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Accelerated tests on failures

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- 1. Failure Rate Age Design and Analysis ofAccelerated Reliability Tests, with Piecewise Linear Failure Rate Functions (PLFR) ASQ SV Statistical Group Sept. 8, 2004 IEEE Reliability Society Silicon Valley Larry George Problem Solving ToolsPST http://www.fieldreliability.com 1
- 2. DART Abstract Part 1 proposes piecewise linear failure rate (PLFR) function models, for modeling simplicity and resemblance to the left-hand end of the bathtub curve. The PLFR is inspired by: Failure rates are not constant, often because of infant mortality Tests have too few samples, are for too short times, and have few failures Need to quantify infant mortality as well as MTBF It shows how to estimate the PLFR parameters, reliability, infant mortality, and MTBF. It proposes acceleration alternatives, including one that accelerates testing greatly without screwing up results. Part 2 describes how to design and analyze accelerated reliability tests, assuming a PLFR and power law acceleration. It shows how to obtain credible results, with limited sample size and test time, at one accelerated stress level. It provides estimators for model parameters, reliability, MTBF, confidence intervals, and it shows how to test model assumptions and verify MTBF. PST http://www.fieldreliability.com 2
- 3. Part 1 Contents Motivation for PLFR MTBF and reliability for PLFR Acceleration of PLFR and RAFPST http://www.fieldreliability.com 3
- 4. DART Objectives Make credible MTBF, reliability, and failure rate function estimates (Credible Reliability Prediction, http://www.asq-rd.org/publications.htm and http://www.fieldreliability.com/Preface.htm) Quantify infant mortality: proportion and duration Verify MTBF Use accelerated tests with only one, high stress level Use available information early in life cyclePST http://www.fieldreliability.com 4
- 5. Today’s Situation? Management wants reliability ASAP How to verify MTBF with tests that end long before MTBF, accelerated, with few if any failures? How to verify P[Life > useful life] > 0.9 with high confidence with small samples and short tests? Has management ever agreed to sample size and test time? Can you extrapolate accelerated tests, at high stress, to working stress, with few failures well before MTBF? NIST, ASQ [Meeker and Hahn], and others [Nelson, Bagdonavicius et al, Viertl] recommend ≥ two acc. stress levels PST http://www.fieldreliability.com 5
- 6. Intel FITS have Infant Mortality Data used to be at http://www.intel.com/support 10000 28F400BX 1000 28F400BV 28F008SA 100 28F016SV 28F001 10 87C196KC 80C51BH 1 80486SXSA 0.1 1 10 80486DX2 Age, years PST http://www.fieldreliability.com 6
- 7. Common, Invalid Assumptions Constant failure rate Infant mortality initially ↓ failure rate. Monotonic ↑ or ↓ failure rate Products often have both (rules out Weibull) [George 1995]. Cite bathtub curve Acceleration doesn’t affect Weibull shape parameter It does, usually, according to Richard Barlow [ http://www.esc.auckland.ac.nz/Organisations/ORSNZ/Newsletter ] Can’t extrapolate to normal stress with only one accelerated stress level (one hand clapping) Yes we can! PST http://www.fieldreliability.com 7
- 8. Piecewise Linear Failure Rate a(t) = a+bt = 0.0001+0.0001(7−t)+ Dotted line is a possibly ↑ failure rate Failure Rate 0.0008 0.0006 0.0004 0.0002 Age 2 4 6 8 10 12 14PST http://www.fieldreliability.com 8
- 9. Test Conconi Aerobic threshold is the heart rate at which the slope of work rate vs. heart rate decreases PST http://www.fieldreliability.com 9
- 10. Reliability with PLFR Reliability function has two parts, IM and after: Exp[(0.0001t2)/2−t(0.0001+0.0001to)] for t < to Exp[−0.0001t−(0.0001to2)/2] for t ≥ to P[Fail in IM] ~bto2/2 MTBF~(1−to2b)/2+to2b/6−ato4b/24 = 9975.5 Reliability Age 2 4 6 8 10 12 14 0.999 0.998 0.997 PST http://www.fieldreliability.com 10
- 11. Acceleration alternatives Constant segment increases to greater constant Constant segment becomes linearly increasing (limit of equal step stress); i.e. acc. induces premature wearout, Infant mortality slope increases and perhaps to, the age at the end of IM, decreases as acceleration exacerbates process defects System acceleration ≠ part accelerations! (unless parts are iid and in series)PST http://www.fieldreliability.com 11
- 12. Acceleration alternativesFailure Rate 0.001 Constant b ↑ Linearly ↑0.00080.00060.0004 Constant a ↑0.0002 Age 12 2 4 6 8 10 12 14 PST http://www.fieldreliability.com
- 13. Reliability Acceleration Factor RAF(t) = (1-RUnacc(t)/(1-Racc(t)) > 1.0 RAF(60) = 1.705 for double constant failure rate 2a from 0.0001 to 0.0002 RAF(60) = 1.288 for double infant mortality, b, increases from 0.0001 to 0.0002 RAF(60) = 11.350 for changing from constant, a, to linearly increasing failure rate, a+0.0005*t!PST http://www.fieldreliability.com 13
- 14. Fairly General AccelerationModel aAcc(t) = aUnAcc[t/θ(x)]/θ(x) [Xiong and Ji] lnθ(x) = α + βx x is stress factor, (stress-normal)/(max stress-normal) Continuous version of equal-step stress Multiplies failure rate by a factor and rescales age t Includes Arrhenius and Eyring models, [Shaked], motivated by Miner’s rule Apply it to constant, IM slope, or entire piecewise linear failure rate functionPST http://www.fieldreliability.com 14
- 15. Part 2 Designs and examples |D|-optimal and other statistical designs fail Exponential, Weibull, and normal designs exist Moderately credible design Contrary to popular recommendations, you need only one acceleration level Examples: estimate parameters, LR test of MTBF Unacc. and acc. FreebiesPST http://www.fieldreliability.com 15
- 16. Alternative Designs |D|-optimal is versatile, but recommends tests at 0, to, and anywhere thereafter DoE expects every design point to yield age at failure. Reliability tests often don’t. Highly censored data. Consider Neyman design for multiple strata [Neyman, George 2002 (DORT)] In minimum variance design, must specify how much variance. [Nelson, Meeker and Hahn] Moderately credible design gives 50% probability of at least one failure in infant mortality and one thereafter, sufficient to estimate piecewise linear parameters PST http://www.fieldreliability.com 16
- 17. Moderately Credible Design Want 50% probability of ≥ 1 failure in IM and ≥ 1 after IM before end of test, tParameters Case 1 Case 2 Case 3a constant (guess) 0.01 0.01 0.01b IM slope (guess) 0.01 0.01 0.01to IM ends (guess) 2 2 2n sample size (choose) 29 34 31t test time (choose) 7 5.6 6.4P[failure < to] 0.039 0.039 0.039P[failure in [to, t)] 0.047 0.034 0.041P[failure < to|n] 0.371 0.356 0.366P[≥ 1 failure in [to, t)|n-1] 0.739 0.680 0.718P[Both, all] 0.501 0.499 0.504 PST http://www.fieldreliability.com 17
- 18. Example Data (Unacc.) Sample Age at failure Survivors’ ages 1 1 2 2 3 15 4 30 5 45 6 45 19 45 20 45PST http://www.fieldreliability.com 18
- 19. Example ResultParameter/M a a+b(t–to) ct b(t–to)+ct a+ct a+b(t–to)odel +cta 0.007 0.004 0.008 0.000b 0.016 0.018 0.018c 0.000 0.000 0.000 0.000to 3.319 3.346 3.346MTBF 154 215 73 83 125 84ln likelihood -30.17 -28.21 -34.97 -27.78 -30.27 -27.78LR statistic 3.919 14.389 4.989Sig level 10% 10% 10%χ2 6.251 6.251 7.779 Best model PST http://www.fieldreliability.com 19
- 20. Put all your eggs in onebasket for acceleration a(t) = xp(a+b(to−t)++ct) Test at highest reasonable stress Predict MTBF or use specified MTBF Find mle of parameters, constrained to specified MTBF at working stress, x=1 Use LR to test specified MTBF -2ln[L(MTBF)/L(unconstrained)]~χ2PST http://www.fieldreliability.com 20
- 21. Example Data (Accel.)Sample Ages at failures Survivors’ age1 12 13 24 25 106 157 208 259 3010 3511 4020 45 PST http://www.fieldreliability.com 21
- 22. Example Result, x = 1.5 Parameter xp(a+ct) xp(a+b(t–to)+ct) a 0.001452 0 b 0.018298 c 7.79E-05 0.000180 to 3.345768 p 5.149690 5 MTBF 125 125 Log likelihood -53.84 -56.17 LR test statistic -4.65 Sig level 10% Chi-square 9.23634 Better modelPST http://www.fieldreliability.com 22
- 23. Switch Example Demonstrate MTBF > 39,500 hours with 75% confidence Test 7 switches for 6 weeks (1008 hours) at 60° C with MTBF AF = 14.6 (Arrhenius) to give χ2 LCL of ~39,000 hours Xcvrs failed at 486 and 660 hours (16 xcvrs per switch), after IMPST http://www.fieldreliability.com 23
- 24. Real Example Data Parameter Value c 3.56E-8 per hour per hour Stdev c c/√(2n) = 2.38E-9 per hr2 MTBF √(π/2c) = 6645 hours 25th %ile of MTBF 6584 hours MTBF of 16 xcvrs acc. √(π/32c) = 1661 hours 25th %ile of 16-xcvr MTBF ~1000 hours 25th %ile of 16-xcvr MTBF, 1000*35 = 35,000 hours unacc.PST http://www.fieldreliability.com 24
- 25. Recommendations For simplicity, use the PLFR to approximate left- hand end of bathtub curve… Approximate acceleration with power law, rescale age if necessary and if Miner’s rule fits Use one, high level of acc. and MTBF to test hypotheses and extrapolate back to working stress Send data to pstlarry@yahoo.com for PLFR analyses, free of chargePST http://www.fieldreliability.com 25
- 26. Freebies athttp://www.fieldreliability.com MTBF prediction a la MIL-HDBK-217F Kaplan-Meier nonparametric reliability estimate from ages at failures and survivors’ ages Redundancy reliability allocation Weibull reliability estimate from ages at failures and survivors’ ages What would you like? PST http://www.fieldreliability.com 26
- 27. References Bagdonavicius, Vilijandas and Mikhail Nikulin, Accelerated Life Models, Modeling and Statistical Analysis, Chapman and Hall, New York, 2002 George, L. L., “Design of Ongoing Reliability Tests (DORT),” ASQ Reliability Review, Vol. 22, No. 4, pp 5-13, 28, Dec. 2002 George, L. L. “Design of Accelerated Reliability Tests,” ASQ Reliability Review, Part 1, Vol. 24, No. 2, pp 11-31, June. 2004 and Part 2, Vol. 24, No. 3, pp 6-28, Sept. 2004. Presentation is at http://www.ewh.ieee.org/r6/scv/rs/articles/DART.pdf Kalbfleisch, John D. and Ross L. Prentice, The Statistical Analysis of Failure Time Data, Second Edition, Wiley, New York, 2002 Meeker, William Q. and Gerald J. Hahn, How to Plan an Accelerated Life, Test: Some Practical Guidelines, Vol. 10, ASQ, 1985 Nelson, Wayne, Accelerated Testing, Wiley, New York, 1990 NIST, Engineering Statistics Handbook, Ch. 8.3.1.4, “Accelerated Life Tests,” http://www.itl.nist.gov/div898/handbook/apr/section3/apr314.htm Shaked, Moshe, “Accelerated life testing for a class of linear hazard rate type distributions,” Technometrics, Vol. 20, No. 4, pp 457-466, November 1978 Viertl, Reinhard, Statistical Methods in Accelerated Life Testing, Vandenhoeck & Ruprecht, Göttingen, 1988 George, L. L., “What MTBF Do You Want?” ASQ Reliability Review, Vol. 15, No. 3, pp 23-25, Sept. 1995 Neyman, J., “On the Two Different Aspects of the Representative Method: The Method of Stratified Sampling and the Method of Purposive Selection,” J. of the Roy. Statist. Soc., Vol. 97, pp 558-606, 1934 Xiong, Chengjie, and Ming Ji, “Analysis of Grouped and Censored Data from Step-Stress Life Test,” IEEE Trans. on Rel., Vol. 53, No. 1, pp. 22-28, March 2004 PST http://www.fieldreliability.com 27

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