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)

• 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,…?

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]1. Components in different sockets are not necessarily alike

and are stochastically independent

2. Every component failure causes equipment failure

3. Each component is replaced immediately at failure

4. The process of failure detection, trouble location, and replacement is assumed to consume no appreciable time…

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(tE[TTFF])/sTTFF)– MTBF growth = E[TBF1|TTFF=t]E[TBF1]=rsTBF1(tE[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– sTBF1 can be changed with redundancy, X2 uncertainty, and ???

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’s atb 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’

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?

How does redundancy change standard deviation?

• k-out-of-n failure rate is not constant despite what NASA, UPS vendors, and FAA say

• Assuming constant failure rate (1/MTBF) partsSystem Redundancy Mean/MTBF s/MTBF

Any 1-out-of-2 3/2 √(5)/2=1.118Any 1-out-of-3 11/6 7/6=1.1667

Any 2-out-of-3 5/6 √(13)/6=0.7211VIMM 4-out-of-5 9/20 √(41)/20=0.321056TCM 2-out-of-4 13/12 √(61)/12=0.650854Dies 113-out-of-128 0.132975 0.0332681

How to Really Measure Reliability Growth?

• Broom charts [Jerry Ackaret]• My favorite broom chart (EEPROMS)

0 3 6 9 12 15 18 21 24 270.97

0.98

0.99

1.00

97–4

98–1

98–2

98–3

98–4

99–1

99–2

99–3

99–4

Age at Return, Months

Reliability

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?

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