International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 1
Number 2
June 2007
199
On Ship Systems Multi-state Safety Analysis
P. Dziula, M. Jurdzinski, K. Kolowrocki & J. Soszynska
Gdynia Maritime University, Gdynia, Poland
ABSTRACT A multi-state approach to defining basic notions of the system safety analysis is proposed. A
system safety function and a system risk function are defined. A basic safety structure of a multi-state series
system of components with degrading safety states is defined. For this system the multi-state safety function is
determined. The proposed approach is applied to the evaluation of a safety function, a risk function and other
safety characteristics of a ship system composed of a number of subsystems having an essential influence on
the ship safety. Further, a semi-markov process for the considered system operation modelling is applied. The
paper also offers an approach to the solution of a practically important problem of linking the multi-state
system safety model and its operation process model. Finally, the proposed approach is applied to the
preliminary evaluation of safety characteristics of a ship system in varying operation conditions.
1 INTRODUCTION
Taking into account the importance of the safety and
operating process effectiveness of technical systems
it seems reasonable to expand the two-state approach
to multi-state approach in their safety analysis
(Dziula, Jurdzinski, Kolowrocki & Soszynska 2007).
The assumption that the systems are composed of
multi-state components with safety states degrading
in time gives the possibility for more precise analysis
and diagnosis of their safety and operational
processes’ effectiveness. This assumption allows us
to distinguish a system safety critical state to exceed
which is either dangerous for the environment or
does not assure the necessary level of its operational
process effectiveness. Then, an important system
safety characteristic is the time to the moment of
exceeding the system safety critical state and its
distribution, which is called the system risk function.
This distribution is strictly related to the system
multi-state safety function that is a basic
characteristic of the multi-state system. Determining
the multi-state safety function and the risk function
of systems on the base of their components’ safety
functions is then the main research problem.
Modelling of complicated systems operations’
processes is difficult mainly because of large number
of operations states and impossibility of precise
describing of changes between these states. One of
the useful approaches in modelling of these
complicated processes is applying the semi-markov
model (Grabski 2002). Modelling of multi-state
systems’ safety and linking it with semi-markov
model of these systems’ operation processes is the
main and practically important research problem of
this paper. The paper is devoted to this research
problem with reference to basic safety structures of
technical systems (Soszynska 2005, 2006) and
particularly to safety analysis of a ship series system
(Jurdzinski, Kolowrocki & Dziula 2006) in variable
operation conditions. This new approach to system
safety investigation is based on the multi-state
system reliability analysis considered for instance in
(Aven 1985, Hudson & Kapur 2985, Kolowrocki
2004, Lisnianski & Levitin 2003, Meng 1993, Xue
& Yang 1995) and on semi-markov processes
modelling discussed for instance in (Grabski 2002).
200
2 BASIC NOTIONS
In the multi-state safety analysis to define systems
with degrading components we assume that:
−
n
is the number of system's components,
− E
i
, i = 1,2,...,n, are components of a system,
− all components and a system under consideration
have the safety state set {0,1,...,z},
,1≥z
− the safety state indexes are ordered, the state 0
is the worst and the state z is the best,
− T
i
(u), i = 1,2,...,n, are independent random
variables representing the lifetimes of components
E
i
in the safety state subset {u,u+1,...,z}, while
they were in the state z at the moment t = 0,
− T(u) is a random variable representing the
lifetime of a system in the safety state subset
{u,u+1,...,z} while it was in the state z at the
moment t = 0,
− the system and its components safety states
degrade with time t,
− E
i
(t) is a component E
i
safety state at the moment
t,
).,
0 ∞∈<t
− S(t) is a system safety state at the moment t,
).,
0 ∞∈<t
The above assumptions mean that the safety states
of the system with degrading components may be
changed in time only from better to worse. The way
in which the components and the system safety states
change is illustrated in Figure 1.
Fig. 1. Illustration of a system and components safety states
changing
The basis of our further considerations is a system
component safety function defined as follows.
Definition 1. A vector
s
i
(t
, ⋅
) = [s
i
(t,0), s
i
(t,1),..., s
i
(t,z)],
),,0 ∞∈<t
,,...,2,1 ni =
where
s
i
(t,u) = P(E
i
(t) ≥ u | E
i
(0) = z) = P(T
i
(u) > t)
for
u = 0,1,...,z,
,,...,2,1 ni =
is the
probability that the component E
i
is in the state
subset
},...,1,{ zuu +
at the moment t,
),,0 ∞∈<t
while it was in the state z at the moment t = 0,
is called the multi-state safety function of a
component E
i
.
Similarly, we can define a system multi-state
safety function.
Definition 2. A vector
s
n
(t
, ⋅
) = [s
n
(t,0), s
n
(t,1),..., s
n
(t,z)],
),,0 ∞∈<t
where
s
n
(t,u) = P(S(t) ≥ u | S(0) = z) = P(T(u) > t) (1)
for
),,0 ∞∈<t
u = 0,1,...,z, is the probability that the
system is in the state subset
},...,1,{ zuu +
at the
moment t,
while it was in the state z at
the moment t = 0, is called the multi-state safety
function of a system.
Under this definition we have
s
n
(t,0) ≥ s
n
(t,1) ≥ . . . ≥ s
n
(t,z),
).,0 ∞∈<t
Further, if we introduce the vector of probabilities
p(t
, ⋅
) = [p(t,0), p(t,1),..., p(t,z)],
),,0 ∞∈<t
where
p(t,u) = P(S(t) = u | S(0) = z)
for
),,0 ∞∈<t
u = 0,1,...,z, is the probability that the
system is in the state u at the moment t,
),,0 ∞∈<t
while it was in the state z at the moment t = 0, then
s
n
(t,0) = 1, s
n
(t,z) = p(t,z),
(2)
and
p(t,u) = s
n
(t,u) – s
n
),1,( +ut
,1,...,1,0 −= zu
(3)
).,0 ∞∈<t
Moreover, if
s
n
(t,u) = 1 for t ≤ 0, u = 1,2,...,z,
then
m(u) =
∫
∞
0
,),( dtut
n
s
u = 1,2,...,z, (4)
is the mean value of the system lifetime in the safety
state subset
},,...,1,{ zuu +
while
,)]([)()(
2
umunu −=
σ
u = 1,2,...,z, (5)
201
where
,),(2)(
0
dtuttun
n
s
∫
=
∞
u = 1,2,...,z, (6)
is the standard deviation of the system lifetime in the
state subset
},...,1,{ zuu +
and moreover
)(um
=
∫
∞
0
,),( dtutp
u = 1,2,...,z, (7)
is the mean value of the system lifetime in the state u
upon that the integrals (4)-(5) and (6) are
convergent.
Additionally, according to (2)-(4) and (7), we get
the following relationship
),1()()( +−= umumum
,1,...,1,0 −= zu
).()( zmzm =
Close to the multi-state system safety function, its
basic characteristic is the system risk function
defined as follows.
Definition 3. A probability
r(t) = P(S(t) < r | S(0) = z) = P(T(r) ≤ t),
),,0 ∞∈<t
that the system is in the subset of states worse than
the critical state r, r ∈{1,...,z} while it was in the
state z at the moment t = 0 is called a risk function of
the multi-state system.
Under this definition, from (1), we have
r(t) = 1 - P(S(t) ≥ r | S(0) = z) = 1 - s
n
(t,r), (8)
and, if
τ
is the moment when the risk exceeds
a permitted level
δ
,
,1,0 >∈<
δ
then
=
τ
r
),(
1
δ
−
where r
)(
1
t
−
, if it exists, is the inverse function of
the risk function r(t) given by (8).
3 BASIC SYSTEM SAFETY STRUCTURES
The proposition of a multi-state approach to
definition of basic notions, analysis and diagnosing
of systems’ safety allows us to define the system
safety function and the system risk function. It also
allows us to define basic structures of the multi-state
systems of components with degrading safety states.
For these basic systems it is possible to determine
their safety functions. Further, as an example, we
will consider a series system. Other safety structures
can be defined and analysed similarly.
Definition 4. A multi-state system is called a series
system if it is in the safety state subset
},...,1,{ zuu +
if and only if all its components are in this subset of
safety states.
Corollary 1. The lifetime T(u) of a multi-state series
system in the state subset
},...,1,{ zuu +
is given by
T(u) =
)}({min
1
uT
i
ni≤≤
, u = 1,2,...,z.
The scheme of a series system is given in Figure 2.
Fig. 2. The scheme of a series system
It is easy to work out the following result.
Corollary 2. The safety function of the multi-state
series system is given by
),( ⋅t
n
s
= [1,
)1,(t
n
s
,...,
),( zt
n
s
],
),,0 ∞∈<t
where
),( ut
n
s
=
∏
=
n
i
i
uts
1
),(
,
),,0 ∞∈<t
u = 1,2,...,z.
From Corollary 2, we immediately get the following
result.
Corollary 3. If components of the multi-state series
system have exponential safety functions, i.e., if
s
i
(t
, ⋅
) = [1, s
i
(t,1),..., s
i
(t,z)],
),,0 ∞∈<t
where
])(exp[),( t
uuts
ii
λ
−=
for
),,0 ∞∈<t
0)( >u
i
λ
,
u = 1,2,...,z,
,,...,2,1 ni =
then its safety function is given by
),( ⋅t
n
s
= [1,
)1,(t
n
s
,...,
),( zt
n
s
],
where
),( ut
n
s
=
])(exp[
1
∑
−
=
n
i
i
tu
λ
for
u = 1,2,...,z.
202
4 SHIP SAFETY IN CONSTANT OPERATION
CONDITIONS
We preliminarily assume that the ship is composed
of a number of main subsystems having an essential
influence on its safety. These subsystems are
illustrated in Figure 3.
Fig. 3. Subsystems having an essential influence on ship’s
safety
On the scheme of the ship presented in Figure 3,
there are distinguished her following subsystems:
1
S
– a navigational subsystem,
2
S
– a propulsion and controlling subsystem,
3
S
– a loading and unloading subsystem,
4
S
– a hull subsystem,
5
S
– a protection and rescue subsystem,
6
S
– an anchoring and mooring subsystem,
7
S
– a social subsystem.
In our further ship safety analysis we will omit
the social subsystem
7
S
and we will consider its
technical subsystems
1
S
,
2
S
,
3
S
,
4
S
,
5
S
and
6
S
only.
According to Definition 1, we mark the safety
functions of these subsystems respectively by
vectors
s
i
(t
, ⋅
) = [s
i
(t,0), s
i
(t,1),..., s
i
(t,z)],
),,0 ∞∈<t
,6,...,2,1=i
with co-ordinates
s
i
(t,u) = P(S
i
(t) ≥ u | S
i
(0) = z) = P(T
i
(u) > t)
for
),,0 ∞∈<t
u = 0,1,...,z,
,6,...,2,1=i
where T
i
(u),
i = 1,2,...,6, are independent random variables
representing the lifetimes of subsystems S
i
in the
safety state subset {u,u+1,...,z}, while they were in
the state z at the moment t = 0 and S
i
(t) is a
subsystem S
i
safety state at the moment t,
).,0 ∞∈<t
Further, assuming that the ship is in the safety
state subset {u,u+1,...,z} if and only if all its
subsystems are in this subset of safety states and
considering Definition 4, we conclude that the ship
is a series system of subsystems
1
S
,
2
S
,
3
S
,
4
S
,
5
S
,
6
S
with a scheme presented in Figure 4.
Fig. 4. The scheme of a structure of ship subsystems
Therefore, the ship safety is defined by the vector
),(
6
⋅ts
= [
)0,(
6
ts
,
)1,(
6
ts
,...,
),(
6
zts
],
),,0 ∞∈<t
with co-ordinates
),(
6
uts
= P(S(t) ≥ u | S(0) = z) = P(T(u) > t)
for
),,0 ∞∈<t
u = 0,1,...,z, where T(u) is a random
variable representing the lifetime of the ship in the
safety state subset {u,u+1,...,z} while it was in the
state z at the moment t = 0 and S(t) is the ship safety
state at the moment t,
),,0 ∞∈<t
and according to
Corollary 2, is given by the formula
),(
6
⋅ts
= [1,
)1,(
6
ts
,...,
),(
6
zts
],
),,0 ∞∈<t
(9)
where
),(
6
uts
=
∏
=
6
1
),(
i
i
uts
,
),,0 ∞∈<t
u = 1,2,...,z. (10)
Applying (9)-(10), we can find
– the mean value of the system lifetime in the safety
state subset
},,...,1,{ zuu +
m(u) =
∫
∞
0
6
,),( dtuts
u = 1,2,...,z,
– the standard deviation of the system lifetime in
the state subset
},...,
1,{ zuu +
,)]([)()(
2
umunu −=
σ
u = 1,2,...,z,
where
,),(2)(
6
0
dtuttun s
∫
=
∞
u = 1,2,...,z,
– the mean values of the ship lifetimes in the
particular states
),1()()( +−= umumum
,1,...,1,0 −= zu
).
()( zmzm =
Moreover, if the safety critical state is r, r
∈{1,...,z}, then the ship risk function is given by
203
r(t) = 1 - P(S(t) ≥ r | S(0) = z)
= 1 –
6
s
(t,r),
),,0 ∞∈<t
(11)
and, if
τ
is the moment when the risk exceeds
a permitted level
δ
,
,1,0 >∈<
δ
then
=
τ
r
),(
1
δ
−
where r
)(
1
t
−
is the inverse function of r(t) given by
(11).
5 SHIP OPERATION PROCESS
Technical subsystems
1
S
,
2
S
,
3
S
,
4
S
,
5
S
,
6
S
indicated in Figure 3 are forming a general ship
safety structure presented in Figure 4. However, the
ship safety structure and the ship subsystems safety
depend on her changing in time operation states.
Considering basic sea transportation processes the
following operation ship states have been specified:
1
z
– loading and unloading of cargo,
2
z
– route planning,
3
z
– leaving and entering the port,
4
z
– navigation at restricted water areas,
5
z
– navigation at open sea waters.
In this case the ship operation process Z(t) may be
described by (Dziula, Jurdzinski, Kolowrocki &
Soszynska 2007):
− the vector of probabilities of the process initial
operation states
,)]0([
51xb
p
− the matrix of the probabilities of the process
transitions between the operation states
55
][
x
bl
p
,
where
0)( =tp
bb
for
,5,...,2,1=b
− the matrix of the conditional distribution
functions
55
)]([
x
bl
tH
of the lifetimes
,
bl
θ
,lb ≠
of
the process lifetimes
,
bl
θ
,lb ≠
in the operation
state
b
z
when the next operation state is
,
l
z
where
)()( tPtH
blbl
<=
θ
for
,5,...,2,1, =lb
,lb ≠
and
0)( =tH
bb
for
.5,...,2,1=b
Under these assumptions, the lifetimes
bl
θ
mean
values are given by
][
blbl
EM
θ
=
∫
=
∞
0
),(ttdH
bl
,5,...,2,1, =lb
.lb ≠
(12)
The unconditional distribution functions of the
lifetimes
b
θ
of the ship operation process
)(tZ
at the
operation states
,
b
z
,5,...,2,1=b
are given by
)(tH
b
=
∑
=
5
1
),(
l
blbl
tHp
.5,...,2,1=b
The mean values E[
b
θ
] of the unconditional
lifetimes
b
θ
are given by
][
bb
EM
θ
=
=
∑
=
v
l
blbl
Mp
1
,
,5,...,2,1=b
where
bl
M
are defined by (12).
Limit values of the transient probabilities at the
operation states
)(tp
b
= P(Z(t) =
b
z
) ,
),,0 +∞∈<t
,5,...,2,1=b
are given by
b
p
=
)(
lim
tp
b
t ∞→
=
,
5
1
∑
=l
ll
bb
M
M
π
π
,5,...,2,1=b
where the probabilities
b
π
of the vector
51
][
xb
π
satisfy the system of equations
∑
=
=
=
5
1
.1
]][[][
l
l
blbb
p
π
ππ
6 SHIP SAFETY IN VARIABLE OPERATION
CONDITIONS
We assume as earlier that that the ship is composed
of
6=n
subsystems
,
i
S
,6,...,2,1
=i
and that the
changes of the process
)(tZ
of ship operation states
have an influence on the ship subsystems safety and
on the ship safety structure as well (Dziula,
Jurdzinski, Kolowrocki & Soszynska 2007). Thus,
we denote the conditional safety function of the ship
subsystem
i
S
while the ship is at the operational
state
,
b
z
,5,...,2,1=b
by
),(
)(
⋅ts
b
i
= [1,
),1,(
)(
ts
b
i
),2,(
)(
ts
b
i
...,
),(
)(
zts
b
i
], (13)
where for
),,0 ∞∈<t
,5,...,2,1=b
,,...,2,1 zu =
),)()((),(
)()(
b
b
i
b
i
ztZtuTPuts =>=
(14)
and the conditional safety function of the ship while
the ship is at the operational state
,
b
z
,5,...,2,1=b
by
204
),(
)(
⋅t
b
b
n
s
= [1,
),1,(
)(
t
b
b
n
s
),2,(
)(
t
b
b
n
s
...,
),(
)(
zt
b
b
n
s
], (15)
where for
),,0 ∞∈<t
,5,...,2,1=b
},6,5,4,3,2,1{∈
b
n
,,...,
2,1 zu =
),,(
)(
ut
b
b
n
s
).)()((
)(
b
b
ztZtuTP =>=
(16)
The co-ordinate
),(
)(
uts
b
i
defined by (14) of the
safety function (13) is the conditional probability
that the subsystem
i
S
lifetime
)(
)(
uT
b
i
in the state
subset
},...,1,{ zuu +
is not less than t, while the
process Z(t) is at the ship operation state
.
b
z
Similarly, the co-ordinate
),(
)(
ut
b
b
n
s
defined by (16)
of the safety function (15) is the conditional
probability that the ship lifetime
)(
)(
uT
b
in the state
subset
},...,1,{ zuu +
is not less than t, while the
process Z(t) is at the ship operation state
.
b
z
In the case when the ship operation time is large
enough, the unconditional reliability function of the
system is given by
),(
6
⋅ts
= [1,
),1,(
6
ts
),2,(
6
ts
...,
),
(
6
zts
],
,0≥t
where
),(
6
uts
))(( tuTP >=
),(
)(
5
1
utp
b
b
n
b
b
s
∑
≅
=
(17)
for
,0≥t
},6,5,4,3,2,1{∈
b
n
,,...,2,1 zu =
and
)(uT
is
the unconditional lifetime of the ship in the safety
state subset
}.,...,1,{ zuu +
The mean values and variances of the ship
lifetimes in the safety state subset
},...,1,{ zuu +
are
,)()}([)(
5
1
)(
∑
≅=
=b
b
b
umpuTEum
,,...,2,1 zu =
(18)
where
∫
=
∞
0
)()(
,),()( dtutum
bb
b
n
s
for
},6,5,4,3,2,1{∈
b
n
,,...,2,1 zu =
and
∫
−=
∞
0
2
6
,)]([),(2)]([ umdtuttuTD s
,
,...,2,1 zu =
and
),(
)(
ut
b
b
n
s
is given by (16) and
),(
6
uts
is given by
(17).
The mean values of the system lifetimes in the
particular safety states
,u
are
),1()()( +−= umumum
,1,...,2,1 −= zu
),()( zm
zm =
where
),(u
m
,,...,2,1 zu =
are given by (18).
Moreover, if the safety critical state is r, r
∈{1,...,z}, then the ship risk function is given by
r(t) = 1 –
6
s
(t,r),
),,0 ∞∈<t
(19)
where
),(
6
rts
a is given by (17), and if
τ
is the
moment when the risk exceeds a permitted level
δ
,
,1,0 >∈<
δ
then
=
τ
r
),(
1
δ
−
where r
)(
1
t
−
is the inverse function of r(t) given by
(19).
7 CONCLUSIONS
In the paper the multi-state approach to the analysis
and evaluation of systems’ safety has been
considered. Theoretical definitions and preliminary
results have been illustrated by the example of their
application in the safety evaluation of a ship system.
The ship safety structure used in the application is
very general and simplified and the subsystems
safety precise data are not know at the moment and
therefore the results may only be considered as an
illustration of the proposed methods possibilities of
applications in ship safety analysis. However, the
obtained evaluation may be a very useful example in
simple and quick ship system safety characteristics
evaluation, especially during the design and when
planning and improving her operation processes
safety and effectiveness.
The results presented in the paper can suggest that
it seems reasonable to continue the investigations
focusing on the methods of safety analysis for other
more complex multi-state systems and the methods
of safety evaluation related to the multi-state systems
in variable operation processes (Soszynska 2005,
2006) and their more adequate applications to the
ship transportation systems and processes (Dziula,
Jurdzinski, Kolowrocki & Soszynska 2007).
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th
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205
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th
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