Study of probabilistic models for peer-to-peer and mobile networks

Abstract : The goal of this thesis is to solve four problems motivated by modern com- munication networks; the appropriate tools to solve these problems belong to the theory of probability. Solving these problems gives insight into the original physi- cal systems, and contributes at the same time to the theory since new theoretical results of independent interest are proved. Two kinds of communication networks are considered. Mobile networks are these networks where customers perform trajectories within the network indepen- dently of the service they receive; in contrast with classical queueing networks, transitions of customers are not triggered by service completions. In peer-to-peer networks the distinction between clients and servers is abolished, since in these net- works a server is a former client that offers a file once it has downloaded it. These last networks are especially efficient in spreading large or popular files. In Chapters I and II, the stationary behavior of such networks is considered. In each case, one describes the network through a discrete state-space, continuous time Markov process, and establishes its ergodicity or transience. A specificity of these two models is that the transition rates of the corresponding Markov processes are unbounded: in the case of the mobile network of Chapter I this is due to the fact that customers move independently of one another, while for the peer-to-peer network of Chapter II this is because the capacity of the system is proportional to the number of customers. Classically, to analyze the stability of a stochastic network, one can study the limits of a sequence of suitably rescaled Markov processes, the so-called fluid limits. This scaling is well suited for “locally additive” processes, i.e., processes which lo- cally behave as random walks; this is however not the case when the transition rates are unbounded. These techniques are nonetheless adapted to study the stability of the mobile network of Chapter I: using fluid limits to study the stability of Markov processes with unbounded transition rates represents one of the contributions of this work. The peer-to-peer network of Chapter II is not amenable to the same techniques, and Lyapounov type arguments are used. Another additional key ingredient is re- lated to a special class of branching processes. These new branching processes are defined and studied in Chapter II, and estimates on their extinction time make it possible, thanks to coupling arguments, to derive stability results on the stochastic network. In addition to the stationary behavior of peer-to-peer networks, their transient behavior can also be studied: this is the object of the simple model of Chapter III.
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Florian Simatos. Study of probabilistic models for peer-to-peer and mobile networks. Probability [math.PR]. Ecole Polytechnique X, 2009. English. ⟨pastel-00005681⟩

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