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:

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)

thus:

(4)

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

and substituting for m in Eqn. (4):

(5)

Eyring Exponential Statistical Properties Summary

Mean or MTTF

The mean, , or mean time to failure (MTTF) for the Eyring-exponential model is given by:

Median

The median, for the Eyring-exponential relationship is given by:

Mode

The mode, for the Eyring-exponential model is = 0.

Standard Deviation

The standard deviation, , for the Eyring-exponential model is given by:

Eyring-Exponential Reliability Function

The Eyring-exponential reliability function is given by:

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

and:

Conditional Reliability

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

Reliable Life

For the Eyring-exponential model , the reliable life, or the mission duration for a desired reliability goal tR is given by:

or:

Parameter Estimation

Maximum Likelihood Estimation Method

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

where:

and:

Fe is the number of groups of exact times-to-failure data points.
Ni is the number of times-to-failure in the ith time-to-failure data group.
Vi is the stress level of the ith 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).
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 and so that = 0 and = 0 where: