Bayesian Reliability Demonstration Test Design
For life tests, especially for systems like missiles,
sample size is always limited. It is a challenge to get
accurate reliability estimates with limited samples, and
Bayesian methods have been used to solve this problem.
Weibull++ 8
now offers nonparametric Bayesian
reliability demonstration test (RDT) design. Unlike traditional nonparametric test design methods, this
new feature allows engineers to integrate engineering knowledge
or subsystem testing results into system reliability test
design. In this article, we will explain the theory of
Bayesian RDT and illustrate how to use Weibull++ to
design an efficient reliability demonstration test.
Theory of Bayesian RDT
The following binomial equation is often used in
reliability demonstration test design:

(1) 
where CL is the required confidence level, r is the
number of failures, n is the total number of units on
test and R is the reliability that needs to be
demonstrated.
Eqn. (1) has four unknowns: CL, r, n and R. Given any
three of these, the remaining one can be solved. For
example, given CL, r and R, the required sample size
can be determined.
Eqn. (1) shows that the reliability is a random number with
a beta distribution. It can be rewritten as:

(2) 
As discussed in [1], Eqn. (2) also can be obtained using
the Bayesian theory when the noninformative prior
distribution 1/R is used. In general, when a beta
distribution is used as the prior distribution for
reliability R, the posterior distribution obtained from
Eqn. (1) is also a beta distribution. For example,
assuming the prior distribution is Beta(R, α_{0}, β_{0}), the posterior
distribution for R is:

(3) 
Therefore, Eqn. (3) can be used for Bayesian RDT.
For a random variable x with beta distribution Beta(x, α, β), its mean and variance are:

(4) 
If the expected value and the variance are known,
parameters α and β in the beta distribution can be solved
by:

(5) 
Example 1: Bayesian Test Design with Prior
Information from Expert Opinion
Suppose you wanted to know the reliability of a
system and the following prior knowledge of the system
is available:
 Lowest possible reliability: a = 0.80
 Most likely reliability: b = 0.85
 Highest possible reliability: c = 0.97
Based on this information, the mean and variance of
the prior system reliability can be estimated as:
Using the above two values and Eqn. (5), the prior
distribution for R is a beta distribution Beta(R, α_{0}, β_{0}) with:

(6) 
Given the above prior information, if there is 1
failure out of 20 test samples, what is the demonstrated
reliability at a confidence level of CL = 0.8? The result
is given in Figure 1.
Figure 1
 Bayesian DRT with expert opinion as prior
Figure 1 shows the demonstrated reliability is
85.103991%.
Without the prior information for system reliability,
the demonstrated reliability is 85.75745%, as given in
Figure 2:
Figure 2  DRT without expert opinion as prior
The prior expert opinion can have a significant effect
on the RDT result. From this example, we can see that
the Bayesian method and the regular nonparametric binomial
method produce similar results. However, if the reliability is very high based on prior expert opinion, the Bayesian
RDT will give very different results than the regular
one. Therefore, you must be very careful when you apply any Bayesian
method.
For example, if we change the expert opinion in this example to:
 Lowest possible reliability: a = 0.90
 Most likely reliability: b = 0.95
 Highest possible reliability: c = 0.97
and keep the rest settings unchanged, the Bayesian
RDT result is:
Figure 3  DRT with modified expert opinion as prior
Example 2: Bayesian Test Design with Prior
Information from Subsystem Tests
For each subsystem i in a system, the reliability can
also be modeled using a beta distribution. If there are
r_{i} failures out of n_{i} test samples,
R_{i} is a beta distribution
with the following cumulative distribution function:

(7) 
where s_{i} = n_{i}  r_{i} is the number of successes. Therefore, the
expected value and the variance of R_{i} are:

(8) 
Assuming that all the subsystems are connected reliabilitywise in
a series configuration, the expected value and
the variance of the system’s reliability R can then be
calculated as:

(9) 
From Eqn. (9), we can get the α and β parameters for the prior
distribution of R in Eqn. (5).
Assume a system of interest is composed of three
subsystems: A, B and C. Prior information from
tests of these subsystems is given in the table below.
Subsystem 
Number of Units (n) 
Number of Failures
(r) 
A 
20 
0 
B 
30 
1 
C 
100 
4 
Given the above information, in order to demonstrate
the system reliability of 0.9 at a confidence level of
0.8, how many samples are needed in the test? Assume the
allowed number of failures is 1.
The result is given in the figure shown next, which shows that at least 49 test units are needed.
Figure
3
 Bayesian DRT with subsystem tests as prior
Conclusion
In this article, we discussed the theory of Bayesian
reliability demonstration test design. Weibull++
was used to
solve two examples. Considering prior information about
system reliability allows us to design a better test and estimate the
system reliability more accurately. This article
shows how both expert opinion and subsystem test
information can be used to incorporate the prior distribution of
the system reliability.
Reference
[1] H. Guo, T. Jin and A. Mettas,
"Designing reliability demonstration tests for oneshot
systems under zero component failure," IEEE Transactions
on Reliability, vol. 60, no. 1, pp. 286294, March
2011.
