#mang #maintainance

$$MTTF = E(t) = \int tf(t) dt = \int R(t)dt$$

repairable system - system where a failed component can be released and the new component is as good as the original
availability - how much of the time the system can be in service

# renewal

fundimental renewal equation - 
$$M(t) = F(t) + \int_0^t M(t-x) f(x) dx$$
renewal density equation - 
$$m(t) = f(t) + \int_0^t m(t-x) f(x) dx$$
$$M(t) = \lambda (t)$$
# discrete time approach
way to approximate the time
 - [ ] #task look at lecture 15 slide 21 #mang


# optimal maintenance & inspection

## cost minimisation
### constant interval
replace if component breaks
replace after a constant time always
$t_p$ is when it is replaces
$$c(t_p) = \frac{\text{total expected cost}}{\text{length of interval}}$$
the total cost is the cost of preventative replacement plus the expected cost of spares
- [ ] #task actually im just going to have fun #mang

if you have a failure right before replacement, you replace the component and then the interval happens and you do it again

### predetermined age

basically the same but you do it based on the age of the component instead of wall clock time