International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 5
Number 4
December 2011
479
1 INTRODUCTION
In operational activity of stevedoring companies, the
many cases may occur related to the situations of
risk. The main of them are listed below:
− damage of a ship’s hull or equipment during the
loading/unloading;
− cargo package’s damage in result of violation of
loading/unloading rules or rules of port’s mecha-
nisms exploitation;
− damage of cargo in result of violations of rules
of its storage at warehouse;
− failures of port’s equipment;
− exceeding of a ship’s laytime.
Appearance of above events leads to some addi-
tional expenditure for the stevedoring company,
charterer or cargo owner. As shows the international
commercial practice, many stevedoring companies
which operate in big seaports insure their responsi-
bility for safe and qualitative transshipment of cargo
(within the framework of contract responsibility) [1].
When the managers of stevedoring company
make decision concerning an insurance of its re-
sponsibility for safe transshipment of cargo it is use-
ful and even necessary to apply the methods of
probability theory and actuarial mathematics [2, 3].
At the same time the standard methods of quantita-
tive evaluation of risk proposed by mathematical
risk theory are mainly aimed at insurance compa-
nies’ profit but not at protection of commercial in-
terests of insurants. Therefore, specifics of seaports
operational activity and interrelations between ste-
vedoring company’s managers and its clients de-
mand the special methods for actuarial calculations.
The purpose of our paper is working out a method
of a risk evaluation of containers damage under their
transshipment at a seaport terminal and substantia-
tion of insurance expediency of this risk by a steve-
doring company.
2 MAIN RESULTS
Our approach is based on representation of port’s
container terminal as a queueing system of GI/G/m
type (m identical servers in parallel, infinite waiting
room, service discipline is FIFO).
We denote:
)(t
ω
be a random number of served ships in time
interval
),(0 t
;
k
γ
be a random number of containers transshipped
on/from the kth ship served in time interval (0, t);
k
ν
be a random number of damaged containers
during loading/unloading of the kth ship;
Method of Evaluation of Insurance Expediency
of Stevedoring Company’s Responsibility for
Cargo Safety
M. Ya. Postan & O. O. Balobanov
Odessa National Maritime University, Ukraine
ABSTRACT: The method of insurance expediency of stevedoring company’s responsibility for safety of con-
tainers under their transshipment at port’s terminal is proposed. This method is based on representation of
terminal as a queueing system of GI/G/m type and on comparison of the stevedoring company’s insurance
expenditures and random value of transshipped containers’ total damage (sum insured) for a given period of
time.
480
ki
Δ
be a random value of damage caused to the ith
container loaded on or unloaded from the kth ship
(estimated in money).
It is assumed that:
1 the random variables
.
,..
2
,
1
γγ
are independent
and identically distributed (i.i.d.) with the discrete
distribution
∑
≥
====
1
;1 ,..,2,1},
1
Pr{
M
M
MM
M
πγπ
)1(
2
.,..
2
,
1
νν
are the i.i.d. random variables with the
conditional binomial distribution
,,...,1,0
),1( }Pr{
Mn
pq
nM
q
n
p
n
M
CM
k
n
k
ν
=
−=
−
===
γ
(2)
where p is the probability that a damage is caused
to arbitrary container through a stevedoring compa-
ny’s fault ;
3
,...
211211
,...,
Δ
,
ΔΔ
are the i.i.d. random varia-
bles with the distribution function (d.f.)
};
11
Pr{)( x
Δ
xD ≤=
(3)
4 the sequences of random variables
...
,
2
,
1
νν
and
.
12
,
11
,..
ΔΔ
are mutually independent.
If
τ
denotes the constant loading/unloading time
of one container, than service time of the kth served
ship is the random variable
.
τγ
k
We shall consider
the steady-state regime of our queueing system func-
tioning and assume that the following stability con-
dition holds true
),
1
E /(
γτλ
m<
(4)
where
1−
λ
is the mean interarrival time of the ships.
Let us evaluate the total damage in time interval
(0, t) caused to containers by stevedoring company
.
))(( t∆
Using the above designations we can write
∑
=
∑
=
=∆
)(
1
1
. )(
t
k
k
i
ki
Δ
t
ω
ν
(5)
The financial managers of a stevedoring company
face the dilemma: to insure or not to insure the pos-
sible total damage (4) with the gross risk premium
rate c (we assume that the sum insured is
)(t∆
).
Note that t we consider as the period of insurance
policy action.
The simplest criterion of insurance expediency is:
the average profit of a stevedoring company in result
of the total damage insurance must be positive, i.e.
.
0))(E(
>−
∆
ct
t
(6)
Taking into account relations (1)-(5) and applying
to right-hand side of (5) theorem of total mathemati-
cal expectation, from (6) we have
,
11
E
1
E)(E ct
Δ
t >
νω
(7)
where
.
0
1
1
E,)(
11
E ∞
∫
∞
<
∑
≥
=∞<=
n
n
npxxdD
Δ πν
(8)
For ergodic queue (see (4))
tt
λω
=)(E
[4].
Therefore, from (7) we obtain
.
11
E
1
E c
Δ
>
νλ
(8)
More precise criterion than (8) is
,
1})(Pr{
ε
−≥>∆ ctt
(9)
where
ε
is a given small probability. For applica-
tion of criterion (9) we need to determine the d.f. of
stochastic process
.
)(t∆
For the sake of simplicity, we suppose that
,.,..2,1, == kN
k
γ
where N may be interpreted as
hold capacity of a ship (in TEU). In other words, we
assume that each ship arrives for loading/unloading
of exactly N containers. Then by theorem of total
probability, taking into account (5), mutual inde-
pendence of
)(t
ω
and
.,..
2
,
1
νν
, we can write
)10(
),(
)...
1
(
0
1
0
1
...
1
})(Pr{}0)(Pr{})(Pr{),(
x
k
nn
D
i
nN
q
i
n
p
N
n
N
k
n
k
i
i
n
N
C
k
kttxttxF
++
−
∑
=
∑
=
∏
=
×
∑
∞
=
×=+==≤∆=
ωω
where
)(
)(
x
n
D
is n-multiple convolution of d.f.
)(xD
with itself,
.1)(
)0(
≡xD
Due to the formula (10) the criterion (9) takes the
form
.)(
)...
1
(
0
1
0
1
...
})(
1
Pr{}0)(Pr{),(
ε
ωω
≤
++
−
∑
=
∑
=
∏
=
×
×=
∑
∞
=
+==
ct
k
nn
D
i
nN
q
i
n
p
N
n
N
k
n
k
i
i
n
N
C
kt
k
ttctF
(11)
In practice, N may be considered as large and p
as small quantities. Therefore, the binomial terms in
481
(11) may approximately be substituted for the Pois-
son distribution. From (11), it follows
)12( )(
)...
1
(
0
1
0
1
!
...
1
})(Pr{}0)(Pr{
,
ε
ωω
≤
++
∑
∞
=
∑
∞
=
∏
=
×
∑
∞
=
×=
−
+=
ct
k
nn
D
n
k
n
k
i
i
n
i
n
a
k
kt
ak
et
where
.Npa =
The Laplace-Stieltjes transform of d.f. (10) on
variable x is given by
(13) ,
0Re ), ,])(([
1
])(}[)(Pr{}0)(Pr{
0
1
0
1
)]([...
1
})(Pr{
}0)(Pr{),(
0
≥+=
=
∑
∞
=
+=+=
∑
=
∑
=
∏
=
−
∑
∞
=
=+
+==
∫
∞
−
=
=
st
N
qsp
Φ
k
kN
qspktt
N
n
N
k
n
k
i
i
nN
q
i
n
sp
i
n
N
C
k
kt
ttxF
x
d
sx
e
δ
δωω
δω
ω
where
∑
∞
=
==
∫
∞
−
=
1
})(Pr{
0
)( and )()(
k
kt
k
yy,t
Φ
xdD
sx
es
ωδ
is
the generating function of stochastic process’
)(t
ω
distribution,
.1≤y
In particular, from (13) we find
).(Var
2
)
11
E()(E]
2
)
11
E(
2
11
E[
2
))(E(
0
),])(([
2
2
)(Var
,)(E
11
E
0
),])(([)(E
t
Δ
Npt
Δ
p
Δ
Np
t
s
t
N
qsp
Φ
s
t
t
Δ
Np
s
t
N
qsp
Φ
s
t
ωω
δ
ωδ
+−=
=∆−
=
+
∂
∂
=∆
=
=
+
∂
∂
−=∆
One more simplification of criterion (9) may be
done by application of the Chebyshev’s inequality.
Applying this inequality, taken in modified form [5],
we obtain (under condition (6))
.
)(
2
E
2
) )((E
})(Pr{
t
ct-t
ctt
∆
∆
≥>∆
(14)
Hence, the criterion (9) may be reduced to the
simple inequality
.1
)(
2
E
2
))((E
ε
−≥
∆
−∆
t
ctt
For application of the criteria (11),(12),(14) it is
necessary to find the probabilistic distribution of
process
.
)(t
ω
It may be found by the methods of
queueing theory [6]. Below, will be considered two
particular cases of queue GI/G/m for which this dis-
tribution is known.
1 Queue of M/D/∞ type, i.e. with infinite number of
servers, the Poisson input with the rate
,
λ
and con-
stant service time. Such queueing system is good
approximation to multi-server queue if
.
mN <<
λτ
As it was shown in [7], for such system (in equilib-
rium)
..,.2,1,0 ,
!
)(
})(Pr{ =
−
==
k
t
e
k
k
t
kt
λ
λ
ω
(15)
and, consequently,
.)(E)(Var ttt
λωω
==
In this
case the condition (11) takes the following form
(16)
.)](
)...
1
(
0
1
0
1
...
0
!
)(
ε
λ
λ
≤
++
−
∑
=
∑
=
∏
=
∑
∞
=
ct
k
nn
D
i
nN
q
i
n
p
N
n
N
k
n
k
i
i
n
N
C
k
k
k
t
t-
e
From (15), it follows also that
]}.))((1[exp{) ,])(([
N
qsptt
N
qsp
Φ
+−−=+
δλδ
For inversion of this expression the known nu-
merical methods of the Laplace transform inversion
may be used [8].
The criterion (16) is too complex for calculations.
Note that in this case
ctt −∆ )(
is the compound
Poisson process with the drift c [9]. Therefore if
∞→t
, we can apply the central limit theorem for
such kind of stochastic processes [9]. Hence, instead
of (9), we have as
∞→t
(17) ,)(N}0)(Pr{
ε
≤≈≤−∆ tRctt
where
R =
×− )
11
E/( Np
Δ
c
λ
];
2
)
11
E()1(
2
11
E[
Δ
pN
Δ
Np −+×
λ
N(x) is the standard normal distribution with zero
mean and variance equals to unity.
2 One-server queue of M/D/1 type, i.e. with the
Poisson input and constant service time. For such
system the following result is valid [6]:
482
(18) ,
]
)
0
1(
0
)
0
1(
0
)
0
1)(1(
1
)1[(
)1(
2
)1(
]
)1(
)1)(
0
1(
1)[,(
*
0
)
0
,(
),(
*
θ
ρ
τλ
τλ
ρ
τθ
τθ
θ
τθ
λ
τθ
θ
τθ
λ
θ
θ
θ
+
+
−−
−
−−
−
−−+
−
−×
×
−
−
−
−
−
−
−
−
−−
+=
=
∫
∞
−
≡
Nz
ez
Nz
yez
z
y
N
ey
N
ye
N
e
N
ye
N
ez
y
Φ
dtty
Φ
t
ey
Φ
where
;1<=
τλρ
N
;]
0
)
0
1(
)1(
)
0
1(
)
0
1(
[
)
0
(
1
),(
*
0
z
Nz
e
N
e
Nz
yez
z
y
Φ
−
−−
−
−
−−
−
+×
×
−+
−
=
τλ
τθ
τλ
λ
θ
λλθθ
ρ
θ
0
z
is the unique root of the equation
])
0
(exp[
0
τλλθ
Nzyz −+−=
in the domain
.,
0Re1
>≤
θ
y
With the help of relation (18) we can determine
)(Var t
ω
and then use the criterion (14).
3 NUMERICAL RESULTS
Let us demonstrate the application of criterion (17)
for real initial data. Put
5=
λ
ships per month, t =
25 months, p = 10
-3
, and assume that
.
2
)
11
E(2
2
11
E
ΔΔ
=
The results of calculations of
probability in the formula (17) for different values of
Np and ratio
11
E/
Δ
c
are given in the Table.
Table
From these results, it follows the expedience of
insurance, for example, if Np = 0,1,
2,0
11
E/ ≤
Δ
c
or Np
>
0,1,
3,0
11
E/ ≤
Δ
c
because probability in
(17) is sufficiently small in these cases.
4 CONCLUSIONS
The real problems of risk-management concerning
the port operator’s (or stevedoring company’s) activ-
ity may be formulated and solved with application of
mathematical risk theory. The main feature of above
problems is: first of all they must be aimed at the
protection of financial state of stevedoring company
but not an insurance firm. In most cases these prob-
lems may not be solved by standard theoretical
methods and require the use of combination of dif-
ferent fields of applied probability, for example, ruin
theory, queueing and reliability theories, theory of
storage processes, etc. This is necessary for model-
ing the port’s operational activity side by side with
the corresponding financial processes [10].
For practical applications of results obtained it is
necessary to use the corresponding statistical data
concerning the cases of containers damage and val-
ues of damage, moments of ships’ arrival, etc. for a
previous period. Such information must be accumu-
lated in the data base of a stevedoring company.
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2. Grandell J (1992) Aspects of Risk Theory, Springer, Berlin
Heidelberg New York
3. Asmussen S (2001) Ruin Probabilities, World Scientific,
Singapore New Jersey London Hong Kong
4. Harris R (1974) The expected number of idle servers in a
queueing system. Operations Research 22, 6: 1258-1259
5. Feller W (1971) An Introduction to Probability Theory and
Its Applications. Vol.II. 2
nd
Ed. Jhon Wiley & Sons, Inc.,
New York London Sydney Toronto
6. Jaiswall NK (1968) Priority Queues, Academic Press, New
York London
7. Mirasol N (1963) The output of an M/G/∞ queueing sys-
tem is Poisson. Operations Research 11, 2: 282-284
8. Krylov VI, Skoblya NS (1968) Handbook on Numerical In-
version of Laplace Transform, Nauka i Tehnika, Minsk (in
Russian)
9. Prabhu NU (1997) Stochastic Storage Processes: Queues,
Insurance Risk, Dams, and Data Communications. 2
nd
Ed.
Springer, Berlin Heidelberg New York
10. Postan MYa (2006) Economic-Mathematical Models of
Multimodal Transport, Astroprint, Odessa (in Russian)
c/EΔ
11
Np
0,15
0,20
0,25
0,30
0,35
0,10
0,0436
0,0721
0,1112
0,1635
0,2327
0,15
0,0092
0,0150
0,0244
0,0384
0,0582
0,20
0,0021
0,0035
0,0058
0,0092
0,0140
0,25
0,0006
0,0009
0,0015
0,0024
0,0037
