Zeszyty Naukowe 20(92) 67
Scientific Journals
Zeszyty Naukowe
Maritime University of Szczecin
Akademia Morska w Szczecinie
2010, 20(92) pp. 67–70 2010, 20(92) s. 67–70
A vessel braking time as a normal random variable
Czas hamowania jako zmienna losowa o rozkładzie normalnym
Lech Kasyk
Maritime University in Szczecin, Department of Mathematics Akademia Morska w Szczecinie, Zakład Matematyki
70-500 Szczecin, ul. Wały Chrobrego 1–2, e-mail: l.kasyk@am.szczecin.pl
Key words: vessel traffic, convolution method, random variable, normal distribution Abstract
In convolution method of the determination of vessel traffic stream parameters, it is indispensable to know the distribution of braking time. In this paper, on base of simulations trials, the fitting the type of the probability distribution to data of vessel braking times has been done. To achieve this goal the chi square Pearson test has been used.
Słowa kluczowe: ruch statków, metoda splotów, zmienna losowa, rozkład normalny Abstrakt
W metodzie splotów, związanej z wyznaczaniem parametrów strumienia ruchu statków, potrzebna jest zna-jomość rozkładu prawdopodobieństwa zmiennej losowej, którą jest czas hamowania. W niniejszym artykule, na podstawie prób symulacyjnych, dopasowano rozkład prawdopodobieństwa czasu hamowania. W tym celu wykorzystano test chi kwadrat Pearsona.
Introduction
The vessel traffic in restricted areas is subject to different restrictions: speed limit, overtaking ban, passing ban and others [1, 2]. When ships adapt to these regulations, they often must reduce their speed. And this manoeuvre takes some time. This time has been called a braking time.
The vessel braking time depends on different factors: vessel velocity, draught, force of wind, force of stream, shape of ship’s hull and many others. So this time can be treated like a random variable. It has some distribution. What is this distribution? The replay to this question is the subject of this paper. To achieve this goal computer simulations of the manoeuvre of the braking have been done.
Simulation trials
Simulation conditions and vessels
Simulations were done in the simulator in the Marine Traffic Engineering Institute of MU in
Szczecin. There are simulators based on “Polaris” System from Kongsberg Maritime AS. The model used to simulations is based on characteristic of Chemical Tanker 6000 DWT “Bow Master” [3]. During simulations wind force, wind direction, stream force and stream direction were changed. There were six values of wind speed: 0 m/s, 1 m/s, 2 m/s, 3 m/s, 4 m/s, 5 m/s. There were six values of wind direction: 0, 30, 90, 120, 180, 270. There were six values of stream direction: 0, 45, 90, 120, 180, 270. There were ten values of stream force: 0 m/s, 0.5 m/s, 0.7 m/s, 1 m/s, 1.2 m/s, 1.5 m/s, 1.7 m/s, 2 m/s, 2.2 m/s, 2.5 m/s. Six sets of simulation trials were obtained.
Simulation results
First set is for the manoeuvre of the speed reduction from 14 knots to 10 knots with the 50% power of ship engines.
Using the chi-square goodness-of-fit test [4, 5] and programme Statistica we find that the test statistic is equal to 6.27. At the 0.05 level of
Lech Kasyk
68 Scientific Journals 20(92)
significance the critical value is equal to 3.84, so we must reject the hypothesis that the braking time has a normal distribution (Fig. 1).
Second data set is for the manoeuvre of the speed reduction from 14 knots to 12 knots with the 50% power of ship engines. Using the chi-square goodness-of-fit test and programme Statistica we find that the test statistic is equal to 2.9. At the 0.05 level of significance the critical value is equal to 3.84, so we are unable to reject the hypothesis that the braking time has a normal distribution.
Fig. 1. Frequency histogram of first set of simulation trials Rys. 1. Histogram częstości dla pierwszego zbioru danych
50,6667 57,0000 63,3333 69,6667 76,0000 82,3333 88,6667 Braking time 0 1 2 3 4 5 6 7 8 9 10 11 N u m b e r o f o b se rva tio n s
Fig. 2. Frequency histogram of second set of simulation trials Rys. 2. Histogram częstości dla drugiego zbioru danych
In this case probability density function of the braking time is the following form:
2
2 t 0.0452exp 0.00642t68.4
f (1)
Third data set is for the manoeuvre of the speed reduction from 12 knots to 8 knots with the 15% power of ship engines.
Using the chi-square goodness-of-fit test and programme Statistica we find that the test statistic is equal to 3.79. At the 0.05 level of significance
the critical value is equal to 3.84, so we are unable to reject the hypothesis that the braking time has a normal distribution.
For this data set, p.d.f. (probability density function) of the braking time has the following form:
2
3t 0.0354exp 0.00393t111.9 f (2) 85,7143 92,8571 100,0000 107,1429 114,2857 121,4286 128,5714 135,7143 Braking time 0 1 2 3 4 5 6 7 8 9 10 11 N u m b e r o f o b se rva tio n sFig. 3. Frequency histogram of third set of simulation trials Rys. 3. Histogram częstości dla trzeciego zbioru danych
Fourth data set is for the manoeuvre of the speed reduction from 12 knots to 10 knots with the 15% power of ship engines. Using the chi-square goodness-of-fit test and programme Statistica we find that the test statistic is equal to 2.38. At the 0.05 level of significance the critical value is equal to 3.84, so we are unable to reject the hypothesis that the braking time has a normal distribution.
44 48 52 56 60 64 Braking time 0 2 4 6 8 10 12 14 16 N u m b e r o f o b se rva tio n s
Fig. 4. Frequency histogram of fourth set of simulation trials Rys. 4. Histogram częstości dla czwartego zbioru danych
In this case p.d.f. of this variable is the following form:
2
4 t 0.0906exp 0.02577t54.9 f (3) Breaking time N umb er o f ob ser va tio ns Breaking time N umb er o f ob ser va tio ns Breaking time N umb er o f ob ser va tio ns Breaking time N umb er o f ob ser va tio nsA vessel braking time as a normal random variable
Zeszyty Naukowe 20(92) 69
Fifth data set is for the manoeuvre of the speed reduction from 12 knots to 8 knots with the 15% power of ship engines at the area with a depth of 12 meters.
Using the chi-square goodness-of-fit test and programme Statistica we find that the test statistic is equal to 2.75. At the 0.05 level of significance the critical value is equal to 3.84, so we are unable to reject the hypothesis that the braking time has a normal distribution. 64,0000 68,5714 73,1429 77,7143 82,2857 86,8571 91,4286 96,0000 Braking time 0 2 4 6 8 10 12 14 16 N u m b e r o f o b se rw a tio n s
Fig. 5. Frequency histogram of fifth set of simulation trials Rys. 5. Histogram częstości dla piątego zbioru danych
For this data set, p.d.f. of the braking time has the following form:
2
5 t 0.07413exp 0.01727t79.7
f (4)
Sixth data set is for the manoeuvre of the speed reduction from 12 knots to 10 knots with the 15% power of ship engines at the area with a depth of 12 meters. Using the chi-square goodness-of-fit test and programme Statistica we find that the test statistic is equal to 0.29. At the 0.05 level of significance the critical value is equal to 3.84, so
we are unable to reject the hypothesis that the braking time has a normal distribution.
For this data set, p.d.f. of the braking time has the following form:
2
6t 0.13915exp 0.0608t41.7
f (5)
Deceleration
Coasting stop test
The manoeuvre of the speed reduction from 14.6 kn to 1.6 kn with change machinery telegraph setting from position 1 to 0, takes 10 minutes for “Bow Master” [3]. And then the average decelera-tion is equal to 0.0111 m/s2.
Comparison between coasting stop test and deceleration from simulations
All simulation trials were done with definite percent of ship engines power. Now we will test the hypothesis that average deceleration corresponds to percent of ship engines power. To achieve this goal, a hypothesis on the mean of normal distribution with unknown variance for large samples, has been tested [5].
For second set of simulation trials the null hypothesis is the following: the mean deceleration is equal to 50% of the deceleration in coasting stop test (as 50% power of ship engines). H0 : = 0.055.
The sample mean x is equal to 0.0153 and the
sample variance s2 = (0.0021)2, so the test statistic
Z0 is equal to 26.3. The critical region, for the
significance level of = 0.05, amounts (–,–1.96) (1.96,). There is strong evidence that the mean deceleration exceeds 0.0055.
For third set of simulation trials the null hypothesis is the following: the mean deceleration is equal to 85% of the deceleration in coasting stop test (as 15% power of ship engines). The sample mean x is equal to 0.0186 and the sample variance
s2 = (0.0019)2, so the test statistic Z
0 is equal to
27.5. There is strong evidence that the mean deceleration exceeds 0.0094.
And in other cases we have similarly situations, we must reject the hypothesis, that the mean deceleration corresponds to percent of ship engines power.
Summary
In five from six examined cases we are unable to reject the hypothesis that the braking time has a normal distribution. So we can found the braking time as a normal random variable. But parameter values of this distribution is a separate problem.
33,3333 35,5556 37,7778 40,0000 42,2222 44,4444 46,6667 48,8889 51,1111 53,3333 Braking time 0 2 4 6 8 10 12 14 N u m b e r o f o b se rva tio n s
Fig. 6. Frequency histogram of sixth set of simulation trials Rys. 6. Histogram częstości dla szóstego zbioru danych
Breaking time N umb er o f ob ser va tio ns Breaking time N umb er o f ob ser va tio ns
Lech Kasyk
70 Scientific Journals 20(92)
As we can see there isn’t a simply relationship between percent of power of ship engines and per-cent of the deceleration in coasting stop test. And there is a difference between deceleration values in traffic at the deep area (II, III, IV) and at the shal-low area (V, VI), but to small number of data sets won’t let to formulate more precision conclusions.
References
1. KASYK L.: Process of Ship Reports after Covering a
Spe-cial Fairway Section, 10th International Conference TRANSCOMP 2006. The Publishing and Printing House of the Institute for Sustainable Technologies, Radom 2006.
2. KASYK L.: Convolutions of Density Functions as a Deter-mination Method of Intensity of Disturbed Vessel Traffic Stream, 12th International Conference TRANSCOMP 2008. The Publishing and Printing House of the Institute for Sustainable Technologies, Radom 2008.
3. ZAIKOV S.: Description of ship model TANK09L Chemical
Tanker 6000 DWT, Kongsberg Maritime Ship System AS, 2002.
4. KASYK L.: Empirical distribution of the number of ship
reports on the fairway Szczecin – Świnoujście, 14th International Scientific and Technical Conference The Part of navigation in Support of Human Activity on the Sea. Naval Academy, Gdynia 2004.
5. MONTGOMERY D.C.,RUNGER G.C.: Applied Statistics and
Probability for Engineers, John Wiley & Sons, Inc., New York 1994.
