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Systems Reliability Growth Modeling
and Analysis
Dr. Jerrell T. Stracener, SAE Fellow
Leadership in Engineering
EMIS 7305/5305
Systems Reliability, Supportability and Availability Analysis
Systems Engineering Program
Department of Engineering Management, Information and Systems

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Reliability Development Test Definitions
• MIL-STD-785 Definition
A series of tests conducted to disclose deficiencies
and to verify that corrective actions will prevent
recurrence in the operational inventory (also
known as TAAF testing)
• Definition - Narrow View
A reliability development test (RDT) is a dedicated
and specifically designed and conducted test under
usage and maintenance environments that the
equipment is expected to experience during its
service life

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Reliability Development Test Definitions - continued
• Definition - General View
A reliability development test is any test where
results are designed to improve equipment
reliability

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Reliability Growth Test Objective and Purpose
• Objective - identify reliability problems early
in the development phase and incorporate
corrective action to preclude recurrence during
service usage
• Purpose - improve the reliability of production
equipment by identifying and correcting
deficiencies in the design, selection of parts,
and manufacturing processes

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•A technique for reliability development and growth that
requires that a series of tests be conducted, problems
identified and analyzed, and corrective actions be taken.
•TAAF and other reliability growth techniques require a
closed loop feedback and corrective action system and
follow a common problem solving process:
1. Detect failures
2. Feed back problems to designer
3. Analyze and redesign to correct problem
4. Incorporate redesign into system
5. Verify fix by operating system
Test, Analyze and Fix (TAAF)

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The TAAF Process
• Test analyze and fix (TAAF) is a philosophy
that can be applied to any test
- Data for all failures/problems should
be used for TAAF
- This approach decreases the likelihood
of major problems in service
• Data from each failure/problem from every
test conducted on an item should be analyzed
for corrective action, positive fix and verification
of the fix

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TAAF Benefits
TAAF provides the means to accelerate design
and reliability maturity through the corrective
actions taken for design performance/reliability
problems identified.
TAAF provides advanced information related
to how the design will work in the field during
the early years of deployment.

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Selection Criteria for RDT
• Major contributor to system (end-item) failure rate
• New development or application
• High unit cost
• Experience indicates need for improvement

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Reliability Test Length
Fixed length
- 400 to 10,000 hours (chamber hours)
various tests over past 10 years mostly avionics
and electronics
- Easy to price
- Avoids numbers game of when to stop testing
Truncated
- Stop testing when corrective action rate
reaches a specified level
- Difficult to price
- Analysis of data becomes more important

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Reliability Growth Management
• Planning the reliability growth necessary to
meet program objectives as a function of
program schedule
• Planning of tasks and program resources
necessary to achieve the planned growth
- Reliability test data
- Failure analysis
- Corrective action
• Assessment and forecast of reliability growth
relative to plan
- Provides early warning of program
reliability problem

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Reliability Growth
• Reliability growth is the positive improvement
of system reliability through the systematic and
permanent removal of the reasons for failure
• Reliability improvements can be achieved
through testing to identify deficiencies and/or
weaknesses, followed by positive actions to
correct them. Test, analyze, and fix (TAAF) is
one process for achieving reliability growth.

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Reliability Growth - continued
• Growth curves are management tools; they
show the manager where the system reliability
is now and where is must go
• Air Force Regulation 800-18, Air Force reliability
and maintainability program, requires the use of
reliability growth curves to track improvements
to achieve reliability requirements

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Reliability Growth Curves
•Decision makers can use reliability growth curves
to assess the need to reallocate resources or
change schedules to achieve R&M requirements

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DOD Policy on Reliability Growth and Testing
• DOD 5000.40 requires that a period of testing
be scheduled with each intermediate milestone
for purposes of finding design and manufacturing
defects
• AFR 800-18 is intended to implement the
requirements of DOD 5000.40 and implies
reliability growth by such statements as ‘terms
are expressed in mature system values along
with interim thresholds’

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DOD Policy on Reliability Growth and Testing
• MIL-STD-785 states that a properly balanced
reliability program will emphasize ESS and
RD/GT. It also notes that RD/GT must not
include accept/reject criteria because the
contractors interest in passing a test would be
in conflict with the purpose of doing a RD/GT
MIL-HDBK-781

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Reliability Growth Models
• AMSAA/Duane
• ARIMA
• ARINC
• Aroef
• Barlow-Scheuer
• Cox-Lewis
• Endless-Burn-In (EBI)
• Exponential single term
• Extended Cox-Lewis
• General Random Point
Process
• Gompertz
• Homogeneous Poisson
process models
• IBM
• Linear
• Lloyd-Lipow
• Logistic
• Modified Duane
• Modified Geometric
• Modified Gompertz
• Simple exponential

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Reliability Growth Model Types
Engineering Statistical
models models
Deterministic Models Poisson Process Time Series
models
Endless-Burn-In AMSAA/Duane ARIMA
Cox-Lewis
Modified Duane

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The Duane Model
•In 1964 J.T Duane, with General Electric Company, published a
paper in the IEEE Transactions on Aerospace (Vol. 2, No. 2, April
1964) titled “Learning Curve Approach to Reliability Monitoring.”
•Duane formulated a mathematical relationship for forecasting and
monitoring reliability improvement as a function of cumulative
time.
•The model was based on concluding, from analysis of data, that a
straight line provided a reasonably good fit to cumulative MTBF
vs. cumulative time when plotted on a log-log scale.

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Reliability Growth Analysis
• To meet reliability goals, a technique must be
used to track reliability and signal when corrective
action should be taken
• J.T. Duane
(1964) noticed
that in many
cases this
relationship
followed a
straight
line on a
log-log plot

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Jet Engine Control System Reliability - Growth

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The Duane Model
The Duane Model can be formulated in two different,
but mathematically equivalent, ways: MTBF as a
Function of time and failure rate as a function of time.
In terms of MTBF, the Duane Model is:
where MTBFc(t) is the cumulative MTBF at cumulative
time t, t is cumulative time, and  and  are
parameters of the model.
  α
C κt
t
MTBF 

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The Duane Model
The parameter  is the reliability improvement, or
growth, rate and  is cumulative MTBF at t=1 hour.
Notice that the graph of the Duane Model on log-log
scale is a straight line since
log MTBF(t) = log k +  log t
which is of the form
y = a + bx

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The Duane Model
Since
where rc=the cumulative number of failures in time t,
 
c
C
r
t
t
MTBF 
 
α
1
α
C
c
t
κ
1
κt
t
t
MTBF
t
r






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The Duane Model
Then
dt
drc
(t)
MTBF
1
κt
α
-
1
α)t
-
(1
κ
1
)
t
κ
1
(
dt
d
i
α
α
α
1






(t)
MTBF
κt c
α

α
1
(t)
MTBF
(t)
MTBF c
I


But since

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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


1
t
MTBFi
(t)
MTBF
1
1
C




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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
where
C(t) is the cumulative failure rate after test time t
k* 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 t
κ
(t)
λ 
(t)
α)λ
(1
t
α)κ
(1
(t)
λ
C
-α
*
i





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The Duane Model
The initial MTBF or failure rate at the beginning of
RDT depends on a number of other factors including
type of equipment (electronics, structure, etc.)
complexity of the design and equipment operation,
technology, in terms of MTBF of failure rate,
depends on the same factors as does the starting
value, and this in addition depends on management
FRACAS and TAAF implementation.

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Failure Rate and MTBF

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AMSAA/Duane Model - plotted with group data

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AMSAA/Duane Model - plotting the data and solving for  and 

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AMSAA/Duane Model - plotting by MTBF