Weibull Analysis of Downtime Data: Shape, Scale and Failure Modes

What Weibull analysis is, on a production line

Classic life data analysis starts from the ages at which parts failed. On a production line the equivalent record is the stop log: for each failure mode, the runs between stops are time-to-failure (TTF) samples and the stop durations are time-to-repair (TTR) samples. The three-parameter Weibull distribution is:

F(t) = 1 − exp(−((t − γ) / λ)^κ)

• Shape κ (often written β): how the failure rate changes with running time.
• Scale λ (often written η): the characteristic life, in the same time units as the data.
• Location γ: an optional failure-free period. With γ = 0 this is the two-parameter Weibull.

ReliaSim stores a Weibull as λ, κ and γ, and ReliaStats uses the same parameters. Weibull is the usual starting point for uptime because one shape parameter spans all three phases of the bathtub curve, but it is not always the best fit, so it is tested against other candidates.

The shape parameter: κ < 1, κ = 1, κ > 1

The shape parameter sets the hazard rate, the rate of stopping for a machine that has run t minutes since restart:

h(t) = (κ / λ) · (t / λ)^{κ − 1}

• κ < 1, falling hazard: early-life failure, or on a line, stops that cluster just after a restart.
• κ = 1, constant hazard: the exponential distribution. Running time says nothing about the next stop.
• κ > 1, rising hazard: wear-out. A stop becomes more likely the longer the machine runs.

Computed from Weibull distributions with a mean of 60 min: λ = 60 / Γ(1 + 1/κ), R(t) = exp(−(t/λ)^κ), and P(stop within 10 min | running t) = 1 − R(t + 10) / R(t).

Same MTBF, opposite risk profiles, and different actions:

• κ > 1: time-based inspection or replacement can prevent stops, because risk builds with running time.
• κ = 1: time-based maintenance does not lower the failure rate, so the effort belongs in removing the cause or shortening the repair.
• κ < 1: every restart puts the machine back into its riskiest period. Look at setup, adjustment and the minutes after a jam clears. The Interrupt Explorer's reliability window (2 minutes by default) measures the share of runs that end that soon after a restart.

A shape estimate describes the data, not the machine. A mature machine that comes back with κ < 1 may have pooled failure modes or stops logged twice, so check the data before the fit goes into a model.

Scale and location

Scale λ is the characteristic life: whatever the shape, 63.2% of runs have ended by t = λ, because F(λ) = 1 − e⁻¹. It is not the mean, which is λ · Γ(1 + 1/κ). With λ = 60 minutes, κ = 0.5 gives a mean of 120 minutes, κ = 1 gives 60 and κ = 2 gives 53.2. Compare fits on shape and mean together.

Location γ shifts the distribution right, so no failure occurs before γ. Use it when the data shows a real failure-free period after restart; otherwise it is one more parameter to overfit.

Fit each failure mode, not each machine

The most common Weibull mistake on a production line is fitting one distribution to all of a machine's stops. The failure modes compete, each run ending at whichever fires first, so the machine's hazard is the sum of the modes' hazards. No single Weibull can follow a falling hazard and a rising one added together.

The pooled column is the maximum-likelihood Weibull for the exact two-mode run distribution, computed numerically and checked against 200,000 sampled runs.

The pooled fit gets the mean about right, so it passes an MTBF check. Its shape of 1.3 reads as mild wear-out, not far from random, and hides both real findings: early re-stops are about a third more common than it predicts, and runs over two hours nearly four times rarer. A maintenance plan built on κ = 1.3 would miss both.

That is why ReliaStats works at the level of the cause group, the place and interrupt code that behave as one failure mode, identified automatically in the desktop app. A mode's run is measured from the previous stop of any cause; same-cause gaps count other modes' stops as running time (see competing risks).

Censored runs

A run ended by a different mode, or still going when the record ends, shows that this mode had not occurred by then, but not when it would have. Reliability texts call these censored observations, or suspensions, and treat them differently from failures. How they are handled changes the fitted parameters, most of all for a rare mode whose runs are usually ended by something else.

Goodness of fit: is Weibull the right model?

A fitting routine returns parameters even for a distribution that describes the data badly. ReliaStats fits all eight distribution types ReliaSim supports (Weibull, LogNormal, Normal, Exponential, Uniform, Triangular, Johnson SU and Fixed), tests each with Kolmogorov-Smirnov and Anderson-Darling, and ranks them.

• Kolmogorov-Smirnov (K-S) measures the largest vertical gap between the empirical and fitted CDFs, and is most sensitive in the middle of the distribution.
• Anderson-Darling (A-D) weights the tails more heavily. In downtime work the tails often matter most: chronic re-stops and repairs that outlast a buffer.

Read the results as a ranking, not a verdict. With thousands of stops a formal test can reject every candidate over differences too small to matter; with a few dozen it may reject none. Two checks help:

• Plot it. On a Weibull probability plot, ln(−ln(1 − F)) against ln t, a two-parameter Weibull sample falls on a straight line with slope κ. A bend or dogleg suggests mixed modes, a location offset or another distribution. Plotting positions usually come from Bernard's median-rank approximation, (i − 0.3) / (n + 0.4), which the Interrupt Explorer also uses for its survival curves.
• Check the numbers the line cares about. The fitted mean should land near the sample mean. Compare the share of very short runs, and the share of stops longer than the downstream buffer covers.

From fitted distributions to a line simulation

In the ReliaStats workflow a fit's larger job is to parameterize a model. Each failure mode becomes an interrupt with uptime and downtime distributions and a maximum downtime, exported as a CSV that ReliaSim imports. ReliaSim draws individual runs and repairs from them, and blocking and starving emerge from how the stops interact with rates and buffers.

The distributions are the inputs; MTBF and MTTR are outputs of the run, summaries you check against history (see MTBF vs MTTR: outputs, not inputs). Two fits with the same mean give the same availability but different numbers of early re-stops and long stops.

The model is then validated interrupt by interrupt against history. A model within 1% of measured OEE is achievable when the model and data are handled correctly, and correctly fitted TTF and TTR distributions for each failure mode are central to that. The food-plant model from Fischel and Lange's WSC 2020 paper was rebuilt in ReliaSim and independently validated by Tom Lange to within 1% of both the plant's measured OEE and the original published model (case study). A validated model can then rank failure modes by production recovered; see the downtime Pareto guide.

Run Weibull analysis on your own stop data

• Interrupt Designer (free): set a Weibull TTF, change κ, and watch the curves, percentiles and availability move.
• Interrupt Explorer: survival curves R(t) by cause from an .xlsx or .csv stop log, plus the reliability-window share.
• ReliaStats desktop app: import event logs, identify cause groups, fit and rank all eight types and export for ReliaSim, as part of a ReliaStats subscription.

For the product, see Weibull & TTF/TTR Distribution Fitting. For the whole path from stop log to validated model, see downtime data analysis.

Frequently asked questions