I have a little question.

All the components of which I speak are identical. I have a system which is composed of 4 of those components. If one of them breaks down the entire system breaks down. I have 7 components in stock.

If I hand in the broken components into repair it takes about three months to get them repaired, but unfortunately only about 1/3 of the pieces get repaired (the other ones are trash).

The mean time between failure of one of the components is about three months.

So:

- 4 components
- 7 stock
- 3 months repair time
- 1/3 repair ratio

Now I want to know two things:

- The repair time can be changed slightly (a matter of given priority); which is the optimal repair time and why?
- How long will my system last (best case)? What’s the formula to determine that?

I tried to figure those out by myself but didn’t manage to compose a formula for this problem because the components of which the system is composed can be changed (once they are repaired they can replace broken ones of the system, and the original can get repaired too…).

Can someone help me to compose the formula?

Where can I find the mathematical background?