The Israel Electric Corporation Ltd.
Optimal Survivability Enhancement in
Complex Vulnerable systems
Gregory Levitin
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
Reliability + vulnerability
different performance rates
analysis
Pr{w>W*}
S(W*)
w
W*
SYSTEM OUTPUT PERFORMANCE DISTRIBUTION
…
P
0.1
0.05
0.1
0.05
0.70
0.85
0.75
W1
W2
G
System survivability enhancement by element separation
Basic Definitions
lowest-level part of system, which is characterized by
its inherent value, availability and performance
distribution
quantitative measure of task performing intensity of
element or system (capacity, productivity, processing
speed, task completion time etc.)
collection of elements with the same functionality
connected in parallel in reliability logic-diagram
sense
Basic Definitions
technical 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)
Optimal element separation problem
...
PARAMETERS OF SYSTEM ELEMENTS
N of
element
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
G
A
1.2
1.4
1.6
1.8
2.0
5.0
5.0
2.0
2.5
3.5
1.1
1.1
1.3
1.3
1.4
1.4
0.97
0.95
0.94
0.93
0.98
0.98
0.98
0.99
0.97
0.98
0.98
0.98
0.99
0.99
0.98
0.98
11
1
2
8
6
4
5
13
9
3
7
12
10
14
15
16
OPTIMAL SEPARATION SOLUTION
FOR v=0.05
11
2
3
1
4
5
15
8
6
7
13
9
14
10
12
16
System survivability enhancement by element protection
Survivability optimization problem
...
Optimal system structure
Csystem  min | S system  S *
Functional scheme of system
List of available elements with
given performance distributions
List of chosen elements
Separation and protection of
elements
Survivability and cost of possible
protections
Desired system performance and
survivability
W, S*
Producing units
No of
No of
Component Version
1
2
1
3
4
5
6
1
2
2
3
1
3
2
3
4
1
4
2
3
4
g
A

1.2
1.6
1.8
2.0
5.0
5.0
1.8
3.6
5.4
1.4
1.6
1.8
2.0
1.4
2.6
3.8
5.0
0.97
0.92
0.94
0.93
0.86
0.91
0.98
0.98
0.96
0.9
0.93
0.91
0.95
0.86
0.91
0.93
0.85
3.1
4.2
4.7
5
11
14.5
3.1
6
8.8
6.6
7
7.9
9.4
2.6
6
7.9
9.4
Protection
No of
Protection
Component
Level
m
1
1
2
3
2
1
1
3
2
3
1
4
2
Vulnerability
v
Cost
c
0.35
0.15
0.05
0.01
0.60
0.35
0.15
0.10
0.03
0.1
4.1
15.7
1.0
1.0
5.5
17.0
1.1
4.2
Optimal structure for W=5, S*=0.85
3
3
3
3
3
1
1
3
3
1
1
2
2
3
3
2
2
3
2
2
SMSS=0. 8504
CMSS=152.2
1
1
1
1
1
1
1
Multilevel protection
s3
s6
5
1
s5
s4
s2
6
3
2
s1
4
7
Protection survivability importance in simplest binary systems
a
s1
s2
s3
...
sn
n
 (1  si )
n
S  a (1   (1  si )), I m  a i 1
,
1  sm
i 1
sm  I m 
si  I m 
 S
a
s1
s2
s3
...
Im  S  a  s ,
 sm
sn
n
n
i 1
i
sm  I m 
a
s1
s2
n
Im 
 si
a n i 1
sm
si  I m 
n
n
 (1  asi )
S  1   (1  asi ), I m  a i 1
,
1  asm
i 1
s3
sm  I m 
...
sn
,
si  I m 
Protection survivability importance and relevancy
in multi-state system
s3
s6
5
1
s4
s2
6
3
2
4
s5
s1
I 2  0 if w  g2
7
I1  0
Optimal multilevel protection problem
C prot  min | S system  S *
Structure of series-parallel system
3
1
7
Performance distribution of
system elements
10
4
5
8
6
9
2
11
12
List of chosen protections
3
7
1
10
4
5
8
6
9
2
11
12
Survivability and cost of possible
protections
Desired system performance and
survivability
cm, sm
W, S*
Parameters of a system to be optimized
No of element (j)
1
2
3
4
5
6
State (k)
pjk
gjk
pjk
gjk
pjk
gjk
pjk
gjk
pjk
gjk
pjk
gjk
1
0.75
7
0.75
7
0.75
4
0.75
4
0.75
4
0.75
4
2
0.15
5
0.15
5
0.15
2
0.15
2
0.15
2
0.15
2
3
0.05
3
0.05
3
0.1
0
0.1
0
0.1
0
0.1
0
4
0.05
0
0.05
0
-
-
-
-
-
-
-
-
No of element (j)
7
8
9
10
11
12
State (k)
pjk
gjk
pjk
gjk
pjk
gjk
pjk
gjk
pjk
gjk
pjk
gjk
1
0.85
5
0.85
5
0.10
6
0.80
8
0.95
8
0.85
10
2
0.05
3
0.05
3
0.70
4
0.15
5
0.05
0
0.10
7
3
0.10
0
0.10
0
0.15
2
0.05
0
-
-
0.05
0
4
-
-
-
-
0.05
0
-
-
-
-
-
-
No of
protection
set of protected
elements
protection
survivability
protection
cost
1
1
0.95
1.5
2
2
0.95
1.5
3
3
0.90
1.0
3
7
1
10
4
5
8
…
2
11
6
12
28
7,8,9,10,11,12
0.65
5.2
29
1,2,3,4,5,6,7,8,9,10,11,
12
0.70
7.0
9
Optimal multilevel protection solutions
3
7
1
10
4
5
8
2
11
12
9
w=7, S*=0.85
Ctot
6
80
70
60
50
40
30
20
10
0.75
0.8
0.85
S
0.9
Protection against multiple factor impacts
Destructive factors
Protections
Complex protections
Example of two different protection configurations
1
1
5
8
3
1
5
7
4
3
S (w )
6
6
SA<SB
0.8
9
4
2
7
0.6
2
A
0.4
4
0.2
3
1
1
A
5
8
5
0
6
3
7
9
2
SA>SB
4
w
6
7
2
B
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
A
B
Unintentional vs. intentional impacts
No impact
strategy
Attacker’s strategy
maximizing the
expected damage
Expected damage model
Protection
vulnerability
v
g
Cumulative
performance of the
group
Failures
Attack probability
p
Equipment
losses
System
performance
reduction
Expected
damage
Defense strategy
Separation
Protection
Damage
Destruction
probability
g
v
False targets
Disinformation
Impact
probability
p
Single attack strategy
Perfect knowledge
about the system
p=1
No knowledge
about the system
p=1/N
Imperfect knowledge
p about the system
p
p
Spi=1
Multiple attack strategy
Unlimited
resource
p=1
Limited resource + perfect
knowledge about the system
p=1
Limited resource + imperfect
knowledge about the system
p
p
p
Spi>1
Tools for solving the problems
Evaluating system performance distribution
Universal generating function technique
uj(z)
ui ( z )  u j ( z ) 
u j ( z)  ui ( z)
ui(z)
Ki
 pik z
k 1
g ik
Kj
  p jh z
 h 1
g jh

Ki K j
  pik p jh z
 ( g ik , g jh )
k 1h 1
Solving optimization problems
Universal simulated evolution technique
Genetic Algorithm
References
1. 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.93104, (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.
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.