Reliability can be Interesting, Especially Repairable Systems

Reliability can be Interesting, Especially Repairable Systems TTFF TBF1 TBF2 Etc. TTFF Kelly AFB engine shop manager said, “Build that sucker so it doesn’t come back for at least 600 hours!”[P&W F100 F15 engine]

Don’t Believe Easy Solutions! • Failure rate is constant: theorem from when hell freezes over [Barlow and Proschan] • Reliability is Weibull • US Patent 7149659 Dec, 12, 2006 “System and Method for Performing Reliability Analysis” Alan Lesmerises, Standard Aerospace • TBFs are independent and identically distributed • Demand is normally distributed • US Patent 5287627 Feb. 15, 1994, “Method for Parts Procurement Quantity where Demand is Uncertain for the Product in which the Parts are Used” Robin Roundy et al. • Examples: • Aircraft recips and gas turbine engines • Emergency diesel generators • Tescosupermarket kitchen appliances • Solyndra workstations

Airline Engines’ pdf first 750 hours • TTFF TBF1 TBF2 TBF3 • Real • Weibull

First set of correlations omits suspensions; second assumes suspensions are TBFs. Both sets are computed from same data. Real correlation is probably somewhere in between. Airline engine correlation estimates

F-18 Engines’ FOD CCDF • TTFF TBF1 TBF2 • Real • Weibull

F-18 Engines FOD pdf(TTFF, TBF1) • Units are usage relative to average?

Conditional TBF1 corr(TTFF, TBF1|TTFF < 5 and TBF1 < 5) F-18 Engine correlations

Emergency Diesel Generators

EDG Sample Distribution Parameters

Tesco Bread Plant cdfs

Tesco Bread Plant Bivariate CDF of TTFF and TBF1 • TTFF is 1-16 months and TBF1 is 1-8 months • Left is nonparametric; right is Gaussian copula

Solyndra Workstations • TBFs and TTRs were correlated within and between • Increase correlation of workstation (TBF(i), TBF(i+1)), (TBF(i), TTR(i)), and correlation of workstation processing times at successive workstations • Increases throughput • Reduces variance of throughput • Reduces time from start to steady state • ARS Symposium, Reno, 2010

Recommendations • Take advantage of dependence to improve throughput, http://pstlarry.home.comcast.net/genie.htm • Optimize opportunistic maintenance depending on objective (Max P[Life > 600 hours]?) • If TBF(i) is small and corr(TBF(i),TBF(i+1|fix) > 0, fix a lot; If TBF(i) is small and corr(TBF(i),TBF(i+1|fix) < 0, fix a little? • What else could you do while fixing something? • Duane-Crow-AMSAA “Reliability [MTBF] Growth” estimation for dependence, joint distributions, and multiple systems starting successively, in perhaps different configurations • Use MCF (mean cumulative function) with caution • Send field reliability data to pstlarry@yahoo.com for nonparametric estimates of joint pdf and correlations, installed base and failure or repair counts are statistically sufficient

References • G. R. Weckman et al., "Modeling the Reliability of Repairable Systems in the Aviation Industry,“ Computers and Industrial Engineering, 40 2001, pp. 51-63 [Weibull] • Jens-UweKluegel, “Investigation of time-dependent trends in failure data of active components,” NPP Goesgen, CH-4658 Daeniken, Switzerland [Weibull] • Moreira, Ana and Luis Meira-Machado, “survivalBIV: Estimation of the Bivariate Distribution Function for Sequentially Ordered Events Under Univariate Censoring,“ J. Statistical Software, Vol. 46, No. 13, March 2012 • Lin, D. Y., W. Sun, and Zhialiang Yin, “Nonparametric estimation fo the gap time distributions for serial events with censored data,“ Biometrika, Vol. 86, no. 1, pp. 59-70, 1999 • Millar, R. C. And David Olwell, “Parametric Models for Aircraft Engine Removals Resulting from Foreign Object Damage,“ Naval Engineers Journal, 2011 #3

Reliability can be Interesting, Especially Repairable Systems