Reliability Engineering: MTBF, Weibull Analysis, Failure Rate, and Accelerated Life Testing

The bathtub curve and failure rate regimes

The bathtub curve is a graphical representation of how the failure rate of a population of products changes over time. It has three distinct phases:

Infant mortality (decreasing failure rate): the failure rate is highest at the start of a product’s life and decreases over time. Infant mortality failures are caused by manufacturing defects, assembly errors, substandard components, and latent material flaws that manifest early in service. The infant mortality region is addressed through burn-in testing (operating the product under normal or elevated stress before delivery to the customer), incoming inspection, and process quality improvements that reduce the rate of latent defects introduced during manufacturing.

Useful life (constant failure rate): after surviving the infant mortality phase, the failure rate stabilizes at a relatively constant level. Failures in the useful life phase are random — they are caused by unpredictable stress events, random component degradation, and environmental extremes rather than by systematic defects. The constant failure rate is the assumption that underlies the exponential distribution and the MTBF metric. During this phase, reliability is improved by robust design (designing the product to survive the full range of environmental and operational stresses it will encounter in service), redundancy, and derating of components.

Wear-out (increasing failure rate): after a product has accumulated sufficient service cycles or time, the failure rate begins to increase again as components reach the end of their fatigue life, lubricants degrade, seals harden, and materials undergo corrosion or creep. The wear-out failure mode is addressed through preventive maintenance programs, scheduled replacement of life-limited components, and design decisions about the service life of the product.

Failure rate and MTBF

The failure rate (lambda, λ) is the expected number of failures per unit of time for a population of items. Under the constant failure rate assumption (the useful life portion of the bathtub curve), the failure rate is related to the mean time between failures (MTBF) by the simple relationship: MTBF = 1/λ. For example, a component with a failure rate of 0.001 failures per hour has an MTBF of 1000 hours.

MTBF applies to repairable systems: it is the average time between successive failures of a system that is repaired and returned to service after each failure. Mean time to failure (MTTF) applies to non-repairable items: it is the average time from the start of operation to the first (and only) failure. For non-repairable items, MTTF and MTBF are numerically equal under the constant failure rate assumption, but conceptually distinct.

A common misunderstanding of MTBF is interpreting it as a guarantee of failure-free operation for the stated duration. MTBF is a population average, not a minimum life. Under the exponential distribution (constant failure rate), approximately 63% of a population of items will have failed by the time equal to one MTBF. A product with an MTBF of 10,000 hours does not mean any individual unit will last 10,000 hours before failing: it means the average time between failures across the entire population is 10,000 hours, and about 37% of units will survive beyond that point.

Mean time to repair (MTTR) and availability

Mean time to repair (MTTR) is the average time required to restore a failed system to operational condition. MTTR includes the time to diagnose the failure, procure replacement parts, perform the repair, and verify that the system is functioning correctly. MTTR is distinct from mean time to restore: restoration time may include logistical delays (waiting for a technician to arrive or for parts to be shipped) that are not inherent to the repair process itself.

System availability combines MTBF and MTTR into a single metric that expresses the proportion of time the system is in a functioning state:

Availability = MTBF / (MTBF + MTTR)

For example, a system with an MTBF of 500 hours and an MTTR of 5 hours has an availability of 500 / (500 + 5) = 0.990, or 99.0%. Availability is the primary reliability metric for production equipment, utilities, and any system where downtime directly affects output. Improving availability requires either increasing MTBF (through better design and preventive maintenance) or decreasing MTTR (through faster diagnosis, better parts access, and improved maintainability design).

The Weibull distribution

The Weibull distribution is the most widely used statistical model in reliability engineering because it can describe all three phases of the bathtub curve using a single distribution by varying its shape parameter (beta, β):

β less than 1: decreasing failure rate (infant mortality). The Weibull distribution with β less than 1 describes a population where failure rate decreases over time, consistent with the burn-in and early-life failure behavior of a product with latent manufacturing defects.

β equal to 1: constant failure rate (random/useful life failures). When β = 1, the Weibull distribution reduces to the exponential distribution, which is the special case of constant failure rate. This is the regime where MTBF calculations are most applicable.

β greater than 1: increasing failure rate (wear-out). The Weibull distribution with β greater than 1 describes wear-out failures, where the probability of failure increases with accumulated service time. When β is approximately 3.5, the Weibull distribution closely approximates the normal distribution, which is often used to model component fatigue life.

The scale parameter of the Weibull distribution (eta, η, also called the characteristic life) is the time at which 63.2% of the population has failed, regardless of the value of β. The characteristic life is related to MTBF through the gamma function: MTBF = η × Γ(1 + 1/β).

Weibull analysis fits the two-parameter (or three-parameter) Weibull distribution to failure data using probability plotting or maximum likelihood estimation. The resulting β and η values characterize the failure behavior of the population and support decisions about design life, warranty policy, preventive maintenance intervals, and spare parts inventory.

Reliability at time t and B-life

The reliability function R(t) gives the probability that a unit will survive until time t without failure. Under the Weibull distribution:

R(t) = exp(-(t/η)^β)

The B-life (or Bx life) is the time at which x percent of the population has failed: B10 life is the time at which 10% of units have failed (and 90% survive), B50 life is the median life (50% survival), and so on. B10 life is widely used as a reliability design target in bearing design, automotive component design, and industrial equipment, because it characterizes the time at which the first significant fraction of the population begins to fail.

Accelerated life testing

Accelerated life testing (ALT) is the practice of subjecting products to elevated stress levels — higher temperature, higher vibration, higher humidity, or higher cycling rates than they will encounter in normal service — to induce failures faster than they would occur in normal use. The goal of ALT is to observe failure modes and estimate the product’s service life in a fraction of the time that normal-use testing would require.

The statistical link between the accelerated test conditions and normal-use conditions is an acceleration model. The most commonly used models are:

Arrhenius model: models temperature-accelerated failures (chemical degradation, corrosion, electromigration). The acceleration factor between test temperature and use temperature is calculated from the activation energy of the dominant failure mechanism and the Boltzmann constant. A 10°C increase in temperature roughly doubles the reaction rate for many degradation mechanisms (the Q10 rule of thumb).

Inverse power law model: models stress-accelerated failures where the failure rate increases as a power of the stress level. Used for voltage-stress, mechanical fatigue (where the stress is cyclic load amplitude), and erosion or wear mechanisms.

Eyring model: a more general model that combines temperature and one or more non-thermal stress factors (humidity, voltage, current) into a single acceleration factor expression.

Highly accelerated life testing (HALT) is a variant of ALT used during product development to find design weaknesses and failure modes, rather than to estimate precise reliability numbers. HALT applies step-stress profiles — increasing stress in discrete steps until the product fails — to find the operating limit, the destruct limit, and the failure modes at each limit. HALT findings drive design improvements that increase the product’s robustness margin before production begins.

Common errors in reliability engineering

• Treating MTBF as a guaranteed minimum life. MTBF is a population average failure rate metric, not a minimum life guarantee. Under exponential (constant failure rate) assumptions, 63% of a population will fail before one MTBF elapses. Using MTBF to set warranty periods or maintenance intervals without understanding the underlying Weibull shape parameter leads to systematic over- or under-maintenance and inaccurate warranty cost projections.
• Applying the exponential distribution to wear-out failures. The exponential distribution (β = 1) is only appropriate for random failures in the useful life phase of the bathtub curve. Applying it to wear-out failure modes, which have increasing failure rate (β greater than 1), will underestimate the failure rate late in product life and produce nonconservative reliability estimates. Weibull analysis, not the exponential distribution, should be used for wear-out failure characterization.
• Reporting system MTBF without specifying what counts as a failure. MTBF has no meaning without a precise definition of what constitutes a failure. Does a degraded-mode operation count? Does a customer-reported no-trouble-found (NTF) return count? Different definitions produce dramatically different MTBF values for the same product. MTBF specifications must always be accompanied by a clear operational definition of failure.
• Extrapolating ALT results beyond the range of the acceleration model. Acceleration models are empirical fits to observed data within a specific stress range. Extrapolating outside the validated stress range — for example, applying an Arrhenius model at temperatures far above those tested — can produce large errors if the dominant failure mechanism changes at higher stresses.

Frequently asked questions

What is the difference between MTBF, MTTF, and MTTR?

MTBF (mean time between failures) applies to repairable systems and is the average time between successive failure events over the operational life of the system. A repairable system is repaired and returned to service after each failure, so it experiences multiple failures over its life; MTBF is the average of the intervals between those failures. MTTF (mean time to failure) applies to non-repairable items — items that are discarded rather than repaired when they fail. MTTF is the average time from when the item is placed in service to when it first fails. Under the constant failure rate assumption (exponential distribution), MTBF and MTTF are numerically equal because the memoryless property of the exponential distribution means that a repaired item behaves statistically the same as a new item. For wear-out failure modes (Weibull β greater than 1), this equivalence does not hold and the distinction matters. MTTR (mean time to repair) is the average time required to restore a failed system to operational condition: it is the repair side of the reliability picture, while MTBF is the failure side. Availability, the fraction of time the system is operational, is calculated as MTBF / (MTBF + MTTR).

What does the Weibull shape parameter beta tell you about a failure mode?

The Weibull shape parameter beta (β) describes the behavior of the failure rate over time for a specific failure mode. When β is less than 1, the failure rate is decreasing: the population is experiencing infant mortality failures, where units with latent defects fail early and the survivors are progressively stronger. When β equals 1, the failure rate is constant: failures are random and independent of the age of the unit (the exponential distribution special case), consistent with externally caused random failures or electronic component failures during the useful life phase. When β is between 1 and 3, the failure rate is moderately increasing, characteristic of early wear-out or fatigue accumulation. When β is approximately 3.5, the distribution approximates a normal distribution, typical of tightly controlled fatigue failures. When β is greater than 4, the failure rate is sharply increasing, consistent with a wear-out mechanism with a well-defined end-of-life (for example, a bearing that is operated past its design load life). Knowing β from field data or test data tells the reliability engineer whether to focus corrective actions on manufacturing process improvement (β less than 1), design robustness (β approximately 1), preventive maintenance scheduling (β greater than 1), or component replacement programs (β much greater than 1).

What is accelerated life testing and how is it used in product development?

Accelerated life testing (ALT) is a reliability testing methodology that subjects products to higher-than-normal operating stresses to induce failures faster than they would occur in normal service, then uses an acceleration model to translate the test results back to equivalent normal-use life predictions. ALT is used in product development to estimate service life and failure rates without waiting the full service life of the product under normal conditions: a product designed for a 10-year service life cannot be tested for 10 years during development, but ALT can provide statistically valid life estimates in weeks or months. The key requirements for valid ALT are: (1) the failure modes induced by the accelerated stress must be the same as those that would occur in normal use (if the elevated stress induces a different failure mechanism, the results do not represent normal-use reliability); (2) the acceleration model must be validated for the relevant stress type and failure mechanism; and (3) the statistical analysis must account for the acceleration factor when converting test results to use-condition predictions. Highly accelerated life testing (HALT) is a design-phase variant that applies step-stress profiles to find and fix design weaknesses before production rather than to produce precise life estimates. HALT is qualitative: its purpose is to find failures, not to measure them.

How is reliability related to quality and why does it require separate treatment?

Quality and reliability both address product conformance to requirements, but they address different dimensions of time. Quality control assesses whether a product meets its specifications at the point of manufacture or delivery: it is a snapshot evaluation. Reliability addresses whether the product will continue to meet its specifications over its service life under the range of conditions it will encounter in actual use: it is a time-domain evaluation. A product can be perfectly conformant at the time of shipment and still have poor reliability if it was designed with inadequate safety margins against the stresses of normal use, or if it was designed for a shorter service life than the customer expects. Reliability requires a separate engineering discipline because predicting and demonstrating time-to-failure behavior requires statistical methods (Weibull analysis, accelerated testing, growth testing), physics-of-failure knowledge (understanding the mechanisms by which specific materials and designs degrade over time), and test protocols that are not part of standard quality control practice. Reliability data from the field is also a critical input to the CAPA process: field failures with increasing failure rate (Weibull β greater than 1) indicate a design or material wear-out problem that production quality control cannot address, because the root cause is in the design, not in manufacturing variation.

What is B10 life and why is it used instead of MTBF for wear-out failure modes?

B10 life is the time at which 10% of a population of units has failed (or equivalently, 90% of units have survived). It is the point on the cumulative failure distribution at 10% cumulative probability, read from the Weibull probability plot. B10 life is preferred over MTBF for wear-out failure modes because MTBF (with its exponential distribution assumption of constant failure rate) is an inappropriate metric when the failure rate is increasing with age. For a wear-out failure mode described by a Weibull distribution with β greater than 1, using MTBF to characterize reliability produces a single average number that obscures the age dependence: it does not tell the designer or the maintenance planner when failures will begin to accelerate. B10 life directly answers the operationally important question: “At what age should I schedule preventive maintenance or replacement to keep the field failure rate below 10%?” It is also a more conservative metric than MTBF for wear-out modes: the B10 life is always shorter than the MTBF for a wear-out Weibull distribution, which means using B10 life as the maintenance interval produces a more conservative (and more reliable) maintenance schedule. In bearing design, the L10 life (equivalent to B10 life) has been the standard bearing selection metric for over 70 years.

How does reliability engineering connect to FMEA and the control plan?

Reliability engineering and FMEA are deeply interconnected. The design FMEA (DFMEA) is the primary tool for identifying potential failure modes in a product design before production, assessing their severity and probability of occurrence, and documenting the design controls and verification actions intended to prevent or detect them. Reliability engineering provides the quantitative failure rate and failure mode data that makes FMEA assessments defensible: occurrence ratings in the DFMEA should ideally be based on field data, reliability test results, or component supplier reliability data rather than subjective expert opinion. Conversely, FMEA findings drive the reliability test program: failure modes identified as high-RPN in the DFMEA should be the primary targets of ALT and HALT testing. The monitoring and system response FMEA (FMEA-MSR), required by IATF 16949, explicitly integrates reliability and safety concepts by addressing how the system monitors its own health and responds to detected failures. The control plan closes the loop by specifying the in-process monitoring and inspection controls that detect the leading indicators of the failure modes identified in the FMEA before they produce non-conforming product or field failures. Field reliability data from warranty returns and customer complaints should be fed back into the DFMEA to update occurrence ratings and verify that the design controls were effective in preventing the field failure modes they were intended to address.