Fluid dynamics models provide a powerful deterministic technique to approximate stochasticity in a variety of application areas. In this paper, we study two classes of fluid models, investigate their relationship as well as some of their applications. This analysis allows us to provide analytical models of travel times as they arise in dynamically evolving environments, such as transportation networks as well as supply chains. In particular, using the laws of hydrodynamic theory, we first propose and examine a general second order fluid model. We consider a first-order approximation of this model and show how it is helpful in analyzing the dynamic traffic equilibrium problem. Furthermore, we present an alternate class of fluid models that are traditionally used in the context of dynamic traffic assignment. By interpreting travel times as price/inventory-sojourn-time relationships, we are also able to connect this approach with a tractable fluid model in the context of dynamic pricing and inventory management. Finally, we investigate the relationship between these two classes of fluid models. / Singapore-MIT Alliance (SMA)

Dans cette thèse, on étudie diverses propriétés dimensionnelles de la régularité de processus de difusions à sauts, solution d’une classe d’équations différentielles stochastiques à sauts. En particulier, on décrit la fluctuation de la régularité höldérienne de ces processus et celle de la dimension locale pour la mesure d’occupation qui leur est associée en calculant leur spectre multifractal. La dimension de Hausdorff de l’image et du graphe de ces processus ont aussi étudiées.Dans le dernier chapitre, on applique une nouvelle notion de dimension de grande échelle pour décrire l’asymptote à l’infini du temps de séjour d’un mouvement brownien en dimension 1 sous des frontières glissantes / In this dissertation, we study various dimension properties of the regularity of jump di usion processes, solution of a class of stochastic di erential equations with jumps. In particular, we de- scribe the uctuation of the Hölder regularity of these processes and that of the local dimensions of the associated occupation measure by computing their multifractal spepctra. e Hausdor dimension of the range and the graph of these processes are also calculated.In the last chapter, we use a new notion of “large scale” dimension in order to describe the asymptotics of the sojourn set of a Brownian motion under moving boundaries

In recent years, the increase in the population of mobile users and the advances in computational capabilities of mobile devices have led to an exponentially increasing traffic load on the wireless networks. This trend is foreseen to continue in the future due to the emerging applications such as cellular Internet of things (IoT) and machine type communications (MTC). Since the spectrum resources are limited, the only promising way to keep pace with the future demand is through aggressive spatial reuse of the available spectrum which can be realized in the networks through dense deployment of small cells. There are many challenges associated with such densely deployed heterogeneous networks (HetNets). The main challenges which are considered in this research work are capacity enhancement, velocity estimation of mobile users, and energy efficiency enhancement. We consider different approaches for capacity enhancement of the network. In the first approach, using stochastic geometry we theoretically analyze time domain inter-cell interference coordination techniques in a two-tier HetNet and optimize the parameters to maximize the capacity of the network. In the second approach, we consider optimization of the locations of aerial bases stations carried by the unmanned aerial vehicles (UAVs) to enhance the capacity of the network for public safety and emergency communications, in case of damaged network infrastructure. In the third approach, we introduce a subsidization scheme for the service providers through which the network capacity can be improved by using regulatory power of the government. Finally, we consider the approach of device-to-device communications and multi-hop transmissions for enhancing the capacity of a network. Velocity estimation of high speed mobile users is important for effective mobility management in densely deployed small cell networks. In this research, we introduce two novel methods for the velocity estimation of mobile users: handover-count based velocity estimation, and sojourn time based velocity estimation. Using the tools from stochastic geometry and estimation theory, we theoretically analyze the accuracy of the two velocity estimation methods through Cramer-Rao lower bounds (CRLBs). With the dense deployment of small cells, energy efficiency becomes crucial for the sustained operation of wireless networks. In this research, we jointly study the energy efficiency and the spectral efficiency in a two-tier HetNet. We optimize the parameters of inter-cell interference coordination technique and study the trade-offs between the energy efficiency and spectral efficiency of the HetNet.

Part I - Level crossings and other level functionals.<br />Part II - Some contributions in statistics of extremes and in statistical mechanics.