Stability of multi-dimensional birth-and-death processes with state-dependent 0-homogeneous jumps

We study the positive recurrence of multi-dimensional birth-and-death processes describing the evolution of a large class of stochastic systems, a typical example being the randomly varying number of flow-level transfers in a telecommunication wire-line or wireless network. We first provide a generic method to construct a Lyapunov function when the drift can be extended to a smooth function on $\mathbb R^N$, using an associated deterministic dynamical system. This approach gives an elementary proof of ergodicity without needing to establish the convergence of the scaled version of the process towards a fluid limit and then proving that the stability of the fluid limit implies the stability of the process. We also provide a counterpart result proving instability conditions. We then show how discontinuous drifts change the nature of the stability conditions and we provide generic sufficient stability conditions having a simple geometric interpretation. These conditions turn out to be necessary (outside a negligible set of the parameter space) for piece-wise constant drifts in dimension 2.Comment: 18 pages, 4 figure

Modeling of historical evolution of salt water distribution in the phreatic aquifer in and around the silted up Zwin estuary mouth (Flanders, Belgium)

The evolution of the salt-water distribution around the Zwin estuary mouth is modeled for a period of about five centuries. The modeled area is situated in the Flemisch coastal plain near the border of The Netherlands and Belgium. The Zwin estuary is the former waterway to the medieval seaports of Bruges and Damme. During the considered period this alluvial estuary silted up and the modeled area changes from an area around a tidal channel, over a mud flat to a rather complex polder dune area. The evolution is simulated by the 3D density depended groundwater flow model MOCDENS3D (Lebbe & Oude Essink, 1999). The row direction of the applied finite-difference grid is parallel to the present coast line. The simulation is based on old paintings and a large number of maps which allow a relatively detailed reconstruction of the evolution of the landscape. The results show the historical evolution of a large number of different inverse density problems in this area

Large deviations for the stationary measure of networks under proportional fair allocations

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; ArgentinaFil: Lopez, S.. Universidad Nacional Autónoma de México; Méxic

Euclidean versus hyperbolic congestion in idealized versus experimental networks

This paper proposes a mathematical justification of the phenomenon of extreme congestion at a very limited number of nodes in very large networks. It is argued that this phenomenon occurs as a combination of the negative curvature property of the network together with minimum length routing. More specifically, it is shown that, in a large n-dimensional hyperbolic ball B of radius R viewed as a roughly similar model of a Gromov hyperbolic network, the proportion of traffic paths transiting through a small ball near the center is independent of the radius R whereas, in a Euclidean ball, the same proportion scales as 1/R^{n-1}. This discrepancy persists for the traffic load, which at the center of the hyperbolic ball scales as the square of the volume, whereas the same traffic load scales as the volume to the power (n+1)/n in the Euclidean ball. This provides a theoretical justification of the experimental exponent discrepancy observed by Narayan and Saniee between traffic loads in Gromov-hyperbolic networks from the Rocketfuel data base and synthetic Euclidean lattice networks. It is further conjectured that for networks that do not enjoy the obvious symmetry of hyperbolic and Euclidean balls, the point of maximum traffic is near the center of mass of the network.Comment: 23 pages, 4 figure

Robustness of energy landscape control for spin networks under decoherence

Quantum spin networks form a generic system to describe a range of quantum devices for quantum information processing and sensing applications. Understanding how to control them is essential to achieve devices with practical functionalities. Energy landscape shaping is a novel control paradigm to achieve selective transfer of excitations in a spin network with surprisingly strong robustness towards uncertainties in the Hamiltonians. Here we study the effect of decoherence, specifically generic pure dephasing, on the robustness of these controllers. Results indicate that while the effectiveness of the controllers is reduced by decoherence, certain controllers remain sufficiently effective, indicating potential to find highly effective controllers without exact knowledge of the decoherence processes.Comment: 6 pages, 6 figure

Robustness of Energy Landscape Controllers for Spin Rings under Coherent Excitation Transport

The design and analysis of controllers to regulate excitation transport in quantum spin rings presents challenges in the application of classical feedback control techniques to synthesize effective control, and generates results in contradiction to the expectations of classical control theory. In this paper, we examine the robustness of controllers designed to optimize the fidelity of an excitation transfer to uncertainty in system and control parameters. We use the logarithmic sensitivity of the fidelity error as the measure of robustness, drawing on the classical control analog of the sensitivity of the tracking error. In our analysis we demonstrate that quantum systems optimized for coherent transport demonstrate significantly different correlation between error and the log-sensitivity depending on whether the controller is optimized for readout at an exact time T or over a time-window about T.Comment: 10 pages, 4 figures, 2 table