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,

Note that this is a 2-parameter model. The failure rate (the parameter of the exponential distribution) of the model is simply λ = KVn and is only a function of stress.

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

IPL-Exponential Statistical Properties Summary

Mean or MTTF

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

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, for the IPL-exponential model is given by:

Mode

The mode, for the IPL-exponential model is given by:

Standard Deviation

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

IPL-Exponential Reliability Function

The IPL-exponential reliability function is given by:

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

or:

Conditional Reliability

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

Reliable Life

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

or:

Parameter Estimation

Maximum Likelihood Parameter Estimation

Substituting the inverse power law relationship into the exponential log-likelihood equation yields:

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.
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).
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 , so that = 0 and = 0, where:

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