9.6.2 Series Systems
Figure 9.3 depicts a system of m elements in series or cascade with individual
reliabilities R 1 , R 2 , . . ., R i , . . ., R m respectively.
If any one component fails, then the complete system fails, because the system
will work only if all the components are working.
Normally, all components are independent of one another. Therefore, the reliability of each component is independent of the reliability of the other components.
In such a (series) system, the probability that the system does not fail is the
probability that every one of the m components does not fail. Each such probability
being R i , the system reliability R SYST is therefore the product of the individual
component reliabilities i.e.
R SYST ¼ R 1 R 2 . . . , R i . . . R m
ð9:2Þ
When infrastructure systems have to be highly reliable, this implies that the
corresponding system and component unreliabilities F need to be very small. The
corresponding system and component reliabilities R are therefore very close to
1, typically 0.998, 0.9999 or similar.
In such cases, using Eq. 9.2 to calculate R SYST may be cumbersome with
calculator decimal accuracy, etc. It may be easier to use a different equation with
unreliabilities.
Since R SYST ¼ 1 – F SYST and Ri ¼ 1 – F i , Eq. 9.2 can be rewritten as:
1 À F SYST ¼ 1 À F 1
ð
Þ 1 À F 2
ð
Þ. . . : 1 À F i
ð
Þ. . . : 1 À F m
ð
Þ
¼ 1 À F 1 þ F 2 þ . . . þ F i þ . . . þ F m
ð
Þ þ terms involving products of the F i
ð9:3Þ
The individual F i being very small, i.e. F i < < 1, the various terms involving the
different products of the F i are even smaller and negligible, resulting in the approximate equation:
F SYST ¼ F 1 þ F 2 þ . . . þ F i þ . . . þ F m
ð9:4Þ
i.e. the unreliability of the entire system is approximately the sum of the individual
component element unreliabilities.
And of course, R SYST ¼ 1 À F SYST will easily provide R SYST .
INPUT
OUTPUT
C 2
C 1
C i
C m
R 1
R 2
R i
R m
Fig. 9.3 Reliability of a series system (e.g. bridges, pumps in series)
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9 Quality and Reliability of Infrastructure
Figure 9.3 depicts a system of m elements in series or cascade with individual
reliabilities R 1 , R 2 , . . ., R i , . . ., R m respectively.
If any one component fails, then the complete system fails, because the system
will work only if all the components are working.
Normally, all components are independent of one another. Therefore, the reliability of each component is independent of the reliability of the other components.
In such a (series) system, the probability that the system does not fail is the
probability that every one of the m components does not fail. Each such probability
being R i , the system reliability R SYST is therefore the product of the individual
component reliabilities i.e.
R SYST ¼ R 1 R 2 . . . , R i . . . R m
ð9:2Þ
When infrastructure systems have to be highly reliable, this implies that the
corresponding system and component unreliabilities F need to be very small. The
corresponding system and component reliabilities R are therefore very close to
1, typically 0.998, 0.9999 or similar.
In such cases, using Eq. 9.2 to calculate R SYST may be cumbersome with
calculator decimal accuracy, etc. It may be easier to use a different equation with
unreliabilities.
Since R SYST ¼ 1 – F SYST and Ri ¼ 1 – F i , Eq. 9.2 can be rewritten as:
1 À F SYST ¼ 1 À F 1
ð
Þ 1 À F 2
ð
Þ. . . : 1 À F i
ð
Þ. . . : 1 À F m
ð
Þ
¼ 1 À F 1 þ F 2 þ . . . þ F i þ . . . þ F m
ð
Þ þ terms involving products of the F i
ð9:3Þ
The individual F i being very small, i.e. F i < < 1, the various terms involving the
different products of the F i are even smaller and negligible, resulting in the approximate equation:
F SYST ¼ F 1 þ F 2 þ . . . þ F i þ . . . þ F m
ð9:4Þ
i.e. the unreliability of the entire system is approximately the sum of the individual
component element unreliabilities.
And of course, R SYST ¼ 1 À F SYST will easily provide R SYST .
INPUT
OUTPUT
C 2
C 1
C i
C m
R 1
R 2
R i
R m
Fig. 9.3 Reliability of a series system (e.g. bridges, pumps in series)
268
9 Quality and Reliability of Infrastructure