A maintainability prediction is an exercise in adding up small times and then dividing correctly. Both halves go wrong in characteristic ways.
The elemental activities
A corrective repair decomposes into a fixed set of activities, and estimating each separately is what makes the total arguable rather than asserted:
| Activity | What it covers |
|---|---|
| Localisation | Determining that something is wrong and roughly where |
| Isolation | Determining which replaceable item it is |
| Disassembly | Gaining access: covers, brackets, harnesses, adjacent items |
| Interchange | Removing the failed item and fitting the replacement |
| Reassembly | Putting back everything disassembly disturbed |
| Alignment | Adjustment, configuration, calibration after fitting |
| Checkout | Verifying the repair and returning the equipment to service |
Two of those are diagnosis and four are mechanical work. Which pair dominates is an empirical question per design, and the answer is usually the mechanical one, which is why maintainability is a packaging discipline as much as a diagnostic one.
The weighting, and why it is the whole method
The mean corrective maintenance time is not the average of the task times. It is the average a technician actually experiences, which weights each repair by how often it arrives:
Mct = Σ (λᵢ · tᵢ) ÷ Σ λᵢ
Every failure rate in that expression comes from the prediction, which is what keeps the two analyses consistent. The unweighted average of the same task times is a different and meaningless number, and it is the one a spreadsheet produces by default.
The same weighting gives the useful ranking. The product λᵢ tᵢ is the technician's time per item per unit of operating time, and sorting by it answers "where does the maintenance budget go" in a way that neither the failure-rate list nor the repair-time list can.
The distribution, and the percentile
Repair times are skewed: a cluster of short jobs and a tail of long ones, because the things that make a repair long (access, alignment, a difficult diagnosis) multiply rather than add. The lognormal is the standard model, and it is fitted on the logarithms:
μ = Σ λᵢ ln tᵢ ÷ Σ λᵢ and σ² = Σ λᵢ (ln tᵢ − μ)² ÷ Σ λᵢ
from which the median is e^μ, the mean is e^(μ + σ²/2), and the percentile that contracts actually specify is
Mmax(p) = e^(μ + zp σ) with z = 1.282 at 90 per cent and 1.645 at 95
The mean and the percentile are far apart and both are needed. In the worked example the mean is 61.7 minutes and the 95th percentile is 116.4, nearly double it: a depot slot booked at the mean is overrun by a third of the work that arrives.
A useful internal check falls out of the two ways of computing the mean. The weighted arithmetic mean of the task times and the fitted lognormal's e^(μ + σ²/2) should agree closely; where they do not, the lognormal is a poor description of that particular mix, which usually means two populations have been averaged into one.
Corrective, preventive, and the measures built on them
| Quantity | What it is |
|---|---|
Mct | Mean corrective maintenance time: the repair-time prediction |
Mpt | Mean preventive maintenance time, from the scheduled task list |
M̄ | Mean active maintenance time, combining both weighted by their frequencies |
MTTR | Used loosely for Mct; a specification should say which it means |
MDT | Mean down time: active maintenance plus logistic and administrative delay |
The last row is where a prediction meets reality. Mct is the technician's time with the equipment; the operator experiences MDT, which adds waiting for a part, a permit and a person. The derating module's operational availability discussion makes the same point from the other side, and the gap is routinely larger than everything this analysis computes.
What the numbers are worth
An elemental time is a judgement about how long a trained person takes to undo six fasteners, on hardware that does not exist, in a posture nobody has tried. Three consequences follow, and the honest presentation states all three:
- Absolute accuracy is modest. A factor of two on any individual task time is unremarkable.
- Comparative accuracy is good. Two arrangements estimated by the same person with the same library rank correctly far more often than either estimate is right.
- The ranking is robust. The contribution list survives large errors in individual times, because it is dominated by the product of two quantities that are both large.