Reliability can be Interesting, Especially Repairable Systems

TTFF TBF1 TBF2 TTFFEtc.

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– Tesco supermarket kitchen appliances– Solyndra workstations

Airline Engines’ pdf first 750 hours

• TTFF TBF1 TBF2 TBF3

• Real

• Weibull

Airline engine correlation estimates

TTFF TBF1 TBF2

TBF1 0.073376

TBF2 0.298017 -0.33782

TBF3 0.896035 -0.52976 0.891329

TBF1 0.073376

TBF2 0.501917 0.039882

TBF3 0.820457 -0.42322 0.077551

• 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.

F-18 Engines’ FOD CCDF

• TTFF TBF1 TBF2

• Real

• Weibull

F-18 Engines FOD pdf(TTFF, TBF1)

01

23

4-0.02

0

0.02

0.04

0.06

0.08

0.1

0

1

2

3

4

5

pdf of TTFF and TBF1

TBF1TTFF

• Units are usage relative to average?

F-18 Engine correlations

TTFF TBF1TBF1 -0.36962TBF2 0.104095 -0.04504Conditional TBF1 -0.2Conditional TBF2 Not enough data Not enough data

• Conditional TBF1 – corr(TTFF, TBF1|TTFF < 5 and TBF1 < 5)

Emergency Diesel GeneratorsJAN FEB MARAPR MAY JUN JUL AUG SSP OCT NOV DEC

0% 6% 6% 6% 13% 13% 6% 13% 6% 0% 13% 25%

EDG Sample Distribution Parameters

Parameter Value, Days Exponential

sum 5571mean 348.1875stdev 375.4963Exponential should have ~ same stdev and meanskew 1.7321Exponential should have skew ~ 2kurt 3.5300Exponential should have excess kurt ~ 6

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-Uwe Kluegel, “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