Distributed computer systems are becoming increasingly prevalent, thanks to modern technology, and this leads to significant challenges for the software developers of these systems. In particular, in order to provide a certain service level agreement with users, the performance characteristics of the system are critical. However, developers today typically consider performance only in the later stages of development, when it may be too late to make major changes to the design. In this thesis, we propose a performance driven approach to development — based around tool support that allows developers to use performance modelling techniques, while still working at the level of program code. There are two central themes to the thesis. The first is to automatically relate performance models to program code. We define the Simple Imperative Remote Invocation Language (SIRIL), and provide a probabilistic semantics that interprets a program as a Markov chain. To make such an interpretation both computable and efficient, we develop an abstract interpretation of the semantics, from which we can derive a Performance Evaluation Process Algebra (PEPA) model of the system. This is based around abstracting the domain of variables to truncated multivariate normal measures. The second theme of the thesis is to analyse large performance models by means of compositional abstraction. We use two abstraction techniques based on aggregation of states — abstract Markov chains, and stochastic bounds — and apply both of them compositionally to PEPA models. This allows us to model check properties in the three-valued Continuous Stochastic Logic (CSL), on abstracted models. We have implemented an extension to the Eclipse plug-in for PEPA, which provides a graphical interface for specifying which states in the model to aggregate, and for performing the model checking.

Gaussian process regression (GPR) is a long-standing technique for statistical interpolation between observed data points. Having originally been applied to spatial analysis in the 1950s, GPR offers highly nonlinear predictions with uncertainty adjusting to the degree of extrapolation -- at the expense of very few model parameters to be fit. Thus GPR has gained considerable popularity in statistical applications such as machine learning and nonparametric density estimation. In this thesis, we explore the potential for GPR to improve the efficiency of parametric inference for continuous-time stochastic processes. For almost all such processes, the likelihood function based on discrete observations cannot be written in closed-form. However, it can be very well approximated if the inter-observation time is small. Therefore, a popular strategy for parametric inference is to introduce missing data between actual observations. In a Bayesian context, samples from the posterior distribution of the parameters and missing data are then typically obtained using Markov chain Monte Carlo (MCMC) methods, which can be computationally very expensive. Here, we consider the possibility of using GPR to impute the marginal distribution of the missing data directly. These imputations could then be leveraged to produce independent draws from the joint posterior by Importance Sampling, for a significant gain in computational efficiency. In order to illustrate the methodology, three continuous processes are examined. The first one is based on a neural excitation model with a non-standard periodic component. The second and third are popular financial models often used for option pricing. While preliminary inferential results are quite promising, we point out several improvements to the methodology which remain to be explored.

Orientador: Rogério Rosenfeld / Banca: Mario José de Oliveira / Banca: Marcos Eugênio da Silva / Resumo: Neste trabalho estudamos a calibração de opções de câmbio no mercado brasileiro utilizando o processo estocástico proposto por Heston [Heston, 1993], como uma alternativa ao modelo de apreçamento de Black e Scholes [Black e Scholes,1973], onde as volatilidades implícitas de opções para diferentes preços de exercícios e prazos são incorporadas ad hoc. Comparamos dois métodos de apreçamento: o método de Carr e Madan [Carr e Madan, 1999], que emprega transfomada rápida de Fourier e função característica, e expansão assintótica para baixos valores de volatilidade da variância. Com a nalidade de analisar o domínio de aplicabilidade deste método, selecionamos períodos de alta volatilidade no mercado, correspondente à crise subprime de 2008, e baixa volatilidade, correspondente ao período subsequente. Adicionalmente, estudamos a incorporação de swaps de variância para melhorar a calibração do modelo / Abstract: In this work we study the calibration of forex call options in the Brazilian market using the stochastic process proposed by Heston [Heston, 1993], as an alternative to the Black and Scholes [Black e Scholes,1973] pricing model, in which the implied option volatilities related to di erent strikes and maturities are incorporated in an ad hoc manner. We compare two pricing methods: one from Carr and Madan [Carr e Madan, 1999], which uses fast Fourier transform and characteristic function, and asymptotic expantion for low values of the volatility of variance. To analyze the applicability of this method, we select periods of high volatility in the market, related to the subprime crisis of 2008, and of low volatility, correspondent to the following period. In addition, we study the use of variance swaps to improve the calibration of the model / Mestre

[pt] Medida da magnitude de flutuação dos preços, a volatilidade é uma métrica importante para definir as estratégias de negociação e de controle de risco mais adequadas. Esse trabalho desenvolve um modelo de volatilidade fenomenológico baseado na rede microscópica heterogenea na qual os agentes especuladores respondem à chegada das informações. A dinâmica das características da volatilidade, modeladas por processos estocásticos, é governada por assimetrias no fluxo de informação através de diferentes resoluções temporais de análise. Entre essas características, destacamos os fatos estilizados de memória longa, clustering e efeito de alavancagem. Essas propostas são elucidadas através da análise empírica das séries de preço de um minuto do índice Ibovespa no período de dez anos. / [en] Volatility, as a metric for price uncertainty, is an important quantity for suitable trade strategy and risk control. This work develops a phenomenological volatility model based on a heterogeneous microstructure framework in which the market agents of speculative activity respond to information arrivals. The dynamic features of volatility, modeled as a stochastic process, is governed by asymmetries in the informational flow across different time resolutions. Among these features, we highlight the stylized facts of long memory, clustering and leverage effect. These proposals are contrasted with our empirical analysis of a ten-year time series of one-minute Brazilian market Index.

Faculty of Engineering and the Built Environment; Master of Science in Engineering; Research Report / The primary objective of this study was to develop a methodology for evaluating how the reliability profile of the typical South African Platinum concentrator plant is affected by firstly the size of the primary milling units incorporated in the circuit and secondly by the way that the primary milling units are configured. A methodology, together with a set of general expressions is presented which considers the Platinum concentrator as a stochastic process where the behaviour of the primary mill is a direct measure of the failure pattern of the overall concentrator. The reliability, availability and maintainability (RAM) of the primary mill, and hence the overall concentrator, is then determined by a combination of three different Markov models where each Markov model is used to evaluate and measure a separate set of reliability parameters. This approach effectively overcomes the computational complexity associated with large Markov models. The results of two case studies used to validate the methodology do indicate that the reliability, availability and maintainability profiles of large single stream Platinum concentrators could be fundamentally different from the conventional multiple stream primary mill configurations.

Neste trabalho, estudamos o passeio aleatório do elefante introduzido em (SCHUTZ; TRIMPER, 2004). Um processo estocástico não Markoviano com memória de alcance ilimitada que apresenta transição de fase. Nosso objetivo é demonstrar a convergência quase certa do passeio aleatório do elefante nos casos subcrítico e crítico. Além destes resultado, também apresentamos a demonstração do Teorema Central do Limite para ambos os regimes. Para o caso supercrítico, vamos demonstrar a convergência do passeio aleatório do elefante para uma variável aleatória não normal com base nos artigos (BAUR; BERTOIN, 2016), (BERCU, 2018) e (COLETTI; GAVA; SCHUTZ, 2017b). / In this work we study the elephant random walk introduced in (SCHUTZ; TRIMPER, 2004), a discrete time, non-Markovian stochastic process with unlimited range memory that presents phase transition. Our objective is to proof the almost sure convergence for the subcritical and critical regimes of the model. We also present a demonstration of the Central Limit Theorem for both regimes. For the supercritical regime we proof the convergence of the elephant random walk to a non-normal random variable based on the articles (BAUR; BERTOIN, 2016), (BERCU, 2018) and (COLETTI; GAVA; SCHUTZ, 2017b).

O comportamento de sistemas epidêmicos é frequentemente descrito de maneira determinística, através do emprego de equações diferenciais ordinárias. Este trabalho visa fornecer uma visão estocástica do problema, traçando um paralelo entre o encontro de indivíduos em uma população e o choque entre partículas de uma reação química. Através dessa abordagem é apresentado o algoritmo de Gillespie, que fornece uma forma simples de simular a evolução de um sistema epidêmico. Fundamentos de processos estocásticos são apresentados para fundamentar uma técnica para a estimação de parâmetros através de dados reais. Apresentamos ainda o modelo de Tau-leaping e o modelo difusivo elaborados através de equações diferenciais estocásticas que são aproximações do modelo proposto por Gillespie. A aplicação dos modelos apresentados é exemplificada através do estudo de dados reais da epidemia de dengue ocorrida no estado do Rio de Janeiro entre os anos de 2012 e 2013. / The behavior of epidemic systems is often described in a deterministic way, through the use of ordinary differential equations. This paper aims to provide a stochastic view of the problem, drawing a parallel between the encounter between individuals in a population and the clash between particles of a chemical reaction. Through this approach is presented the Gillespie algorithm, which provides a simple way to simulate the evolution of an epidemic system. Fundamentals of stochastic process theory are presented to support a technique for estimating parameters through real data. We present the model of Tau-leaping and the diffusive model elaborated by stochastic differential equations that are approximations of the model proposed by Gillespie. The application of the presented models is exemplified through the study of real data of the dengue epidemic occurred in the state of Rio de Janeiro between the years of 2012 and 2013.

Nesta tese desenvolvemos um modelo estocástico para a transcrição do gene even-skipped de Drosophila melanogaster no qual a variável estocástica é o número de moléculas de mRNA transcritas. Nesse modelo, consideramos um gene com dois níveis de ativação sendo regulado externamente. As probabilidades de se encontrar o gene ligado ou desligado e com determinado número de moléculas de mRNA transcritas obedecem equações lineares dadas por processos markovianos de nascimento-e-morte (taxas de produção e degradação) e termos de chaveamento entre os níveis cujas dependências temporais são dadas por funções de Heaviside. Notamos que tal dependência é suficiente para garantir uma atividade transcricional inicial intensa seguida de uma súbita interrupção, conforme sugerem os dados experimentais. Desconsiderando efeitos difusivos e fenômenos de transporte, estendemos esse constructo às outras regiões do embrião, permitindo dependências espaciais apenas às termos de chaveamento, e o resultado gerado descreve os dados experimentais com boa concordância, indicando também que o aspecto binário do gene é suficiente para uma descrição semiquantitativa do fenômeno. Notavelmente, na região onde a listra se forma e concomitantemente a sua formação, o modelo prevê a redução do desvio quadrático (flutuação) e do ruído. Calculando a distribuição de probabilidade, verificamos que o regime estacionário é atingido antes da listra começar a desaparecer. Também estudamos uma conexão entre parâmetros do modelo e as proteínas envolvidas na regulação e, baseado em resultados da literatura, obtemos uma função com aproximadamente o mesmo efeito regulatório considerando gradientes de seis fatores de transcrição (Bcd, Hb, Gt, Kr, Kni e Tll) e apenas quatro sítios de ligação, o que sugere que a informação transcricional pode estar concentrada na regulação de poucos sítios. / In this thesis we develop a stochastic model to transcription of Drosophila melanogaster even-skipped gene in which the stochastic variable is the number of mRNA molecules transcribed. In this model we considered a gene with two activation levels being regulated externally. The probabilities of gene being on or off when there is a certain number of transcripts obey linear equations given by Markovian birth-and-death processes (production and degradation rates) and terms of switch between levels whose time-dependence is given by Heaviside functions. We note that is sufficient to ensure a strong transcriptional activity followed by a sudden disruption, as suggested by the experimental data. Disregarding diffusion effects and transport phenomena, we extend this construct to the others regions ot the embryo, allowing space-dependence only to terms of switch, and the results describe the experimental data with good agreement, indicating also that binary character of gene is sufficient to a semiquantitative description of the phenomenon. Notably, in the region where the stripe 2 is formed and simultaneously with its formation, the model predicts the reduction in standard deviation (fluctuation) and noise. By calculating the probability distribution, we find that stationary state is reached before stripe 2 starts to fade. We also study a connection between the parameters of the model and proteins involved in regulation and, based on results from the literature, we obtain a function with approximately the same regulatory effect considering six transcription factors (Bcd, Hb, Gt, Kr, Kni e Tll) and only four binding sites, suggesting that transcriptional information may be concentrated in regulation of few sites.

This work is a study of the relationship between Brownian motion and elementary, linear partial differential equations. In the text, I have shown that Brownian motion is a Markov process, and that Brownian motion itself, and certain Stochastic processes involving Brownian motion are also martingales. In particular, Dynkin's formula for Brownian motion was shown. Using Dynkin's formula and Brownian motion, I then constructed solutions for the classical Dirichlet problem and the heat equation, given by Δu=0 and ut= 1/2Δu+g, respectively. I have shown that the bounded solution is unique if Brownian motion will always exit the domain of the function once it has started at a point in the domain. The heat equation also has a unique bounded solution.

Timely identification of undesirable pavement crack conditions has been a major task in pavement management. Up to date, myriads of pavement performance models have been developed for forecasting pavement crack condition with the traditional preferred techniques being the use of regression relationships developed from laboratory and/or field statistical data. However, it becomes difficult for regression techniques to predict the crack performance accurately and robustly in the presence of a variety of tributary factors, high nonlinearity, and uncertainty. With the advancement of modeling techniques, two innovative breeds of models, Artificial Neural Networks and Markov Chains, have drawn increasing attention from researchers for modeling complex phenomena like the pavement crack performance. In this study, two distinct models, a recurrent Markov chain, and an Artificial Neural Network (ANN), were developed for modeling the performance of pavement crack condition with time. A logistic model was used to establish a dynamic relationship between transition probabilities associated with the pavement crack condition and the applicable tributary variables. The logistic model was then used conveniently to construct a recurrent Markov chain for use in predicting the crack performance of asphalt pavements in Florida. Florida pavement condition survey database were utilized to perform a case study of the proposed methodologies. For comparison purpose, a currently popular static Markov chain was also developed based on a homogeneous transition probability matrix that was derived from the crack index statistics of Florida pavement survey database. To evaluate the model performance, two comparisons were made; (1) between the recurrent Markov chain and the static Markov chain; and (2) between the recurrent Markov chain and the ANN. It is shown that the recurrent Markov chain outperforms both the static Markov chain and the ANN in terms of one-year forecasting accuracy. Therefore, with high uncertainty typically experienced in the pavement condition deterioration process, the probabilistic dynamic modeling approach as embodied in the recurrent Markov chain provides a more appropriate and applicable methodology for modeling the pavement deterioration process with respect to cracks.