EUROPEAN CONFERENCE ON QUEUEING THEORY 2016

International audienceThis booklet contains the proceedings of the second European Conference in Queueing Theory (ECQT) that was held from the 18th to the 20th of July 2016 at the engineering school ENSEEIHT, Toulouse, France. ECQT is a biannual event where scientists and technicians in queueing theory and related areas get together to promote research, encourage interaction and exchange ideas. The spirit of the conference is to be a queueing event organized from within Europe, but open to participants from all over the world. The technical program of the 2016 edition consisted of 112 presentations organized in 29 sessions covering all trends in queueing theory, including the development of the theory, methodology advances, computational aspects and applications. Another exciting feature of ECQT2016 was the institution of the Takács Award for outstanding PhD thesis on "Queueing Theory and its Applications"

Non-homogeneous random walks: from the transport properties to the statistics of occupation times

This dissertation details our research on random walks seen as simple mathematical models useful to describe the complex dynamics of many physical systems. In particular, we focus on the role of spatial inhomogeneity in determining the deviations from the standard behaviour. In the first chapter we present a general method that can be used to obtain the continuum limit for the evolution equations of a random walk with nearest neighbour jumps, from which one can derive the asymptotic properties and deduce the physical interpretation of the walk itself. Then in the following we adopt this method to discuss two particular models. The first model, which we call Gillis random walk, is treated in the second chapter and consists in a random walk with space-dependent drift. Although lacking translational invariance, it provides one of the very few examples of a stochastic system allowing for a number of exact results. From the continuum limit, one deduces that this model provides a microscopical description for the problem of Brownian diffusion in a logarithmic potential, and indeed we compare the results regarding the diffusion problem already present in the literature with the behaviour of the Gillis random walk, finding good agreement. The second model, which we have originally introduced in the literature and deal with in the third chapter, is a correlated random walk closely related to the L\ue9vy-Lorentz gas, a stochastic system where a particle is scattered by static points arranged on a line in such a way that the distances between first neighbour scatterers are independent and identically distributed random variables, drawn from a heavy-tailed distribution. Our model results from a particular procedure of average over all possible arrangements of scatterers and it is mathematically described as a correlated random walk on the integer lattice, where at each step the particle can be either reflected or transmitted according to a space-dependent probability. We apply the continuum limit and derive the long-time properties of the system, which to some extent match those of the original L\ue9vy-Lorentz gas. In the fourth and last chapter we consider the problem of occupation times for onedimensional random walks, showing that for a wide class of processes a single exponent related to a local property of the system is sufficient to describe the distributions of the variables of interests. We test our findings using the two stochastic models presented in the previous chapters and obtain good agreement with our theory. However, our result breaks down, for example, if we consider continuous time random walk models in which the distribution of waiting times between steps does not possess finite mean. Nevertheless, we show how also in this case the theory developed in the first part of the chapter is useful to obtain the statistics of occupation times. We revise some of the results already present in the literature in terms of our theory and test the results on a novel continuous time model based on the dynamics of the Gillis random walk, finding good agreement with both the literature and our theory

Renewal structure and local time for diffusions in random environment

We study a one-dimensional diffusion $X$ in a drifted Brownian potential $W\_\kappa$, with 0\textless{}\kappa\textless{}1, and focus on the behavior of the local times $(\mathcal{L}(t,x),x)$ of $X$ before time t\textgreater{}0.In particular we characterize the limit law of the supremum of the local time, as well as the position of the favorite sites. These limits can be written explicitly from a two dimensional stable L{\'e}vy process. Our analysis is based on the study of an extension of the renewal structure which is deeply involved in the asymptotic behavior of $X$.Comment: 61 page

Piecewise deterministic processes following two alternating patterns

We propose a wide generalization of known results related to the telegraph process. Functionals of the simple telegraph process on a straight line and their generalizations on an arbitrary state space are studied. © Applied Probability Trust 2019

Recent Advances in Single-Particle Tracking: Experiment and Analysis

This Special Issue of Entropy, titled “Recent Advances in Single-Particle Tracking: Experiment and Analysis”, contains a collection of 13 papers concerning different aspects of single-particle tracking, a popular experimental technique that has deeply penetrated molecular biology and statistical and chemical physics. Presenting original research, yet written in an accessible style, this collection will be useful for both newcomers to the field and more experienced researchers looking for some reference. Several papers are written by authorities in the field, and the topics cover aspects of experimental setups, analytical methods of tracking data analysis, a machine learning approach to data and, finally, some more general issues related to diffusion

Generalized master equation for first-passage problems in partitioned spaces

Abstract Motivated by a range of biological applications related to the transport of molecules in cells, we present a modular framework to treat first-passage problems for diffusion in partitioned spaces. The spatial domains can differ with respect to their diffusivity, geometry, and dimensionality, but can also refer to transport modes alternating between diffusive, driven, or anomalous motion. The approach relies on a coarsegraining of the motion by dissecting the trajectories on domain boundaries or when the mode of transport changes, yielding a small set of states. The time evolution of the reduced model follows a generalized master equation, which takes the form of a set of linear integro-differential equations in the occupation probabilities of the states and the corresponding probability fluxes. Further building blocks of the model are partial first-passage time (FPT) densities, which encode the transport behavior in each domain or state. The approach is exemplified and validated for a target search problem with two domains in one- and three-dimensional space, first by exactly reproducing known results for an artificially divided, homogeneous space, and second by considering the situation of domains with distinct diffusivities. Analytical solutions for the FPT densities are given in Laplace domain and are complemented by numerical backtransform yielding FPT densities over many decades in time, confirming that the geometry and heterogeneity of the space can introduce additional characteristic timescales

Anomalous Stochastic Transport of Particles with Self-Reinforcement and Mittag-Leffler Distributed Rest Times

We introduce a persistent random walk model for the stochastic transport of particles involving self-reinforcement and a rest state with Mittag–Leffler distributed residence times. The model involves a system of hyperbolic partial differential equations with a non-local switching term described by the Riemann–Liouville derivative. From Monte Carlo simulations, we found that this model generates superdiffusion at intermediate times but reverts to subdiffusion in the long time asymptotic limit. To confirm this result, we derived the equation for the second moment and find that it is subdiffusive in the long time limit. Analyses of two simpler models are also included, which demonstrate the dominance of the Mittag–Leffler rest state leading to subdiffusion. The observation that transient superdiffusion occurs in an eventually subdiffusive system is a useful feature for applications in stochastic biological transport.</jats:p