In the uncertainty growth projection process, we utilize information about the calibration history of the subject parameter to develop a reliability model. This reliability model provides a means for determining how the subject parameter bias uncertainty grows with time since calibration.
This means that the uncertainty surrounding the measured value we report will increase with time until the next calibration.
If we have access to a reliability modeling application, we can identify the appropriate reliability model and acquire the model’s characteristics and enter this information into UncertaintyAnalyzer’s Subject Parameter Reliability Model Worksheet.
Alternatively, we can enter an elapsed time, a beginning-of-period (BOP) reliability and an end-of-period (EOP) reliability for the calibration interval. For certain models, we must also enter an average-over-period (AOP) reliability. These values apply to the subject parameter's population and are based on service history records or engineering knowledge.
The Subject Parameter Reliability Model Worksheet has eight reliability models to choose.
Each model is defined by a mathematical equation with characteristic coefficients. An
applicable reliability model must be chosen based on knowledge about the stability of the subject parameter over time.
The application of each of the reliability models available in UncertaintyAnalyzer is de-scribed below along with information needed to implement each of them.
7.4.1 Exponential Model. The exponential reliability model is defined by the mathematical equation
(7-29) ae bt
t R( )= −
where R(t) is the in-tolerance probability at time t and a and b are the model coefficients.
The exponential model is useful for parameters whose failure probability is not a function of time interval T, beginning at some time t, is the same as the probability of going out-of-tolerance in the same time interval T, beginning at some other time t'.
To implement the exponential model you need to know either of the following:
1. The value of the model coefficients, a and b.
2. The beginning of period (BOP) in-tolerance probability and the end of period (EOP) in-tolerance probability.
7.4.2 Mixed Exponential Model. The mixed exponential reliability model is defined by the mathematical equation
b
b t at
R
⎟⎠
⎜ ⎞
⎝⎛ +
= 1 ) 1
( (7-30)
where R(t) is the in-tolerance probability at time t and a and b are the model coefficients.
The mixed exponential model is useful for parameters whose out-of-tolerance behavior depends on a number of constituent parameters, each of which can be modeled with the exponential model.
To implement the mixed exponential model you need to know either of the following:
1. The value of the model coefficients, a and b.
2. The BOP and EOP in-tolerance probabilities.
7.4.3 Weibull Model. The Weibull reliability model is defined by the mathematical equation
(7-31)
( )btc
ae t R( )= −
where R(t) is the in-tolerance probability at time t and a, b and c are the model coefficients.
The Weibull model is useful for parameters that go out-of-tolerance as a result of gradual wear or decay.
To implement the Weibull model you need to know either of the following:
1. The value of the model coefficients, a, b and c.
2. The BOP and EOP tolerance probabilities and the average-over period (AOP) in-tolerance probability.
7.4.4 Gamma Model. The gamma reliability model is defined by the mathematical equation
( ) ( )
6 1 2
)
( 2 3
bt bt bt
t ae
R bt
+ +
+
= − (7-32)
where R(t) is the in-tolerance probability at time t and a and b are the model coefficients.
The gamma model is useful for parameters that go out-of-tolerance in response to some number of events, such as being activated and deactivated.
To implement the gamma model you need to know either of the following:
1. The value of the model coefficients, a and b.
2. The BOP and EOP in-tolerance probabilities.
7.4.5 Mortality Drift Model. The mortality drift reliability model is defined by the mathematical equation
( 2) )
(t ae bt ct
R = − + (7-33)
where R(t) is the in-tolerance probability at time t and a, b and c are the model coefficients.
The mortality drift model is useful for parameters that are characterized by a slowly varying out-of-tolerance rate.
To implement the mortality drift model you need to know either of the following:
1. The value of the model coefficients, a, b and c.
2. The BOP, AOP, and EOP in-tolerances.
7.4.6 Warranty Model. The warranty reliability model is defined by the mathematical equation
)
1 (
) 1
( a t b
t e
R −
= + (7-34)
where R(t) is the in-tolerance probability at time t, and a and b are the model coefficients.
The warranty model is useful for parameters that tend to stay in-tolerance until reaching a well-defined cut-off time, at which point, they go out-of-tolerance.
To implement the warranty model you need to know either of the following:
1. The value of the model coefficients, a and b.
2. The BOP and EOP in-tolerance probabilities.
7.4.7 Random Walk Model. The random walk reliability model is defined by the mathematical equation
⎟⎠
⎜ ⎞
⎝
⎛
= +
bt erf a
t
R 1
)
( (7-35)
where R(t) is the in-tolerance probability at time t, and a and b are the model coefficients.
The random walk model is useful for parameters whose values fluctuate in a purely random way with respect to magnitude and direction (positive or negative).
To implement the random walk model you need to know either of the following:
1. The value of the model coefficients, a and b.
2. The BOP and EOP in-tolerance probabilities.
7.4.8 Restricted Random Walk Model. The restricted random walk reliability model is defined by the mathematical equation
( )
⎟⎟⎠⎞
⎜⎜
⎝
⎛
−
= +
e−ct
b erf a
t
R 1
) 1
( (7-36)
where R(t) is the in-tolerance probability at time t, and a, b, and c are the model coefficients.
The restricted random walk model is similar to the random walk model, except that parame-ter fluctuations are confined within a restricted region around a mean or nominal value.
To implement the restricted random walk model you need to know either of the following:
1. The value of the model coefficients, a, b, and c.
2. The BOP, AOP, and EOP in-tolerances.
CHAPTER 8