Gap Analysis

This article also appears in the Reliability Growth and Repairable System Analysis Reference book.

Most of the reliability growth models used for estimating and tracking reliability growth based on test data assume that the data set represents all actual system failure times consistent with a uniform definition of failure (complete data). In practice, this may not always be the case and may result in too few or too many failures being reported over some interval of test time. This may result in distorted estimates of the growth rate and current system reliability. This section discusses a practical reliability growth estimation and analysis procedure based on the assumption that anomalies may exist within the data over some interval of the test period but the remaining failure data follows the Crow-AMSAA reliability growth model. In particular, it is assumed that the beginning and ending points in which the anomalies lie are generated independently of the underlying reliability growth process. The approach for estimating the parameters of the growth model with problem data over some interval of time is basically to not use this failure information. The analysis retains the contribution of the interval to the total test time, but no assumptions are made regarding the actual number of failures over the interval. This is often referred to as gap analysis.

Consider the case where a system is tested for time $$T\,\!$$ and the actual failure times are recorded. The time $$T\,\!$$ may possibly be an observed failure time. Also, the end points of the gap interval may or may not correspond to a recorded failure time. The underlying assumption is that the data used in the maximum likelihood estimation follows the Crow-AMSAA model with a Weibull intensity function $$\lambda \beta {{t}^{\beta -1}}\,\!$$. It is not assumed that zero failures occurred during the gap interval, rather, it is assumed that the actual number of failures is unknown, and hence no information at all regarding these failure is used to estimate $$\lambda \,\!$$ and $$\beta \,\!$$.

Let $${{S}_{1}}\,\!$$, $${{S}_{2}}\,\!$$ denote the end points of the gap interval, $${{S}_{1}}<{{S}_{2}}.\,\!$$ Let $$0<{{X}_{1}}<{{X}_{2}}<\ldots <{{X}_}\le {{S}_{1}}\,\!$$ be the failure times over $$(0,\,{{S}_{1}})\,\!$$ and let $${{S}_{2}}<{{Y}_{1}}<{{Y}_{2}}<\ldots <{{Y}_}\le T\,\!$$ be the failure times over $$({{S}_{2}},\,T)\,\!$$. The maximum likelihood estimates of $$\lambda \,\!$$ and $$\beta \,\!$$ are values $$\widehat{\lambda }\,\!$$ and $$\widehat{\beta }\,\!$$ satisfying the following equations.


 * $$\widehat{\lambda }=\frac{{{N}_{1}}+{{N}_{2}}}{S\widehat{_{1}^{\beta }}+{{T}^{\widehat{\beta }}}-S_{2}^{\widehat{\beta }}}\,\!$$


 * $$\widehat{\beta }=\frac{{{N}_{1}}+{{N}_{2}}}{\widehat{\lambda }\left[ S\widehat{_{1}^{\beta }}\ln {{S}_{1}}+{{T}^{\widehat{\beta }}}\ln T-S_{2}^{\widehat{\beta }}\ln {{S}_{2}} \right]-\left[ \underset{i=1}{\overset{\mathop{\sum }}}\,\ln {{X}_{i}}+\underset{i=1}{\overset{\mathop{\sum }}}\,\ln {{Y}_{i}} \right]}\,\!$$

In general, these equations cannot be solved explicitly for $$\widehat{\lambda }\,\!$$ and $$\widehat{\beta }\,\!$$, but must be solved by an iterative procedure.