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The aim of this article is to study the forest composed by point-to-line geodesics in the last-passage percolation model with exponential weights. We will show that the location of the root can be described in terms of the maxima of a random walk, whose distribution will depend on the geometry of the substrate (line). For flat substrates, we will get power law behaviour of the height function, study its scaling limit, and describe it in terms of variational problems involving the Airy process.

We address a conjecture introduced by Massouli´e (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. Fil: Jonckheere, Matthieu Thimothy Samson. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires; Argentina Fil: Lopez, S.. Universidad Nacional Autónoma de México; México