Civil Engineering Reference
In-Depth Information
17.4.1 Reliability assessment
Reliability analysis provides evidence that can be used to make decisions
as to which actions should be taken by ensuring that network performance
is kept above a certain acceptable level (Konak
et al.
, 2004; Murray
et al.
,
2007). Reliability is often calculated as the complement of the failure prob-
ability by evaluating a multivariate integral of probability density functions
of relevant random variables over the failure space. Both probability func-
tions and failure space are extremely diffi cult to obtain for most realistic
networks. Therefore, an analytical solution is not available. In this section,
the basic reliability problem for network systems is presented.
Let
β
(
X
j
) be a measure of performance under state
X
j
=
{
y
1
, . . . ,
y
n
,
z
1
, . . . ,
z
m
}; with
y
i
=
1 if node
i
is in operation (
z
for the case of edges) and 0 other-
be the set of all possible scenarios (i.e., failure and full operation),
then, the system reliability under state
X
j
is defi ned as the probability of the
performance measure
wise. Let
Ω
).
A comprehensive reliability analysis requires testing the network for
every possible failure scenario. Defi ne a scenario as the vector state of the
system
X
β
(
X
j
) exceeding a threshold
Ψ
, i.e.,
P
(
β
(
X
j
) >
Ψ
{
x
1
,
x
2
, . . . ,
x
n
} in which
x
i
corresponds to the state of element
i
(i.e., full operation or failure of either a node or link). Then, if the set of
independent scenarios that causes the system to fail is
=
Φ
F
, the probability
of failure
P
f
can be computed as:
⎛
⎜
⎞
⎟
∑
∏∏
(
)
P
=
1
−
p
p
[17.1]
f
i
i
xH
∉
xH
∈
X
⊂
Φ
j
F
i
j
i
j
1 (component survives), and
p
i
is
the failure probability of the network element
x
i
. Note that the term in
parentheses is the probability of occurrence of the
j
-th failure scenario. Equa-
tion 17.1 is generic and assumes that elements fail independently, which is
not necessarily true in practical cases; more rigorous formulations including
correlation can be found elsewhere (Song and Der Kiureghian (2003)).
Computing network reliability is a diffi cult problem for which there are
two basic approaches: exact or approximate methods. Exact methods can
be further classifi ed into those focused on the enumeration of all minimal
paths or cut sets and those based on network partitioning. Some exact
methods for computing reliability are the inclusion-exclusion formula and
other enumeration methods (Bell and Ilida, 1997), optimization approaches
(Ahuja
et al.
, 1993), network reduction methods (Todinov, 2007), and closed
form recursive approaches for specifi c topologies (Duenas-Osorio and
Rojo, 2011).
Simulation algorithms provide alternative methods. Crude Monte Carlo
is a popular choice but too expensive for large networks, in particular, when
failure probabilities are very small. Better estimates can be obtained from:
where
H
j
is the set of
x
i
in
X
j
such that
x
i
=
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