Three failure mechanisms, one curve
Plot the failure rate of a large population of identical components against time and, for a lot of real hardware, you get a shape that looks like a bathtub in cross-section: high at the very start, dropping to a long flat (or slightly rising) middle, then climbing again at the end. That shape isn't one physical process — it's three independent failure mechanisms superimposed, each dominant in its own window of the component's life. The standard mistake is treating "the failure rate" as a single number instead of a sum of three competing curves.
The math behind each phase
The early region is driven by manufacturing defects — components with latent flaws (a bad solder joint, a contaminated wafer, a weak weld) that fail almost immediately under load and then are gone from the surviving population. This contribution decays exponentially with an initial level λ0 and a decay time constant τ:
λ_infant(t) = λ0 × exp(-t / τ)
At t = 0 this is at its peak λ0; by t = 3τ or so it has dropped to under 5% of its starting value, because the weak units have already failed and been screened out (or, in the field, have already broken and been noticed).
The middle region is dominated by failures that arrive at a roughly constant rate regardless of age — random overstress events, cosmic-ray-induced upsets, unpredictable external shocks. This contribution is just a constant:
λ_random(t) = λc
It doesn't decay and it doesn't grow; it's the floor the whole curve approaches once infant mortality has burned off and wear-out hasn't started yet.
The late region is wear-out: fatigue, corrosion, insulation breakdown, bearing wear — mechanisms that are essentially absent early in life and then accelerate once the component has accumulated enough cycles or enough calendar time past an onset point t_w. A simple representation ramps this contribution up once t exceeds t_w.
The total hazard rate at any evaluation time t is the sum of all three:
λ(t) = λ_infant(t) + λ_random(t) + λ_wearout(t)
That sum is the key idea. There's no single "failure rate" for a component — there's a total hazard λ(t) that changes character depending on where t sits relative to τ and t_w, and the minimum hazard rate the curve reaches (roughly λc, once infant mortality has decayed and before wear-out has ramped up) tells you how good your best-case reliability actually is.
It's worth being precise about what λ(t) actually represents, because it's easy to conflate with a probability. The hazard rate is a conditional rate — the instantaneous likelihood that a unit still surviving at time t fails in the next small interval, given that it has survived this long. It is not the same as the fraction of the original population that has failed by time t (that's the cumulative distribution function), and it is not the probability density of failure times either (that's the derivative of the CDF). A component can have a falling hazard rate during infant mortality while its cumulative failure count is still climbing — the two curves are related but they answer different questions, and mixing them up is a common source of confusion when reading a reliability report.
Worked example: qualifying a new relay design
Consider a relay design going through reliability qualification. Historical data on a similar part family gives an infant-failure level λ0 of 8 failures per 1000 hours, decaying with a time constant τ of 200 hours — meaning early defects mostly clear out within the first few hundred hours of operation. The random failure rate λc, from environmental stress testing, comes out to 0.15 per 1000 hours. Wear-out, driven by contact erosion, has an onset time t_w of 20,000 hours based on accelerated life testing.
Evaluate at t = 50 hours, right after first power-on. The infant contribution is λ0 × exp(-50/200) = 8 × exp(-0.25) ≈ 8 × 0.779 ≈ 6.2 per 1000h. The random contribution is a flat 0.15. Wear-out hasn't started (t is well below t_w), so its contribution is essentially zero. Total hazard λ(t) ≈ 6.4 per 1000h, and the current life phase here is clearly infant mortality — the dominant term by a wide margin.
Move to t = 1000 hours, roughly 5τ past start. The infant contribution has decayed to 8 × exp(-5) ≈ 8 × 0.0067 ≈ 0.05 per 1000h — essentially negligible. The random contribution is still 0.15. Wear-out is still zero. Total hazard ≈ 0.20 per 1000h, close to the minimum hazard rate the curve will reach, and the current life phase reads as useful life — the flat bottom of the tub.
Now push out to t = 25,000 hours, past the wear-out onset of 20,000. Infant contribution is zero (long since decayed). Random contribution is still 0.15. Wear-out contribution, now 5000 hours past onset and accelerating, might climb to something like 2–3 per 1000h depending on the ramp steepness assumed. Total hazard λ(t) jumps to roughly 2.2–3.2 per 1000h — an order of magnitude above the useful-life floor — and the phase reads wear-out. That transition is the entire point of a bathtub-curve analysis: it tells you when to schedule replacement or overhaul, not just what the average failure rate looks like over the part's whole life.
Reading the curve correctly
The most common misuse is quoting a single MTBF (mean time between failures) figure derived from early-life data and applying it to a component's entire service life. A relay tested for 500 hours during qualification will show a failure rate dominated by infant mortality — extrapolate that number out to year five of deployment and you'll wildly overestimate the failure rate during useful life while completely missing the wear-out cliff at the other end.
The second mistake is assuming every component follows a proper bathtub shape at all. Some parts — well-screened electronics with burn-in already performed — start directly in the flat useful-life region with negligible infant mortality. Others, particularly mechanical components with fatigue-dominated failure modes, may show wear-out onset so early that there's barely a flat middle region. Always check where τ and t_w actually sit relative to your intended service life before assuming the classic three-phase shape applies cleanly.
Third, screening (burn-in testing before shipment) is specifically a strategy to consume the infant mortality region on the manufacturer's clock instead of the customer's. If your λ0 and τ suggest a significant early-failure population, deliberately running units for a few multiples of τ before delivery — and discarding units that fail during that window — pushes the shipped population's effective starting point well past the initial λ0 peak.
There's a cost side to this trade-off too, and it's often what actually drives the burn-in duration decision rather than the reliability math alone. Every hour of burn-in is an hour of factory test capacity and energy spent on units that mostly aren't going to fail — you're paying to screen out a shrinking minority. Running burn-in out to 3τ typically clears the vast majority of the exploitable infant-mortality population; pushing to 5τ or 6τ chases diminishing returns for a rapidly shrinking number of additional units caught, and at some point the marginal cost of extra burn-in time exceeds the expected warranty cost of letting a few infant failures reach the field. Knowing τ precisely is what turns that decision from a guess into an actual optimization.
Model it yourself
Because the three phases interact only by addition, not multiplication, it's easy to underestimate how sharply the total hazard rate can shift near t_w or how long infant mortality actually persists relative to τ. The Bathtub Curve Simulator lets you set λ0, τ, λc, and t_w, sweep the evaluation time t, and watch each contribution and the total hazard λ(t) update together — a fast way to sanity-check a reliability qualification plan before committing to burn-in durations or warranty periods.
For related reliability work, the fault tree tool is a useful next step for tracing how individual component failures combine into system-level failure modes.
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