IPL Exponential

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

The IPL-exponential model can be derived by setting m = L(V) in Eqn. (31), yielding the following IPL-exponential pdf,

$images\8_3_1.gif$

Note that this is a 2-parameter model. The failure rate (the parameter of the exponential distribution) of the model is simply $images\lambdal.gif$ = $images\kvn.gif$ and is only a function of stress.

$images\ipl-exponential_failure_rate.gif$

Fig. 4: IPL-Exponential Failure Rate function at different stress levels.

IPL-Exponential Statistical Properties Summary

Mean or MTTF

The mean, $images\tline.gif$ , or mean time to failure (MTTF) for the IPL-exponential relationship is given by:

$images\8_311_1.gif$

Note that the MTTF is a function of stress only and is simply equal to the IPL relationship (which is the original assumption), when using the exponential distribution.

Median

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

$images\8_312_1.gif$

Mode

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

$images\8_313_1.gif$

Standard Deviation

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

$images\8_314_1.gif$

IPL-Exponential Reliability Function

The IPL-exponential reliability function is given by:

$images\8_315_1.gif$

This function is the complement of the IPL-exponential cumulative distribution function,

$images\8_315_2.gif$

or,

$images\8_315_3.gif$

Conditional Reliability

The conditional reliability function for the IPL-exponential relationship is given by:

$images\8_316_1.gif$

Reliable Life

For the IPL-exponential relationship, the reliable life or the mission duration for a desired reliability goal, $images\tr2.gif$ is given by:

$images\8_317_1.gif$

or,

$images\8_317_2.gif$

Parameter Estimation

Maximum Likelihood Parameter Estimation

Substituting the inverse power law model into the exponential log-likelihood equation yields,

$images\8_321_1.gif$

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 in the $images\ith.gif$ time-to-failure data group.

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

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

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

· $images\tsubi.gif$ 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 the parameters $images\khat.gif$ , $images\nhat.gif$ so that $images\vk.gif$ = 0 and $images\vn2.gif$ = 0, where,

$images\8_321_2.gif$