No / The batch renewal process is the least-biased choice of process given only the measures of count correlation and interval correlation at all lags.This paper reviews the batch renewal process, both for LRD (long-range-dependent) traffic and for SRD (short-range-dependent) traffic in the discrete space-discrete time domain, and in the wider context of general traffic in that domain. It shows some applications of the batch renewal process in simple queues and in queueing network models. The paper concludes with open research problems and issues arising from the discussion.

In this thesis we present some related problems about the renewal processes. More precisely, let $gamma_{t}$ be the residual life at time $t$ of the renewal process $A={A(t),t geq 0}$, $F$ be the common distribution function of the inter-arrival times. Under suitable conditions, we prove that if $Var(gamma_{t})=E^2(gamma_{t})-E(gamma_{t}),forall t=0,1 ho,2 ho,3 ho,... $, then $F$ will be geometrically distributed under the assumption $F$ is discrete. We also discuss the tails of random sums for the renewal process. We prove that the $k$ power of random sum is always new worse than used ($NWU$).

This thesis explores methodologies of modeling and evaluating the operational performance of a typical aeroengine having field experience over two decades. Upon failure, the engine is repaired and restored to flight worthy condition and hence comes under the purview of repairable systems. Operational performance of the engine is being measured in terms of five functions of time, namely, M(t), which is the expected number of system failures in the time interval [0,t]; system failure rate m(t), which is an unconditional quantity and is simply the derivative of M(t); ρ(t), the conditional failure intensity given the history of a system Ht, which is nothing but limdt→1 Prob(System fails in [t,t + dt] |Ht); and M′(t) and m′(t), which are 0 dt conditional entities analogous to M(t) and m(t) defined in the same spirit as that of ρ(t), the details of which are given in the third chapter of the thesis. These functions are being estimated using field failure-repair data of 418 aeroengines, where the observations on time between failures are being measured in number of flying hours logged in between failures, and the corresponding repair duration is being measured in number of calendar days. To start with, using the superimposed renewal process model the above quantities M(t), m(t), m′(t), M′(t) and ρ(t) are estimated both in the frequentist as well as the Bayesian framework. Subsequently repair times have been incorporated into the model and analysed using both frequentist and Bayesian approaches. Next, the model of Lawless and Thiagarajah (1996) which incorporates both renewal and time trend, has been generalized to include repair time as well, and a comprehensive methodology of Bayesian model selection under this model has been developed. After introducing the research problem in the first chapter, the engineering system description leading to the identification of the failure modes, repair practice and the variables of interest is taken up in the following chapter at the outset, as a pre-requisite to the stochastic modeling and the statistical analysis that to follow in the remainder of the thesis. As the first stochastic model, the number of system failures in a given time interval is modeled as a superimposed renewal process with the constituent independent renewal processes running in different component sockets having Weibull inter failure times. This model is first empirically validated using the field failure data and then using this model, the five quantities of interest as mentioned above viz. M(t), m(t), ρ(t), M′(t) and m′(t) are analysed from a frequentist maximum likelihood perspective. A Bayesian analysis of the same follows in the subsequent chapter. Next, the repair effect is incorporated into the superimposed renewal process model by considering the Weibull parameters of inter failure times of the constituent renewal processes running in independent component sockets as a polynomial in the last repair time. The nature of this polynomial relationships are empirically deter-mined and the Weibull assumption is validated through a test of hypothesis. Different polynomial relationships lead to consideration of several models, with the correct ones chosen through a series of likelihood ratio tests. Next based on the appropriate models a maximum likelihood analysis of M(t), ρ(t) and M′(t) has been carried out. Like the simple superimposed renewal process model, Bayesian analysis of this model incorporating repair times is carried out in the following chapter. In the Bayesian setup however, the problem of model selection could be kept unrestricted to non-nested models as well (unlike the previous chapter, where only nested models could be considered), and a comprehensive model selection exercise has been carried out with the aid of intrinsic Bayes factors and training data sets. The last but one chapter presents a generalised model of Lawless and Thiagarajah (1996) for performance evaluation of aeroengines that incorporate renewals, time trends and the repair characteristics. Here also since the primary problem is one of model selection, the entire analysis like in the preceding chapter has been carried out under the Bayesian frame-work. The final chapter concludes the thesis by comparing the empirical results obtained in the previous five chapters, summarising the main contributions of the thesis and providing directions for future research.

Analysis of functional connectivity of simultaneously recorded multiple spike trains is one of the major issues in the neuroscience. The progress of the statistical methods to the analysis of functional connectivity of multiple spike trains is relatively slow. In this thesis two statistical techniques are presented to the analysis of functional connectivity of multiple spike trains. The first method is known as the modified correlation grid (MCG). This method is based on the calculation of cross-correlation function of all possible pair-wise spike trains. The second technique is known as the Cox method. This method is based on the modulated renewal process (MRP). The original paper on the application of the Cox method (Borisyuk et al., 1985) to neuroscience data was used to analyse only pairs and triplets of spike trains. This method is further developed in this thesis to support simultaneously recorded of any possible set of multiple spike trains. A probabilistic model is developed to test the Cox method. This probabilistic model is based on the MRP. Due to the common probabilistic basis of the probabilistic model and the Cox method, the probabilistic model is a convenient technique to test the Cox method. A new technique based on a pair-wise analysis of Cox method known as the Cox metric is presented to find the groups of coupled spike trains. Another new technique known as motif analysis is introduced which is useful in identifying interconnections among the spike trains. This technique is based on the triplet-wise analysis of the Cox method. All these methods are applied to several sets of spike trains generated by the Enhanced Leaky and Integrate Fire (ELIF) model. The results suggest that these methods are successful for analysing functional connectivity of simultaneously recorded multiple spike trains. These methods are also applied to an experimental data recorded from cat’s visual cortex. The connection matrix derived from the experimental data by the Cox method is further applied to the graph theoretical methods.

Processsos de renovação são um caso especial de processos pontuais envolvendo eventos recorrentes nos quais um item ou unidade, após a ocorrência de uma falha, é recolocado na mesma condição de novo. Devido a essa propriedade os tempos entre ocorrências para um processo de renovação são independentes e a sua função intensidade é dada pela função de risco. Fatores que interferem nos tempos de recorrência de unidades distintas, ou indivíduos, e que não são observados, podem ser modelados com a inclusão de um termo de fragilidade no modelo. Neste trabalho é apresentado o desenvolvimento de um modelo de regressão para um processo de renovação com tempos entre ocorrências com distribuição de Weibull. Na modelagem foi considerada, ainda, a presença de censuras e a inclusão de um termo de fragilidade para explicar a relação existente entre os tempos de recorrências de uma unidade. A metodologia é desenvolvida para o caso em que várias unidades são acometidas por eventos recorrentes. Nas simulações realizadas foram analisadas as probabilidades de cobertura empíricas do intervalo de confiança normal assintótico e também o comportamento das variâncias dos estimadores. A presença de censuras na amostra inflacionou as variâncias dos estimadores de máxima verossimilhança além de produzir estimativas viciadas para um dos parâmetros da regressão, sendo que o vício do estimador foi corrigido por meio de um processo "bootstrap". Na modelagem sem termo de fragilidade, os resultados das análises das probabilidades de cobertura empírica dos intervalos de confiança assintóticos mostraram uma boa aproximação com os valores esperados, mas com certos cuidados a serem tomados, especialmente nos procedimentos baseados na simetria das distribuições empíricas. A inclusão de um termo de fragilidade na modelagem, por sua vez, causou uma perturbação na estimação máxima verossimilhança com um aumento nas variâncias dos estimadores diretamente associados à variabilidade do termo de fragilidade. Além disso, as coberturas empíricas dos intervalos de confiança assintóticos foram, na grande maioria superestimadas, com resultados satisfatórios apenas para o parâmetro de forma da distribuição Weibull. / Renewal Processes are a special case of point processes involving recurrent events in which a unit, after a failure, is restored to the like new condition. Due to that property the times between occurrences for a renewal process are independent and its intensity function is given by the hazard function. Random factors not observed, that afects the recurrence times of the units, can be explained by a frailty term added in the model. In this work a regression model is presented for a renewal process with Weibull distribution for the times between occurrences. The modeling considers censored times and a frailty variable to explain the relationship among the recurrence times of a unit. The methodology was developed for the situation where several units are submitted by recurrent events. The empirical probabilities of coverage of the asymptotic normal confidence interval and the behavior of the variances of the estimators were analyzed in the simulations performed. The presence of censures in the sample inflated the variances of the maximum likelihood estimators besides to produce biased estimates for the regression parameters. The bias of the estimator was corrected by "bootstrap" procedure. The analysis of the probability of empirical coverage of the asymptotic confidence intervals, without frailty, presented a good approximation to the nominal values, but some observations about procedures have to be made on the symmetry of the empirical distributions. The frailty term incorporated at the modeling disturbed the maximum likelihood estimation increasing estimators' variability, directly associated to the variance of the fragility term. In the most of the cases, the empirical coverages of the asymptotic confidence intervals were overestimated, with satisfactory results just for the shape parameter of the Weibull distribution.

Processsos de renovação são um caso especial de processos pontuais envolvendo eventos recorrentes nos quais um item ou unidade, após a ocorrência de uma falha, é recolocado na mesma condição de novo. Devido a essa propriedade os tempos entre ocorrências para um processo de renovação são independentes e a sua função intensidade é dada pela função de risco. Fatores que interferem nos tempos de recorrência de unidades distintas, ou indivíduos, e que não são observados, podem ser modelados com a inclusão de um termo de fragilidade no modelo. Neste trabalho é apresentado o desenvolvimento de um modelo de regressão para um processo de renovação com tempos entre ocorrências com distribuição de Weibull. Na modelagem foi considerada, ainda, a presença de censuras e a inclusão de um termo de fragilidade para explicar a relação existente entre os tempos de recorrências de uma unidade. A metodologia é desenvolvida para o caso em que várias unidades são acometidas por eventos recorrentes. Nas simulações realizadas foram analisadas as probabilidades de cobertura empíricas do intervalo de confiança normal assintótico e também o comportamento das variâncias dos estimadores. A presença de censuras na amostra inflacionou as variâncias dos estimadores de máxima verossimilhança além de produzir estimativas viciadas para um dos parâmetros da regressão, sendo que o vício do estimador foi corrigido por meio de um processo "bootstrap". Na modelagem sem termo de fragilidade, os resultados das análises das probabilidades de cobertura empírica dos intervalos de confiança assintóticos mostraram uma boa aproximação com os valores esperados, mas com certos cuidados a serem tomados, especialmente nos procedimentos baseados na simetria das distribuições empíricas. A inclusão de um termo de fragilidade na modelagem, por sua vez, causou uma perturbação na estimação máxima verossimilhança com um aumento nas variâncias dos estimadores diretamente associados à variabilidade do termo de fragilidade. Além disso, as coberturas empíricas dos intervalos de confiança assintóticos foram, na grande maioria superestimadas, com resultados satisfatórios apenas para o parâmetro de forma da distribuição Weibull. / Renewal Processes are a special case of point processes involving recurrent events in which a unit, after a failure, is restored to the like new condition. Due to that property the times between occurrences for a renewal process are independent and its intensity function is given by the hazard function. Random factors not observed, that afects the recurrence times of the units, can be explained by a frailty term added in the model. In this work a regression model is presented for a renewal process with Weibull distribution for the times between occurrences. The modeling considers censored times and a frailty variable to explain the relationship among the recurrence times of a unit. The methodology was developed for the situation where several units are submitted by recurrent events. The empirical probabilities of coverage of the asymptotic normal confidence interval and the behavior of the variances of the estimators were analyzed in the simulations performed. The presence of censures in the sample inflated the variances of the maximum likelihood estimators besides to produce biased estimates for the regression parameters. The bias of the estimator was corrected by "bootstrap" procedure. The analysis of the probability of empirical coverage of the asymptotic confidence intervals, without frailty, presented a good approximation to the nominal values, but some observations about procedures have to be made on the symmetry of the empirical distributions. The frailty term incorporated at the modeling disturbed the maximum likelihood estimation increasing estimators' variability, directly associated to the variance of the fragility term. In the most of the cases, the empirical coverages of the asymptotic confidence intervals were overestimated, with satisfactory results just for the shape parameter of the Weibull distribution.

The maintenance cost for an engineering system is an uncertain quantity due to uncertainties associated with occurrence of failure and the time taken to restore the system. The problem of probabilistic analysis of maintenance cost can be modeled as a stochastic renewal-reward process, which is a complex problem. Assuming that the time horizon of the maintenance policy approaches infinity, simple asymptotic formulas have been derived for the failure rate and the cost per unit time. These asymptotic formulas are widely utilized in the reliability literature for the optimization of a maintenance policy. However, in the finite life of highly reliable systems, such as safety systems used in a nuclear plant, the applicability of asymptotic approximations is questionable. Thus, the development of methods for accurate evaluation of expected maintenance cost, failure rate, and availability of engineering systems is the subject matter of this thesis. In this thesis, an accurate derivation of any m-th order statistical moment of maintenance cost is presented. The proposed formulation can be used to derive results for a specific maintenance policy. The cost of condition-based maintenance (CBM) of a system is analyzed in detail, in which the system degradation is modeled as a stochastic gamma process. The CBM model is generalized by considering the random repair time and delay in degradation initiation. Since the expected cost is not informative enough to estimate the financial risk measures, such as Value-at-Risk, the probability distribution of the maintenance cost is derived. This derivation is based on an interesting idea that the characteristic function of the cost can be computed from a renewal-type integral equation, and its Fourier transform leads to the probability distribution. A sequential inspection and replacement strategy is presented for the asset management of a large population of components. The finite-time analyses presented in this thesis can be combined to compute the reliability and availability at the system level. Practical case studies involving the maintenance of the heat transport piping system in a nuclear plant and a breakwater are presented. A general conclusion is that finite time cost analysis should be used for a realistic evaluation and optimization of maintenance policies for critical infrastructure systems.

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.

The aim of the present study is to investigate a probability distribution that can be derived from the laplace probability distribution and can be used to model various real world problems. In the last few decades, there has been a growing interest in the construction of flexible parametric classes of probability distributions. Various forms of the skewed and kurtotic distributions have appeared in the literature for data analysis and modeling. In particular, various forms of the skew laplace distribution have been introduced and applied in several areas including medical science, environmental science, communications, economics, engineering and finance, among others. In the present study we will investigate the skew laplace distribution based on the definition of skewed distributions introduced by O'Hagan and extensively studied by Azzalini. A random variable X is said to have the skew-symmetric distribution if its probability density function is f(x) = 2g(x)G(lambda x), where g and G are the probability density function and the cumulative distribution function of a symmetric distribution around 0 respectively and lambda is the skewness parameter. We will investigate the mathematical properties of this distribution and apply it to real applications. In particular, we will consider the exchange rate data for six different currencies namely, Australian Dollar,Canadian Dollar, European Euro, Japanese Yen, Switzerland Franc and United Kingdom Pound versus United States Dollar. To describe a life phenomenon we will be mostly interested when the random variableis positive. Thus, we will consider the case when the skew Laplace pdf is truncated to the left at 0 and we will study its mathematical properties. Comparisons with other life time distributions will be presented. In particular we will compare the truncated skew laplace (TSL) distribution with the two parameter Gamma probability distribution with simulated and real data with respect to its reliability behavior. We also study the hypoexponential pdf and compare it with the TSL distribution. Since the TSL pdf has increasing failure rate (IFR) we will investigate a possible application in system maintenance. In particular we study the problem related to the preventive maintenance.

Darbe nagrinėjamas dvimatis atstatymo procesas. Gaunami jo integralinių lygčių ir laplaso transformacijų pavidalai, bei jų asimptotikos. / In graduate research a two-dimentional renewal process. The integral differential equation of renewal function, its Laplase transform and asymptotic's was found.