Virtualisation is one of the underlying technologies that led to the success of cloud computing platforms (CCPs). The technology, along with other features such as multitenancy allows delivering of computing resources in the form of service through efficient sharing of physical resources. As these resources are provided through virtualisation, a robust agreement is outlined for both the quantity and quality-of-service (QoS) in a service level agreement (SLA) documents. QoS is one of the essential components of SLA, where performance is one of its primary aspects. As the technology is progressively maturing and receiving massive acceptance, researchers from industry and academia continue to carry out novel theoretical and practical studies of various essential aspects of CCPs with significant levels of success. This thesis starts with the assessment of the current level of knowledge in the literature of cloud computing in general and CCPs in particular. In this context, a substantive literature review was carried out focusing on performance modelling, testing, analysis and evaluation of Infrastructure as a Service (IaaS), methodologies. To this end, a systematic mapping study (SMSs) of the literature was conducted. SMS guided the choice and direction of this research. The SMS was followed by the development of a novel open queueing network model (QNM) at equilibrium for the performance modelling and analysis of an OpenStack IaaS CCP. Moreover, it was assumed that an external arrival pattern is Poisson while the queueing stations provided exponentially distributed service times. Based on Jackson’s theorem, the model was exactly decomposed into individual M/M/c (c ≥ 1) stations. Each of these queueing stations was analysed in isolation, and closed-form expressions for key performance metrics, such as mean response time, throughput, server (resource) utilisation as well as bottleneck device were determined. Moreover, the research was extended with a proposed open QNM with a bursty external arrival pattern represented by a Compound Poisson Process (CPP) with geometrically distributed batches, or equivalently, variable Generalised Exponential (GE) interarrival and service times. Each queueing station had c (c ≥ 1) GE-type servers. Based on a generic maximum entropy (ME) product form approximation, the proposed open GE-type QNM was decomposed into individual GE/GE/c queueing stations with GE-type interarrival and service times. The evaluation of the performance metrics and bottleneck analysis of the QNM were determined, which provided vital insights for the capacity planning of existing CCP architectures as well as the design and development of new ones. The results also revealed, due to a significant impact on the burstiness of interarrival and service time processes, resulted in worst-case performance bounds scenarios, as appropriate. Finally, an investigation was carried out into modelling and analysis of performance and security trade-offs for a CCP architecture, based on a proposed generalised stochastic Petri net (GSPN) model with security-detection control model (SDCM). In this context, ‘optimal’ combined performance and security metrics were defined with both M-type or GE-type arrival and service times and the impact of security incidents on performance was assessed. Typical numerical experiments on the GSPN model were conducted and implemented using the Möbius package, and an ‘optimal’ trade-offs were determined between performance and security, which are crucial in the SLA of the cloud computing services. / Petroleum technology development fund (PTDF) of the government of Nigeria Usmanu Danfodiyo University, Sokoto

L'objet de cette thèse est d'étudier divers problèmes de statistique non-paramétrique dans le cadre d'un modèle d'interactions poissoniennes. De tels modèles sont, par exemple, utilisés en neurosciences pour analyser les interactions entre deux neurones au travers leur émission de potentiels d'action au cours de l'enregistrement de l'activité cérébrale ou encore en génomique pour étudier les distances favorisées ou évitées entre deux motifs le long du génome. Dans ce cadre, nous introduisons une fonction dite de reproduction qui permet de quantifier les positions préférentielles des motifs et qui peut être modélisée par l'intensité d'un processus de Poisson. Dans un premier temps, nous nous intéressons à l'estimation de cette fonction que l'on suppose très localisée. Nous proposons une procédure d'estimation adaptative par seuillage de coefficients d'ondelettes qui est optimale des points de vue oracle et minimax. Des simulations et une application en génomique sur des données réelles provenant de la bactérie E. coli nous permettent de montrer le bon comportement pratique de notre procédure. Puis, nous traitons les problèmes de test associés qui consistent à tester la nullité de la fonction de reproduction. Pour cela, nous construisons une procédure de test optimale du point de vue minimax sur des espaces de Besov faibles, qui a également montré ses performances du point de vue pratique. Enfin, nous prolongeons ces travaux par l'étude d'une version discrète en grande dimension du modèle précédent en proposant une procédure adaptative de type Lasso. / The subject of this thesis is the study of some adaptive nonparametric statistical problems in the framework of a Poisson interactions model. Such models are used, for instance, in neurosciences to analyze interactions between two neurons through their spikes emission during the recording of the brain activity or in genomics to study favored or avoided distances between two motifs along a genome. In this setting, we naturally introduce a so-called reproduction function that allows to quantify the favored positions of the motifs and which is considered as the intensity of a Poisson process. Our first interest is the estimation of this function assumed to be well localized. We propose a data-driven wavelet thresholding estimation procedure that is optimal from oracle and minimax points of view. Simulations and an application to genomic data from the bacterium E. coli allow us to show the good practical behavior of our procedure. Then, we deal with associated problems on tests which consist in testing the nullity of the reproduction function. For this purpose, we build a minimax optimal testing procedure on weak Besov spaces and we provide some simulations showing good practical performances of our procedure. Finally, we extend this work with the study of a high-dimensional discrete setting of our previous model by proposing an adaptive Lasso-type procedure.

As redes ópticas estão passando por mudanças significativas, impulsionadas pelo crescimento exponencial do tráfego, principalmente advindo de serviços multimídia e armazenamento em nuvem. Esta demanda exigirá aumento da capacidade da taxa de transmissão para padrões como 400 Gb/s e 1 Tb/s. Nesse contexto, foi proposta uma arquitetura de rede com grade de frequências granular flexível chamada elastic optical network (EON). A EON divide o espectro de frequências em fatias (slots) de tamanho fixo e aloca grupos de slots contíguos estritamente de acordo com os requisitos de banda das demandas de conexão, implicando eficiência de uso do espectro. Com o aumento significativo da taxa de transmissão, acentuou-se a preocupação em manter a sobrevivência da rede, já que pouco tempo de queda no serviço pode acarretar uma imensa perda de dados. Neste trabalho, investigamos esquemas de proteção baseados em caminhos compartilhados (shared-path protection, SPP) e esquemas de restauração de tráfego. Avaliamos esquemas divulgados na literatura como o dynamic load balancing shared-path protection (DLBSPP) e esquemas de restauração como o traffic aware restoration (TAR) e bandwidth squeezed restoration (BSR). Avaliamos também uma heurística de alocação de slots chamada inverted dual stack (IDS). O DLBSPP utiliza balanceamento dinâmico de carga para computar os caminhos primários e de proteção compartilhados. O TAR executa a restauração dinâmica ordenando as conexões por granularidade de banda. O BSR utiliza a capacidade de contração de banda do EON para restaurar conexões por meio da política de melhor esforço ou de banda garantida, dependendo do acordo de níveis de serviço do cliente. O esquema IDS concentra o maior número possível de slots compartilhados em uma região do espectro. As medidas de desempenho dos algoritmos são avaliadas segundo as métricas: probabilidade de bloqueio, taxa de utilização do espectro, número médio de hops e taxa de restauração falha. As simulações computacionais mostram o bom desempenho da utilização do esquema IDS com DLBSPP. / Optical networks are undergoing significant changes driven by the exponentially growing traffic, especially coming from multimedia and cloud storage services. This demand will require increasing of the transmission rate capacity as high as 400 Gb/s and 1 Tb/s. Within this context, it was proposed the elastic optical network (EON), which is a network architecture with flexible granular frequency grid. EON divides the frequency spectrum into slices (slots) of fixed size and allocates groups of contiguous slots strictly according to the bandwidth requirement of the connection demands, providing high spectrum use efficiency. The significant increase in transmission rate put emphasis on the need to maintain the survival of the network, since the occurrence of faults in the network nodes or links can cause huge loss of data. In this work, we investigate protection schemes based on shared-path protection (SPP) and traffic restoration schemes. We evaluate schemes related in the literature, such as the dynamic load balancing shared-path protection (DLBSPP), and restoration schemes such as the traffic aware restoration (TAR) and the bandwidth squeezed restoration (BSR). The DLBP scheme uses dynamic load balancing to compute primary and shared protection paths. The TAR performs dynamic restoration ordering the connections based on band granularity. The BSR uses EON\'s band squeezing feature to restore connections by means of the best effort or guaranteed bandwidth strategy, depending on the customer\'s service level agreement. IDS scheme concentrates the maximum possible number of shared slots in a given region of the spectrum. Performance of the algorithms are evaluated according to metrics: blocking probability, spectrum utilization rate, average number of hops and failure restoration rate. Computer simulations show that the use of the IDS scheme improves the performance of the investigated algorithms.

Multivariate mixed Poisson processes are special multivariate counting processes whose coordinates are, in general, dependent. The first part of this thesis is devoted to properties which multivariate counting processes may possess. Such properties are, for example, the Markov property, the multinomial property and regularity. With regard to regularity we study the properties of transition probabilities and intensities. The second part of this thesis restricts the class of all multivariate counting processes by additional assumptions leading to different types of multivariate mixed Poisson processes which, however, are connected with each other. Using a multivariate version of the Bernstein-Widder theorem, it is shown that multivariate mixed Poisson processes are characterized by the multinomial property. Furthermore, regularity of multivariate mixed Poisson processes and properties of their moments are studied in detail. Throughout this thesis, two types of stability of properties of multivariate counting processes are studied: It is shown that most properties of a multivariate counting process are stable under certain linear transformations including the selection of single coordinates and summation of all coordinates. It is also shown that the different types of multivariate mixed Poisson processes under consideration are in a certain sense stable in time.

As redes ópticas estão passando por mudanças significativas, impulsionadas pelo crescimento exponencial do tráfego, principalmente advindo de serviços multimídia e armazenamento em nuvem. Esta demanda exigirá aumento da capacidade da taxa de transmissão para padrões como 400 Gb/s e 1 Tb/s. Nesse contexto, foi proposta uma arquitetura de rede com grade de frequências granular flexível chamada elastic optical network (EON). A EON divide o espectro de frequências em fatias (slots) de tamanho fixo e aloca grupos de slots contíguos estritamente de acordo com os requisitos de banda das demandas de conexão, implicando eficiência de uso do espectro. Com o aumento significativo da taxa de transmissão, acentuou-se a preocupação em manter a sobrevivência da rede, já que pouco tempo de queda no serviço pode acarretar uma imensa perda de dados. Neste trabalho, investigamos esquemas de proteção baseados em caminhos compartilhados (shared-path protection, SPP) e esquemas de restauração de tráfego. Avaliamos esquemas divulgados na literatura como o dynamic load balancing shared-path protection (DLBSPP) e esquemas de restauração como o traffic aware restoration (TAR) e bandwidth squeezed restoration (BSR). Avaliamos também uma heurística de alocação de slots chamada inverted dual stack (IDS). O DLBSPP utiliza balanceamento dinâmico de carga para computar os caminhos primários e de proteção compartilhados. O TAR executa a restauração dinâmica ordenando as conexões por granularidade de banda. O BSR utiliza a capacidade de contração de banda do EON para restaurar conexões por meio da política de melhor esforço ou de banda garantida, dependendo do acordo de níveis de serviço do cliente. O esquema IDS concentra o maior número possível de slots compartilhados em uma região do espectro. As medidas de desempenho dos algoritmos são avaliadas segundo as métricas: probabilidade de bloqueio, taxa de utilização do espectro, número médio de hops e taxa de restauração falha. As simulações computacionais mostram o bom desempenho da utilização do esquema IDS com DLBSPP. / Optical networks are undergoing significant changes driven by the exponentially growing traffic, especially coming from multimedia and cloud storage services. This demand will require increasing of the transmission rate capacity as high as 400 Gb/s and 1 Tb/s. Within this context, it was proposed the elastic optical network (EON), which is a network architecture with flexible granular frequency grid. EON divides the frequency spectrum into slices (slots) of fixed size and allocates groups of contiguous slots strictly according to the bandwidth requirement of the connection demands, providing high spectrum use efficiency. The significant increase in transmission rate put emphasis on the need to maintain the survival of the network, since the occurrence of faults in the network nodes or links can cause huge loss of data. In this work, we investigate protection schemes based on shared-path protection (SPP) and traffic restoration schemes. We evaluate schemes related in the literature, such as the dynamic load balancing shared-path protection (DLBSPP), and restoration schemes such as the traffic aware restoration (TAR) and the bandwidth squeezed restoration (BSR). The DLBP scheme uses dynamic load balancing to compute primary and shared protection paths. The TAR performs dynamic restoration ordering the connections based on band granularity. The BSR uses EON\'s band squeezing feature to restore connections by means of the best effort or guaranteed bandwidth strategy, depending on the customer\'s service level agreement. IDS scheme concentrates the maximum possible number of shared slots in a given region of the spectrum. Performance of the algorithms are evaluated according to metrics: blocking probability, spectrum utilization rate, average number of hops and failure restoration rate. Computer simulations show that the use of the IDS scheme improves the performance of the investigated algorithms.

Multivariate mixed Poisson processes are special multivariate counting processes whose coordinates are, in general, dependent. The first part of this thesis is devoted to properties which multivariate counting processes may possess. Such properties are, for example, the Markov property, the multinomial property and regularity. With regard to regularity we study the properties of transition probabilities and intensities. The second part of this thesis restricts the class of all multivariate counting processes by additional assumptions leading to different types of multivariate mixed Poisson processes which, however, are connected with each other. Using a multivariate version of the Bernstein-Widder theorem, it is shown that multivariate mixed Poisson processes are characterized by the multinomial property. Furthermore, regularity of multivariate mixed Poisson processes and properties of their moments are studied in detail. Throughout this thesis, two types of stability of properties of multivariate counting processes are studied: It is shown that most properties of a multivariate counting process are stable under certain linear transformations including the selection of single coordinates and summation of all coordinates. It is also shown that the different types of multivariate mixed Poisson processes under consideration are in a certain sense stable in time.

This master's thesis deals with queueing theory and its application in designing node models in packet-switched network. There are described general principles of designing queueing theory models and its mathematical background. Further simulator of packet delay in network was created. This application implements two described models - M/M/1 and M/G/1. Application can be used for simulating network nodes and obtaining basic network characteristics like packet delay or packet loss. Next, lab exercise was created, in that exercise students familiarize themselves with basic concepts of queueing theory and examine both analytical and simulation approach to solving queueing systems.

Cette thèse traite de problèmes de dépendance entre processus stochastiques en temps continu. Ces résultats sont appliqués à la modélisation et à la gestion des risques des marchés de l'électricité.Dans une première partie, de nouvelles copules sont établies pour modéliser la dépendance entre deux mouvements Browniens et contrôler la distribution de leur différence. On montre que la classe des copules admissibles pour les Browniens contient des copules asymétriques. Avec ces copules, la fonction de survie de la différence des deux Browniens est plus élevée dans sa partie positive qu'avec une dépendance gaussienne. Les résultats sont appliqués à la modélisation jointe des prix de l'électricité et d'autres commodités énergétiques. Dans une seconde partie, nous considérons un processus stochastique observé de manière discrète et défini par la somme d'une semi-martingale continue et d'un processus de Poisson composé avec retour à la moyenne. Une procédure d'estimation pour le paramètre de retour à la moyenne est proposée lorsque celui-ci est élevé dans un cadre de statistique haute fréquence en horizon fini. Ces résultats sont utilisés pour la modélisation des pics dans les prix de l'électricité.Dans une troisième partie, on considère un processus de Poisson doublement stochastique dont l'intensité stochastique est une fonction d'une semi-martingale continue. Pour estimer cette fonction, un estimateur à polynômes locaux est utilisé et une méthode de sélection de la fenêtre est proposée menant à une inégalité oracle. Un test est proposé pour déterminer si la fonction d'intensité appartient à une certaine famille paramétrique. Grâce à ces résultats, on modélise la dépendance entre l'intensité des pics de prix de l'électricité et de facteurs exogènes tels que la production éolienne. / In this thesis, we study some dependence modeling problems between continuous time stochastic processes. These results are applied to the modeling and risk management of electricity markets. In a first part, we propose new copulae to model the dependence between two Brownian motions and to control the distribution of their difference. We show that the class of admissible copulae for the Brownian motions contains asymmetric copulae. These copulae allow for the survival function of the difference between two Brownian motions to have higher value in the right tail than in the Gaussian copula case. Results are applied to the joint modeling of electricity and other energy commodity prices. In a second part, we consider a stochastic process which is a sum of a continuous semimartingale and a mean reverting compound Poisson process and which is discretely observed. An estimation procedure is proposed for the mean reversion parameter of the Poisson process in a high frequency framework with finite time horizon, assuming this parameter is large. Results are applied to the modeling of the spikes in electricity prices time series. In a third part, we consider a doubly stochastic Poisson process with stochastic intensity function of a continuous semimartingale. A local polynomial estimator is considered in order to infer the intensity function and a method is given to select the optimal bandwidth. An oracle inequality is derived. Furthermore, a test is proposed in order to determine if the intensity function belongs to some parametrical family. Using these results, we model the dependence between the intensity of electricity spikes and exogenous factors such as the wind production.

No / A review is carried out on the characterisation and algorithmic implementation of an extended product-form approximation, based on the principle of maximum entropy (ME), for a wide class of arbitrary finite capacity open queueing network models (QNMs) with service and space priorities. A single server finite capacity GE/GE/1/N queue with R (R>1) distinct priority classes, compound Poisson arrival processes (CPPs) with geometrically distributed batches and generalised exponential (GE) service times is analysed via entropy maximisation, subject to suitable GE-type queueing theoretic constraints, under preemptive resume (PR) and head-of-line (HOL) scheduling rules combined with complete buffer sharing (CBS) and partial buffer sharing (PBS) management schemes stipulating a sequence of buffer thresholds {N=(N1,¿,NR),0<Ni¿Ni¿1,i=2,¿,R}. The GE/GE/1/N queue is utilised, in conjunction with GE-type first two moment flow approximation formulae, as a cost-effective building block towards the establishment of a generic ME queue-by-queue decomposition algorithm for arbitrary open QNMs with space and service priorities under repetitive service blocking with random destination (RS-RD). Typical numerical results are included to illustrate the credibility of the ME algorithm against simulation for various network topologies and define experimentally pessimistic GE-type performance bounds. Remarks on the extensions of the ME algorithm to other types of blocking mechanisms, such as repetitive service blocking with fixed destination (RS-FD) and blocking-after-service (BAS), are included.

In this paper, we first present the basic principles of set theory and combinatorial analysis which are the most useful tools in computing probabilities. Then, we show some important properties derived from axioms of probability. Conditional probabilities come into play not only when some partial information is available, but also as a tool to compute probabilities more easily, even when partial information is unavailable. Then, the concept of random variable and its some related properties are introduced. For univariate random variables, we introduce the basic properties of some common discrete and continuous distributions. The important properties of jointly distributed random variables are also considered. Some inequalities, the law of large numbers and the central limit theorem are discussed. Finally, we introduce additional topics the Poisson process.