8.
Assessing Product Reliability
8.1. Introduction 8.1.8. How can you evaluate reliability from the "bottomup" (component failure mode to system failure rate)?


Use the competing risk model when the failure mechanisms are independent and the first mechanism failure causes the component to fail 
Assume a (replaceable) component or unit has \(k\)
different ways it can fail. These are called failure modes
and underlying each failure mode is a failure mechanism.
The Competing Risk Model evaluates component reliability by "building up" from the reliability models for each failure mode. The following three assumptions are needed.


Multiply reliabilities and add failure rates 
$$ \begin{eqnarray}
R_c(t) & = & \prod_{i=1}^k R_i(t) \\
& & \\
F_c(t) & = & 1  \prod_{i=1}^k [1  F_i(t)] \\
& & \\
h_c(t) & = & \sum_{i=1}^k h_i(t)
\end{eqnarray} $$
Think of the competing risk model in the following way:
All the failure mechanisms are having a race to see which can reach failure first. They are not allowed to "look over their shoulder or sideways" at the progress the other ones are making. They just go their own way as fast as they can and the first to reach "failure" causes the component to fail.Note that the above holds for any arbitrary life distribution model, as long as "independence" and "first mechanism failure causes the component to fail" holds. When we learn how to plot and analyze reliability data in later sections, the methods will be applied separately to each failure mode within the data set (considering failures due to all other modes as "censored run times"). With this approach, the competing risk model provides the glue to put the pieces back together again. 