Core Reliability
Reliability Calculator R(t)
The probability that a system survives a given mission time without failure, from its MTBF or failure rate.
R(t) = e^(−t ÷ MTBF)What is reliability R(t)?
Reliability R(t) is the probability that a system will perform its intended function without failure for a specified period of time under stated conditions. It ranges from 0 (certain failure) to 1 (certain survival).
- Quantifies mission success probability
- Drives design decisions and redundancy requirements
- Used in safety-critical system analysis
- Basis for warranty period determination
- Key input for availability calculations
Reliability formula
Exponential distribution
R(t) = e^(−λt) = e^(−t ÷ MTBF)Unreliability (CDF)
F(t) = 1 − R(t) = 1 − e^(−λt)λ is the failure rate, t the mission time, and MTBF = 1/λ. This assumes a constant failure rate (the useful-life period).
Reliability quick reference
| Mission time | R(t) | Meaning |
|---|---|---|
| 0.1 × MTBF | 90.5% | ~9 out of 10 survive |
| 0.25 × MTBF | 77.9% | ~8 out of 10 survive |
| 0.5 × MTBF | 60.7% | ~6 out of 10 survive |
| 1 × MTBF | 36.8% | Only ~4 out of 10 survive |
| 2 × MTBF | 13.5% | ~1 out of 7 survives |
| 3 × MTBF | 5.0% | ~1 out of 20 survives |
Reliability calculation example
An emergency shutdown system has an MTBF of 50,000 hours. What is the probability it works when needed after 8,760 hours (1 year) of standby?
Failure rate
λ = 1 ÷ 50,000 = 0.00002 per hourReliability over one year
R(8,760) = e^(−0.00002 × 8,760) = e^(−0.1752) = 83.93%About an 84% chance of working when needed after a year without testing.
Frequently asked questions
Why doesn’t MTBF guarantee survival?
MTBF is an average, not a guarantee. With the exponential distribution, only 36.8% of systems survive to the MTBF. If you need 90% reliability, your mission time must be only about 10% of the MTBF. This is why high-reliability systems need very high MTBF values relative to their mission time.
What reliability level is considered acceptable?
It depends on the application. Consumer products typically target 90–95%. Industrial equipment aims for 95–99%. Safety-critical systems (aerospace, medical, nuclear) require 99.9% or higher. The required reliability level drives the MTBF specification.
Does this calculator work for all failure distributions?
This calculator uses the exponential distribution (constant failure rate), which is appropriate for electronic components and the useful-life period. For wear-out failures (mechanical parts), use the Weibull distribution — see the Weibull calculator.
How do I improve system reliability?
Three main approaches: (1) use higher-reliability components with a higher MTBF; (2) add redundancy (parallel systems); (3) for standby and protective systems, proof-test more often so hidden failures are found sooner. The System Reliability calculator can help analyse redundancy benefits.