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How Much Reliability Growth Do You Want?

How Much Reliability Growth Do You Want?. Larry George, May 5, 2013. Outline of Problem. Theorem from when hell freezes over [Many] Contracts specify MTBF only Quanterion Newsletter reviews Duane, AMSAA-Crow, and Crow “Extended” reliability growth models (really MTBF)

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How Much Reliability Growth Do You Want?

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  1. How Much Reliability Growth Do You Want? Larry George, May 5, 2013

  2. Outline of Problem • Theorem from when hell freezes over [Many] • Contracts specify MTBF only • Quanterion Newsletter reviews Duane, AMSAA-Crow, and Crow “Extended” reliability growth models (really MTBF) • What if more than one machine is on test? Sequentially? Different configurations? • Good-As-Old:Suppose X1, X2,… are part failure times and E[TBFj] = E[Xj]-E[Xj-1] is decreasing? Dependent? • Good-As-Or-Better-Than-New: TAAF may eliminate X1 from repeating? What about improving X2, X3,…?

  3. Theorem from when hell freezes over • “Consider a system consisting of many components,… Under some reasonably general conditions, the distribution of times between equipment failures tends to the exponential as the complexity and the time of operation increases.” [Barlow and Proschan page 18] • Components in different sockets are not necessarily alike and are stochastically independent • Every component failure causes equipment failure • Each component is replaced immediately at failure • The process of failure detection, trouble location, and replacement is assumed to consume no appreciable time…

  4. Outline of Solutions • Change TBFs conditional on prior failure times • For example, suppose (TTFF, TBF1)~N[m,S] perhaps correlated • E[TBF1|TTFF=t] =E[TBF1]+rsTBF1(tE[TTFF])/sTTFF) • MTBF growth = E[TBF1|TTFF=t]E[TBF1]=rsTBF1(tE[TTFF])/sTTFF) • Make r and sTBF1 what you want! • r± depends on X1, X2,…, service, and what to do when prior TTFF = t relative to its mean • sTBF1can be changed with redundancy, X2 uncertainty, and ???

  5. How to quantify MTBF growth with more than one machine on test? • Start first machine at 0 and another at t1 • Assume “Weibull” process MTBF(t) = atb • If configurations are same? • First machine’satb is second machine’s for t > t1 • If configurations differ? • Hysterecal? Second machine is at’b for some 0 < t’< t1 • Improved? Second machine is a’tb’

  6. How does correlation happen? • Suppose X1, X2,… are part failure times? • TTFF = X1, and TBFj = Xj+1−Xj • Cov(TBFj,TBFJ+1) = −Var[Xj] if Xj are indep • Corr(Tbfj,TBFj+1) = −1/2 if variances are same • Suppose service between TTFF and TBF1 introduces it? • Improvements make Corr(TTFF, TBF1) > 0? • Screwups make Corr(TTFF, TBF1) < 0?

  7. How does redundancy change standard deviation? • k-out-of-nfailure rate is not constant despite what NASA, UPS vendors, and FAA say • Assuming constant failure rate (1/MTBF) parts

  8. How to Really Measure Reliability Growth? • Broom charts [Jerry Ackaret] • My favorite broom chart (EEPROMS)

  9. Recommendations • Specify more than MTBF in contracts • Send data on system(s) failure times and parts’ field reliabilities (pstlarry@yahoo.com) • Review redundancy reliability allocation. Are you getting the best bang for buck?

  10. References • Quanterion, “Models Commonly Used to Measure Reliability Growth,” Vol. 8, No. 1, April 2013, http://quanterion.com/ReliabilityQues/V8N1.html • “Broom Charts,” ERI Newsletter, Vol. 7, May 2002, page 3, http://www.equipment-reliability.com/newsletters/newslt7/nl7.htm • Barlow, Richard S. and Frank Proschan, “Mathematical Theory of Reliability,” Wiley, 1965

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