Library / Reliability / REL-02
Reliability
Series system reliability
Reliability of a chain of n identical parts that must all work.
- Ref
- REL-02
- Rev
- v1
- Field
- Reliability
- Result
- Rs
r is the reliability of one part, the probability it works for the mission, and n the number of identical parts in series, all of which must work. The system works only if every part does, so the probabilities multiply. For parts with different reliabilities, multiply their individual values.
Reliability of a chain of n identical parts that must all work. Reliability falls quickly as parts are added.
Calculator
- Number of parts at which Rs falls to 50 %13.51
- Reliability each part needs for Rs = 0.990.998
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.
- Five parts at 95 % each give a system reliability of only 77 %, and 14 such parts would give less than 50 %.
- To reach 99 % overall with five parts, each needs 99.8 %. Complex systems with many parts in series therefore need very reliable components, or redundancy.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| r Reliability of one part | 0.5 to 0.9999 | 0.03125 | 0.9995 | 0.9683 | +10% | +61.1 % |
| n Number of parts in series | 1 to 100 | 0.005921 | 0.95 | 0.9441 | +1 | −5 % |
Sweep of Number of parts in series
| Number of parts in series | System reliability |
|---|---|
| 1 | 0.95 |
| 13.38 | 0.5036 |
| 25.75 | 0.2669 |
| 38.13 | 0.1415 |
| 50.5 | 0.075 |
| 62.88 | 0.03975 |
| 75.25 | 0.02107 |
| 87.63 | 0.01117 |
| 100 | 0.005921 |
Sweep one variable
Variables and how each one contributes
System reliability
What the equation gives you. Everything else feeds it.
Reliability of one part
The probability that one part works for the whole mission. Raised to the power n: the effect is magnified n times. A 1 % rise in r raises Rs by about 5 % at n = 5.
Number of parts in series
The number of parts, all of which must work. Each extra part multiplies the result by r again: a 10 % rise (from 5 to 5.5) lowers Rs by about 2.5 %, and the effect grows with n.
Worked example with the values above
- Reliability of one part: r = 0.95.
- Multiply it by itself 5 times: 0.955.
- Rs = 0.77378, or 77.38 %.
Assumptions and limits
- Failures are independent: one part’s failure does not make others more likely.
- All parts have the same reliability and the same mission time, and the system needs every part.
Common mistakes
- Using it for redundant parts, where the system works if any one works. That needs the parallel formula.
- Averaging the failure probabilities instead of multiplying the reliabilities.
Statistics and uncertainty
ln Rs = n ln r, so a relative error in r is magnified n times: with n = 5 a 1 % error in r gives a 5 % error in Rs, and with n = 50 a 50 % error. Part reliabilities estimated from a small number of tests are uncertain, and that uncertainty is multiplied in the system figure.
Where it is used
- Why complex systems need very reliable individual parts.
- Setting a target reliability for each component from a system requirement.
Maths you need
- Probability: independent events multiply
- Powers: rⁿ