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METHODS AND ALGORITHMS FOR EVALUATING UNKNOWN PARAMETERS OF OPERATION PROCESSES OF COMPLEX TECHNICAL SYSTEMS
K. Kolowrocki, J. Soszynska.
Gdynia Maritime University, Gdynia, Poland e-mail: katmatkk@am.gdynia.pl, joannas@am.gdynia.pl
ABSTRACT
The paper objectives are to present the methods and tools useful in the statistical identifying unknown parameters of the operation models of complex technical systems and to apply them in the maritime industry. There are presented statistical methods of determining unknown parameters of the semi-markov model of the complex technical system operation processes. There is also presented the chi-square goodness-of-fit test applied to verifying the distributions of the system operation process conditional sojourn times in the particular operation states. Applications of these tools to identifying and predicting the operation characteristics of a ferry operating at the Baltic Sea waters are presented as well.
1 INTRODUCTION
Many real transportation systems belong to the class of complex systems. It is concerned with the large numbers of components and subsystems they are built and with their operating complexity. Modeling the complicated system operation processes, is difficult because of the large number of the operation states, impossibility of their precise defining and because of the impossibility of the exact describing the transitions between these states. The changes of the operation states of the system operations processes cause the changes of these systems reliability structures and also the changes of their components reliability functions (Blokus-Roszkowska et all. 2008b). The models of various multistate complex systems are considered in (Blokus-Roszkowska et all. 2008a). The general joint models linking these system reliability models with the models of their operation processes (Kolowrocki, Soszynska 2008), allowing us for the evaluation of the reliability and safety of the complex technical systems in variable operations conditions, are constructed in (Blokus-Roszkowska et all. 2008b).
In order to be able to apply these general models practically in the evaluation and prediction of the reliability of real complex technical it is necessary to elaborate the statistical methods concerned with determining the unknown parameters of the proposed models, namely the probabilities of the initials system operation states, the probabilities of transitions between the system operation states and the distributions of the sojourn times of the system operation process in the particular operation states and also the unknown parameters of the conditional multistate reliability functions of the system components in various operation states. It is also necessary the elaborating the methods of testing the hypotheses concerned with the conditional sojourn times of the system operations process in particular operations states and the hypotheses concerned with the conditional multistate reliability functions of the system components in the system various operation states. The model of the operation process of the complex technical system with the distinguished their operation states is proposed in (Kolowrocki, Soszynska 2008). The semi-markov process is used to construct a general probabilistic model of the considered complex industrial system operation process. To construct this model there were defined the vector of the probabilities of the system initial operation states, the
matrix of the probabilities of transitions between the operation states, the matrix of the distribution functions and the matrix of the density functions of the conditional sojourn times in the particular operation states. To describe the system operation process conditional sojourn times in the particular operation states the uniform distribution, the triangular distribution, the double trapezium distribution, the quasi-trapezium distribution, the exponential distribution, the Weibull's distribution, the normal distribution and the chimney distribution are suggested in (Kolowrocki, Soszynska 2008). In this paper, the formulae estimating unknown parameters of these distributions are given and the chi-square test is applied to verifying the hypotheses about these distributions validity. Moreover these tools are applied to unknown parameters estimation and characteristics prediction of the Stena Baltica ferry operation process.
2 IDENTYFICATION OF THE OPERATION PROCESS OF THE COMPLEX TECHNICAL SYSTEM
2.1. Estimation of unknown parameters of the semi-markov model of the operation process
We assume, similarly as in (Blokus-Roszkowska et all 2008b, Kolowrocki, Soszynska 2008) that a system during its operation at the fixed moment t, t e <0, + » >, may be in one of v, v e N, different operations states zb, b = 1,2, ..., v. Next, we mark by Z(t), t e <0, +<»>, the system operation process, that is a function of a continuous variable t, taking discrete values in the set Z = {z1,z2,...zv } of the operation states. We assume a semi-markov model (Blokus-Roszkowska et all 2008b, Kolowrocki, Soszynska 2008) of the system operation process Z(t) and we mark by 6bl its random conditional sojourn times at the operation states zb when its next operation state is z,, b, l = 1,2,..., v, b * l.
Under these assumption, the operation process may be described by the vector [pb (0)]1xv of probabilities of the system operation process staying in particular operations states at the initial moment t = 0, the matrix [pbl (t)]vxv of the probabilities of the system operation process transitions
between the operation states and the matrix [Hbl (t)]vxv of the distribution functions of the conditional sojourn times 9bl of the system operation process at the operation states or equivalently by the matrix [hbl (t)]vxv of the distribution functions of the conditional sojourn times 9bl of the system operation process at the operation states.
To estimate the unknown parameters of the system operations process, firstly during the experiment, we should collect necessary statistical data performing the following steps:
i) to analyze the system operation process and either to fix or to define its following general parameters:
- the number of the operation states o the system operation process v,
- the operation states o the system operation process z1, z2, ..., zv,
- the duration time of the experiment 0 ;
ii) to fix and to collect the following statistical data necessary to evaluating the probabilities of the initial states of the system operations process:
- the number of the investigated (observed) realizations of the system operation process n(0),
- the numbers of staying of the operation process respectively in the operations states z1, z2, ..., zv, at the initials moment t = 0 of all n(0) observed realizations of the system operation process
n1 (0), «2(0), «v(0);
iii) to fix and to collect the following statistical data necessary to evaluating the probabilities of transitions between the system operation states:
- the numbers nbl, b, l = 1,2,...,v, b^l, of the transitions of the system operation process from the operation state zb to the operation state zl during all observed realizations of the system operation process;
- the numbers nb, b = 1,2,...,v, of departures of the system operation process from the operation states zb ;
iv) to fix and to collect the following statistical data necessary to evaluating the unknown parameters of the distributions of the conditional sojourn times of the system operation process in the particular operation states:
- the realizations 9kbI, k = 1,2, ..., nbl, for each b, l = 1,2,...,v, b * l of the conditional sojourn times 9bI of the system operations process at the operation state zb when the next transition is to the operation state zl during the observation time;
After collecting the above statistical data it is possible to estimate the unknown parameters of the system operation process performing the following steps: i) to determine the vector
[ p(0)] = [ pY (0), p2(0),..., pv (0)],
of the realizations of the probabilities pb (0), b = 1,2,..., v, of the initial states of the system operation process, according to the formula
(0)=
«(0)
for b = 1,2,...,
where n(0) = £ nb (0), is the number of the realizations of the system operation process starting at the
4=1
initial moment t = 0; ii) to determine the matrix
[ Pi ] =
Pll Pl2 ••• Plv
P21 p22 . . . p2v
Pvl Pv2 ... Pvv
of the realizations of the probabilities pu, b, l = 1,2,..., v, of the system operations process transitions from the operations state zb to the operations state zl during the experiment time according to the formula
n
pbl =-iL for b,l = 1,2,..., v, b * l, pbb = 0 for b = 1,2,..., v,
where nb = £ nbl, b = 1,2,..., v, is the realization of the total number of the system operations process
b*l
departures from the operations state zb during the experiment time ©;
iii) to determine the following empirical characteristics of the realizations of the conditional sojourn time of the system operation process in the particular operation states:
V
b
- the realizations of the mean values 9bl of the conditional sojourn times 9bl of the system operations process at the operations state Hbl (t) when the next transition is to the operation state 9bl , according to the formula
— 1 "bl , , , ebl = — i ekbl b,i = 1,2,..., v, b * I,
- the number 9kM, of the disjoint intervals k = 1,2,...,nbl,, 9bl, that include the realizations 9^, k = 1,2,...,nbl, of the conditional sojourn times 9bl at the operation state zb when the next transition is to the operation state zl, according to the formula
the length d of the intervals Ij =< aj,bj), j = 1,2,..., r , according to the formula
R
d = , where R = max 6^ — min 6^ .
r — 1 1<k <nbl 1Sk Snbl
the ends aj, bj, of the intervals Ij =< aJbl,bj), j = 1,2,..., r , according to the formulae
ab = min 6 — d, K = ah + jd, j = 1,2,...,r , abi = bb— , j = 2,3,..., r,
1<k <nbl 2
in the way such that
/1 uI2 u... uIr =< ah,bbi
and
It n Ij = 0 for all i * j, i, j e {1,2,..., r},
- the numbers nj of the realizations 9^ in particular intervals Ij, j = 1,2,...,r , according to the formula
nj = #{k : 9k e Ij, k e {1,2,...,"bl}}, j = 1,2,..., r ,
where I nj = nbl, whereas the symbol # means the number of elements of the set;
j=1
iv) to estimate the parameters of the distributions of the conditional sojourn times of the system operation process in the particular operation states for the following distinguished distributions respectively in the following way: - the uniform distribution with a density function
nbl k=1
r = Vnbl
hbl (t) =
t < x.
bl
1
> xbl < t < ybi
ybl xbl 0, t > ybi,
where 0 < xbl < ybl <
the estimates of the unknown parameters of this distribution are:
xbl = abl , ybl = Xbl + rd ; - the triangular distribution with a density function
i0,
hbl (t) =
t < x
2
t-xb
ybl xbl zbl xbl 2 ybl-t
ybl xbl ybl zbl 0,
bl
xbl < t < zbl
zbl < t < ybl
t > ybl,
where 0 < xbl < zbl < ybl <+<»,
the estimates of the unknown parameters of this distribution are:
xbl = abi, ybl = xbl + rd, i = r;
- the double trapezium distribution with a density function
f0,
hu (t) =
t < x
bl
qu +
wb, +
Cbl -qbl
zbl ■ xbl
Cbl -Wbl
ybl -zbl
(t-xblX xb, < t < Zbl -(yu-tX zu < t < ybl
t > ybl :
where
Cbl =
2- qu (zu -xbl)- wbl(yu-zbi) ybl - xbl
, 0 < Xbl < Zbl < ybi < +<», 0 < qu < +», 0 < Wbl < +<»,
0 < qu (zbi ~xbt) + wbl(ybl ~zbl) < 2;
the estimates of the unknown parameters of this distribution are:
n n
xbl = ah, ybi = xbl +rd, qbl = , =, = Ob
nbld ' nbld'
-bl "bl'
0
0
- the quasi-trapezium distribution with a density function
hbl (t) =
qbl + (t~xbl),
zbl xbl
Wbl +
ybl zbl
t < xbl xbl < t < zh
1 ^ -f ^ 2 zbl < t < zbl
2
zbl < t < ybl
t > ybl,
where
A _
2 qbl (zbl xbl ) wbl (ybl zbl ) 0 < X < z 1 < z 2 < y < 0 < q <
-T-z---,0 < Xbl < zbl < zbl < ybl > 0 < qbl <+'x>,
zbl zbl + ybl Xbl
0 < Wbl < 0 < qbl(4 ~xbl) + wbl (zh -ybl) < 2 ,
the estimates of the unknown parameters of this distribution are:
xbl = abl , ybl = xbl + rd, qbl =
nbl w _ nbl _ 1 _ Q\ _ 2 _ 02
- wbl _-7' zbl _ 0bl, zbl _
nbld
nbld'
where
— 1 1 n(me) . — 2 °bi _- E , _
nbl n,, + 1
E0. , nme) _ [^];
n(me) 1 _1 nbl - n(me) J_n(me) +1 2
- the exponential distribution with a density function
0,
hbl (t) _\ '
t < Xbl, Xbl > 0
abl exp[-abl (t-xbl)], t > xbl,
where 0 < abl < +<x>, 0 < xbl = aj,,
the estimates of the unknown parameters of this distribution are:
1
Xbl _ abl , abl _ —
0bl Xbl
- the Weibull's distribution with a density function
h (t) _j°' t < Xbl'Xbl > 0' " \außu (t - Xbl)ßbl-1 exp[-abl (t - Xbl)ßbl ], t > Xbl,
where 0 < abl < +<x>, 0 < /3bl < +<x>, 0 < xbl = albl,,
the estimates of the unknown parameters of this distribution are:
0
bl
0
1
«m «bl .
bl- + X Inidj-Xbr )
Xbl = abl, abl = «
abl =
ßbl i=1
X (ßj )
j=l
j \ßbl
nbl X
j=l
Xßl)ßbl lnßbl -Xbl)
- the normal distribution with a density function
hbl (t) =
exp[-(t mb\)2], t e (-œ, œ), 2a,,
where - <x> < mu < +<x>, 0 < abi < +<x>,
the estimates of the unknown parameters of this distribution are:
— 2 _2 l nbl . 2
mbi = ßbl, abl = abl =-X (ß -mbl ) ,
- the chimney distribution with a density function
hbl (t) =
zbl Xbl
,2 1
db
ybl zbl 0,
t < X
bl
Xbl < t < zbl
1 ^ ^ 2 zbl < t < zbl
zh < t < ybl
t > ybl,
where 0 < xbl < z\, < zl, < ybl < 0 < qbl < 0 < wbl < +M, abl > cbl > 0, dbl > 0, abl + cbl + dbl = b
The estimates of the unknown parameters of this distribution are:
xbl = abl , ybl = xbl + rd,
and moreover, if
nbl = max{njl}, i = j, where j e{12,..., r},
1< j<r
is such a number of the interval for which nj = nbl, then: for i = 1
« ii
zh = Xbl + (i - 1)d, 4 = Xbl +id, abl = 0, cbl =—, dbl = bl
«b+1 +...+«b
n
bl
1
0
a
bl
c
bl
n
bl
bl
n
when n'b+1 _ 0 or n'b+1 * 0 and > 3,
n' + nM ri+2 + + ri
zlbl _ Xbl + (i - 1)d, 4 _ Xl + (i + 1)d, abl _ 0, cbl _ -, dbl _-»—■■ b'
n
bl
when rib+1 * 0 and < 3,
bl n'+l
nbl
for i _ 2,3,..., r-1
nl, +... + nb-1 ni , nb+1 +... + nb
zu _ Xbl + (i 1)d, zbl _ Xbl +id, abl _ , cbl _ , dbl
nbl nbl nbl
n
when n'b-1 _ 0 or n'b-1 * 0 and —^ > 3 and
nb-
nb +... + rib-1 n[, + ni+1 nb+2 +... + nb
zu _ Xb, + (i - 1)d, 4 _ XU + (i + 1)d, abl _ b' , cu -^, du _ b'
nbl nbl nbl
when n'bl 1 _ 0 or ribl 1 * 0 and —-- > 3 and
n'bl
4 _ Xbl + (i - 2)d, zh _ Xbl +id, abl _
n\, +... + n'u2 ^ n'b- + nl , _nb+1 +... + nh
bl ' "bl ' "bl nbl nbl
nn when nb - * 0 and -b- < 3 and when n'b+1 _ 0 or ri^1 * 0 and > 3,
nb- nb+
nb +... + nb-1 nbl1 + ni, + n':1 , nb,2 +... + nb
„2 „ , /-,. , i\ j bl ' ■■■ ' "bl "bl ' ,lbl ' /7 ,lbl 1 ••• 1 ,lbl
zu _ Xbl + (i- 2)d, zu _ Xbl + (i + 1)d, abl _-, cbl _-, dbl _■
nbl nbl nbl
when n'bi1 * 0 and —-- < 3 and when n'b+1 * 0 and —+1 < 3,
nb, nb+
for i _ r
n\, +... + nb1
4 _ Xbl + (i - 1)d, zh _ Xbl + id, abl _ -—, cbl _—, dbl
nn
nbl
n
when nb- _ 0 or nb- * 0 and -JbL- > 3 ,
n'bl
= xbl + (i - 2)d,
Zbl = xbl +
nhl +... + n
bl
nb-1 + n
Cu, =■
bl
dM = 0,
i-2
abl =
z
n
n
bl
bl
n
when nl-1 ^ 0 and < 3
nbl
2.2. Identification of distributions of conditional sojourn times in operation states
To formulate and next to verify the non-parametric hypothesis concerning the form of the distribution function Hbl (t) of the system conditional sojourn time 9bl at the operation state zb
when the next transition is to the operation state zl, on the basis of its realizations 9kbi, k = 1,2,...,nbl, it is necessary to proceed according to the following scheme:
- to construct and to plot the realization of the histogram of the system conditional sojourn time 9bl at the operation states, defined by the following formula
— nJ
K (t) = -f for t elj,
nbi
- to analyze the realization of the histogram, comparing it with the graphs of the density functions hbl (t) of the previously distinguished distributions, to select one of them and to formulate the null hypothesis H0 and the alternative hypothesis HA, concerning the unknown form of the distribution function Hbl(t) of the conditional sojourn time 9bl in the following form:
H0 : The system conditional sojourn time 9bl at the operation state zb when the next transition is to the operations state zl, has the distribution function Hbl (t),
HA : The system conditional sojourn time 9bl at the operation state zb when the next transition is to the operations state zl, has the distribution function different from Hbl (t),
- to join each of the intervals Ij that has the number nj of realizations is less than 4 either with the neighbour interval Ij+1 or with the neighbour interval Ij-1 this way that the numbers of realizations
in all intervals are not less than 4,
- to fix a new number of intervals r ,
- to determine new intervals Ij =< aj,bj), j = 1,2,.., r,
- to fix the numbers nj of realizations in new intervals ^, j = 1,2,.., r,
- to calculate the hypothetical probabilities that the variable 9bl takes values from the interval Ij, under the assumption that the hypothesis H0 is true, i.e. the probabilities
Pj = P(9bl e Ij) = P(aj < 9bl < bj ) = Hbl (bbj) -Hbl (aj), j = 1,2,..., f,
where Hbl (bbj) and Hbl (aj) are the values of the distribution function Hbl (t) of the random variable 9bl defined in the null hypothesis H0,
- to calculate the realization of the %2 (chi-square)-Pearson's statistics Unu, according to the formula
= £ (nbl -nblPj f J=1 n., p j
u
- to assume the significance level a (a = 0.01, a = 0.02, a = 0.05 or a = 0.10) of the test,
- to fix the number f-l-1 of degrees of freedom, substituting for l for the distinguished distributions respectively the following values: l = 0 for the uniform, triangular, double trapezium, quasi-trapezium and chimney distributions, l = 1 for the exponential distribution, l = 2 for the Weibull's and normal distributions,
- to read from the Tables of the %2 - Pearson's distribution the value ua for the fixed values of the significance level a and the number of degrees of freedom f-l -1 such that the following equality holds
P(Unu > Ua) = 1 -a,
and next to determine the critical domain in the form of the interval (ua,+<x>) and the acceptance domain in the form of the interval < 0, ua >,
- to compare the obtained value unu of the realization of the statistics U with the red from the Tables critical value ua of the chi-square random variable and to verify previously formulated the null hypothesis H0 in the following way: if the value unbi does not belong to the critical domain, i.e. when unu < ua,then we do not reject the hypothesis H0, otherwise if the value unbi belongs to the critical domain, i.e. when unbi > ua, then we reject the hypothesis H0 in favor of the hypothesis HA .
3 APPLICATION IN MARITIME TRANSPORT
3.1. The Stena Baltica ferry operation process and its statistical identification
Taking into account the operation process of the considered ferry we distinguish the following as its eighteen operation states:
• an operation state z1 - loading at Gdynia Port,
• an operation state z2 -unmooring operations at Gdynia Port,
• an operation state z3 - leaving Gdynia Port and navigation to "GD" buoy,
• an operation statez4-navigation at restricted waters from "GD" buoy to the end of Traffic Separation Scheme,
• an operation state z5 - navigation at open waters from the end of Traffic Separation Scheme to "Angoring" buoy,
• an operation state z6 - navigation at restricted waters from "Angoring" buoy to "Verko" Berth at Karlskrona,
• an operation state z7 - mooring operations at Karlskrona Port,
• an operation state z8 - unloading at Karlskrona Port,
• an operation state z9 - loading at Karlskrona Port,
• an operation state z10 - unmooring operations at Karlskrone Port,
• an operation state z11 - ship turning at Karlskrone Port,
• an operation state z12 - leaving Karlskrone Port and navigation at restricted waters to "Angoring" buoy,
• an operation state z13 - navigation at open waters from "Angoring" buoy to the entering Traffic Separation Scheme,
• an operation state z14 -navigation at restricted waters from the entering Traffic Separation Scheme to "GD" buoy,
• an operation state z15 - navigation from "GD" buoy to turning area,
• an operation state z16 - ship turning at Gdynia Port,
• an operation state z17 - mooring operations at Gdynia Port,
• an operation state z18 - unloading at Gdynia Port.
To identify all parameters of Stena Baltica ferry operation process the statistical data about this process is needed. The statistical data that has been collected up to now is given in Tables 1-7 in (Soszynska et all 2009, Appendix 5A). In the Tables 1-7 there are presented the realizations 9kbI, k = 1,2,...,42, for each b = 1,2,...,17, l = b +1 and b = 18, l = 1 of the ship operation process conditional sojourn times 9bl, b = 1,2,...,17, l = b +1 and b = 18, l = 1 in the state zb while the next transition is the state zt during the experiment time 0 = 42 days.
These statistical data allow us, applying the methods and procedures given in the section 2, to formulate and to verify the hypotheses about the conditional distribution functions Hbl (t) of the Stena Baltica ferry operation process sojourn times 9bl, b = 1,2,...,17, l = b +1 and b = 18, l = 1 in the state zb while the next transition is to the state zl on the base of their realizations 9kbI, k = 1,2,...,42. On the basis of the statistical data, given in the Appendix 5A in [5], the vector of the probabilities of the system initial operation states was evaluated in the following form
[ Pb (0)] = [1, 0, 0,..., 0, 0].
The matrix of the probabilities pbl of transitions from the operation state zb into the operation state zl were evaluated as well. Their evaluation are given in the matrix below
[ Pbl ]1
010...00 001...00
000...01 100...00
Next the matrix [hbl (t)]18118 of conditional density functions of the system operation process Z(t)
conditional sojourn times 9bl (Soszynska et all 2009, Appendix 5A) were evaluated.
The results of the distributions unknown parameters estimation and the hypotheses testing are as
follows:
- the conditional sojourn time 912 have a triangular distribution with the density function
¿12 (t) = \
0, t < 7,
0.000441- 0.0031, 7 < t < 54, 0.044 -0.000431; 54 < t < 103, 0, t > 103;
- the conditional sojourn time 923 have an exponential distribution with the density function
Mt) =
0,
t < 1.6
1.03exp[-1.03(t-1.6)], t > 1.6;
- the conditional sojourn time d34 have a steep chimney distribution with the density function
MO =
0, t < 29,
0.0278, 29 < t < 35,
0.1984, 35 < t < 38,
0.0266, 38 < t < 47,
0, t > 47;
- the conditional sojourn time 6>45 have a chimney distribution with the density function
MO =
0, t < 41,
0.0095, 41 < t < 46,
0.0762, 46 < t < 56,
0.0127, 56 < t < 71,
0, t > 71;
- the conditional sojourn time d56 have a double trapezium distribution with the density function
MO =
0, t < 467.8,
- 0.00004t + 0.0277, 467.8 < t < 525.95,
- 0.00006t + 0.0397, 525.95 < t < 650.2, 0, t < 650.2;
- the conditional sojourn time 067 have a double trapezium distribution with the density function
Mt) =
0, t < 31.9,
0.0067t-0.1747, 31.9 < t < 37.17, 0.0031t-0.0395, 37.7 < t < 45.1, 0, t > 45.1;
- the conditional sojourn time 6>78 have a double trapezium distribution with the density function
K(t) =
0, t < 4.5,
- 0.0183t + 0.2922, 4.5 < t < 7.02,
- 0.0069t + 0.2122, 7.02 < t < 10.5, 0, t > 10.5;
- the conditional sojourn time 6>89 have a triangular distribution with the density function
h»(t) =
0, t < 0,
0.0021t, 0 < t < 21.4
- 0.002t + 0.087, 21.4 < t < 44.4,
0, t > 44.4;
- the conditional sojourn time 6>910 have a double trapezium distribution with the density function
h910(t) =
0, t < 14.6,
0.0001t + 0.0109, 14.6 < t < 52.26,
0.0062, 52.2 < t < 127.4,
0, t > 127.4;
- the conditional sojourn time 6>1011 have a double trapezium distribution with the density function
h1011(t)
0, t < 1.6, - 0.3071t +1.0014, 1.6 < t < 2.93, 0.0398t -0.0145, 2.93 < t < 6.4, 0, t > 6.4;
the conditional sojourn time 9nn have a quasi-trapezium distribution with the density function
h1112(t) =
0, t < 3.8,
-148.899t + 567.4863, 3.8 < t < 3.81, 0.181, 3.81 < t < 4.48,
0.3773t -1.5094, 4.48 < t < 6.2, 0, t > 6.2;
- the conditional sojourn time 01213 have a triangular distribution with the density function
h1213(t) =
0, t < 18.7,
0.025t -0.4675, 18.7 < t < 23.86,
- 0.012t + 0.4116, 23.86 < t < 34.3,
0, t > 34.3;
- the conditional sojourn time 6>1314 have a chimney distribution with the density function
h1314(t) =
0, t < 410,
0.0017, 410 < t < 478,
0.0189, 478 < t < 512,
0.0024, 512 < t < 614,
0, t > 614;
- the conditional sojourn time 6>1415 have a double trapezium distribution with the density function
h1415(t) =
0, t < 36.8,
-0.0006t + 0.0518, 36.8 < t < 50.14, 0.0003t + 0.0062, 50.14 < t < 75.2, 0, t > 75.2;
- the conditional sojourn time 91516 have a chimney distribution with the density function
h1516(t) =
0, t < 30,
0.0317, 30 < t < 33,
0.2698, 33 < t < 36,
0.0084, 36 < t < 48,
0, t > 48;
the conditional sojourn time 91617 have a triangular distribution with the density function
10,
h1617(t) =
t < 2.7,
0.305t -0.823, 2.7 < t < 4.52, -0.313t +1.9719, 4.52 < t < 6.3, 0, t > 6.3;
- the conditional sojourn time 91718 have a double trapezium distribution with the density function
h1718(t) =
0, t < 2.3,
- 0.1134t + 0.6707, 2.3 < t < 5.62, 0.0071t - 0.0063, 5.62 < t < 10.7, 0, t > 10.7;
- the conditional sojourn time 9181 have a triangular distribution with the density function
h181(t) =
0, t < 0,
0.0023t, 0 < t < 18.74,
-0.0016t + 0.0729, 18.74 < t < 45.59,
0, t > 45.59.
3.2. The Stena Baltica ferry operation process prediction
On the basis of the previous section, the mean values Mbl = E[9bl ], b,l = 1,2,...,18, b ^ l, (12) in (Kolowrocki, Soszynska 2008) of the system operation process Z(t) conditional sojourn times in particular operation states were determined and there are given by:
M12 = 54.33, M23 = 2.57, M34 = 36.57,
M45 = 52.5, M56 = 525.95, M67 = 37.16,
M78 = 7.02, M89 = 21.43, M910 = 53.69,
M1011 = 2.93, M1112 = 4.38, M1213 = 23.86,
M1314 = 509.69, M1415 = 50.14, M1516 = 34.28,
M1617 = 4.52, M1718 = 5.62, M181 = 18.74.
Hence, by (21) in (Kolowrocki, Soszynska 2008), the unconditional mean sojourn time in the particular operation states are given by:
M1 = E[91 ] = p12 M12 = 1 • 54.33 = 54.33,
M 2 = E[92 ] = p23M 23 = 1 • 2.57 = 2.57,
M 3 = E[93] = p34 M 34 = 1 • 36.57 = 36.57,
M 4 = E [94] = p45M45 = 1 • 52.5 = 52.5,
M 5 = E[95 ] = p56 M 56 = 1 • 525.95 = 525.95,
M 6 = E9] = P67 M 67 = 1 • 37.16 = 37.16,
M 7 = E[97] = p78 M 78 = 1 • 7.02 = 7.02,
M 8 = E9] = pggMs9 = 1 • 21.43 = 21.43,
M 9 = E[99] = P910 M- 910 = 1 • 53.69 = 53.69,
M!0 = E[9j0] = pV)nMvm = 1 • 2.93 = 2.93,
M11 = E[9n] = pm2 M1112 = 1 • 4.38 = 4.38,
M12 = E[912] = Pj213 M1213 = 1 • 23.86 = 23.86,
M13 = E[913] = p1314M1314 = 1 • 509.69 = 509.69,
M14 = E[914] = p1415 M1415 = 1 • 50.14 = 50.14,
M !5 = E[9^5 ] = p1516 M ,5,6 = 1 • 34.28 = 34.28,
M16 = E[9|6 ] = P1617 1617 = 1 • 4.52 = 4.52,
M17 = E[917] = p1718 M1718 = 1 • 5.62 = 5.62,
M18 = E[918] = P181M181 = 1 • 18.74 = 18.74.
Since from the system of equations below (23) in (Kolowrocki, Soszynska 2008) that takes the form
n1 = n2 = ... = n18 = 0.056,
then the limit values of the transient probabilities (the portions of time of a week, as the operation process is periodic) pb (t) at the operational states zb, according to (22) in (Kolowrocki, Soszynska 2008), are given by
p1 = 0.037, p2 = 0.002, p3 = 0.025,
p4 = 0.036, p5 = 0.364, p6 = 0.025,
p7 = 0.005, p8 = 0.014, p9 = 0.037,
p10 = 0.002, p11 = 0.003, p12 = 0.017,
p13 = 0.354, p14 = 0.035, p15 = 0.024,
p16 = 0.003, p17 = 0.004, p18 = 0.013.
Hence by (26) in (Kolowrocki, Soszynska 2008), the mean values of the system operation process total sojourn times 9b in the particular operation states zb, for the operation time 0 = 1 month = 720 hours are approximately given by
we get
E[9J = pl9 = 26.64, E[92] = p29 = 1.44,
E[93] = p9 = 18.00, E[94] = p9 = 25.92,
E[95] = p9 = 262.08, E[96] = p69 = 18.00,
E[97] = p9 = 3.6, E[98] = p9 = 10.08,
E[99] = p99 = 26.64, E[910] = pw9 = 1.44,
E[9„] = pn9 = 2.16, E[912] = pn9 = 12.24,
E[9j3] = p139 = 25.88, E[9,4] = pu9 = 25.20,
E[915] = p159 = 17.28, E[916] = p169 = 2.16,
E[0„] = pxie = 2.88, E[0J = pKd = 9.36.
4 CONCLUSION
The statistical methods and algorithms for the unknown parameters of the operation process of complex technical systems in variable operation conditions are proposed. Next, these methods are applied to estimating the operation process of Stena Baltica ferry operating between Gdynia Port in Poland and Karsklone Port in Sweden. The proposed methods other very wide applications to port and shipyard transportation systems operation processes characteristics evaluation are obvious. The results are expected to be the basis to the reliability and safety of complex technical systems optimization and their operation processes effectiveness and cost analysis
Acknowledgements
The paper describes part of the work in the Poland-Singapore Joint Research Project titled "Safety and Reliability of Complex Industrial Systems and Processes" supported by grants from the Poland's Ministry of Science and Higher Education (MSHE grant No. 63/N-Singapore/2007/0) and the Agency for Science, Technology and Research of Singapore (A*STAR SERC grant No. 072 1340050).
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