This thesis studies a spin-star model and an open spin-star model. The spin-star model is exactly solvable, and the approximation methods can be applied and compared with the exact solution. As this model is shown to be non-Gaussian and non-Markovian, Born-Markovian approximation is not valid. The comparison of Nakajima-Zwanzig and time-convolutionless methods show that the performances of those two techniques depend on the specific property of the model. The open spin-star model is not exactly solvable and the analytical solution under Gaussian approximation is obtained.

This thesis studies a spin-star model and an open spin-star model. The spin-star model is exactly solvable, and the approximation methods can be applied and compared with the exact solution. As this model is shown to be non-Gaussian and non-Markovian, Born-Markovian approximation is not valid. The comparison of Nakajima-Zwanzig and time-convolutionless methods show that the performances of those two techniques depend on the specific property of the model. The open spin-star model is not exactly solvable and the analytical solution under Gaussian approximation is obtained.

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It has been almost eighty years since quantum mechanics emerged as a complete theory, yet debates about how should quantum mechanics be interpreted still occur. Interpretations are many and varied, some taking us as fundamental in determining reality (orthodox interpretation), while others proposing that reality exists outside of us, but it is a lot more complicated than that implied by classical mechanics. In this thesis I am going to try to provide new light on this debate by investigating dynamics under both the orthodox and modal interpretation. In particular I will answer the question what is the interpretation of non-Markovian stochastic Schrodinger equations? I conclude that under the orthodox view these equations have only a numerical interpretation. They provide a rule for calculating the state of the system at time t if we made a measurement on the bath (a collection of oscillators {ak}) at that time, yielding results {zk}. However in the modal view they have a meaning: non-Markovian stochastic Schrodinger equations represent the evolution of the system part of the property state of the universe (bath + system).

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With the arrival of the New Basel Capital Accord, which was acknowledged by most of Czech banks during the years 2007 and 2008, the importance of internal ratings for the assessment of the health of the whole financial sector has grown tremendously. Internal ratings are now used for the calculation and allocation of capital, as well as for the determination of interest rates and margins. It is the changes of internal ratings which are obvious applications of the multi-states models. Through the use of methods usual for the Semimarkovian chains analysis, it is possible to analyze the structure of the internal ratings changes, to monitor the periods between successive changes, and to focus also on the transition matrices themselves. The important part of this work is the comparison of given parameters as observed during steady times, and during the nancial crises, which dates from the fall of the Lehman Brothers in September 2008.

L’objet de cette thèse est d’étudier la probabilité de survie d’un processus de branchement en environnement aléatoire markovien et d’étendre dans ce cadre les résultats connus en milieu aléatoire i.i.d.. le cœur de l’étude repose sur l’utilisation des théorèmes limites locaux pour une marche aléatoire centrée (Sn)n≥0 sur R à pas markoviens et pour (rnn)n≥0, où mn = min (0, S1,... , Sn). Pour traiter le cas d’un environnement aléatoire markovien, nous développons dans un premier temps une étude des théorèmes locaux pour une chaîne semi-markovienne à valeurs réelles en améliorant certains résultats déjà connus et développés initialement par E. L. Presman (voir aussi [21]). Nous utilisons ensuite ces résultats pour l’étude du comportement asymptotique de la probabilité de survie d’un processus de branchement critique en environnement aléatoire markovien. Les résultats principaux de cette thèse (théorème limite local et son application au processus de branchement critique eu milieu aléatoire) ont été acceptés et publiés dans le Comptes Rendus de l‘Académie des Sciences ([20]). Le texte principal de cette mémoire de thèse consisite les détails des preuves. / The purpose of this thesis is to study the survival probability of a branching process in markovian random environment and expand in this framework some known results which have been developed for a branching processus in i.i.d. random environment, the core of the study is based on the use of the local limit theorem for a centered random walk (Sn)n≥o on R with markovian increasements and for (mn)n≥0. where mn = min (O. S1,……. , Sn). In order to treat the case of a markovian random environment, we establish firstly a local limit theorem for a semi-markovian chain on R. which improves certain results developed initially by E. P. Presman (see also [21]). And then we use these results to study the asymptotic behavior of a critical branching process in markovian environment. The main results et this thesis (local limit theorem and its application to the critical branching process in random environment) are accepted and published in Comptes Rendus de l’Académie des Sciences ([20]). The principal text et this thesis contains the details of the proofs.

The purpose of this research is to investigate and analyse some batch arrival queueing systems with Bernoulli scheduled vacation process and single server providing service. The study aims to explore and extend the work done on vacation and unreliable queues with a combination of assumptions like balking and re-service, reneging during vacations, time homogeneous random breakdowns and fluctuating modes of service. We study the steady state properties, and also transient behaviour of such queueing systems. Due to vacations the arriving units already in the system may abandon the system without receiving any service (reneging). Customers may decide not to join the queue when the server is in either working or vacation state (balking). We study this phenomenon in the framework of two models; a single server with two types of parallel services and two stages of service. The model is further extended with re-service offered instantaneously. Units which join the queue but leave without service upon the absence of the server; especially due to vacation is quite a natural phenomenon. We study this reneging behaviour in a queueing process with a single server in the context of Markovian and non-Markovian service time distribution. Arrivals are in batches while each customer can take the decision to renege independently. The non-Markovian model is further extended considering service time to follow a Gamma distribution and arrivals are due to Geometric distribution. The closed-form solutions are derived in all the cases. Among other causes of service interruptions, one prime cause is breakdowns. We consider breakdowns to occur both in idle and working state of the server. In this queueing system the transient and steady state analysis are both investigated. Applying the supplementary variable technique, we obtain the probability generating function of queue size at random epoch for the different states of the system and also derive some performance measures like probability of server‟s idle time, utilization factor, mean queue length and mean waiting time. The effect of the parameters on some of the main performance measures is illustrated by numerical examples to validate the analytical results obtained in the study. The Mathematica 10 software has been used to provide the numerical results and presentation of the effects of some performance measures through plots and graphs.

When a ferromagnet is heated, it loses its magnetism. Stars and planets have magnetic fields, as does the Earth. But it is known that the center of the Earth is very hot. Therefore, to sustain the large magnetic field of a planet, we cannot look to simple ferromagnetism like that of a bar magnet, but we have to look at the movement of electric charges within the Earth’s molten core to generate magnetic field. This magnetic field sustainment against ohmic dissipation by turbulent flow is referred to as the turbulent dynamo effect. Theoretical research into the mechanisms that create the dynamo has been actively pursued for several decades, culminating recently in massive computer simulations of the Earth’s core. Most of these studies have employed the equations of magnetohydrodynamics (MHD), a nonlinear theory of electrically conducting fluids. The EDQNM (Eddy-Damped Quasi-Normal Markovian) closure is a statistical model designed so that the turbulence equations derived from Navier-Stokes dynamics can be closed and satisfy the realizability condition of positivity of the kinetic energy spectrum. In case of MHD turbulence, realizability requires more work. We have proved in an earlier work that equations analogous to those expected of the EDQNM closure for MHD without mean fields satisfy the appropriate realizability conditions (Turner and Pratt 1999). In this work, we discuss requirements needed to make the MHD equations realizable with mean fields, extending those of neutral fluid turbulence by Turner [1]. Finally, we discuss direct numerical simulations and the correspondence of the statistical theories with simulation results.

An Origin-Destination (OD) traffic matrix provides a major input to the design, planning and management of a telecommunications network. Since the Internet is being proposed as the principal delivery mechanism for telecommunications traffic at the present time, and this network is not owned or managed by a single entity, there are significant challenges for network planners and managers needing to determine equipment and topology configurations for the various sections of the Internet that are currently the responsibility of ISPs and traditional telcos. Planning of these sub-networks typically requires a traffic matrix of demands that is then used to infer the flows on the administrator's network. Unfortunately, computation of the traffic matrix from measurements of individual flows is extremely difficult due to the fact that the problem formulation generally leads to the need to solve an under-determined system of equations. Thus, there has been a major effort f rom among researchers to obtain the traffic matrix using various inference techniques. The major contribution of this thesis is the development of inference techniques for traffic matrix estimation problem according to three different approaches, viz: (1) deterministic, (2) statistical, and (3) dynamic approaches. Firstly, for the deterministic approach, the traffic matrix estimation problem is formulated as a nonlinear optimization problem based on the generalized Kruithof approach which uses the Kullback distance to measure the probabilistic distance between two traffic matrices. In addition, an algorithm using the Affine scaling method is developed to solve the constrained optimization problem. Secondly, for the statistical approach, a series of traffic matrices are obtained by applying a standard deterministic approach. The components of these matrices represent estimates of the volumes of flows being exchanged between all pairs of nodes at the respective measurement points and they form a stochastic counting process. Then, a Markovian Arrival Process of order two (MAP-2) is applied to model the counting processes formed from this series of estimated traffic matrices. Thirdly, for the dynamic approach, the dual problem of the multi-commodity flow problem is formulated to obtain a set of link weights. The new weight set enables flows to be rerouted along new paths, which create new constraints to overcome the under-determined nature of traffic matrix estimation. Since a weight change disturbs a network, the impact of weight changes on the network is investigated by using simulation based on the well-known ns2 simulator package. Finally, we introduce two network applications that make use of the deterministic and the statistical approaches to obtain a traffic matrix respectively and also describe a scenario for the use of the dynamic approach.