Machines: predicting when hardware fails

Reliability engineering is the oldest of the seven fields surveyed here, and its central tool — the Weibull distribution — turns up, in disguise, throughout the rest of this site.

The core idea

Every physical component has a hazard rate: the instantaneous probability that it fails right now, given that it has survived until now. Reliability engineers noticed early on that this hazard rate isn't constant across a product's life — it typically traces a "bathtub" shape. Early in life, defective units fail quickly (infant mortality); the hazard rate then flattens out for a long stretch of random, age-independent failures (useful life); and finally it rises again as parts wear out (wear-out phase).

The bathtub curve

The three phases are described by the shape parameter of a single distribution family, the Weibull distribution. A shape parameter below 1 produces a decreasing hazard rate (infant mortality); a shape parameter of exactly 1 produces a constant hazard rate (the useful-life phase, mathematically identical to the simpler exponential distribution); a shape parameter above 1 produces an increasing hazard rate (wear-out).

The formula

The Weibull probability density function, with shape parameter k and scale parameter λ:

Its cumulative distribution function (the probability of having failed by time x):

And the hazard/failure-rate function, which is what actually produces the bathtub shape:

The mean time to failure (MTTF, or MTBF for repairable systems) follows directly from the shape and scale parameters via the gamma function Γ:

When k = 1, the Weibull distribution reduces to the simpler exponential distribution, whose reliability function has a clean, memoryless form:

"Memoryless" means the probability a component survives an additional interval of time is independent of how long it has already run — a property unique to the constant-hazard (k=1) case, and one of the reasons the useful-life phase of the bathtub curve is so mathematically convenient.

How this field validates its predictions

Reliability engineering never reports a bare "% accurate" figure. The deliverable is always a point estimate plus a confidence interval around it, with the interval width itself treated as the quality signal — a narrower interval at the same confidence level means a better- characterized system, not simply "more accurate."

For a time-truncated life test, the standard method uses the chi-square distribution with 2n+2 degrees of freedom (n = number of observed failures, T = total test time, α = significance level):

The 2n+2 form (rather than the simpler 2n form) is preferred because it still produces a valid upper bound even in a zero-failure test — for example, a 1,000-hour zero-failure test yields a 90%-confidence upper bound of roughly 19,496 hours on the true MTBF.

A related technique, accelerated life testing, runs components at elevated stress (heat, voltage, load) to estimate their time-to-failure distribution at normal-use conditions faster than real-time testing would allow. Parameters are extrapolated back to use conditions via a log-linear regression model, and the field explicitly flags that the farther the extrapolation is from actual-use stress levels, the wider — i.e., less certain — the resulting confidence bounds become.