Library / Reliability / REL-01
Reliability
Exponential reliability
Probability that an item with a constant failure rate survives a mission of a given length.
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- REL-01
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- v1
- Field
- Reliability
- Result
- R(t) [%]
where 1 / MTBF is the constant failure rate λ, so the same equation is often written R(t) = e−λt.
If failures strike at random at a steady rate, the chance of getting through a mission falls off exponentially with its length. MTBF is the average time between failures. It is not a promise that an item lasts that long.
Calculator
- Failure rate λ0.001 per hour
- Failure rate in FIT1,000,000 per 10⁹ h
- Median life693.1 h
- Survival to t = MTBF36.79 %
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- At t = MTBF only about 37% of items are still working (e−1). MTBF is not a guaranteed life.
- For short missions R ≈ 1 − t / MTBF, so halving the mission length halves the chance of failure. Median life is about 0.69 × MTBF.
- In a series system with constant failure rates, add the failure rates (1 / MTBF), not the MTBFs.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (%) | Local step | Change in result |
|---|---|---|---|---|---|---|
| t Mission duration | 1 to 10,000 h | 0.00454 | 99.9 | 99.9 | +10% | −0.9 pp |
| MTBF Mean time between failures | 100 to 100,000 h | 36.79 | 99.9 | 63.11 | +10% | +0.826 pp |
Sweep of Mission duration
| Mission duration (h) | Reliability (survival probability) (%) |
|---|---|
| 1 | 99.9 |
| 3.162 | 99.68 |
| 10 | 99 |
| 31.62 | 96.89 |
| 100 | 90.48 |
| 316.2 | 72.89 |
| 1,000 | 36.79 |
| 3,162 | 4.233 |
| 10,000 | 0.00454 |
Sweep one variable
Variables and how each one contributes
Reliability (survival probability)
What the equation gives you. Everything else feeds it.
Mean time between failures
Average operating time between failures, for an item with a constant failure rate. Reliability depends only on t / MTBF. Ten times the MTBF cuts the failure risk on a short mission by about ten times.
Mission duration
Length of the mission the item must survive. Longer missions fall off exponentially. At t = MTBF only about 37% of items are still working.
Worked example with the values above
- Ratio of mission time to MTBF: t / MTBF = 100 / 1,000 = 0.1.
- Exponential: e−0.1 = 0.9048.
- As a percentage: R(t) = 90.48 %.
Assumptions and limits
- Constant failure rate: the flat part of the bathtub curve, with no infant mortality or wear-out.
- Failures are random and independent.
- The item is not repaired during the mission.
- MTBF is treated as known. In practice it is estimated from test or field data.
Common mistakes
- Reading MTBF as expected life or a warranty period.
- Applying it to wear-out items such as bearings and batteries, whose failure rate rises with age.
- Adding the MTBFs of components in series. Add failure rates instead.
- Mixing operating hours, calendar hours and cycles.
Statistics and uncertainty
This is where statistics enters directly. MTBF is an estimate from a finite amount of test time and a few failures, so it should be quoted with a confidence bound. For a time-terminated test with total test time T and r failures, a one-sided lower bound uses the chi-squared distribution: MTBFlower = 2T / χ²(α; 2r+2). With few failures the interval is wide, so a high point estimate can still be a weak claim.
Where it is used
- Mission reliability and availability planning for equipment and platforms.
- Spares and maintenance provisioning from failure rates.
- Comparing a redundant configuration with a single item (in parallel, R = 1 − Π(1 − Ri)).
Maths you need
- Exponentials: e−x and how it behaves for small and large x
- Probability basics: reliability is a survival probability between 0 and 1
- Failure rate: λ = 1 / MTBF for a constant rate