François Baccelli

Pour toute suite stationnaire X=(Xn) de variables aléatoires à valeurs entières avec moyenne m∈[−∞,∞], soit S=(Sn) le processus stochastique dont les incréments sont égaux à X et tel que S0=0. Le graphe des records de X a pour noeuds les éléments de Z et, pour tout noeud i, une arête de (i,Si) à (j,Sj), où j>i est le plus petit entier tel que Sj≥Si, si un tel j existe. Si X est ergodique, alors son graphe des records présente les transitions de phase suivantes lorsque m varie de −∞ à ∞ : pour m0, c’est un arbre à deux extrémités. La distribution de la composante de 0 dans le graphe des records est analysée lorsque X est une suite de variables aléatoires i.i.d. dont la distribution commune a pour support {−1,0,1,…}, ce qui fait de S une marche aléatoire sans discontinuités vers la gauche. Pour cette marche aléatoire, si m0, alors le graphe des records enraciné en 0 est un arbre de Kesten éternel bivarié unimodularisé. Un arbre aléatoire dirigé unimodulaire est dit être représentable par des records s’il est la composante de 0 dans le graphe des records d’une suite stationnaire à valeurs entières. On démontre que chaque arbre dirigé unimodulaire infini ordonné avec une ligne de succession unique est représentable par des records. En particulier, chaque arbre dirigé unimodulaire à une extrémité possède une ligne de succession unique et est donc représentable par des records.

We study the Multiserver-Job Queuing Model (MJQM) with general independent arrivals and service times under FCFS scheduling, using stochastic recurrence equations (SREs) and ergodic theory. We prove the monotonicity and separability properties of the MJQM SRE, enabling the application of the monotone-separable extension of Loynes' theorem and the formal definition of the MJQM stability condition. Based on these results, we introduce and implement two algorithms: one for drawing sub-perfect samples (SPS) of the system's workload and the second one to estimate the system's stability condition given the statistics of the jobs' input stream. The SPS algorithm allows for a massive GPU parallelization, greatly improving the efficiency of performance metrics evaluation. We also show that this approach extends to more complex systems, including MJQMs with typed resources.

Cloud computing data centers handle highly variable workloads: job resource requirements can range from just one or few cores to thousands, and job service times can range from milliseconds to hours or days. This variability significantly limits the maximum achievable utilization of infrastructures. Queuing theory has addressed the study of these systems with the definition of the Multiserver-Job Queuing Model (MJQM), where s identical servers are present and job n requires α _n of the s servers for a random amount of time σ _n . The α _n servers are occupied and released simultaneously. Unfortunately, despite its simple formulation, the MJQM remains elusive. For example, the MJQM stability condition has been derived only in particular cases. As a consequence, even applying Discrete-Event Simulation (DES) under high load becomes challenging, because stability cannot be determined a priori. In this paper, we analyze the MJQM with general independent arrival processes and service times under FCFS scheduling, using stochastic recurrence equations (SREs) and ergodic theory. Starting from the definition of the MJQM SRE, we prove the monotonicity and separability properties that allow us to apply an extension of Loynes' theorem, known as the monotone-separable framework, and formally define the MJQM stability condition. From these results, we introduce and implement two algorithms: the first one is used to draw sub-perfect samples (SPS) of the system's workload and the second one estimates the system's stability condition given the statistics of the jobs' input stream. The nature of the SPS algorithm allows for a massive GPU parallelization, thus greatly improving the efficiency in the estimation of performance metrics. The algorithm for the estimation of the stability condition solves an important problem for the analysis of MJQMs. We also define new metrics that capture the synchronization loss in MJQM systems and we show how these metrics can be efficiently evaluated using the SRE approach. Finally, we show that the approach proposed in this paper can be extended to more complicated systems, including MJQMs where resources have types.

Providing performance guarantees is one of the critical objectives of recent and future communication networks, toward which regulations, i.e., constraints on key system parameters, have played an indispensable role. This is the case for large wireless communication networks, where spatial regulations (e.g., constraints on intercell distance) have recently been shown, through a spatial network calculus, to be essential for establishing provable wireless link-level guarantees. In this work, we focus on performance guarantees for the downlink of cellular networks where we impose a hardcore (spatial) regulation on base station (BS) locations and evaluate how BS scheduling (which controls which BSs can transmit at a given time) impacts performance. Hardcore regulation is the simplest form of spatial regulation that enforces a minimal distance between any pair of transmitters in the network. Within this framework of spatial network calculus, we first provide an upper bound on the power of total interference for a spatially regulated cellular network, and then, identify the regimes where scheduling BSs yields better link-level rate guarantees compared to scenarios where base stations are always active. The hexagonal cellular network is analyzed as a special case. The results offer insights into what spatial regulations are needed, when to choose scheduling, and how to potentially reduce the network power consumption to provide a certain target performance guarantee.

We study the Multiserver-Job Queuing Model (MJQM) with general independent arrivals and service times under FCFS scheduling, using stochastic recurrence equations (SREs) and ergodic theory. We prove the monotonicity and separability properties of the MJQM SRE, enabling the application of the monotone-separable extension of Loynes' theorem and the formal definition of the MJQM stability condition. Based on these results, we introduce and implement two algorithms: one for drawing sub-perfect samples (SPS) of the system's workload and the second one to estimate the system's stability condition given the statistics of the jobs' input stream. The SPS algorithm allows for a massive GPU parallelization, greatly improving the efficiency of performance metrics evaluation. We also show that this approach extends to more complex systems, including MJQMs with typed resources.

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

The reconfigurable intelligent surface (RIS) technology allows one to engineer spatial diversity in complex cellular networks. This paper provides a stochastic geometry framework for the system-level performance assessment of RIS-assisted networks. To account for the inherent randomness in the spatial deployments of base stations (BSs) and RISs, we model the RIS placements as point processes (PPs) conditioned on the associated BSs, which are modeled by a Poisson point process (PPP). We assume that the system uses the orthogonal frequency division multiplexing (OFDM) technique to exploit the multipath diversity provided by RISs. The downlink coverage probability and ergodic rate can be evaluated when RISs operate as batched powerless beamformers. The resulting analytical expressions provide a general methodology for assessing the impact of a parameterized RIS model on system performance. These RIS PPs can be adapted based on the deployment strategy. We focus on modeling the RISs as a Matérn cluster process (MCP), where each RIS cluster is a finite PPP within a ring centered on its associated BS. This model connects link-level knowledge to system-level impacts, such as overall interference and the effects of imperfect channel state information (CSI). It also evaluates key RIS deployment parameters, including batch size and RIS density. Furthermore, we analyze a variant of RIS placement in which RISs are deployed around coverage holes to demonstrate the framework’s flexibility and applicability. Numerical evaluations of the analytical expressions and Monte-Carlo simulations jointly validate the proposed analytical approach and provide valuable insights into the design of future RIS-assisted cellular networks.

Reconfigurable intelligent surfaces (RISs) enhance wireless communication by creating engineered signal reflection paths in addition to direct links. This work presents a stochastic geometry framework using point processes (PPs) to model multiple randomly deployed RISs conditioned on their associated base station (BS) locations. By characterizing aggregated reflections from multiple RISs using the Laplace transform, we analytically assess the performance impact of RIS-reflected signals by integrating this characterization into well-established stochastic geometry frameworks. Specifically, we derive closed-form expressions for the Laplace transform of the reflected signal power in several deployment scenarios. These analytical results facilitate performance evaluation of RIS-enabled enhancements. Numerical simulations validate that optimal RIS placement favors proximity to BSs or user equipment (UEs), and further quantify the impact of reflected interference, various fading assumptions, and diverse spatial deployment strategies. Importantly, our analytical approach shows superior computational efficiency compared to Monte Carlo simulations.

Even when driven by the same stimulus, neuronal responses are well-known to exhibit a striking level of spiking variability. In-vivo electrophysiological recordings also reveal a surprisingly large degree of variability at the subthreshold level. In prior work, we considered biophysically relevant neuronal models to account for the observed magnitude of membrane voltage fluctuations. We found that accounting for these fluctuations requires weak but nonzero synchrony in the spiking activity, in amount that are consistent with experimentally measured spiking correlations. Here we investigate whether such synchrony can explain additional statistical features of the measured neural activity, including neuronal voltage covariability and voltage skewness. Addressing this question involves conducting a generalized moment analysis of conductance-based neurons in response to input drives modeled as correlated jump processes. Technically, we perform such an analysis using fixed-point techniques from queuing theory that are applicable in the stationary regime of activity. We found that weak but nonzero synchrony can consistently explain the experimentally reported voltage covariance and skewness. This confirms the role of synchrony as a primary driver of cortical variability and supports that physiological neural activity emerges as a population-level phenomenon, especially in the spontaneous regime. Author summary: Owing to the sheer complexity of biological networks, identifying the design principles for neural computations will only be possible via the simplifying lens of theory. However, to be accepted as valid explanations, theories need to be implemented in idealized neuronal models that can reproduce key aspects of the measured neural activity. Only then can these theories be subjected to experimental validation. In this manuscript, we address this requirement by asking: under which conditions can biophysically relevant neuronal models reproduce physiologically realistic subthreshold activity? We answer this question by focusing on the membrane voltage correlation and skewness, two key statistical signatures of the variable neuronal responses that have been well characterized in behaving mammals. As our core result, we show that the presence of weak but nonzero spiking synchrony is necessary to elicit physiological neuronal responses. The identification of synchrony as a primary driver of neural activity runs counter to the currently prevailing asynchronous state hypothesis, which serves as the basis for many leading neural network theories. Recognizing a central role for synchrony supports that neural computations fundamentally emerge at the collective level rather than as the result of independent parallel processing in neural circuits.