Bayesian sampling schemes for estimating the reliability of a series system

Determining the reliability of a system is an important aspect of system design. In this dissertation we develop a model based on a series system with two independent components, each with its own unknown reliability, &thetas; being the probability that the first component succeeds and ω the probability that the second component is successful. The purpose of this dissertation is to estimate the reliability of the series system, namely that both components pass, or in a mathematical sense, R = &thetas;ω where R is the reliability that the overall system will succeed. Unfortunately, each of these values is unknown. To satisfy our objective, we estimate the reliability of each component using a Bayesian approach by estimating the product of means of two independent Bernoulli populations, since the components can either pass or fail. Using a squared error loss, we apply statistical procedures and methods to create an estimate based on observations of test cases. However, each unit we take to define the estimate on observations of test cases. However, each unit we take to define the estimate is done at some cost, namely c₁ for each unit taken from the first population and c₂ for each unit taken from the second population. This cost factor will be included in our model for estimating the reliability of the system. We will find the minimum Bayes risk to obtain our best possible estimate. We will determine the lower bound of the Bayes risk with associated costs and develop a procedure for determining when we satisfy optimality. We take advantage of information from previous testing by utilizing the Bayesian approach, with conjugate beta prior distributions, to enhance our knowledge of the estimate. We will look at examples of different allocation schemes; both non-random fixed cases and dynamically random sequential cases and determine if these allocations satisfy the desired asymptotical optimality property.