Eyring-Exponential

The pdf for the Eyring relationship and the exponential distribution is given next.

The pdf of the 1-parameter exponential distribution is given by:

$images\7_3_1.gif$

It can be easily shown that the mean life for the 1-parameter exponential distribution, presented in detail in the Life Distributions chapter, is given by:

(3)

$images\7_3_3.gif$

thus,

(4)

$images\7_3_4.gif$

The Eyring-exponential model pdf can then be obtained by setting m = L(V) in Eqn. (1),

$images\7_3_2.gif$

and substituting for m in Eqn. (4),

(5)

$images\7_3_5.gif$

Eyring Exponential Statistical Properties Summary

Mean or MTTF

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

$images\7_311_1.gif$

Median

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

$images\7_312_1.gif$

Mode

The mode, $images\twave.gif$ for the Eyring-exponential relationship is $images\twave.gif$ = 0.

Standard Deviation

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

$images\7_314_1.gif$

Eyring-Exponential Reliability Function

The Eyring-exponential reliability function is given by:

$images\7_315_1.gif$

This function is the complement of the Eyring-exponential cumulative distribution function or,

$images\7_315_2.gif$

and,

$images\7_315_3.gif$

Conditional Reliability

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

$images\7_316_1.gif$

Reliable Life

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

$images\7_317_1.gif$

or,

$images\7_317_2.gif$

Parameter Estimation

Maximum Likelihood Estimation Method

The complete exponential log-likelihood function of the Eyring model is composed of two summation portions,

$images\7_321_1.gif$

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.

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

· B is the second Eyring 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 $images\ahat.gif$ and $images\bhat.gif$ so that $images\va.gif$ = 0 and $images\vb2.gif$ = 0 where:

$images\7_321_2.gif$