Bathtub curve

The formula

The curve plots the hazard rate — the instantaneous failure rate among survivors:

h(t) = f(t) / R(t)

Reading: of the units still alive at time t, what fraction per hour are failing right now. f(t) is the failure-time density, R(t) the fraction surviving.

The hazard rate determines reliability completely:

R(t) = exp( −∫₀ᵗ h(u) du )

Reading: survival probability is the exponential of the accumulated hazard. Any hazard shape you can draw, this integral turns into a survival curve.

The workhorse model for each region is the Weibull hazard ^[2]^[4]:

h(t) = (β/η) · (t/η)^(β−1)

Reading: η sets the time scale, β sets the shape. β < 1 gives a falling hazard (infant mortality). β = 1 gives a flat hazard (useful life). β > 1 gives a rising hazard (wearout). One knob picks your region of the tub.

The flat middle is the exponential distribution, h(t) = λ constant:

R(t) = e^(−λt) MTBF = 1/λ

Reading: during useful life, failures arrive at random with no memory. A unit that has run 10,000 clean hours has exactly the same chance of dying in the next hour as one fresh out of burn-in.

A full bathtub is a sum of the three: a β < 1 term, a constant λ, and a β > 1 term, added together as competing risks.

Where you meet it

The burn-in rack. Every ESS or burn-in requirement in a test procedure — 168 hours at temperature cycling before acceptance — exists to push production units past the steep left wall of the tub so the customer receives hardware from the flat region.
The proposal and the review board. When a reliability engineer briefs an MTBF of 10⁶ hours from a parts-count prediction, the entire number rests on the flat-region assumption of MIL-HDBK-217-style constant failure rates ^[12]^[13]. Know that before you repeat it.
The warranty meeting. Warranty length is a bet on where the left wall ends and the right wall begins. Return-rate data plotted as a hazard curve tells you whether early failures are escaping screening or the warranty is reaching into wearout.
Fleet sustainment. Depot-level decisions to life-limit a component — bearings, seals, TWTs, batteries — are decisions that the population has entered the β > 1 region, where replace-on-schedule beats replace-on-failure.

How it works

The curve is a statement about a population, plotted as hazard rate — not a failure histogram ^[1]^[3]. More failures land in the flat middle than anywhere else simply because most of the population's operating hours are spent there; the rate per survivor is what's low. Confusing the hazard curve with the density curve is the most common reading error.

The three regions have three different physics. Infant mortality is mostly a defect subpopulation — cold solder joints, contaminated wire bonds, cracked ceramic — a weak minority mixed in with sound units. The falling hazard early on isn't units getting stronger; it's the weak ones dying out of the mix. Useful life is externally driven: transients, ESD events, cosmic-ray upsets, handling — arrivals that don't care how old the unit is. Wearout is accumulated damage: fatigue, electromigration, lubricant depletion, capacitor dry-out.

Numbers to keep you honest during the flat region. With λ = 1 failure per 10⁶ hours, a 5,000-hour mission gives R = e^(−0.005) ≈ 0.995. But run one unit out to its full MTBF and R = e^(−1) ≈ 0.368 — 63% of the population is dead by the MTBF. MTBF is not a lifetime. A 10⁶-hour MTBF (about 114 years) on a fan tray does not promise 114 years of fan; the fan's wearout mechanisms exit the flat region long before the exponential math stops applying.

The β sensitivity is worth feeling once. A β = 0.5, η = 1000 h infant-mortality term has a hazard at 10 hours roughly 7 times its hazard at 500 hours — that steep left wall is what a 168-hour burn-in buys down. A β = 3 wearout term's hazard quadruples between 500 and 1,000 hours — that steep right wall is why life limits are hard cutoffs rather than suggestions.

Limits of validity. NIST's handbook notes the curve also applies — on much empirical evidence — to repairable systems, where the vertical axis becomes the rate of occurrence of failures (ROCOF), not a component hazard function ^[1]. Mechanical parts under fatigue often have no flat region at all — the hazard rises from day one, which is why bearing life is quoted as L10, not MTBF. And Klutke, Kiessler, and Wortman showed in 2003 that a decreasing observed hazard near t = 0 can be produced by mixtures of perfectly ordinary increasing-hazard units — so a falling early failure rate does not by itself prove a burn-in program is removing defects ^[8]^[9]. Plot your own field data before you assume your product owns a bathtub.

The mistake people make: designing the reliability program for one region while the hardware lives in another. Burn-in applied to a wearout-dominated part consumes life instead of removing defects. Scheduled replacement applied during the flat region buys nothing — the replacement unit has the same hazard as the one you pulled, plus a fresh chance at infant mortality from the maintenance action itself.

History

The shape came from the insurance business before it came from engineering. Benjamin Gompertz, actuary at the Alliance Assurance Company in London, showed in 1825 that adult human death rates climb in geometric progression with age ^[6]^[7] — the right-hand wall of the tub. Actuaries already knew the left-hand wall from infant death statistics, which is why reliability engineers still call early hardware failures "infant mortality."

Engineering picked the shape up when World War II electronics made unreliability a battlefield problem. The Department of Defense stood up the Advisory Group on Reliability of Electronic Equipment, whose 1957 report — Reliability of Military Electronic Equipment — gave the field its formal definition of reliability and its testing framework ^[10]^[11]. Six years earlier the math had arrived from Sweden: Waloddi Weibull's 1951 paper "A statistical distribution function of wide applicability" in the Journal of Applied Mechanics supplied the one-parameter-family hazard that models all three regions ^[4]^[5].

The flat middle then got institutionalized, arguably too well. By the 1960s the DoD's electronics prediction handbook, MIL-HDBK-217, had built its methodology on exponentially distributed failure times — constant failure rate everywhere — and field data showing otherwise tended to get explained away ^[12]. The pushback took decades; the 2003 Klutke–Kiessler–Wortman paper in IEEE Transactions on Reliability is the standard modern caution that real hazard functions rarely behave the way the tidy textbook tub suggests ^[8]^[9].

Related tools

/tools/bearing-life-l10 — the wearout wall in its purest form; L10 life is a Weibull percentile, and rolling elements largely skip the flat region.
/tools/arrhenius-rate — the acceleration model behind burn-in and HTOL screening that compresses the infant-mortality region into days at elevated temperature.
/tools/battery-life — runtime estimates assume a healthy cell; capacity fade is a wearout mechanism that walks the pack up the right-hand wall.
/tools/half-life-decay — the same constant-hazard exponential math as the useful-life region, wearing its nuclear-physics uniform.

Sources

Written by HE in our own words from the cited sources — engineering judgment included, your stamp still required. All entries →