Reliability HotWire Issue 106, December 2009
Hot Topics

Reliability Demonstration Tests for Repairable Systems

Reliability demonstration tests for non-repairable systems are well developed and regularly used by reliability engineers. A tool called the Design of Reliability Tests (DRT) that helps users design such tests is included in Weibull++. For non-repairable systems or components, reliability demonstration tests are used to demonstrate the reliability at a given time with a given confidence level. For repairable systems, users are more interested in demonstrating the MTBF or the number of failures/repairs during a given time period. In RGA 7, a new DRT tool for repairable systems has been designed for this purpose.

Before we discuss the details of the DRT for repairable systems, we will briefly introduce some concepts regarding reliability growth analysis during product development and repairable system analysis for fielded systems. Before a product is launched to the field, it usually needs to go through several development stages. The following figure shows these stages.

Figure 1: Product development stages

For the stages in Figure 1, customers only care about the final achieved MTBF at the final development stage and the MTBF curve during field use. Even when the achieved MTBF at the end of development is high, if it decreases very quickly during field use, customers will not be satisfied with the performance of the product. Therefore, operational tests and accelerated tests are usually conducted after the development stages to study the field reliability. Usually, a variation of the NHPP model, such as the Crow-AMSAA model, is used to model the repairable systems in the field. The Crow-AMSAA model is:


  • : is the instantaneous failure intensity at time t.

  • and : are the model parameters.

The instantaneous MTBF is:

The cumulative failure intensity and MTBF are:

Based on the Crow-AMSAA model and the cumulative binomial equation, a DRT tool for repairable system has been developed.[1] From the above equations, you can see that the instantaneous and cumulative failure intensity and MTBF are correlated by the model parameters. In the DRT tool in RGA 7, either of them can be used as the demonstrated reliability metric. We will use an example to illustrate how to use the DRT tool in RGA 7 to design a demonstration test for a repairable system.

A customer is planning to buy a fleet of equipment. Before he makes the decision to purchase the equipment from a particular manufacturer, he wants the manufacturer to demonstrate that the number of failures per system in a 10 year period will not be greater than 20, with a confidence level of 80%.

This requirement can be explained in terms of the cumulative MTBF. The manufacturer must demonstrate that the cumulative MTBF is 10/20 = 0.5 year with a confidence level of 80%. Assume the manufacturer knows that the failure process in the field can be modeled by a Crow-AMSAA model with β = 1.5 and only three months are available for the test. How many samples are necessary to demonstrate the required MTBF?

The DRT tool in RGA 7 can be used to design the test. Enter all the above information into the DRT tool as shown below.

Figure 2: Using the DRT to determine the number of test units

In Figure 2, we can see that the assumed beta is 1.5. The required cumulative MTBF is 0.5 at the end of 10 years. The confidence level is 80%. The available test time per unit is 0.25 year. Another input called "Allowable Failures" also affects the final result. If the total number of failures or repairs is assumed to be 0, the calculated number of units is 21.

Therefore, 21 samples must be tested for 3 months to demonstrate the required MTBF if no failures are allowed.

If the number of units is known and the test time needs to be determined, the DRT also can show results as test time per unit. The required test time with 2 allowable failures and 20 available test units is shown below.

Figure 3: Using the DRT to determine the test time per unit

The DRT also provides a plot to show how the test time changes with different numbers of allowable failures. Click the Plot button and enter the required information as seen in the Table/Plot Setup area on the right side of the Repairable Systems DRT Results window.

Figure 4: Plot settings and table for test time

After entering the data in the Table/Plot Setup area, click the Plot icon and the following plot appears.

Figure 5: Plot for Test Time vs. Number of Test Units

Figure 5 has six different curves corresponding to six different numbers of allowable failures. From these curves, users easily can identify a test time and test unit combination that can meet the design requirement. For example, if there are 20 units available for testing and only 2 failures are allowed, the required test time per unit is 0.4855 years. This is the same as the result in Figure 3.


In this article, we discussed an example using a new utility in RGA 7 to design a reliability demonstration test for repairable systems. This tool is similar to the DRT in Weibull++ for non-repairable systems. More details, including the theory behind the calculations, can be found in ReliaSoft's reference books.[1, 2]


[1] ReliaSoft Corporation, Reliability Growth and Repairable System Data Analysis Reference. Tucson: ReliaSoft Publishing, Version 7, 2009.
[2] ReliaSoft Corporation, Life Data Analysis Reference. Tucson: ReliaSoft Publishing, Version 7, 2005.

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