Using the Power Law Model for Data Analysis in RGA
[Editor's Note: This article has been updated
since its original publication to reflect a more recent
version of the software interface.]
The Power Law model is a popular method for analyzing the reliability of
complex repairable systems in the field. In this article we first give a brief
introduction to the Power Law model and we then give an example
that shows how to use Power Law model
in RGA to estimate the conditional
reliability of a group of systems.
Introduction to the Power Law
In the real world, some systems consist of many components. A failure of one
critical component would bring down the whole system. After the component is
repaired, the system has been repaired. However, because there are many other
components still operating with various ages, the system is not put back into a
like new condition after the repair of the single component. The repair of a
single component is only enough to get the system operational again, which means
the system reliability is almost the same as that before it failed. For example, a
car is not as good as new after the replacement of a failed water pump. This kind
of repair is called minimal repair. Distribution theory does not
apply to the failures of a complex system; the intervals between failures of a
complex repairable system do not follow the same distribution. Rather, the sequence
of failures at the system levels follows a nonhomogeneous Poisson process
(NHPP).
When the system is first put into service, its age is 0. Under the NHPP, the
first failure is governed by a
distribution F(x)
with failure rate r(x). Each succeeding
failure is
governed by the intensity function u(x) of
the process. Let
t be the age of the system and assume
that Δt is very small. The probability that a
system of age t fails between
t
and t + Δt is given by the intensity
function u(t)Δt; the failure
intensity u(t) for the NHPP has the same
functional form as the failure rate governing the first system failure. If the
first system failure follows the Weibull distribution, the failure rate is:
and the system intensity function is:
This is the Power Law model. The Weibull distribution governs the first system
failure and the Power Law model governs each succeeding system failure. The Power
Law mean value function is:
Here T is the system operation end
time and N(T) is the number of failures
over time 0 to T.
Parameter Estimation
For the Power Law model, there are two parameters
λ
and β. The MLE estimation for them is given by:
where:
 K is the number of systems under
study.
 q is the index of
the q^{th} system under
observation from time S_{q}
to T_{q}.
 N_{q} is the number of failures
experienced by the q^{th} system.
 X_{i,q} is the age of this system
at the i^{th} occurrence of
failure.
If X_{N}_{i,q} = T_{q}
then the data on the q^{th} system is said
to be failure terminated and T_{q}
is a random variable with N_{q} fixed. If
X_{N}_{i,q} < T_{q}
then the data on the q^{th} system is said
to be time terminated and N_{q} is
a random variable.
In general, these equations cannot be solved explicitly
for
and ,
but must be solved by iterative procedures.
If S_{1} = S_{2} = ... = S_{
q} = 0 and
T_{1} = T_{2} = ... =
T_{q}(q = 1,2,...,K), then the MLE
for
and
are in closed form:
Conditional Reliability
By using the Power Law model, it is easy to estimate the probability that the
system will survive to age t + d without failure
given that the current system age is t. That is,
the equation to get the mission reliability for a system of
age t and mission
time d is:
Confidence Bounds for Reliability
There are two kinds of reliability confidence bounds available
in RGA: the Fisher Matrix confidence bounds and the Crow confidence
bounds.
Fisher Matrix Confidence Bounds
The Fisher Matrix confidence bounds on reliability are given by:
where:
The variance can be calculated using the Fisher Information Matrix
where Λ is the natural loglikelihood function.
Crow Confidence Bounds
The Crow confidence bounds on reliability with failure terminated data are given
by:
where:
The values of ρ_{1}
and ρ_{2} can be obtained by finding the
solution c
to
for
and , respectively.
where:
The Crow confidence bounds on reliability with time terminated data are given
by:
where:
The values of Π_{1}
and Π_{2} can be obtained by finding the
solution x
to
and , respectively.
I_{1}(.) is the modified Bessel
function of order 1.
Example
The following example shows how to use the Power Law model
in RGA. Table 1 shows the failure times for each
unit in a sample of 11 systems in a fleet. The end time is the last recorded known
age when the analysis was performed. The end time for each unit is less than the
last failure time (if the unit has failed), thus the data set is time terminated
data.
Table 1  Failure Times Data
System ID 
End Time 
Failure Times 
System 1 
1268 
68, 1137, 1167 
System 2 
1300 
682, 744, 831 
System 3 
1593 
845 
System 4 
1421 
263, 399 
System 5 
1574 
No Failures 
System 6 
1415 
No Failures 
System 7 
1290 
598 
System 8 
1556 
No Failures 
System 9 
1426 
No Failures 
System 10 
1124 
730 
System 11 
1568 
No Failures 
Step 1: In the Data Sheet Setup window, choose the "Repairable" data type in the "Fielded"
category.
Step 2: Add the failure data to the data sheet, then click the
Calculate icon to estimate the parameters using
the MLE method with Crow
confidence bounds. As shown below, the estimated parameters
are ,
and .
Step 3: Click the QCP icon to open the Quick Calculation Pad.
Select the Reliability option and enter the inputs as shown below. Using
a Time/Stage value of 2000 and a Mission Time value of 200, and
using twosided confidence bounds with a confidence level of 0.9, the
conditional reliability is 0.8577 with an upper confidence bound of 0.9389 and a
lower confidence bound of 0.7322.
Conclusion
This article briefly introduced the Power Law model, and then described the
equations for estimating the conditional reliability and its bounds. Finally an
example using RGA provided an illustration of using the Power Law
model.
