A Bayesian Model for Complex System Reliability Growth Under Arbitrary Corrective Actions

Abstract

This paper presents a new method for projecting the reliability growth of a complex continuously operating system. The model allows for arbitrary corrective action strategies, and it differs from other models of this type by using all available data rather than failure mode first occurrence times only. It also differs from other reliability growth projection models in that it provides a complete inference framework via the posterior distribution on the system failure intensity. A unique feature of this approach relative to other Bayesian techniques is the analytic expression for the failure intensity contribution from unobserved failure modes. Expressions for estimating the initial failure intensity, growth potential failure intensity, and the cumulative number of failure modes expected in future testing are also developed. Extensions to the basic framework are also developed. The first accounts for multiple systems under test, and the second develops the posterior distribution while allowing for uncertainty on the Fix Effectiveness Factor values that are assessed. Two separate goodness-of-fit procedures are presented for assessing the appropriateness of the underlying model assumptions.Index Terms-Bayesian reliability, fix effectiveness, reliability growth, reliability growth projection.

Assessing reliability using developmental and operational test data

“…The approach builds upon the reliability growth model developed in [11], where one or more failure modes may be mitigated through the implementation of a corrective action during developmental testing. The operational test is assumed to be a constant configuration test, and the difference in the two test environments is modeled explicitly through the application of a scale parameter.…”

“…The results from this procedure are then used as prior information when assessing the reliability from the Operational Test (OT) in Section 2.2. The approach is the same as that developed in [11]. Further details are available therein, but an outline of the method is presented here for completeness.…”

“…The posterior estimate for the system level failure intensity can be found by summing the individual mode posterior estimates and taking the limit as the number of modes becomes large [11]. The result can be shown to be well approximated by a Gamma distribution.…”

“…We use the notation to denote the system level failure intensity from developmental testing and distinguish between later assessments of the desired operational failure intensity, and the conditional variable refers to the vector containing the number of failures for each of the observed failure modes. In taking the limit with respect to the number of modes, the mean and variance in (4) and (5) are expressed in terms of the prior system level mean and the original parameter from the prior Gamma distribution [11].…”

“…This allows for ease in modeling multiple DT phases within the framework already established. Updating in this manner assumes consistency between successive test phases, and model goodness of fit procedures described in [11] should be employed to ensure that the results are reasonable. Equations for Empirical Bayes estimation of the parameters in (4) and (5) are also provided in [11].…”