New survival distributions that quantify the gain from eliminating flawed components

A general method for deriving new survival distributions from old is presented. This yields a class of useful mixture distributions. Fitting such distributions to failure-time data allows estimation of the improvement in reliability that could be gained from eliminating ‘frail’ components. One model parameter is the proportional increase of expected survival time that could be achieved. Some 2 and 3 parameter distributions in this class are described, which are extensions of the Weibull, exponential, gamma and lognormal distributions. The methodology is illustrated by fitting some well travelled datasets. Keywords: Weibull distribution, gamma distribution, mixture distribution, hazard function, partial integration, frailty mode

Modeling Multimodal Failure Effects of Complex Systems Using Polyweibull Distribution

The Department of Defense (DoD) enlists multiple complex systems across each of their departments. Between the aging systems going through an overhaul and emerging new systems, quality assurance to complete the mission and secure the nation‘s objectives is an absolute necessity. The U.S. Air Force‘s increased interest in Remotely Piloted Aircraft (RPA) and the Space Warfighting domain are current examples of complex systems that must maintain high reliability and sustainability in order to complete missions moving forward. DoD systems continue to grow in complexity with an increasing number of components and parts in more complex arrangements. Bathtub-shaped hazard functions arise from the existence of multiple competing failure modes which dominate at different periods in a systems lifecycle. The standard method for modeling the infant mortality, useful-life, and end-of-life wear-out failures depicted in a bathtub-curve is the Weibull distribution. However, this will only model one or the other, and not all three at once. The poly-Weibull distribution arises naturally in scenarios of competing risks as it describes the minimum of several independent random variables where each follows a distinct Weibull law. Little is currently known or has been developed for the poly-Weibull distribution. In this report, the poly-Weibull is compared against other goodness-of-fit models to model these completing multimodal failures. An equation to determine the moments for the poly-Weibull is derived leading to the development of properties such as the mean, variance, skewness, and kurtosis using Maximum Likelihood Estimation (MLE) parameters obtained from a data set with known bathtub shaped hazard function

Modeling Reliability Growth in Accelerated Stress Testing

Qualitative accelerated test methods improve system reliability by identifying and removing initial design flaws. However, schedule and cost constraints often preclude sufficient testing to generate a meaningful reliability estimate from the data obtained in these tests. In this dissertation a modified accelerated life test is proposed to assess the likelihood of attaining a reliability requirement based on tests of early system prototypes. Assuming each prototype contains an unknown number of independent competing failure modes whose respective times to occurrence are governed by a distinct Weibull law, the observed failure data from this qualitative test are shown to follow a poly-Weibull distribution. However, using an agent-based Monte Carlo simulation, it is shown that for typical products subjected to qualitative testing, the failure observations result from a homogenous subset of the total number of latent failure modes and the failure data can be adequately modeled with a Weibull distribution. Thus, the projected system reliability after implementing corrective action to remove one or more failure modes can be estimated using established quantitative accelerated test data analysis methods. Our results suggest that a significant cost and time savings may be realized using the proposed method to signal the need to reassess a product’s design or reallocate test resources to avoid unnecessary maintenance or redesigns. Further, the proposed approach allows a significant reduction in the test time and sample size required to estimate the risk of meeting a reliability requirement over current quantitative accelerated life test techniques. Additional contributions include a numerical and analytical procedure for obtaining the maximum likelihood parameter estimates and observed Fisher information matrix components for the generalized poly-Weibull distribution. Using this procedure, we show that the poly-Weibull distribution outperforms the best-fit modified Weibull alternatives in the literature with respect to their fit of reference data sets for which the hazard rate functions are non-monotone

Reliability Analysis of Discrete Lifetime Distributions

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The exponentiated discrete Weibull Distribution

In this paper, the exponentiated discrete Weibull distribution is introduced. This new generalization of the discrete Weibull distribution can also be considered as a discrete analogue of the exponentiated Weibull distribution. A special case of this exponentiated discrete Weibull distribution defines a new generalization of the discrete Rayleigh distribution for the first time in the literature. In addition, discrete generalized exponential and geometric distributions are some special sub-models of the new distribution. Here, some basic distributional properties, moments, and order statistics of this new discrete distribution are studied. We will see that the hazard rate function can be in- creasing, decreasing, bathtub, and upside-down bathtub shaped. Estimation of the parameters is illustrated using the maximum likelihood method. The model with a real data set is also examine

Extended Weibull distributions in reliability engineering

Ph.DDOCTOR OF PHILOSOPH