of parameters than [START_REF] Ali | 3GPP NR V2X Mode 2: Overview, Models and System-level Evaluation[END_REF], as will be required by 5G V2X services in the future. Specifically, we evaluate the impact of the MCS value on the performance of 5G V2X broadcast communication as a function of three parameters: (i) message transmit rate, (ii) transmit density, (iii) packet size. Our objective is not to identify various optimal MCS values as a function of a particular context, but a globally optimal MCS value applicable to any context.

The rest of the paper is organized as follows: Section II introduces 5G NR V2X and MCS. Section III describes the methodology used in this study, while Section IV provides evaluation results. Section V summarizes the key findings.

II. BACKGROUND

A. 5G NR V2X Sidelink Overview

Sidelink (SL) is an extension of 5G NR communication [1] to support device-to-device (D2D) communication. Defined in 3GPP since LTE rel. 12 for Proximity Services (ProSe), SL has been specified for 5G NR in rel. 16 for V2X communication. Further releases (rel.17 and 18) define SL enhancements, aiming to support more stringent requirements and operation scenarios for V2X or ProSe, such as wider coverage, reliability improvement, latency reduction or power saving [START_REF] Harounabadi | V2X in 3GPP Standardization: NR Sidelink in Release-16 and Beyond[END_REF]. Altogether, a 15 kHz PRB of 1ms corresponds to one slot and represents the default smallest resource unit in 5G NR. 5G NR further defines mini-slots to enable < 1ms transmit delays. Keeping the number of resources per slot equal, minislots proportionally increase the required frequency resources as depicted on Fig. 1. The number of slots available per 5G NR sub-frame vary according to the applied numerology. Finally, a sub-channel is a variable amount of slots (e.g. [START_REF] Zugno | Implementation of a Spatial Channel Model for ns-3[END_REF]12,15,20,25,50,75, or 100) in the frequency domain corresponding to the smallest packet allocation unit for the 5G NR scheduler.

B. 5G NR MCS for V2X SL

3GPP enables flexible 5G NR SL data rates through a numerous MCS values, which are defined on Table 5.1.3.1-1 of TS 138 214 [7] and reproduced on Table I. Impact of various MCS values on a same packet size is illustrated in the bottom right side of Fig. 1. A higher MCS value can effectively compress the packet into fewer sub-channels and consequently improve potential sub-frame packet multiplexing using 5G NR V2X SL schedulers.

The default 5G NR V2X SL mode 2(a) listen-beforetalk (LBT) scheduler is, however, not adapted to PDCPlevel packet multiplexing to fit to sub-channel resources for broadcast communications. Specifically, the scheduler is not capable of differentiating between a fully or partially occupied sub-channel in its resource allocation. If a particular MCS and packet size can optimally occupy all resources belonging to one or more sub-channels, different MCS values and packet sizes lead to a partial sub-channel occupation and to resource wastage as depicted respectively on the right side of Fig. 1. Accordingly, increasing the MCS may reduce the required sub-channels per packet, in turn enabling more channel resources to be distributed to other transmitters, this benefit may be lost if sub-channels end up being partially occupied. As shown on the top right side of Fig. 1, the subframe has enough absolute resources to multiplex T X 1 and T X 3 together in one sub-frame, but the scheduler does not succeed as both T X 1 and T X 3 waste one sub-channel each due to partial sub-channel usage. If the impact of the data rate (i.e MCS) on the performance of 5G V2X broadcast communication is traditionally understood to be sensitive to channel conditions, we can see that for 5G NR V2X SL, it also depends on how efficiently sub-channels are occupied and packets multiplexed.

C. Optimal V2X Broadcast MCS

Most V2X services are based on broadcast communication and on ad-hoc (infrastructure-less) resource allocation ( [1]). Accordingly, vehicles need to select a default V2X MCS. Although V2X MCS has been studied for other radio access technologies over the years, comparatively limited analysis on broadcast MCS 5G NR SL can be found in recent research. Jiang et al. [START_REF] Jiang | Communication Density: A Channel Load Metric for Vehicular Communications Research[END_REF] suggested a QPSK 1/2 MCS value as optimal for ITS-G5/DSRC technology. Although widely adopted by subsequent studies, standards, and even for the Without loss of generalities, we consider a simple two-lane traffic topology, where all transmitting vehicles are located at a configurable uniform inter-distance on the first lane, the receiving vehicle is located at the center of the topology on the second lane, an average communication performance is to be examined on receiving vehicle from all transmitters. Mobility is not considered in this study, assuming either an static or a mutually static topology. We evaluate MCS values considering three varying representative parameters: message transmit rate, the number of transmitters, and the packet size.

As we want to avoid the impact of other parameters than the previous three on the evaluation of the MCS, we rely on a harmonizing metric called Communication Density and defined as follows:

Dens comm = T x range × M sg Rate Dist v2v (1)
Varying parameters such as transmit range, traffic density, or message rate maintaining a constant communication density will lead to similar broadcast communication performances as discussed in [START_REF] Jiang | Communication Density: A Channel Load Metric for Vehicular Communications Research[END_REF]. A,B,C) and for each level, we consider three topology scenarios (sparse, normal and dense). As the communication density is constant in all three topology scenarios, only the metric under study will impact the MCS performance. In order to validate our methodology, we also add a reference group (Level Reference) configured similarly to [START_REF] Ali | 3GPP NR V2X Mode 2: Overview, Models and System-level Evaluation[END_REF] and designed to verify that we reach the same conclusions under the same conditions.

1) Message Transmit Rate: In this first approach, we fix the total number of transmitters, then adjust their interdistance (Dist v2v ) so as to let communication density be fully determined by the message transmit rate. The topology is depicted in Fig. 2, where light red vehicles are transmitters and the dark blue vehicle is the receiver. We configure our four levels and three scenarios according to parameters depicted in Table II. According to Eq. 1, the three scenarios employ a globally similar communication density.

2) Number of Transmitters: As the density of transmitters plays a key role in the performance of the 5G NR V2X SL mode 2(a) scheduler, in this step, we let the communication density be fully determined by the density of transmitters by adjusting the message transmit rate according to Eq. 1.

As before, we configure the four levels and three scenarios according to parameters depicted in Table III. According to Eq. 1, the three scenarios are categorized by the vehicle density. The methodology is designed to analyse the impact brought by the number of transmitters allocated in each scenario depicted on Fig. 3 within the same or different communication density levels.

3) Packet size: Diverse packet sizes require different number of resources allocated per packet, which may not match the predefined structure as described in Section II. This can typically happen when changing the required number of resources by varying the MCS value. In that case, resources are wasted, and the consequent impact must also be investigated. We propose here a methodology to evaluate the efficiency of fitting to the 5G NR V2X SL subchannel We first introduce a concept of effective utilization ratio ρ as expressed Eq. 2, which represents the ratio between the actual occupied bytes over the total assigned bytes.

ρ = S γ × M (2)
S represents the packet size in bytes, γ denotes the number of required sub-channels per packet, as the larger the packet the more sub-channels are required. M represents the capacity of each resource block within a sub-channel, and is specified by the 5G NR numerology and the MCS. ρ, however, should not be considered alone, as within a 5G NR sub-frame, multiple packets can be multiplexed, especially at higher MCS values and under heavy traffic conditions. Therefore Eq. 3 describes a multiplexing level parameter δ defined as the f loor function of the total number of sub-channels within bandwidth N over γ.

δ = N γ (3)
Combining Eq. 2 and Eq. 3, we can evaluate the exact influence of various packet sizes typically generated by V2X services on the optimal MCS value.

B. Simulation Environment

We perform a system-level analysis relying on the ns3 simulator, enhanced with a 5G NR V2X SL stack [START_REF] Patriciello | An E2E simulator for 5G NR networks[END_REF]. Without loss of generalities, the wireless channel is modelled according to a standard 3GPP Model [START_REF] Zugno | Implementation of a Spatial Channel Model for ns-3[END_REF] for V2X highway communications, we leave a detailed investigation of the impact of more stringent channel conditions to future studies. The major determining parameters are listed in Table IV. We are calculating the average packet reception rate (PRR) among all transmitters as a key performance indicator(KPI) to determine the impact of MCS values. Each result is obtained over an average of 200 simulation runs with random seeds. This section analyses the impact of the message transmit rate on the packet reception rate (PRR) considering four message rate levels described on Table II. Fig. 4(a) shows the reference group considering the exact same parameter setting as in [START_REF] Ali | 3GPP NR V2X Mode 2: Overview, Models and System-level Evaluation[END_REF]. We can confirm that MCS 14 is the ideal modulation scheme value as shown in [START_REF] Ali | 3GPP NR V2X Mode 2: Overview, Models and System-level Evaluation[END_REF] under all scenarios. As the PRR drops after MCS 14 for the sparse scenario, it remains stable for the normal and dense scenarios. This can be explained by a comparatively reduced transmission range in higher traffic densities according to Eq. 1, and an increased traffic capacity offered by higher MCS values mitigating resource exhaustion and packet collisions.

The next three figures of Fig. 4 consider different communication conditions than [START_REF] Ali | 3GPP NR V2X Mode 2: Overview, Models and System-level Evaluation[END_REF]. On Fig. 4(b), the message transmit rate is slightly increased, moving the optimal MCS value to MCS 20. Although a higher transmission rate generates a higher chance of collision, the MCS 24 does not provide a significantly reduced performance compared to MCS 20, in particular with normal or sparse scenarios. When increasing the message transmission rate, we can observe that a consistent optimal PRR is reached by MCS 24, as illustrated in Fig. 4(c) and Fig. 4(d). The improved performance is particularly visible for normal to sparse scenarios, which corresponds to a representative traffic context with V2X services generating a large amount of data, such as joint Cooperative Awareness (CA) or Collaborative Perception (CP) services 1 .

All in all, we can observe that only under a low message rate and communication density MCS 14 outperforms other MCS values as shown in Fig. 4(a), confirming results from [START_REF] Ali | 3GPP NR V2X Mode 2: Overview, Models and System-level Evaluation[END_REF]. Under any other scenario, MCS 24 provides consistently better performances than any other MCS value.

B. Transmitter Density Impact

This section analyses the impact of the density of transmitters on the packet reception rate (PRR) considering four communication density levels described in Table III. As before, Fig. 5(a) shows the reference group, considering the exact same parameter setting as in [START_REF] Ali | 3GPP NR V2X Mode 2: Overview, Models and System-level Evaluation[END_REF] and again proves that MCS 14 is the optimal choice under all scenarios. With regards to MCS 14, the PRR achieves an optimum at 0.98 in sparse scenarios, while it reaches only 0.65 for dense scenarios. Higher MCS values degrade the PRR, but this is not significant, as MCS 24 only experiences a PRR reduction between 1% to 3%. The next three figures of Fig. 5 consider an increased and more realistic communication density compared to the comparatively low value modelled by the reference group in [START_REF] Ali | 3GPP NR V2X Mode 2: Overview, Models and System-level Evaluation[END_REF]. Fig. 5(b) shows that with an already mild increase in communication density, a PRR optimum is no longer reached by MCS 14 but rather by MCS 20. Quite interestingly, this outperforms the negative impact of a reduced Signal-to-Noise Ratio (SNR) on higher MCS values due to the increased communication density. This can be explained by a stronger benefit of an improved channel availability for the 5G NR V2X scheduler due to reduced resource requirements by higher MCS values. When increasing the communication density, we can observe that a consistent optimal PRR is reached for MCS 24 as illustrated in Fig. 5(c) and Fig. 5(d). Moreover, normal and dense scenarios outperform sparse scenarios, this is due to the fact that more transmitters are within the reachable range with higher transmission densities, resulting in a better reception rate. When combining the previous analyses, while in certain conditions other MCS values outperform MCS 24, we observe that MCS 24 remains consistently optimum or only experiencing minor loss compared to other MCS values, and it remains definitively better than MCS 14.

C. Packet Size Impact

This section investigates the impact of packet size on optimum MCS values. We considered realistic packet size ranges between 200 bytes and 1500 bytes according to actual V2X packets sizes measured and reported in [12] [START_REF][END_REF]. We applied five MCS indices from Table. I, namely 8, 12, 14 (as reference value), 16, and 24.

Referring to Eq. 2 and considering the basic NR numerology 0, M fully depends on MCS, which means that a higher modulation scheme allows more bytes to be packed into a single resource block. Fig. 6 depicts the impact of the MCS on 5G NR V2X SL resource usage efficiency according to ρ (Eq. 2) and γ (Eq. 3). The zigzagging lines with left side of the y-axis indicators represent ρ, showing periodical changes in the percentage of resource utilization. From these lines we can observe that certain packet sizes and MCS values result in a perfect utilisation of the 5G NR V2X SL subchannels. However, we can also observe from the periodical gradual increase that, for most of other packet size values, resources are wasted by not fully using all resources granted by each 5G NR V2X SL subchannel. Besides the available channel resources not actually being used, this also impacts the performance of the 5G NR V2X SL scheduler, as partially available subchannel resources can not be individually allocated to other transmitters. Still on Fig. 6, the block values representing γ shows the number of occupied 5G NR V2X SL subchannels, as can be read on the right side of the y-axis. A subchannel is considered

TABLE V: MCS impact on 5G NR V2X Multiplexing

To better illustrate this point, a theoretical calculation of packet multiplexing numbers is shown in Table V. As expected, a higher MCS supports more packets being multiplexed than lower MCS. We can observe that MCS 24 enables a 5 packets multiplexing per sub-frame up to 600 bytes, compared to 200 bytes supported by MCS 14. This corresponds to a 200% improvement in capacity and in delay, as more packets can be multiplexed in a 1ms sub-frame.

V. CONCLUSION In this paper, we demonstrate that a 64QAM (MCS 24) corresponds to an optimal modulation and coding rate for 5G NR V2X sidelink broadcast communication, which represents a major change from the previously admitted values.The difference between this study and previous studies is that we considered in our methodology heterogeneous types of transmit characteristics and densities, representing more realistic communication patterns expected for 5G V2X services. This study shows that packets requiring fewer V2X physical resources (with small packets or under higher MCS) permit a more efficient spectral efficiency under 5G NR strict subchannel numerology. In future work, we intend to generalize this study to include higher numerology as well as investigating optimal modulation and coding for NR V2X sidelink groupcast communications.

Fig. 1 :

 1 Fig. 1: 5G NR Sidelink Physical Structure 5G NR SL V2X physical resources are similar to 5G NR and span across the metrics of time and frequency domain. In the frequency domain, the bandwidth is split into 15kHz Physical Resource Blocks (PRB). In the time

Fig. 2 :

 2 Fig. 2: Topology varying the message transmit rate. ... ...

Fig. 3 :

 3 Fig. 3: Topology varying the number of transmitters. ... ...

Fig. 4 :Fig. 5 :

 45 Fig. 4: Message Rate Impact Result

Fig. 6 :

 6 Fig. 6: Sub-channel Occupation over different Packet sizeMultiplexity Capacity within One Sub-Frame P P P P P P MCS Size 200 400 600 800 1000 1200 1400 12 5 2 1 1 1 1 -14 5 2 2 1 1 1 1 16 5 5 2 2 1 1 1 20 5 5 2 2 2 1 1 24 5 5 5 2 2 2 2

TABLE I :

 I Modulation and Coding Scheme Index

	MCS Index	Modulation Order	Target code Rate	Spectral efficiency
	0	2(QPSK)	120	0.2344
	1	2	157	0.3066
	2	2	193	0.3770
	3	2	251	0.4902
	4	2	308	0.6016
	5	2	379	0.7402
	6	2	449	0.8770
	7	2	526	1.0273
	8	2	602	1.1758
	9	2	679	1.3262
	10	4(16 QAM)	340	1.3281
	11	4	378	1.4766
	12	4	434	1.6953
	13	4	490	1.9141
	14	4	553	2.1602
	15	4	616	2.4063
	16	4	658	2.5703
	17	6(64 QAM)	438	2.5664
	18	6	466	2.7305
	19	6	517	3.0293
	20	6	567	3.3223
	21	6	616	3.6094
	22	6	666	3.9023
	23	6	719	4.2129
	24	6	772	4.5234
	25	6	822	4.8164
	26	6	873	5.1152
	27	6	910	5.3320
	28	6	948	5.5547
	Under these conditions, 5G NR MCS 14 appeared to be
	optimal. However, Day-2 V2X scenarios include multiple
	V2X services involving packets of various sizes and more
	stringent topology conditions, which requires a generalized
	investigation.
		III. METHODOLOGY
	A. Topology and Evaluation Design

LTE-V2X technology, recent works (

[START_REF] Sepulcre | Why 6 Mbps is Not (Always) the Optimum Data Rate for Beaconing in Vehicular Networks[END_REF]

,

[3]

) shed light on the potential benefit to increase it. Burbano-Abril et al.

[START_REF] Burbano-Abril | MCS Adaptation within the Cellular V2X Sidelink[END_REF]

propose a dynamic adaptive MCS methodology based on diverse traffic scenarios in order to optimize overall performance for the LTE-V2X technology. Recently, Ali et al.

[START_REF] Ali | 3GPP NR V2X Mode 2: Overview, Models and System-level Evaluation[END_REF]

utilise various MCS values for 5G NR V2X SL under a single packet, fixed packet size, and periodic traffic, corresponding to a Day-1 Cooperative Awareness (CA) service.

TABLE II :

 II Settings for varying the message transmit rateWe therefore investigate in three levels of communication densities (Level

TABLE III :

 III Settings for varying the number of transmitters.

structure as a function of the MCS value.

TABLE IV :

 IV Simulation Baseline Parameters

	Parameter	Value
	Randomness	Seeds: 30; Run: 200
	Performed Frequency	5.89GHz
	Bandwidth	10MHz
	Numerology	0
	subchannel size	10 PRBs
	Available SL symbol per slot	8/14
	Sensing Window	100
	Selection Window	30
	Reservation Period	20
	Re-transmission	Disabled
	Propagation Model	3GPP LoS Channel Model [10]
	Tx Power	23dBm
	Antenna Setting	Array of 1x2 antenna elements
	IV. EVALUATION
	A. Message Rate Impact

At high vehicular density, the vehicular speed would be proportionally reduced, which would in turn also reduce the message generation rate.

ACKNOWLEDGMENT This work has received funding from the European Union's Horizon 2020 R&D program under grant agreement No: 101015405 for Safe4RAIL-3 (https://safe4rail-3.eu/) and No: 957218 for Intelliot (https://intelliot.eu/).