Calculator Inputs
Use one consistent time unit for mission time, MTBF, and Weibull scale values.
Formula Used
Exponential model
Reliability: R(t) = e-λt
Cumulative failure probability: F(t) = 1 - R(t)
Hazard rate: h(t) = λ
Mean time to failure: MTTF = 1 / λ
Time to threshold probability p: tp = -ln(1 - p) / λ
Weibull model
Reliability: R(t) = e-(t/η)β
Cumulative failure probability: F(t) = 1 - R(t)
Hazard rate: h(t) = (β/η)(t/η)β-1
Mean time to failure: MTTF = ηΓ(1 + 1/β)
Time to threshold probability p: tp = η[-ln(1 - p)]1/β
Population metrics
Probability of at least one failure among n independent units: Fpopulation(t) = 1 - [R(t)]n
Expected failed units: E[failed] = n × F(t)
These population equations assume statistically independent and identically modeled units.
How to Use This Calculator
- Choose either the Weibull model or the exponential model.
- Select one time unit and keep all entered values consistent with it.
- Enter mission time, number of units, threshold probability, and graph settings.
- For exponential analysis, enter either failure rate or MTBF.
- For Weibull analysis, enter scale η and shape β.
- Press the calculate button to see summary metrics and the graph.
- Use the CSV or PDF button to export the current result set.
Example Data Table
| Case | Model | Mission Time | Key Inputs | Units | Unit Failure Probability | Population Failure Probability |
|---|---|---|---|---|---|---|
| Pump fleet | Exponential | 1,000 hours | λ = 0.00025/hour | 25 | 0.221199 | 0.998070 |
| Bearing group | Weibull | 2,000 hours | η = 5,000, β = 1.8 | 10 | 0.174842 | 0.853657 |
| Seal assembly | Weibull | 3,000 cycles | η = 7,000, β = 2.3 | 5 | 0.132767 | 0.509453 |
FAQs
1. What does cumulative failure probability mean?
It is the probability that a unit has failed by a specified mission time. It accumulates over time, so the value never decreases as time increases.
2. When should I use the exponential model?
Use it when the hazard rate is approximately constant over time. It is common for electronic parts during their useful life period and for quick screening studies.
3. When is the Weibull model better?
Use Weibull when failure behavior changes over time. It handles infant mortality, random failures, and wear-out patterns by adjusting the shape parameter β.
4. What does the shape parameter β tell me?
If β is less than 1, hazard decreases over time. If β equals 1, hazard is constant. If β is greater than 1, hazard increases with age.
5. Why does the calculator ask for population size?
Population size lets you estimate the risk that at least one unit fails in a group and the expected number of failed units over the same mission time.
6. What is threshold failure probability time?
It is the time required to reach a chosen cumulative failure probability, such as 10% or 50%. It helps with maintenance and inspection planning.
7. Can I use cycles instead of hours?
Yes. The calculator uses whichever time label you choose, provided your mission time, MTBF, failure rate basis, and Weibull scale use the same unit consistently.
8. Are these results exact for real equipment?
They are model-based estimates. Accuracy depends on good input data, correct distribution choice, and whether assumptions like independence and stable operating conditions are reasonable.