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.
FIis 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:
See Also:
Inverse Power Law Relationship
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