14: Analysis of Test Data

“Testing leads to failure, and failure leads to understanding."

Definitions

• Failure rate = the number of failures per million hours of operation λ =1/MTBF

• Failure: the state of inability of an item to perform its required function

• Reliability = the probability that an item will perform a required function, under specified conditions, for a specified period of time, at a desired confidence level. (MTBF or Failure rate, λ)

• Minimum life = time to first failure

Definitions, Continued

• Confidence level = probability that a given statement is correct

• Confidence limits = the extremes of a confidence interval within which the unknown has a designated probability of being included

• The minimum life of a device = the time of occurrence of the first failure.

• Pareto analysis = plot of individual failures versus the frequency of the failures.

Example 1: MTBF

• An EEG machine has a MTBF of 4380 hours. What is the failure rate?

• λ = 1/MTBF• λ = 1/4380 failures per hour• λ = 0.000228 failures per hour• λ = 228 failures per million hours

Example 2: MTBF• 10 power supplies are put on test, to be terminated after

each has completed 1000 hours of operation. Two power supplies fail, one at 420 hours and the other at 665 hours. What is the failure rate of the power supplies?

• Eight units completed 1000 hours.• Total test time = 8(1000) + 420 + 665 = 9085 hours• • λ = number of failures/total test time• λ = 2/9085 = 0.000220 failures per hour• • λ = 220 failures per million hours

General MTBF Tests in practice:

• Time terminated, failed parts replaced• Time terminated, no replacement (last…)• Failure terminated, failed parts replaced• Failure terminated, no replacement (easy)• No failures observed during the test

…SOME EXAMPLES FOLLOW

Time terminated, failed parts replaced:

• MTBF = N(td)/r• where• N = number of units tested• td = test duration• r = number of failures

…continued• The performance of ten pressure monitors is monitored while

operating for a period of 1200 hours. The test results are listed below. Every failed unit is replaced immediately. What is the MTBF?

• Unit Number Time of Failure (hours) #failures• 1 650 1• 2 420 1• 3 130 and 725 2• 4 585 1• 5 630 and 950 2• 6 390 1• 7 No Failure 0• 8 880 1• 9 No Failure 0• 10 220 and 675 2• 11

finished …

• N = 10 sets of tests• r = 11 failues• td = 1200 hours (test duration)• • MTBF = N(td)/r• = 10(1200)/11• = 1091 hours

Failure Terminated, Failed Parts Replaced

• Six TENS units were placed on test until all units failed, the last occurring at 850 hours. The test results are listed below. Every failed unit, except the last one, is replaced immediately. What is the MTBF?

• Unit Number Time of Failure (hours) Failures• 1 130 1• 2 850 1• 3 120 and 655 2• 4 440 1• 5 725 1• 6 580 1• • MTBF = N(td)/r• = 6(850)/7 = 729 hours

No failures observed during the test

• For the case where no failures are observed, an MTBF value cannot be calculated. A lower one sided ‑confidence limit must be calculated and the MTBF stated to be greater than that value.

• ml = 2(Ta)/Χ2 α;2

• where• ml = lower one sided confidence limit‑• Ta = total test time• Χ2 α;2= the chi square value from the table in ‑

Appendix• 1, where α is the risk level and 2 is the # of• degrees of freedom

continued …• 10 ventilators are tested for 1000 hours without failure. What is

the MTBF at a 90% confidence level?• N = 10 test units• td = 1000 = test duration• r = 0 = failures• 1 % = 0.90 = confidence level‑• % = 0.10 = risk level• Ta = N(td) = 10(1000) = 10000 = total test time• ml = 2(Ta)/Χ2 α;2

• = 2(10000)/Χ210;2

• = 20000/4.605• = 4343 hours = lower 1 sided confidence limit…• We can then state that the MTBF > 4343 hours, with 90% confidence.

Reliability

• Reliability = exp( λt) = ‑ exp ( t/MTBF)‑

• Example: test time =3200 hours, λ = 220 failures/ million hours

• Reliability = exp(-3200*220/1000000)• = exp(-.704) = .495

• or ~ half will fail in 3200 hours….

Confidence Limits

• Time Terminated Confidence Limits• mL = 2(Ta)/Χ2

α/2;2r+2

• where• mL = lower confidence limit• Ta = total test time• Χ2

α/2;2r+2 = Chi square value from Appendix 1 for α risk level α and 2r+2 degrees of freedom

• mU = 2(Ta)/Χ21-α/2;2r

Confidence Limits, continued

• mL = 2(Ta)/Χ2α/2;2r

• and

• mU = 2(Ta)/Χ21-α/2;2r

• … left to the reader -

Pareto Analysis

Field Data Plots:

Final comments

The data examples given are for estimate purposes only, most of the reliability estimates will likely be slightly on the low side, which can be a good thing…