Component Importance Reference Example

This example validates the component importance results for BlockSim diagrams.

The data set is from example 4.14 on page 100 in the book Life Cycle Reliability Engineering by Dr. Guangbin Yang, John Wiley & Sons, 2007.

A computing system with four computers is configured as given in the BlockSim flowchart diagram below. (Please see Figure 4.29 on page 100 in the reference book for the original configuration.) Each computer follows an exponentially distributed failure rate with the given failure rate parameters in the table below. The Birnbaum’s importance measures of each computer in the system at 4000 and 8000 hours of operation are sought in this example. The system reliability equation, the reliabilities of each computer at the given times and the reliability importance at different times are also scrutinized.

The system reliability equation is computed from Equation 4.40 on page 87 as:


 * $$\begin{align}

R&=Pr(system good|A)Pr(A)+Pr(system good|\bar{A})Pr(\bar{A})\\ R(t)&=[1-(1-R_{C1})(1-R_{C3})]R_{C2}+R_{C3}R_{C4}(1-R_{C2})\\ R(t)&=R_{C1}R_{C2}+R_{C2}R_{C3}+R_{C3}R_{C4}-R_{C1}R_{C2}R_{C3}-R_{C2}R_{C3}R_{C4}\\ \end{align}\,\!$$

Since the times to failure of the computers are exponentially distributed and the reliability can be calculated via $$R_{i} (t)=e^{-\lambda_{i} t}\,\!$$ for i = C1, C2, C3, and C4, the reliabilities of each computer at 4000 and 8000 hours are calculated as:


 * $$\begin{align}

R_{C1}(4000)&=0.9782; R_{C2}(4000) = 0.7711\\ R_{C3}(4000)&=0.8420; R_{C2}(4000) = 0.9712\\ \end{align}\,\!$$

and


 * $$\begin{align}

R_{C1}(8000)&=0.9570; R_{C2}(8000) = 0.5945\\ R_{C3}(8000)&=0.7089; R_{C2}(8000) = 0.9433\\ \end{align}\,\!$$

The system reliabilities at the specified times are calculated as:


 * $$R(4000)=0.9556; R(8000) = 0.8582\,\!$$

Brinbaum’s importance measures for each computer can be formulated via Equation 4.59 on page 100 as:


 * $$I_{B}(i|t)= \frac{\partial R(t)}{\partial R_{i}(t)}\,\!$$


 * $$I_{B}(C1|t)= R_{2}(1-R_{3}); I_{B}(C2|t) = R_{1}+R_{3}-R_{1}R_{3}-R_{3}R_{4}\,\!$$


 * $$I_{B}(C3|t)= R_{2}+ R_{4}- R_{1}R_{2} - R_{2}R_{4}; I_{B}(C4|t) = R_{3}(1 - R_{2})\,\!$$

Therefore, the values of the importance measures can be calculated at 4000 and 8000 hours of operation as:


 * $$\begin{align}

I_{B}(C1|4000)&= 0.1218; I_{B}(C2|4000)= 0.1788\\ I_{B}(C3|4000)&= 0.2391; I_{B}(C4|4000)= 0.1928\\ \end{align}\,\!$$

and


 * $$\begin{align}

I_{B}(C1|8000)&= 0.1730; I_{B}(C2|8000)= 0.3188\\ I_{B}(C3|8000)&= 0.4081; I_{B}(C4|8000)= 0.2875\\ \end{align}\,\!$$

Please note that there is a printing error in the reference book and the importance measure for Computer 4 at 8000 hours is printed as 0.2975 instead of 0.2875.