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

),,0 ∞∈<t

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,

),,0 ∞∈<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),

),,0 ∞∈<t

(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)

),,0 ∞∈<t

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

),,0 ∞∈<t

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).

REFERENCES

Aven T., 1985. Reliability evaluation of multi-state systems

with multi-state components. IEEE Transactions on

Reliability 34, 473-479.

Dziula P., Jurdzinski M., Kolowrocki K. & Soszynska J., 2007.

On multi-state approach to ship systems safety analysis.

Proc. 12

th

International Congress of the International

Maritime Association of the Mediterranean, IMAM 2007.

205

A.A. Balkema Publishers: Leiden - London - New York -

Philadelphia - Singapore.

Dziula P., Jurdzinski M., Kolowrocki K. & Soszynska J., 2007.

Multi-state safety analysis of ships in variable operation

conditions. Proc. 12

th

International Congress of the Inter-

national Maritime Association of the Mediterranean, IMAM

2007. A.A. Balkema Publishers: Leiden - London - New

York - Philadelphia - Singapore.

Grabski F., 2002. Semi-Markov Models of Systems Reliability

and Operations. Warsaw: Systems Research Institute,

Polish Academy of Science.

Hudson J. & Kapur K., 1985. Reliability bounds for multi-state

systems with multi-state components. Operations Research

33, 735- 744.

Jurdzinski M., Kolowrocki K. & Dziula P. 2006. Modelling

maritime transportation systems and processes. Report

335/DS/2006. Gdynia Maritime University.

Kolowrocki K. 2004. Reliability of large Systems. Elsevier:

Amsterdam - Boston - Heidelberg - London - New York -

Oxford - Paris - San Diego - San Francisco - Singapore -

Sydney - Tokyo.

Lisnianski A. & Levitin G., 2003. Multi-state System Reliability.

Assessment, Optimisation and Applications. World Scientific

Publishing Co., New Jersey, London, Singapore , Hong Kong.

Meng F., 1993. Component - relevancy and characterisation in

multi-state systems. IEEE Transactions on reliability 42,

478-483.

Soszynska J., 2005. Reliability of large series-parallel system in

variable operation conditions. Proc. European Safety and

Reliability Conference, ESREL 2005, 27-30 June, 2005,

Tri City, Poland. Advances in Safety and Reliability, Edited

by K. Kolowrocki, Volume 2, 1869-1876, A.A. Balkema

Publishers: Leiden - London - New York - Philadelphia -

Singapore.

Soszynska J., 2006. Reliability evaluation of a port oil

transportation system in variable operation conditions.

International Journal of Pressure Vessels and Piping,

Vol. 83, Issue 4, 304-310.

Xue J. & Yang K., 1995. Dynamic reliability analysis of

coherent multi-state systems. IEEE Transactions on

Reliability 4, 44, 683-688.