IV. A. 2. e. Reliability Block Diagrams and Models – Keynote

(D) 0.966

There three (maybe more) ways to solve a keystone RBD. Keystone configurations do not resolve by simplifying parallel and series constructions alone.

The best way I know to quickly solve this type of RBD for system reliability is by first identifying the keystone element. That means if you say a block will work (or not work) the remaining elements are easy to reduce and resolve.

In this case component E is the keystone element.

Using Bayes’ Theorem permits us to determine the system reliability given E is working, then add the system reliability given E is not working. Here’s the formula:

$$ {{R}_{sys}}=P({{R}_{sys-E}}|E)\times P(E)+P({{R}_{sys-\bar{E}}}|\bar{E})\times P(\bar{E})$$

where

$$ \bar{E}=1-{E}$$

First let’s sort out the reliability of the system when component E is working. This creates a structure

Start with reducing the two parallel elements

$$ \begin{array}{l}{{R}_{A||C}}=(1-(1-{{R}_{A}})(1-{{R}_{C}}))\\{{R}_{A||C}}=(1-(1-0.85)(1-0.88))\\{{R}_{A||C}}=0.982\end{array}$$

and, the other parallel element

$$ \begin{array}{l}{{R}_{B||D}}=(1-(1-{{R}_{B}})(1-{{R}_{D}}))\\{{R}_{B||D}}=(1-(1-0.96)(1-0.81))\\{{R}_{B||D}}=0.9924\end{array}$$

The solve for the situation when component E is working

$$ \begin{array}{l}{{R}_{sys-E}}={{R}_{A||C}}\times {{R}_{B||D}}\times {{R}_{E}}\\{{R}_{B||D}}=0.982\times 0.9924\times 0.7\\{{R}_{B||D}}=0.6822\end{array}$$

Then the system reliability given E is not working starting with redrawing the RBD structure

Start with reducing the pairs of series structures

$$ \begin{array}{l}{{R}_{AB}}={{R}_{A}}\times {{R}_{B}}=0.85\times 0.96=0.816\\{{R}_{CD}}={{R}_{C}}\times {{R}_{D}}=0.88\times 0.81=0.7128\end{array}$$

The reduce the remaining parallel structure

$$ \begin{array}{l}{{R}_{AB||CD}}=1-\left( 1-{{R}_{AB}} \right)\left( 1-{{R}_{CD}} \right)\\{{R}_{AB||CD}}=1-\left( 1-0.816 \right)\left( 1-0.7128 \right)\\{{R}_{AB||CD}}=0.2841\end{array}$$

The system reliability is then

$$ \begin{array}{l}{{R}_{sys}}=P({{R}_{sys-E}}|E)\times P(E)+P({{R}_{sys-\bar{E}}}|\bar{E})\times P(\bar{E})\\{{R}_{sys}}={{R}_{sys-E}}+{{R}_{sys-\bar{E}}}\\{{R}_{sys}}=0.6822+0.2841=0.9663\end{array}$$