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1. Extreme Value Laws for non stationary processes generated by sequential and random dynamical systems

Author

Ana C. Freitas, Sandro Vaienti, Jorge Milhazes Freitas, Centro de Matemática - Universidade do Porto (CMUP), Universidade do Porto = University of Porto, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), CPT - Ex E7 Systèmes dynamiques et théorie ergodique, Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), ACMF was partially supported by FCT (Portugal) grant SFRH/BPD/66174/2009. JMF was partially supported by FCT grant SFRH/BPD/66040/2009. All these grants are financially supported by the program POPH/FSE. ACMF and JMF were partially supported by FCT project FAPESP/19805/2014 and by CMUP (UID/MAT/00144/2013), which is funded by FCT with national (MEC) and European structural funds through the programs ,FEDER, under the partnership agreement PT2020. SV was supported by the ANRProject Perturbations and by the project Atracción de Capital Humano Avanzado del Extranjero MEC 80130047, CONICYT, at the CIMFAV, University of Valparaiso. All authors were partially supported by FCT project PTDC/MAT/12034 /2010, which is funded by national and European structural funds through the programs FEDER and COMPETE. SV is grateful to N. Haydn, M. Nicol and A. Török for several and fruitful discussions on sequential systems, especially in the framework of indifferent maps. JMF is grateful to M. Todd for fruitful discussions and careful reading of a preliminary version of this paper. All authors acknowledge the Isaac Newton Institute for Mathematical Sciences, where this work was initiated during the program Mathematics for the Fluid Earth., ANR-10-BLAN-0106,PERTURBATIONS,Perturbations aléatoires de systèmes dynamiques: applications non-uniformément dilatantes, isométries, billards et systèmes de fonctions itérées. Grandes déviations et valeurs extrêmes.(2010), Faculdade de Economia, Faculdade de Ciências, and Universidade do Porto

Subjects
Hitting Times, Statistics and Probability, Pure mathematics, Random dynamical systems, 60G70, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], FOS: Physical sciences, Dynamical Systems (math.DS), 01 natural sciences, 010104 statistics & probability, Sequential dynamical systems, FOS: Mathematics, 37A50, Mathematics - Dynamical Systems, 0101 mathematics, Extreme value theory, Mathematical Physics, Mathematics, 37A25, 37B20, Non-stationarity, Probability (math.PR), 010102 general mathematics, Mathematical Physics (math-ph), Non stationarity, Statistics, Probability and Uncertainty, Mathematics - Probability
Abstract

International audience; We develop and generalize the theory of extreme value for non-stationary stochastic processes, mostly by weakening the uniform mixing condition that was previously used in this setting. We apply our results to non-autonomous dynamical systems, in particular to sequential dynamical systems, both given by uniformly expanding maps and by maps with a neutral fixed point, and to a few classes of random dynamical systems. Some examples are presented and worked out in detail.

Published
2017

2. Quality of Real-Time Streaming in Wireless Cellular Networks - Stochastic Modeling and Analysis

Author

Bartlomiej Blaszczyszyn, Miodrag Jovanovic, Mohamed Kadhem Karray, Dynamics of Geometric Networks (DYOGENE), Département d'informatique - ENS Paris (DI-ENS), École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Inria Paris-Rocquencourt, Institut National de Recherche en Informatique et en Automatique (Inria), France Télécom Recherche & Développement (FT R&D), France Télécom, This paper reports the results of the research undertaken under CRE-CIFRE thesis co-advising agreement between Inria and Orange Labs., Département d'informatique de l'École normale supérieure (DI-ENS), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-École normale supérieure - Paris (ENS Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Inria Paris-Rocquencourt

Subjects
FOS: Computer and information sciences, Computer science, Throughput, Computer Science - Networking and Internet Architecture, Base station, [INFO.INFO-NI]Computer Science [cs]/Networking and Internet Architecture [cs.NI], outage, FOS: Mathematics, Wireless, Electrical and Electronic Engineering, Radio resource management, Dimensioning, deep outage, stochastic model, Networking and Internet Architecture (cs.NI), business.industry, Wireless network, Applied Mathematics, Quality of service, Probability (math.PR), Real-time streaming, capacity-sharing, Poisson process, Computer Science Applications, LTE, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], interruptions, Cellular network, quality of service, Resource allocation, business, Mathematics - Probability, mobile TV, Computer network
Abstract

We present a new stochastic service model with capacity sharing and interruptions, appropriate for the evaluation of the quality of real-time streaming (RTS), like e.g. mobile TV, in wireless cellular networks. The general model takes into account multi-class Markovian process of call arrivals, (to capture different radio channel conditions, requested streaming bit-rates and durations) and allows for a general resource allocation policy saying which users are temporarily denied the requested fixed streaming bit-rates (put in outage) due to resource constraints. We give expressions for several important performance characteristics of the model, including mean time spent in outage and mean number of outage incidents for a typical user of a given class. These expressions involve only stationary probabilities of the (free) traffic demand process, which is a vector of independent Poisson random variables describing the number of users of different classes. In order to analyze RTS in 3GPP Long Term Evolution (LTE) cellular networks, we specify our general model assuming orthogonal user channels with the peak bit-rates close to the theoretical Shannon's bound in the additive white Gaussian noise (AWGN) channel, which leads to the resource constraints in a multi-rate linear form. In this setting we consider a natural class of least-effort-served-first resource allocation policies, for which the characteristics of the model can be further evaluated using Fourier analysis of Poisson variables. Within this class we identify and evaluate an optimal and a fair policy, the latter being suggested by LTE implementations. We also propose some intermediate policies, which allow to solve the optimality/fairness tradeoff caused by unequal user radio-channel conditions. Our results can be used for the evaluation of the quality of RTS in LTE networks and dimensioning of these networks., Comment: (06/2012)

Published
2014

3. Quality of Service in Wireless Cellular Networks Subject to Log-Normal Shadowing

Author

Bartlomiej Blaszczyszyn, M. K. Karray, Theory of networks and communications (TREC), Département d'informatique - ENS Paris (DI-ENS), Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche en Informatique et en Automatique (Inria)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Inria Paris-Rocquencourt, Institut National de Recherche en Informatique et en Automatique (Inria), Orange Labs [Issy les Moulineaux], France Télécom, This paper reports the results of the research undertaken under 2010 contract number CRE~46146063-A012 between INRIA and France Telecom., Département d'informatique de l'École normale supérieure (DI-ENS), École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Inria Paris-Rocquencourt, École normale supérieure - Paris (ENS-PSL), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS-PSL)

Subjects
FOS: Computer and information sciences, Engineering, geometry, 02 engineering and technology, Interference (wave propagation), Blocking (statistics), interference factor, path-loss, Computer Science - Networking and Internet Architecture, Base station, [INFO.INFO-NI]Computer Science [cs]/Networking and Internet Architecture [cs.NI], honeycomb, 0203 mechanical engineering, FOS: Mathematics, 0202 electrical engineering, electronic engineering, information engineering, Path loss, stochastic resonance, Electrical and Electronic Engineering, indoor, Networking and Internet Architecture (cs.NI), business.industry, Probability (math.PR), 020302 automobile design & engineering, 020206 networking & telecommunications, Poisson, Erlang (unit), [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Handover, blocking probability, Cellular network, Wireless cellular networks, shadowing, Log-distance path loss model, business, Mathematics - Probability, Computer network
Abstract

International audience; Shadowing is believed to degrade the quality of service (QoS) in wireless cellular networks. Assuming log-normal shadowing, and studying mobile's path-loss with respect to the serving base station (BS) and the corresponding interference factor (the ratio of the sum of the path-gains form interfering BS's to the path-gain from the serving BS), which are two key ingredients of the analysis and design of the cellular networks, we discovered a more subtle reality. We observe, as commonly expected, that a strong variance of the shadowing increases the mean path-loss with respect to the serving BS, which in consequence, may compromise QoS. However, in some cases, an increase of the variance of the shadowing can significantly reduce the mean interference factor and, in consequence, improve some QoS metrics in interference limited systems, provided the handover policy selects the BS with the smallest path loss as the serving one. We exemplify this phenomenon, similar to stochastic resonance and related to the ''single big jump principle'' of the heavy-tailed log-nornal distribution, studying the blocking probability in regular, hexagonal networks in a semi-analytic manner, using a spatial version of the Erlang's loss formula combined with Kaufman-Roberts algorithm. More detailed probabilistic analysis explains that increasing variance of the log-normal shadowing amplifies the ratio between the strongest signal and all other signals thus reducing the interference. The above observations might shed new light, in particular on the design of indoor communication scenarios.

Published
2012

4. On discretization schemes for stochastic evolution equations

Author

Annie Millet, Istvan Gyongy, School of Mathematics - University of Edinburgh, University of Edinburgh, Laboratoire de Probabilités et Modèles Aléatoires (LPMA), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Statistique Appliquée et MOdélisation Stochastique (SAMOS), Université Paris 1 Panthéon-Sorbonne (UP1), Modélisation Appliquée, Trajectoires Institutionnelles et Stratégies Socio-Économiques (MATISSE - UMR 8595), Université Paris 1 Panthéon-Sorbonne (UP1)-Centre National de la Recherche Scientifique (CNRS), This paper was written while I. Gyongy was visiting the University of Paris X. The research of I. Gyongy is partially supported byEU Network HARP. The research of A. Millet is partially supported by the research project BMF2003-01345, and Centre National de la Recherche Scientifique (CNRS)-Université Paris Diderot - Paris 7 (UPD7)-Université Pierre et Marie Curie - Paris 6 (UPMC)

Subjects
Discretization, 010102 general mathematics, Probability (math.PR), Banach space, MathematicsofComputing_NUMERICALANALYSIS, Stochastic evolution, 01 natural sciences, 010101 applied mathematics, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Monotone polygon, Convergence (routing), Monotone operators, FOS: Mathematics, Applied mathematics, finite elements, 60H15 65M60, coercivity, 0101 mathematics, Analysis, Stochastic evolution equations, Mathematics - Probability, Mathematics
Abstract

International audience; Stochastic evolutional equations with monotone operators are considered in Banach spaces. Explicit and implicit numerical schemes are presented. The convergence of the approximations to the solution of the equations is proved.

Published
2006