Results 1  10
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19
Interacting MultiClass Transmissions in Large Stochastic Networks
, 2008
"... The meanfield limit of a Markovian model describing the interaction of several classes of permanent connections in a network is analyzed. In the same way as for the TCP algorithm, each of the connections has a selfadaptive behavior in that its transmission rate along its route depends on the level ..."
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Cited by 19 (9 self)
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The meanfield limit of a Markovian model describing the interaction of several classes of permanent connections in a network is analyzed. In the same way as for the TCP algorithm, each of the connections has a selfadaptive behavior in that its transmission rate along its route depends on the level of congestion of the nodes of the route. Since several classes of connections going through the nodes of the network are considered, an original meanfield result in a multiclass context is established. It is shown that, as the number of connections goes to infinity, the behavior of the different classes of connections can be represented by the solution of an unusual nonlinear stochastic differential equation depending not only on the sample paths of the process, but also on its distribution. Existence and uniqueness results for the solutions of these equations are derived. Properties of their invariant distributions are investigated and it is shown that, under some natural assumptions, they are determined by the solutions of a fixed point equation in a finite dimensional space.
Limit theorems for some branching measurevalued processes. ArXiv eprints
, 2011
"... ABSTRACT. We consider a particles system, where, the particles move independently according to a Markov process and branching event occurs at an inhomogeneous time. The offspring locations and their number may depend on the position of the mother. Our setting capture, for instance, the processes ind ..."
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Cited by 11 (4 self)
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ABSTRACT. We consider a particles system, where, the particles move independently according to a Markov process and branching event occurs at an inhomogeneous time. The offspring locations and their number may depend on the position of the mother. Our setting capture, for instance, the processes indexed by GaltonWatson tree. We first determine the asymptotic behaviour of the empirical measure. The proof is based on an expression of the empirical measure using an auxiliary process. This latter is not distributed as a one cell lineage, there is a biased phenomenon. Our model is a microscopic description of a random (discrete) population of individuals. We then obtain a large population approximation as weak solution of a growthfragmentation equation. We illustrate our result with two examples. The first one is a sizestructured population model which describes the mitosis and the second one can model a parasite infection.. CONTENTS
Wasserstein decay of one dimensional jumpdiffusions. ArXiv eprints
, 2012
"... ABSTRACT. 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 FeynmanKac semigroup, we prove a bound in ..."
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Cited by 10 (2 self)
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ABSTRACT. 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 FeynmanKac 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 KolmogorovLangevin
Optimal stopping for partially observed piecewisedeterministic Markov processes. Stochastic Process
 Appl
"... Benôıte de Saporta François Dufour This paper deals with the optimal stopping problem under partial observation for piecewisedeterministic Markov processes. We first obtain a recursive formulation of the optimal filter process and derive the dynamic programming equation of the partially observed ..."
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Cited by 6 (1 self)
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Benôıte de Saporta François Dufour This paper deals with the optimal stopping problem under partial observation for piecewisedeterministic Markov processes. We first obtain a recursive formulation of the optimal filter process and derive the dynamic programming equation of the partially observed optimal stopping problem. Then, we propose a numerical method, based on the quantization of the discretetime filter process and the interjump times, to approximate the value function and to compute an actual optimal stopping time. We prove the convergence of the algorithms and bound the rates of convergence.
A recursive nonparametric estimator for the transition kernel of a piecewisedeterministic markov process
, 2014
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Piecewise deterministic Markov process — recent results
, 2013
"... We give a short overview of recent results on a specific class of Markov process: the Piecewise Deterministic Markov Processes (PDMPs). We first recall the definition of these processes and give some general results. On more specific cases such as the TCP model or a model of switched vector fields, ..."
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Cited by 4 (1 self)
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We give a short overview of recent results on a specific class of Markov process: the Piecewise Deterministic Markov Processes (PDMPs). We first recall the definition of these processes and give some general results. On more specific cases such as the TCP model or a model of switched vector fields, better results can be proved, especially as regards long time behaviour. We continue our review with an infinite dimensional example of neuronal activity. From the statistical point of view, these models provide specific challenges: we illustrate this point with the example of the estimation of the distribution of the interjumping times. We conclude with a short overview on numerical methods used for simulating PDMPs. 1 General introduction The piecewise deterministic Markov processes (denoted PDMPs) were first introduced in the literature by Davis ([Dav84, Dav93]). Already at this time, the theory of diffusions had such powerful tools as the theory of Itō calculus and stochastic differential equations at its disposal. Davis’s goal was to endow the PDMP with rather general tools. The main reason for that was to provide a general framework, since up to then only very particular cases had been dealt with, which turned out not to be easily generalizable.
Selfadaptive congestion control for multiclass intermittent connections in a communication network
, 2010
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Stability Properties of Networks with Interacting TCP Flows
 in "Proceedings of NETCOOP
, 2009
"... Abstract. The equilibrium distributions of a Markovian model describing the interaction of several classes of permanent connections in a network are analyzed. It has been introduced by Graham and Robert [5]. For this model each of the connections has a selfadaptive behavior in that its transmission ..."
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Abstract. The equilibrium distributions of a Markovian model describing the interaction of several classes of permanent connections in a network are analyzed. It has been introduced by Graham and Robert [5]. For this model each of the connections has a selfadaptive behavior in that its transmission rate along its route depends on the level of congestion of the nodes on its route. It has been shown in [5] that the invariant distributions are determined by the solutions of a fixed point equation in a finite dimensional space. In this paper, several examples of these fixed point equations are studied. The topologies investigated are rings, trees and a linear network, with various sets of routes through the nodes. 1