Change of Slope Analysis

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

The assumption of the Crow-AMSAA (NHPP) model is that the failure intensity is monotonically increasing, decreasing or remaining constant over time. However, there might be cases in which the system design or the operational environment experiences major changes during the observation period and, therefore, a single model will not be appropriate to describe the failure behavior for the entire timeline. RGA incorporates a methodology that can be applied to scenarios where a major change occurs during a reliability growth test. The test data can be broken into two segments with a separate Crow-AMSAA (NHPP) model applied to each segment.

Consider the data in the following plot from a reliability growth test.



As discussed above, the cumulative number of failures vs. the cumulative time should be linear on logarithmic scales. The next figure shows the data plotted on logarithmic scales.



One can easily recognize that the failure behavior is not constant throughout the duration of the test. Just by observing the data, it can be asserted that a major change occurred at around 140 hours that resulted in a change in the rate of failures. Therefore, using a single model to analyze this data set likely will not be appropriate.

The Change of Slope methodology proposes to split the data into two segments and apply a Crow-AMSAA (NHPP) model to each segment. The time of change that will be used to split the data into the two segments (it will be referred to as $${{T}_{1}}\,\!$$ ) could be estimated just by observing the data, but will most likely be dictated by engineering knowledge of the specific change to the system design or operating conditions. It is important to note that although two separate models will be applied to each segment, the information collected in the first segment (i.e., data up to $${{T}_{1}}\,\!$$ ) will be considered when creating the model for the second segment (i.e., data after $${{T}_{1}}\,\!$$ ). The models presented next can be applied to the reliability growth analysis of a single system or multiple systems.

Model for First Segment (Data up to T1)
The data up to the point of the change that occurs at $${{T}_{1}}\,\!$$ will be analyzed using the Crow-AMSAA (NHPP) model. Based on the ML equations for $$\lambda \,\!$$ and $$\beta \,\!$$ (in the section Maximum Likelihood Estimators), the ML estimators of the model are:


 * $$\widehat=\frac{T_{1}^}\,\!$$


 * and


 * $${{\widehat{\beta }}_{1}}=\frac{{{n}_{1}}\ln {{T}_{1}}-\underset{i=1}{\overset{\mathop{\sum }}}\,\ln {{t}_{i}}}\,\!$$


 * where:


 * $${{T}_{1}}\,\!$$ is the time when the change occurs
 * $${{n}_{1}}\,\!$$ is the number of failures observed up to time $${{T}_{1}}\,\!$$
 * $${{t}_{i}}\,\!$$ is the time at which each corresponding failure was observed

The equation for $$\widehat{\beta_{1}}\,\!$$ can be rewritten as follows:


 * $$\begin{align}

{{\widehat{\beta }}_{1}}= & \frac{{{n}_{1}}\ln {{T}_{1}}-\left( \ln {{t}_{1}}+\ln {{t}_{2}}+...+\ln {{t}_} \right)} \\ = & \frac{(\ln {{T}_{1}}-\ln {{t}_{1}})+(\ln {{T}_{1}}-\ln {{t}_{2}})+(...)+(\ln {{T}_{1}}-\ln {{t}_})} \\ = & \frac{\ln \tfrac+\ln \tfrac+...+\ln \tfrac} \end{align}\,\!$$


 * or


 * $${{\widehat{\beta }}_{1}}=\frac{\underset{i=1}{\overset{\mathop{\sum }}}\,\ln \tfrac}\,\!$$

Model for Second Segment (Data after T1)
The Crow-AMSAA (NHPP) model will be used again to analyze the data after $${{T}_{1}}\,\!$$. However, the information collected during the first segment will be used when creating the model for the second segment. Given that, the ML estimators of the model parameters in the second segment are:


 * $$\widehat=\frac{T_{2}^}\,\!$$

and:


 * $${{\widehat{\beta }}_{2}}=\frac{{{n}_{1}}\ln \tfrac+\underset{i={{n}_{1}}+1}{\overset{n}{\mathop{\sum }}}\,\ln \tfrac}\,\!$$


 * where:


 * $${{n}_{2}}\,\!$$ is the number of failures that were observed after $${{T}_{1}}\,\!$$
 * $$n={{n}_{1}}+{{n}_{2}}\,\!$$ is the total number of failures observed throughout the test
 * $${{T}_{2}}\,\!$$ is the end time of the test. The test can either be failure terminated or time terminated