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Modeling critical infrastructure accident consequences
Chapter 3
Modeling critical infrastructure accident consequences Chapter Outline 3.1 Modeling process of initiating events 3.2 Modeling process of environment threats
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3.3 Modeling process of environment degradation
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The probabilistic models of these processes of initiating events, environment threats, and environment degradation are designed. The vectors of initial probabilities of these processes staying at their particular states, the matrices of probabilities of these processes transitions between their particular states, and the matrices of conditional distribution functions and density functions of these processes conditional sojourn times at their particular states are defined. The distributions such as uniform, exponential, chimney, double trapezium, and quasitrapezium are suggested and introduced to describe the processes conditional sojourn times at the particular states. Moreover, the mean values of particular processes’ conditional sojourn times having these distributions at their states are defined and the unconditional distribution functions of sojourn times at particular states of these processes, the mean values of their unconditional sojourn times and limit values of transient probabilities at particular states are defined. Further, the superposition of these processes of initiating events, environment threats, and environment degradation is done to create the joint probabilistic general model of critical infrastructure accident consequences. Finally, the unconditional transient probabilities and the mean values of sojourn total times of the joint general process at its particular states for the sufficiently large time are determined.
3.1
Modeling process of initiating events
To model the process of initiating events, we fix the time interval tAh0; 1 NÞ as the time of a critical infrastructure operation and distinguish n1 ; n1 AN; events initiating the dangerous situation for the critical infrastructure operating environment and mark them by E1 ; E2 ; . . .; En1 : Further, we introduce a set of vectors (3.1) E 5 e: e 5 e1 ; e2 ; . . .; en1 ; ei Af0; 1g ; where
ei 5
1; 0;
if the initiating event Ei occurs; if the initiating event Ei does not occur;
(3.2)
for i 5 1; 2; . . .; n1 : We call vectors (3.1) the initiating events state. We may eliminate vectors that cannot occur, and we number the remaining states of the set E from l 5 1 up to ω; ωAN; where ω is the number of different elements of the set E 5 fe1 ; e2 ; . . .; eω g; where
h i el 5 el1 ; el2 ; . . .; eln1 ; l 5 1; 2; . . .; ω; and eli Af0; 1g; i 5 1; 2; . . .; n1 :
Next, we can define the process of initiating events EðtÞ on the time interval tAh0; 1 NÞ with its discrete states from the set Consequences of Maritime Critical Infrastructure Accidents. DOI: https://doi.org/10.1016/B978-0-12-819675-5.00003-6 © 2020 Elsevier Inc. All rights reserved.
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Consequences of Maritime Critical Infrastructure Accidents
E 5 e1 ; e2 ; . . .; eω : After that, we assume a semi-Markov model (Grabski, 2015; Kołowrocki 2004, 2014; Kołowrocki and Soszy´nskaBudny, 2011; Limnios and Oprisan, 2005; Macci, 2008; Mercier, 2008) of the process of initiating events EðtÞ and denote by θlj its random conditional sojourn time at the state el , while its next transition will be done to the state ej ; l; j 5 1; 2; . . .; ω; l 6¼ j: This way, the process of initiating events can be described by G
the vector of the probabilities pl ð0Þ 5 PðEð0Þ 5 el Þ; l 5 1; 2; . . .; ω; of its initial states at the moment t 5 0 ½pl ð0Þ1 3 ω 5 ½p1 ð0Þ; p2 ð0Þ; . . .; pω ð0Þ;
G
(3.3)
the matrix of probabilities of transitions between the states el and ej ; l; j 5 1; 2; . . .; ω; l 6¼ j; 2 11 3 p p12 : : : p1ω 6 p21 p22 : : : p2ω 7 lj 7; p ω3ω 5 6 4 : : : : : : 5 pω1 pω2 : : : pωω
(3.4)
where by formal agreement ’l 5 1; 2; . . .; ω; pll 5 0; G
the matrix of conditional distribution functions tAh0; 1 NÞ;
H lj ðtÞ 5 Pðθlj , tÞ;
l; j 5 1; 2; . . .; ω;
l 6¼ j;
of conditional sojourn times θlj of the process EðtÞ at the state el , while its next transition will be done to the state ej ; l; j 5 1; 2; . . .; ω; 2 11 3 H ðtÞ H 12 ðtÞ : : : H 1ω ðtÞ 6 H 21 ðtÞ H 22 ðtÞ : : : H 2ω ðtÞ 7 lj 7; H ðtÞ ω 3 ω 5 6 (3.5) 4 : : : : : : 5 ω1 ω2 ωω H ðtÞ H ðtÞ : : : H ðtÞ where by formal agreement ’l 5 1; 2; . . .; ω; H ll ðtÞ 5 0; which is strictly related to the matrix of corresponding conditional density functions of conditional sojourn times θlj of the process EðtÞ at the state el , while its next transition will be done to the state ej ; l; j 5 1; 2; . . .; ω; l 6¼ j; 2 11 3 h ðtÞ h12 ðtÞ : : : h1ω ðtÞ 6 h21 ðtÞ h22 ðtÞ : : : h2ω ðtÞ 7 lj 7; h ðtÞ ω 3 ω 5 6 (3.6) 4 : : : : : : 5 hω1 ðtÞ hω2 ðtÞ : : : hωω ðtÞ where by formal agreement ’l 5 1; 2; . . .; ω; hll ðtÞ 5 0: We assume that the appropriate and typical distributions suitable to describe the process EðtÞ conditional sojourn times θlj , l; j 5 1; 2; . . .; ω; l 6¼ j at the particular states are (Kołowrocki and Soszy´nska-Budny, 2011) G
the exponential distribution with a density function (Fig. 3.1) h ðtÞ 5 lj
0; αlj exp½ 2 αlj ðt 2 xlj Þ;
where 0 # αlj , 1 N;
t , xlj t $ xlj ;
(3.7)
Modeling critical infrastructure accident consequences Chapter | 3
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´ FIGURE 3.1 The graph of the exponential distribution’s density function. Based on Kołowrocki, K., Soszynska-Budny, J., 2011. Reliability and Safety of Complex Technical Systems and Processes: Modeling Identification Prediction Optimization. Springer, London, Dordrecht, Heidelberg, New York. G
the chimney distribution with a density function (Fig. 3.2) 8 0; > > > > > Alj > > ; > > lj > z1 2 xlj > > > > > > < Clj lj ; h ðtÞ 5 zlj2 2 zlj1 > > > > > > > Dlj > > ; > > > ylj 2 zlj2 > > > > : 0;
t , xlj xlj # t , zlj1 zlj1 # t , zlj2
(3.8)
zlj2 # t , ylj t $ ylj ;
where 0 # xlj # zlj1 # zlj2 # ylj , 1 N; Alj $ 0; C lj $ 0; Dlj $ 0; Alj 1 Clj 1 Dlj 5 1: As the mean values of the conditional sojourn times θlj are given by (Kołowrocki and Soszy´nska-Budny, 2011) ðN ðN M lj 5 E½θlj 5 tdH lj ðtÞ 5 thlj ðtÞdt; l; j 5 1; 2; . . .; ω; l 6¼ j; (3.9) 0
0
then for the distinguished distributions (3.7) and (3.8), the mean values of the process of initiating events EðtÞ conditional sojourn times θlj ; l; j 5 1; 2; . . .; ω; l 6¼ j; at the particular states are respectively given by the following (Kołowrocki and Soszy´nska-Budny, 2011): G
G
for the exponential distribution 1 M lj 5 E θlj 5 xlj 1 lj ; α
(3.10)
i 1h M lj 5 E θlj 5 Alj xlj 1 zlj1 1 Clj zlj1 1 zlj2 1 Dlj zlj2 1 ylj : 2
(3.11)
for the chimney distribution
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Consequences of Maritime Critical Infrastructure Accidents
´ FIGURE 3.2 The graphs of the chimney distribution’s density functions. Based on Kołowrocki, K., Soszynska-Budny, J., 2011. Reliability and Safety of Complex Technical Systems and Processes: Modeling Identification Prediction Optimization. Springer, London, Dordrecht, Heidelberg, New York.
From the formula for total probability, it follows that the unconditional distribution functions of the sojourn times θl ; l 5 1; 2; . . .; ω of the process of initiating events EðtÞ at the states el ; l 5 1; 2; . . .; ω are determined by (Kołowrocki and Soszy´nska-Budny, 2011) H l ðtÞ 5
ω X
plj H lj ðtÞ; l 5 1; 2; . . .; ω:
(3.12)
j51
Therefore the mean values E½θl of the process of initiating events EðtÞ unconditional sojourn times θl ; l 5 1; 2; . . .; ω at the states are given by M l 5 E½θl 5
ω X
plj M lj ; l 5 1; 2; . . .; ω;
(3.13)
j51
where M lj is defined by formula (3.9) in the case of any distribution of sojourn times θlj and by formulae (3.10), (3.11) in the cases of particular distribution of these sojourn times defined, respectively, by (3.7), (3.8). The limit values of the process of initiating events EðtÞ transient probabilities at the particular states pl ðtÞ 5 PðEðtÞ 5 el Þ; tAh0; 1 NÞ; l 5 1; 2; . . .; ω;
(3.14)
Modeling critical infrastructure accident consequences Chapter | 3
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are given by (Kołowrocki and Soszy´nska-Budny, 2011) pl 5 lim pl ðtÞ 5 t- 1 N
πl M l
ω P
πj M j
; l 5 1; 2; . . .; ω;
(3.15)
j51
where M is given by (3.13), while the steady probabilities πl of the vector ½πl 1 3 ω satisfy the system of equations 8 l l lj > < ½π 5 ½π ½p ω X (3.16) πj 5 1 > : l
j51
and ½plj ω 3 ω is given by (3.4). l The asymptotic distribution of the sojourn total time θ^ of the process of initiating events EðtÞ in the time interval l h0; θi; θ . 0; at the state e is normal with the expected value l l M^ 5 E½θ^ Dpl θ; l 5 1; 2; . . .; ω;
(3.17)
where pl is given by (3.15).
3.2
Modeling process of environment threats
To construct the general model of the environment threats caused by the process of initiating events generated by critical infrastructure loss of required safety critical level, we distinguish the set of n2 ; n2 AN kinds of threats as the consequences of initiating events that may cause the environment degradation and denote them by H1 ; H2 ; . . .; Hn2 : We also distinguish n3 ; n3 AN environment subareas D1 ; D2 ; . . .; Dn3 of the considered critical infrastructure operating in the environment area D 5 D1 , D2 , ? , Dn3 that may be degraded by the environment threats Hi ; i 5 1; 2; . . .; n2 : The environment threats possibility of influence on the distinguished subareas is presented in Fig. 3.3. We assume that the operating area D can be affected by some of threats Hi ; i 5 1; 2; . . .; n2 , and that a particular environment threat Hi ; i 5 1; 2; . . .; n2 ; can be characterized by the parameter f i ; i 5 1; 2; . . .; n2 : Moreover, we assume that the scale of the threat Hi ; i 5 1; 2; . . .; n2 influence on area D depends on the range of its parameter value and for i particular parameter f i ; i 5 1; 2; . . .; n2 . We distinguish li ranges f i1 ; f i2 ; . . .; f il of its values. After that, we introduce a set of vectors S 5 fs: s 5 ½f 1 ; f 2 ; . . .; f n2 g; where
8 > < 0; if a threat Hi does not appear in the area D; i f 5 f ij ; if a threat Hi appears in the area D > : and its parameter is in the range f ij ; j 5 1; 2; . . .; li ;
for i 5 1; 2; . . .; n2 :
FIGURE 3.3 Illustration of environment threats possibility of influence on the critical infrastructure operating subareas.
(3.18)
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Consequences of Maritime Critical Infrastructure Accidents
We call vectors (3.18) the environment threat state in the area D. Simultaneously, we proceed for the particular subareas Dk ; k 5 1; 2; . . .; n3 : The vector n2 1 2 sðkÞ 5 ½fðkÞ ; fðkÞ ; . . .; fðkÞ ;
k 5 1; 2; . . .; n3 ;
(3.19)
where 8 0; > < ij i fðkÞ 5 fðkÞ ; > :
if a threat Hi does not appear in the subarea Dk ; if a threat Hi appears in the subarea Dk and
(3.20)
ij its parameter is in the range fðkÞ ; j 5 1; 2; . . .; li ;
for i 5 1; 2; . . .; n2 ; k 5 1; 2; . . .; n3 is called the environment threat state in the subarea Dk : From the abovementioned definition the maximum number of the environment threat states for the subarea Dk ; k 5 1; 2; . . .; n3 is equal to υk 5 l1ðkÞ 1 1 ; l2ðkÞ 1 1 ; . . .; lnðkÞ2 1 1 ; k 5 1; 2; . . .; n3 : Further, we number the subarea environment threat states defined by (3.19) and (3.20) and mark them by sυðkÞ for υ 5 1; 2; . . .; υk ; and form the set
k 5 1; 2; . . .; n3 ;
n o SðkÞ 5 sυðkÞ ; υ 5 1; 2; . . .; υk ;
k 5 1; 2; . . .; n3 ;
where siðkÞ 6¼ sjðkÞ
for i 6¼ j;
i; jAf1; 2; . . .; υk g:
The set SðkÞ ; k 5 1; 2; . . .; n3 is called the set of the environment threat states in the subarea Dk ; k 5 1; 2; . . .; n3 ; while a number υk is called the number of the environment threat states of this subarea. A function SðkÞ ðtÞ; k 5 1; 2; . . .; n3 ; defined on the time interval tAh0; 1 NÞ and having values in the environment threat states set, SðkÞ ; k 5 1; 2; . . .; n3 ; is called the process of environment threats in the subarea Dk ; k 5 1; 2; . . .; n3 : Next, to involve the process of environment threats in the subarea Dk ; k 5 1; 2; . . .; n3 with the process of initiating events, we introduced the function Sðk=lÞ ðtÞ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω;
(3.21)
defined on the time interval tAh0; 1 NÞ; depending on the states el ; l 5 1; 2; . . .; ω of the process of initiating events EðtÞ and taking its values in the set of the environment threat states SðkÞ ; k 5 1; 2; . . .; n3 : This function is called the conditional process of environment threats in the subarea Dk ; k 5 1; 2; . . .; n3 ; while the process of initiating events EðtÞ is at the state el ; l 5 1; 2; . . .; ω: We assume a semi-Markov model (Grabski, 2015; Kołowrocki 2004, 2014; Kołowrocki and Soszy´nska-Budny, 2011; Limnios and Oprisan, 2005; Macci, 2008; Mercier, 2008) of the process Sðk=lÞ ðtÞ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω and denote by ηijðk=lÞ its random conditional sojourn times at the state siðk=lÞ , while its next transition will be done to the state sjðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω: The process of environment threats is defined by G
the vector of probabilities
Modeling critical infrastructure accident consequences Chapter | 3
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piðk=lÞ ð0Þ 5 P Sðk=lÞ ð0Þ 5 siðk=lÞ ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω; of its initial states at the moment t 5 0 h i piðk=lÞ ð0Þ G
1 3 υk
h i k 5 p1ðk=lÞ ð0Þ; p2ðk=lÞ ð0Þ; . . .; pυðk=lÞ ð0Þ ;
(3.22)
the matrix of probabilities pijðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; :::; n3 ; l 5 1; 2; :::; ω; of transitions between the states siðk=lÞ and sjðk=lÞ ; i; j 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; :::; ω; 2 3 k p11 p12 : : : p1υ ðk=lÞ ðk=lÞ ðk=lÞ 6 21 2υk 7 22 6 7 ½pijðk=lÞ υk 3 υk 5 6 pðk=lÞ pðk=lÞ : : : pðk=lÞ 7; 4 : : : : : : 5 k1 k2 k υk pυðk=lÞ pυðk=lÞ : : : pυðk=lÞ
(3.23)
where piiðk=lÞ 5 0 for i 5 1; 2; . . .; υk ; G
the matrix of conditional distribution functions
ij ðtÞ 5 P ηijðk=lÞ , t ; tAh0; 1 NÞ; Hðk=lÞ
i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω; of conditional sojourn times ηijðk=lÞ of the process Sðk=lÞ ðtÞ at the state siðk=lÞ , the state sjðk=lÞ , i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω; 2 11 12 Hðk=lÞ ðtÞ Hðk=lÞ ðtÞ : : : 6 21 22 6 ij ½Hðk=lÞ ðtÞυk 3 υk 5 6 Hðk=lÞ ðtÞ Hðk=lÞ ðtÞ : : : 4 : : : : : υk 1 υk 2 Hðk=lÞ ðtÞ Hðk=lÞ ðtÞ : : :
while the next transition will be done to 1υk Hðk=lÞ ðtÞ
3
7 2υk Hðk=lÞ ðtÞ 7; 7 5 : υk υk Hðk=lÞ ðtÞ
(3.24)
where ii Hðk=lÞ ðtÞ 5 0
for i 5 1; 2; . . .; υk ;
or equivalently by corresponding to it the matrix of conditional density functions of the conditional sojourn times ηijðk=lÞ of the process Sðk=lÞ ðtÞ at the state siðk=lÞ , while the next transition will be done to the state sjðk=lÞ , i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω; 2 3 1υk 12 h11 ðk=lÞ ðtÞ hðk=lÞ ðtÞ : : : hðk=lÞ ðtÞ 6 21 7 2υk 22 6 7 ½hijðk=lÞ ðtÞυk 3 υk 5 6 hðk=lÞ ðtÞ hðk=lÞ ðtÞ : : : hðk=lÞ ðtÞ 7; (3.25) 4 : : : : : : 5 υk 1 υk 2 υk υk hðk=lÞ ðtÞ hðk=lÞ ðtÞ : : : hðk=lÞ ðtÞ where hiiðk=lÞ ðtÞ 5 0 for i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω: We assume that the appropriate and typical distributions suitable to describe the process Sðk=lÞ ðtÞ conditional sojourn times ηijðk=lÞ , i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω at the particular states are (Kołowrocki and Soszy´nska-Budny, 2011)
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Consequences of Maritime Critical Infrastructure Accidents
´ FIGURE 3.4 The graphs of the double trapezium distribution’s density functions. Based on Kołowrocki, K., Soszynska-Budny, J., 2011. Reliability and Safety of Complex Technical Systems and Processes: Modeling Identification Prediction Optimization. Springer, London, Dordrecht, Heidelberg, New York. G
the double trapezium distribution with a density function (Fig. 3.4) 8 > 0; > > 2 3 > > ij ij ij ij ij ij > > 2 2 q z 2 x y 2 z 2 w t 2 xijðk=lÞ > ðk=lÞ ðk=lÞ ðk=lÞ ðk=lÞ ðk=lÞ ðk=lÞ > ij ij 4 5 > q 1 2 qðk=lÞ U ij ; > > < ðk=lÞ zðk=lÞ 2 xijðk=lÞ yijðk=lÞ 2 xijðk=lÞ ij 2 3 hðk=lÞ ðtÞ 5 > > 2 2 qijðk=lÞ zijðk=lÞ 2 xijðk=lÞ 2 wijðk=lÞ yijðk=lÞ 2 zijðk=lÞ yijðk=lÞ 2 t > ij ij 5 > 4 > U w 1 2 w ; > ðk=lÞ ðk=lÞ > > yijðk=lÞ 2 zijðk=lÞ yijðk=lÞ 2 xijðk=lÞ > > > > : 0; where
G
t , xijðk=lÞ xijðk=lÞ # t , zijðk=lÞ (3.26) zijðk=lÞ # t , yijðk=lÞ t $ yijðk=lÞ ;
0 # xijðk=lÞ # zijðk=lÞ # yijðk=lÞ , 1 N; 0 # qijðk=lÞ , 1 N; 0 # wijðk=lÞ , 1 N; 0 # qijðk=lÞ zijðk=lÞ 2 xijðk=lÞ 1 wijðk=lÞ yijðk=lÞ 2 zijðk=lÞ # 2;
the quasitrapezium distribution with a density function (Fig. 3.5) 8 0; > > > > > Aij 2 qijðk=lÞ > > > qij 1 ðk=lÞ > t 2 xijðk=lÞ ; > ij ij ðk=lÞ > zðk=lÞ1 2 xðk=lÞ > > < ij ij hðk=lÞ ðtÞ 5 Aðk=lÞ ; > > > Aijðk=lÞ 2 wijðk=lÞ ij > > ij > > w 1 y 2 t ; > ðk=lÞ > yijðk=lÞ 2 zijðk=lÞ2 ðk=lÞ > > > > : 0;
t , xijðk=lÞ xijðk=lÞ # t , zijðk=lÞ1 zijðk=lÞ1 # t , zijðk=lÞ2 zijðk=lÞ2 # t , yijðk=lÞ t $ yijðk=lÞ ;
(3.27)
Modeling critical infrastructure accident consequences Chapter | 3
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´ FIGURE 3.5 The graphs of the quasitrapezium distribution’s density functions. Based on Kołowrocki, K., Soszynska-Budny, J., 2011. Reliability and Safety of Complex Technical Systems and Processes: Modeling Identification Prediction Optimization. Springer, London, Dordrecht, Heidelberg, New York.
where Aijðk=lÞ 5
2 2 qijðk=lÞ zijðk=lÞ1 2 xijðk=lÞ 2 wijðk=lÞ yijðk=lÞ 2 zijðk=lÞ2 zijðk=lÞ2 2 zijðk=lÞ1 1 yijðk=lÞ 2 xijðk=lÞ
;
0 # xijðk=lÞ # zijðk=lÞ1 # zijðk=lÞ2 # yijðk=lÞ , 1 N; 0 # qijðk=lÞ , 1 N; 0 # wijðk=lÞ , 1 N; G
the exponential distribution with a density function (the graph is similar to the one presented in Fig. 3.1 than we omit it)
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Consequences of Maritime Critical Infrastructure Accidents
8 < 0;
hijðk=lÞ ðtÞ 5 :
h
αijðk=lÞ exp 2αijðk=lÞ t 2 xijðk=lÞ
i
t , xijðk=lÞ ;
t $ xijðk=lÞ ;
(3.28)
where 0 # αijðk=lÞ , 1 N; G
the chimney distribution with a density function (the graphs omit it) 8 > 0; > > > > > Aijðk=lÞ > > > ; > > > zijðk=lÞ1 2 xijðk=lÞ > > > > ij > < Cðk=lÞ ij ; hðk=lÞ ðtÞ 5 > zijðk=lÞ2 2 zijðk=lÞ1 > > > > > > Dijðk=lÞ > > > ; > > > yijðk=lÞ 2 zijðk=lÞ2 > > > > : 0;
are similar to the ones presented in Fig. 3.2 than we t , xijðk=lÞ xijðk=lÞ # t , zijðk=lÞ1 zijðk=lÞ1 # t , zijðk=lÞ2
(3.29)
zijðk=lÞ2 # t , yijðk=lÞ t $ yijðk=lÞ ;
where 0 # xijðk=lÞ # zijðk=lÞ1 # zijðk=lÞ2 # yijðk=lÞ , 1 N; ij ij Aijðk=lÞ $ 0; Cðk=lÞ $ 0; Dijðk=lÞ $ 0; Aijðk=lÞ 1 Cðk=lÞ 1 Dijðk=lÞ 5 1:
As the mean values of the conditional sojourn times ηijðk=lÞ are given by (Kołowrocki and Soszy´nska-Budny, 2011) ðN h i ðN ij ij Mðk=lÞ 5 E ηijðk=lÞ 5 tdHðk=lÞ ðtÞ 5 thijðk=lÞ ðtÞdt; (3.30) 0 0 i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω; then for the distinguished distributions (3.26)(3.29), the mean values of the process of environment threat Sðk=lÞ ðtÞ conditional sojourn times ηijðk=lÞ , i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω at the particular states are respectively given by the following (Kołowrocki and Soszy´nska-Budny, 2011): G
for the double trapezium distribution ij Mðk=lÞ
5E
h
ηijðk=lÞ
1
i
5
wijðk=lÞ yijðk=lÞ
2
xijðk=lÞ 1 yijðk=lÞ 1 zijðk=lÞ 3 2 2 qijðk=lÞ xijðk=lÞ 2
3 xijðk=lÞ yijðk=lÞ xijðk=lÞ 1 yijðk=lÞ wijðk=lÞ 1 qijðk=lÞ ij ij ij ij 4 x z 5 1 ðk=lÞ ðk=lÞ 2 yðk=lÞ zðk=lÞ 1 6 xijðk=lÞ 2 yijðk=lÞ 3 3 xijðk=lÞ qijðk=lÞ 1 yijðk=lÞ wijðk=lÞ 2 ; 3 yijðk=lÞ 2 xijðk=lÞ 2
(3.31)
Modeling critical infrastructure accident consequences Chapter | 3
G
for the quasitrapezium distribution h i qij 2 2
ðk=lÞ ij zijðk=lÞ1 2 xijðk=lÞ Mðk=lÞ 5 E ηijðk=lÞ 5 2 ij ij 2 2
Aðk=lÞ 2 qðk=lÞ 2 2 zijðk=lÞ1 2 5xijðk=lÞ zijðk=lÞ1 2 xijðk=lÞ 6 ij 2 2 wij 2 2
Aðk=lÞ ðk=lÞ ij ij ij ij 1 zðk=lÞ2 2 zðk=lÞ1 yðk=lÞ 2 zðk=lÞ2 1 2 2 2
wijðk=lÞ 2 Aijðk=lÞ ij 2 ij ij ij 2 2 zðk=lÞ2 2 5yðk=lÞ zðk=lÞ2 2 yðk=lÞ ; 6
G
(3.32)
for the exponential distribution h i ij Mðk=lÞ 5 E ηijðk=lÞ 5 xijðk=lÞ 1
G
31
1 αijðk=lÞ
;
(3.33)
for the chimney distribution h i 1h i ij ij Mðk=lÞ 5 E ηijðk=lÞ 5 Aijðk=lÞ xijðk=lÞ 1 zijðk=lÞ1 1 Cðk=lÞ zijðk=lÞ1 1 zijðk=lÞ2 1 Dijðk=lÞ zijðk=lÞ2 1 yijðk=lÞ : 2
(3.34)
From the formula for total probability, it follows that the unconditional distribution functions of the sojourn times ηiðk=lÞ ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω of the process of environment threats Sðk=lÞ ðtÞ at the states siðk=lÞ ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω are determined by (Kołowrocki and Soszy´nska-Budny, 2011) i Hðk=lÞ ðtÞ 5
υk X
ij pijðk=lÞ Hðk=lÞ ðtÞ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω:
(3.35)
j51
Therefore the mean values E½ηiðk=lÞ of the process of environment threats Sðk=lÞ ðtÞ unconditional sojourn times ηiðk=lÞ ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω at the states are given by υk X ij i Mðk=lÞ 5 E ηiðk=lÞ 5 pijðk=lÞ Mðk=lÞ ; j51
(3.36)
i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω; ij is defined by formula (3.30) in the case of any distribution of sojourn times ηijðk=lÞ and by formulae (3.31) where Mðk=lÞ (3.34) in the cases of particular distributions of these sojourn times defined, respectively, by (3.26)(3.29). The limit values of the process of environment threats Sðk=lÞ ðtÞ transient probabilities at the particular states piðk=lÞ ðtÞ 5 P Sðk=lÞ ðtÞ 5 siðk=lÞ ; (3.37) tAh0; 1 NÞ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω;
are given by (Kołowrocki and Soszy´nska-Budny, 2011) piðk=lÞ 5 lim piðk=lÞ ðtÞ 5 t- 1 N
i πiðk=lÞ Mðk=lÞ ; υk X j j πðk=lÞ Mðk=lÞ
(3.38)
j51
i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω; i Mðk=lÞ
where equations
is given by (3.36), while the steady probabilities πiðk=lÞ of the vector ½πiðk=lÞ 1 3 υk satisfy the system of
32
Consequences of Maritime Critical Infrastructure Accidents
8 i ij i > < ½πðk=lÞ 5 ½πðk=lÞ ½pðk=lÞ υk X > πjðk=lÞ 5 1: :
(3.39)
j51
½pijðk=lÞ υk 3 υk
and is given by (3.23). The asymptotic distribution of the sojourn total time η^ iðk=lÞ of the process of environment threats Sðk=lÞ ðtÞ in the time interval h0; ηi; η . 0 at the state siðk=lÞ is normal with the expected value i M^ ðk=lÞ 5 E η^ iðk=lÞ Dpiðk=lÞ η; (3.40) where piðk=lÞ is given by (3.38). Thus according to the formula for total probability and (3.14) and (3.37), the probabilities piðkÞ ðtÞ 5 PðSðtÞ 5 siðkÞ Þ;
tAh0; 1 NÞ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ;
(3.41)
are defined by piðkÞ ðtÞ 5
ω ω X
X P EðtÞ 5 el UP SðkÞ ðtÞ 5 siðkÞ EðtÞ 5 el 5 pl ðtÞUpiðk=lÞ ðtÞ; l51
i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ;
l51
and according to (3.15) and (3.38), their limit forms are piðkÞ 5
ω X
pl Upiðk=lÞ ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 :
(3.42)
l51
3.3
Modeling process of environment degradation
The particular states of the process of the environment threats SðkÞ ðtÞ in the subarea Dk ; k 5 1; 2; . . .; n3 may lead to dangerous effects degrading the environment at this subarea. Thus we assume that there are mk different dangerous degradation effects for the environment subarea Dk ; k 5 1; 2; . . .; n3 , and we mark them by k R1ðkÞ ; R2ðkÞ ; . . .; Rm ðkÞ :
This way, the set
n o k RðkÞ 5 R1ðkÞ ; R2ðkÞ ; . . .; Rm ðkÞ ; k 5 1; 2; . . .; n3 ;
is the set of degradation effects for the environment in the subarea Dk : These degradation effects may attain different levels. Namely, the degradation effect Rm ðkÞ ; m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 ; may reach ν m ðkÞ levels mν m
ðkÞ m2 Rm1 ðkÞ ; RðkÞ ; . . .; RðkÞ ; m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 ;
that are called the states of this degradation effect. The set n o mν m ðkÞ m1 m2 Rm ; m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 ; ðkÞ 5 RðkÞ ; RðkÞ ; . . .; RðkÞ is called the set of states of the degradation effect Rm ðkÞ ; m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 for the environment in the subarea Dk ; k 5 1; 2; . . .; n3 : Under the abovementioned assumptions, we can introduce the environment subarea degradation process as a vector h i k RðkÞ ðtÞ 5 R1ðkÞ ðtÞ; R2ðkÞ ðtÞ; . . .; Rm (3.43) ðkÞ ðtÞ ;
Modeling critical infrastructure accident consequences Chapter | 3
33
where Rm ðkÞ ðtÞ; m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 ; are the processes of degradation effects for the environment in the subarea Dk ; k 5 1; 2; . . .; n3 defined on the time interval tAh0; 1 NÞ and having their values in the degradation effect state sets
The vector
Rm ðkÞ ; m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 :
(3.44)
h i mk m 1 2 5 dðkÞ ; dðkÞ ; . . .; dðkÞ rðkÞ ; k 5 1; 2; . . .; n3 ;
(3.45)
where 8 0; if a degradation effect Rm ðkÞ does not appear in the subarea Dk ; > > < m dðkÞ 5 Rmj ; if a degradation effect Rm appears in the subarea D k > ðkÞ ðkÞ > : mj m and its level is equal to RðkÞ ; j 5 1; 2; . . .; ν ðkÞ ;
(3.46)
for m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 is called the degradation state in the subarea Dk ; k 5 1; 2; . . .; n3 : From the abovementioned definition the maximum number of the environment degradation states for the subarea Dk ; k 5 1; 2; . . .; n3 is equal to k ‘k 5 ν 1ðkÞ 1 1 ; ν 2ðkÞ 1 1 ; . . .; ν m ðkÞ 1 1 ; k 5 1; 2; . . .; n3 : Further, we number the subarea degradation states defined by (3.45) and (3.46) and mark them by ‘ rðkÞ
for ‘ 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ;
and form the set of degradation states
n o ‘ ; ‘ 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; RðkÞ 5 rðkÞ
where j i rðkÞ 6¼ rðkÞ
for i 6¼ j; i; jAf1; 2; . . .; ‘k g:
The set RðkÞ ; k 5 1; 2; . . .; n3 is called the set of the environment degradation states in the subarea Dk ; k 5 1; 2; . . .; n3 ; while a number ‘k is called the number of the environment degradation states of this subarea. A function RðkÞ ðtÞ; k 5 1; 2; . . .; n3 ; defined on the time interval tAh0; 1 NÞ and having values in the environment degradation states set RðkÞ ; k 5 1; 2; . . .; n3 ; is called the process of the environment degradation in the subarea Dk ; k 5 1; 2; . . .; n3 : Next, to involve the process of environment degradation with the process of the environment threats, we define the conditional process of environment degradation Rðk=υÞ ðtÞ, while the process of the environment threats SðkÞ ðtÞ in the subarea Dk ; k 5 1; 2; . . .; n3 is at the state sυðkÞ ; υ 5 1; 2; . . .; υk as a vector h i k ðtÞ ; (3.47) Rðk=υÞ ðtÞ 5 R1ðk=υÞ ðtÞ; R2ðk=υÞ ðtÞ; . . .; Rm ðk=υÞ where Rm ðk=υÞ ðtÞ; m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ; defined on the time interval tAh0; 1 NÞ and having values in the degradation effect states set Rm ðkÞ ; m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 :
34
Consequences of Maritime Critical Infrastructure Accidents
The abovementioned definition means that the conditional process of environment degradation Rðk=υÞ ðtÞ; tAh0; 1 NÞ also takes the degradation states from the set RðkÞ of the unconditional process of environment degradation RðkÞ ðtÞ; tAh0; 1 NÞ defined by (3.43). We assume a semi-Markov model (Grabski, 2015; Kołowrocki 2004, 2014; Kołowrocki and Soszy´nska-Budny, 2011; Limnios and Oprisan, 2005; Macci, 2008; Mercier, 2008) of the conditional process of environment degradation in the subarea Dk Rðk=υÞ ðtÞ; tAh0; 1 NÞ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ; i and denote by ζ ijðk=υÞ its random conditional sojourn times in the state rðk=υÞ , while its next transition will be done to the j state rðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk : The process of environment degradation is defined by G
the vector of probabilities
i qiðk=υÞ ð0Þ 5 P Rðk=υÞ ð0Þ 5 rðkÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ;
of its initial states at the moment t 5 0 h i qiðk=υÞ ð0Þ G
1 3 ‘k
h i k 5 q1ðk=υÞ ð0Þ; q2ðk=υÞ ð0Þ; . . .; q‘ðk=υÞ ð0Þ ;
(3.48)
the matrix of probabilities qijðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ; j i of transitions between the states rðk=υÞ and rðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; 2 11 3 k qðk=υÞ q12 : : : q1‘ ðk=υÞ ðk=υÞ 6 21 h i 2‘k 7 22 6 7 qijðk=υÞ 5 6 qðk=υÞ qðk=υÞ : : : qðk=υÞ 7; 4 : ‘k 3 ‘k : : : : : 5 ‘k 1 ‘k 2 ‘k ‘k qðk=υÞ qðk=υÞ : : : qðk=υÞ
(3.49)
where qiiðk=υÞ 5 0 G
for i 5 1; 2; . . .; ‘k ;
the matrix of conditional distribution functions
Gijðk=υÞ ðtÞ 5 P ζ ijðk=υÞ , t ; tAh0; 1 NÞ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ;
i , while its next transition will of conditional sojourn times ζ ijðk=υÞ of the degradation process Rðk=υÞ ðtÞ in the state rðk=υÞ j be done to the state rðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ; 2 3 1‘k 12 G11 ðk=υÞ ðtÞ Gðk=υÞ ðtÞ : : : Gðk=υÞ ðtÞ 6 21 7 h i 2‘k 22 6 7 Gijðk=υÞ ðtÞ 5 6 Gðk=υÞ ðtÞ Gðk=υÞ ðtÞ : : : Gðk=υÞ ðtÞ 7; (3.50) 4 5 ‘k 3 ‘k : : : : : : ‘k 1 k2 k ‘k Gðk=υÞ ðtÞ G‘ðk=υÞ ðtÞ : : : G‘ðk=υÞ ðtÞ
where Giiðk=υÞ ðtÞ 5 0
for i 5 1; 2; . . .; ‘k ;
or equivalently by corresponding to it the matrix of conditional density functions of conditional sojourn times ζ ijðk=υÞ j i of the process Rðk=υÞ ðtÞ at the state rðk=υÞ ; while its next transition will be done to the state rðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ;
Modeling critical infrastructure accident consequences Chapter | 3
2 h
i gijðk=υÞ ðtÞ
12 g11 ðk=υÞ ðtÞ gðk=υÞ ðtÞ :
:
6 21 22 6 5 6 gðk=υÞ ðtÞ gðk=υÞ ðtÞ : 4 ‘k 3 ‘k : : : ‘k 1 ‘k 2 gðk=υÞ ðtÞ gðk=υÞ ðtÞ :
: : :
k : g1‘ ðk=υÞ ðtÞ
35
3
7 k 7 : g2‘ ðk=υÞ ðtÞ 7; 5 : : k ‘k : g‘ðk=υÞ ðtÞ
(3.51)
where giiðk=υÞ ðtÞ 5 0
G
for i 5 1; 2; . . .; ‘k :
We assume that the appropriate and typical distributions suitable to describe the process Rðk=υÞ ðtÞ conditional sojourn times ζ ijðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk at the particular states are similar to distributions describing the process of initiating events EðtÞ that graphs are presented in Section 3.1 in Figs. 3.1 and 3.2 than we omit them and in Kołowrocki and Soszy´nska-Budny (2011). There are two types of distribution with a density function: the exponential distribution with a density function 8 < 0; t , xijðk=υÞ ij h i gðk=υÞ ðtÞ 5 (3.52) ij ij : αij exp 2αij t 2 x ; ; t $ x ðk=υÞ ðk=υÞ ðk=υÞ ðk=υÞ where 0 # αijðk=υÞ , 1 N;
G
the chimney distribution with a density function 8 > 0; > > > > > Aijðk=υÞ > > > ; > > > zijðk=υÞ1 2 xijðk=υÞ > > > > ij > < Cðk=υÞ ij ; gðk=υÞ ðtÞ 5 > zijðk=υÞ2 2 zijðk=υÞ1 > > > > > > Dijðk=υÞ > > > ; > > > yijðk=υÞ 2 zijðk=υÞ2 > > > > : 0;
t , xijðk=υÞ xijðk=υÞ # t , zijðk=υÞ1 zijðk=υÞ1 # t , zijðk=υÞ2
(3.53)
zijðk=υÞ2 # t , yijðk=υÞ t $ yijðk=υÞ ;
where 0 # xijðk=υÞ # zijðk=υÞ1 # zijðk=υÞ2 # yijðk=υÞ , 1 N; ij ij Aijðk=υÞ $ 0; Cðk=υÞ $ 0; Dijðk=υÞ $ 0; Aijðk=υÞ 1 Cðk=υÞ 1 Dijðk=υÞ 5 1:
As the mean values of the conditional sojourn times ζ ijðk=υÞ are given by (Kołowrocki and Soszy´nska-Budny, 2011) h i Ð ÐN N ij 5 E ζ ijðk=υÞ 5 0 tdGijðk=υÞ ðtÞ 5 0 tgijðk=υÞ ðtÞdt; Mðk=υÞ (3.54) i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ; then for the distinguished distributions (3.52) and (3.53), the mean values of the process of environment degradation Rðk=υÞ ðtÞ conditional sojourn times ζ ijðk=υÞ i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk at the particular states are respectively given by the following (Kołowrocki and Soszy´nska-Budny, 2011): G
for the exponential distribution h i ij Mðk=υÞ 5 E ζ ijðk=υÞ 5 xijðk=υÞ 1
1 αijðk=υÞ
;
(3.55)
36
G
Consequences of Maritime Critical Infrastructure Accidents
for the chimney distribution h i 1h i ij ij Mðk=υÞ 5 E ζ ijðk=υÞ 5 Aijðk=υÞ xijðk=υÞ 1 zijðk=υÞ1 1 Cðk=υÞ zijðk=υÞ1 1 zijðk=υÞ2 1 Dijðk=υÞ zijðk=υÞ2 1 yijðk=υÞ : 2
(3.56)
From the formula for total probability, it follows that the unconditional distribution functions of the sojourn times ζ iðk=υÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk of the process of environment degradation Rðk=υÞ ðtÞ at the states i rðk=υÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk are determined by (Kołowrocki and Soszy´nska-Budny, 2011) Giðk=υÞ ðtÞ 5
‘k X qijðk=υÞ Gijðk=υÞ ðtÞ;
(3.57)
j51
i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk : E½ζ iðk=υÞ
of the process of environment degradation Rðk=υÞ ðtÞ unconditional sojourn times Therefore the mean values ζ iðk=υÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk at the states are given by ‘k h i X ij i 5 E ζ iðk=υÞ 5 qijðk=υÞ Mðk=υÞ ; Mðk=υÞ
(3.58)
j51
i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ; ij is defined by the formula (3.54) in a case of any distribution of sojourn times ζ ijðk=υÞ and by formulae where Mðk=υÞ (3.55) and (3.56) in the cases of particular distributions of these sojourn times defined, respectively, by (3.52) and (3.53). The limit values of the process of environment degradation Rðk=υÞ ðtÞ transient probabilities at the particular states i qiðk=υÞ ðtÞ 5 P Rðk=υÞ ðtÞ 5 rðk=υÞ ; (3.59) tAh0; 1 NÞ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ;
are given by (Kołowrocki and Soszy´nska-Budny, 2011) qiðk=υÞ 5 lim qiðk=υÞ ðtÞ 5 t- 1 N
i πiðk=υÞ Mðk=υÞ ‘k X j πjðk=υÞ Mðk=υÞ
; (3.60)
j51
i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ;
h i i where Mðk=υÞ is given by (3.58), while the steady probabilities πiðk=υÞ of the vector πiðk=υÞ satisfy the system of 1 3 ‘k equations i h ih i 8h ij i i > π 5 π q > ðk=υÞ ðk=υÞ < ðk=υÞ > > :
‘k X
πjðk=υÞ 5 1
j51
(3.61)
and qijðk=υÞ ‘k 3 ‘k is given by (3.49). i The asymptotic distribution of the sojourn total time ζ^ ðk=υÞ of the process of environment degradation Rðk=υÞ ðtÞ in the i time interval h0; ζi; ζ . 0; at the state rðk=υÞ is normal with the expected value h i i i M^ ðk=υÞ 5 E ζ^ ðk=υÞ Dqiðk=υÞ ζ; (3.62) where qiðk=υÞ is given by (3.60). Thus according to the formula for total probability and (3.41) and (3.59), the probabilities, i qiðkÞ ðtÞ 5 P RðtÞ 5 rðkÞ ; tAh0; 1 NÞ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ;
Modeling critical infrastructure accident consequences Chapter | 3
37
are defined by qiðkÞ ðtÞ 5
υk υk
X X i P SðtÞ 5 sυðkÞ UP RðkÞ ðtÞ 5 rðkÞ pυðkÞ ðtÞUqiðk=υÞ ðtÞ;
SðtÞ 5 sυðkÞ 5 υ51
i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 :
υ51
Therefore according to (3.42) and (3.60), for sufficiently large t, the limit transient probabilities of the unconditional process of the environment degradation RðkÞ ðtÞ at its particular states are given by " # υk υk X ω X X υ i l υ p Upðk=lÞ qiðk=υÞ qiðkÞ D pðkÞ Uqðk=υÞ 5 (3.63) υ51 υ51 l51 i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; where pl ; pυðk=lÞ , and qiðk=υÞ are defined, respectively, by (3.15), (3.38), and (3.60). i Therefore the sojourn total time ζ^ ðkÞ of the unconditional process of the environment degradation RðkÞ ðtÞ i k 5 1; 2; . . .; n3 ; in the time interval h0; θi; θ . 0; at the state rðkÞ has the normal distribution with the expected value h i i i M^ ðkÞ 5 E ζ^ ðkÞ DqiðkÞ θ; i 5 1; 2; . . .; ‘k ; (3.64) where qiðkÞ is given by (3.63).
Modeling critical infrastructure accident consequences Chapter Outline 3.1 Modeling process of initiating events 3.2 Modeling process of environment threats
21 25
3.3 Modeling process of environment degradation
32
The probabilistic models of these processes of initiating events, environment threats, and environment degradation are designed. The vectors of initial probabilities of these processes staying at their particular states, the matrices of probabilities of these processes transitions between their particular states, and the matrices of conditional distribution functions and density functions of these processes conditional sojourn times at their particular states are defined. The distributions such as uniform, exponential, chimney, double trapezium, and quasitrapezium are suggested and introduced to describe the processes conditional sojourn times at the particular states. Moreover, the mean values of particular processes’ conditional sojourn times having these distributions at their states are defined and the unconditional distribution functions of sojourn times at particular states of these processes, the mean values of their unconditional sojourn times and limit values of transient probabilities at particular states are defined. Further, the superposition of these processes of initiating events, environment threats, and environment degradation is done to create the joint probabilistic general model of critical infrastructure accident consequences. Finally, the unconditional transient probabilities and the mean values of sojourn total times of the joint general process at its particular states for the sufficiently large time are determined.
3.1
Modeling process of initiating events
To model the process of initiating events, we fix the time interval tAh0; 1 NÞ as the time of a critical infrastructure operation and distinguish n1 ; n1 AN; events initiating the dangerous situation for the critical infrastructure operating environment and mark them by E1 ; E2 ; . . .; En1 : Further, we introduce a set of vectors (3.1) E 5 e: e 5 e1 ; e2 ; . . .; en1 ; ei Af0; 1g ; where
ei 5
1; 0;
if the initiating event Ei occurs; if the initiating event Ei does not occur;
(3.2)
for i 5 1; 2; . . .; n1 : We call vectors (3.1) the initiating events state. We may eliminate vectors that cannot occur, and we number the remaining states of the set E from l 5 1 up to ω; ωAN; where ω is the number of different elements of the set E 5 fe1 ; e2 ; . . .; eω g; where
h i el 5 el1 ; el2 ; . . .; eln1 ; l 5 1; 2; . . .; ω; and eli Af0; 1g; i 5 1; 2; . . .; n1 :
Next, we can define the process of initiating events EðtÞ on the time interval tAh0; 1 NÞ with its discrete states from the set Consequences of Maritime Critical Infrastructure Accidents. DOI: https://doi.org/10.1016/B978-0-12-819675-5.00003-6 © 2020 Elsevier Inc. All rights reserved.
21
22
Consequences of Maritime Critical Infrastructure Accidents
E 5 e1 ; e2 ; . . .; eω : After that, we assume a semi-Markov model (Grabski, 2015; Kołowrocki 2004, 2014; Kołowrocki and Soszy´nskaBudny, 2011; Limnios and Oprisan, 2005; Macci, 2008; Mercier, 2008) of the process of initiating events EðtÞ and denote by θlj its random conditional sojourn time at the state el , while its next transition will be done to the state ej ; l; j 5 1; 2; . . .; ω; l 6¼ j: This way, the process of initiating events can be described by G
the vector of the probabilities pl ð0Þ 5 PðEð0Þ 5 el Þ; l 5 1; 2; . . .; ω; of its initial states at the moment t 5 0 ½pl ð0Þ1 3 ω 5 ½p1 ð0Þ; p2 ð0Þ; . . .; pω ð0Þ;
G
(3.3)
the matrix of probabilities of transitions between the states el and ej ; l; j 5 1; 2; . . .; ω; l 6¼ j; 2 11 3 p p12 : : : p1ω 6 p21 p22 : : : p2ω 7 lj 7; p ω3ω 5 6 4 : : : : : : 5 pω1 pω2 : : : pωω
(3.4)
where by formal agreement ’l 5 1; 2; . . .; ω; pll 5 0; G
the matrix of conditional distribution functions tAh0; 1 NÞ;
H lj ðtÞ 5 Pðθlj , tÞ;
l; j 5 1; 2; . . .; ω;
l 6¼ j;
of conditional sojourn times θlj of the process EðtÞ at the state el , while its next transition will be done to the state ej ; l; j 5 1; 2; . . .; ω; 2 11 3 H ðtÞ H 12 ðtÞ : : : H 1ω ðtÞ 6 H 21 ðtÞ H 22 ðtÞ : : : H 2ω ðtÞ 7 lj 7; H ðtÞ ω 3 ω 5 6 (3.5) 4 : : : : : : 5 ω1 ω2 ωω H ðtÞ H ðtÞ : : : H ðtÞ where by formal agreement ’l 5 1; 2; . . .; ω; H ll ðtÞ 5 0; which is strictly related to the matrix of corresponding conditional density functions of conditional sojourn times θlj of the process EðtÞ at the state el , while its next transition will be done to the state ej ; l; j 5 1; 2; . . .; ω; l 6¼ j; 2 11 3 h ðtÞ h12 ðtÞ : : : h1ω ðtÞ 6 h21 ðtÞ h22 ðtÞ : : : h2ω ðtÞ 7 lj 7; h ðtÞ ω 3 ω 5 6 (3.6) 4 : : : : : : 5 hω1 ðtÞ hω2 ðtÞ : : : hωω ðtÞ where by formal agreement ’l 5 1; 2; . . .; ω; hll ðtÞ 5 0: We assume that the appropriate and typical distributions suitable to describe the process EðtÞ conditional sojourn times θlj , l; j 5 1; 2; . . .; ω; l 6¼ j at the particular states are (Kołowrocki and Soszy´nska-Budny, 2011) G
the exponential distribution with a density function (Fig. 3.1) h ðtÞ 5 lj
0; αlj exp½ 2 αlj ðt 2 xlj Þ;
where 0 # αlj , 1 N;
t , xlj t $ xlj ;
(3.7)
Modeling critical infrastructure accident consequences Chapter | 3
23
´ FIGURE 3.1 The graph of the exponential distribution’s density function. Based on Kołowrocki, K., Soszynska-Budny, J., 2011. Reliability and Safety of Complex Technical Systems and Processes: Modeling Identification Prediction Optimization. Springer, London, Dordrecht, Heidelberg, New York. G
the chimney distribution with a density function (Fig. 3.2) 8 0; > > > > > Alj > > ; > > lj > z1 2 xlj > > > > > > < Clj lj ; h ðtÞ 5 zlj2 2 zlj1 > > > > > > > Dlj > > ; > > > ylj 2 zlj2 > > > > : 0;
t , xlj xlj # t , zlj1 zlj1 # t , zlj2
(3.8)
zlj2 # t , ylj t $ ylj ;
where 0 # xlj # zlj1 # zlj2 # ylj , 1 N; Alj $ 0; C lj $ 0; Dlj $ 0; Alj 1 Clj 1 Dlj 5 1: As the mean values of the conditional sojourn times θlj are given by (Kołowrocki and Soszy´nska-Budny, 2011) ðN ðN M lj 5 E½θlj 5 tdH lj ðtÞ 5 thlj ðtÞdt; l; j 5 1; 2; . . .; ω; l 6¼ j; (3.9) 0
0
then for the distinguished distributions (3.7) and (3.8), the mean values of the process of initiating events EðtÞ conditional sojourn times θlj ; l; j 5 1; 2; . . .; ω; l 6¼ j; at the particular states are respectively given by the following (Kołowrocki and Soszy´nska-Budny, 2011): G
G
for the exponential distribution 1 M lj 5 E θlj 5 xlj 1 lj ; α
(3.10)
i 1h M lj 5 E θlj 5 Alj xlj 1 zlj1 1 Clj zlj1 1 zlj2 1 Dlj zlj2 1 ylj : 2
(3.11)
for the chimney distribution
24
Consequences of Maritime Critical Infrastructure Accidents
´ FIGURE 3.2 The graphs of the chimney distribution’s density functions. Based on Kołowrocki, K., Soszynska-Budny, J., 2011. Reliability and Safety of Complex Technical Systems and Processes: Modeling Identification Prediction Optimization. Springer, London, Dordrecht, Heidelberg, New York.
From the formula for total probability, it follows that the unconditional distribution functions of the sojourn times θl ; l 5 1; 2; . . .; ω of the process of initiating events EðtÞ at the states el ; l 5 1; 2; . . .; ω are determined by (Kołowrocki and Soszy´nska-Budny, 2011) H l ðtÞ 5
ω X
plj H lj ðtÞ; l 5 1; 2; . . .; ω:
(3.12)
j51
Therefore the mean values E½θl of the process of initiating events EðtÞ unconditional sojourn times θl ; l 5 1; 2; . . .; ω at the states are given by M l 5 E½θl 5
ω X
plj M lj ; l 5 1; 2; . . .; ω;
(3.13)
j51
where M lj is defined by formula (3.9) in the case of any distribution of sojourn times θlj and by formulae (3.10), (3.11) in the cases of particular distribution of these sojourn times defined, respectively, by (3.7), (3.8). The limit values of the process of initiating events EðtÞ transient probabilities at the particular states pl ðtÞ 5 PðEðtÞ 5 el Þ; tAh0; 1 NÞ; l 5 1; 2; . . .; ω;
(3.14)
Modeling critical infrastructure accident consequences Chapter | 3
25
are given by (Kołowrocki and Soszy´nska-Budny, 2011) pl 5 lim pl ðtÞ 5 t- 1 N
πl M l
ω P
πj M j
; l 5 1; 2; . . .; ω;
(3.15)
j51
where M is given by (3.13), while the steady probabilities πl of the vector ½πl 1 3 ω satisfy the system of equations 8 l l lj > < ½π 5 ½π ½p ω X (3.16) πj 5 1 > : l
j51
and ½plj ω 3 ω is given by (3.4). l The asymptotic distribution of the sojourn total time θ^ of the process of initiating events EðtÞ in the time interval l h0; θi; θ . 0; at the state e is normal with the expected value l l M^ 5 E½θ^ Dpl θ; l 5 1; 2; . . .; ω;
(3.17)
where pl is given by (3.15).
3.2
Modeling process of environment threats
To construct the general model of the environment threats caused by the process of initiating events generated by critical infrastructure loss of required safety critical level, we distinguish the set of n2 ; n2 AN kinds of threats as the consequences of initiating events that may cause the environment degradation and denote them by H1 ; H2 ; . . .; Hn2 : We also distinguish n3 ; n3 AN environment subareas D1 ; D2 ; . . .; Dn3 of the considered critical infrastructure operating in the environment area D 5 D1 , D2 , ? , Dn3 that may be degraded by the environment threats Hi ; i 5 1; 2; . . .; n2 : The environment threats possibility of influence on the distinguished subareas is presented in Fig. 3.3. We assume that the operating area D can be affected by some of threats Hi ; i 5 1; 2; . . .; n2 , and that a particular environment threat Hi ; i 5 1; 2; . . .; n2 ; can be characterized by the parameter f i ; i 5 1; 2; . . .; n2 : Moreover, we assume that the scale of the threat Hi ; i 5 1; 2; . . .; n2 influence on area D depends on the range of its parameter value and for i particular parameter f i ; i 5 1; 2; . . .; n2 . We distinguish li ranges f i1 ; f i2 ; . . .; f il of its values. After that, we introduce a set of vectors S 5 fs: s 5 ½f 1 ; f 2 ; . . .; f n2 g; where
8 > < 0; if a threat Hi does not appear in the area D; i f 5 f ij ; if a threat Hi appears in the area D > : and its parameter is in the range f ij ; j 5 1; 2; . . .; li ;
for i 5 1; 2; . . .; n2 :
FIGURE 3.3 Illustration of environment threats possibility of influence on the critical infrastructure operating subareas.
(3.18)
26
Consequences of Maritime Critical Infrastructure Accidents
We call vectors (3.18) the environment threat state in the area D. Simultaneously, we proceed for the particular subareas Dk ; k 5 1; 2; . . .; n3 : The vector n2 1 2 sðkÞ 5 ½fðkÞ ; fðkÞ ; . . .; fðkÞ ;
k 5 1; 2; . . .; n3 ;
(3.19)
where 8 0; > < ij i fðkÞ 5 fðkÞ ; > :
if a threat Hi does not appear in the subarea Dk ; if a threat Hi appears in the subarea Dk and
(3.20)
ij its parameter is in the range fðkÞ ; j 5 1; 2; . . .; li ;
for i 5 1; 2; . . .; n2 ; k 5 1; 2; . . .; n3 is called the environment threat state in the subarea Dk : From the abovementioned definition the maximum number of the environment threat states for the subarea Dk ; k 5 1; 2; . . .; n3 is equal to υk 5 l1ðkÞ 1 1 ; l2ðkÞ 1 1 ; . . .; lnðkÞ2 1 1 ; k 5 1; 2; . . .; n3 : Further, we number the subarea environment threat states defined by (3.19) and (3.20) and mark them by sυðkÞ for υ 5 1; 2; . . .; υk ; and form the set
k 5 1; 2; . . .; n3 ;
n o SðkÞ 5 sυðkÞ ; υ 5 1; 2; . . .; υk ;
k 5 1; 2; . . .; n3 ;
where siðkÞ 6¼ sjðkÞ
for i 6¼ j;
i; jAf1; 2; . . .; υk g:
The set SðkÞ ; k 5 1; 2; . . .; n3 is called the set of the environment threat states in the subarea Dk ; k 5 1; 2; . . .; n3 ; while a number υk is called the number of the environment threat states of this subarea. A function SðkÞ ðtÞ; k 5 1; 2; . . .; n3 ; defined on the time interval tAh0; 1 NÞ and having values in the environment threat states set, SðkÞ ; k 5 1; 2; . . .; n3 ; is called the process of environment threats in the subarea Dk ; k 5 1; 2; . . .; n3 : Next, to involve the process of environment threats in the subarea Dk ; k 5 1; 2; . . .; n3 with the process of initiating events, we introduced the function Sðk=lÞ ðtÞ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω;
(3.21)
defined on the time interval tAh0; 1 NÞ; depending on the states el ; l 5 1; 2; . . .; ω of the process of initiating events EðtÞ and taking its values in the set of the environment threat states SðkÞ ; k 5 1; 2; . . .; n3 : This function is called the conditional process of environment threats in the subarea Dk ; k 5 1; 2; . . .; n3 ; while the process of initiating events EðtÞ is at the state el ; l 5 1; 2; . . .; ω: We assume a semi-Markov model (Grabski, 2015; Kołowrocki 2004, 2014; Kołowrocki and Soszy´nska-Budny, 2011; Limnios and Oprisan, 2005; Macci, 2008; Mercier, 2008) of the process Sðk=lÞ ðtÞ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω and denote by ηijðk=lÞ its random conditional sojourn times at the state siðk=lÞ , while its next transition will be done to the state sjðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω: The process of environment threats is defined by G
the vector of probabilities
Modeling critical infrastructure accident consequences Chapter | 3
27
piðk=lÞ ð0Þ 5 P Sðk=lÞ ð0Þ 5 siðk=lÞ ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω; of its initial states at the moment t 5 0 h i piðk=lÞ ð0Þ G
1 3 υk
h i k 5 p1ðk=lÞ ð0Þ; p2ðk=lÞ ð0Þ; . . .; pυðk=lÞ ð0Þ ;
(3.22)
the matrix of probabilities pijðk=lÞ ; i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; :::; n3 ; l 5 1; 2; :::; ω; of transitions between the states siðk=lÞ and sjðk=lÞ ; i; j 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; :::; ω; 2 3 k p11 p12 : : : p1υ ðk=lÞ ðk=lÞ ðk=lÞ 6 21 2υk 7 22 6 7 ½pijðk=lÞ υk 3 υk 5 6 pðk=lÞ pðk=lÞ : : : pðk=lÞ 7; 4 : : : : : : 5 k1 k2 k υk pυðk=lÞ pυðk=lÞ : : : pυðk=lÞ
(3.23)
where piiðk=lÞ 5 0 for i 5 1; 2; . . .; υk ; G
the matrix of conditional distribution functions
ij ðtÞ 5 P ηijðk=lÞ , t ; tAh0; 1 NÞ; Hðk=lÞ
i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω; of conditional sojourn times ηijðk=lÞ of the process Sðk=lÞ ðtÞ at the state siðk=lÞ , the state sjðk=lÞ , i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω; 2 11 12 Hðk=lÞ ðtÞ Hðk=lÞ ðtÞ : : : 6 21 22 6 ij ½Hðk=lÞ ðtÞυk 3 υk 5 6 Hðk=lÞ ðtÞ Hðk=lÞ ðtÞ : : : 4 : : : : : υk 1 υk 2 Hðk=lÞ ðtÞ Hðk=lÞ ðtÞ : : :
while the next transition will be done to 1υk Hðk=lÞ ðtÞ
3
7 2υk Hðk=lÞ ðtÞ 7; 7 5 : υk υk Hðk=lÞ ðtÞ
(3.24)
where ii Hðk=lÞ ðtÞ 5 0
for i 5 1; 2; . . .; υk ;
or equivalently by corresponding to it the matrix of conditional density functions of the conditional sojourn times ηijðk=lÞ of the process Sðk=lÞ ðtÞ at the state siðk=lÞ , while the next transition will be done to the state sjðk=lÞ , i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω; 2 3 1υk 12 h11 ðk=lÞ ðtÞ hðk=lÞ ðtÞ : : : hðk=lÞ ðtÞ 6 21 7 2υk 22 6 7 ½hijðk=lÞ ðtÞυk 3 υk 5 6 hðk=lÞ ðtÞ hðk=lÞ ðtÞ : : : hðk=lÞ ðtÞ 7; (3.25) 4 : : : : : : 5 υk 1 υk 2 υk υk hðk=lÞ ðtÞ hðk=lÞ ðtÞ : : : hðk=lÞ ðtÞ where hiiðk=lÞ ðtÞ 5 0 for i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω: We assume that the appropriate and typical distributions suitable to describe the process Sðk=lÞ ðtÞ conditional sojourn times ηijðk=lÞ , i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω at the particular states are (Kołowrocki and Soszy´nska-Budny, 2011)
28
Consequences of Maritime Critical Infrastructure Accidents
´ FIGURE 3.4 The graphs of the double trapezium distribution’s density functions. Based on Kołowrocki, K., Soszynska-Budny, J., 2011. Reliability and Safety of Complex Technical Systems and Processes: Modeling Identification Prediction Optimization. Springer, London, Dordrecht, Heidelberg, New York. G
the double trapezium distribution with a density function (Fig. 3.4) 8 > 0; > > 2 3 > > ij ij ij ij ij ij > > 2 2 q z 2 x y 2 z 2 w t 2 xijðk=lÞ > ðk=lÞ ðk=lÞ ðk=lÞ ðk=lÞ ðk=lÞ ðk=lÞ > ij ij 4 5 > q 1 2 qðk=lÞ U ij ; > > < ðk=lÞ zðk=lÞ 2 xijðk=lÞ yijðk=lÞ 2 xijðk=lÞ ij 2 3 hðk=lÞ ðtÞ 5 > > 2 2 qijðk=lÞ zijðk=lÞ 2 xijðk=lÞ 2 wijðk=lÞ yijðk=lÞ 2 zijðk=lÞ yijðk=lÞ 2 t > ij ij 5 > 4 > U w 1 2 w ; > ðk=lÞ ðk=lÞ > > yijðk=lÞ 2 zijðk=lÞ yijðk=lÞ 2 xijðk=lÞ > > > > : 0; where
G
t , xijðk=lÞ xijðk=lÞ # t , zijðk=lÞ (3.26) zijðk=lÞ # t , yijðk=lÞ t $ yijðk=lÞ ;
0 # xijðk=lÞ # zijðk=lÞ # yijðk=lÞ , 1 N; 0 # qijðk=lÞ , 1 N; 0 # wijðk=lÞ , 1 N; 0 # qijðk=lÞ zijðk=lÞ 2 xijðk=lÞ 1 wijðk=lÞ yijðk=lÞ 2 zijðk=lÞ # 2;
the quasitrapezium distribution with a density function (Fig. 3.5) 8 0; > > > > > Aij 2 qijðk=lÞ > > > qij 1 ðk=lÞ > t 2 xijðk=lÞ ; > ij ij ðk=lÞ > zðk=lÞ1 2 xðk=lÞ > > < ij ij hðk=lÞ ðtÞ 5 Aðk=lÞ ; > > > Aijðk=lÞ 2 wijðk=lÞ ij > > ij > > w 1 y 2 t ; > ðk=lÞ > yijðk=lÞ 2 zijðk=lÞ2 ðk=lÞ > > > > : 0;
t , xijðk=lÞ xijðk=lÞ # t , zijðk=lÞ1 zijðk=lÞ1 # t , zijðk=lÞ2 zijðk=lÞ2 # t , yijðk=lÞ t $ yijðk=lÞ ;
(3.27)
Modeling critical infrastructure accident consequences Chapter | 3
29
´ FIGURE 3.5 The graphs of the quasitrapezium distribution’s density functions. Based on Kołowrocki, K., Soszynska-Budny, J., 2011. Reliability and Safety of Complex Technical Systems and Processes: Modeling Identification Prediction Optimization. Springer, London, Dordrecht, Heidelberg, New York.
where Aijðk=lÞ 5
2 2 qijðk=lÞ zijðk=lÞ1 2 xijðk=lÞ 2 wijðk=lÞ yijðk=lÞ 2 zijðk=lÞ2 zijðk=lÞ2 2 zijðk=lÞ1 1 yijðk=lÞ 2 xijðk=lÞ
;
0 # xijðk=lÞ # zijðk=lÞ1 # zijðk=lÞ2 # yijðk=lÞ , 1 N; 0 # qijðk=lÞ , 1 N; 0 # wijðk=lÞ , 1 N; G
the exponential distribution with a density function (the graph is similar to the one presented in Fig. 3.1 than we omit it)
30
Consequences of Maritime Critical Infrastructure Accidents
8 < 0;
hijðk=lÞ ðtÞ 5 :
h
αijðk=lÞ exp 2αijðk=lÞ t 2 xijðk=lÞ
i
t , xijðk=lÞ ;
t $ xijðk=lÞ ;
(3.28)
where 0 # αijðk=lÞ , 1 N; G
the chimney distribution with a density function (the graphs omit it) 8 > 0; > > > > > Aijðk=lÞ > > > ; > > > zijðk=lÞ1 2 xijðk=lÞ > > > > ij > < Cðk=lÞ ij ; hðk=lÞ ðtÞ 5 > zijðk=lÞ2 2 zijðk=lÞ1 > > > > > > Dijðk=lÞ > > > ; > > > yijðk=lÞ 2 zijðk=lÞ2 > > > > : 0;
are similar to the ones presented in Fig. 3.2 than we t , xijðk=lÞ xijðk=lÞ # t , zijðk=lÞ1 zijðk=lÞ1 # t , zijðk=lÞ2
(3.29)
zijðk=lÞ2 # t , yijðk=lÞ t $ yijðk=lÞ ;
where 0 # xijðk=lÞ # zijðk=lÞ1 # zijðk=lÞ2 # yijðk=lÞ , 1 N; ij ij Aijðk=lÞ $ 0; Cðk=lÞ $ 0; Dijðk=lÞ $ 0; Aijðk=lÞ 1 Cðk=lÞ 1 Dijðk=lÞ 5 1:
As the mean values of the conditional sojourn times ηijðk=lÞ are given by (Kołowrocki and Soszy´nska-Budny, 2011) ðN h i ðN ij ij Mðk=lÞ 5 E ηijðk=lÞ 5 tdHðk=lÞ ðtÞ 5 thijðk=lÞ ðtÞdt; (3.30) 0 0 i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω; then for the distinguished distributions (3.26)(3.29), the mean values of the process of environment threat Sðk=lÞ ðtÞ conditional sojourn times ηijðk=lÞ , i; j 5 1; 2; . . .; υk ; i 6¼ j; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω at the particular states are respectively given by the following (Kołowrocki and Soszy´nska-Budny, 2011): G
for the double trapezium distribution ij Mðk=lÞ
5E
h
ηijðk=lÞ
1
i
5
wijðk=lÞ yijðk=lÞ
2
xijðk=lÞ 1 yijðk=lÞ 1 zijðk=lÞ 3 2 2 qijðk=lÞ xijðk=lÞ 2
3 xijðk=lÞ yijðk=lÞ xijðk=lÞ 1 yijðk=lÞ wijðk=lÞ 1 qijðk=lÞ ij ij ij ij 4 x z 5 1 ðk=lÞ ðk=lÞ 2 yðk=lÞ zðk=lÞ 1 6 xijðk=lÞ 2 yijðk=lÞ 3 3 xijðk=lÞ qijðk=lÞ 1 yijðk=lÞ wijðk=lÞ 2 ; 3 yijðk=lÞ 2 xijðk=lÞ 2
(3.31)
Modeling critical infrastructure accident consequences Chapter | 3
G
for the quasitrapezium distribution h i qij 2 2
ðk=lÞ ij zijðk=lÞ1 2 xijðk=lÞ Mðk=lÞ 5 E ηijðk=lÞ 5 2 ij ij 2 2
Aðk=lÞ 2 qðk=lÞ 2 2 zijðk=lÞ1 2 5xijðk=lÞ zijðk=lÞ1 2 xijðk=lÞ 6 ij 2 2 wij 2 2
Aðk=lÞ ðk=lÞ ij ij ij ij 1 zðk=lÞ2 2 zðk=lÞ1 yðk=lÞ 2 zðk=lÞ2 1 2 2 2
wijðk=lÞ 2 Aijðk=lÞ ij 2 ij ij ij 2 2 zðk=lÞ2 2 5yðk=lÞ zðk=lÞ2 2 yðk=lÞ ; 6
G
(3.32)
for the exponential distribution h i ij Mðk=lÞ 5 E ηijðk=lÞ 5 xijðk=lÞ 1
G
31
1 αijðk=lÞ
;
(3.33)
for the chimney distribution h i 1h i ij ij Mðk=lÞ 5 E ηijðk=lÞ 5 Aijðk=lÞ xijðk=lÞ 1 zijðk=lÞ1 1 Cðk=lÞ zijðk=lÞ1 1 zijðk=lÞ2 1 Dijðk=lÞ zijðk=lÞ2 1 yijðk=lÞ : 2
(3.34)
From the formula for total probability, it follows that the unconditional distribution functions of the sojourn times ηiðk=lÞ ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω of the process of environment threats Sðk=lÞ ðtÞ at the states siðk=lÞ ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω are determined by (Kołowrocki and Soszy´nska-Budny, 2011) i Hðk=lÞ ðtÞ 5
υk X
ij pijðk=lÞ Hðk=lÞ ðtÞ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω:
(3.35)
j51
Therefore the mean values E½ηiðk=lÞ of the process of environment threats Sðk=lÞ ðtÞ unconditional sojourn times ηiðk=lÞ ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω at the states are given by υk X ij i Mðk=lÞ 5 E ηiðk=lÞ 5 pijðk=lÞ Mðk=lÞ ; j51
(3.36)
i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω; ij is defined by formula (3.30) in the case of any distribution of sojourn times ηijðk=lÞ and by formulae (3.31) where Mðk=lÞ (3.34) in the cases of particular distributions of these sojourn times defined, respectively, by (3.26)(3.29). The limit values of the process of environment threats Sðk=lÞ ðtÞ transient probabilities at the particular states piðk=lÞ ðtÞ 5 P Sðk=lÞ ðtÞ 5 siðk=lÞ ; (3.37) tAh0; 1 NÞ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω;
are given by (Kołowrocki and Soszy´nska-Budny, 2011) piðk=lÞ 5 lim piðk=lÞ ðtÞ 5 t- 1 N
i πiðk=lÞ Mðk=lÞ ; υk X j j πðk=lÞ Mðk=lÞ
(3.38)
j51
i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ; l 5 1; 2; . . .; ω; i Mðk=lÞ
where equations
is given by (3.36), while the steady probabilities πiðk=lÞ of the vector ½πiðk=lÞ 1 3 υk satisfy the system of
32
Consequences of Maritime Critical Infrastructure Accidents
8 i ij i > < ½πðk=lÞ 5 ½πðk=lÞ ½pðk=lÞ υk X > πjðk=lÞ 5 1: :
(3.39)
j51
½pijðk=lÞ υk 3 υk
and is given by (3.23). The asymptotic distribution of the sojourn total time η^ iðk=lÞ of the process of environment threats Sðk=lÞ ðtÞ in the time interval h0; ηi; η . 0 at the state siðk=lÞ is normal with the expected value i M^ ðk=lÞ 5 E η^ iðk=lÞ Dpiðk=lÞ η; (3.40) where piðk=lÞ is given by (3.38). Thus according to the formula for total probability and (3.14) and (3.37), the probabilities piðkÞ ðtÞ 5 PðSðtÞ 5 siðkÞ Þ;
tAh0; 1 NÞ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ;
(3.41)
are defined by piðkÞ ðtÞ 5
ω ω X
X P EðtÞ 5 el UP SðkÞ ðtÞ 5 siðkÞ EðtÞ 5 el 5 pl ðtÞUpiðk=lÞ ðtÞ; l51
i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 ;
l51
and according to (3.15) and (3.38), their limit forms are piðkÞ 5
ω X
pl Upiðk=lÞ ; i 5 1; 2; . . .; υk ; k 5 1; 2; . . .; n3 :
(3.42)
l51
3.3
Modeling process of environment degradation
The particular states of the process of the environment threats SðkÞ ðtÞ in the subarea Dk ; k 5 1; 2; . . .; n3 may lead to dangerous effects degrading the environment at this subarea. Thus we assume that there are mk different dangerous degradation effects for the environment subarea Dk ; k 5 1; 2; . . .; n3 , and we mark them by k R1ðkÞ ; R2ðkÞ ; . . .; Rm ðkÞ :
This way, the set
n o k RðkÞ 5 R1ðkÞ ; R2ðkÞ ; . . .; Rm ðkÞ ; k 5 1; 2; . . .; n3 ;
is the set of degradation effects for the environment in the subarea Dk : These degradation effects may attain different levels. Namely, the degradation effect Rm ðkÞ ; m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 ; may reach ν m ðkÞ levels mν m
ðkÞ m2 Rm1 ðkÞ ; RðkÞ ; . . .; RðkÞ ; m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 ;
that are called the states of this degradation effect. The set n o mν m ðkÞ m1 m2 Rm ; m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 ; ðkÞ 5 RðkÞ ; RðkÞ ; . . .; RðkÞ is called the set of states of the degradation effect Rm ðkÞ ; m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 for the environment in the subarea Dk ; k 5 1; 2; . . .; n3 : Under the abovementioned assumptions, we can introduce the environment subarea degradation process as a vector h i k RðkÞ ðtÞ 5 R1ðkÞ ðtÞ; R2ðkÞ ðtÞ; . . .; Rm (3.43) ðkÞ ðtÞ ;
Modeling critical infrastructure accident consequences Chapter | 3
33
where Rm ðkÞ ðtÞ; m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 ; are the processes of degradation effects for the environment in the subarea Dk ; k 5 1; 2; . . .; n3 defined on the time interval tAh0; 1 NÞ and having their values in the degradation effect state sets
The vector
Rm ðkÞ ; m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 :
(3.44)
h i mk m 1 2 5 dðkÞ ; dðkÞ ; . . .; dðkÞ rðkÞ ; k 5 1; 2; . . .; n3 ;
(3.45)
where 8 0; if a degradation effect Rm ðkÞ does not appear in the subarea Dk ; > > < m dðkÞ 5 Rmj ; if a degradation effect Rm appears in the subarea D k > ðkÞ ðkÞ > : mj m and its level is equal to RðkÞ ; j 5 1; 2; . . .; ν ðkÞ ;
(3.46)
for m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 is called the degradation state in the subarea Dk ; k 5 1; 2; . . .; n3 : From the abovementioned definition the maximum number of the environment degradation states for the subarea Dk ; k 5 1; 2; . . .; n3 is equal to k ‘k 5 ν 1ðkÞ 1 1 ; ν 2ðkÞ 1 1 ; . . .; ν m ðkÞ 1 1 ; k 5 1; 2; . . .; n3 : Further, we number the subarea degradation states defined by (3.45) and (3.46) and mark them by ‘ rðkÞ
for ‘ 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ;
and form the set of degradation states
n o ‘ ; ‘ 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; RðkÞ 5 rðkÞ
where j i rðkÞ 6¼ rðkÞ
for i 6¼ j; i; jAf1; 2; . . .; ‘k g:
The set RðkÞ ; k 5 1; 2; . . .; n3 is called the set of the environment degradation states in the subarea Dk ; k 5 1; 2; . . .; n3 ; while a number ‘k is called the number of the environment degradation states of this subarea. A function RðkÞ ðtÞ; k 5 1; 2; . . .; n3 ; defined on the time interval tAh0; 1 NÞ and having values in the environment degradation states set RðkÞ ; k 5 1; 2; . . .; n3 ; is called the process of the environment degradation in the subarea Dk ; k 5 1; 2; . . .; n3 : Next, to involve the process of environment degradation with the process of the environment threats, we define the conditional process of environment degradation Rðk=υÞ ðtÞ, while the process of the environment threats SðkÞ ðtÞ in the subarea Dk ; k 5 1; 2; . . .; n3 is at the state sυðkÞ ; υ 5 1; 2; . . .; υk as a vector h i k ðtÞ ; (3.47) Rðk=υÞ ðtÞ 5 R1ðk=υÞ ðtÞ; R2ðk=υÞ ðtÞ; . . .; Rm ðk=υÞ where Rm ðk=υÞ ðtÞ; m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ; defined on the time interval tAh0; 1 NÞ and having values in the degradation effect states set Rm ðkÞ ; m 5 1; 2; . . .; mk ; k 5 1; 2; . . .; n3 :
34
Consequences of Maritime Critical Infrastructure Accidents
The abovementioned definition means that the conditional process of environment degradation Rðk=υÞ ðtÞ; tAh0; 1 NÞ also takes the degradation states from the set RðkÞ of the unconditional process of environment degradation RðkÞ ðtÞ; tAh0; 1 NÞ defined by (3.43). We assume a semi-Markov model (Grabski, 2015; Kołowrocki 2004, 2014; Kołowrocki and Soszy´nska-Budny, 2011; Limnios and Oprisan, 2005; Macci, 2008; Mercier, 2008) of the conditional process of environment degradation in the subarea Dk Rðk=υÞ ðtÞ; tAh0; 1 NÞ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ; i and denote by ζ ijðk=υÞ its random conditional sojourn times in the state rðk=υÞ , while its next transition will be done to the j state rðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk : The process of environment degradation is defined by G
the vector of probabilities
i qiðk=υÞ ð0Þ 5 P Rðk=υÞ ð0Þ 5 rðkÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ;
of its initial states at the moment t 5 0 h i qiðk=υÞ ð0Þ G
1 3 ‘k
h i k 5 q1ðk=υÞ ð0Þ; q2ðk=υÞ ð0Þ; . . .; q‘ðk=υÞ ð0Þ ;
(3.48)
the matrix of probabilities qijðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ; j i of transitions between the states rðk=υÞ and rðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; 2 11 3 k qðk=υÞ q12 : : : q1‘ ðk=υÞ ðk=υÞ 6 21 h i 2‘k 7 22 6 7 qijðk=υÞ 5 6 qðk=υÞ qðk=υÞ : : : qðk=υÞ 7; 4 : ‘k 3 ‘k : : : : : 5 ‘k 1 ‘k 2 ‘k ‘k qðk=υÞ qðk=υÞ : : : qðk=υÞ
(3.49)
where qiiðk=υÞ 5 0 G
for i 5 1; 2; . . .; ‘k ;
the matrix of conditional distribution functions
Gijðk=υÞ ðtÞ 5 P ζ ijðk=υÞ , t ; tAh0; 1 NÞ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ;
i , while its next transition will of conditional sojourn times ζ ijðk=υÞ of the degradation process Rðk=υÞ ðtÞ in the state rðk=υÞ j be done to the state rðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ; 2 3 1‘k 12 G11 ðk=υÞ ðtÞ Gðk=υÞ ðtÞ : : : Gðk=υÞ ðtÞ 6 21 7 h i 2‘k 22 6 7 Gijðk=υÞ ðtÞ 5 6 Gðk=υÞ ðtÞ Gðk=υÞ ðtÞ : : : Gðk=υÞ ðtÞ 7; (3.50) 4 5 ‘k 3 ‘k : : : : : : ‘k 1 k2 k ‘k Gðk=υÞ ðtÞ G‘ðk=υÞ ðtÞ : : : G‘ðk=υÞ ðtÞ
where Giiðk=υÞ ðtÞ 5 0
for i 5 1; 2; . . .; ‘k ;
or equivalently by corresponding to it the matrix of conditional density functions of conditional sojourn times ζ ijðk=υÞ j i of the process Rðk=υÞ ðtÞ at the state rðk=υÞ ; while its next transition will be done to the state rðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ;
Modeling critical infrastructure accident consequences Chapter | 3
2 h
i gijðk=υÞ ðtÞ
12 g11 ðk=υÞ ðtÞ gðk=υÞ ðtÞ :
:
6 21 22 6 5 6 gðk=υÞ ðtÞ gðk=υÞ ðtÞ : 4 ‘k 3 ‘k : : : ‘k 1 ‘k 2 gðk=υÞ ðtÞ gðk=υÞ ðtÞ :
: : :
k : g1‘ ðk=υÞ ðtÞ
35
3
7 k 7 : g2‘ ðk=υÞ ðtÞ 7; 5 : : k ‘k : g‘ðk=υÞ ðtÞ
(3.51)
where giiðk=υÞ ðtÞ 5 0
G
for i 5 1; 2; . . .; ‘k :
We assume that the appropriate and typical distributions suitable to describe the process Rðk=υÞ ðtÞ conditional sojourn times ζ ijðk=υÞ ; i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk at the particular states are similar to distributions describing the process of initiating events EðtÞ that graphs are presented in Section 3.1 in Figs. 3.1 and 3.2 than we omit them and in Kołowrocki and Soszy´nska-Budny (2011). There are two types of distribution with a density function: the exponential distribution with a density function 8 < 0; t , xijðk=υÞ ij h i gðk=υÞ ðtÞ 5 (3.52) ij ij : αij exp 2αij t 2 x ; ; t $ x ðk=υÞ ðk=υÞ ðk=υÞ ðk=υÞ where 0 # αijðk=υÞ , 1 N;
G
the chimney distribution with a density function 8 > 0; > > > > > Aijðk=υÞ > > > ; > > > zijðk=υÞ1 2 xijðk=υÞ > > > > ij > < Cðk=υÞ ij ; gðk=υÞ ðtÞ 5 > zijðk=υÞ2 2 zijðk=υÞ1 > > > > > > Dijðk=υÞ > > > ; > > > yijðk=υÞ 2 zijðk=υÞ2 > > > > : 0;
t , xijðk=υÞ xijðk=υÞ # t , zijðk=υÞ1 zijðk=υÞ1 # t , zijðk=υÞ2
(3.53)
zijðk=υÞ2 # t , yijðk=υÞ t $ yijðk=υÞ ;
where 0 # xijðk=υÞ # zijðk=υÞ1 # zijðk=υÞ2 # yijðk=υÞ , 1 N; ij ij Aijðk=υÞ $ 0; Cðk=υÞ $ 0; Dijðk=υÞ $ 0; Aijðk=υÞ 1 Cðk=υÞ 1 Dijðk=υÞ 5 1:
As the mean values of the conditional sojourn times ζ ijðk=υÞ are given by (Kołowrocki and Soszy´nska-Budny, 2011) h i Ð ÐN N ij 5 E ζ ijðk=υÞ 5 0 tdGijðk=υÞ ðtÞ 5 0 tgijðk=υÞ ðtÞdt; Mðk=υÞ (3.54) i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ; then for the distinguished distributions (3.52) and (3.53), the mean values of the process of environment degradation Rðk=υÞ ðtÞ conditional sojourn times ζ ijðk=υÞ i; j 5 1; 2; . . .; ‘k ; i 6¼ j; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk at the particular states are respectively given by the following (Kołowrocki and Soszy´nska-Budny, 2011): G
for the exponential distribution h i ij Mðk=υÞ 5 E ζ ijðk=υÞ 5 xijðk=υÞ 1
1 αijðk=υÞ
;
(3.55)
36
G
Consequences of Maritime Critical Infrastructure Accidents
for the chimney distribution h i 1h i ij ij Mðk=υÞ 5 E ζ ijðk=υÞ 5 Aijðk=υÞ xijðk=υÞ 1 zijðk=υÞ1 1 Cðk=υÞ zijðk=υÞ1 1 zijðk=υÞ2 1 Dijðk=υÞ zijðk=υÞ2 1 yijðk=υÞ : 2
(3.56)
From the formula for total probability, it follows that the unconditional distribution functions of the sojourn times ζ iðk=υÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk of the process of environment degradation Rðk=υÞ ðtÞ at the states i rðk=υÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk are determined by (Kołowrocki and Soszy´nska-Budny, 2011) Giðk=υÞ ðtÞ 5
‘k X qijðk=υÞ Gijðk=υÞ ðtÞ;
(3.57)
j51
i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk : E½ζ iðk=υÞ
of the process of environment degradation Rðk=υÞ ðtÞ unconditional sojourn times Therefore the mean values ζ iðk=υÞ ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk at the states are given by ‘k h i X ij i 5 E ζ iðk=υÞ 5 qijðk=υÞ Mðk=υÞ ; Mðk=υÞ
(3.58)
j51
i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ; ij is defined by the formula (3.54) in a case of any distribution of sojourn times ζ ijðk=υÞ and by formulae where Mðk=υÞ (3.55) and (3.56) in the cases of particular distributions of these sojourn times defined, respectively, by (3.52) and (3.53). The limit values of the process of environment degradation Rðk=υÞ ðtÞ transient probabilities at the particular states i qiðk=υÞ ðtÞ 5 P Rðk=υÞ ðtÞ 5 rðk=υÞ ; (3.59) tAh0; 1 NÞ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ;
are given by (Kołowrocki and Soszy´nska-Budny, 2011) qiðk=υÞ 5 lim qiðk=υÞ ðtÞ 5 t- 1 N
i πiðk=υÞ Mðk=υÞ ‘k X j πjðk=υÞ Mðk=υÞ
; (3.60)
j51
i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; υ 5 1; 2; . . .; υk ;
h i i where Mðk=υÞ is given by (3.58), while the steady probabilities πiðk=υÞ of the vector πiðk=υÞ satisfy the system of 1 3 ‘k equations i h ih i 8h ij i i > π 5 π q > ðk=υÞ ðk=υÞ < ðk=υÞ > > :
‘k X
πjðk=υÞ 5 1
j51
(3.61)
and qijðk=υÞ ‘k 3 ‘k is given by (3.49). i The asymptotic distribution of the sojourn total time ζ^ ðk=υÞ of the process of environment degradation Rðk=υÞ ðtÞ in the i time interval h0; ζi; ζ . 0; at the state rðk=υÞ is normal with the expected value h i i i M^ ðk=υÞ 5 E ζ^ ðk=υÞ Dqiðk=υÞ ζ; (3.62) where qiðk=υÞ is given by (3.60). Thus according to the formula for total probability and (3.41) and (3.59), the probabilities, i qiðkÞ ðtÞ 5 P RðtÞ 5 rðkÞ ; tAh0; 1 NÞ; i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ;
Modeling critical infrastructure accident consequences Chapter | 3
37
are defined by qiðkÞ ðtÞ 5
υk υk
X X i P SðtÞ 5 sυðkÞ UP RðkÞ ðtÞ 5 rðkÞ pυðkÞ ðtÞUqiðk=υÞ ðtÞ;
SðtÞ 5 sυðkÞ 5 υ51
i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 :
υ51
Therefore according to (3.42) and (3.60), for sufficiently large t, the limit transient probabilities of the unconditional process of the environment degradation RðkÞ ðtÞ at its particular states are given by " # υk υk X ω X X υ i l υ p Upðk=lÞ qiðk=υÞ qiðkÞ D pðkÞ Uqðk=υÞ 5 (3.63) υ51 υ51 l51 i 5 1; 2; . . .; ‘k ; k 5 1; 2; . . .; n3 ; where pl ; pυðk=lÞ , and qiðk=υÞ are defined, respectively, by (3.15), (3.38), and (3.60). i Therefore the sojourn total time ζ^ ðkÞ of the unconditional process of the environment degradation RðkÞ ðtÞ i k 5 1; 2; . . .; n3 ; in the time interval h0; θi; θ . 0; at the state rðkÞ has the normal distribution with the expected value h i i i M^ ðkÞ 5 E ζ^ ðkÞ DqiðkÞ θ; i 5 1; 2; . . .; ‘k ; (3.64) where qiðkÞ is given by (3.63).