IPL Lognormal

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

The pdf of the lognormal distribution is given by:

(6)

$images\8_6_6.gif$

where,

$images\tdash2.gif$ = ln(T),

T = times-to-failure,

and,

$images\tdash.gif$ = mean of the natural logarithms of the times-to-failure,

$images\otdash.gif$ = standard deviation of the natural logarithms of the times-to-failure.

The median of the lognormal distribution is given by:

(7)

$images\8_6_7.gif$

The IPL-lognormal model pdf can be obtained first by setting $images\tu.gif$ = L(V) in Eqn. (30). Therefore,

$images\8_6_1.gif$

$images\8_6_3.gif$

Thus,

(8)

$images\8_6_8.gif$

Substituting Eqn. (8) into Eqn. (6) yields the IPL- lognormal model pdf or,

$images\8_6_2.gif$

IPL-Lognormal Statistical Properties Summary

The Mean

· The mean life of the IPL-lognormal model (mean of the times-to-failure), $images\tline.gif$ , is given by:

(9)

$images\8_611_9.gif$

· The mean of the natural logarithms of the times-to-failure, $images\tdash.gif$ , in terms of $images\tline.gif$ and $images\ot.gif$ is given by:

$images\8_611_1.gif$

The Standard Deviation

· The standard deviation of the IPL-lognormal model (standard deviation of the times-to-failure), $images\ot.gif$ , is given by:

(10)

$images\8_612_10.gif$

· The standard deviation of the natural logarithms of the times-to-failure, $images\otdash.gif$ , in terms of $images\tline.gif$ and $images\ot.gif$ is given by:

$images\8_612_1.gif$

The Mode

· The mode of the IPL-lognormal is given by:

$images\8_613_1.gif$

IPL-Lognormal Reliability

The reliability for a mission of time T, starting at age 0, for the IPL-lognormal model is determined by:

$images\8_614_1.gif$

or,

$images\8_614_2.gif$

Reliable Life

The reliable life, or the mission duration for a desired reliability goal, $images\tr2.gif$ is estimated by first solving the reliability equation with respect to time, as follows,

$images\8_615_1.gif$

where,

$images\8_615_2.gif$

and,

$images\8_615_3.gif$

Since $images\tdash2.gif$ = ln(T) the reliable life, $images\tr2.gif$ , is given by:

$images\8_615_4.gif$

Lognormal Failure Rate

The lognormal failure rate is given by:

$images\8_62_1.gif$

Parameter Estimation

Maximum Likelihood Estimation Method

The complete IPL-lognormal log-likelihood function is composed of two summation portions,

$images\8_631_1.gif$

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.

· $images\otdash.gif$ is the standard deviation of the natural logarithm of the times-to-failure (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 $images\otdash2.gif$ , $images\khat.gif$ , $images\nhat.gif$ so that $images\votline.gif$ = 0, $images\vk.gif$ = 0 and $images\vn2.gif$ = 0, where (for FI = 0),

$images\8_631_2.gif$

and,

$images\8_631_3.gif$