Systems Reliability Growth

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Systems Engineering Program
Department of Engineering Management, Information
and Systems
EMIS 7305/5305 Systems Reliability,
Supportability and Availability Analysis
Systems Reliability Growth Planning and Data
Analysis
Dr. Jerrell T. Stracener, SAE Fellow
Leadership in Engineering
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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,
• B 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
• Confidence Bounds
• Plotting the estimated MTBF Growth Curves and
  Confidence Bounds
• Procedures from MIL-HDBK-189, Feb. 13. 1981

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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

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Least Squares for the Duane
Model The Least squares estimates of ?0 and ?1
are
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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
tc MTBFC
12.5 2.0
21 2.8
35 3.5
60 4.8
100 6.0
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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?