BIZOMAT MAKEOS · Maintenance
System Reliability from Weibull Components (Series and Redundancy)
What is the chance this machine runs a mission without failure when its components wear out on Weibull curves, and does redundancy on one item get it to target?
Reliability of a system of up to four component blocks in series, each with its own Weibull shape and characteristic life (as the Weibull reliability tool) and a number of identical units in active redundancy: component and block reliability at the mission time, system reliability, the weakest block, and the system mean time to failure by numerical integration of the system reliability curve.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Mission time | h | Operating time the system must survive. |
| Target system reliability | % | Reliability the system must reach over the mission. |
| Block 1 | ||
| Block 1 Weibull shape β | Weibull shape of the units in block 1 (below 1 early failures, 1 random, above 1 wear-out). | |
| Block 1 characteristic life η (0 = not used) | h | Characteristic life of the units in block 1, where 63.2% have failed; 0 if the block is not used. |
| Block 1 identical units in parallel | Number of identical units sharing the job in block 1 (1 for a single unit; 2 for a duty/standby pair running together). | |
| Block 2 | ||
| Block 2 Weibull shape β | Weibull shape of the units in block 2 (below 1 early failures, 1 random, above 1 wear-out). | |
| Block 2 characteristic life η (0 = not used) | h | Characteristic life of the units in block 2, where 63.2% have failed; 0 if the block is not used. |
| Block 2 identical units in parallel | Number of identical units sharing the job in block 2 (1 for a single unit; 2 for a duty/standby pair running together). | |
| Block 3 | ||
| Block 3 Weibull shape β | Weibull shape of the units in block 3 (below 1 early failures, 1 random, above 1 wear-out). | |
| Block 3 characteristic life η (0 = not used) | h | Characteristic life of the units in block 3, where 63.2% have failed; 0 if the block is not used. |
| Block 3 identical units in parallel | Number of identical units sharing the job in block 3 (1 for a single unit; 2 for a duty/standby pair running together). | |
| Block 4 | ||
| Block 4 Weibull shape β | Weibull shape of the units in block 4 (below 1 early failures, 1 random, above 1 wear-out). | |
| Block 4 characteristic life η (0 = not used) | h | Characteristic life of the units in block 4, where 63.2% have failed; 0 if the block is not used. |
| Block 4 identical units in parallel | Number of identical units sharing the job in block 4 (1 for a single unit; 2 for a duty/standby pair running together). | |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Mission time
- What it is
- The operating time the system must run without failure; reliability is calculated for this mission length.
- Units and format
- Enter a number in h: hours. The form accepts 0.000001 h to 10000000 h. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2000 h as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
What this means: Target system reliability
- What it is
- The probability of completing the whole mission without a system failure that the design must reach.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 1% to 99.9999%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 90% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Block 1
What this means: Block 1 Weibull shape β
- What it is
- The Weibull shape parameter of each unit in this block, fitted from failure records or given by the supplier; it describes how the failure rate changes with age.
- Units and format
- Enter a plain number with no unit. The form accepts 0.2 to 10. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Ignored when the block’s characteristic life is set to 0.
What this means: Block 1 characteristic life η (0 = not used)
- What it is
- The characteristic life of each unit in this block: the operating time by which 63.2 % of such units have failed. Fit it from failure records or supplier reliability data.
- Units and format
- Enter a number in h: hours. The form accepts 0 h or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10000 h as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Zero leaves the whole block out of the system.
What this means: Block 1 identical units in parallel
- What it is
- The number of identical units in this block running together in active redundancy, where any one of them alone can do the job.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 10. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use 1 for a single unit. Standby units switched in on failure need a different model.
Block 2
What this means: Block 2 Weibull shape β
- What it is
- The Weibull shape parameter of each unit in this block, fitted from failure records or given by the supplier; it describes how the failure rate changes with age.
- Units and format
- Enter a plain number with no unit. The form accepts 0.2 to 10. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1.5 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Ignored when the block’s characteristic life is set to 0.
What this means: Block 2 characteristic life η (0 = not used)
- What it is
- The characteristic life of each unit in this block: the operating time by which 63.2 % of such units have failed. Fit it from failure records or supplier reliability data.
- Units and format
- Enter a number in h: hours. The form accepts 0 h or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20000 h as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Zero leaves the whole block out of the system.
What this means: Block 2 identical units in parallel
- What it is
- The number of identical units in this block running together in active redundancy, where any one of them alone can do the job.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 10. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use 1 for a single unit. Standby units switched in on failure need a different model.
Block 3
What this means: Block 3 Weibull shape β
- What it is
- The Weibull shape parameter of each unit in this block, fitted from failure records or given by the supplier; it describes how the failure rate changes with age.
- Units and format
- Enter a plain number with no unit. The form accepts 0.2 to 10. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Ignored when the block’s characteristic life is set to 0.
What this means: Block 3 characteristic life η (0 = not used)
- What it is
- The characteristic life of each unit in this block: the operating time by which 63.2 % of such units have failed. Fit it from failure records or supplier reliability data.
- Units and format
- Enter a number in h: hours. The form accepts 0 h or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50000 h as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Zero leaves the whole block out of the system.
What this means: Block 3 identical units in parallel
- What it is
- The number of identical units in this block running together in active redundancy, where any one of them alone can do the job.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 10. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use 1 for a single unit. Standby units switched in on failure need a different model.
Block 4
What this means: Block 4 Weibull shape β
- What it is
- The Weibull shape parameter of each unit in this block, fitted from failure records or given by the supplier; it describes how the failure rate changes with age.
- Units and format
- Enter a plain number with no unit. The form accepts 0.2 to 10. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Ignored when the block’s characteristic life is set to 0.
What this means: Block 4 characteristic life η (0 = not used)
- What it is
- The characteristic life of each unit in this block: the operating time by which 63.2 % of such units have failed. Fit it from failure records or supplier reliability data.
- Units and format
- Enter a number in h: hours. The form accepts 0 h or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 h as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Zero leaves the whole block out of the system.
What this means: Block 4 identical units in parallel
- What it is
- The number of identical units in this block running together in active redundancy, where any one of them alone can do the job.
- Units and format
- Enter a plain number with no unit. The form accepts 1 to 10. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Use 1 for a single unit. Standby units switched in on failure need a different model.
Method and formulas
- Unit reliability R = exp[−(t ÷ η)^β] (as the Weibull reliability tool).
- Block with k units in active redundancy R_b = 1 − (1 − R)^k; system in series R_s = Π R_b.
- System MTTF = ∫ R_s(t) dt, integrated numerically.
- Example: over 2000 h the system reliability is 92.945%; block 1 is the weakest; MTTF about 7,426.2 h.
Limitations
- Independent failures; common-cause failures defeat redundancy.
- Active redundancy with perfect switching; standby units need a different model.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Check a machine against a reliability target
- See whether a redundant pump or fan is worth it
- Find the component that limits reliability
Who it is for
- Reliability engineers
- Maintenance engineers
- Design engineers
What you get
- Unit and block reliability at the mission time
- System reliability and the weakest block
- System mean time to failure
Interpreting the result
- Redundancy helps most on the weakest block; doubling an already reliable block adds little.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 128–133. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These BIZOMAT Library readings explain the method this tool uses, with worked examples.
Method explained here · BIZOMAT original book
Reliability and life
In Thermoelectric Cooling by the Numbers
What it covers: How thermoelectric coolers wear out, Weibull and series-system reliability for a cooler, field MTBF with confidence bounds, spares, and a thermostat that cycled a module to death.
Method explained here · BIZOMAT original book
Drive life: capacitors, fans and power cycling
In Variable-Speed Drives by the Numbers
What it covers: How temperature sets the life of a drive's DC link capacitors and cooling fans, how power cycling wears out power modules, and how to combine the parts into a drive reliability and a maintenance plan.
Useful alongside · BIZOMAT original book
Redundancy and the time to first failure
In Breakdowns and Buffers by the Numbers
What it covers: Two compressors, one needed: mean first passage time to losing both, active and standby redundancy, why repair makes redundancy work, failures to start, and common causes.
Useful alongside · BIZOMAT original book
Voting between sensors
In Trips and Alarms by the Numbers
What it covers: What one-out-of-two, two-out-of-two and two-out-of-three voting do to spurious trips and failures on demand, why voting does not cure process noise, common-cause failures, and deviation alarms.
BIZOMAT tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.
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