Visualise the bathtub curve, the classic model of how a product's failure rate changes over its operating life. Adjust the infant-mortality, random and wear-out components to see the total hazard rate λ(t) at any evaluation time and which life phase the product is in right now.
Parameters
Infant-failure level λ0
/1000h
Height of the infant failures right after start-up
Infant-failure decay time τ
h
Time for the weak units to be weeded out
Random failure rate λc (constant)
/1000h
Height of the flat bottom of the useful life
Wear-out onset time t_w
h
Time at which cumulative damage takes over
Evaluation time t
h
Operating time at which to evaluate the failure rate
Results
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Infant contribution (/1000h)
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Random contribution (/1000h)
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Wear-out contribution (/1000h)
—
Total hazard λ(t) (/1000h)
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Minimum hazard rate (/1000h)
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Current life phase
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Bathtub curve — three regions and evaluation marker
The vertical axis is the hazard rate, the horizontal axis is operating time. The infant-mortality (blue), useful-life (green) and wear-out (red) regions are shaded, with a marker at the hazard rate of the evaluation time.
Bathtub curve λ(t) (0 to 120,000 h)
The three failure-rate components (infant, random, wear-out)
The total hazard rate λ(t) of the bathtub curve. The three terms are the infant-mortality (exponential decay), constant-random and wear-out (rising quadratically, only for t>t_w) contributions.
The bottom of the bathtub (minimum hazard rate) equals the random failure rate λc. The wear-out component is set by the wear-out coefficient k=2.0/1000h and the wear-out onset time t_w.
What is the Bathtub Curve?
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A "bathtub curve" — is that literally a curve shaped like a bathtub? Cute name for an engineering topic.
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The name is exactly what it looks like. Take a large batch of identical components and record, for each one, how many operating hours it lasted before it failed. Plot operating time on the horizontal axis and the failure rate on the vertical axis, and the curve takes the shape of a bathtub cross-section: a high left shoulder, a long flat middle, and a rising right shoulder. It is the most famous curve in reliability engineering.
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I see. But why is the left shoulder high right after you start using something? Brand-new parts breaking — isn't that strange?
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Good question — this is the "infant-mortality" period. Among the new units, a small minority are poorly made, the "weak ones": a cold solder joint, an invisible microscopic crack, a contaminated batch. Only those weak ones fail off early, so the failure rate drops fast from a high level. That is why manufacturers do a "burn-in" before shipping: they deliberately run the products under stress to provoke and use up the infant failures before they ever reach the customer.
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What about that flat bottom in the middle? It looks like almost nothing fails there.
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That is the "random-failure period", also called the "useful life". The failure rate is low and roughly constant, and the failures that occur are essentially random — an unlucky overload, an unfortunate transient — nothing to do with the product's age. In fact, the exponential reliability calculation that assumes a constant failure rate is only valid in this flat region. Raise the random failure rate λc on the left and you will see the whole bottom of the bathtub lift.
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So the right shoulder rising again means... the product has reached the end of its life?
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Exactly — the "wear-out period". As something is used for a long time, cumulative damage like fatigue, corrosion, insulation breakdown and bearing wear starts to take over, and the failure rate climbs steeply. A factory motor's bearings, for instance, definitely wear down after tens of thousands of hours of running. This rise is the signal that "preventive maintenance is now due": replace the part on a schedule before wear-out failures become common. It is safer than waiting for it to break, and cheaper in the end.
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So the countermeasure is completely different in each of the three periods. Even though it is all "failure", the cause is a different thing.
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That is the central lesson of the bathtub curve. Infant failures call for burn-in and better manufacturing quality; random failures call for design margin and protection circuits; wear-out failures call for preventive maintenance. Different cause, different remedy. This tool shows the three components on separate charts, so slide the evaluation time t and check which failure mode dominates right now. Once you know that, you know where to invest.
Frequently Asked Questions
The bathtub curve is the classic reliability-engineering model of how the failure rate of a population of components changes over its operating life. Plotting operating time on the horizontal axis and the hazard (failure) rate on the vertical axis, the curve very often takes a shape like the cross-section of a bathtub. The high left shoulder is the infant-mortality period, the long flat bottom is the useful life (random-failure period), and the rising right shoulder is the wear-out period. This tool visualises these three failure modes as three separate components.
Right after manufacture a minority of substandard, weak units are mixed into the population — a cold solder joint, a microscopic crack, a contaminated batch. These weak units fail early and weed themselves out, so the failure rate falls steadily from a high level. This is why manufacturers of critical electronics run a burn-in: they deliberately operate products under stress before shipping to provoke and discard the infant failures. This tool models the infant-mortality contribution as an exponential decay λ0·exp(−t/τ).
The useful life is the long, flat bottom of the bathtub, where the failure rate is low and roughly constant. The failures that occur here are essentially random — an unlucky overload, a stray transient — unrelated to the age of the product. It is in this region, and only this region, that the simple constant-failure-rate exponential reliability model genuinely applies. The minimum hazard rate of the bathtub equals this random failure rate.
In the wear-out period cumulative damage — fatigue, corrosion, insulation breakdown, bearing wear — takes over and the failure rate climbs steeply. This rising rate is the signal that preventive replacement or overhaul has begun to pay off: replacing a part on a schedule, before wear-out failures become common, is cheaper and safer than waiting for it to break. In this tool, once the evaluation time exceeds the wear-out onset, the wear-out component rises quadratically with time.
Real-World Applications
Electronics quality control and burn-in: For high-reliability electronics — servers, medical devices, automotive ECUs, industrial controllers — the heart of quality control is "killing off" the infant-mortality period before shipping. A burn-in (an ageing test under high temperature and power) is run for a set time so that the weak units fail inside the factory rather than in the field. The infant-failure decay time τ in this tool can be read as a guide to how long that burn-in needs to be: a larger τ means the infant failures have a longer tail, so the burn-in must be longer too.
Preventive maintenance planning for rotating machinery: Rotating machines such as pumps, fans, gearboxes and compressors develop a wear-out period from the wear of bearings and gears. In a maintenance plan the wear-out onset time t_w is estimated from life tests and operating history, and parts are replaced or overhauled on a schedule before it. Reacting only after the wear-out period has begun leads to large production losses from unplanned stoppages. Move the evaluation time t around t_w in this tool to feel how steeply the failure rate climbs.
Setting warranty periods and MTBF: The basic rule for a product warranty is to set it so that it absorbs the infant-mortality period yet stays within the random-failure period. Too short a warranty pushes infant failures onto the user; too long a warranty reaches into the wear-out period and complaints spike. MTBF (mean time between failures) is also computed on the assumption of the constant failure rate of the useful life, so you must be aware which region of the bathtub curve it refers to.
Interpreting reliability-test data: When analysing accelerated life-test or field-failure data, the distribution you should fit depends on which region of the bathtub curve the data come from. The infant-mortality and wear-out periods are modelled with the Weibull distribution (shape parameter m<1 and m>1), while the random-failure period is handled with the exponential distribution. The three-component decomposition in this tool helps you grasp this "a different model for each region" idea visually.
Common Misconceptions and Pitfalls
The biggest misconception is assuming the constant-failure-rate exponential distribution can be used at any time. MTBF and exponential reliability calculations are only valid on the flat bottom of the bathtub curve (the random-failure period). The infant-mortality period is still in the middle of falling, and the wear-out period is in the middle of rising — neither is "constant". Applying the useful-life MTBF to an old piece of equipment that has entered the wear-out period gives a far too optimistic prediction and invites unplanned stoppages. Always confirm which region the product is currently in.
Next is the belief that every component produces a clean bathtub shape. In reality, some components show almost no infant mortality (well-controlled semiconductors, for example), and others have no clear wear-out period (most electronic parts become obsolete before they wear out). The bathtub curve is only a typical model; the relative size of the three components varies greatly with the failure mechanism of the component. Even in this tool, setting the infant-failure level to zero produces an L-shaped curve with no infant-mortality period. Before fitting it to real data, understand the dominant failure mode of that component.
Finally, "the longer the burn-in, the better" is a misconception. Burn-in is an effective way to remove infant failures, but overdoing it backfires. The burn-in itself imposes operating time and stress on the product, so an excessively long burn-in eats into the useful life and can even bring wear-out failures forward. The optimal burn-in time should stop where "the infant failures are essentially gone", watching the infant-failure decay time τ; anything longer is wasted cost and wasted life. Remember that removing infant failures and preserving life are a trade-off.
How to Use
Enter infant mortality rate (failures/1000h) and decay constant to model early-life defects in electronics or bearings
Set random failure rate (constant hazard during useful life) and wear-out onset time (e.g., 5000 operating hours for a pump seal)
Adjust wear-out acceleration factor to simulate degradation kinetics; simulator calculates total hazard λ(t) across all three phases and identifies current reliability zone
Worked Example
Industrial bearing assembly: infant mortality 0.8/1000h with decay rate 0.0005/hour, random failures 0.12/1000h, wear-out begins at 8000 hours. At t=2000h (early phase), λ(t)≈0.35/1000h dominated by infant term. At t=5000h (steady state), λ(t)≈0.12/1000h (random only). At t=10000h (wear-out active), λ(t) rises to 0.48/1000h as degradation accelerates. Minimum hazard 0.12/1000h occurs during useful-life plateau.
Practical Notes
Infant mortality dominates first 500–2000 operating hours; design burn-in testing duration to reach 90% confidence that defective units fail before field deployment
Random failure phase (flat hazard) typically spans 30%–70% of design life; MTBF = 1/λ_random; use this to set preventive maintenance intervals
Wear-out phase acceleration depends on environmental stress (temperature, vibration, corrosion); adjust onset time and wear factor based on Weibull shape parameter (Weibull k > 1 confirms wear-out behavior)
For safety-critical systems (aircraft engines, medical devices), target infant contribution <10% of random rate through rigorous incoming inspection and environmental stress screening (ESS)