1. Using D-spectra in Network Monte Carlo:
Estimation of System Reliability and Component
Importance
Ilya Gertsbakh
Department of Mathematics
Ben Gurion University
Yoseph Shpungin
Department of Software Engineering
Shamoon College of Engineering

2. Contents I. Historical Remarks II. The Network
III. D-Spectrum (Signatures)
IV. The BIM Spectrum
V. Numerical Evaluation
VI. Network Optimal Design
VII. Concluding Remarks
VIII. References

3. I. Brief historical remarks about
signatures and spectra.
1985 – Samaniego, IEEE, signatures.
1991 – Elperin, Gertsbakh, Lomonosov, IEEE,
Internal Distribution (ID) equiv. to signature,
examples of ID for family of networks.
2004, 2005 – Gertsbakh, Shpungin – D-spectra,
which is a combinatorial version of signature,
network reliability.
2008 – Gertsbakh, Shpungin – BIM-spectra,
use of BIM-spectra in network synthesis.

4. I. Brief historical remarks about
signatures and spectra.
2009 – Models of Network Reliability, textbook.
D-spectra, BIM-spectra, Glossary of spectra,
algorithms.
Overall review of theoretical work on various
aspects of using signatures can be found in works
of Balakrishnan, Navarro, Spizzichino.

5. II. The Network The network and its elements The Network N(V, E, T)b 4
The Network N(V, E, T) 1 9
V – the set of vertices (nodes) s 3 5 t 7 8
E – the set of edges 2 10
T – the set of terminals 6 d c
Each node/edge may be in two states:
up (operational) and down (not operational).
N is UP iff T is connected.

6. II. The Network The network and its UP and DOWN States
We deal with the k-connectivity operational criteria up a 4 b
terminal
terminal 1 8 s t
down 6 d c
This state is UP

7. II. The network Static Network Reliability
Each node v is associated with probabilities
and
of being up and down respectively.
For edges these probabilities are
and
The element failures are independent events.
The network reliability R(N) is the probability that
N is in the UP state.

8. III. D-Spectrum (Signatures)4 b 1 9 s 3 5 t 7 8 2 10 6 d c
={4, 1, 3, 9, 10, 8, 7, 6, 5, 2}
Network fails on the 5th step
Anchor
r ( )=5

9. III. D-Spectrum (Signatures)
Associate with every permutation π, p(π)=1/n!
Denote:
The distribution
is called D-spectrum (n – number of elements)
The CDF
is called cumulative D-spectrum

10. III. D-Spectrum (Signatures)
The principal combinatorial property of the
cumulative D-spectrum:
C(x) – the #failure sets with exactly x components down
Components are independent. Pr(up)=p.

11. III. D-Spectrum (Signatures)
It is a topological feature of a network and does not
depend on the up probabilities of the nodes and edges.
How to estimate the spectrum?
The crucial feature of the appropriate algorithm
is that a single spanning tree allows finding
the anchor of the π.

12. III. D-Spectrum (Example)
H4 – Network with 16 nodes and 32 edges.
Nodes 1, 7, 14 are terminals Edges are unreliable. 7 14 1

13. III. D-Spectrum (Example)
D-spectrum for H4 (106 replications – 30sec.) yi i 25 17 9 1 26 18 10 2 27 19 11 3 28 20 12 4 29 21 13 5 30 22 14 6 31 23 15 7 32 24 16 8

14. IV. The BIM Spectrum The Birnbaum Importance Measure (BIM)
is defined as:
It expresses the rate of increase of the network reliability
with respect to the component's reliability increase.
It can be proved that
where Φ=1-R.
The question is: how to compute this measure?

15. IV. The BIM Spectrum. Example
={4, 1, 3, 9, 10, 8, 7, 6, 5, 2} s a b c d 1 2 4 3 5 6 7 9 10 8 t
Add 1 to the counter of zi,α if r(π)≤i and component
α is on one of the first i positions.

16. IV. The BIM Spectrum Denote by the number of all permutations
such that:
(1) The anchor of π, r(π)≤i.
(2) Element α is among the first i elements of π .
The collection of values
is called the cumulative BIM – spectrum
of the element α.

17. IV. The BIM Spectrum Theorem.
D-spectrum
The important combinatorial property of the
BIM-spectrum:
where C(x,α) is a number of failure sets with
exactly x down elements, containing α.

18. The BIM Spectrum. Comparing components.
In many cases we do not need to calculate the BIM’s
but only range them. We can do it by using the following
Theorem.
Let us take
for
(1) If for all
then
for all p values.
(2) If the condition of (i) takes place starting from some i=k
then there exists p0 such that for p> p0

19. IV. The BIM Spectrum. Example - H415 16
1,7,14-terminals
Edges - nonreliable 11 12 7 8 3 4 13 14 5 6 1 2
The first 4 groups by BIM’s ranking: 9 10
1-st group: (1,5), (5,7) 1 2
3,4
2-nd group: (6,14), (13,14)
Partial ranking
3-rd group: (1,3), (3,7), (5,6), (5,13)
4-rd group: (1,2), (1,9), (7,8), (7,15)

20. V. Numerical Evaluation
Algorithm - Computing BIM and D - spectra.
1. Initialize all ai, bij to be zero, i = 1, ..., n; j = 1, ..., n.
2. Simulate permutation π of edges.
3. Find out the permutation anchor r(π).
4. Put ar := ar + 1.
5. Find all j such that ej occupies one of the first r positions
in π, and for each such j put brj := brj + 1.

21. V. Numerical Evaluation
6. Put r := r + 1. If r ≤ n, GOTO 4.
7. Repeat 2 -6 M times.
8. Estimate F(i), yij via
The crucial feature of the algorithm is that a single
spanning tree allows finding the anchor of the π.

22. VI. Network optimal Design
Consider the following problem:
Suppose that for each element Pr(up)=p.
Choose k elements for replacing each by
more reliable with p*>p, in order to maximize
the network reliability.
Heuristic algorithm:
1. Estimate BIM spectrum for all elements.
2. Range the elements by their BIM spectrum.
3. Take the first k most important elements
and replace them by more reliable.

23. VI. Network optimal Design. Example.
R(p=0.6)=0.8693
R(p*group=0.9)
R(p*group=0.8)
R(p*group=0.7)
Edges group
0.9681
0.9379
0.9045
(1)+(2)
0.9402
0.9174
0.8936
(3)
0.9420
0.9235
0.8999
(4)

24. Knowing them is important for computing system
VII. Concluding Remarks
Networks (and coherent systems in general) have
several very useful combinatorial invariants: D-spectra
(signatures), BIM –spectra, etc.
Knowing them is important for computing system
reliability, component importance, system
comparison, optimal synthesis.
Our research concentrates mainly on efficient
Monte Carlo procedures and algorithms for
numerical evaluation of these invariants.

25. VII. Concluding remarks
More details , data, examples, algorithms, and
a catalog of D-spectra can be found in
“MODELS OF NETWORK RELIABILITY:
Analysis, Combinatorics, Monte Carlo”, CRC Press, 2009

26. VIII. References
1. Barlow, R.E. and F. Proschan. Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, Inc, 1975.
2. Elperin, T., Gertsbakh, I. and M. Lomonosov. Estimation of Network Reliability Using Graph Evolution Models. IEEE Trans. Reliab., R-40, 1991, pp
3. Gertsbakh, I. and Y. Shpungin. Combinatorial Approaches to Monte Carlo Estimation of Network Lifetime Distribution. Applied Stochastic Models in Business and Industry, 20, 2004, pp
4. Gertsbakh, I. and Y. Shpungin. Network Reliability Importance Measures: Combinatorics andMonte Carlo Based Computations,WSEAS Transactions on Computers, Issue 4, 7, 2008, pp

27. VIII. References
5. Ilya B. Gertsbakh and Yoseph Shpungin. Models of Network Reliability: Analysis, Combinatorics, and Monte Carlo, CRC Press, 2009, 217 p.
6. Samaniego, F. J. On Closure of the IFR Under Formation of Coherent Systems, IEEE Trans. Reliability, 34, 1985,
7. Samaniego, Francisko J. System Signatures and Their Applications in Engineering Reliability, 2007, Springer-New York, Berlin

28. Thank you for your attention.