Consider a single-server queueing system with an unreliable server and service repetition. In this system, if the service is interrupted, the service is restarted anew immediately and ends whenever the service period is failure free. In this paper, we give some characterization of the exponential distribution by the constancy of total service time in this system. The result can be viewed as a kind of memoryless property of the exponential distribution.

Many physical, biological, ecological and behavioral events occur at times and rates that are exponentially distributed. Modeling these systems requires simulators that can accurately generate a large quantity of exponentially distributed random numbers, which is a computationally intensive task. To improve the performance of these simulators, one approach is to move portions of the computationally inefficient simulation tasks from software to custom hardware implemented in Field Programmable Gate Arrays (FPGAs). In this work, we study efficient FPGA implementations of exponentially distributed random number generators to improve simulator performance. Our approach is to generate uniformly distributed random numbers using standard techniques and scale them using the inverse cumulative distribution function (CDF). Scaling is implemented by curve fitting piecewise linear, quadratic, cubic, and higher order functions to solve for the inverse CDF. As the complexity of the scaling function increases (in terms of order and the number of pieces), number accuracy increases and additional FPGA resources (logic cells and block RAMs) are consumed. We analyze these tradeoffs and show how a designer with particular accuracy requirements and FPGA resource constraints can implement an accurate and efficient exponentially distributed random number generator.

Η παρούσα διατριβή εντάσσεται ερευνητικά στην περιοχή της στατιστικής θεωρίας αποφάσεων. Αντικείμενο της είναι η μελέτη προβλημάτων εκτίμησης ποσοστιαίου σημείου για το μοντέλο της παραμετρικής εκθετικής κατανομής. Στο πρώτο κεφάλαιο περιέχονται κάποιοι ορισμοί και παρουσιάζονται γνωστά αποτελέσματα. Στο δεύτερο κεφάλαιο παρατίθεται το μοντέλο της διπ. εκθετικής κατανομής και κατασκευάζονται γνωστοί εκτιμητές για το ποσοστιαίο σημείο της. Στο τρίτο κεφάλαιο εξετάζονται οι εκτιμητές τύπου stein, στο τέταρτο κεφάλαιο οι εκτιμητές τύπου bayes, ενώ στο τελευταίο κεφάλαιο δίνονται οι γραφικές παραστάσεις οι οποίες δείχνουν το ποσοστό βελτίωσης που πετυχαίνεται με την χρήση των πιο πάνω εκτιμητών. / This paper is enlisted in the area of the statistical decision theory. Its objective is the problem of estimation of the fraction point in the model of the exponential distribution with two parameters. In the first chapter, we propose simple and well known theorems, the second chapter comprises of the model of the exponential distribution and well known estimators for her fraction point, in the third chapter the stein estimators are presented, the fourth chapter has the bayes estimators and finally we present the graphical presentations.

A mixture model approach for clustering longitudinal data is introduced. The approach, which is based on mixtures of multivariate power exponential distributions, allows for varying tail-weight and peakedness in data. In the longitudinal setting, this corresponds to more or less concentration around the most central time course in a component. The models utilize a modified Cholesky decomposition of the component scale matrices and the associated maximum likelihood estimators are derived via a generalized expectation-maximization algorithm. / Thesis / Master of Science (MSc)

Master of Science / Department of Statistics / James J. Higgins / The time for an event to take place in an individual is called a survival time. Examples include the time that an individual survives after being diagnosed with a terminal illness or the time that an electronic component functions before failing. A popular parametric model for this type of data is the Weibull model, which is a flexible model that allows for the inclusion of covariates of the survival times. If distributional assumptions are not met or cannot be verified, researchers may turn to the semi-parametric Cox proportional hazards model. This model also allows for the inclusion of covariates of survival times but with less restrictive assumptions. This report compares estimates of the slope of the covariate in the proportional hazards model using the parametric Weibull model and the semi-parametric Cox proportional hazards model to estimate the slope. Properties of these models are discussed in Chapter 1. Numerical examples and a comparison of the mean square errors of the estimates of the slope of the covariate for various sample sizes and for uncensored and censored data are discussed in Chapter 2. When the shape parameter is known, the Weibull model far out performs the Cox proportional hazards model, but when the shape parameter is unknown, the Cox proportional hazards model and the Weibull model give comparable results.

Submitted by Filipe dos Santos (fsantos@pucsp.br) on 2016-11-29T11:25:22Z No. of bitstreams: 1 Carlos Willians Paschoal.pdf: 3019426 bytes, checksum: e3288d4fbd7054a639ed1029f451317e (MD5) / Made available in DSpace on 2016-11-29T11:25:22Z (GMT). No. of bitstreams: 1 Carlos Willians Paschoal.pdf: 3019426 bytes, checksum: e3288d4fbd7054a639ed1029f451317e (MD5) Previous issue date: 2016-08-17 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / This research had as aim investigates how and if the Mathematical Model can be a facilitator of teaching Probability of Exponential Distribution to engineering. We had as an aim answer the follow question: Will Mathematical Model be a favorable teaching methodology to learn Probability of Exponential Distribution in an engineering course? We Used the methodology, Mathematical Model, as described by Bassanezi (2004), Biembengut and Hein (2008), Almeida, Silva and Vertuan (2012), and others who showed the steps to create activities sequence. The theoretical framework was based on the Theory of Didactic Situations from Brousseau (2008), Silva (2008), Freitas (2008), and Teixeira and Passos (2013), which was organized in parallel with the Mathematical Model and the Theory of Didactic Situations based on the work of Dias and Santo (2014), the intention was built an analysis grid to analyze the activities sequence. This sequence was applied to 2º year students from Chemistry Production Engineering from a private institution at São Paulo state. It was realized two meetings, one to organize the experiment and other to obtain the mathematical model from Exponential Distribution. From the data collected we realized its validation in the source problem and in questions from the didactic book proposed in the amendment. This sequence was divided in four activities, which steps of mathematical model, typology of didactics situations and the means structures were identified and analyzed in each of them. This analysis found that the steps foreseen were met, but the students did not achieve a general model, therefore it was needed a step of institutionalization on the validation step. The use of the model was assimilated by the participants’ students who understood the methodology as helpful and motivating. They were able to apply the Exponential Distribution in other problems from didactic books. This research was limited to analyze how the Exponential Distribution can be taught by the Mathematical Model methodology, presenting as future perspective the possibility of reproduction the activity in several topics that include teaching random variable beyond the surveys from types of state of art or metaanalysis about what was researched on the topic / Essa pesquisa teve como objetivo investigar se a Modelagem Matemática pode ser um facilitador do ensino da Distribuição de Probabilidade Exponencial para Engenharias. Tivemos como objetivo responder à seguinte questão de pesquisa: Será a Modelagem Matemática, uma metodologia de ensino favorável à aprendizagem da Distribuição de Probabilidade Exponencial, em um curso de Engenharia? Utilizamos a metodologia Modelagem Matemática, conforme descrita por Bassanezi (2004), Biembengut e Hein (2008), Almeida, Silva e Vertuan (2012) entre outros, que nos inspiraram nos passos para criação de uma sequência de atividades. O referencial teórico está fundamentado na Teoria das Situações Didáticas de Brousseau (2008), Silva (2008), Freitas (2008) e Teixeira e Passos (2013). Foi organizado uma comparação entre a Modelagem Matemática e a Teoria das Situações didáticas com base no trabalho de Dias e Santo (2014), com o intuito de criar uma grade de análise da sequência de atividades. Esta foi aplicada para alunos do 2º ano de Engenharia de Produção Química de uma instituição de ensino particular do Estado de São Paulo, em dois encontros, um para a organização da experimentação, e outro para obtenção do modelo matemático da Distribuição Exponencial. A partir dos dados coletados efetuamos sua validação no problema de origem e em questões retiradas de um livro didático proposto na ementa do referido curso. Esta sequência, foi dividida em quatro atividades, sendo que as etapas da Modelagem Matemática, a Tipologia das Situações Didáticas e as estruturas do meio foram identificadas e analisadas em cada uma. Nesta análise constatou-se que as etapas previstas foram cumpridas, mas que os alunos não chegaram a um modelo geral, sendo necessária uma etapa de institucionalização no meio da etapa de validação. O uso do modelo foi assimilado pelos alunos participantes que entenderam a metodologia como útil e motivadora, sendo capazes de aplicar a Distribuição Exponencial em outros problemas, retirados de livros didáticos. Esta pesquisa se limitou à análise de como a Distribuição Exponencial pode ser ensinada por meio da metodologia da Modelagem Matemática, apresentando como perspectiva futura a possibilidade de reprodução da atividade em diversos tópicos que englobam o ensino de variáveis aleatórias, além de levantamentos de estados da arte ou metanálises sobre o que já foi pesquisado no tema

In the development of any product to perform a specific function the first concern of the engineer is to design for satisfactory operation. Engineers originally approached the reliability problem by using excessive safety factors to be assured the structure or material would withstand the calculated loads and stresses. The engineer also learned from operating or testing the equipment until failures occurred and then redesigning as mistakes became apparent. These methods were time consuming and often resulted in bulky over designed products. These approaches became impractical with the advent of new technological advancements. The accelerated industrial development of aircraft, missiles, and modern electronics coupled with a need for a drastic reduction in weight and size magnified the problem. As products became more complex the problem of building a reliable product was intensified. An appreciation for the increase in complexity can be gained from considering that in a period of fifteen years the requirements for electronic tubes on a U.S. Navy destroyer changed from sixty to thirty-six hundred (14). During World War II new equipment was developed that had to be operational for extended period of time if the military mission was to be accomplished. The addition of a time requirement added to the already difficult problem caused by the increasing complexity of equipment. It soon became obvious that new techniques had to be developed that would assist the manufacturer in designing a reliable product.

As applications like IPTV and VoD (Video on demand) are gaining popularity, it is becoming more important to study the behavior of video signals in the Internet access infrastructures such as ADSL and cable networks. Average delay, average jitter and packet loss in these networks affect the quality of service, hence transmission and access speeds need to be determined such that these parameters are minimized. In this study the behavior of the above mentioned IP networks under variable bit rate (VBR) video traffic is investigated. ns-2 simulator is used for this purpose and actual as well as artificially generated signals are applied to the networks under test. Variable bit rate (VBR) traffic is generated synthetically using ON/OFF sources with ON/OFF times taken from exponential or Pareto distributions. As VBR video shows long range dependence with a Hurst parameter between 0.5 and 1, this parameter was used as a metric to measure the accuracy of the synthetic sources. Two different topologies were simulated in this study: one similar to ADSL access networks and the other behaving like cable distribution network. The performance of the networks (delay, jitter and packet loss) under VBR video traffic and different access speeds were measured. According to the obtained results, minimum access speeds in order achieve acceptable quality video delivery to the customers were suggested.

In this thesis we investigate thinning of the renewal process. After multinomial thinning from a renewal process A, we obtain the k thinned processes, A_i , i =1,¡K, k. Based on some characterizations of the Poisson process as a renewal process, we give another characterizations of the Poisson process from some relations of expectation, variance, covariance, residual life of the k thinned processes. Secondly, we consider that at each arrival time we allow the number of arrivals to be i.i.d. random variables, also the mass of each unit atom can be split into k new atoms with the i-th new atom assigned to the process D_i , i =1,¡K, k. We also have characterizations of the Poisson process from some relations of expectation, variance of the process D_i , i =1,¡K, k.

It is essential for the managers to make investment on hardware based on the utilization information of the equipment. From December 2014, a pool of hardware and a scheduling and resource sharing system is implemented by one of the software testing sections in Ericsson. To monitor the efficiency of these equipment and the workflow, a model of non-homogeneous M/M/c queue is developed that successfully captures the main aspects of the system. The model is decomposed into arrival, service, failure and each part is estimated. Mixture exponential is estimated with EM algorithm and the impact of scheduling change is also examined. Finally a simulation of workflow is done with Python module and the optimized number of hardware is proposed based on this M/M/c queue system.