Bathtub Curve⚠ unverified

Understanding the Parameters

• The three failure regimes — the bathtub superposes three mechanisms with distinct causes:

• Infant mortality ($\lambda_{early}$): early failures from manufacturing defects, weak units, assembly errors. In a true bathtub this rate decreases as defective units are weeded out; the model here uses a constant offset (see the note).

• Random / useful life ($\lambda_{random}$): a constant background rate from unpredictable overstress events — the flat bottom, well described by the exponential model.

• Wear-out ($\lambda_{wear}$, onset $t_{wear}$): an increasing rate as aging mechanisms (fatigue, corrosion, erosion) take over. Modelled here as a linear ramp starting at $t_{wear}$.

• Wear-out onset $t_{wear}$ — the age at which the rising wear-out term switches on ($\max(0, t - t_{wear})$ keeps it zero before then). It marks the end of useful life and the start of accelerating failures.

• The output $h(t)$ — the total instantaneous failure rate, the sum of whichever contributions are active at age $t$. It is the composite hazard that a real fielded population exhibits.

• The design responses. Each region has a countermeasure: burn-in (running units to clear infant mortality before shipping) attacks the early region; redundancy mitigates the random region; preventive replacement before $t_{wear}$ handles wear-out. The bathtub is the map that dictates these strategies.

Derivation (Approaching a Proof)

The bathtub curve is a superposition model rather than a single derived law: the observed failure rate of a mixed population is the sum of independent competing failure mechanisms, each with its own time dependence. For independent competing causes, hazard rates add (the population's chance of failing in the next instant is the sum of the chances from each mechanism):

$$h(t) = h_{early}(t) + h_{random}(t) + h_{wear}(t).$$

The model assigns each a simple form: a constant $\lambda_{early}$ for early life, a constant $\lambda_{random}$ for the useful-life floor, and a ramp $\lambda_{wear}\max(0, t - t_{wear})$ that switches on at wear-out onset, giving

$$h(t) = \lambda_{early} + \lambda_{random} + \lambda_{wear}\max(0,\,t - t_{wear}). \qquad\blacksquare$$

The additive-competing-risks structure is exact; the shapes are simplifications. A fully faithful bathtub uses a decreasing early term (e.g. a Weibull $\beta < 1$ hazard, Weibull Failure Rate) so the curve actually descends through infancy, a constant middle, and an increasing ($\beta > 1$) wear-out term — three Weibulls superposed. The registry's constant early term reproduces the flat and rising parts but not the initial decline.

Dimensional check. $\lambda_{early}$, $\lambda_{random}$ are rates ($1/\text{h}$); $\lambda_{wear}$ is $1/\text{h}^2$ times $(t - t_{wear})$ in hours $= 1/\text{h}$, so every term is a rate and $h(t)$ is $1/\text{h}$. $\checkmark$

History and Development

• A reliability icon. The bathtub curve became the organising picture of reliability engineering in the mid-20th century, distilling the observation that failure mechanisms dominate at different life stages into one memorable shape.

• Burn-in and screening. The infant-mortality region justified burn-in and environmental stress screening — deliberately aging or stressing products before shipment to precipitate early failures in the factory rather than the field, a standard practice in electronics and aerospace.

• Critiques and refinements. Modern reliability data (notably for electronics) often show that the clean bathtub is an idealisation — some populations lack a distinct wear-out region within their service life, others show "roller-coaster" hazards. The curve remains an invaluable conceptual model, best implemented as superposed Weibulls (Weibull Failure Rate) rather than the simplified constant-early form here.

Related Concepts: Weibull Failure Rate, Weibull Reliability, Exponential Reliability, Hazard Rate Ratio, Temperature Derating, Reliability Growth, B10 Life

Notes: Registry calculator bathtub-curve (unverified). Piecewise sum: $\lambda_{early}+\lambda_{random}$ for $t<t_{wear}$, plus $\lambda_{wear}(t-t_{wear})$ after (latex shows only the flat branch). The early term is constant, not decreasing, so it does not reproduce the declining infant-mortality slope — a faithful bathtub superposes decreasing/constant/increasing Weibulls. Hazards add for competing risks. Registry labels inputs dimensionless (are times/rates). Flagged in Known Issues.