An element in a manufacturing system has a mean time between failures of 25 h. When it fails it takes, on average, 2 h to restore it to an operating condition. It has been suggested that the problems caused by the unreliability of this element could be avoided by installing a second identical element as a standby:

1. Calculate the probability of completing an 8 h shift without a system failure both for the single-element and for the redundant-element system.
2. If a repair team is made available such that repairs can be started on a failed element immediately after failure, calculate the long-term system availability (assuming continuous operation) for both the single-element and the redundant-element system. State all assumptions made and comment on whether you consider them reasonable.
3. If the additional element costs 25 000 and downtime costs 100 per hour, what is the payback period for the additional element? (Ignore discounting of cash flows.)