International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 4

Number 4

December 2010

379

1 INTRODUCTION

1.1 Narrow fairways

The vessel traffic on narrow fairways is subject to

different restrictions: speed limit, overtaking ban,

passing ban and others. When ships must go one by

one they must maintain minimum distance between

each other. This distance is specific for each basin,

for example on the Świnoujście – Szczecin fairway,

the minimum distance between successive vessels is

equal to 2 cable.

1.2 Vessel traffic intensity

The intensity of vessel traffic is measured by a num-

ber of vessels passing in a time unit (Jagniszczak &

Uchacz 2002, Gucma 2003). When ships report in-

dividually and independently of one another, the in-

tensity can be describing by Poisson distribution

(Ciletti 1978, Fujii 1977, Montgomery & Runger

1994). In the case when vessel traffic is disturbed,

the density can be determined by using the convolu-

tion method. In earlier works (Kasyk 2006) author

presented solutions of different problems using par-

ticular parts of the convolution method. And this

paper is the first application of full convolution

method worked out by author (Kasyk 2008).

2 DETERMINATION OF INTENSITY

2.1 Component random variables

According with the convolutions method (Kasyk

2008, Nowak 2002) it’s necessary to isolate particu-

lar random variables. The time difference between

leavings the fairway section with the disturbance, by

successive ships is equal to:

( ) ( )

BA B A

DT X Y Y W W=+−+ −

(1)

where X denotes the waiting time for the reporting

of the successive fairway unit in none disturbance

traffic; Y denotes the time necessary to change of

vessel traffic parameters; W is the time necessary to

cover the fairway section on which the order to

maintain minimum distance between successive ves-

sels exist. The indexes A and B by names of random

variables denotes realisations of particular variables

for different successive units.

The variable X has an exponential distribution

(Ciletti 1978, Fujii 1977, Gucma 2003, Kasyk 2004,

Nelson 1995). In this paper the variable Y has a

normal distribution (Kasyk 2006). When the ship is

forced to sail after the more slowly unit, she must

reduce her own speed. The longest time necessary to

cover the fairway section on which the order to

maintain minimum distance exist is equal to d/v

where d is the length of this section and v

is the

average velocity in this section. While the shortest

time of covering this fairway section amounts d/v

max

where v

max

is the highest velocity in this section. On

narrow fairways, usually the average velocity

An Influence of the Order to Maintain

Minimum Distance Between Successive Vessels

on the Vessel Traffic Intensity in the Narrow

Fairways

L. Kasyk

Maritime University of Szczecin, Szczecin, Poland

ABSTRACT: All vessel traffic regulations disturb the randomness of the vessel traffic stream. In this paper

the disturbing factor is the order to maintain minimum distance between successive vessels. The intensity of

the disturbed vessel traffic has been determined. To achieve this goal the convolution method has been used.

Next the connection between traffic stream parameters and this disturbed intensity has been analysed.

380

doesn’t differ much from the maximum velocity.

Hence the variable W can be described by an uni-

form distribution on the interval from d/v

max

to d/v

2.2 Probability distribution of vessel traffic

intensity

Using all operations of the convolution method

(Kasyk 2008), p.d.f. of variable 1/T has been deter-

mined. This variable, as the inverse of the time be-

tween leavings the fairway section by successive

ships, denotes the number of ships leaving the spe-

cial section in the time unit. This is a continuous var-

iable and its probability density function f(x) is giv-

en by the form presented below. In this form the

function erf(z) appears. It is the integral of the

Gaussian distribution, given by:

( )

erf e

z dt

−

∫

(2)

The function erfc(z) is given by:

( ) ( )

erfc 1 erfzz= −

( )

22 2 2

22 22

14 4

4 exp

8 14 1

exp exp erf

1 11 1

2 erf 2 erfc

exp exp 2 erfc

fx r

rx x

rx rx

x xx x

σσ

ππ

σσ

λσ λ λ





−−





= + +⋅











−+

−

  



⋅ − −+

  



  



−+

     

+− − + +

     

     



+ −− ⋅





( )

2 122 1

erfc exp erfc

2 21

exp 2 erfc 2 exp

1 21

erfc 2 exp erfc

rx rx

xx x

λσ

λσ λ λσ

σσ

λ λσ

λλ

λσ λ λσ

σσ





−











 

−− −+



++ +

 





 



++ − ⋅







    

⋅ −− + +



    

    



(3)

Integrating the function f(x) in corresponding limits

we obtain the probability mass function of the varia-

ble I (the vessel traffic intensity after leaving the

fairway section with the order to maintain minimum

distance):

( )

0,5

()

P I n f x dx

−

= =

∫

(4)

3 ANALYSIS OF DEPENDENCE DENSITY

FUNCTION ON TRAFFIC PARAMETERS

3.1 Traffic parameters

Function f(x) depends on three parameters: λ, σ and

the difference r = (b – a). 1/λ is the mean of the var-

iable X. σ is the standard deviation of the variable Y

and the interval [a , b] is the range of the variable

W. Figure 1 presents the dependence of f(x) on the

parameter λ, with established σ and r.

All parameters have been examined in ranges cor-

responding with real conditions. Hence r is located

between 0.1 hour and 2 hours, σ stays within the

range from 0.01 hour to 1 hour and λ is from the in-

terval [0.1/h, 10/h].

Figure 1. Dependence of function f(x) on parameter λ.

Fig. 2 presents the dependence of the function f(x)

on the parameter σ, with established λ and r.

Figure 2. Dependence of function f(x) on parameter σ.

Figure 3 presents the dependence of the function f(x)

on the parameter r, with established λ and σ.

381

Figure 3. Dependence of function f(x) on parameter r.

Function f(x) changes little for different values σ

and r (a bit more for σ). With the change of value of

λ the function f(x) changes a lot. Especially when λ

closes to 0, the curve f(x) has greater values and it

has maximum for the argument closer 0.

3.2 Comparison between disturbed intensity and

random intensity

The vessel traffic intensity on the exit of the fairway

section with the order to maintain minimum distance

is different than the vessel traffic intensity on the en-

trance to this section. The greatest differences ap-

pear in the case when the exponential distribution

parameter has value greater than 1 (the higher value

of λ the bigger differences between intensities) and

values of parameters σ and r are high (Fig.4). The

closer 0 λ, the less differences between intensities.

And when σ and r close to 0, then density function

curves of intensities almost coincide (Fig.5).

Figure 4. Difference between intensities for large λ

Figure 5. Difference between intensities for λ closing to 0

In above figures the probability density function of

the vessel traffic intensity on the entrance to the

fairway section on which the order to maintain min-

imum distance between successive vessels exist, is

marked by dashed line.

3.3 Extreme case

In the case, when there are so many ships that they

sail one by one with the minimum distance d

min

be-

tween each other, then the intensity is equal to:

min

3600

= ⋅

(5)

where d

min

is expressed in metres; the average vessel

speed v

is expressed in metres per second.

4 CONCLUSIONS

Intensity of the disturbed vessel traffic, as a number

of reports in a time unit, has been approximated by

continuous random variable 1/T. Applying the con-

volution method the density function of variable 1/T

has been determined.

If disturbances in fairway vessel traffic are big

(values of parameters σ and r are high), then there

are large differences between the vessel traffic inten-

sity on the exit of the fairway section with the order

to maintain minimum distance and the vessel traffic

intensity on the entrance to this section.

For practical uses, the random variables separated

in this model, should be verified with measurements

or simulations.

REFERENCES

Ciletti M. 1978. Traffic Models for use in Vessel Traffic Sys-

tems, The Journal of Navigation 31

Fujii Y. 1977. Development of Marine Traffic Engineering in

Japan, The Journal of Navigation 30

0.2

0.4

0.6

0.8

1.0

0.5

1.0

1.5

2.0

382

Jagniszczak I. & Uchacz W. 2002. Model of simulated barge

traffic at the Lower Odra. Scientific Papers of Maritime

University of Szczecin 65

Gucma L. 2003. Ships Traffic Intensities in Szczecin Port and

on the Szczecin – Swinoujscie Waterway (in Polish), Scien-

tific Papers of Maritime University of Szczecin 70: 95-115.

Kasyk L. 2004. Empirical distribution of the number of ship

reports on the fairway Szczecin – Swinoujscie, XIV-th In-

ternational Scientific and Technical Conference The Part of

navigation in Support of Human Activity on the Sea. Gdy-

nia: Naval Academy.

Kasyk L. 2006. Process of Ship Reports after Covering a Spe-

cial Fairway Section, 10

International Conference

TRANSCOMP 2006. Radom: The Publishing and Printing

House of the Institute for Sustainable Technologies

Kasyk L. 2008. Convolutions of Density Functions as a Deter-

mination Method of Intensity of Disturbed Vessel Traffic

Stream, 12

International Conference TRANSCOMP 2008.

Radom: The Publishing and Printing House of the Institute

for Sustainable Technologies

Nowak R. 2002. Statystyka dla fizyków. Warszawa: Wydawnic-

two Naukowe PWN

Montgomery D. C. & Runger G.C. 1994. Applied Statistics and

Probability for Engineers, New York: John Wiley & Sons,

Inc.

Nelson P. 1995. On deterministic developments of traffic

stream models, Transportation Research Part B 29: 297-

302