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Queueing theory is the mathematical study of queues or waiting lines, which are formed whenever demand for service exceeds the capacity to provide service. A queueing system is composed of customers, packets or calls that need some kind of service. These entities arrive at queueing system, join a queue if service is not immediately available and leave system after receiving service. There are also cases when customers, packets or calls leave system without joining queue or drop out without receiving service even after waiting for some time. Queueing network models with finite capacity have facilitated the analysis of discrete flow systems, such as computer systems, transportation networks, manufacturing systems and telecommunication networks, by providing powerful and realistic tools for performance evaluation and prediction. In wireless cellular systems mobility is the most important feature and continuous service is achieved by supporting handoff from one cell to another. Hand off is the process of changing channel associated with the current connection while a call is in progress. A handoff is required when a mobile terminal moves from one cell to another or the signal quality deteriorates in current cell. Since neighbouring cells use disjoint subset of frequency bands therefore negotiation must take place between mobile terminal, the current base station and next potential base station. A poorly designed handoff scheme significantly decreases quality of service (QOS). Different schemes have been devised and in these schemes handoff calls are prioritize. Also most of the performance evaluation techniques consider the case where the arrival process is poisson and service is exponential i.e. there is single arrival and single departure. Whereas in practice there is burstiness in cellular traffic i.e. there can be bulk arrivals and bulk departures. Other issue is that, assumptions made by stochastic process models are not satisfied. Most of the effort is concentrated on providing different interpretations of M/M queues rather than attempting to provide a new methodology. In this thesis performance evaluation of multi traffic cellular models i.e. non priority, priority and hand off calling scheme for bursty traffic are devised. Moreover extensions are carried out towards the analysis of a multi-traffic M/M queueing system and state probabilities are calculated analytically.

<p>Recently, the Conway-Maxwell Poisson (COM-Poisson) cure rate model has been proposed which includes as special cases some of the well-known cure rate models discussed in the literature. Data obtained from cancer clinical trials are often right censored and the expectation maximization (EM) algorithm can be efficiently used for the determination of the maximum likelihood estimates (MLEs) of the model parameters based on right censored data.</p> <p>By assuming the lifetime distribution to be exponential, lognormal, Weibull, and gamma, the necessary steps of the EM algorithm are developed for the COM-Poisson cure rate model and some of its special cases. The inferential method is examined by means of an extensive simulation study. Model discrimination within the COM-Poisson family is carried out by likelihood ratio test as well as by information-based criteria. Finally, the proposed method is illustrated with a cutaneous melanoma data on cancer recurrence. As the lifetime distributions considered are not nested, it is not possible to carry out a formal statistical test to determine which among these provides an adequate fit to the data. For this reason, the wider class of generalized gamma distributions is considered which contains all of the above mentioned lifetime distributions as special cases. The steps of the EM algorithm are then developed for this general class of distributions and a simulation study is carried out to evaluate the performance of the proposed estimation method. Model discrimination within the generalized gamma family is carried out by likelihood ratio test and information-based criteria. Finally, for the considered cutaneous melanoma data, the two-way flexibility of the COM-Poisson family and the generalized gamma family is utilized to carry out a two-way model discrimination to select a parsimonious competing cause distribution along with a suitable choice of a lifetime distribution that provides the best fit to the data.</p> / Doctor of Philosophy (PhD)

The term aggregation refers to overdispersion and both are used interchangeably in this thesis. In addressing the problem of prevalence of infectious parasite species faced by most rural livestock farmers, we model the distribution of faecal egg counts of 15 parasite species (13 internal parasites and 2 ticks) common in sheep and goats. Aggregation and excess zeroes is addressed through the use of generalised linear models. The abundance of each species was modelled using six different distributions: the Poisson, negative binomial (NB), zero-inflated Poisson (ZIP), zero-inflated negative binomial (ZINB), zero-altered Poisson (ZAP) and zero-altered negative binomial (ZANB) and their fit was later compared. Excess zero models (ZIP, ZINB, ZAP and ZANB) were found to be a better fit compared to standard count models (Poisson and negative binomial) in all 15 cases. We further investigated how distributional assumption a↵ects aggregation and zero inflation. Aggregation and zero inflation (measured by the dispersion parameter k and the zero inflation probability) were found to vary greatly with distributional assumption; this in turn changed the fixed-effects structure. Serial autocorrelation between adjacent observations was later taken into account by fitting observation driven time series models to the data. Simultaneously taking into account autocorrelation, overdispersion and zero inflation proved to be successful as zero inflated autoregressive models performed better than zero inflated models in most cases. Apart from contribution to the knowledge of science, predictability of parasite burden will help farmers with effective disease management interventions. Researchers confronted with the task of analysing count data with excess zeroes can use the findings of this illustrative study as a guideline irrespective of their research discipline. Statistical methods from model selection, quantifying of zero inflation through to accounting for serial autocorrelation are described and illustrated. / Statistics / M.Sc. (Statistics)

Cure rate models, introduced by Boag (1949), are very commonly used while modelling lifetime data involving long time survivors. Applications of cure rate models can be seen in biomedical science, industrial reliability, finance, manufacturing, demography and criminology. In this thesis, cure rate models are discussed under a competing cause scenario, with the assumption of proportional odds (PO) lifetime distributions for the susceptibles, and statistical inferential methods are then developed based on right-censored data. In Chapter 2, a flexible cure rate model is discussed by assuming the number of competing causes for the event of interest following the Conway-Maxwell (COM) Poisson distribution, and their corresponding lifetimes of non-cured or susceptible individuals can be described by PO model. This provides a natural extension of the work of Gu et al. (2011) who had considered a geometric number of competing causes. Under right censoring, maximum likelihood estimators (MLEs) are obtained by the use of expectation-maximization (EM) algorithm. An extensive Monte Carlo simulation study is carried out for various scenarios, and model discrimination between some well-known cure models like geometric, Poisson and Bernoulli is also examined. The goodness-of-fit and model diagnostics of the model are also discussed. A cutaneous melanoma dataset example is used to illustrate the models as well as the inferential methods. Next, in Chapter 3, the destructive cure rate models, introduced by Rodrigues et al. (2011), are discussed under the PO assumption. Here, the initial number of competing causes is modelled by a weighted Poisson distribution with special focus on exponentially weighted Poisson, length-biased Poisson and negative binomial distributions. Then, a damage distribution is introduced for the number of initial causes which do not get destroyed. An EM-type algorithm for computing the MLEs is developed. An extensive simulation study is carried out for various scenarios, and model discrimination between the three weighted Poisson distributions is also examined. All the models and methods of estimation are evaluated through a simulation study. A cutaneous melanoma dataset example is used to illustrate the models as well as the inferential methods. In Chapter 4, frailty cure rate models are discussed under a gamma frailty wherein the initial number of competing causes is described by a Conway-Maxwell (COM) Poisson distribution in which the lifetimes of non-cured individuals can be described by PO model. The detailed steps of the EM algorithm are then developed for this model and an extensive simulation study is carried out to evaluate the performance of the proposed model and the estimation method. A cutaneous melanoma dataset as well as a simulated data are used for illustrative purposes. Finally, Chapter 5 outlines the work carried out in the thesis and also suggests some problems of further research interest. / Thesis / Doctor of Philosophy (PhD)

Les tests d’adéquation font partie des pratiques qu’ont les statisticiens pour prendre une décision concernant l’hypothèse de l’utilisation d’une distribution paramétrique pour un échantillon. Dans ce mémoire, une application du test d’adéquation basé sur la fonction caractéristique proposé par Jiménez-Gamero et al. (2009) est faite dans le cas de la loi de Poisson bivariée. Dans un premier temps, le test est élaboré dans le cas de l’adéquation à une loi de Poisson univariée et nous avons trouvé son niveau bon. Ensuite cette élaboration est étendue au cas de la loi de Poisson bivariée et la puissance du test est calculée et comparée à celle des tests de l’indice de dispersion, du Quick test de Crockett et des deux familles de tests proposés par Novoa-Muñoz et Jiménez-Gamero (2014). Les résultats de la simulation ont permis de constater que le test avait un bon niveau comparativement aux tests de l’indice de dispersion et au Quick test de Crockett et qu’il était généralement moins puissant que les autres tests. Nous avons également découvert que le test de l’indice de dispersion devrait être bilatéral alors qu’il ne rejette que pour de grandes valeurs de la statistique de test. Finalement, la valeur-p de tous ces tests a été calculée sur un jeu de données de soccer et les conclusions comparées. Avec une valeur-p de 0,009, le test a rejeté l’hypothèse que les données provenaient d’une loi de Poisson bivariée alors que les tests proposés par Novoa-Muñoz et Jiménez-Gamero (2014) donnaient une conclusion différente. / Our aim in this thesis is to conduct the goodness-of-fit test based on empirical characteristic functions proposed by Jiménez-Gamero et al. (2009) in the case of the bivariate Poisson distribution. We first evaluate the test’s behaviour in the case of the univariate Poisson distribution and find that the estimated type I error probabilities are close to the nominal values. Next, we extend it to the bivariate case and calculate and compare its power with the dispersion index test for the bivariate Poisson, Crockett’s Quick test for the bivariate Poisson and the two test families proposed by Novoa-Muñoz et Jiménez-Gamero (2014). Simulation results show that the probability of type I error is close to the claimed level and that it is generally less powerful than other tests. We also discovered that the dispersion index test should be bilateral whereas it rejects for large values only. Finally, the p-value of all these tests is calculated on a real dataset from soccer. The p-value of the test is 0,009 and we reject the hypothesis that the data come from a Poisson bivariate while the tests proposed by Novoa-Muñoz et Jiménez-Gamero (2014) leads to a different conclusion.

In this work, we study the so-called combined models, generalized linear mixed models with extension to allow for overdispersion, in the context of genetics and breeding. Such flexible models accommodates cluster-induced correlation and overdispersion through two separate sets of random effects and contain as special cases the generalized linear mixed models (GLMM) on the one hand, and commonly known overdispersion models on the other. We use such models while obtaining heritability coefficients for non-Gaussian characters. Heritability is one of the many important concepts that are often quantified upon fitting a model to hierarchical data. It is often of importance in plant and animal breeding. Knowledge of this attribute is useful to quantify the magnitude of improvement in the population. For data where linear models can be used, this attribute is conveniently defined as a ratio of variance components. Matters are less simple for non-Gaussian outcomes. The focus is on time-to-event and count traits, where the Weibull-Gamma-Normal and Poisson-Gamma-Normal models are used. The resulting expressions are sufficiently simple and appealing, in particular in special cases, to be of practical value. The proposed methodologies are illustrated using data from animal and plant breeding. Furthermore, attention is given to the occurrence of negative estimates of variance components in the Poisson-Gamma-Normal model. The occurrence of negative variance components in linear mixed models (LMM) has received a certain amount of attention in the literature whereas almost no work has been done for GLMM. This phenomenon can be confusing at first sight because, by definition, variances themselves are non-negative quantities. However, this is a well understood phenomenon in the context of linear mixed modeling, where one will have to make a choice between a hierarchical and a marginal view. The variance components of the combined model for count outcomes are studied theoretically and the plant breeding study used as illustration underscores that this phenomenon can be common in applied research. We also call attention to the performance of different estimation methods, because not all available methods are capable of extending the parameter space of the variance components. Then, when there is a need for inference on such components and they are expected to be negative, the accuracy of the method is not the only characteristic to be considered. / Neste trabalho foram estudados os chamados modelos combinados, modelos lineares generalizados mistos com extensão para acomodar superdispersão, no contexto de genética e melhoramento. Esses modelos flexíveis acomodam correlação induzida por agrupamento e superdispersão por meio de dois conjuntos separados de efeitos aleatórios e contem como casos especiais os modelos lineares generalizados mistos (MLGM) e os modelos de superdispersão comumente conhecidos. Tais modelos são usados na obtenção do coeficiente de herdabilidade para caracteres não Gaussianos. Herdabilidade é um dos vários importantes conceitos que são frequentemente quantificados com o ajuste de um modelo a dados hierárquicos. Ela é usualmente importante no melhoramento vegetal e animal. Conhecer esse atributo é útil para quantificar a magnitude do ganho na população. Para dados em que modelos lineares podem ser usados, esse atributo é convenientemente definido como uma razão de componentes de variância. Os problemas são menos simples para respostas não Gaussianas. O foco aqui é em características do tipo tempo-até-evento e contagem, em que os modelosWeibull-Gama-Normal e Poisson-Gama-Normal são usados. As expressões resultantes são suficientemente simples e atrativas, em particular nos casos especiais, pelo valor prático. As metodologias propostas são ilustradas usando dados de melhoramento animal e vegetal. Além disso, a atenção é voltada à ocorrência de estimativas negativas de componentes de variância no modelo Poisson-Gama- Normal. A ocorrência de componentes de variância negativos em modelos lineares mistos (MLM) tem recebido certa atenção na literatura enquanto quase nenhum trabalho tem sido feito para MLGM. Esse fenômeno pode ser confuso a princípio porque, por definição, variâncias são quantidades não-negativas. Entretanto, este é um fenômeno bem compreendido no contexto de modelagem linear mista, em que a escolha deverá ser feita entre uma interpretação hierárquica ou marginal. Os componentes de variância do modelo combinado para respostas de contagem são estudados teoricamente e o estudo de melhoramento vegetal usado como ilustração confirma que esse fenômeno pode ser comum em pesquisas aplicadas. A atenção também é voltada ao desempenho de diferentes métodos de estimação, porque nem todos aqueles disponíveis são capazes de estender o espaço paramétrico dos componentes de variância. Então, quando há a necessidade de inferência de tais componentes e é esperado que eles sejam negativos, a acurácia do método de estimação não é a única característica a ser considerada.

A hanseníase, doença crônica estigmatizante com potencial de causar danos neurológicos, resulta da infecção pelo Mycobacterium leprae. Análises epidemiológicas atuais têm utilizado ferramentas clínicas e de análise espacial para o mapeamento dos principais focos de ocorrência de doenças e de áreas de alto risco. Analisar os municípios maranhenses quanto à distribuição dos casos de hanseníase torna-se uma ferramenta a mais na prevenção e controle da Hanseníase no estado por inúmeros fatores: comporta-se como área hiperendêmica de hanseníase; apresenta fluxo migratório intenso com outras cidades de forma interestadual; e tem grandes contrastes sociais marcados por pouca, ou nenhuma, infraestrutura básica em algumas áreas dos vários municípios deste. Objetivos: Analisar a distribuição espaço-temporal da hanseníase para o estado do Maranhão, no período de 2001 a 2013. Identificar a ocorrência de agrupamentos espaços-temporais de provável alta transmissão (risco) e verificar se há associação dessa distribuição de taxas de detecção de risco relativo (RR) da doença com as variáveis do contexto geográfico como socioeconômicas e ambientais. Metodologia: A fonte de coleta dos dados clínicos e epidemiológicos foi o Sistema de Informação Nacional de Agravos Notificáveis do Ministério da Saúde e dos dados demográficos, ambientais e bases cartográficas digitais do Instituto Brasileiro de Geografia e Estatística. Foi adotada uma abordagem ecológica sobre tendências dos padrões espaçostemporais de transmissibilidade, com utilização dos métodos: varredura espacial (scan), para a identificação dos agregados (clusters) de risco, considerando o modelo de distribuição de probabilidade Discreto de Poisson; Estimador Bayesiano Empírico para a suavização local de taxas, a partir de informações de municípios vizinhos tendo como estratégia de construção o critério da contiguidade; regressão múltipla espacial considerando uma modelagem com distribuição de Poisson no contexto Bayesiano, levando em conta a dependência espacial, com o propósito de avaliar a relação entre a ocorrência da variável dependente com as variáveis demográficas, socioeconômicas e ambientais. Resultados: A taxa média de detecção foi de 6,73 casos por 10.000 hab., com 53.826 casos notificados no período. O estudo revelou que a distribuição dos casos de sexo masculino (57,75%) apresentou maior proporção em relação ao feminino (42,25%), havendo predominância da doença na faixa etária >15 anos (89,87%). A alta ocorrência na classificação operacional multibacilar (60,10%) é um forte indicativo decorrente do longo período de incubação da doença somado ao não diagnóstico precoce. A análise da distribuição dos agregados espaciais identificou 14 (7 de risco alto e 7 de risco baixo) e 6 (3 de risco alto e 3 de risco baixo) agrupamentos espaciais, considerando-se 10% e 50% da população em risco, respectivamente, em áreas com taxas de detecção alta e que possuem baixa qualidade de vida. O estimador Bayesiano empírico local possibilitou gerar índices corrigidos e com menores instabilidades. A análise de regressão múltipla espacial mostrou que as variáveis índice Gini, bioma predominante cerrado/caatinga e percentual de população urbana tiveram associação positiva e significativa para explicar o risco relativo (RR) no estado do Maranhão. Conclusões: O estudo mostrou que existem aglomerados com elevado risco para transmissão da hanseníase no estado do Maranhão. A associação entre o risco relativo da hanseníase e o percentual de população urbana indica que a hipótese que associa o M. leprae e a população que vive em condições de acentuada desigualdade socioeconômica ainda é forte. Essa hiperendemicidade pode demonstrar que o crescimento da população urbana é um preditor de incidência da hanseníase, face à urbanização descontrolada e ao fluxo de migrantes advindos de diferentes espaços rurais. Foi possível identificar áreas prioritárias para implementação de programas eficazes de controle de hanseníase no estado do Maranhão. / Leprosy, a chronic stigmatizing disease with the potential to cause neurological damage resulting from infection by Mycobacterium leprae. Current epidemiological studies have used clinical and spatial analysis for mapping of the main occurrence of disease outbreaks and high-risk areas. Analyze the municipalities of Maranhão state regarding the distribution of leprosy cases becomes another tool in the prevention and control of leprosy in the state by numerous factors like behaves as hyper-endemic area of leprosy; It presents intense migration to other interstate cities; and has great social contrasts marked by little or no basic infrastructure in some areas of several municipalities.. Objectives: To analyze the spatiotemporal distribution of leprosy in the Maranhão state, from 2001 to 2013. To identify the spatiotemporal clusters occurrence of probable high transmission (risk) and check for association of this distribution of relative risk (RR) detection rates of the disease with the variables of geographic context as socioeconomic and environmental. Methodology: Clinical and epidemiological data was obtained from the Ministry of Healths Disease Reporting System and demographic data, environmental and digital cartographic bases were obtained from the Brazilian Geography and Statistics Institute. An ecological approach to trends transmissibility of spatiotemporal patterns, using the methods: spatial scan to identification the clusters of risk, considering the Discrete Poisson probability distribution model; empirical Bayesian method was applied for local rate flattening, using data from municipalities having as building strategy the criterion of contiguity; ecological regression modeling with considering a Poisson distribution in the Bayesian context, taking into account the spatial dependence, in order to evaluate the relationship between the occurrence of the dependent variable with demographic, socioeconomic and environmental variables. Results: The mean detection rate was 6.73 cases per 10,000 inhabitants, with 53,826 reported cases. The study revealed that the distribution of male cases (57.75%) showed a predominance over female (42.25%), with predominance of the disease in the age group upper than 15 years (89.87%). The high occurrence in operational classification multibacillary (60.10%) is a strong indication due to the long incubation period of the disease added to no early diagnosis. The analysis of the distribution of spatial clusters identified 14 (7 high risk and 7 low risk) and 6 (3 high risk and 3 low risk) spatial clusters, considering 10% and 50% of the population at risk in areas with high detection rates and which have low quality of life. Local empirical Bayes estimator allowed to generate fixed and minor instabilities indexes. The best results of modeling to spatial multiple regression analysis for the relative risk (RR) presented for the variables Gini index, cerrado/caatinga biome and percentage of urban population. Conclusions: The study showed that there are clusters at high risk for transmission of leprosy in the Maranhao state. The association between the relative risk of leprosy and the percentage of urban population indicates that the hypothesis that associates M. leprae and the population living in severe socioeconomic inequality is still strong. This hyperendemicity can demonstrate that the growth of the urban population is a predictor incidence of leprosy due to uncontrolled urbanization and the influx of migrants coming from different rural areas.It was possible to identify priority areas for implementation of effective leprosy control programs in the Maranhão state.

Metodologias para obtenção do tamanho amostral para estimar a concentração de organismos em água de lastro e verificar normas internacionais são desenvolvidas sob uma abordagem bayesiana. Consideramos os critérios da cobertura média, do tamanho médio e da minimização do custo total sob os modelos Poisson com distribuição a priori gama e binomial negativo com distribuição a priori Pearson Tipo VI. Além disso, consideramos um processo Dirichlet como distribuição a priori no modelo Poisson com o propósito de obter maior flexibilidade e robustez. Para fins de aplicação, implementamos rotinas computacionais usando a linguagem R. / Sample size methodologies for estimating the organism concentration in ballast water and for verifying international standards are developed under a Bayesian approach. We consider the criteria of average coverage, of average length and of total cost minimization under the Poisson model with a gamma prior distribution and the negative binomial model with a Pearson type VI prior distribution. Furthermore, we consider a Dirichlet process as a prior distribution in the Poisson model with the purpose to gain more flexibility and robustness. For practical applications, we implemented computational routines using the R language.

Our aim is to investigate Poisson type approximations to the sums of dependent integer-valued random variables. In this thesis, only one type of dependence is considered, namely m-dependent random variables. The accuracy of approximation is measured in the total variation, local, uniform (Kolmogorov) and Wasserstein metrics. Results can be divided into four parts. The first part is devoted to 2-runs, when pi=p. We generalize Theorem 5.2 from A.D. Barbour and A. Xia “Poisson perturbations” in two directions: by estimating the second order asymptotic expansion and asymptotic expansion in the exponent. Moreover, lower bound estimates are established, proving the optimality of upper bound estimates. Since, the method of proof does not allow to get small constants, in certain cases, we calculate asymptotically sharp constants. In the second part, we consider sums of 1-dependent random variables, concentrated on nonnegative integers and satisfying analogue of Franken's condition. All results of this part are comparable to the known results for independent summands. In the third part, we consider Poisson type approximations for sums of 1-dependent symmetric three-point distributions. We are unaware about any Poisson-type approximation result for dependent random variables, when symmetry of the distribution is taken into account. In the last part, we consider 1-dependent non-identically distributed Bernoulli random variables. It is shown, that even for this simple... [to full text] / Disertacijoje tiriamas diskrečių m-priklausomų atsitiktinių dydžių aproksimavimo Puasono tipo matais tikslumas. Silpnai priklausomų atsitiktinių dydžių sumos yra natūralus nepriklausomų atsitiktinių dydžių sumų apibendrinimas. Vis dėlto atsitiktinių dydžių priklausomybė žymiai pasunkina tokių sumų tyrimą. Disertacijoje pagrindinis dėmesys skiriamas dviparametrėms ir triparametrėms diskrečiosioms aproksimacijoms. Gautus rezultatus galima suskirstyti į keturias dalis. Pirmoje dalyje nagrinėjant dviejų narių serijų statistikos aproksimaciją Puasono ir sudėtiniais Puasono skirstiniais buvo nustatyta, kad dviparametrė sudėtinė Puasono aproksimacija yra tikslesnė už Puasono dėsnio asimptotinį skleidinį su vienu asimptotikos nariu. Aproksimacijos tikslumas įvertintas pilnosios variacijos ir lokalioje metrikoje. Specialiu atveju apskaičiuotos asimptotiškai tikslios konstantos. Taip pat nustatyta, kad gautieji įverčiai iš apačios yra tos pačios eilės, kaip ir įverčiai iš viršaus. Antroje dalyje buvo gauta, kad sveikaskaičiai atsitiktiniai dydžiai, tenkinantys Frankeno sąlygos analogą, gali būti naudojami perėjimui nuo m-priklausomų prie 1-priklausomų atsitiktinių dydžių. Nustatyta, kad ženklą keičiančios sudėtinės Puasono aproksimacijos yra tokios pačios tikslumo eilės, kaip žinomi rezultatai nepriklausomų atsitiktinių dydžių sumoms. Trečioje dalyje nustatyta, kad kai atsitiktiniai dydžiai yra simetriniai, tuomet sudėtinio Puasono aproksimacijos tikslumas yra daug geresnis nei... [toliau žr. visą tekstą]