Other Issues• Time – just because it is hours…• Between – note the duration of the failure free period!• Failure – use the customer definition
History of Use• Early Parts Count based on adding failure rates of components (60’s and early 70’s) − λ1t − λ 2t − λnt R(t ) = e •e •• e − ( λ1 + λ 2 ++ λ n ) t R(t ) = e
History of Use• Remember Slide Rule and Mechanical Adding Machines• Victor Adding Machine
Beta = 0.63 Depth Cut Response data Weibull Probability Plot .5 Weibull Distribution ML Fit Exponential Distribution ML Fit .3 95% Pointwise Confidence Intervals .2 .1 .05 .03Fraction Failing .02 .01 .005 .003 .001 .0005 .0003 .0002 .0001 10^-01 10^00 10^01 10^02 10^03 10^04 DEPTH.CUT
Beta = 1.97 test7.df data Weibull Probability Plot .7 Weibull Distribution ML Fit .3 Exponential Distribution ML Fit 95% Pointwise Confidence Intervals .1 .03 .01 .003 Fraction Failing .001 .0003 .0001 .00003 .00001 .000003 .000001 .0000003 .0000001.00000003.00000001 1 10 100 1000 10000 100000 Depth In
Use Reliability• R(t) is the probability that a random unit drawn from the population will still be operating by t hours• R(t) is the fraction of all units in the population that will survive t hours Applied Reliability, 2nd Ed., pg 29
The four (five) elements• Function• Duration• Probability• Environment• They all change over time
Use better models/distributions −( t ) β• Weibull RWeibull (t ) = e η• Type I Gumbel − ( et )• Exponential RGumbel (t ) = e• Log Normal − λt Rexp onential (t ) = e• Etc. t ln T 50 Rlog normal (t ) = Φ σ
Other Measures• What is the cost of a field failure?• Warranty $ per unit shipped• Returns/field failure $ per unit shipped• What else could you use?
Actually…• MTBF is or should be used for repairable systems• MTTF is what I’ve been talking about• MTTF is calculated the same as MTBF when we assume – negligible repair time – Interarrival times as from an independent sample of nonrepairable parts – Expontential distribution for lifetime of parts• See Chap 10, Applied Reliability for more info
Where to Get More Information• Tobias, Paul A. and Trindade, David C., Applied Reliability, 2nd Ed. Chapman & Hall, New York, 1995.• “The Limitations of Using the MTTF as a Reliability Specification” Reliability Edge, Qtr 2, 2000, Vol 1, Issue 1.
Presenter’s Biographical Sketch• Fred Schenkelberg, Consultant• Independent Reliability Engineering and Management Consultant for past 5 years. Previously at HP Corporate Reliability Engineering Program for 5 years.• MS Statistics Stanford, BS Physics USMA• firstname.lastname@example.org• (408) 710-8248• www.opsalacarte.com