8.4.1 Introduction
The single purpose of a reliability demonstration test is to
determine conformance to specified, quantitative reliability requirements as a
basis for qualification or acceptance; this is to answer the question, “Does the item meet or exceed (not by how
much) the specified minimum reliability requirement?”
Reliability testing involves an
empirical measurement of timetofailure during equipment operation for the
purpose of determining whether an equipment meets the established reliability
requirements. A reliability test is effectively a “sampling” test in the sense
that it is a test involving a sample of objects selected from a population. In
reliability testing, the population being measured encompasses all failures
that will occur during the life span of the equipment. A test sample is drawn
from this population by observing those failures occurring during a small
portion of the equipment's life. In reliability testing, as in any sampling
test, the sample is assumed to be representative of the population, and the
mean value of the various elements of the sample (e.g., timestofailure) is
assumed to be a measure of the true mean (MTBF, etc.) of the
population.
A sample in a reliability test
consists of a number of timestofailure, and the population is all the
timestofailure that could occur either from the one equipment or the more
than one equipment on test. The “test” equipments (assuming more than one
equipment) are considered identical and, thus, their populations are also
identical. Under the assumption of an exponential failure model
(constant
l), a test of
10 devices for 100 hours each is mathematically equivalent to a test of 1
device for 1000 hours. If all possible samples of the same number of
timestofailure were drawn from the same or identical equipment, the
resulting set of sample means would be distributed about the true MTBF
(q) of the
equipment, following a normal distribution as is shown in Figure
8.41.
Since it is not economically
feasible to test the complete population, we have to be satisfied with a
sample of the population. From the data in the sample we then make some
statement about the population parameter.
FIGURE 8.41: NORMAL
DISTRIBUTION
What we are doing is testing a statistical hypothesis: For
example, we might test
H_{0}: (null
hypothesis) q_{0} ³ 200 hours
H_{1}: (alternate hypothesis) q_{1} £ 100 hours
Based upon the test results, we either accept
H_{0} or reject it. In making our decision we have to keep
several risks in mind.
Producer’s risk
(a)
is the probability of rejecting
H_{0} when it is true (probability of rejecting good
equipment)
Consumer’s risk
(b)
is the probability of accepting
H_{0} when it is false (probability of accepting bad
equipment)
Looking at it another way, if
q_{0} and q_{1} represent the hypotheses, then the a and b errors are the hatched areas
shown in Figure 8.42A. Of course, if we could take enough samples, then the
standard deviation about each of the means would be reduced and the
a and
b errors would
also be reduced.
However, this is usually
impractical so the sample size is set as low as possible to reduce costs by
specifying the maximum acceptable a and b risks that can be associated with q_{0} and the smallest acceptable
q_{1}.
Why two values? Let’s look at our decision rule, or accept/reject criteria. We
would like it to look like Figure 8.43A.
FIGURE 8.44:
ACTUAL OPERATING
CHARACTERISTIC
CURVE
The decision rule “tends” to give the right decision, but won't
always result in an accept decision for m > 200 or a reject decision for m
< 200. Remember P_{A}
+ P_{R} = 1. Thus,
we can see that we have almost a fiftyfifty chance of accepting an m of 167
hours (0.446) and a greater than 20% chance of rejecting an m = 250 hours.
Neither the producer or consumer would be happy with this. Each would like a
lower risk probability. But since P_{A} = 1 
P_{R}, if we lower
P_{A} for m £ 200 to 0.1, we raise P_{R} for m >
200 to 1  0.1 = 0.9. What do we do now?
In order to overcome this
difficulty it is necessary to specify the reliability requirements, either
explicitly or implicitly, in terms of two MTBF values rather than a single
MTBF value. The lower value is defined as the lower test MTBF
(M_{m} or q_{1}) and the higher value is defined as the upper test
MTBF (M_{R} or q_{0}). The test plan can then be designed to give a low
probability of an accept decision
for equipment with an MTBF of m
< Mm (or q_{1}) and a low probability of reject decision when m > M_{R}.
P_{A} at m = M_{m} (or q_{1}) is the
consumers risk (b); P_{R} at m = M_{R} (or
q_{0}) is the
producer’s risk (a). Thus, specifying the two MTBF values
M_{m}(q_{1}) and
M_{R}(q_{0}) and the two
risks (a and
b) defines two
points on the OC curve as shown in Figure 8.45.
The curve on the right is the OC curve for failure rate
(a) rather than
for MTBF. l_{m} = 1/M_{m} is the maximum acceptable failure rate. l_{R} = 1/M_{R} is the designrequired (specified) failure rate with l_{R} < l_{m}.
The method used to design a
fixed time reliability (R) demonstration test is mathematically
equivalent to the method used to construct confidence limits for MTBF.
Therefore, if a fixed time R demonstration involving a test time T and an
accept number r0 provides a consumer risk of b
with respect to a minimum acceptable MTBF
(M_{m} or q_{1}), it
will be found that if the maximum allowable number of failures, r_{0},
actually occurs, the lower 100(1  b)% confidence limit for MTBF as calculated from the
test data is exactly M_{m}. For
this reason, the value (1  b), or 100(1  b)% is often called the confidence level of the demonstration test. Thus, a fixed time
R demonstration test providing a 10%
consumer risk is called “a demonstration test at a 90% confidence level,” or
is said to “demonstrate with 90% confidence that the lower test MTBF is
achieved.” This is not really correct since, technically, confidence level is
used in the estimation of a parameter while an R demonstration test is testing
a hypothesis about the parameter, m, rather than constructing an interval
estimate for m.
There are six characteristics of
any reliability demonstration test that must be specified:
(1) 

The reliability deemed to be acceptable,
R_{0}, “upper test
MTBF”

(2) 

A value of reliability deemed to be
unacceptable, R_{1}, “lower test
MTBF”

(3) 

Producer's risk, or a

(4) 

Consumer's risk, or b

(5) 

The probability distribution to be used
for number of failures or for timetofailure

(6) 

The sampling
scheme 
Another term frequently used
in connection with reliability demonstration tests should be defined here
although it is derived from two of the six characteristics. The discrimination
ratio is the ratio of upper test reliability to the lower test reliability.
R0/R1 is an additional method of specifying certain test
plans.
There are, of course, an infinite
number of possible values for the actual reliability. In the specification of
two numerical values, R_{0}
and R_{1}, the experimenter achieves the producer's risk,
a, and consumer's
risk, b, only for
those specific reliabilities.
For other values, the relationship
is:
(a) 

Probability of Acceptance ³ 1a
for R ³ R_{0}

(b) 

Probability of Acceptance
£ b
for R
£ R_{1}

(c) 

Probability of Acceptance
> b for R_{1} £ R £
R_{0} 