Systems Reliability Growth Planning and Data Analysis

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Systems Reliability GrowthPlanning and Data
Analysis

• Systems Reliability, Supportability and
  Availability Analysis

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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.

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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, a
• TAAF implementation and FRACAS Management
• type of equipment
• complexity of the design and equipment operation
• Maturity

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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
• where
• ?C(t) is the cumulative failure rate after test
  time t
• k is the initial failure rate and k1/k,
• a is the failure rate growth (decrease rate)
• ?i(t) is the instantaneous failure rate at time t

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Determination of Reliability Growth Test Time

• Specified MTBF at system maturity, ?0
• Solve Duane Model for t

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Determination of Reliability Growth Test Time -
Example

• Determine required test time to achieve specified
  MTBF, ?01000, if
• a0.5 and K10
• Solution

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Determination of Reliability Growth Test Time
Example (continued)

• Interpretation of ?01000 hours at t2499.88
  hours
• If no further reliability growth testing is
  conducted, the systems failure rate at t2499.88
  hours is
• 
  Failures per hour,
• and is constant for time beyond t2499.88 hours

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Determination of Reliability Growth Test Time
Example (continued)

• Check
• since
• Cumulative MTBF at t 2499.88 hours
• Since

1000
500
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Determination of Reliability Growth Test Time
Example

• Test Time
• 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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Determination of Reliability Growth Test Time
Example (continued)

• The test time required depends on the starting
  point. The usual convention is to start the
  growth curve at t100. 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 0.2?0 is at t1 hour.

Required MTBF
a0.3
20 of Required MTBF
Required Test Time
1 100 tR1 tR100
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Determination of Reliability Growth Test Time
Example (continued)

• Since
• Using the Duane Model
• since ?0.3

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Determination of Reliability Growth Test Time
Example (continued)

• But
• and
• so that
• or
• so that the model is

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Determination of Reliability Growth Test Time
Example (continued)

• To find the test time required to grow the MTBF
  to ?0
• set
• so that
• and

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Determination of Reliability Growth Test Time
Example (continued)

• Since
• at t21373.4 hours,

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Determination of Reliability Growth Test Time
Example (continued)

• If the initial MTBFI(t) is interpreted to be at
  t1 hour, then
• and
• so that
• or
• so that the model is

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Determination of Reliability Growth Test Time
Example (continued)

• To find the test time required to grow the MTBF
  to ?0
• set
• so that
• and

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Determination of Reliability Growth Test Time
Example (continued)

• Since
• at t213.747 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
• Plotting the estimated MTBF Growth Curves
• Procedures from MIL-HDBK-189, Feb. 13. 1981

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Least Squares for the Duane Model

• Since MTBFc(t)Kt?,
• And for simplicity in the calculations, let
• so that
• Transforming the data (ti, MTBFc(ti)) to (xi, yi)
  for i1, 2, , n use the method of least squares
  to estimate the equation y?0 ?1x?

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Least Squares for the Duane Model

• Then the best fit Duane model is
• MTBFc(t)Kt?
• where ke and ab1

b0
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Least Squares for estimating the Duane Model
Parameter

• The Least squares estimates of ?0 and ?1 are

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Example - 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

tc MTBFC
12.5 2.0
21 2.8
35 3.5
60 4.8
100 6.0
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Example - Reliability Growth Data Analysis -
continued

• 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?

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Solution
First plot the calculated cumulative MTBF values
vs failure times
MTBFC(t)
MTBFi(t)
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Solution
tc xi xi2 MTBFc yi xiyi
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 17.83 66.26 6.34 24.01
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Solution
Using the calculations from chart 27
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Solution
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Solution

• At t200 hours, the system failure rate becomes a
  constant

Failures per hour