Tesis sobre el tema "Invariant distribution of Markov processes"

Cette thèse étudie le comportement en temps long des particules run-and-tumble (RTPs), un modèle pour les bactéries en physique statistique hors équilibre, en utilisant des processus de Markov déterministes par morceaux (PDMPs). La motivation est d'améliorer la compréhension au niveau particulaire des phénomènes actifs, en particulier la séparation de phase induite par la motilité (MIPS). La mesure invariante pour deux RTPs avec jamming sur un tore 1D est déterminée pour mécanismes de tumble et jamming généraux, révélant deux classes d'universalité hors équilibre. De plus, la dépendance du temps de mélange en fonction des paramètres du modèle est déterminée en utilisant des techniques de couplage et le modèle continu PDMP est rigoureusement relié à un modèle sur réseau connu. Dans le cas de deux RTPs avec jamming sur la droite réelle et interagissant à travers un potentiel attractif, la mesure invariante présente des différences qualitatives en fonction des paramètres du modèle, rappelant des transitions de forme et des classes d'universalité. Des taux de convergence fins sont à nouveau obtenus par des techniques de couplage. Par ailleurs, la mesure invariante explicite de trois RTPs se bloquant sur le tore 1D est calculée. Enfin, les résultats de convergence hypocoercive sont étendus aux RTPs, obtenant ainsi des taux de convergence \( L^2 \) fins dans un cadre général qui couvre également les PDMPs utilisés pour l'échantillonnage et Langevin cinétique
This thesis investigates the long-time behavior of run-and-tumble particles (RTPs), a model for bacteria's moves and interactions in out-of-equilibrium statistical mechanics, using piecewise deterministic Markov processes (PDMPs). The motivation is to improve the particle-level understanding of active phenomena, in particular motility induced phase separation (MIPS). The invariant measure for two jamming RTPs on a 1D torus is determined for general tumbling and jamming, revealing two out-of-equilibrium universality classes. Furthermore, the dependence of the mixing time on model parameters is established using coupling techniques and the continuous PDMP model is rigorously linked to a known on-lattice model. In the case of two jamming RTPs on the real line interacting through an attractive potential, the invariant measure displays qualitative differences based on model parameters, reminiscent of shape transitions and universality classes. Sharp quantitative convergence bounds are again obtained through coupling techniques. Additionally, the explicit invariant measure of three jamming RTPs on the 1D torus is computed. Finally, hypocoercive convergence results are extended to RTPs, achieving sharp \( L^2 \) convergence rates in a general setting that also covers kinetic Langevin and sampling PDMPs

La première partie de cette thèse porte sur les automates cellulaires probabilistes (ACP) sur la ligne et à deux voisins. Pour un ACP donné, nous cherchons l'ensemble de ces lois invariantes. Pour des raisons expliquées en détail dans la thèse, ceci est à l'heure actuelle inenvisageable de toutes les obtenir et nous nous concentrons, dans cette thèse, surles lois invariantes markoviennes. Nous établissons, tout d'abord, un théorème de nature algébrique qui donne des conditions nécessaires et suffisantes pour qu'un ACP admette une ou plusieurs lois invariantes markoviennes dans le cas où l'alphabet E est fini. Par la suite, nous généralisons ce résultat au cas d'un alphabet E polonais après avoir clarifié les difficultés topologiques rencontrées. Enfin, nous calculons la fonction de corrélation du modèleà 8 sommets pour certaines valeurs des paramètres du modèle en utilisant une partie desrésultats précédents
The first part of this thesis is about probabilistic cellular automata (PCA) on the line and with two neighbors. For a given PCA, we look for the set of its invariant distributions. Due to reasons explained in detail in this thesis, it is nowadays unthinkable to get all of them and we concentrate our reections on the invariant Markovian distributions. We establish, first, an algebraic theorem that gives a necessary and sufficient condition for a PCA to have one or more invariant Markovian distributions when the alphabet E is finite. Then, we generalize this result to the case of a polish alphabet E once we have clarified the encountered topological difficulties. Finally, we calculate the 8-vertex model's correlation function for some parameters values using previous results.The second part of this thesis is about infinite iterations of stochastic processes. We establish the convergence of the finite dimensional distributions of the α-stable processes iterated n times, when n goes to infinite, according to parameter of stability and to drift r. Then, we describe the limit distributions. In the iterated Brownian motion case, we show that the limit distributions are linked with iterated functions system

We studied invariant measures and invariant densities for dynamical systems with random switching (switching systems, in short). These switching systems can be described by a two-component Markov process whose first component is a stochastic process on a finite-dimensional smooth manifold and whose second component is a stochastic process on a finite collection of smooth vector fields that are defined on the manifold. We identified sufficient conditions for uniqueness and absolute continuity of the invariant measure associated to this Markov process. These conditions consist of a Hoermander-type hypoellipticity condition and a recurrence condition. In the case where the manifold is the real line or a subset of the real line, we studied regularity properties of the invariant densities of absolutely continuous invariant measures. We showed that invariant densities are smooth away from critical points of the vector fields. Assuming in addition that the vector fields are analytic, we derived the asymptotically dominant term for invariant densities at critical points.

A Hidden Markov Model generates two basic stochastic processes, a Markov chain, which is hidden, and an observation sequence. The filtering process of a Hidden Markov Model is, roughly speaking, the sequence of conditional distributions of the hidden Markov chain that is obtained as new observations are received. It is well-known, that the filtering process itself, is also a Markov chain. A classical, theoretical problem is to find conditions which implies that the distributions of the filtering process converge towards a unique limit measure. This problem goes back to a paper of D Blackwell for the case when the Markov chain takes its values in a finite set and it goes back to a paper of H Kunita for the case when the state space of the Markov chain is a compact Hausdor space. Recently, due to work by F Kochmann, J Reeds, P Chigansky and R van Handel, a necessary and sucient condition for the convergence of the distributions of the filtering process has been found for the case when the state space is finite. This condition has since been generalised to the case when the state space is denumerable. In this paper we generalise some of the previous results on convergence in distribution to the case when the Markov chain and the observation sequence of a Hidden Markov Model take their values in complete, separable, metric spaces; it has though been necessary to assume that both the transition probability function of the Markov chain and the transition probability function that generates the observation sequence have densities.

Bibliography: leaves 145-150. Electronic publication; Full text available in PDF format; abstract in HTML format. A MAP/PH/1 queue is a queue having a Markov arrival process (MAP), and a single server with phase-type (PH-type) distributed service time. This thesis considers the departure process of these types of queues, using matrix analytic methods, the Jordan canonical form of matrices, non-linear filtering and approximation techniques. Electronic reproduction.[Australia] :Australian Digital Theses Program,2001.

This dissertation addresses a fundamental question of multi-agent coordination: what infor- mation should be sent to whom and when, with the limited resources available to each agent? Communication requirements for multi-agent systems can be rather high when an accurate picture of the environment and the state of other agents must be maintained. To reduce the impact of multi-agent coordination on networked systems, e.g., power and bandwidth, this dissertation introduces new concepts to enable Intelligent Knowledge Distribution (IKD), including Constrained-action POMDPs (CA-POMDP) and concurrent decentralized (CoDec) POMDPs for an agnostic plug-and-play capability for fully autonomous systems. Each agent runs a CoDec POMDP where all the decision making (motion planning, task allocation, asset monitoring, and communication) are separated into concurrent individual MDPs to reduce the combinatorial explosion of the action and state space while maintaining dependencies between the models. We also introduce the CA-POMDP with action-based constraints on partially observable Markov decision processes, rewards driven by the value of information, and probabilistic constraint satisfaction through discrete optimization and Markov chain Monte Carlo analysis. IKD is adapted real-time through machine learning of the actual environmental impacts on the behavior of the system, including collaboration strategies between autonomous agents, the true value of information between heterogeneous systems, observation probabilities and resource utilization.
Doctor of Philosophy
This dissertation addresses a fundamental question behind when multiple autonomous sys- tems, like drone swarms, in the field need to coordinate and share data: what information should be sent to whom and when, with the limited resources available to each agent? Intelligent Knowledge Distribution is a framework that answers these questions. Communication requirements for multi-agent systems can be rather high when an accurate picture of the environment and the state of other agents must be maintained. To reduce the impact of multi-agent coordination on networked systems, e.g., power and bandwidth, this dissertation introduces new concepts to enable Intelligent Knowledge Distribution (IKD), including Constrained-action POMDPs and concurrent decentralized (CoDec) POMDPs for an agnostic plug-and-play capability for fully autonomous systems. The IKD model was able to demonstrate its validity as a "plug-and-play" library that manages communications between agents that ensures the right information is being transmitted at the right time to the right agent to ensure mission success.

Autonomous vehicles is soon a reality in the every-day life. Though before it is used commercially the vehicles need to be proven safe. The current standard for functional safety on roads, ISO 26262, does not include autonomous vehicles at the moment, which is why in this project an approach using semi-Markov models is used to assess safety. A semi-Markov process is a stochastic process modelled by a state space model where the transitions between the states of the model can be arbitrarily distributed. The approach is realized as a MATLAB tool where the user can use a steady-state based analysis called a Loss and Risk based measure of safety to assess safety. The tool works and can assess safety of semi-Markov systems as long as they are irreducible and positive recurrent. For systems that fulfill these properties, it is possible to draw conclusions about the safety of the system through a risk analysis and also about which autonomous driving level the system is in through a sensitivity analysis. The developed tool, or the approach with the semi-Markov model, might be a good complement to ISO 26262.

Thesis (M.S. in Operations Research and M.S. in Applied Mathematics)--Naval Postgraduate School, June 2004.
Thesis advisor(s): Craig W. Rasmussen, Thomas M. Cioppa. Includes bibliographical references (p. 131-132). Also available online.

We introduce a simple tree growth process that gives rise to a new two-parameter family of discrete fragmentation trees that extends Ford's alpha model to multifurcating trees and includes the trees obtained by uniform sampling from Duquesne and Le Gall's stable continuum random tree. We call these new trees the alpha-gamma trees. In this thesis, we obtain their splitting rules, dislocation measures both in ranked order and in sized-biased order, and we study their limiting behaviour. We further extend the underlying exchangeable fragmentation processes of such trees into partly exchangeable fragmentation processes by weakening the exchangeability. We obtain the integral representations for the measures associated with partly exchangeable fragmentation processes and subordinator of the tagged fragments. We also embed the trees associated with such processes into continuum random trees and study their limiting behaviour. In the end, we generate a three-parameter family of partly exchangeable trees which contains the family of the alpha-gamma trees and another important two-parameter family based on Poisson-Dirichlet distributions.

This dissertation is concerned with the potential multistability of protein concentrations in the cell that can arise in biochemical networks. That is, situations where one, or a family of, proteins may sit at one of two or more different steady state concentrations in otherwise identical cells, and in spite of them being in the same environment. Models of multisite protein phosphorylation have shown that this mechanism is able to exhibit unlimited multistability. Nevertheless, these models have not considered enzyme docking, the binding of the enzymes to one or more substrate docking sites, which are separate from the motif that is chemically modified. Enzyme docking is, however, increasingly being recognised as a method to achieve specificity in protein phosphorylation and dephosphorylation cycles. Most models in the literature for these systems are deterministic i.e. based on Ordinary Differential Equations, despite the fact that these are accurate only in the limit of large molecule numbers. For small molecule numbers, a discrete probabilistic, stochastic, approach is more suitable. However, when compared to the tools available in the deterministic framework, the tools available for stochastic analysis offer inadequate visualisation and intuition. We firstly try to bridge that gap, by developing three tools: a) a discrete `nullclines' construct applicable to stochastic systems - an analogue to the ODE nullcines, b) a stochastic tool based on a Weakly Chained Diagonally Dominant M-matrix formulation of the Chemical Master Equation and c) an algorithm that is able to construct non-reversible Markov chains with desired stationary probability distributions. We subsequently prove that, for multisite protein phosphorylation and similar models, in the deterministic domain, enzyme docking and the consequent substrate enzyme-sequestration must inevitably limit the extent of multistability, ultimately to one steady state. In contrast, bimodality can be obtained in the stochastic domain even in situations where bistability is not possible for large molecule numbers. We finally extend our results to cases where we have an autophosphorylating kinase, as for example is the case with $Ca^{2+}$/calmodulin-dependent protein kinase II (CaMKII), a key enzyme in synaptic plasticity.

Cette thèse traite de rupture localisée de structures construites en matériau composite hétérogène, comme le béton, à deux échelles différentes. Ces deux échelles sont connectées par le biais de la mise à l'échelle stochastique, où toute information obtenue à l'échelle méso est utilisée comme connaissance préalable à l'échelle macro. À l'échelle méso, le modèle de réseau est utilisé pour représenter la structure multiphasique du béton, à savoir le ciment et les granulats. L'élément de poutre représenté par une poutre Timoshenko 3D intégrée avec de fortes discontinuités assure un maillage complet indépendance de la propagation des fissures. La géométrie de la taille des agrégats est prise en accord avec la courbe EMPA et Fuller tandis que la distribution de Poisson est utilisée pour la distribution spatiale. Les propriétés des matériaux de chaque phase sont obtenues avec une distribution gaussienne qui prend en compte la zone de transition d'interface (ITZ) par l'affaiblissement du béton. À l'échelle macro, un modèle de plasticité multisurface est choisi qui prend en compte à la fois la contribution d'un écrouissage sous contrainte avec une règle d'écoulement non associative ainsi que des composants d'un modèle d'adoucissement de déformation pour un ensemble complet de différents modes de défaillance 3D. Le modèle de plasticité est représenté par le critère de rendement Drucker-Prager, avec une fonction potentielle plastique similaire régissant le comportement de durcissement tandis que le comportement de ramollissement des contraintes est représenté par le critère de St. Venant. La procédure d'identification du modèle macro-échelle est réalisée de manière séquentielle. En raison du fait que tous les ingrédients du modèle à l'échelle macro ont une interprétation physique, nous avons fait l'étalonnage des paramètres du matériau en fonction de l'étape particulière. Cette approche est utilisée pour la réduction du modèle du modèle méso-échelle au modèle macro-échelle où toutes les échelles sont considérées comme incertaines et un calcul de probabilité est effectué. Lorsque nous modélisons un matériau homogène, chaque paramètre inconnu du modèle réduit est modélisé comme une variable aléatoire tandis que pour un matériau hétérogène, ces paramètres de matériau sont décrits comme des champs aléatoires. Afin de faire des discrétisations appropriées, nous choisissons le raffinement du maillage de méthode p sur le domaine de probabilité et la méthode h sur le domaine spatial. Les sorties du modèle avancé sont construites en utilisant la méthode de Galerkin stochastique fournissant des sorties plus rapidement le modèle avancé complet. La procédure probabiliste d'identification est réalisée avec deux méthodes différentes basées sur le théorème de Bayes qui permet d'incorporer de nouvelles bservations générées dans un programme de chargement particulier. La première méthode Markov Chain Monte Carlo (MCMC) est identifiée comme mettant à jour la mesure, tandis que la deuxième méthode Polynomial Chaos Kalman Filter (PceKF) met à jour la fonction mesurable. Les aspects de mise en œuvre des modèles présentés sont donnés en détail ainsi que leur validation à travers les exemples numériques par rapport aux résultats expérimentaux ou par rapport aux références disponibles dans la littérature
This thesis deals with the localized failure for structures built of heterogeneous composite material, such as concrete, at two different scale. These two scale are latter connected through the stochastic upscaling, where any information obtained at meso-scale are used as prior knowledge at macro-scale. At meso scale, lattice model is used to represent the multi-phase structure of concrete, namely cement and aggregates. The beam element represented by 3D Timoshenko beam embedded with strong discontinuities ensures complete mesh independency of crack propagation. Geometry of aggregate size is taken in agreement with EMPA and Fuller curve while Poisson distribution is used for spatial distribution. Material properties of each phase is obtained with Gaussian distribution which takes into account the Interface Transition Zone (ITZ) through the weakening of concrete. At macro scale multisurface plasticity model is chosen that takes into account both the contribution of a strain hardening with non-associative flow rule as well as a strain softening model components for full set of different 3D failure modes. The plasticity model is represented with Drucker-Prager yield criterion, with similar plastic potential function governing hardening behavior while strain softening behavior is represented with St. Venant criterion. The identification procedure for macro-scale model is perfomed in sequential way. Due to the fact that all ingredients of macro-scale model have physical interpretation we made calibration of material parameters relevant to particular stage. This approach is latter used for model reduction from meso-scale model to macro-scale model where all scales are considered as uncertain and probability computation is performed. When we are modeling homogeneous material each unknown parameter of reduced model is modeled as a random variable while for heterogeneous material, these material parameters are described as random fields. In order to make appropriate discretizations we choose p-method mesh refinement over probability domain and h-method over spatial domain. The forward model outputs are constructed by using Stochastic Galerkin method providing outputs more quickly the the full forward model. The probabilistic procedure of identification is performed with two different methods based on Bayes’s theorem that allows incorporating new observation generated in a particular loading program. The first method Markov Chain Monte Carlo (MCMC) is identified as updating the measure, whereas the second method Polynomial Chaos Kalman Filter (PceKF) is updating the measurable function. The implementation aspects of presented models are given in full detail as well as their validation throughthe numerical examples against the experimental results or against the benchmarks available from literature

Je décris le comportement en temps long de plusieurs processus qui illustrent les mécanismes de sélection naturelle. Il arrive que ces effets de sélection s’interprètent comme un conditionnement qui biaise la dynamique d’un processus aléatoire "neutre". Ce processus évolue sur un espace potentiellement très général, notamment continu et non borné. On peut ainsi caractériser aussi bien la dynamique du profil complet de la population que celle du profil d’un individu choisi uniformément dans la population. On voit naturellement apparaître dans ces modèles des transitions brutales de ces lois qui rendent l’analyse plus délicate que pour les modèles sans conditionnement. La première partie de ce travail consiste à mettre en place des conditions permettant d’obtenir des résultats analogues à la stationnarité lorsqu’on prend en compte un tel conditionnement. La seconde porte sur l’application de ces critères à différents modèles de population sous sélection et sur leur interprétation. Les exemples d'applications concernent le modèle de l'optimum mobile pour l'adaptation à un changement environnemental, un modèle de sélection de groupes et celui du cliquet de Müller sur le maintien d'une sous-population préservée des mutations délétères
I describe the long term behavior of several processes that model the mechanisms of natural selection. In the cases under consideration, one can interpret those selective effects as a conditioning which introduces a bias on the dynamics of some « neutral » stochastic process. This process evolves in a potentially very general space, notably continuous and unbounded. By these means, one can characterize the dynamics of the whole profile of individuals in the population of study as well as the profile of an individual uniformly chosen in the population. One can naturally see in these descriptions some brutal transitions of the distribution laws as time evolves, which makes the analysis much trickier than for the models without conditioning. The first part of this thesis consists in setting the conditions under which one can prove results analogous to the notion of stationarity while taking the conditioning into account. The second part is dedicated to the application of these criteria for a diversity of models of populations under selection and their interpretation. Examples of application include the mobile optimum model for adaptation to environmental change, a group selection model and Müller's ratchet model that describes how the sub-population unaffected by deleterious mutations can maintain itself

En la actualidad, la Internet de las Cosas (Internet of Things, IoT) es una tecnología esencial para la próxima generación de sistemas inalámbricos. La conectividad es la base de IoT, y el tipo de acceso requerido dependerá de la naturaleza de la aplicación. Uno de los principales facilitadores del entorno IoT es la comunicación machine-to-machine (M2M) y, en particular, su enorme potencial para ofrecer conectividad ubicua entre dispositivos inteligentes. Las redes celulares son la elección natural para las aplicaciones emergentes de IoT y M2M. Un desafío importante en las redes celulares es conseguir que la red sea capaz de manejar escenarios de acceso masivo en los que numerosos dispositivos utilizan comunicaciones M2M. Por otro lado, los sistemas celulares han experimentado un tremendo desarrollo en las últimas décadas: incorporan tecnología sofisticada y nuevos algoritmos para ofrecer una amplia gama de servicios. El modelado y análisis del rendimiento de estas redes multiservicio es también una tarea desafiante que podría requerir un gran esfuerzo computacional. Para abordar los desafíos anteriores, nos centramos en primer lugar en el diseño y la evaluación de las prestaciones de nuevos mecanismos de control de acceso para hacer frente a las comunicaciones masivas M2M en redes celulares. Posteriormente nos ocupamos de la evaluación de prestaciones de redes multiservicio y proponemos una nueva técnica analítica que ofrece precisión y eficiencia computacional. Nuestro principal objetivo es proporcionar soluciones para aliviar la congestión en la red de acceso radio cuando un gran número de dispositivos M2M intentan conectarse a la red. Consideramos los siguientes tipos de escenarios: (i) los dispositivos M2M se conectan directamente a las estaciones base celulares, y (ii) forman grupos y los datos se envían a concentradores de tráfico (gateways) que les proporcionan acceso a la infraestructura. En el primer escenario, dado que el número de dispositivos añadidos a la red aumenta continuamente, esta debería ser capaz de manejar el considerable incremento en las solicitudes de acceso. El 3rd Generation Partnership Project (3GPP) ha propuesto el access class barring (ACB) como una solución práctica para el control de congestión en la red de acceso radio y la red troncal. El ajuste correcto de los parámetros de ACB de acuerdo con la intensidad del tráfico es crítico, pero cómo hacerlo de forma dinámica y autónoma es un problema complejo cuya solución no está recogida en las especificaciones del 3GPP. Esta tesis doctoral contribuye al análisis del rendimiento y al diseño de nuevos algoritmos que implementen efectivamente este mecanismo, y así superar los desafíos introducidos por las comunicaciones masivas M2M. En el segundo escenario, dado que la heterogeneidad de los dispositivos IoT y las arquitecturas celulares basadas en hardware imponen desafíos aún mayores para permitir una comunicación flexible y eficiente en los sistemas inalámbricos 5G, esta tesis doctoral también contribuye al diseño de software-defined gateways (SD-GWs) en una nueva arquitectura propuesta para redes inalámbricas definidas por software que se denomina SoftAir. Esto permite manejar tanto un gran número de dispositivos como el volumen de datos que estarán vertiendo en la red. Otra contribución de esta tesis doctoral es la propuesta de una técnica novedosa para el análisis de prestaciones de redes multiservicio de alta capacidad que se basa en un nuevo enfoque del modelizado analítico de sistemas que operan a diferentes escalas temporales. Este enfoque utiliza el análisis del transitorio de una serie de subcadenas absorbentes y lo denominamos absorbing Markov chain approximation (AMCA). Nuestros resultados muestran que para un coste computacional dado, AMCA calcula los parámetros de prestaciones habituales de un sistema con mayor precisión, en comparación con los resultados obtenidos por otr
Nowadays, Internet of Things (IoT) is an essential technology for the upcoming generation of wireless systems. Connectivity is the foundation for IoT, and the type of access required will depend on the nature of the application. One of the leading facilitators of the IoT environment is machine-to-machine (M2M) communication, and particularly, its tremendous potential to offer ubiquitous connectivity among intelligent devices. Cellular networks are the natural choice for emerging IoT and M2M applications. A major challenge in cellular networks is to make the network capable of handling massive access scenarios in which myriad devices deploy M2M communications. On the other hand, cellular systems have seen a tremendous development in recent decades; they incorporate sophisticated technology and algorithms to offer a broad range of services. The modeling and performance analysis of these large multi-service networks is also a challenging task that might require high computational effort. To address the above challenges, we first concentrate on the design and performance evaluation of novel access control schemes to deal with massive M2M communications. Then, we focus on the performance evaluation of large multi-service networks and propose a novel analytical technique that features accuracy and computational efficiency. Our main objective is to provide solutions to ease the congestion in the radio access or core network when massive M2M devices try to connect to the network. We consider the following two types of scenarios: (i) massive M2M devices connect directly to cellular base stations, and (ii) they form clusters and the data is forwarded to gateways that provide them with access to the infrastructure. In the first scenario, as the number of devices added to the network is constantly increasing, the network should handle the considerable increment in access requests. Access class barring (ACB) is proposed by the 3rd Generation Partnership Project (3GPP) as a practical congestion control solution in the radio access and core network. The proper tuning of the ACB parameters according to the traffic intensity is critical, but how to do so dynamically and autonomously is a challenging task that has not been specified. Thus, this dissertation contributes to the performance analysis and optimal design of novel algorithms to implement effectively this barring scheme and overcome the challenges introduced by massive M2M communications. In the second scenario, since the heterogeneity of IoT devices and the hardware-based cellular architectures impose even greater challenges to enable flexible and efficient communication in 5G wireless systems, this dissertation also contributes to the design of software-defined gateways (SD-GWs) in a new architecture proposed for wireless software-defined networks called SoftAir. The deployment of these SD-GWs represents an alternative solution aiming at handling both a vast number of devices and the volume of data they will be pouring into the network. Another contribution of this dissertation is to propose a novel technique for the performance analysis of large multi-service networks. The underlying complexity of the network, particularly concerning its size and the ample range of configuration options, makes the solution of the analytical models computationally costly. However, a typical characteristic of these networks is that they support multiple types of traffic flows operating at different time-scales. This time-scale separation can be exploited to reduce considerably the computational cost associated to determine the key performance indicators. Thus, we propose a novel analytical modeling approach based on the transient regime analysis, that we name absorbing Markov chain approximation (AMCA). For a given computational cost, AMCA finds common performance indicators with greater accuracy, when compared to the results obtained by other approximate methods proposed in the literature.
En l'actualitat, la Internet de les Coses (Internet of Things, IoT) és una tecnologia essencial per a la propera generació de sistemes sense fil. La connectivitat és la base d'IoT, i el tipus d'accés requerit dependrà de la naturalesa de l'aplicació. Un dels principals facilitadors de l'entorn IoT és la comunicació machine-to-machine (M2M) i, en particular, el seu enorme potencial per oferir connectivitat ubiqua entre dispositius intel · ligents. Les xarxes mòbils són l'elecció natural per a les aplicacions emergents de IoT i M2M. Un desafiament important en les xarxes mòbils que actualment está rebent molta atenció és aconseguir que la xarxa siga capaç de gestionar escenaris d'accés massiu en què una gran quantitat de dispositius utilitzen comunicacions M2M. D'altra banda, els sistemes mòbils han experimentat un gran desenvolupament en les últimes dècades: incorporen tecnologia sofisticada i nous algoritmes per oferir una àmplia gamma de serveis. El modelatge i análisi del rendiment d'aquestes xarxes multiservei és també un desafiament important que podria requerir un gran esforç computacional. Per abordar els desafiaments anteriors, en aquesta tesi doctoral ens centrem en primer lloc en el disseny i l'avaluació de les prestacions de nous mecanismes de control d'accés per fer front a les comunicacions massives M2M en xarxes cel · lulars. Posteriorment ens ocupem de l'avaluació de prestacions de xarxes multiservei i proposem una nova tècnica analítica que ofereix precisió i eficiència computacional. El nostre principal objectiu és proporcionar solucions per a alleujar la congestió a la xarxa d'accés ràdio quan un gran nombre de dispositius M2M intenten connectar-se a la xarxa. Considerem els dos tipus d'escenaris següents: (i) els dispositius M2M es connecten directament a les estacions base cel · lulars, i (ii) formen grups i les dades s'envien a concentradors de trànsit (gateways) que els proporcionen accés a la infraestructura. En el primer escenari, atès que el nombre de dispositius afegits a la xarxa augmenta contínuament, aquesta hauria de ser capaç de gestionar el considerable increment en les sol · licituds d'accés. El 3rd Generation Partnership Project (3GPP) ha proposat l'access class barring (ACB) com una solució pràctica per al control de congestió a la xarxa d'accès ràdio i la xarxa troncal. L'ajust correcte dels paràmetres d'ACB d'acord amb la intensitat del trànsit és crític, però com fer-ho de forma dinàmica i autònoma és un problema complex, la solució del qual no està recollida en les especificacions del 3GPP. Aquesta tesi doctoral contribueix a l'anàlisi del rendiment i al disseny de nous algoritmes que implementen efectivament aquest mecanisme, i així superar els desafiaments introduïts per les comunicacions massives M2M en les xarxes mòbils actuals i futures. En el segon escenari, atès que l'heterogeneïtat dels dispositius IoT i les arquitectures cel · lulars basades en hardware imposen desafiaments encara més grans per permetre una comunicació flexible i eficient en els sistemes sense fil 5G, aquesta tesi doctoral també contribueix al disseny de software-defined gateways (SD-GWS) en una nova arquitectura proposada per a xarxes sense fils definides per programari que s'anomena SoftAir. Això permet gestionar tant un gran nombre de dispositius com el volum de dades que estaran abocant a la xarxa. Una altra contribució d'aquesta tesi doctoral és la proposta d'una tècnica innovadora per a l'anàlisi de prestacions de xarxes multiservei d'alta capacitat que es basa en un nou enfocament del modelitzat analític de sistemes que operen a diferents escales temporals. Aquest enfocament utilitza l'anàlisi del transitori d'una sèrie de subcadenes absorbents i l'anomenem absorbing Markov chain Approximation (AMCA). Els nostres resultats mostren que per a un cost computacional donat, AMCA calcula els paràmetres de prestacions habituals d
Tello Oquendo, LP. (2018). Design and Performance Analysis of Access Control Mechanisms for Massive Machine-to-Machine Communications in Wireless Cellular Networks [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/107946
TESIS

Thesis (Ph. D.)--University of Wisconsin--Madison, 1999.
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Many problems within the biological sciences, such as DNA sequencing, protein structure prediction, and molecular docking, are being approached computationally. These problems require sophisticated solution methods that understand the complex natures of biological domains. Traditionally, such solution methods are problem specific, but recent advances in generic problem-solvers furnish hope for a new breed of computational tools. The challenge is to develop methods that can automatically learn or acquire an understanding of a complex problem domain. Estimation of Distribution Algorithms (EDAs) are generic search methods that use statistical models to learn the structure of a problem domain. EDAs have been successfully applied to many difficult search problems, such as circuit design, optimizing Ising spin glasses, and various scheduling tasks. However, current EDAs contain ad hoc limitations that reduce their capacity to solve hard problems. This dissertation presents a new EDA method, the Markovian Learning Estimation of Distribution Algorithm (MARLEDA), that employs a Markov random field model. The model is learned in a novel way that overcomes previous ad hoc limitations. MARLEDA is shown to perform well on standard benchmark search tasks. A multiobjective extension of MARLEDA is developed for use in predicting the secondary structure of RNA molecules. The extension is shown to produce high-quality predictions in comparison with several contemporary methods, laying the groundwork for a new computational tool for RNA researchers.

A MAP/PH/1 queue is a queue having a Markov arrival process (MAP), and a single server with phase-type (PH -type) distributed service time. This thesis considers the departure process from these type of queues. We use matrix analytic methods, the Jordan canonical form of matrices, non-linear filtering and approximation techniques. The departure process of a queue is important in the analysis of networks of queues, as it may be the arrival process to another queue in the network. If a simple description were to exist for a departure process, the analysis of at least feed-forward networks of these queues would then be analytically tractable. Chapter 1 is an introduction to some of the literature and ideas surrounding the departure process from MAP/PH/1 queues. Chapter 2 sets up the basic notation and establishes some results which are used throughout the thesis. It contains a preliminary consideration of PH -type distributions, PH -renewal processes, MAP s, MAP/PH/1 queues, non-linear filtering and the Jordan canonical form. Chapter 3 is an expansion of "The Output process of an MMPP/M/1 queue", where the question of whether a MAP description can exist for the departure process of a non-trivial MAP/M/1 queue is considered. In a 1994 paper, Olivier and Walrand conjectured that the departure process of a MAP/PH/1 queue is not a MAP unless the queue is a stationary M/M/1 queue. This conjecture was prompted by their claim that the departure process of an MMPP/M/1 queue is not MAP unless the queue is a stationary M/M/1 queue. We show that their proof has an algebraic error, which leaves open the above question of whether the departure process of an MMPP/PH/1 queue is a MAP or not. In Chapter 4, the more fundamental problem of identifying stationary M/M/1 queues in the class of MAP/PH/1 queues is considered. It is essential to be able to determine from its generator when a stationary MAP is a Poisson process. This does not appear to have been discussed in the literature prior to the author's paper, where this deficiency was remedied using ideas from non-linear filtering theory, to give a characterisation as to when a stationary MAP is a Poisson process. Chapter 4 expands upon "When is a MAP Poisson". This investigation of higher order representations of the Poisson process is motivated by first considering when a higher order PH -type distribution is just negative exponential. In Chapter 5, we consider the related question of minimal order representations for PH -type distributions, an issue which has attracted much interest in the literature. A discussion of other authors' ideas is given and these ideas are then inter-related to the work presented in Chapter 4 on the PH -type distributions. The MAP/M/1 queue is then considered in Chapter 6 from the perspective of whether having an exact level and phase independent stationary distribution of the geometric form [Formula - Not available: see pdf version of the abstract] implies that the MAP is Poisson. The answer is in the affirmative for this question, but the converse is not strictly true. Apart from showing the ubiquitous asymptotic form of level and phase independence exhibited by all stable MAP/M/1 queues, we prove that a very large class of stable queues, exhibits what we have termed shift-one level and phase independence. Stable MAP/M/1 queues exhibiting shift-one level and phase independence, are characterised by a stationary distribution of the following form: [Formula - Not Available: see pdf version of the abstract] In Chapter 7, a family of approximations is proposed for the output process of a stationary MAP/PH/1 queue. To check the viability of these approximations, they are used as input to another single server queue. Performance measures for the second server are obtained analytically in both the tandem and approximation cases, thus eliminating the need for simulation to compare results. Comparison of these approximations is also made against other approximation methods in the literature. In Chapter 8, we show that our approximations from Chapter 7 have the property of exactly matching the inter-departure time distribution. Our kth approximation also accurately captures the first k-1 lag-correlation coefficients of the stationary departure process. The proofs of this direct association between lag-correlation coefficients and the level of complexity k are given.
Thesis (Ph.D.)--School of Applied Mathematics, 1999.

Recently the independent multinomial selections model (IMS) with the multinomial logit link has been suggested as an analysis tool for radio-telemetry habitat selection data. This model assumes independence between animals, independence between sightings within an animal, and identical multinomial habitat selection probabilities for all animals. We propose two generalizations to the IMS model. The first generalization is to allow a Markov chain dependence between consecutive sightings of the same animal. This generalization allows for both positive correlation (individuals persisting in the same habitat class in which they were previously sighted) and negative correlation (individual vacating the habitat class in which they were previously sighted). The second generalization is to allow for heterogeneity. Here, a hierarchical Dirichlet-multinomial distribution is used to allow for variability in selection probabilities between animals. This generalization accounts for over-dispersion of selection probabilities and allows for inference to the population of animals, assuming that the animals studied constitute a random sample from that population.. Both generalizations are one parameter extensions to the multinomial logit model and allow for testing the assumptions of identical multinomial selection probabilities and independence. These tests are performed using the score, Wald, and asymptotic likelihood ratio statistics. Estimates of model parameters are obtained using maximum likelihood techniques, and habitat characteristics are tested using drop-in-deviance statistics. Using example data, we show that persistence and heterogeneity exist in habitat selection data and illustrate the difference in analysis results between the IMS model and the persistence and heterogeneity models. Through simulation, we show that analyzing persistence data assuming independence between sightings within an animal gives liberal tests of significance for habitat characteristics when the data are generated with positive correlation and conservative tests of significance when the data are generated with negative correlation. Similarly, we show that analyzing heterogeneous data, assuming identical multinomial selection probabilities, gives liberal tests of significance for habitat characteristics.
Graduation date: 2001

Nous y introduisons une nouvelle classe de distributions bivariées de type Marshall-Olkin, la distribution Erlang bivariée. La transformée de Laplace, les moments et les densités conditionnelles y sont obtenus. Les applications potentielles en assurance-vie et en finance sont prises en considération. Les estimateurs du maximum de vraisemblance des paramètres sont calculés par l'algorithme Espérance-Maximisation. Ensuite, notre projet de recherche est consacré à l'étude des processus de risque multivariés, qui peuvent être utiles dans l'étude des problèmes de la ruine des compagnies d'assurance avec des classes dépendantes. Nous appliquons les résultats de la théorie des processus de Markov déterministes par morceaux afin d'obtenir les martingales exponentielles, nécessaires pour établir des bornes supérieures calculables pour la probabilité de ruine, dont les expressions sont intraitables.
In this contribution, we introduce a new class of bivariate distributions of Marshall-Olkin type, called bivariate Erlang distributions. The Laplace transform, product moments and conditional densities are derived. Potential applications of bivariate Erlang distributions in life insurance and finance are considered. Further, our research project is devoted to the study of multivariate risk processes, which may be useful in analyzing ruin problems for insurance companies with a portfolio of dependent classes of business. We apply results from the theory of piecewise deterministic Markov processes in order to derive exponential martingales needed to establish computable upper bounds of the ruin probabilities, as their exact expressions are intractable.

While decision-making under uncertainty is a major universal problem, its implications in the field of transportation systems are especially enormous; where the benefits of right decisions are tremendous, the consequences of wrong ones are potentially disastrous. In the realm of highway systems, decisions related to the highway configuration (number of lanes, right of way, etc.) need to incorporate both the traffic demand and land price uncertainties. In the literature, these uncertainties have generally been modeled using the Geometric Brownian Motion (GBM) process, which has been used extensively in modeling many other real life phenomena. But few scholars, including those who used the GBM in highway configuration decisions, have offered any rigorous justification for the use of this model. This thesis attempts to offer a detailed analysis of various aspects of transportation systems in relation to decision-making. It reveals some general insights as well as a new concept that extends the notion of opportunity cost to situations where wrong decisions could be made. Claiming deficiency of the GBM model, it also introduces a new formulation that utilizes a large and flexible parametric family of jump models (i.e., Lévy processes). To validate this claim, data related to traffic demand and land prices were collected and analyzed to reveal that their distributions, heavy-tailed and asymmetric, do not match well with the GBM model. As a remedy, this research used the Merton, Kou, and negative inverse Gaussian Lévy processes as possible alternatives. Though the results show indifference in relation to final decisions among the models, mathematically, they improve the precision of uncertainty models and the decision-making process. This furthers the quest for optimality in highway projects and beyond.