8.1.8.1. Competing risk model

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

8.1.8.1. Competing risk model

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.

Each failure mechanism leading to a particular type of failure (i.e., failure mode) proceeds independently of every other one, at least until a failure occurs.
The component fails when the first of all the competing failure mechanisms reaches a failure state.
Each of the $k$ failure modes has a known life distribution model $F_i(t)$.

The competing risk model can be used when all three assumptions hold. If $R_c(t)$, $F_c(t)$, and $h_c(t)$ denote the reliability, CDF and failure rate for the component, respectively, and $R_i(t)$, $F_i(t)$, and $h_i(t)$ are the reliability, CDF and failure rate for the $i$-th failure mode, respectively, then the competing risk model formulas are:

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.
Under these conditions the component reliability is the product of the failure mode reliabilities and the component failure rate is just the sum of the failure mode failure rates.

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.