The bathtub curve and MTBF, explained
Short answer: the bathtub curve plots failure rate over a product's life, and it has three phases:
- Infant mortality: the failure rate starts high and falls. Causes are defects from manufacturing or assembly. Countermeasures are burn-in, screening and better process control. (In Weibull terms, β < 1.)
- Useful life: the failure rate is low and roughly constant, and failures are random. This is where the exponential model and MTBF apply (β ≈ 1).
- Wear-out: the failure rate rises as parts fatigue, corrode or wear. Countermeasures are preventive replacement and design life margin (β > 1).
MTBF and the exponential model
In the useful-life phase, failure rate λ = failures / total operating time, MTBF = 1/λ, and the chance of surviving to time t is R(t) = e−λt.
Worked example
20 units run for 500 hours each, and 4 fail. Failed units are replaced right away, so total operating time is 20 × 500 hours.
What it means: a single unit has about a 96% chance of running 100 hours without failing. Only about 37% of units survive to the MTBF. MTBF is not "how long a typical unit lasts."
Common exam traps
- Thinking MTBF means half the units survive to that time. Under the exponential model, only 36.8% survive to t = MTBF.
- Using MTBF for wear-out. The constant-rate model doesn't apply once the failure rate rises. Use Weibull instead.
- MTBF vs MTTF. MTBF is for repairable items, and MTTF is for items that are replaced rather than repaired.
- Series systems. Reliabilities multiply (Rsys = R₁ × R₂ × …), so a series system is less reliable than its weakest part.
Enter failures and total operating time to get λ and MTBF, then compute R(t) for any mission time.
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