#mang #reliability 

# importance metrics
$n$ components in a system
$i$th component works when $X_i = 1$, otherwise failed
probability of a component failing is $q_i = P(X_i = 0)$
$\phi(\textbf{x}) = 0$ when the system is failed, and otherwise $1$

System unreliability funciton $G(\textbf q) = 1-P(\phi(\textbf X) = 1)$
basically $1-R_{sys}$
$G(0_i; \textbf q)$ is the system unabiliability whe component $i$ is working ($q_i = 0$)
$G(1_i; \textbf q)$ is the system unabiliability whe component $i$ is working ($q_i = 1$)
important to remember that $q_i$ is the probability that $i$ has failed

**no single metric is the best in every circumstance**

## birnbaum;s important metric

the probability that $i$ is critical to the funciton of the system at time $t$
$I^i_B(t) = G(1_i, \textbf q) - G(0_i, \textbf q)$

### example
4 batteries, $\lambda = 0.005, 0.009, 0.003, 0.05$
they can either be put in series or parallel

#### series

$q_1 = 1-R_1(t) = 1-e^{-\lambda_1t} = 0.181$
similar for the others


to get to the system unreliability function, we need the probability that the system works. for a series system, this is the probability that none have failed
$$P\left(\phi(\textbf X) = 1\right) = (1-q_1)(1-q_2)(1-q_3)(1-q_4)$$
as defined above, the system unreliability function is that the system doesnt work
$$G(\textbf q) = 1-P\left(\phi(\textbf X) = 1\right) = 1-(1-q_1)(1-q_2)(1-q_3)(1-q_4)$$

the importance metric is the difference in unreliability when $i$ is broken vs when it isnt

$$G(1_1, \textbf q) = 1-(1-1)(1-q_2)(1-q_3)(1-q_4)$$
$$G(0_1, \textbf q) = 1-(1-0)(1-q_2)(1-q_3)(1-q_4)$$
and then the importance metric is $$I_B^i = G(1_i, \textbf q) - G(0_i, \textbf q)$$
this simplifies to $$I_B^1 = (1-q_2)(1-q_3)(1-q_4) = 0.084$$
the others come out to $0.098, 0.077, 0.507$
this shows that battery 4 is the most important to improv

- [ ] #task slides 8-10 he does this shit too fast 📅 2025-10-25 

4 has the highest failure rate so it is most likely to break the system

#### parallel
$I(40) = (1)q_2q_3q_4 - (0)q_2q_3q_4=0.029$
and similar for the others

3 has the lowest failure rate so it is most likely to fail last, this makes it most important

## criticality importance
the chance that $i$ is critical **and** that it has failed by a given time
![[Pasted image 20251023162429.png]]

because there is the conditional element, there is a chance that everything else has failed. This means that it cant do parallel systems???

## upgrading functions
basically only useful for exponental functions
can be used to determine the optimal choice of upgrade