This thesis was conducted in cooperation with the Swedish bank SEB, who expressed interest in getting an increased understanding of how the marketing measure Customer Lifetime Value could be implemented and used in the retail banking industry. Accordingly, the purpose of this thesis was to provide insight into how Customer Lifetime Value could be modelled in an appropriate way in the retail banking industry and provide an increased understanding of necessary considerations for the modelling process. First, performance requirements for models of Customer Lifetime Value in the retail banking industry were identified through literary analysis and interviews with SEB. These requirements were then used to evaluate six general modelling approaches: RFM, Probability, Econometric, Persistance, Diffusion/Growth and Computer science. Based on the evaluation, the computer science approach and the econometric approach were identified as suitable for further investigation. This was achieved by implementing and analysing the performance of two models chosen as examples of respective approach. Specifically, a computer science model based on the \textit{random forest} algorithm and an econometric model based on \textit{Markov chains} were chosen. The results indicate that both approaches could be appropriate for the retail banking industry, but that an econometric approach could have the advantage of higher interpretability while a computer science approach can have the advantage of higher predictive accuracy.  In conclusion, the results indicate that the specific considerations and performance requirements for models of Customer Lifetime Value in the retail banking context should be based on a specific use case and area of business application. However, the discussions, considerations and examples of implementations provided in this thesis could serve as a foundation for future research and model development in this context. / Detta arbete genomfördes i samarbete med SEB, som uttryckt ett intresse för att öka sin kunskap kring hur marknadsföringsmåttet Customer Lifetime Value skulle kunna implementeras och användas i retail banking-branschen. Syftet med denna uppsats var följaktligen att ge en ökad förståelse för vad som är en lämplig modell av Customer Lifetime Value i branschen, samt ge en ökad förståelse för nödvändiga hänsynstaganden i modelleringsprocessen. Detta gjordes genom att först identifiera existerande modellkrav genom litteraturanalyys och intervjuer med SEB. Kraven användes sedan för att utvärdera sex generella modelltyper: RFM, Probability, Econometric, Persistance, Diffusion/Growth and Computer science. Baserat på utvärderingen identifierades Econometric och Computer science som lämpliga modelltyper för vidare undersökning, vilken gjordes genom att implementera en modell från respektive modelltyp. Specifikt valdes en Computer science-metod baserad på algoritmen random forest och en Econometric-metod baserad på Markovkedjor. Resultaten indikerade att båda modelltyper är lämpliga för implementering i retail banking-branschen, men att en Econometric-metod skulle kunna ha större tolkbarhet och att en Computer science-metod skulle kunna ha bättre precision. Sammanfattningsvis konstateras att hänsynstaganden och modellkrav på modeller av Customer Lifetime Value i retail banking-branschen bör utformas utifrån det specifika tilltänkta användningsområdet. De diskussioner, hänsynstaganden och implementationsexempel som presenteras i detta arbete kan dock fungera som grund för vidare forskning och modellutveckling i kontexten.

Current models that assess infrastructure failure risk are "linear," and therefore, only consider the direct influence attributed to each factor that defines risk. These models do not consider the undeniable relationships that exist among these parameters. In reality, factors that define risk are interdependent and influence each other in a "non-linear" fashion through feedback effects. Current infrastructure failure risk assessment models are also static, and do not allow infrastructure managers and decision makers to evaluate the impacts over time, especially the long-term impact of risk mitigation actions. Factors that define infrastructure failure risk are in constant change. In a strategic manner, this research proposes a new risk-based infrastructure management framework and supporting system, Risk-Based Dynamic Infrastructure Management System (RiskDIMS), which moves from linear to non-linear risk assessment by applying systems engineering methods and analogs developed to address non-linear complex problems. The approach suggests dynamically integrating principal factors that define infrastructure failure risk using a unique platform that leverages Geospatial Information System services and extensions in an unprecedented manner. RiskDIMS is expected to produce results that are often counterintuitive and unexpected, but aligned to our complex reality, suggesting that the combination of geospatial and temporal analyses is required for sustainable risk-based decision making. To better illustrate the value added of temporal analysis in risk assessment, this study also develops and implements a non-linear dynamic model to simulate the behavior over time of infrastructure failure risk associated with an existing network of levees in New Orleans due to diverse infrastructure management investments. Although, the framework and RiskDIMS are discussed here in the context of levees, the concept applies to other critical infrastructure assets and systems. This research aims to become the foundation for future risk analysis system implementation. / Master of Science

[ES] Las redes de sensores inalámbricas (WSN) han experimentado un resurgimiento debido al desarrollo de la Internet de las Cosas (IoT). Una de las características de las aplicaciones de la IoT es la necesidad de hacer uso de dispositivos sensores y actuadores. En aplicaciones como automatización de edificios, de gestión energética, industriales o de salud, los nodos sensores que componen la WSN, transmiten información a un colector central o sink. La información es posteriormente procesada, analizada y utilizada para propósitos específicos. En cada una de estas aplicaciones, los dispositivos sensores pueden considerarse como parte de una WSN. En ese sentido el modelado y la evaluación de las prestaciones en las WSN es importante, ya que permite obtener una visión más clara de su comportamiento, facilitando un adecuado diseño y una exitosa puesta en operación. En el presente trabajo de tesis se han desarrollado modelos matemáticos para evaluar las prestaciones de WSN, los cuales están basados en Cadenas de Markov en Tiempo Discreto (DTMC). Los parámetros de prestaciones elegidos para la evaluación son: energía consumida promedio, eficiencia energética, caudal cursado y retardo promedio de los paquetes. Los resultados que se han obtenido han sido validados por medio de simulación basada en eventos discretos (DES). Existen estudios de WSN en escenarios homogéneos, donde los nodos que componen la red inalámbrica son del mismo tipo y tienen las mismas características de operación. En estos análisis se definen WSN homogéneas compuestas por un nodo central o sumidero (sink), que recibe la información de los nodos sensores localizados alrededor, formando una célula o cluster. Estos nodos realizan las transmisiones en SPT (Single Packet Transmission), enviando un solo paquete por ciclo de transmisión. Sin embargo, es posible encontrar, más ahora con el desarrollo de la IoT, escenarios donde coexisten distintos tipos de nodos, con características diferentes y, por tanto, con requerimientos de operación específicos. Esto da lugar a la formación de clusters cuyos nodos tienen aplicaciones distintas, desigual consumo de energía, diversas tasas de trasmisión de datos, e incluso diferentes prioridades de acceso al canal de transmisión. Este tipo de escenarios, que denominamos heterogéneos, forman parte de los escenarios estudiados en el presente trabajo de tesis. En una primera parte, se ha desarrollado un modelo para evaluar las prestaciones de una WSN heterogénea y con prioridades de acceso al medio. El modelado incluye un par de DTMC de dos dimensiones (2D-DTMC) cada una, cuya solución en términos de la distribución de probabilidad estacionaria, es utilizada para determinar los parámetros de prestaciones. Se desarrollan, por tanto, expresiones cerradas para los parámetros de prestaciones, en función de la distribución estacionaria que se ha obtenido a partir de la solución de las 2D-DTMC. En una segunda parte, se desarrolla un modelo analítico también pensado para escenarios heterogéneos y con prioridades, pero en el que los nodos de la WSN, cuando consiguen acceso al canal, transmiten un conjunto de paquetes en vez de uno solo como en el modelo de la primera parte. Estos dos modos de operación de los sensores los denominamos aggregated packet trans- mission (APT) y single packet transmission (SPT), respectivamente. El número de paquetes que un nodo funcionando en APT trasmite cuando accede al canal es el menor entre un parámetro configurable y el número de paquetes que tuviera en la cola en ese momento. Este modo de operación consigue una mayor eficiencia energética y un aumento en el caudal cursado, además de una disminución en el retardo promedio de los paquetes. En una tercera parte, se propone un nuevo procedimiento analítico para la determinación del consumo energético de los nodos que conforman una WSN. A diferencia de los métodos de cálculo anteriores, la nueva prop / [CA] Les xarxes de sensors sense fils (WSN) han experimentat un ressorgiment causa de al desenvolupament de la Internet de les Coses (IoT). Una de les característiques de IoT és la inclusió, en les seves aplicacions, de dispositius sensors i actuadors. En aplicacions com automatització d'edificis, de gestió energètica, industrials o de salut, els nodes sensors que componen la WSN, transmeten informació a un col·lector central o sink. La informació és posteriorment processada, analitzada i utilitzada per a propòsits específics. En cadascuna d'aquestes aplicacions, els dispositius sensors poden considerar com a part d'una WSN. En aquest sentit el modelitzat i l'avaluació de l'acompliment en les WSN és important, ja que permet obtenir una visió més clara del seu comportament, facilitant un adequat disseny i una exitosa posada en operació. En el present treball de tesi s'han desenvolupat models matemàtics per avaluar l'acompliment de WSN, els quals estan basats en Cadenes de Markov en Temps Discret (DTMC). Els paràmetres d'acompliment obtinguts per a l'avaluació són: energia consumida mitjana, eficiència energètica, cabal cursat i retard mitjà dels paquets. Els resultats que s'han obtingut, han estat validats per mitjà de simulació basada en esdeveniments discrets (DES). Existeixen estudis de WSN en escenaris homogenis, on els nodes que componen la xarxa sense fils són de el mateix tipus i tenen les mateixes característiques d'operació. En aquests anàlisis prèvies es defineixen WSN homogènies compostes per un node central o embornal (sink), que rep la informació dels nodes sensors localitzats al voltant, formant una cèl·lula o cluster. Aquests nodes realitzen les transmissió en SPT (Single Packet Transmission), és a dir, enviant un sol paquet cada vegada que transmeten. No obstant això, és possible trobar, més ara amb el desenvolupament de la IOT, escenaris on hi ha una coexistència de distints tipus de nodes, amb característiques diferents i, per tant, amb requeriments d'operació específics. Això dona lloc a formació de clusters els nodes tenen aplicacions diferents, desigual consum d'energia, diverses taxes de transmissió de dades, i fins i tot diferent prioritats d'accés a canal de transmissió. Aquest tipus d'escenaris, que anomenem heterogenis, formen part dels escenaris estudiats en el present treball de tesi. En una primera part, s'ha desenvolupat un model per avaluar l'acompliment d'una WSN heterogènia i amb prioritats d'accés al medi. El modelitzat inclou un parell DTMC de dues dimensions (2D-DTMC), la solució en termes de la distribució estacionària de probabilitat, és utilitzada per obtenir posteriorment els paràmetres d'acompliment. Es desenvolupen, per tant, expressions tancades per a la determinació dels paràmetres d'acompliment, on és substituïda la distribució estacionària que s'ha obtingut a partir de la solució de les 2D-DTMC. En una segona part, es desenvolupa un model, en el qual els nodes pertanyents a la WSN, poden transmetre els seus paquets en agregat (APT) en escenaris heterogenis i amb prioritats. A diferència del model anterior, on els nodes transmeten un paquet per cicle (SPT), en APT els nodes poden transmetre més d'un paquet. Això porta com a conseqüència una major eficiència energètica, a més d'un augment en el cabal cursat i disminució en el retard mitjana. En una tercera part, es proposa un nou desenvolupament analític per a la determinació del consum energètic dels nodes que conformen una WSN. A diferència de les expressions utilitzades anteriorment per al càlcul del consum energètic, aquesta proposta alternativa permet obtenir resultats més precisos a través del desenvolupament d'expressions més intuïtives i sistemàtiques. Amb aquest nou procediment, es realitzen estudis energètics per WSN en escenaris homogenis i heterogenis. / [EN] Wireless sensor networks (WSN) have experienced a resurgence due to the development of the Internet of Things (IoT). One of the characteristics of IoT is the deployment of applications that require sensor devices and actuators. In applications such as building automation, energy management, industrial or health, the sensor nodes that make up the WSN transmit information to a central collector or sink. The information is processed, analyzed, and used for specific purposes. In each of these applications, the sensor devices can be considered part of a WSN. In this sense, the modeling and performance evaluation of WSN is important, since it allows obtaining a clearer vision of their behavior, facilitating an adequate design and a successful operation. In the present thesis, analytical models based on Discrete Time Markov Chains (DTMC) have been developed to evaluate the performance of WSN. The parameters defined for the performance evaluation are: average consumed energy, energy efficiency, throughput and average packet delay. The obtained results have been validated by means of discrete event simulation (DES). There are studies of WSN in homogeneous scenarios, where the nodes that compose the WSN are of the same type and have the same operating characteristics. In these previous studies, homogeneous WSN are defined as a cell or cluster composed of a central node or sink, which receives the information from the sensor nodes located around it. These nodes operate in SPT (Single Packet Transmission), sending a single packet per transmission cycle. However, it is possible to find, especially now with the development of the IoT, scenarios where different types of nodes coexist, although they have different characteristics or specific operational requirements. This results in the formation of clusters whose nodes have different applications, uneven power consumption, different data transmission rates, and even different priorities for access to the transmission channel. These types of scenarios, which we call heterogeneous, are part of the scenarios studied in this thesis work. In the first part, a model has been developed to evaluate the performance of a heterogeneous WSN and with priorities to access a common channel. The model includes a two-dimensional DTMC pair (2D-DTMC), whose solution in terms of the stationary probability distribution is used to obtain the performance parameters. Closed expressions are provided for the determination of performance parameters of interest, given in terms of the stationary distribution of the 2D-DTMC. In a second part, an analytical model is developed to evaluate the performance of a heterogeneous WSN, where nodes operate in aggregate packet transmission (APT) mode and deploy different channel access priorities. Un like the previous model, where the nodes transmit one packet per cycle (SPT) when they gain access to the channel, in APT the nodes can transmit a number of packets larger than one, that is the minimum between a configurable parameter and the number of packets in the packet queue of the node. This results in greater energy efficiency and throughput, while decreases the average packet delay. In a third part, a new analytical model is proposed to determine the energy consumption of the nodes that make up a WSN. Unlike previous computation procedures, this alternative proposal is based on more intuitive and systematic expressions and allows to obtain more accurate results. With this new procedure, energy studies are performed for WSN in homogeneous and heterogeneous scenarios. / Este trabajo se ha desarrollado en el marco de los siguientes proyectos de investigación: Platform of Services for Smart Cities with Dense Machine to Machine Networks, PLASMA, TIN2013-47272-C2-1-R and New Paradigms of Elastic Networks for a World Radically Based on Cloud and Fog Computing, Elastic Networks, TEC2015-71932-REDT. También quisiera agradecer el apoyo recibido por parte de the European Union under the program Erasmus Mundus Partnerships, project EuroinkaNet, GRANT AGREEMENT NUMBER - 2014-0870/001/001 y La Secretaria de Educación Pública (México), bajo el Programa para el Desarrollo Profesional Docente: SEP-SES (DSA/103.5/15/6629). / Portillo Jiménez, C. (2021). Modelado y evaluación de prestaciones de redes de sensores inalámbricos heterogéneos con ciclo de trabajo síncrono [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/171275

The electric power systems are currently being transformed through the integration of intermittent renewable energy resources and new types of electric loads. These developments run the risk of increasing mismatches between electricity supply and demand, and may cause non-favorable utilization rates of some power system components. Using Demand Response (DR) from flexible loads in the building stock is a promising solution to overcome these challenges for electricity market actors. However, as DR is not used at a large scale today, there are validity concerns regarding its cost-benefit and reliability when compared to traditional investment options in the power sector, e.g. network refurbishment. To analyze the potential in DR solutions, bottom-up simulation models which capture consumption processes in buildings is an alternative. These models must be simple enough to allow aggregations of buildings to be instantiated and at the same time intricate enough to include variations in individual behaviors of end-users. This is done so the electricity market actor can analyze how large volumes of flexibility acts in various market and power system operation contexts, but also can appreciate how individual end-users are affected by DR actions in terms of cost and comfort. The contribution of this thesis is bottom-up simulation models for generating load profiles in detached houses and office buildings. The models connect end-user behavior with the usage of appliances and hot water loads through non-homogenous Markov chains, along with physical modeling of the indoor environment and consumption of heating and cooling loads through lumped capacitance models. The modeling is based on a simplified approach where openly available data and statistics are used, i.e. data that is subject to privacy limitations, such as smart meter measurements are excluded. The models have been validated using real load data from detached houses and office buildings, related models in literature, along with energy-use statistics from national databases. The validation shows that the modeling approach is sound and can provide reasonably accurate load profiles as the error results are in alignment with related models from other research groups. This thesis is a composite thesis of five papers. Paper 1 presents a bottom-up simulation model to generate load profiles from space heating, hot water and appliances in detached houses. Paper 2 presents a data analytic framework for analyzing electricity-use from heating ventilation and air conditioning (HVAC) loads and appliance loads in an office building. Paper 3 presents a non-homogeneous Markov chain model to simulate representative occupancy profiles in single office rooms. Paper 4 utilizes the results in paper 2 and 3 to describe a bottom-up simulation model that generates load profiles in office buildings including HVAC loads and appliances. Paper 5 uses the model in paper 1 to analyze the technical feasibility of using DR to solve congestion problems in a distribution grid. / Integrering av förnybara energikällor och nya typer av laster i de elektriska energisystemen är möjliga svar till klimatförändringar och uttömning av ändliga naturresurser. Denna integration kan dock öka obalanserna mellan utbud och efterfrågan av elektricitet, och orsaka en ogynnsam utnyttjandegrad av vissa kraftsystemkomponenter. Att använda efterfrågeflexibilitet (Demand Response) i byggnadsbeståndet är en möjlig lösning till dessa problem för olika elmarknadsaktörer. Men eftersom efterfrågeflexibilitet inte används i stor skala idag finns det obesvarade frågor gällande lösningens kostnadsnytta och tillförlitlighet jämfört med traditionella investeringsalternativ i kraftsektorn. För att analysera efterfrågeflexibilitetslösningar är botten-upp-simuleringsmodeller som fångar elförbrukningsprocesser i byggnaderna ett alternativ. Dessa modeller måste vara enkla nog för att kunna representera aggregeringar av många byggnader men samtidigt tillräckligt komplicerade för att kunna inkludera unika slutanvändarbeteenden. Detta är nödvändigt när elmarknadsaktören vill analysera hur stora volymer efterfrågeflexibilitet påverkar elmarknaden och kraftsystemen, men samtidigt förstå hur styrningen inverkar på den enskilda slutanvändaren.  Bidraget från denna avhandling är botten-upp-simuleringsmodeller för generering av elförbrukningsprofiler i småhus och kontorsbyggnader. Modellerna kopplar slutanvändarbeteende med elförbrukning från apparater och varmvattenanvändning tillsammans med fysikaliska modeller av värmedynamiken i byggnaderna. Modellerna är byggda på en förenklad approach som använder öppen data och statistisk, där data som har integritetsproblem har exkluderats. Simuleringsresultat har validerats mot elförbrukningsdata från småhus och kontorsbyggnader,  relaterade modeller från andra forskargrupper samt energistatistik från nationella databaser. Valideringen visar att modellerna kan generera elförbrukningsprofiler med rimlig noggrannhet. Denna avhandling är en sammanläggningsavhandling bestående av fem artiklar. Artikel 1 presenterar botten-upp-simuleringsmodellen för genereringen av elförbrukningsprofiler från uppvärmning, varmvatten och apparater i småhus. Artikel 2 presenterar ett dataanalytiskt ramverk för analys av elanvändningen från uppvärmning, ventilation, och luftkonditioneringslaster (HVAC) och apparatlaster i en kontorsbyggnad. Artikel 3 presenterar en icke-homogen Markovkedjemodell för simulering av representativa närvaroprofiler i enskilda kontorsrum. Artikel  4 använder resultaten i artiklarna  2 och 3 för att beskriva en botten-upp-simuleringsmodell för generering av elförbrukningsprofiler från HVAC-laster och apparater i kontorsbyggnader. Artikel  5 använder modellen i artikel 1 för att analysera den tekniska möjligheten att använda efterfrågeflexibilitet för att lösa överbelastningsproblem i ett eldistributionsnät. / <p>QC 20160329</p>

Cette thèse étudie deux modèles paramétriques et non paramétriques pour le changement de représentation. L'objectif des deux modèles diffère. Le premier cherche une représentation en plus grande dimension pour gagner en robustesse. L'objectif est de répartir uniformément l’information d’un signal sur toutes les composantes de sa représentation en plus grande dimension. La recherche d'un tel code s'exprime comme un problème inverse impliquant une régularisation de type norme infinie. Nous proposons une formulation bayésienne du problème impliquant une nouvelle loi de probabilité baptisée démocratique, qui pénalise les fortes amplitudes. Deux algorithmes MCMC proximaux sont présentés pour approcher des estimateurs bayésiens. La méthode non supervisée présentée est appelée BAC-1. Des expériences numériques illustrent les performances de l’approche pour la réduction de facteur de crête. Le second modèle identifie un sous-espace pertinent de dimension réduite à des fins de modélisation. Mais les méthodes probabilistes proposées nécessitent généralement de fixer à l'avance la dimension du sous-espace. Ce travail introduit BNP-PCA, une version bayésienne non paramétrique de l'analyse en composantes principales. La méthode couple une loi uniforme sur les bases orthonormales à un a priori non paramétrique de type buffet indien pour favoriser une utilisation parcimonieuse des composantes principales et aucun réglage n'est nécessaire. L'inférence est réalisée à l'aide des méthodes MCMC. L'estimation de la dimension du sous-espace et le comportement numérique de BNP-PCA sont étudiés. Nous montrons la flexibilité de BNP-PCA sur deux applications / This thesis proposes Bayesian parametric and nonparametric models for signal representation. The first model infers a higher dimensional representation of a signal for sake of robustness by enforcing the information to be spread uniformly. These so called anti-sparse representations are obtained by solving a linear inverse problem with an infinite-norm penalty. We propose in this thesis a Bayesian formulation of anti-sparse coding involving a new probability distribution, referred to as the democratic prior. A Gibbs and two proximal samplers are proposed to approximate Bayesian estimators. The algorithm is called BAC-1. Simulations on synthetic data illustrate the performances of the two proposed samplers and the results are compared with state-of-the art methods. The second model identifies a lower dimensional representation of a signal for modelisation and model selection. Principal component analysis is very popular to perform dimension reduction. The selection of the number of significant components is essential but often based on some practical heuristics depending on the application. Few works have proposed a probabilistic approach to infer the number of significant components. We propose a Bayesian nonparametric principal component analysis called BNP-PCA. The proposed model involves an Indian buffet process to promote a parsimonious use of principal components, which is assigned a prior distribution defined on the manifold of orthonormal basis. Inference is done using MCMC methods. The estimators of the latent dimension are theoretically and empirically studied. The relevance of the approach is assessed on two applications

<p>Computer programs keep finding their ways into new safety-critical applications, while at the same time growing more complex. This calls for new and better methods to verify the correctness of software. We focus on one approach to verifying systems, namely that of <i>model checking</i>. At first, we investigate two categories of problems related to model checking: <i>games</i> and <i>stochastic infinite-state systems</i>. In the end, we join these two lines of research, by studying <i>stochastic infinite-state games</i>.</p><p>Game theory has been used in verification for a long time. We focus on finite-state 2-player parity and limit-average (mean payoff) games. These problems have applications in model checking for the <i>μ</i>-calculus, one of the most expressive logics for programs. We give a simplified proof of memoryless determinacy. The proof applies <i>both</i> to parity and limit-average games. Moreover, we suggest a strategy improvement algorithm for limit-average games. The algorithm is discrete and strongly subexponential.</p><p>We also consider probabilistic infinite-state systems (Markov chains) induced by three types of models. <i>Lossy channel systems (LCS)</i> have been used to model processes that communicate over an unreliable medium. <i>Petri nets</i> model systems with unboundedly many parallel processes. <i>Noisy Turing machines</i> can model computers where the memory may be corrupted in a stochastic manner. We introduce the notion of <i>eagerness</i> and prove that all these systems are eager. We give a scheme to approximate the value of a reward function defined on paths. Eagerness allows us to prove that the scheme terminates. For probabilistic LCS, we also give an algorithm that approximates the limit-average reward. This quantity describes the long-run behavior of the system.</p><p>Finally, we investigate Büchi games on probabilistic LCS. Such games can be used to model a malicious cracker trying to break a network protocol. We give an algorithm to solve these games.</p>

Computer programs keep finding their ways into new safety-critical applications, while at the same time growing more complex. This calls for new and better methods to verify the correctness of software. We focus on one approach to verifying systems, namely that of model checking. At first, we investigate two categories of problems related to model checking: games and stochastic infinite-state systems. In the end, we join these two lines of research, by studying stochastic infinite-state games. Game theory has been used in verification for a long time. We focus on finite-state 2-player parity and limit-average (mean payoff) games. These problems have applications in model checking for the μ-calculus, one of the most expressive logics for programs. We give a simplified proof of memoryless determinacy. The proof applies both to parity and limit-average games. Moreover, we suggest a strategy improvement algorithm for limit-average games. The algorithm is discrete and strongly subexponential. We also consider probabilistic infinite-state systems (Markov chains) induced by three types of models. Lossy channel systems (LCS) have been used to model processes that communicate over an unreliable medium. Petri nets model systems with unboundedly many parallel processes. Noisy Turing machines can model computers where the memory may be corrupted in a stochastic manner. We introduce the notion of eagerness and prove that all these systems are eager. We give a scheme to approximate the value of a reward function defined on paths. Eagerness allows us to prove that the scheme terminates. For probabilistic LCS, we also give an algorithm that approximates the limit-average reward. This quantity describes the long-run behavior of the system. Finally, we investigate Büchi games on probabilistic LCS. Such games can be used to model a malicious cracker trying to break a network protocol. We give an algorithm to solve these games.

Nous considérons des processus de diffusion, définis par des équations différentielles stochastiques, et puis nous nous intéressons à des problèmes de premier passage pour les chaînes de Markov en temps discret correspon- dant à ces processus de diffusion. Comme il est connu dans la littérature, ces chaînes convergent en loi vers la solution des équations différentielles stochas- tiques considérées. Notre contribution consiste à trouver des formules expli- cites pour la probabilité de premier passage et la durée de la partie pour ces chaînes de Markov à temps discret. Nous montrons aussi que les résultats ob- tenus convergent selon la métrique euclidienne (i.e topologie euclidienne) vers les quantités correspondantes pour les processus de diffusion. En dernier lieu, nous étudions un problème de commande optimale pour des chaînes de Markov en temps discret. L’objectif est de trouver la valeur qui mi- nimise l’espérance mathématique d’une certaine fonction de coût. Contraire- ment au cas continu, il n’existe pas de formule explicite pour cette valeur op- timale dans le cas discret. Ainsi, nous avons étudié dans cette thèse quelques cas particuliers pour lesquels nous avons trouvé cette valeur optimale. / We consider diffusion processes, defined by stochastic differential equa- tions, and then we focus on first passage problems for Markov chains in dis- crete time that correspond to these diffusion processes. As it is known in the literature, these Markov chains converge in distribution to the solution of the stochastic differential equations considered. Our contribution is to obtain ex- plicit formulas for the first passage probability and the duration of the game for the discrete-time Markov chains. We also show that the results obtained converge in the Euclidean metric to the corresponding quantities for the diffu- sion processes. Finally we study an optimal control problem for Markov chains in discrete time. The objective is to find the value which minimizes the expected value of a certain cost function. Unlike the continuous case, an explicit formula for this optimal value does not exist in the discrete case. Thus we study in this thesis some particular cases for which we found this optimal value.

L’objet de cette thèse est d’établir des théorèmes limites locaux pour des marches aléatoires réfléchies sur N. La théorie des fluctuations des marches aléatoires et la factorisation de Wiener- Hopf y jouent un rôle important. On développera dans la première partie une approche classique que l’on appliquera à l’étude des marches aléatoires sur R+ avec réflexions non élastiques en 0. Dans la deuxième partie, on explicitera une méthode différente qui fait intervenir des outils algébriques, d’analyse complexe et des techniques de factorisation utilisant de manière essentielle les fonctions génératrices. Cette approche a été développée il y a une cinquantaine d’année pour l’étude de marches de Markov, elle sera présentée dans cette partie dans le cas des marches aléatoires à pas i.i.d. où un certain nombre de simplifications apparaissent et sera ensuite utilisée pour étudier les marches aléatoires sur N avec réflexions élastiques ou non élastiques en zéro. Finalement, dans la dernière partie, nous mettons en place les outils nécessaires pour établir une factorisation de Wiener-Hopf dans un cadre markovien afin d’étudier les fluctuations des marches de Markov sur Z; nous reprenons des travaux anciens dont les démonstrations méritaient d’être détaillées, l’objectif à moyen terme étant d’appliquer les méthodes algébriques décrites ci-dessus pour l’étude de marches de Markov réfléchies sur N. / The purpose of this thesis is to establish some local limit theorems for reflected random walks on N. The fluctuations theory and the Wiener-Hopf factorization play a crucial role. We will develop in the first part a classical approach that we will apply to the study of random walks on R+ with non-elastic reflections at zero. In the second part, we will explicit a different method which involves algebraic tools, complex analysis and factorization techniques, using in an essential way generating functions. These approach was developed 50 years ago to cover Markov walks, it will be presented in this part in the case of random walks with i.i.d jumps where many simplifications appear and will be then used to study random walks on N with either elastic or non-elastic reflections at zero. Finally, in the last part, we will introduce the useful tools to establish a Wiener-Hopf factorization in a markovian framework in order to study the fluctuations of Markov walks on Z. We investigate some previous work, especially some proofs that warranted to be more detailed, with a mediumterm objective of applying the algebraic tools described above to study reflected Markov walks on N.

Made available in DSpace on 2016-06-02T20:06:06Z (GMT). No. of bitstreams: 1 4331.pdf: 1908865 bytes, checksum: d564b46a6111fdca6f7cc9f4d5596637 (MD5) Previous issue date: 2009-03-18 / Universidade Federal de Minas Gerais / The Weibull distribuition is a common initial choice for modeling data with monotone hazard rates. However, such distribution fails to provide a reasonable parametric _t when the hazard function is unimodal or bathtub-shaped. In this context, Cooray (2006) proposed a generalization of the Weibull family by considering the distributions of the odds of Weibull and inverse Weibull families, referred as the odd Weibull family which is not just useful for modeling unimodal and bathtub-shaped hazards, but it is also convenient for testing goodness-of-_t of Weibull and inverse Weibull as submodels. In this project we have systematically studied the odd Weibull family along with its properties, showing motivations for its utilization, inserting covariates in the model, pointing out some troubles associated with the maximum likelihood estimation and proposing interval estimation and hypothesis test construction methodologies for the model parameters. We have also compared resampling results with asymptotic ones. Coverage probability from proposed con_dence intervals and size and power of considered hypothesis tests were both analyzed as well via Monte Carlo simulation. Furthermore, we have proposed a Bayesian estimation methodology for the model parameters based in Monte Carlo Markov Chain (MCMC) simulation techniques. / A distribuição Weibull é uma escolha inicial freqüente para modelagem de dados com taxas de risco monótonas. Entretanto, esta distribuição não fornece um ajuste paramétrico razoável quando as funções de risco assumem um formato unimodal ou em forma de banheira. Neste contexto, Cooray (2006) propôs uma generalização da família Weibull considerando a distribuição da razão de chances das famílias Weibull e Weibull inversa, referida como família Weibull de razão de chances. Esta família não é apenas conveniente para modelar taxas de risco unimodal e banheira, mas também é adequada para testar a adequabilidade do ajuste das famílias Weibull e Weibull inversa como submodelos. Neste trabalho, estudamos sistematicamente a família Weibull de razão de chances e suas propriedades, apontando as motivações para o seu uso, inserindo covariáveis no modelo, veri_cando as di_culdades referentes ao problema da estimação de máxima verossimilhança dos parâmetros do modelo e propondo metodologia de estimação intervalar e construção de testes de hipóteses para os parâmetros do modelo. Comparamos os resultados obtidos por meio dos métodos de reamostragem com os resultados obtidos via teoria assintótica. Tanto a probabilidade de cobertura dos intervalos de con_ança propostos quanto o tamanho e poder dos testes de hipóteses considerados foram estudados via simulação de Monte Carlo. Além disso, propusemos uma metodologia Bayesiana de estimação para os parâmetros do modelo baseados em técnicas de simulação de Monte Carlo via Cadeias de Markov.