Core Reliability Mathematics
Under the classical assumption of a constant failure rate (λ), time-to-failure follows an exponential distribution. Mean Time Between Failures (MTBF) represents the expected operating time between successive breakdown events in repairable systems, while Mean Time to Failure (MTTF) applies to non-repairable components.
Formulas & Statistical Derivations
Worked Industrial Example
A plant runs 5 identical packaging machines for 200 hours each, accumulating a total of 1,000 operating hours. During this campaign, 2 breakdown events occur. Management requests a 90% confidence assessment for mission reliability across a 100-hour production run.
Results:
- Point MTBF: 1,000 / 2 = 500 hours.
- Mission Reliability R(100h): exp(-100 / 500) = 81.87%.
- 90% Confidence Bounds: Lower = 158.8 hours, Upper = 2,814.1 hours.
- Engineering Takeaway: With only 2 failures, statistical uncertainty is broad. Relying solely on the point estimate (500h) overestimates lower-tail reliability.
Frequently Asked Questions
What is the difference between Type I and Type II testing?
Type I (time-terminated) testing stops at a predetermined test duration T, regardless of how many failures occur. Type II (failure-terminated) testing continues until a predetermined number of failures r occurs. Because the stopping boundary differs, the degrees of freedom in the denominator χ² differ by 2.
Can I use MTBF for components that wear out?
No. MTBF assumes a constant failure rate. Mechanical wear-out (bearings, gears, valves) exhibits an increasing failure rate that requires Weibull analysis.