Reliability HotWire

Reliability HotWire

Issue 131, January 2012

Reliability Basics

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 non-homogeneous 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(tt; 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:

Equation

and the system intensity function is:

Equation

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:

Equation

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:

Equation

Equation

where:

  • K is the number of systems under study.
  • q is the index of the qth system under observation from time Sq to Tq.
  • Nq is the number of failures experienced by the qth system.
  • Xi,q is the age of this system at the ith occurrence of failure.
  • 0ln0 is defined to be 0.

If XNi,q = Tq then the data on the qth system is said to be failure terminated and Tq is a random variable with Nq fixed. If XNi,q < Tq then the data on the qth system is said to be time terminated and Nq is a random variable.

In general, these equations cannot be solved explicitly for Equation and Equation, but must be solved by iterative procedures. If S1 = S2 = ... = S q = 0 and T1 = T2 = ... = Tq(q = 1,2,...,K), then the MLE for Equation and Equation are in closed form:

Equation

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:

Equation

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:

Equation

where:

Equation

The variance can be calculated using the Fisher Information Matrix where Λ is the natural log-likelihood function.

Equation

Crow Confidence Bounds

The Crow confidence bounds on reliability with failure terminated data are given by:

Equation

where:

Equation

The values of ρ1 and ρ2 can be obtained by finding the solution c to Equation for Equation and Equation, respectively.

where:

Equation

The Crow confidence bounds on reliability with time terminated data are given by:

Equation

where:

Equation

The values of Π1 and Π2 can be obtained by finding the solution x to Equation and Equation, respectively.

Equation

I1(.) 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.

New Data Sheet Setup

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 Equation, and Equation.

RGA Folio

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 two-sided 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.

RGA QCP

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.