Modelling and Application on the Extension Scale of Port Based on Queuing Theory

Minyuan Zhang

College of Logistics Engineering Wuhan University of Technology

Wuhan, P.R.China e-mail: clover_1@163.com

Kan Zhou

College of Logistics Engineering Wuhan University of Technology

Wuhan, P.R.China e-mail: jlyqxhzk@126.com

Sanyou Ji College of Logistics Engineering Wuhan University of Technology

Wuhan, P.R.China e-mail: jisanyou@126.com

ABSTRACT—The extension scale of port is analyzed in the perspective of quantitative analysis. The optimization model for increasing the number of berths is built and then solved by using the queuing theory. The optimal extension scale of port is resulted. This model is applied to the study on the extension scale of Chongqing Cuntan Port.

Keywords-Port Scale; Port Service System; Queuing Theory

I. INTRODUCTION Along with the integration and globalization of economic

and the continuous improvement of market economy, national economic and foreign trade are growing rapidly, cargo handling capacity of ports is developing continuously too. The cargo handling capacity of ports above the designated size reaching 6.91 billion ton in 2009, growing 8.2%, but growth rate declined 1.4% from that of last year. Among them, coastal port reaching 4.73 billion ton, up 7.7%; inland port reaching 2.18 billion ton， up 8.8%. Meanwhile, a serious shortage of port capacity has also become increasingly prominent. Then how to extend the existing port scale reasonablely while facing increasing cargo handling capacity, to eliminate the overload problem of the port with the least input is an urgent problem [1].

II. MATHEMATICAL MODEL OF PORT SERVICE SYSTEM Terminal service system is a typical queuing system, the

major process among it are vessel's arrival, cargo handling and other activities. The object in this system is vessel; the service equipment is all facilities in port. According to queuing theory, if the entry process, service mechanism and queuing rules are different, the queuing model is different too. According to a large number of internal and external statistical data, most of the port service process can be considered as a k/ E /M S model.

A. Input Process The arrival process of vessels follows a Poisson

distribution:

( ) , 1,2,3,!

n

nP P n e nn

λλ −= = = ⋅⋅ （1）

Wherein n is the number of daily arrival vessels; λ is the daily average number of arrival vessels; ( )P n is the probability of n vessels arrival in a day.

B. Service process After arrival, handling process of vessels will be done in

port，the service time needed for arrival vessels follows a k -Erlang distribution:

1( )( ) 0( 1)!

kk t

kk ktf t e tk

μμ μ −−= >

− （2）

When k → ∞ ，Erlang distribution is a fixed-length distribution, that is / /M M S queuing model which has complete computing equations and is very convenient to use; when 1k = ，Erlang distribution turned to be a negative exponential distribution; μ is the number of vessels served in a single berth every day.(unit: vessel / day)[2~3]。

C. Service desk numbe In port service system, service desk number is the

number of berths owned by the port, we designate it by S . Set the strength factor of system loading

/ ( * )Sρ λ μ= ，according to the / /M M S queuing model:

11

00

1 1 1{ ( ) ( ) }! ! 1

Sn S

n

Pn S

λ λμ ρ μ

−−

=

= +−∑ i （3）

0

0

1 ( ) ( )!

( )1 ( ) ( )

!

n

nn

n S

P n Sn

P P nP n S

S S

λμ

λμ−

⎧ <⎪⎪= = ⎨⎪ ≥⎪⎩

（4）

2010 Ninth International Symposium on Distributed Computing and Applications to Business, Engineering and Science

978-0-7695-4110-5/10 $26.00 © 2010 IEEE

DOI 10.1109/DCABES.2010.137

663

Wherein 0P is the probability of no vessels arrival in a day, namely the probability that every berth of the port is idle.

D. Performance indicators The main performance indicators in port services system

including: 1) The average number of vessels waiting in port

021

( * ) *( )!(1 )

S

q nn S

SL n S P PS

ρ ρρ

∞

= +

= − =−∑ （5）

2) The average number of arrival vessels

0

*s n qn

L nP L S ρ∞

=

= = +∑ （6）

3) The average residence time of vessels /s sW L λ= （7）

4) The mean waiting time of vessels /q qW L λ= （8）

5) The delay probability of arrival vessels When all berths are occupied, the probability that vessels

have to wait for service is

0 0

( )

1 ( )( )! !(1 )

nn S

n S

n Sn S

P P n S P

SP PS S S

λ ρμ ρ

∞

=

∞

−=

= ≥ =

= =−

∑

∑

∼

i （9）

The computational process of / /kM E S queuing model is complicated, the average queue length and mean waiting time of vessels can be computed according to the approximate equations [4~6].

*qE qL Lα= （10）

*qE qW Wα= （11） wherein

1 1 4 5 1(1 )( 1)2 32

k k a SSk k S a

α + − + −= + − −

III. EXTENSION SCALE COST MODEL OF PORT

A. Build the cost model In port construction, an excessive number of berths will

increase the investment, at the same time, the economic benefits of the port will be reduced for low utilization rate of berth. On the contrary, an insufficient number of berths will increase the residence time which will cause heavy losses to shipping companies and cargo owners, and then erode the competitiveness of port. The number of berths should benefit both the port and shipping companies during the planning process, to reach the level that “meet the needs without any waste”.

Considering that the port operation is complicated, the following assumptions are made:

（1）With the increasing number of berths, handling efficiency of port berths stays still;

（2）The construction and operating costs of berths, purchasing and maintenance expense of cargo handling equipment and the delay cost of vessel will not change over time.

Set S as he original number of berths, s as the added number of berths，then the existing number of berths is m S s= + , the total cost model is:

*min ( ) * *zb p s

Cf m C m C Lλμ

= + + （12）

Wherein λ is the average number of daily arrival vessels; μ is the average handling efficiency of berths;

bC is the average fixed cost of depreciation charge;

zC is the average daily operating costs of berths, including berth charges, operating costs of cargo handling equipment and wages for workmen;

pC is the delay cost for one day;

sL is the daily average number of vessels in port where the number of berths is S ;

sW is the average residence time of vessels where the number of berths is S , unit: day;

qW is the average waiting time for vessels where the

number of berths is S , unit: day;

B. Solving the model Build the queuing model according to the basic feature

of this system: input process (vessel arrival), service mechanism and queuing discipline. This model can be

solved by the queuing theory, set 1 *kλρ

μ= , then

The average utilization rate of port berths:

*mλρ

μ= （13）

The average queue length：

10 12

1 1[ (1 )( 1)2

4 5 1 *]*[ ]32 !(1 )

m

k k aLs Sk k S

S Pa m

ρ ρ ρρ

+ −= + − −

+ − +−

（14）

The average waiting time for vessels： /s sW L λ= （15）

Wherein 1

110 1

0

1[ ]! !(1 )

nmS

n

Pn mρ ρ

ρ

−−

=

= +−∑

664

By the equations , ( )f m under different numbers of

berths can be calculated. *S is the optimum solution when S satisfy the following equation.

( *) ( * 1)( * 1) ( *)

f s f sf s f s

> −⎧⎨ + <⎩

（16）

IV. APPLICATION CASE Chongqing is the western materials distribution center of

China，and the transport hub of the southwest region and the upper areas along the Yangtze River. Chongqing’s international container terminal Cuntan port is an important waterway transportation infrastructure of Chongqing’s construction. Currently, Cuntan port has seven berths in operation, including five 3000-ton container berths and two automobile Ro-Ro terminals; the designed throughput capacity of container is 700,000 TEUs / year. It is predicted that in the year 2015, container throughput in port of Chongqing will reach to 1.46 million TEUs, greater than the current designed throughput capacity, so the port extension is imperative.

According to the statistics related to port of Chongqing, The arrival events of vessels follows a Poisson distribution，the service time needed for arrival vessels follows a 3-Erlang distribution, other parameters shown in TableⅠ.

TABLE I. PARAMETERS OF TOTAL COST MODEL

Parameter λ μ bC zC pC

Value 23 25 TEU/H

46.379 thousand

Yuan

12.547 thousand

Yuan

33.312 thousand

Yuan Choose 5S = , substitute the correlation parameters in

Table Ⅰ in the equation , then solve it which can get the average utilization rate of berths: 0.95ρ = ， the daily

average number of vessels in port 16.9591sL = , the

average residence time of vessels 0.7374sW = day, the

average waiting time for vessels 0.5985qW = day. Table Ⅱ shows the total cost of berths.

TABLE II. LIST OF THE TOTAL COST

Title Fixed cost of depreciation

charge

Operating cost

Delay cost

Total cost

Value（thousand Yuan）

231.895 72.145 564.942 868.982

Assuming that s berths are added, then the current number of berths is S s+ , set m instead. According to (12), the total cost model is:

1.2547*min ( ) 4.6379* 3.3312* sf m m Lλμ

= + + (17)

The optimal number of berths can be got and this program can be calculated through marginal analysis. The final results are shown in Fig. 1 to Fig. 3.

16.9591

4.55583.5944 3.3323 3.246

0

2

4

6

8

10

12

14

16

18

5 6 7 8 9

Number of berths

Daily average number of

ships in port

Figure 1. Daily average number of vessels in port

0.7374

0.19810.1563 0.1449 0.1411

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

5 6 7 8 9

Number of berthsAverage residence time

of vessels

Figure 2. Average residence time of vessels

0.9583

0.7986

0.6845

0.5990

0.5324

0.4

0.5

0.6

0.7

0.8

0.9

1

5 6 7 8 9

Number of berths

Average utilization

rate of berths

Figure 3. Average utilization rate of berths:

As shown in the above pictures, the average utilization rate of berths, residence time of vessels and the daily average number of vessels in port decrease as the number of berths increases, when the handling efficiency is fixed. The arrival of the vessel can be served without queuing while the berths are idle. Obviously, it is beneficial to the shipping companies but consume a lot of resources of the port. Table Ⅲ shows the total cost of pot under different numbers of berths.

TABLE III. TOTAL COST OF PORT

Number of berths. 5 6 7 8 9

Total cost（

thousand Yuan）

868.982 500.159 494.512 542.159 585.6633

665

The optimal number of berths * 7S = from the above table . That is to say 2 berths should be added into the current condition to bring the lowest total cost which will be 494.512 thousand Yuan/a day. For both port and shipping companies, the cost is reduced. For the port side, the exorbitant utilization rate of berths is eased, and for the shipping companies, residence time of vessels is reduced so that the delay cost is cut down, a win-win situation is achieved.

V. CONCLUSION Using queuing theory in the process of port extension .

Build the mathematical model of the extension scale while taking the common benefit of both port and shipping companies into consideration in order to achieve the lowest total cost target. The extension scale problem is optimized to provide relatively reliable basis for determine a reasonable scale for the construction and development of port.

REFERENCES [1] Ma Zhuang. “Extended Construction of Tianjin Container Terminal”,

Shipping Management, Vol.29, No.3, April 2005, pp.1~3. [2] Zhang Jin, Liu Cuilian. “Influence Factors and Simulation on Port

Berth Service System” Journal of Dalian Maritime University (Social Sciences Edition), Vol.7, No.6, December 2008, pp.109~111.

[3] Tang Yinghui, Tang Xiaowo. Queuing Theory: Basis and Analysis, Beijing: Science Press, 2006.

[4] Chen Hongmei, Xiao Qian. “Analysis on Operating Effect of Liner Ship Based on Queuing Theory”, Vol.31, No.11, Shipping Management, November 2009, pp.20~22.

[5] Wang Yanyan Yang Haidong. “Optimal Berth Scale Based on the Math Model of the Integrated Expenses”, Vol.24, No.8, Port Engineering Technology, March 2007, pp.14~17.

[6] Lu Ziai Lin Minbiao. “On Numerical Simulation of Port Service System”, Journal of Hehai University (Natural Sciences), Vol.27, No.3, May 1999, pp.17~21

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