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