HAL Id: hal-00968778
https://hal-enac.archives-ouvertes.fr/hal-00968778
Submitted on 1 Apr 2014
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Accident Risk Assessment and Monte Carlo Simulation Methods
Pascal Lezaud, Jaroslav Krystul, Henk Blom
To cite this version:
Pascal Lezaud, Jaroslav Krystul, Henk Blom. Accident Risk Assessment and Monte Carlo Simulation Methods. 2003. �hal-00968778�
Distributed Control and Stochastic Analysis of Hybrid Systems Supporting Safety Critical Real-Time Systems Design
WP8: Accident risk decomposition
Accident Risk Assessment and Monte Carlo Simulation Methods
Pascal Lezaud
1, Jaroslav Krystul
2, Henk Blom
39
thSeptember 2003
Version: 0.4 Task number: 8.2 Deliverable number: D8.2
Contract: IST-2001-32460 of European Commission
1 Centre d’Etudes de la Navigation Aérienne, France 2 Twente University, NL
3 NLR, NL
2
DOCUMENT CONTROL SHEET
Title of document: Accident Risk Assessment Methods Authors of document: P.Lezaud, J.Krystul, H.Blom
Deliverable number: D8.2
Contract: IST-2001-32460 of European Commission
Project: Distributed Control and Stochastic Analysis of Hybrid Systems Supporting Safety Critical Real-Time Systems Design (HYBRIDGE)
DOCUMENT CHANGE LOG
Version # Issue Date Sections affected Relevant information
0.1 29 Jun. 2003 All 1st draft
0.2 15 Aug. 2003 1, 5, A. 2nd draft 0.3 26 Aug. 2003 1, 2, 5, 6. 3rd draft 0.4 9 Sept. 2003 3, 4. 4th draft
Version 0.4 Organisation Signature/Date
Authors Pascal Lezaud CENA
Jaroslav Krystul TWEN
Henk Blom NLR
Internal reviewers
! " #
# $ % & $' (
# ) *
!
+ " " ,-. /) 0 1 " 20 +3( "$
" " " * 4 2 $
" " " "& 5 4" "
2 " * & " "
" 4 " +3( 6 7
( $ 4 * " 1 2 " $
& 1 5 " 2 " 4
" 4 " " 2 4 " & 2 " "
"
( $ / 4 " 8 "
& 2 * " "
" " " " &
"
( $ / 5 5 "
" "
( 1 " 4 " 4 " " * 9
(
( $ * " " 4 " . &
" 4" " 2 " "
4" 4 2 2
" ( " 2 :' & ; "
" 2 " (
! " 2 4 & &$
1 " " " 1 <
=! > & " ! "
2 2 4 7 " 2 &
" " 2 " "
" " & *
2 " & 4 2 2 & " $
" 2 " * 2 "
) 2 1 % ? " + 2 4 : @A; " " 9 &
6 " & & 2& " $
2 " 4 +3(
& " "
" " 2 2 2
+ " " & " 7 & 1 $
" ! & 1
5 " " 2 ( " 4 +3( ! $
& " 2 1 2 &
& = :' & ;> " 9 4 "
2 ! * / = :. ;>
" 5 2 4" " $
& 4 & , 4 1 2 $
2 2 1 & 2 1
" 4 5 " 1 " 20
( & 5 $ " 4" "
& $ 2& 2 5
"
& " " 2 5 $
" 2 1 " 2
% &1 2 " & "
& 2 2 & 2 " 4 & " "
" & . 1 4 "
4 " & 1 " 1
& & $ ! 4 & " $
2 " 0 & 1 2 "
" " * 4 9 2 & " $
" " " " 2 " 4 &
" " : 4 @(;
" 6 4 7 2 4
9 " 5 " *
" 5
" " " $
" 5 " $
" " ! " 2 ! 4
" 2 & "
# % &1 B " " 4
" " " 2& " " 4 "
,-. /) 7 1 2 1 $ 1
1 $ 1 2 1 1 $
1 2 9 % " " " " *
2 4 1 2 " 4 4" " " 2& "
" 4 " ,-. /)
! " 4 4 : = >; " & 2 2 $
& & 2 = > "
" " 2 2 & & = > +" & "
2 & = >1 4 "
: = >; C = > = > C = > = >
= > = >
, 1
: = >; C : = > = >
= > ;
4" " 2 " " 2 2 & & = > + "
" " " : = >; 2& 2&
* : = >; 2&
= > = >
= >
) " = > 4 " 4 "
42 " " " " = >1 "
" " & " 2
" #
+ 4 : = >;
: = >; C : = > = >; D : = >;
4" : = >; 2 & & : = > = >; 2 &
" : = >; + " 5 " : = >
= >;
" 4 4
: = >; C = > = >
4" = > " & " = > 4 7
= > C
= > = > = > C : = > ;
4 " = > C = > 1 & " : = >; C : = >; 1
" 2 = > " 1
: = >; C = > = > C = > = > C : = >;
8 "
= = >> = = >>
4 * & " = >1 2
5 = >
$% &
" 4 " 2
" 1 7
C C = >
+ 2& C " 2 & " "
1 " " = > " " &
2 " " " " 2 = > = C D >
#
" 2 2 2&
" " "
= " > % * 1 " 1
" & " 4 " 7
: = >; C = > = > C = > = >
C = > = > = >
C : = > C ; = >
" 4 " " & = > * &
" 6
$
E " " & " $ 5 $
* " & = > 2&
1 4" " " 4 " 1 1 " "
4 " & " & = > 2 * 2& "
2 7
= > = >
4" " / 1 = > C " 4
/ $ & 4 " $
2 2& "
" 2 4 " 6 4 " 1 " 4 "
9 4 "
#
1 4" " & " 6 2&
" " = > 2& 1 " " 0 &
" & 2 2 0
" 42 7 < 9 "
= 2 >
'# & (
E " " 2 & 2 " 2
1 " " 9 & = > 9 . & " 7
= > C = > = >
= >
" & = > * 2& " &
1 4" " " 4 " 1
1 " " 4 " &1 " . & " " 4 "
4 7
C = >
4" 6
B
Table 1 Techniques used by Monte Carlo simulation methods so far considered within Hybridge WP2 and WP8 Monte Carlo approach
A B C D E.1 E.2 F G
Distinguishing features
§2.1 §2.2 §3.1 §3.2 §4.2 §4.2 WP2.2 [Blom&
Bloem03]
Applicability to
Hybrid processes – Yes – – – – Yes Yes
Importance
sampling Yes Yes – – – – – –
Splitting – – Yes Yes Yes – –
Multi-Level crossing – – Yes Yes Yes Yes – –
Stopping time based
decomposition – – – – Yes* Yes* Yes* –
N particles – Yes – – Yes Yes – Yes
Control variables – – – – – – – –
Resampling – Yes – – Yes – – Yes
Observation filtering – – – – Yes Yes – Yes
Objective Rare event Rare event Rare event Rare event Rare event Rare event Rare event Bayesian filtering
*) applicable only for strong Markov processes
A
" 5 " 2 " 4 $
2 4" " " & 2 = >
4 " " 2 " 2 " " = >
9 " 1 " 4 " 2 2 &
= > C = D D >
1 4 * " * 2& "
C = D D >
F " 1 " 4 2 & " 2 " 7 , 4
2 " " 2 "
C = >
" ! G
" 5 E " =5E > &
1 4" = > E " " " 2 1
"
"
4 ) 2 4 " 1
) 2 E " ) 2 1
"
= @B> @#
+ @#H$ 9 " 4 &
: @B "
D @B "
;
E * " 2 2 & C =#>
# + " C = >1 4" . 2
" # 6 " 4 1 2 *
1 4 " 2 "
#1 " 4 & " C $ ! " 1
" 5E " 4 9 @#
@B "
C @B = >
@B (
1 # " "
% * 1 4 " 1 & 2 2
2 ! " $ "
"
" 2 2 " 2
* E " 4 4 =%= >> 4"
2 4 " 2 2 & & = > " 1 " &
2&
=%= >> C %= > = >
F 41 " 2 2 & & = >1 " " = >! 5 &1
=%= >> 2 4
=%= >> C %= > = >
= > = >
" "
=%= >> C %=&> =&>
=&>
4" " 2 & " " 2 2 & & = > " "
" 2 2 & 4 " 4 1
" = > C%= > = >$= = >%= > > & 6
1 " 4 1 " 2 "
4 =%= >> . &1 " 2&
" = >$ = > & " , &1 "
" = > %= > = > 6 "
" 6 1 4 " 4
I
% 7C %= > = >$ = >
= >$ = > C %= > = >
4" 2 4 " & "
2&
C = >$ = >
= >$ = >
% 9 1 " 2 2 & &1 4 $
1 " 4 2 1 " %I =%= >>
J 1 5E 2 2
E 1 " %I = > " 2 " %= > 4 "
" 1 4" " /
= >
@
5 " , 4 1
1 & 4" 4
0 & C = > ! " $
" 4 " " 4 2 2 1
" "
+ $ = '> " "
C(= > D"= >
4" (= > 1 "= > $ . 4
" 2 2 & + " " ( "
*" & " " " *
" " :' ;
) 4 " & 4 "1 4 9 " 4
= > C = = > C > '
! 5 * =' > 2&
= > = > = >
4" = ) C > 6 "
' 4" C " 6 2 2 2& "
"
I C I D(= D I > D"= D I >= >
E " " 2 = ) >
) 2 4 " * * =* 2 "
& * >
) 2 $ " = >1
" . 4 1 4 " 4 +
+ C * =, > D =, > ,
=+ > C 1 " 4 2 2 & K 2 9 2& " &7
K C =+ >
+ " " 4 1 " = > 2 4
= > C K= = >+ C > = >
.& ) L " "
K C D =, > ,
$ . 4 " 2 2 &
K1 " 9 " . 4 " 4
. 4 " + 2 4 7
C =(= > "= > = >> D"= > K
+ C+ = > K
4" " 4 2 " " * = > 2&
= > = >+ =#>
+ 4 " = > 4" "
" 5 * =#> " " = > % &1
" " = > 9 " 5 "& 2 :' ;: "
# A B;
C M = > : '>
=' > C = >M =B>
4" " $
= > C = > = >
D (= > = >
4" = > " " *1 9 2& C""
=" " "> F 41 I 1 4 2 "
= = >+> C = >+ = > D+ " = > = > K
C+=" = > = > D >= > K
4" " 4 " " 2 ,
2 4 '1 4 2
=' >+ C = >+ C = >D +=" = > = >D >= > K
" " " 4 5 2&
N = = >+ > C K + " D
N = = >> C "
= > 4 4 1 " 2 2 " "
1 4" " 6 1 4 2 7
C "
" 1 4 * " 4 "
4" " " " " ) L "
" + 2
" :% @A; 4" " * 2 2&
" 1 :% ; *
E 0 & C = > 4" "
" & + 4 "
* = = >> 4 " " &- = > "
" 2& " 0 4 "
= >
I C = >
C-= >
= > =A>
, 1 " 9 " 0 & , 4 1 4
" 0 & 4 " &
" 1 = > C = > " 1 4 " " 4 &
- = > C- = >- = > - = >
" " " 4 7 3 C " - $ &
C " - $ & "
" 1 4 " " 2 2 & 2&
4 & - &1
" 4 & = >1 "
4 & " " & " 0 & 2&
= > C = > = > = >
" 4 " C- = >$ = > 2 1 $
" " 2 & 1
C - = >
= >
E " " 4 " & 2 4 $
&
" 2 " "
" 1 1 " 2 " 4 " 2
4 3 &1 4 1 & 0 &
" $6 4 " " &1
:/ ; " &
0 " 4 4 " & 0 "
" " 4 " % &1 4 " 4 " =A> 2&
" 4 "
C = >
4" " 2 0 & M $
2 " " C " " " "
I " & 4 & 1 * 2&
$ 2 C "
" " 4 :/ ;
1 C
% C 1 = > C
% C 1 K = > K C = K >1
% C " 4 "
C - = >
= > =(>
F 6 " 4 "
4 " 0 = M C >
" =K M C > " 4 "
D
F " =(>1 2 " 0 &
" 4 " "
+ " " " "
" 0 & 2 " "
" * 4 9 2 & "
" " " 0 & 1 " $2&$
" & 4 " 2 2 &
" 1 4" " & " 6 2& " " 2&
1 " " 0 & " &
2 2 0
" : 4 @(;
% * 1 D " " " #C
"
=#> C =# > = > = > = > =@>
4" " " " " =@> O
P " " 4 : ;7 .
" " " 1 4 " .
. 4 " 4 " "
" 1 4 . " " 2&
" 1 " #1 4 "
J . " 1 4 " "
. . . 1 * " 4 " "
" $ " 1 " . . 2 " ! 2
=#>
I C . . C
. .
4" " " $ " . / "
2 0 " " " # 2 :N @A;
" " 2
C $ 0%=1=#>> 1= > C2 . $1= > C2
F " 1 " 0 & " & 2 <
& % * 1 " " " 2 2 &
& & *1 " < & 9 " &
" . " % " 1 " 2 2 & "
& 4 " "
F 41 4 " " & " & "
" 3 " & 1 & " &
3 9 1 = " & 3 9 & >
A=B4
B2 B1
O
B3
% 7 * 4 ". C "
+ 4 1 " 1 " 2 2 & "
# 2 3 % * 1 ' " 9
" # . " 9 31 " C =' .>
=' > C = =. ' >>$ 1 4 % 7
" 2 2 31 4 " " . 1 " 4 1 4
2 39 1 4 " % " " "
1 4 " 2 31 " 4 " " .
1 " 4 4 " " + " " " 1
9 &1 " 4" " " # 2 3
1 4 " " ! " "
D 2 " 4 "
: ;1 " '1 " 4
=' =. '>> 1 2 3 &
1 4 2& "
" 4" 2 4 .& " 1 4
" 4 " 1 " " 2 4" " <
!" !
+ 2 Q& 2 " " ! " 2&
R S ? N ? $! :N @ ;1 2 " !
2 1 " 2 2 & =#> &
#
T 2 T
3
T 1
C 2
C 1
C 3
C 2
C 1 B
1
B 1 B
2
B 2
B 3
D 1
D 1
D 2 D 2
D 3 L
A X(t)
% 7 5
& 1 & " 2 2 & " #2
2 3
# 4" 2 2 & 2 %
&1 4 " # & 4" &
= > $ " " % * &
F 41 4 5 = & # 5 5 > 4 " $
" " ' ' + 9 " "
5 5 = " " " '> "
5 5 = 4 " " " '>
" ! " % " 4 $
:N ;7
! "1 1 " 4 &
4 = " " % > "
9 O P = 9 "
>
" " 1 " &
1 " .
" " " "
9 " 4"
! " . " 2 1 "
" F " " 2
" : > . = " > " 4" "
4
B
T 2 T3
T 1
B 1
B 2
B
3 B
3
B 1 D
1 D
1 D
1
D2 D2 D2
D3 D
3
D 1 D3 L
A
t X(t)
% 7 4 " !
& : >1
7 . 9 "
1 . : > " : > 4" "
& 4 4
. : > = 4> "
: > " 2 .
"
! 9 " " " O O
2 "
" 4 " " " " 4
" " 9 " " 4 = " 4"
4 " " 4 " C " >1 *
" ! 1 " " 6
"
F " " 4 " " " 2&. " 4
2 1 2& " 4 " 4
"
E 2 " 2 # " " = "
& > 2 " 2
" 1 " " 2 " =#>
1IC
.
A
:N ; " " 2 " 4
&7 5 4 " 4 " " "
= " $ > 1 4
2 4 "4 " " =' ' >
:N ;1 " " " 2 "
* 6 " 2 4 " ! = " 9
" " " 2& " "
2& " 2 !
> F " 1 " "
" 2 9
# $ % "
+ 4 & & "
1 4 :5? ;
+ $ 4" "
. 34 " 2 2 & 2 6 %
. #1 4 9 " 9 " " #1 &
' C 7 # ' C " &
+ 4 "
=' '> E 4= ' ' '> = >
4" ' " 9 " & 9
. . . 3C " 9 3
.C31 &
' C ! 7 .
# & ' % &(
! &1 " 1 2 . #1 " "
.
#C
4 "3 C . C " 2 " 2 4 "
1 4 " "
" " 2 4 " 1 " " " "
= % >1
C = M' ' ' '> C
4" (C = (> "' " 9 " 1 "
' C 7 D ' C
(
B2 B1
B3 T1
TR
T2 T3
T4
X0 X 2 X1
X3
X4
R
O
A
% 7 2 5"
' ' 1 " C C $ 1 4" ' '
' 1 " C = ' '> C $ % &1 ' '1
" C = ' ' > " " 2 4 "
C + 2 "
=' '> C =' '> C =' ' ' '>
.& " &1 4 " " " "
= > " 4 " " "$ 1 2&7
" 4 " " " " " " 1 4 "
% 9 " " 7C = > 1 4 ", 2&
% = > C
+ " " 1 4 " "
=' '> C =% = > C % = > C >
" 1 " 4 % & $' " = > " 2
:5? ;
= = >' '> C = = > % = >>
= % = >> = >
=' '> C = % = >> = >
4" & 2 9 " "$ 7 1
& D $ 2 1 4 " "
= = >' '> C = = > % = >>
= % = >>
@
# ) *
. " % & $' 1 4 $
" * " = > " $
& *
% & $' " 2 :/ ;1 :/ ;
:/ 2;
2 " * 1 " 2 2 &
2 6 =7> " 71 9 " 2
72&
6 = > 7C = >6 = > C 8 = >
8 = > 4 "8 = > C = = > % = >>
" 1 " " $" = > " " " 6 = % >$6 =% >
+ " 1 " 6 2& $
& &
6 C T =6 > = >
4" " T "
=7> C 6 =7>M6=% >!
=7> 9 2&
T =6>= > C U =6>= > = >
4"
U =6>= > C 6= % >
6=% >
= > " " " 1
= > C = C >
" 1 " = > 4 V $
6 =7> 6I C U =6 > =7> 6 =7> = >
E " " 8 7 2 * "
=6 M >1 " & "
8 = > C6 = > 6 =% >
" 1 4 2 "
8 =% > C =' .>
I
6 = > C = = >' '>
" & $ & 4 " 2
" = > 2 " * 6
2&
6 C = #>
4 " & 9 C =9 9 >
" 7 " 4 1 4 * " 4 $ "
& = > 2& 4 $ "
6 I6 C 6
4" " I9 2 4 " 2
U = > C % =9 >
% =9 >
J " * 6 1 4 4 " 8 " * $
8 9 2&
8 = > C6 = > 6 =% > = B>
5 &1 =' '> " " 8 =% >
&1 " $ & 4 " " & &
= > " " 9 C =9 9 > "
7 W 1 4" W &
9 4 " & 9 C =9 9 >
2 4 " 46 1 " 4 " 1
& 9 C = > 7 = 0 >1
" " " 2 =7>
" 2 2 & " 9 9 C "
T = >
" 2 " " " " 9 & 4 "
" 4 T =6 >
+" " & 9 9 C " " $
=7> = " .>1 " " 4 9 C W
% " 1 * " " 2 " "
" " = D >$ "
* C 79 =' >
* & & " $ =7>1 "
" "
" " 2 4 M " V
I9 9 " V 9 I9
4" " I9 2 4 " 2
U = > C % =9 >
% =9 > C
* = A>
4" * " " *
" " " 4 7
1 C
% C 1 9 6 I9 C9
I9 9
I9 C W1 " 9 C W = " " >1
" 4 1 C 1 & 4 *
" D $ " " .1 "
" " D 1
C 1 " 2 * "
" " D $ "
9 I9
" 9 "
= A>1 D
/ " V 1 " &
2 2 & / " V $
" 1 " 9 2
& 2 " " 4 " &
2 " " * " 1 4" "
First situation
Bn+1
Bn+1
Alternative model N=10 particles
% #7 / 2 4 " 4 V
" " & 1 " 2
:5? ; 2& & " VJ $
4 " " = A>1 4 " 2
4" " " 4 ! D 1 " "
" " D $ " 1 4 1 I9 C9 " 1 " $
$ " "V 9 2
" 1 I9 " & & " "
" " " 1 " " 4" "
2 " * " 4 " 4" "
" = D >$ " & "
4 = % # " 2 4 " 4
>
E 2& " $ " $
C 7 $ =7>
% " 1 4 " * 8 = B>1 4
2
8 =% > C *
I 6 C
*
" 8 =% > 2
=8 =% > > C =' '>
" & 2 " " $
2 :5? ;
+
& " 2 2 " 2 4
" 2 & ! 4 " 2 $
9 7 * " $ $
5 2 " " * $
" " 2 " 4
+3( :' & ; " 5 " "
1 &
" " 2 "
" 4 2 "
, 4 1 " 4 " & 2 & 4" " " $
* & 2 " !
" " 1 2 " 2 2 & $
2 1 " 2 2 $
< & 2 2 2 &
& . " " " & " 2 &
9 " " " &1 "
& 4 " " 4 %
" " " & & " "
& 2 2 & " & & 2 2
* &
" " 4 9 "
7 2 " " = >1
" 2 L " " 2 < 4" "
* &
" & " 4 & "$
2 " " 4 " " 2 " $
& * % & $' 2 "
& 1 " " "
1 " 4 " 2 1 " 2 " 2
Q " & %
" & " 2 2
% " 4 " & 2 1
* " 5 " "&2
8 " " " "&2
9 " " " & 2 4 " +3(
$ :. R. ; ! &
2 2 & " 1
2 1 4 &
! ) 7
!
#
:! (@; ! 1 / " # $ %
& 1 AA #
N 1 @(@
:. ; . 1 , ! 3 1 . 1 ) S 1 0 , 5 1 F S
3 1 " & 2 $
! < 1 ,&2 +3 1
S &
:. R. ; , ! 3 . 1 ! . 1 3 B " 5 $
% 1 5 1 X 1 ! 1 ($ S & 1
A(#$A@
:5? ; 5? 1 % R / 1 3 R E ) 1 % R E 6 1 3 '
' ( ) # " 3 1
:/ ; / 1 3 1 E * %
" $ " %+
% , ? 3 2 2 ?
YYY N E F " F A @1 S !6 1
? &1 E *1 - = >1 $ #1
:/ ; / 1 3 1 E ' %
" %+ !
! 3 2 2 &1 1 F 1 BB$ @(1
:/ 2; / 1 3 S 1 S
- . 5 "$
3 N 1 / 1 ! 1 % 1 F 1
) 1 F = >1
:/ ; / 1 ! % 1 F ) 1 F #
! 5
" 3 N 1 / 1 ! 1 % 1
F 1 ) 1 F = >1
:) @@; ) 3 1 , 2 3 1 " " 2 3 1 Z 0 1
2 2 1 8 $
"1 A1 =S $! 1 @@@>1 #(#$
B
:% ; % 1 S 3 1 ! / (
/ " ) !
% 5 1 " "
% 1 ! 1 1 5 1 = >1
B
:% @A; % 1 E 2 " *1 S 6 1F 0
)$ ! & ! & 1
= >1 @@A1 B $ AB
:' ; ' 6 1 " 1 *
1
:' & ; S ' & 1 ! . " 1 , . 1 $
" 1 ,&2 +3( 1 &
:' & 2; S ' & 1 + * 9 * 1 4
1
:E ; E & . 1 3 * ? 1 $
5 " / 1 8*
J & 3 1
: @A; 1 / R " 1 # " )$
/ . "[ 1 @@A
: 4 @(; 4 1 S / R , 6 1 Z R % 2 & 1 S ! R / $
1 ( )
0 ))) 1 # 2334
: ; < 1 . R 1 ' %
5 0
5 3 "
N 1 E F 5 1
F A(B1 . , 1 , 5 . " 1 5 J "1
B$ @1
:N @ ; N ? $! 1 N ? $! 1 S () 5#(56
# ! 27
! 5 8 1 5 27 9: 1
#5 ;1 5 " 1 / 1 @@ 1 A \AB
:N @A; N ? $! 1 N ? $! 1 S () 5#(56
# )8 ' (
) ! " A1 / ] ! $
1 J L 1 J 3 ? 1
@@A
:N ; N ? $! 1 N ? $! 1 S = > # %
" () 5#(5 6 5 * %
" " 7 ( ) 1 ()
<===1 8 2 #$B1 3 1 &
A
