OptimalAvailability Studio™ Wiki / RBD, RAM and Weibull
Method and tool
RBD, RAM and Weibull
This is the quantitative core: how component reliabilities combine into a system, how availability is defined and measured, how failure history is read with Weibull, and how a Monte Carlo RAM model turns all of it into a predicted availability. A live calculator is at the end.
Reliability block diagrams
An RBD wires the reliability logic of a system as blocks. The system works if a path of working blocks connects input to output. Three patterns cover most cases:
Series Rsys = R1 × R2 × … × Rn = ∏ Ri
Active parallel Rsys = 1 − ∏ (1 − Ri)
k-out-of-n works if at least k of n identical blocks work
Series is the weakest link: every block must work, so reliability falls as components are added. Parallel is redundancy: the system survives while any one path holds, so it climbs fast. Most real systems are a nest of both, plus standby and k-out-of-n sets.
Availability, three definitions
Availability is not one number. Which one you quote depends on what downtime you count:
| Inherent, Ai | Ai = MTBF ÷ (MTBF + MTTR). Corrective repair only. A pure design measure, the ceiling a perfect operation could reach. |
|---|---|
| Achieved, Aa | Aa = MTBM ÷ (MTBM + M). Adds preventive maintenance. MTBM is mean time between maintenance, M the mean active maintenance time. |
| Operational, Ao | Ao = MTBM ÷ (MTBM + MDT) = uptime ÷ (uptime + downtime). Adds logistic and administrative delay through the mean down time MDT. The number the plant actually lives. |
The gap between inherent and operational availability is where reliability engineering earns its keep: it is mostly logistics, spares and delay, not the equipment.
Weibull life-data analysis
Weibull reads the failure pattern and the characteristic life out of a set of failure and suspension times, which is what tells you whether age-based maintenance can even work.
R(t) = e−(t/η)^β h(t) = (β/η)(t/η)β−1 MTTF = η · Γ(1 + 1/β)
β is the shape parameter and the failure pattern: β < 1 infant mortality with a falling hazard, β = 1 random with a constant hazard (the exponential case), β > 1 wear-out with a rising hazard. η is the characteristic life, the age by which 63.2% have failed. The B-life, for example B10, is the age at which 10% have failed. Fit by median-rank regression or maximum likelihood, and always account for censored, still-running items, or the life is understated.
The practical payoff: only β > 1 justifies scheduled restoration or discard. For the random and infant-mortality patterns, which dominate modern equipment, on-condition detection is the only lever, which is the bridge to OptimalTREND™.
RAM modelling and Monte Carlo
Once redundancy, maintenance, spares holding and logistic delay are in play, the algebra stops being tractable and reliability engineers simulate. A RAM model runs the system thousands of times over its life, sampling failures from the fitted distributions and repairs from the maintainability data, to produce:
- System and production availability, the latter weighting partial-capacity states, consistent with ISO 20815.
- Expected number of failures and downtime by cause, so the critical few are ranked.
- Spares demand and criticality, which feed OptimalSPARES™.
- The effect of a change before it is bought: add a spare, add a redundant unit, shorten a repair, and see the availability move.
Fault tree analysis complements the RBD for safety and protective systems: a top event resolved through AND and OR gates to basic events, giving the minimal cut sets, the smallest combinations of failures that cause the top event, and their probability.
Try it: availability and redundancy
Set MTBF and MTTR for one unit, then choose how two units are arranged, to see the system availability. Watch how redundancy and a shorter repair move the number.
Unit inherent availability 99.21%. Redundancy multiplies uptime, series compounds risk.
System availability
Ai = MTBF ÷ (MTBF + MTTR). Green at or above 99%, amber 95–99%, red below.