Bayesian methods for evaluating discrete reliability growth

Abstract

Bayesian estimation procedures are derived herein that may be utilized to evaluate reliability growth of discrete systems, such as guns, rockets, missile systems, torpedoes, etc. One of the advantages of these Bayesian procedures is that they directly quantify the epistemic uncertainties in model parameters (i.e., the shape parameters of the beta distribution), as well as six reliability growth metrics of basic interest to program management. These metrics include: (1) the initial system reliability; (2) the projected reliability following failure mode mitigation; (3) reliability growth potential (i.e., the theoretical upper-limit on reliability achieved by finding and fixing all failure modes via a specified level of fix effectiveness); (4) the expected number of failure modes observed during testing; (5) the probability of observing a new failure mode; and (6) the fraction of the initial system probability of failure associated with failure modes for which program management is aware. These metrics [18] give reliability practitioners the means to estimate the reliability of discrete systems undergoing development, address model goodness-of-fit concerns, quantify programmatic risk, and assess system maturity. Analytical results are presented to obtain Bayes' estimates of the beta shape parameters under a delayed corrective action strategy (i.e., when corrective actions are installed on system prototypes at the end of the current test phase). A Monte Carlo simulation approach is given for constructing uncertainty distributions on each of the aforementioned reliability growth management metrics. Bayesian probability limits for inference on interval estimation are obtained in the usual manner (i.e., via desired percentiles of the uncertainty distributions). These uncertainty distributions are found to be approximated very well by beta and/or Gaussian random variables. These methods are illustrated by simple numerical examples. In particular Bayes' estimates the beta shape parameters are obtained from a small dataset, and compared against the true parameter values. Bayesian epistemic uncertainty distributions are also constructed for the reliability growth management metrics via the proposed Monte Carlo approach. This methodology is useful to program managers and reliability practitioners that wish to quantitatively assess the reliability growth program of one-shot systems developed under a delayed corrective action strategy.