Optimal Survivability Enhancement inOptimal Survivability Enhancement in Complex Vulnerable systemsComplex Vulnerable systems

Gregory LevitinGregory Levitin

The Israel Electric Corporation LtdThe Israel Electric Corporation Ltd..

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*)

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

System survivability enhancement by element separationSystem survivability enhancement by element separation

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.)

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)

...

Optimal element separation problem

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

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

System survivability enhancement by element protectionSystem survivability enhancement by element protection

...

Survivability optimization problem

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

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

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

5

7

s2 63

4

s1

s3

s5

s4

s6

2

1

Multilevel protection

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

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

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

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

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

Protection against multiple factor impacts

Destructive factors

Protections

Complex protections

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

Unintentional vs. intentional impacts

No impact strategy

Attacker’s strategy maximizing the

expected damage

Expected damage model

Cumulative performance of the

group

Attack probabilityProtection vulnerability

System performance

reduction

Equipment losses

Expected damage

Failures

ppvv

gg

Defense strategyDamageSeparation

Protection

Destruction probability

False targets

Impact probability

Disinformationpp

gg

vv

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

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

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

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.

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.