Some simple but challenging Markov processes

In this note, we present few examples of Piecewise Deterministic Markov Processes and their long time behavior. They share two important features: they are related to concrete models (in biology, networks, chemistry,. . .) and they are mathematically rich. Their math-ematical study relies on coupling method, spectral decomposition, PDE technics, functional inequalities. We also relate these simple examples to recent and open problems

Wasserstein decay of one dimensional jump-diffusions

This work is devoted to the Lipschitz contraction and the long time behavior of certain Markov processes. These processes diffuse and jump. They can represent some natural phenomena like size of cell or data transmission over the Internet. Using a Feynman-Kac semigroup, we prove a bound in Wasserstein metric. This bound is explicit and optimal in the sense of Wasserstein curvature. This notion of curvature is relatively close to the notion of (coarse) Ricci curvature or spectral gap. Several consequences and examples are developed, including an $L^2$ spectral for general Markov processes, explicit formulas for the integrals of compound Poisson processes with respect to a Brownian motion, quantitative bounds for Kolmogorov-Langevin processes and some total variation bounds for piecewise deterministic Markov processes

Shot-noise queueing models

We provide a survey of so-called shot-noise queues: queueing models with the special feature that the server speed is proportional to the amount of work it faces. Several results are derived for the workload in an M/G/1 shot-noise queue and some of its variants. Furthermore, we give some attention to queues with general workload-dependent service speed. We also discuss linear stochastic fluid networks, and queues in which the input process is a shot-noise process

TCP and iso-stationary transformations

We consider a class of piecewise deterministic Markov processes that occur as scaling limits of Markov chains describing the Transmission Control Protocol (TCP). The class of processes allows for general increase and decrease profiles. Our key observation is that stationary results for the general class follow directly from the stationary results for the idealized Markov process in [15]. The latter is a Markov process that increases linearly and experiences downward jumps at times governed by a Poisson process. To establish this connection, we introduce space-time transformations that preserve the properties of the class of Markov processes

TCP and iso-stationary transformations

We consider piecewise deterministic Markov processes that occur as scaling limits of discrete-time Markov chains that describe the Transmission Control Protocol (TCP). The class of processes allows for general increase and decrease profiles. Our key observation is that stationary results for the general class follow directly from the stationary results for the idealized TCP process in [22]. The latter is a Markov process that increases linearly and experiences downward jumps at times governed by a Poisson process. To establish this connection, we apply space-time transformations that preserve the properties of the class of Markov processes