Civil Engineering Reference
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state changes is exponential; and, therefore, the Markov property holds
(Howard, 2007). The third group of models is based on the full stochastic
description of the process. In these models, time intervals are not necessarily
exponential and shock distributions may be also arbitrary; e.g., see Sánchez-
Silva
et al.
(2011). These models are described in the following sections.
15.3.3 Hazard rate-based stochastic model
,
P
) with respect to which all random vari-
ables that will be used in this model are measurable. Consider that the
performance of the model throughout its lifetime is defi ned by a unique
threshold,
k*
. Then, the system will be abandoned or intervened every time
the remaining capacity/resistence,
V
(
t
), falls below
k*
, i.e.,
V
(
t
)
Defi ne a probability space (
Ω
,
k
* (Fig.
15.1). Furthermore, in a system that is in operation, failure (reconstruction)
or preventive maintenance can be modeled as a fi rst passage problem
defi ned in terms of the instantaneous intervention rate. This is, the probabil-
ity that if the system has survived for a time
t
(i.e., has not being intervened),
it will not survive an additional time d
t
; this is also commonly referred to
as the
hazard function
. The hazard function, expressed as the rate
<
λ
(
t
), is a
renewal counting process:
(
)
ft T
−
∑
()
=
n
λ
t
⋅
1
[15.5]
Tt
<≤
T
tT
−
n
n
+
1
n
∫
()
1
−
fu u
d
n
≥
0
0
where
T
n
is the sum of times between shocks (Fig. 15.2). At this point, it is
assumed that interventions occur only at shock times. The rate defi ned in
equation 15.5 does not provide information about the remaining capacity/
resistence of the system. Then, assuming that the magnitude of damage and
the shocks' occurrence times are independent, the
instantaneous interven-
tion rate
can be computed as (Sánchez-Silva
et al.
, 2011):
()
=
()
[
()
>−
]
d
Pt
λ
tPD t
v
k
* d
t
[15.6]
s
0
where
D
s
(
t
) is the accumulated damage up to time
t
. Thus, if d
G
(
x
) describes
the probability of having a shock size (i.e., damage size) between
x
and
x
+
d
x
, the instantaneous intervention rate after
n
shocks is:
∞
⎡
⎣
⎤
⎦
(
)
=
∫
()
( )
d
Ptn
,
λ
t Gx
d
d
t
[15.7]
(
)
Vtk n
,
*
,
k
* and
V
(
t
,
n
) is the capacity of the system after
n
shocks. The complexity of the solution of equation 15.7 lies in computing
the lower limit of the integral, which depends on the number of shocks
n
and the distribution of the sum of shock sizes (i.e., accumulated damage)
where
V
(
t
,
k*
,
n
)
=
V
(
t
,
n
)
−
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