Reliability Demonstration Test Design for Repairable Systems

The RGA software provides a utility that allows you to design reliability demonstration tests for repairable systems. For example, you may want to design a test to demonstrate a specific cumulative MTBF for a specific operating period at a specific confidence level for a repairable system. The same applies for demonstrating an instantaneous MTBF or cumulative and instantaneous failure intensity at a given time $$t.\,\!$$

Underlying Theory
The failure process in a repairable system is considered to be a non-homogeneous Poisson process (NHPP) with power law failure intensity. The instantaneous failure intensity at time $$t\,\!$$ is:


 * $${{\lambda }_{i}}\left( t \right)=\lambda \beta {{t}^{\beta -1}}={{\lambda }_{c}}\left( t \right)\beta \,\!$$

So, the cumulative failure intensity at time $$t\,\!$$ is:


 * $${{\lambda }_{c}}\left( t \right)=\lambda {{t}^{\beta -1}}=\frac{{{\lambda }_{i}}\left( t \right)}{\beta }\,\!$$

The instantaneous MTBF is:


 * $$\begin{align}

MTB{{F}_{i}}\left( t \right)= & \frac{1} \\ = & \frac{1}{\lambda \beta }{{t}^{1-\beta }} \\ = & \frac{MTB{{F}_{c}}\left( t \right)}{\beta } \end{align}\,\!$$

The cumulative MTBF at time $$t\,\!$$ is:


 * $$MTB{{F}_{c}}\left( t \right)=\frac{1}{{{\lambda }_{c}}\left( t \right)}=\frac{1}{\lambda }{{t}^{1-\beta }}=MTB{{F}_{i}}\left( t \right)\beta \,\!$$

The relation between the confidence level, required test time, number of systems under test and allowed total number of failures in the test is:


 * $$1-CL=\underset{i=0}{\overset{r}{\mathop \sum }}\,\frac{{{\left( m\lambda {{T}^{\beta }} \right)}^{i}}\exp (-m\lambda {{T}^{\beta }})}{i!}\,\!$$

where:


 * $$T\,\!$$ is the total test time for each system.
 * $$m\,\!$$ is the number of systems under test.
 * $$r\,\!$$ is the number of allowed failures in the test.
 * $$CL\,\!$$ is the confidence level.

Given any three of the parameters, the equation above can be solved for the fourth unknown parameter. Note that when $$\beta =1,\,\!$$ the number of failures is a homogeneous Poisson process, and the time between failures is given by the exponential distribution.