Repository - Scientific Journals of the Maritime University of Szczecin - Non-Poisson character of vessel traffic...

(1)

Zeszyty Naukowe 32(104) z. 2 77

Scientific Journals

Zeszyty Naukowe

Maritime University of Szczecin

Akademia Morska w Szczecinie

2012, 32(104) z. 2 pp. 77–80 2012, 32(104) z. 2 s. 77–80

Non-Poisson character of vessel traffic

on the Szczecin−Świnoujście fairway

Lech Kasyk

Maritime University in Szczecin, Department of Mathematics

70-500 Szczecin, ul. Wały Chrobrego 1–2, e-mail: l.kasyk@am.szczecin.pl Key words: vessel traffic, random variable, mixed distributions, exponential distribution Abstract

In this paper the hypothesis about Poissonian character of vessel traffic stream from Świnoujście to Szczecin has been verified. And the hypothesis about Non-Poissonian character of vessel traffic stream from Szczecin to Świnoujście has been verified. To reach this goal, VTS data from second half of 2009 has been used. To description of non- Poissonian vessel traffic stream, for the first time, mixed distributions have been used.

Introduction

Process of vessel reporting is a Poisson process, when the vessel traffic is completely random (isn’t disturbed). But this is a theoretical situation. In fact, the vessel traffic is always more or less disturbed. Each vessel regulation disturbs the vessel traffic stream [1, 2]. There are many regulations on the Szczecin–Świnoujście fairway [1, 3] (a speed limit, an order to maintain minimum distance between successive vessels, a passing ban and other). But these disturbances not always disturb so strongly the randomness of the vessel traffic stream. As author has been shown in [4], in the first half of 2009 the vessel traffic stream from Świnoujście to Szczecin has Poissonian character and the vessel traffic stream from Szczecin to Świnoujście hasn’t Poissonian character. In this paper the vessel traffic in second half of 2009 has been considered and a form of probability density function used to description of non-Poissonian vessel traffic stream has been determined.

To find that a given process is Poisson process, a hypothesis about exponential distribution of time between reporting successive fairway units must be verified.

Vessel traffic stream from Świnoujście to Szczecin

Szczecin–Świnoujście VTS system has 11 points where times of vessel reports are registered.

In this paper reporting times at 4 points: 11_KILO-METR, CHELMINEK_N, KREPA_DOLNA, INO-UJSCIE have been considered. Let X denotes the waiting time for the reporting of the successive vessel at the given reporting point. In the second half of year 2009, at 11_KILOMETR point, 1320 vessels have been registered. Using the chi-square goodness-of-fit test [2, 5, 6] and programme

Statis-tica we find that the test statistic is equal to 7.9 and

probability p = 0.09. So we are unable to reject the hypothesis that the time between reporting of fair-way vessels has an exponential distribution. Pa-rameter  of this exponential distribution is equal to 0.0049. Figure 1 presents frequency histogram of random variable X at 11_KILOMETR point.

Fig. 1. Frequency histogram of the time between reporting at 11_KILOMETR point 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 Time [min] 0 100 200 300 400 500 600 700 800 900 1000 N u m b e r o f o b se rva tio n s Time [min] Nu m be r of ob se rv ati on s

(2)

Lech Kasyk

78 Scientific Journals 32(104) z. 2

The similar situation is at the rest considering points (Figs 2 and 3).

Fig. 2. Frequency histogram of the time between reporting at CHELMINEK_N point

Fig. 3. Frequency histogram of the time between reporting at: a) KREPA_DOLNA point and b) INOUJSCIE point

Vessel traffic stream from Szczecin to Świnoujście

Using the chi-square goodness-of-fit test and programme Statistica for all four points we must

reject the hypothesis that the time between report-ing of fairway vessels has an exponential distribu-tion. Any typical distribution used in Statistica haven’t been correspond to variable X. To find the form of probability density function of variable X, mixed distributions have been used. Probability density function of a mixed distribution has the following form:

 

x a pdf

 

D1 b pdf

 

D2

f     (1)

where:

a, b – such real numbers, that a + b = 1; pdf(Di) − probability density function of

variable with Di distribution. Procedure of determination good fitted mixed distribution to variable X consists in two stages: Stage 1: 3graphic fitting a pdf curve to graph of

data of variable X (using function

Ma-nipulate in programme Mathematica) [7];

Stage 2: using the chi-square goodness-of-fit test for mixed distribution, determined in graphic fitting.

Vessel traffic at 11_KILOMETR point

In the second half of year 2009, 1309 vessels have been registered at this point. These data on 16 intervals have been divided (each contained over 10 elements) [6]. And the chi-square goodness-of-fit test has been used. Figure 4 presents the pdf curve fitted to data at 11_KILOMETR point.

Fig. 4. Data and fitted pdf graph at 11_KILOMETR point

Using trial and error approach the probability density function of variable X has been determined. It is the mixed distribution: 0.9 (pdf of exponential distribution) +0.1 (pdf of uniform distribution). It has the following form:

 

            1100 for 0054 . 0 1100 0 for 000091 . 0 0054 . 0 006 . 0 006 . 0 x e x e x f x x (2) 0 ,0 0 0 0 8 5 ,7 1 4 3 1 7 1 ,4 2 8 6 2 5 7 ,1 4 2 9 3 4 2 ,8 5 7 1 4 2 8 ,5 7 1 4 5 1 4 ,2 8 5 7 6 0 0 ,0 0 0 0 6 8 5 ,7 1 4 3 7 7 1 ,4 2 8 6 8 5 7 ,1 4 2 9 9 4 2 ,8 5 7 1 1 0 2 8 ,5 7 1 4 1 1 1 4 ,2 8 5 7 1 2 0 0 ,0 0 0 0 1 2 8 5 ,7 1 4 3 1 3 7 1 ,4 2 8 6 1 4 5 7 ,1 4 2 9 1 5 4 2 ,8 5 7 1 1 6 2 8 ,5 7 1 4 1 7 1 4 ,2 8 5 7 1 8 0 0 ,0 0 0 0 Time [min] 0 50 100 150 200 250 300 350 400 450 500 550 N u m b e r o f o b se rva tio n s 0 ,0 0 0 0 8 1 ,8 1 8 2 1 6 3 ,6 3 6 4 2 4 5 ,4 5 4 5 3 2 7 ,2 7 2 7 4 0 9 ,0 9 0 9 4 9 0 ,9 0 9 1 5 7 2 ,7 2 7 3 6 5 4 ,5 4 5 5 7 3 6 ,3 6 3 6 8 1 8 ,1 8 1 8 9 0 0 ,0 0 0 0 9 8 1 ,8 1 8 2 1 0 6 3 ,6 3 6 4 1 1 4 5 ,4 5 4 5 1 2 2 7 ,2 7 2 7 1 3 0 9 ,0 9 0 9 1 3 9 0 ,9 0 9 1 1 4 7 2 ,7 2 7 3 1 5 5 4 ,5 4 5 5 1 6 3 6 ,3 6 3 6 1 7 1 8 ,1 8 1 8 1 8 0 0 ,0 0 0 0 Time [min] 0 50 100 150 200 250 300 350 400 450 500 N u m b e r o f o b se rva tio n s 0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 Time [min] 0 100 200 300 400 500 600 700 N u m b e r o f o b se rva tio n s 0 200 400 600 800 1000T min 0.000 0.002 0.004 0.006 0.008 0.010 p Time [min] Nu m be r of ob se rv ati on s Time [min] Nu m be r of ob se rv ati on s Time [min] Nu m be r of ob se rv ati on s a) b) p T [min]

(3)

Non-Poisson character of vessel traffic on the Szczecin−Świnoujście fairway

Zeszyty Naukowe 32(104) z. 2 79

The chi-square test statistic is equal to 17.27 and the critical value, at the 0.05 level of significance, is equal to 24.99. So there is no evidence to indicate that time between reporting successive fairway units is not distributed by probability density func-tion (2).

Vessel traffic at CHELMINEK_N point

In the second half of year 2009, 1310 vessels have been registered at this point. Figure 5 presents the data at CHELMINEK_N point.

Fig. 5. Data and fitted pdf graph at CHELMINEK_N point

Using the chi-square goodness-of-fit test and trial and error approach, the probability density function of variable X has been determined. It is the mixed distribution: 0.9 (pdf of exponential distribu-tion) +0.1 (pdf of uniform distribudistribu-tion). It has the following form:

 

            1130 for 0054 . 0 1130 0 for 000088 . 0 0054 . 0 006 . 0 006 . 0 x e x e x f x x (3)

The chi-square test statistic is equal to 17.1 and the critical value, at the 0.05 level of significance, is equal to 24.99.

Vessel traffic at KREPA_DOLNA point

In the second half of year 2009, 1276 vessels have been registered at this point. Figure 6 presents the data at KREPA_DOLNA point.

Using the procedure of determination good fitted mixed distribution to variable X, its tion has been determined. It is the mixed distribu-tion: 0.92 (pdf of exponential distribution) +0.08 (pdf of uniform distribution). It has the following form:

 

            1450 for 005336 . 0 1450 0 for 0000552 . 0 005336 . 0 0058 . 0 0058 . 0 x e x e x f x x (4)

Fig. 6. Data and fitted pdf graph at KREPA_DOLNA point

Vessel traffic at INOUJSCIE point

In the second half of year 2009, 1210 vessels have been registered at this point. Figure 7 presents the pdf curve fitted to data at INOUJSCIE point.

Using the chi-square goodness-of-fit test and trial and error approach, we obtained the probabil-ity densprobabil-ity function of variable X. It is the mixed distribution: 0.9 (pdf of exponential distribution) +0.1 (pdf of uniform distribution). It has the follow-ing form:

 

            1300 for 005157 . 0 1300 0 for 000077 . 0 005157 . 0 0057 . 0 0057 . 0 x e x e x f x x (5)

Fig. 7. Data and fitted pdf graph at INOUJSCIE point

0 200 400 600 800 1000T min 0.000 0.002 0.004 0.006 0.008 0.010 p 0 200 400 600 800 1000T min 0.000 0.002 0.004 0.006 0.008 0.010 p 0 200 400 600 800 1000T min 0.000 0.002 0.004 0.006 0.008 0.010 p p T [min] p T [min] p T [min]

(4)

Lech Kasyk

80 Scientific Journals 32(104) z. 2

Coclusions

Based on the results presented herein, the fol-lowing conclusions can be drawn:

• At each VTS reporting point for vessel traffic stream from Świnoujście to Szczecin the report-ing process is Poisson process.

• All vessel traffic regulations on the Świnoujście– Szczecin fairway aren’t disturbing factors of the randomness of the vessel traffic stream from Świnoujście to Szczecin.

• At each VTS reporting point for vessel traffic stream from Szczecin to Świnoujście the report process isn’t Poisson process.

• A departure from Szczecin seaport turned out to be the most disturbing factor on the fairway Szcze-cin–Świnoujście.

• For south vessel traffic stream, to description of reporting process, mixed exponential-uniform distribution has been used.

• Determination of probability density function form of time between reporting successive

fairway units allows studying ships traffic more exactly.

Rferences

1. KASYK L.: Disturbances in vessel traffic stream due to fairway regulations. Journal of KONBiN 2008, Kraków 2008.

2. KASYK L.: Process of Ship Reports after Covering a Special Fairway Section. 10th_{International Conference}

TRANSCOMP 2006, The Publishing and Printing House of the Institute for Sustainable Technologies, Radom 2006. 3. Port regulations. Marine Office in Szczecin, Szczecin 2008. 4. KASYK L.: Analysis of Vessel Stream Parameters at the Fairway Szczecin–Świnoujście. Conference LOGITRANS 2011, Szczyrk 2011.

5. MONTGOMERY D.C.,RUNGER G.C.: Applied Statistics and Probability for Engineers. John Wiley & Sons Inc., New York 1994.

6. SOBCZYK M.: Statystyka. Wydawnictwo Naukowe PWN, Warszawa 2004.

7. DRWAL G.,GRZYMKOWSKI R.,KAPUSTA A.,KUBOSZEK T., SŁOTA D.: Mathematica 6. Wydawnictwo Pracowni Kom-puterowej Jacka Skalmierskiego, Gliwice 2008.