A Framework for Network Survivability Characterization Soung C. Liew and Kevin W. Lu IEEE Journal on...

A Framework for Network Survivability CharacterizationSoung C. Liew and Kevin W. LuIEEE Journal on Selected Areas in Communications, January 1994 (ICC, 1992)

Wendy Wen

OPlab, IM, NTU

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OUTLINE1. Introduction2. Survivability of a Centralized Ring

Network under Link Failures3. General Procedure for Finding

Survivability Function4. Finding Survivability Function for a

Network5. Conclusions

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OUTLINE1. Introduction2. Survivability of a Centralized Ring Network

under Link Failures3. General Procedure for Finding Survivability

Function4. Finding Survivability Function for a Network5. Conclusions

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Objective This paper attempts to formulate a general

framework that both includes and extends the existing definitions for network survivability.

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Survivability Function The probability that a fraction s of the nodes are

connected to the central node.

S : network survivability, which is a random variable e : sample point E : sample space, E = {e} Pe: probability of each e Se: probability of nodes connected to the central node

C:\Documents and Settings\Wendy Wen\My Documents\Framework-Sur\Equation1.xls

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Advantage A number of different quantities of interest can be

derived from the function. E[S] , s* , sr , p0

C:\Documents and Settings\Wendy Wen\My Documents\Framework-Sur\Equation1.xls

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1. S = fraction of nodes connected to the central node under

2. All links are bidirectional. 3. The number of Node is very large. 4. link failures = n

A link failure is equally likely to be located anywhere.The locations of the n failures are independent.

n ≧2

Assumption

self-healing

s = Con.

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Objective Derive the corresponding survivability

function:

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Derivation (1) Dividing into many

small segments, each of length △s.

Size of the sample space is

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Derivation (2) Each of these sample points is equally likely,

Ns : the number of ways to make n cuts that result in

a survivability of s.

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Derivation (3): Find Ns 1. n(n-1) ways of choosin

g two cuts to be Cl and Cr

2. s/ s△ ways of putting the two adjacent cuts, from Cl to Cr

3. Ways of the remaining (n-2) cuts is .

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Derivation (4)

By definition,

Remind that,

Derivation (6): Find ps(S) Let Nc be the r.v. associated with the numbe

r of cuts.

where δ(x) is the impulse function

總和定理

Pr of n=0 and n=1

Pr of n≧2

P[S=s | n] , from eq.12

1, x=00, otherwise

δ(x) =

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Derivation (7): Find ps(S) Let MGF of Nc is

Then,

Assume Poisson distribution,

Thus,

pdf of Poisson

MGF of Poisson

P(N=0)

P(N=1)

P(N 2)=mgf≧ (2)(1-s)

Fr.eq. 17:

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Procedure1. Specify disaster type2. Define “goodness” of networks3. List the sample points {e}, or all

combinations of events 4. Determine the survivability Se

5. Determine or assign probability of each event e

6. Calculate survivability function

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Step 1. Specify disaster type Different disaster types may have different

effects on networks. severe thunderstorm cable cut

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Step 2. Define “goodness” of networks We may obtain results depending on the fe

atures of the network for which we are calculating survivability. the number of subscribers connected to a centr

al node the revenue collected by the network operator

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Step 3. List all combinations ofevents Sample space may simply be too large. It maybe can only be done effectively by a

computer.

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Step 4. Determine Se

This calculation will depend on our definition of survivability.

If as example above, then we would need an efficient algorithm.

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Step 5. Determine Pe This should be based on past observations

or experience. If the disaster happens so rarely, one will n

eed to use one’s judgement when assigning probabilities.

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Step 6. Calculate survivability function By summing the probabilities of all sample

points with the same survivability.

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24 nodes 26 links the number associated

with each link is its length 69 DS3 fiber-optic

transmission systems between 29 node pairs

Assumption

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Table I. DS3 demands between nodes of n network.

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4

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Step 1. Specify disaster type Hurricanes

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Step 2. Define “goodness” of networks Total number of surviving DS3’s under link f

ailures

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Step 3. List all combinations ofevents Localized disasters: the network survivability can

be easily found.

Hurricane: 226 = 67,108,864 possible combinations of link failures.

Assume that more than four link failures are highly unlikely.

179011495026003252626

4

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3

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2

26

1 CCCC

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Step 4. Determine Se

For each event e, the survivability is Se = (surviving DS3’s) / 69

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Step 5. Determine Pe (1) Assume that,

link failures are independent probability of a link failure is proportional to its l

ength

ε: to reflect the extent of damage expected of the hurricane,

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Step 5. Determine Pe (2) Probabilities of single, double, triple, and

quadruple link failures are:

Pr of a link failure Pr of the others

Pr of two link failures Pr of the others

Pr of three link failures Pr of the others

Pr of four link failures Pr of the others

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Step 6. Calculate survivability function (1)

For illustration, we condense all survivability within the intervals,

(0.02i -0.1 , 0.02i +0.1] i=1, 2 ,..., 50

Show survivability functions for ρ= 0.1 and ρ= 0.2.

Remind that,

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ρ = 0.1E [ S ] ≈ 0.923S* ≈ 0.28S10 ≈ 0.8Po ≈ 0

ρ = 0.2E [ S ] ≈ 0.822S* ≈ 0.28S10 ≈ 0.72Po ≈ 0

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When increasing ρ from 0.1 to 0.2, P(s=1) : decreases from 0.341 to 0.11 E[S] : decreases from 0.923 to 0.822 S10 : decreases from 0.8 to 0.72 S* : unchanged

Step 6. Calculate survivability function (2)

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Step 6. Calculate survivability function (3) Although the worst-case survivability is 0.2

8, it corresponds to two events of quadruple link failures links 3-6, 5-6, 6-7 and 6-8 links 3-6, 5-6, 6-7 and 8-15

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97

2510

links 3-6, 5-6, 6-7, 6-897

25

8

links 3-6, 5-6, 6-7, 8-15

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Conclusions Network survivability is characterized by a

survivability function.

Various quantities of interest can be derived from the survivability function.

This framework provides a unified and practical approach to analyzing and designing highly survivable communications networks.

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The End

Thank you for your listening~

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Fig. 1. A ring network with node failures due to a thunderstorm

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Table II. DS3’s lost due to link failures of a network.

3+2

3+ 2+4

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4

46Fig. 6. A network for survivability characterization.

max li

2.18max10 il

126.03.22.181][min min lfailslP i

min li

12.182.181][max max lfailslP i

A Framework for Network Survivability Characterization Soung C. Liew and Kevin W. Lu IEEE Journal on...

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