stracener_emis 7305/5305_spr08_02.28.08 1 systems reliability growth planning and data analysis dr....

Stracener_EMIS 7305/5305_Spr08_02.28.08

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Systems Reliability GrowthPlanning and Data Analysis

Dr. Jerrell T. Stracener, SAE Fellow

Leadership in Engineering

EMIS 7305/5305Systems Reliability, Supportability and Availability Analysis

Systems Engineering ProgramDepartment of Engineering Management, Information and Systems


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Systems Reliability Growth Planning


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The Duane Model

• The instantaneous MTBF as a function of cumulative test time is obtained mathematically from MTBFC(t) and is given by

where MTBFi(t) is the instantaneous MTBF at time t and is interpreted as the equipment MTBF if reliability development testing was terminated after a cumulative amount of testing time, t.

t

K

1tMTBFi

(t)MTBF1

1C


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Reliability Growth Factors

•Initial MTBF, K, depends on–type of equipment –complexity of the design and equipment operation–Maturity

•Growth rate, α–TAAF implementation and FRACAS Management–type of equipment –complexity of the design and equipment operation–Maturity


5

10 100 1000 10000

100

1000

10

Cumulative Test Hours

MTBF

Cumulative

Instantaneous

MTBF Growth Curves


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The Duane Model

• The Duane Model may also be formulated in terms of equipment failure rate as a function of cumulative test time as follows:

and

whereC(t) is the cumulative failure rate after test time tk* is the initial failure rate and k*=1/k,B is the failure rate growth (decrease rate)i(t) is the instantaneous failure rate at time t

-α*C tκ(t)λ

(t)α)λ(1

tα)κ(1(t)λ

C

-α*i


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Determination of Reliability Growth Test Time

•Specified MTBF at system maturity, θ0

•Solve Duane Model for t

01)(

tK

tMTBFI

])1(

ln[1

0

0

0

]1

ln[1

ln

1

Ket

Kt

Kt


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Determination of Reliability Growth Test Time - Example

•Determine required test time to achieve specified MTBF, θ0=1000, if

α=0.5 and K=10

•Solution

88.2499

)]1000)(10

5.01ln[(

5.0

1

e

])1(

ln[1

0 Ket


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Determination of Reliability Growth Test Time – Example (continued)

Interpretation of θ0=1000 hours at t=2499.88 hours

If no further reliability growth testing is conducted, the systems failure rate at t=2499.88 hours is

0

1

001.01000

1

Failures per hour,

and is constant for time beyond t=2499.88 hours


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•Check

since

•Cumulative MTBF at t = 2499.88 hours

Since

5.0)88.2499(5.01

10)88.2499(

1)(

I

I

MTBF

tK

tMTBF

=1000

1000)5.01()88.2499(

)()1()(

C

IC

MTBF

tMTBFtMTBF

=500


Determination of Reliability Growth Test Time – Example

Test Time

a) Determine the test time required to develop (grow) the reliability of a product to 0 if the required reliability is 0.9 based on a 100-hour mission and the initial MTBF is 20% of 0 and =0.3. How many failures would you expect to occur during the test?

Investigate the effect on the test time needed to achieve the required MTBF of deviations in initial MTBF and growth rate.

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The test time required depends on the starting point. The usual convention is to start the growth curve at t=100. We will determine the test time based on this. Also, we will show the effect on the requirement test time if the starting point of 0.20 is at t=1 hour.

1 100 tR1 tR100

=0.3Required MTBF

Required Test Time

20% ofRequiredMTBF

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Since

Using the Duane Model

since =0.3

122.9490.9ln

100θ

0.9,

e(100)R

0

θ

100

00

0.3

0.3

α

I

1.4286κ

0.7

κt

α1

κt(t)MTBF

t

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But

and

so thator

so that the model is

189.82

949.1220.2

0.2θ(100)MTBF 0I

0.3I t682.47(t)MTBF

5.687κtκ4286.1(100)MTBF 3.0I

378.33κ

82.189κ687.5

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To find the test time required to grow the MTBF to 0

set

so that

and

122.949

t682.47(t)MTBF 0.3I

905.19t0.3

hours 21373.4t

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,failures cum

t time,cum(t)MTBFC

664.39r

21373.4

r

t

664.3973.4)33.378(213κt(21373.4)MTBF

c

c

0.3αc

failures 17.32rc


Since

at t=21373.4 hours,

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If the initial MTBFI(t) is interpreted to be at t=1hour, then

and

so thator

so that the model is

189.82

949.1220.2

0.2θ(1)MTBF 0I

0.3I t82.891(t)MTBF

1.4286κ.(1)MTBFI

87.132

82.1894286.1

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To find the test time required to grow the MTBF to 0

set

so that

and

122.949

t82.891(t)MTBF 0.3I

5t0.3

hours 213.747t

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,failures cum

t time,cum(t)MTBFC

664.35r

213.747

r

t

664.35.747)132.87(213κt(213.747)MTBF

c

c

0.3αc

failures 32.0rc


Since

at t=213.747 hours,

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Systems Reliability Growth Data Analysis


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Reliability Growth Test Data Analysis

Duane Model

• Parameter Estimation

• Maximum Likelihood Estimation

• Least Squares Estimation

• Confidence Bounds

• Plotting the estimated MTBF Growth Curves and Confidence Bounds

• Procedures from MIL-HDBK-189, Feb. 13. 1981


Least Squares for the Duane Model

Since MTBFc(t)=t,

And for simplicity in the calculations, let:

so that

Transforming the data (ti, MTBFc(ti)) to (xi, yi) for i=1, 2, …, n use the method of least squares to estimate the equation y=0+ 1x+

ln t αln κ(t)MTBFln c

ii

1

0

ici

ln t xand

αb

ln κb

)(tMTBFln y

n. ..., 2, 1,ifor xbby i10i

22


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Least Squares for the Duane Model The Least squares estimates of 0 and 1 are:

2

11

2

111ˆ

n

ii

n

ii

n

ii

n

ii

n

iii

xxn

yxyxn

n

1ii

n

1ii xˆy

n

1

K̂

e


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Reliability Growth Data Analysis

A test is conducted to growth the reliability of a system. At the end of 100 hours of testing the results are as follows:

12.5 2.0

21 2.8

35 3.5

60 4.8

100 6.0

tc MTBFC


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•Estimate MTBFI(t) and MTBFC(t) as a function test time t and plot.

•What is the estimated MTBF of the system if testing is stopped at 200 hours?

Reliability Growth Data Analysis - continued


Solution


Solution

tc Xi Xi^2 　 MTBFc Yi 　 Xi*Yi

12.5 2.53 6.38 　 2 0.69 　 1.75

21 3.04 9.27 　 2.8 1.03 　 3.13

35 3.56 12.64 　 3.5 1.25 　 4.45

60 4.09 16.76 　 4.8 1.57 　 6.42

100 4.61 21.21 　 6 1.79 　 8.25

　　　　　　　　　 Sum(Xi) Sum(Xi^2) 　　 Sum(Yi) 　 Sum(Xi*Yi)

　 17.83 66.26 　　 6.34 　 24.01


Solution

53.0ˆ2

11

2

111

n

ii

n

ii

n

ii

n

ii

n

iii

xxn

yxyxn 55.0K̂

n

1ii

n

1ii xˆy

n

1

e

53.053.0

53.0ˆ

17.153.01

55.0ˆ1

ˆ)(ˆ

55.0ˆ)(ˆ

ttFBMT

tFBMT

ttKtFBMT

CI

C


Solution

200


Solution

44.1953.01

1182.9)200(

12.9200*55.0)200( 53.0

I

C

MTBF

MTBF

At t=200 hours, the system failure rate becomes a constant

05144.044.19

1

=Failures per hour


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Duane Model Parameter Estimation

• Estimate the “Best Fit” reliability growth equation by using the observed failure times t1, t2, …, tn to estimate the Duane Model parameters

• Use the failure rate version of the Duane Model since most of the theory is based on it

stracener_emis 7305/5305_spr08_02.28.08 1 systems reliability growth planning and data analysis dr....

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