T-H Weibull

The pdf for the Temperature Humidity relationship and the Weibull distribution is given next.

By setting η = L(U,V) as given in Eqn. (1), the T-H Weibull model's pdf is given by:

T-H Weibull Statistical Properties Summary

Mean or MTTF

The mean, (also called MTTF), of the T-H Weibull model is given by:

where is the gamma function evaluated at the value of .

Median

The median, of the T-H Weibull model is given by:

(4)

Mode

The mode, of the T-H Weibull model is given by:

(5)

Standard Deviation

The standard deviation, of the T-H Weibull model is given by:

T-H Weibull Reliability Function

The T-H Weibull reliability function is given by:

Conditional Reliability Function

The T-H Weibull conditional reliability function at a specified stress level is given by:

or:

Reliable Life

For the T-H Weibull model, the reliable life, TR, of a unit for a specified reliability and starting the mission at age zero is given by:

(6)

T-H Weibull Failure Rate Function

The T-H Weibull failure rate function, λ(T, V, U), is given by:

Parameter Estimation

Maximum Likelihood Estimation Method

Substituting the T-H model into the Weibull log-likelihood function yields:

where:

and:

Fe is the number of groups of exact times-to-failure data points.
Ni is the number of times-to-failure data points in the ith time-to-failure data group.
β is the Weibull shape parameter (unknown, the first of four parameters to be estimated).
A is the T-H parameter (unknown, the second of four parameters to be estimated).
is the second T-H parameter (unknown, the third of four parameters to be estimated).
b is the third T-H parameter (unknown, the fourth of four parameters to be estimated).
Vi is the temperature level of the ith group.
Ui is the relative humidity level of the ith group.
Ti is the exact failure time of the ith group.
S is the number of groups of suspension data points.
is the number of suspensions in the ith group of suspension data points.
is the running time of the ith suspension data group.
FI is the number of interval data groups.
is the number of intervals in the ith group of data intervals.
is the beginning of the ith interval.
is the ending of the ith interval.

The solution (parameter estimates) will be found by solving for the parameters A, , b and β so that = 0, = 0, = 0 and = 0.

Example

The following data were collected after testing twelve electronic devices at different temperature and humidity conditions:

Using ALTA, the following results were obtained:

= 5.87439512,
= 0.0000597,
= 0.2805985,
= 5630.329851.

A probability plot for the entered data is shown next.

Note that three lines are plotted because there are three combinations of stresses, namely, (398K, 0.4), (378K, 0.8) and (378K, 0.4).

Given the use stress levels, time estimates can be obtained for specified probability. A life vs. stress plot can be obtained if one of the stresses is kept constant. For example, the following picture is a life vs. humidity plot at a constant temperature of 338K.

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