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Matthieu Jonckheere

Publications and source records attributed to Matthieu Jonckheere.

51 records · Page 3Linked to original sources
arXiv2017

On the Kesten-Stigum theorem in $L^2$ beyond $λ$-positivity

We study supercritical branching processes in which all particles evolve according to some general Markovian motion (which may possess absorbing states) and branch independently at a fixed constant rate. Under fairly natural assumptions on the asymptotic distribution of the underlying motion, we first show using only probablistic tools that there is convergence in $L^2$ of the empirical measure (normalized by the mean number of particles) if and only if an associated additive martingale is bounded in $L^2$. This is a significant improvement over previous results, which were mainly restricted to $λ$-positive motions. We then investigate under which conditions this limit is strictly positive on the event of non-extinction and show that this occurs whenever, on the latter event, particles do not accumulate on the boundary of the state space. In particular, this property also yields the convergence of the real empirical measure (normalized by the number of particles). Moreover, building on previous results we prove that if the motion is $λ$-positive then these limits hold also almost surely whenever the Doob's $h$-transform of this motion admits a suitable Lyapunov functional. Finally, we illustrate our results for a variety of different motions: ergodic motions without absorption, $λ$-positive systems either transient or with absorption, and also certain non $λ$-positive systems such as the Brownian motion with negative drift killed at $0$. A strong law of large numbers for the empirical measure in this last example was announced by Kesten in the late seventies, although a proof of it has remained undisclosed so far. Our results allow us to give a partial proof to Kesten's claim.

math.PR↗
arXiv2016

Scaling Limits and Generic Bounds for Exploration Processes

We consider exploration algorithms of the random sequential adsorption type both for homogeneous random graphs and random geometric graphs based on spatial Poisson processes. At each step, a vertex of the graph becomes active and its neighboring nodes become explored. Given an initial number of vertices $N$ growing to infinity, we study statistical properties of the proportion of explored nodes in time using scaling limits. We obtain exact limits for homogeneous graphs and prove an explicit central limit theorem for the final proportion of active nodes, known as the \emph{jamming constant}, through a diffusion approximation for the exploration process. We then focus on bounding the trajectories of such exploration processes on random geometric graphs, i.e. random sequential adsorption. As opposed to homogeneous random graphs, these do not allow for a reduction in dimensionality. Instead we build on a fundamental relationship between the number of explored nodes and the discovered volume in the spatial process, and obtain generic bounds: bounds that are independent of the dimension of space and the detailed shape of the volume associated to the discovered node. Lastly, we give two trajectorial interpretations of our bounds by constructing two coupled processes that have the same fluid limits.

math.PR↗
arXiv2016

Front propagation and quasi-stationary distributions for one-dimensional Lévy processes

We jointly investigate the existence of quasi-stationary distributions for one dimensional Lévy processes and the existence of traveling waves for the Fisher-Kolmogorov-Petrovskii-Piskunov (F-KPP) equation associated with the same motion. Using probabilistic ideas developed by S. Harris, we show that the existence of a traveling wave for the F-KPP equation associated with a centered Lévy processes that branches at rate $r$ and travels at velocity $c$ is equivalent to the existence of a quasi-stationary distribution for a Lévy process with the same movement but drifted by $-c$ and killed at zero, with mean absorption time $1/r$. This also extends the known existence conditions in both contexts. As it is discussed in a companion article, this is not just a coincidence but the consequence of a relation between these two phenomena.

math.PR↗
arXiv2016

Asymptotics of Insensitive Load Balancing and Blocking Phases

We address the problem of giving robust performance bounds based on the study of the asymptotic behavior of the insensitive load balancing schemes when the number of servers and the load scales jointly. These schemes have the desirable property that the stationary distribution of the resulting stochastic network depends on the distribution of job sizes only through its mean. It was shown that they give good estimates of performance indicators for systems with finite buffers, generalizing henceforth Erlang's formula whereas optimal policies are already theoretically and computationally out of reach for networks of moderate size. We study a single class of traffic acting on a symmetric set of processor sharing queues with finite buffers and we consider the case where the load scales with the number of servers. We characterize central limit theorems and large deviations, the response of symmetric systems under those schemes at different scales and show that three amplitudes of deviations can be identified. A central limit scaling takes place for a sub-critical load; for $ρ=1$, the number of free servers scales like $n^{ {θ\over θ+1}}$ ($θ$ being the buffer depth and $n$ being the number of servers) and is of order 1 for super-critical loads. This further implies the existence of different phases for the blocking probability, Before a (refined) critical load $ρ_c(n)=1-a n^{- {θ\over θ+1}}$, the blocking is exponentially small and becomes of order $ n^{- {θ\over θ+1}}$ at $ρ_c(n)$. This generalizes the well-known Quality and Efficiency Driven (QED) regime or Halfin-Whitt regime for a one-dimensional queue, and leads to a generalized staffing rule for a given target blocking probability.

cs.PF↗
arXiv2015

The Jamming Constant of Uniform Random Graphs

By constructing jointly a random graph and an associated exploration process, we define the dynamics of a "parking process" on a class of uniform random graphs as a measure-valued Markov process, representing the empirical degree distribution of non-explored nodes. We then establish a functional law of large numbers for this process as the number of vertices grows to infinity, allowing us to assess the jamming constant of the considered random graphs, i.e. the size of the maximal independent set discovered by the exploration algorithm. This technique, which can be applied to any uniform random graph with a given degree distribution, can be seen as a generalization in the space of measures, of the differential equation method introduced by Wormald.

math.PR↗
arXiv2015

Sub-Poissonian Statistics of Jamming Limits in Ultracold Rydberg Gases

Several recent experiments have established by measuring the Mandel Q parameter that the number of Rydberg excitations in ultracold gases exhibits sub-Poissonian statistics. This effect is attributed to the Rydberg blockade that occurs due to the strong interatomic interactions between highly-excited atoms. Because of this blockade effect, the system can end up in a state in which all particles are either excited or blocked: a jamming limit. We analyze appropriately constructed random-graph models that capture the blockade effect, and derive formulae for the mean and variance of the number of Rydberg excitations in jamming limits. This yields an explicit relationship between the Mandel Q parameter and the blockade effect, and comparison to measurement data shows strong agreement between theory and experiment.

quant-ph↗
arXiv2015

Scaling limits for exploration algorithms

We consider an exploration algorithm where at each step, a random number of items become active while related items get explored. Given an initial number of items $N$ growing to infinity and building on a strong homogeneity assumption, we study using scaling limits of Markovian processes statistical properties of the proportion of active nodes in time. This is a companion paper that rigorously establishes the claims and heuristics presented in [5]. [5] Jaron Sanders, Matthieu Jonckheere, and Servaas Kokkelmans. Sub-Poissonian statistics of jamming limits in Rydberg gases. 2015. To appear.

math.PR↗
arXiv2014

Estimating the Spatial Reuse with Configuration Models

We propose a new methodology to estimate the spatial reuse of CSMA-like scheduling. Instead of focusing on spatial configurations of users, we model the interferences between users as a random graph. Using configuration models for random graphs, we show how the properties of the medium access mechanism are captured by some deterministic differential equations, when the size of the graph gets large. Performance indicators such as the probability of connection of a given node can then be efficiently computed from these equations. We also perform simulations to illustrate the results on different types of random graphs. Even on spatial structures, these estimates get very accurate as soon as the variance of the interference is not negligible.

cs.NI↗
arXiv2012

Large deviations of the stationary measure of networks under proportional fair allocations

We address a conjecture introduced by Massoulié (2007), concerning the large deviations of the stationary measure of bandwidth-sharing networks functioning under the Proportional fair allocation. For Markovian networks, we prove that Proportional fair and an associated reversible allocation are geometrically ergodic and have the same large deviations characteristics using Lyapunov functions and martingale arguments. For monotone networks, we give a more direct proof of the same result relying on stochastic comparisons that hold for general service requirement distribution. These results comfort the intuition that Proportional fairness is 'close' to allocations of service being insensitive to the service time requirement.

math.OC↗
arXiv2012

Simulation of quasi-stationary distributions on countable spaces

Quasi-stationary distributions (QSD) have been widely studied since the pioneering work of Kolmogorov (1938), Yaglom (1947) and Sevastyanov (1951). They appear as a natural object when considering Markov processes that are certainly absorbed since they are invariant for the evolution of the distribution of the process conditioned on not being absorbed. They hence appropriately describe the state of the process at large times for non absorbed paths. Unlike invariant distributions for Markov processes, QSD are solutions of a non-linear equation and there can be 0, 1 or an infinity of them. Also, they cannot be obtained as Cesàro limits of Markovian dynamics. These facts make the computation of QSDs a nontrivial matter. We review different approximation methods for QSD that are useful for simulation purposes, mainly focused on Fleming-Viot dynamics. We also give some alternative proofs and extensions of known results.

math.PR↗
arXiv2012

Fleming-Viot selects the minimal quasi-stationary distribution: The Galton-Watson case

Consider N particles moving independently, each one according to a subcritical continuous-time Galton-Watson process unless it hits 0, at which time it jumps instantaneously to the position of one of the other particles chosen uniformly at random. The resulting dynamics is called Fleming-Viot process. We show that for each N there exists a unique invariant measure for the Fleming-Viot process, and that its stationary empirical distribution converges, as N goes to infinity, to the minimal quasi-stationary distribution of the Galton-Watson process conditioned on non-extinction.

math.PR↗
arXiv2011

Bandwidth sharing networks with priority scaling

In multi-class communication networks, traffic surges due to one class of users can significantly degrade the performance for other classes. During these transient periods, it is thus of crucial importance to implement priority mechanisms that conserve the quality of service experienced by the affected classes, while ensuring that the temporarily unstable class is not entirely neglected. In this paper, we examine the complex interaction occurring between several classes of traffic when classes obtain bandwidth proportionally to their incoming traffic. We characterize the evolution of the network from the moment the initial surge takes place until the system reaches its equilibrium. Using an appropriate scaling, we show that the trajectories of the temporarily unstable class can be described by a differential equation, while those of the stable classes retain their stochastic nature. A stochastic averaging phenomenon occurs and the dynamics of the temporarily unstable and the stable classes continue to influence one another. We further proceed to characterize the obtained differential equations and the stability region under this scaling for monotone networks. We illustrate these result on several toy examples and we finally build a penalization rule using these results for a network integrating streaming and elastic traffic.

cs.PF↗
arXiv2010

Stability of parallel queueing systems with coupled service rates

This paper considers a parallel system of queues fed by independent arrival streams, where the service rate of each queue depends on the number of customers in all of the queues. Necessary and sufficient conditions for the stability of the system are derived, based on stochastic monotonicity and marginal drift properties of multiclass birth and death processes. These conditions yield a sharp characterization of stability for systems, where the service rate of each queue is decreasing in the number of customers in other queues, and has uniform limits as the queue lengths tend to infinity. The results are illustrated with applications where the stability region may be nonconvex.

math.PR↗
arXiv2008

Stochastic bounds for two-layer loss systems

This paper studies multiclass loss systems with two layers of servers, where each server at the first layer is dedicated to a certain customer class, while the servers at the second layer can handle all customer classes. The routing of customers follows an overflow scheme, where arriving customers are preferentially directed to the first layer. Stochastic comparison and coupling techniques are developed for studying how the system is affected by packing of customers, altered service rates, and altered server configurations. This analysis leads to easily computable upper and lower bounds for the performance of the system.

math.PR↗