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Chapter 04S - Reliability
4S-17
Enrichment Module: Measurement of Reliability
In practice, there are two basic ways of measuring reliability. The most common measure of reliability
is called the failure rate. The failure rate is defined as the number of failures for a given time period
and is generally denoted by the Greek letter (lambda).
A second common measure is the time per failure generally denoted as
1
, which is the reciprocal of
failure rate. Because the failure rate is the number of failures per hour, the reciprocal measure is the
amount of time per failure. However, this reciprocal measure is a function of the type of product
manufactured. If the product is repairable (car engine) and can be reused, the appropriate measure is
the Mean Time Between Failures (MTBF). If because of the failure the product must be replaced (light
bulb), then the appropriate measure is the Mean Time to Failure (MTF).
For most products, failure rates change over time. As can be observed from Figure 4S–1 below, the
failure rates tend to be high during the infant mortality phase (early stages of the product life cycle)
due to defective parts and lack of testing. There are fewer failures in the middle stage of the product
life cycle, while there are high failure rates late in the product life cycle due to worn out components
and aging. In the textbook, we were able to estimate the probabilities and the associated failure time
periods during early and middle stages using the exponential distribution. In this section, we will try to
estimate the failure rate with limited empirical information.
Failure Rate as a Function of Time
Infant
mortality
Few (random)
failures
Time, T
Failures due
to wear-out
0
Failure rate
Chapter 04S - Reliability
4S-18
To determine the failure rate, we collect a random sample of components, determine the time of failure
for each component, and determine the estimated failure rate using the following equation:
sample in the components allfor time testingTotal
period time testinge within thfailures ofNumber
According to the above equation, we would expect the (failure rate) to be fairly high during the
infant mortality phase, lower during the middle phase, and higher during the late phase of the product
life cycle.
The denominator of the above equation (total testing time) is divided into two parts. Summing the two
parts will provide us with the total testing time. The first part is based on the sampled components that
have not failed during the testing period. The time for the first part is calculated by multiplying the
number of components that have not failed during the testing period by the total testing time. The
second part is based on the sampled components that have failed during the testing period. The time
for the second part is calculated by summing the time of failure for each component in the sample.
Summing the two times will provide us with the total testing time for all components in the sample.
Once we obtain , we can take the reciprocal to obtain either MTBF or MTF depending on whether
the product is repairable or replaceable.
To understand the computation of and MTBF/MTF, solve the following problems.
Problem 1
In an attempt to measure the reliability of a new brand of light bulbs, the manufacturer selected a
random sample of 14 light bulbs. The time of failure for these light bulbs is organized from lowest to
highest and is presented in the following table.
Bulb
Hours of Operation
before Failure
1
35
2
90
3
100
4
150
5
700
6
1010
7
1040
8
1190
9
1200
10
1250
11
1270
12
1280
13
1300
14
1450
a) What is the failure rate for a testing period of 450 hours?
b) What is the failure rate for a testing period of 900 hours?
c) What is the failure rate for a testing period of 1350 hours?
Chapter 04S - Reliability
4S-19
d) Match the failure rates determined in parts a, b, and c with the three basic stages of the
product life cycle (early, middle, and late). Do the failure rates correspond to the general idea
behind a bathtub function?
e) Which of the two time measures (MTBF or MTF) would be more appropriate in this instance
and why?
f) Determine the appropriate average amount of time per failure (either MTBF or MTF) for each
of the testing periods in parts a through c.
Problem 2
In a test of reliability of a computer component for an airplane, the reliability engineer collected a
random sample of 15 components. The following table provides a summary of the results on the
number of hours of operation. The values in the table indicate the useful operating hours before
failure. The computer manufacturer has defined the first 300 hours as the early (burn-in) period. The
useful life up to 2000 hours is classified as the middle period. Useful life from 2000 to 2500 hours is
classified as the late (wear-out) period.
Component
Hours of Operation
before Failure
1
28
2
2350
3
2370
4
1450
5
2200
6
97
7
2440
8
2490
9
2380
10
120
11
870
12
>2500
13
>2500
14
2200
15
200
a. Compute the failure rates for 300 hours, 2000 hours, and 2500 hours. Are your results
consistent with the concept of higher failure rates during burn-in and wear-out periods as
compared to the middle period?
b. Determine the MTBF for each of the three testing periods.
Chapter 04S - Reliability
4S-20
Solution to Problem 1
a.
./000821.
1501009035.)450x10(
4
450 hrfailures
hrs components
5
The failure rates correspond to the general shape of the “bathtub” function.
e. MTF is the appropriate measure because a light bulb is not a repairable product.
f.
1
MTF
.hrs 909
0011.
1
MTF
000545.
1
.hrs 218,1
000821.
1
MTF
1350
900
450
Solution to Problem 2
a.
195,24
4
)300(112001209728
4
2500
300
Chapter 04S - Reliability
4S-21
Because
300 and
2500 are larger than
2000, the results are consistent with the concept of higher
failure rates during burn-in and wear-out periods.
b.
.hrs 2.862,1
000537.
11
MTBF
.hrs 2.460,3
000289.
11
MTBF
.hrs 4.936
001068.
11
MTBF
2500
2000
300
