IPL Weibull

The pdf for the Inverse Power Law relationship and the Weibull distribution is given next.

The IPL-Weibull model can be derived by setting $images\etan.gif$ = L(V), yielding the following IPL-Weibull pdf,

$images\8_4_1.gif$

This is a three-parameter model. Therefore it is more flexible but it also requires more laborious techniques for parameter estimation. The IPL-Weibull model yields the IPL-exponential model for $images\betab.gif$ = 1.

IPL-Weibull Statistical Properties Summary

Mean or MTTF

The mean, $images\tline.gif$ , (also called MTTF), of the IPL-Weibull relationship is given by:

$images\8_411_1.gif$

where $images\gamma.gif$ is the gamma function evaluated at the value of $images\1b.gif$ .

Median

The median, $images\tu.gif$ of the IPL-Weibull relationship is given by:

(3)

$images\8_412_3.gif$

Mode

The mode, $images\twave.gif$ of the IPL-Weibull relationship is given by:

(4)

$images\8_413_4.gif$

Standard Deviation

The standard deviation, $images\ot.gif$ of the IPL-Weibull relationship is given by:

$images\8_414_1.gif$

IPL-Weibull Reliability Function

The IPL-Weibull reliability function is given by:

$images\8_415_1.gif$

Conditional Reliability Function

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

$images\8_416_1.gif$

or,

$images\8_416_2.gif$

Reliable Life

For the IPL-Weibull relationship, the reliable life, $images\tr3.gif$ , of a unit for a specified reliability and starting the mission at age zero is given by:

(5)

$images\8_417_5.gif$

IPL-Weibull Failure Rate Function

The IPL-Weibull failure rate function, $images\lambdal.gif$ (T), is given by:

$images\8_418_1.gif$

Parameter Estimation

Maximum Likelihood Estimation Method

Substituting the inverse power law model into the Weibull log-likelihood function yields,

$images\8_421_1.gif$

where:

· $images\fe.gif$ is the number of groups of exact times-to-failure data points.

· $images\ni2.gif$ is the number of times-to-failure data points in the $images\ith.gif$ time-to-failure data group.

· $images\betab.gif$ is the Weibull shape parameter (unknown, the first of three parameters to be estimated).

· K is the IPL parameter (unknown, the second of three parameters to be estimated).

· n is the second IPL parameter (unknown, the third of three parameters to be estimated).

· $images\vi.gif$ is the stress level of the $images\ith.gif$ group.

· $images\tsubi.gif$ is the exact failure time of the $images\ith.gif$ group.

· S is the number of groups of suspension data points.

· $images\nlinei.gif$ is the number of suspensions in the $images\ith.gif$ group of suspension data points.

· $images\tlinei.gif$ is the running time of the $images\ith.gif$ suspension data group.

The solution (parameter estimates) will be found by solving for $images\betab.gif$ , K, n so that $images\vbeta.gif$ = 0, $images\vk.gif$ = 0 and $images\vn2.gif$ = 0, where,

$images\8_421_2.gif$

Example 1

Consider the following times-to-failure data at two different stress levels.

$images\8_5example.gif$

The data set was analyzed jointly and with a complete MLE solution over the entire data set using ReliaSoft's ALTA. The analysis yields,

$images\betahat.gif$ = 2.61647,

$images\khat.gif$ = 0.00102241,

$images\nhat.gif$ = 1.32729123.