Engineering & Technology · Reliability, Safety, Failure & Maintenance

Reliability, failure rates, lifetime data and assumptions

Six small data loggers begin a life test. Two stop working; four are still working when the laboratory closes the test. The four unfinished lifetimes contain useful information. Calling them failures at the stopping time would change the question and distort the answer.

Use Quality Engineering, Experiments & Process Evidence; Engineering Design Methods & Trade-offs; and Material Selection, Degradation & Failure. Read probability as a fraction of comparable cases, distinguish a rate from a count and retain units in energy and inventory balances. Models use invented teaching records and supplied thresholds; they do not approve a real safety or maintenance decision.

Back to the chapter

After reading, you can

Interpret lifetime records and a conditional failure rate before using a reliability formula.

Define success before collecting times

Reliability R(t) is the probability of performing a stated required function through a stated duration t under stated conditions. A logger that still turns on but loses required data may already have failed its mission. Record the failure criterion, environment, duty cycle and population; survival of a convenient indicator is not automatically survival of the service.

A non-repairable lifetime ends at the first defined failure. A repairable asset can fail, return to service and fail again. Its event history and downtime need a repair-process account. Mixing first lifetimes with repeated repair intervals under one label such as “average life” hides that distinction.

Keep exposure and censoring visible

Right censoring means observation ended while the unit had not yet failed. We know its lifetime exceeds the observed duration, not the date of its future failure. The figure preserves those records as circles rather than failure crosses. Units withdrawn because they were deteriorating would require attention to why observation stopped.

The hazard rate h(t) describes failure per unit time among units that have survived to t. It is conditional on survival and has units such as h⁻¹; it is not the fraction of the original population that fails at a particular instant. A constant hazard does not mean a constant cumulative probability of failure. That probability accumulates over time.

Six independent non-repairable units begin observation together. U 1 fails at 200 h, U 2 at 600 h, and U 3–U 6 remain working when observation ends at 1000 h. Solid timelines use a common 0–1000 h horizontal scale; crosses mark failures and hollow circles mark right censoring. Exposure is 200+600+4×1000=4800 unit-hours. Under a separately assumed exponential lifetime model, rate estimate is 2/4800=1/2400 per hour and estimated survival to 1000 h is exp(−1000/2400)=0.65924. End circles do not mean failure at 1000 h.

A distribution is an assumption to examine

An exponential model assumes constant hazard λ and gives R(t)=exp(−λt). Here exp is the exponential function, and λt is dimensionless. Its mean first lifetime is 1/λ. The formula needs a defensible population and conditions; it does not establish that wear, installation defects or changing duty are absent.

Compare patterns and mechanisms as well as a fitted number. Time plots can expose changing conditions or mixed populations. Sparse failures leave substantial statistical uncertainty, while an excellent fit inside one test still leaves uncertainty about another environment. Accelerated testing needs a justified connection between test stress and service, not simply a shorter calendar.

Estimate a rate without discarding survivors

Supply independent, comparable units with exact failures at 200 and 600 h and four right-censored records at 1000 h. Total exposure is 200+600+4×1000=4800 unit-hours. Under the exponential assumption, the maximum-likelihood rate estimate is observed failures divided by total exposure: λ̂=2/4800=1/2400 h⁻¹.

Estimated survival through 1000 h is exp(−1000/2400)=0.65924, about 66%. The fitted mean is 2400 h, not a guaranteed replacement age or an observed lifetime for every unit. With only two failures the estimate is uncertain; these invented data do not validate constant hazard.

Try the reasoning

Why must the four surviving units not be assigned failures at 1000 h?
Show the answer and reasoning
Their future failure times are unknown. The records establish survival through 1000 h and contribute exposure, not four failures.
Does a hazard of 0.001 h⁻¹ mean 100% failure after 1000 h in an exponential model?
Show the answer and reasoning
No. R(1000)=exp(−1), so cumulative failure probability is about 63.2%, not 100%.
Can an excellent laboratory fit alone establish field reliability?
Show the answer and reasoning
No. Field population, duty, environment, failure criteria and the transfer model also need evidence.

What to carry forward

Keep the service definition, survivor population, observation process and model assumptions attached to every reliability number.

References

Teaching resources and research supporting this lesson’s explanations and examples.

National Institute of Standards and Technology (NIST)4 sources