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Optimal Survivability Enhancement inOptimal Survivability Enhancement in Complex Vulnerable systemsComplex Vulnerable systems
Gregory LevitinGregory Levitin
The Israel Electric Corporation LtdThe Israel Electric Corporation Ltd..
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Survivable system - system that is able to “complete its mission in a timely manner, even if significant portions are incapacitated by attack or accident”.
Multi-state system with
different performance rates
Reliability + vulnerability
analysis
w
Pr{w>W*}
W*
S(W*)
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SYSTEM OUTPUT PERFORMANCE DISTRIBUTIONSYSTEM OUTPUT PERFORMANCE DISTRIBUTION
GG
PP
0.70
0.05
0.1
0.05
0.10.1
WW22
0.75
…
WW11
0.85
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System survivability enhancement by element separationSystem survivability enhancement by element separation
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Basic Definitions
lowest-level part of system, which is characterized by its inherent value, availability and performance distribution
collection of elements with the same functionality connected in parallel in reliability logic-diagram sense
quantitative measure of task performing intensity of element or system (capacity, productivity, processing speed, task completion time etc.)
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Basic Definitionstechnical or organizational measure aimed at reduction of destruction probability of a group of system elements in the case of attack
action aimed at preventing simultaneous destruction of several elements in the case of single attack (can be performed by spatial dispersion, by encapsulating different elements into different protective casings, by using different power sources etc.)
group of system elements separated from other elements (and possibly protected) so that a single external impact destroying elements belonging to a certain group cannot destroy elements
from other groups
object that imitates protected group of system elements, but does not contain any element (the total damage caused by the destruction of any false target is much lower than the damage
caused by the destruction of any protection group)
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...
Optimal element separation problem
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PARAMETERS OF SYSTEM ELEMENTS
1
34
2
5
9
8
10
6
7
11
1314
12
15
16
N ofelement
G A
1 1.2 0.972 1.4 0.953 1.6 0.944 1.8 0.935 2.0 0.986 5.0 0.987 5.0 0.988 2.0 0.999 2.5 0.97
10 3.5 0.9811 1.1 0.9812 1.1 0.9813 1.3 0.9914 1.3 0.9915 1.4 0.9816 1.4 0.98
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1
3
4
2
5
9
8
10
11
13
14
12
15
16
6
7
OPTIMAL SEPARATION SOLUTIONFOR v=0.05
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7 8
W
S
Opt separation No separation
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System survivability enhancement by element protectionSystem survivability enhancement by element protection
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Survivability optimization problem
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Functional scheme of system
Desired system performance and survivability
W, S*
Survivability and cost of possible protections
List of available elements with given performance distributions
List of chosen elements
Separation and protection of elements
Optimal system structure
*|min SSC systemsystem
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No ofComponent
No ofVersion
g A
1 1.2 0.97 3.12 1.6 0.92 4.2
1 3 1.8 0.94 4.74 2.0 0.93 55 5.0 0.86 116 5.0 0.91 14.51 1.8 0.98 3.1
2 2 3.6 0.98 63 5.4 0.96 8.81 1.4 0.9 6.6
3 2 1.6 0.93 73 1.8 0.91 7.94 2.0 0.95 9.41 1.4 0.86 2.6
4 2 2.6 0.91 63 3.8 0.93 7.94 5.0 0.85 9.4
No ofComponent
m
ProtectionLevel
Vulnerabilityv
Costc
1 0.35 0.11 2 0.15 4.1
3 0.05 15.72 1 0.01 1.0
1 0.60 1.03 2 0.35 5.5
3 0.15 17.01 0.10 1.1
4 2 0.03 4.2
Producing units
Protection
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1
1
1
1
3
3
3
2
2
3
3
3
3
3
3
22
22
3
111111
1
SMSS=0. 8504 CMSS=152.2
Optimal structure for W=5, S*=0.85
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5
7
s2 63
4
s1
s3
s5
s4
s6
2
1
Multilevel protection
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Protection survivability importance in simplest binary systems
s1
s2
s3
...sn
,))((
n
iisaS
111 ,
)(
m
n
ii
m s
saI
1
11
mm Is mi Is
a
s1 s2 s3...
sn ,
n
ii
n saS1
,m
n
ii
nm s
saI
1
mm Is mi Is
a
s2
s1
sn
s3
...
,)(
n
iiasS
111 ,
)(
m
n
ii
m as
asaI
1
11
mm Is mi Is
a
mm s
SI
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0
0.1
0.2
0.3
0.4
0.5
0.6
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15w
Ij
j=2 j=3 j=6
Protection survivability importance and relevancy
in multi-state system
5
7
s2 63
4
1
2 s1
s3
s5
s4
s6
01 I22 if0 gwI
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Optimal multilevel protection problem
*|min SSC systemprot Structure of series-parallel system
7
9
85
62
3
41
12
10
11
7
9
85
62
3
41
12
10
11
Performance distribution of system elements
List of chosen protections
Desired system performance and survivability
W, S*
Survivability and cost of possible protections
cm, sm
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Parameters of a system to be optimized
7
9
85
6
2
3
4
1
12
10
11
No of element (j) 123456
State (k)pjkgjkpjkgjkpjkgjkpjkgjkpjkgjkpjkgjk
10.7570.7570.7540.7540.7540.754
20.1550.1550.1520.1520.1520.152
30.0530.0530.100.100.100.10
40.0500.050--------
No of element (j) 789101112
State (k)pjkgjkpjkgjkpjkgjkpjkgjkpjkgjkpjkgjk
10.8550.8550.1060.8080.9580.8510
20.0530.0530.7040.1550.0500.107
30.1000.1000.1520.050--0.050
4----0.050------
No of protection
set of protected elements
protection survivability
protection cost
110.951.5
220.951.5
330.901.0
…
287,8,9,10,11,120.655.2
291,2,3,4,5,6,7,8,9,10,11,12
0.707.0
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7
9
85
6
2
3
4
1
12
10
11
w=7, S*=0.85
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
w
S
Max protection S*=0.85 S*=0.80 S*=0.75
10
20
30
40
50
60
70
80
0.75 0.8 0.85 0.9S
Cto
t
Optimal multilevel protection solutions
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Protection against multiple factor impacts
Destructive factors
Protections
Complex protections
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A
5
7
63
4
1
2
1
2
3
4
5
6
7
8
9
5
7
63
4
1
2
4
2
31
5
67
8
9
A
B
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15w
S (w )
A B
Example of two different protection configurations
SA>SB
SA<SB
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Unintentional vs. intentional impacts
No impact strategy
Attacker’s strategy maximizing the
expected damage
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Expected damage model
Cumulative performance of the
group
Attack probabilityProtection vulnerability
System performance
reduction
Equipment losses
Expected damage
Failures
ppvv
gg
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Defense strategyDamageSeparation
Protection
Destruction probability
False targets
Impact probability
Disinformationpp
gg
vv
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Single attack strategy
Perfect knowledge about the system
No knowledge about the system
p=p=11/N/N
p=p=11
pp
Imperfect knowledge about the system
pppp
pi=1
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Multiple attack strategyUnlimited resource
p=p=11
pppp
pp
Limited resource + perfect knowledge about the system
p=p=11
Limited resource + imperfect knowledge about the system
pi>1
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Tools for solving the problems
Evaluating system performance distribution
uj(z) ui(z) )()( zuzu ij
j i j
jhikjhi
ik
K
h
K
k
K
h
ggjhik
gjh
K
k
gikji zppzpzpzuzu
1 1 1
),(
1
)()(
Universal generating function technique
Universal simulated evolution technique
Genetic Algorithm
Solving optimization problems
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References1 .Optimal separation of elements in vulnerable multi-state systems, G. Levitin, A. Lisnianski,Reliability
Engineering & System Safety, vol. 73, pp. 55-66, (2001) .2 .Optimizing survivability of vulnerable series-parallel multi-state systems, G. Levitin, A. Lisnianski ,
Reliability Engineering & System Safety, vol. 79, pp.319-331, (2003) .3 .Optimal multilevel protection in series-parallel systems, G. Levitin, Reliability Engineering & System
Safety, vol. 81, pp.93-102, (2003) .4 .Optimizing survivability of multi-state systems with multi-level protection by multi-processor genetic
algorithm, G. Levitin, Y. Dai, M. Xie, K. L. Poh, Reliability Engineering & System Safety, vol. 82, pp.93-104, (2003).
5 .Protection survivability importance in systems with multilevel protection, G. Levitin, Quality and Reliability Engineering International, vol. 20, pp.727-738, (2004) .
6 .Survivability of series-parallel systems with multilevel protection, E. Korczak, G. Levitin, H. Ben Haim, Reliability Engineering & System Safety, vol. 90, pp.45-54, (2005).
7 .Incorporating common-cause failures into series-parallel multi-state system analysis, G. Levitin, IEEE Transactions on Reliability, vol. 50, No. 4, pp. 380-388 (2001).
8 .Maximizing survivability of vulnerable weighted voting systems, G. Levitin, Reliability Engineering & System Safety, vol. 83, pp.17-26, (2003).
9 .Maximizing survivability of acyclic transmission networks with multi-state retransmitters and vulnerable nodes, G. Levitin, Reliability Engineering & System Safety, vol. 77, pp.189-199, (2002) .
10 .Survivability maximization for vulnerable multi-state system with bridge topology, G. Levitin, A. Lisnianski, Reliability Engineering & System Safety, vol. 70, pp. 125-140, (2000).
11 .Universal generating function in reliability analysis and optimization, G. Levitin, Springer-Verlag, 2005 .12 .Multi-state system reliability. Assessment, optimization and applications, A. Lisnianski, G. Levitin, World
Scientific, 2003.
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Contents:
-Basic Tools and Techniques.
-UGF in Reliability Analysis of Binary Systems.
-Introduction to Multi-state Systems.
-UGF in Analysis of Series-parallel MSS.
-UGF in Optimization of Series-parallel MSS.
-UGF in Analysis and Optimization of Special Types of MSS.
-UGF in Analysis and Optimization of Consecutively Connected Systems and Networks.
-UGF in Analysis and Optimization of Fault-tolerant Software.