A Framework for Network Survivability Characterization Soung C. Liew and Kevin W. Lu IEEE Journal on Selected Areas in Communications, January 1994 (ICC, 1992) Wendy Wen OPlab, IM, NTU

2 OUTLINE 1. Introduction 2. Survivability of a Centralized Ring Network under Link Failures 3. General Procedure for Finding Survivability Function 4. Finding Survivability Function for a Network 5. Conclusions

3 OUTLINE 1. Introduction 2. Survivability of a Centralized Ring Network under Link Failures 3. General Procedure for Finding Survivability Function 4. Finding Survivability Function for a Network 5. Conclusions

4 Objective This paper attempts to formulate a general framework that both includes and extends the existing definitions for network survivability.

5 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}  P e : probability of each e  S e : probability of nodes connected to the central node

6 Advantage A number of different quantities of interest can be derived from the function.  E[S], s*, s r, p 0

7 OUTLINE 1. Introduction 2. Survivability of a Centralized Ring Network under Link Failures 3. General Procedure for Finding Survivability Function 4. Finding Survivability Function for a Network 5. Conclusions

8 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.

9 Objective Derive the corresponding survivability function:

10 Derivation (1) Dividing into many small segments, each of length △ s. Size of the sample space is

11 Derivation (2) Each of these sample points is equally likely, N s : the number of ways to make n cuts that result in a survivability of s.

12 Derivation (3): Find N s 1. n(n-1) ways of choosing two cuts to be C l and C r 2. s/ △ s ways of putting the two adjacent cuts, from C l to C r 3. Ways of the remaining (n-2) cuts is.

13 Derivation (4) By definition, Remind that,

14 Fig. 4. Survivability density functions for a ring network. Fig. 5. survivability distribution functions for a ring network.

15 Derivation (5) The various single-value survivability measures can also be easily derived:

Derivation (6): Find p s (S) Let N c be the r.v. associated with the number 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=0 0, otherwise δ(x) =

17 Derivation (7): Find p s (S) Let MGF of N c 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:

18 OUTLINE 1. Introduction 2. Survivability of a Centralized Ring Network under Link Failures 3. General Procedure for Finding Survivability Function 4. Finding Survivability Function for a Network 5. Conclusions

19 Procedure 1. Specify disaster type 2. Define “goodness” of networks 3. List the sample points {e}, or all combinations of events 4. Determine the survivability S e 5. Determine or assign probability of each event e 6. Calculate survivability function

20 Step 1. Specify disaster type Different disaster types may have different effects on networks.  severe thunderstorm  cable cut

21 Step 2. Define “ goodness ” of networks We may obtain results depending on the features of the network for which we are calculating survivability.  the number of subscribers connected to a central node  the revenue collected by the network operator

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

23 Step 4. Determine S e This calculation will depend on our definition of survivability. If as example above, then we would need an efficient algorithm.

24 Step 5. Determine P e This should be based on past observations or experience. If the disaster happens so rarely, one will need to use one’s judgement when assigning probabilities.

25 Step 6. Calculate survivability function By summing the probabilities of all sample points with the same survivability.

26 OUTLINE 1. Introduction 2. Survivability of a Centralized Ring Network under Link Failures 3. General Procedure for Finding Survivability Function 4. Finding Survivability Function for a Network 5. Conclusions

27 24 nodes 26 links the number associated with each link is its length 69 DS3 fiber-optic transmission systems between 29 node pairs Assumption

28 Table I. DS3 demands between nodes of n network

29 Step 1. Specify disaster type Hurricanes

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

31 Step 3. List all combinations of events Localized disasters: the network survivability can be easily found. Hurricane: 2 26 = 67,108,864 possible combinations of link failures. Assume that more than four link failures are highly unlikely.

32 Step 4. Determine S e For each event e, the survivability is S e = (surviving DS3’s) / 69

33 Step 5. Determine P e (1) Assume that,  link failures are independent  probability of a link failure is proportional to its length ε: to reflect the extent of damage expected of the hurricane,

34 Step 5. Determine P e (2) Probabilities of single, double, triple, and quadruple link failures are: Pr of a link failurePr 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

35 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,

36 ρ = 0.1 E [ S ] ≈ S * ≈ 0.28 S 10 ≈ 0.8 P o ≈ 0 ρ = 0.2 E [ S ] ≈ S * ≈ 0.28 S 10 ≈ 0.72 P o ≈ 0

37 When increasing ρ from 0.1 to 0.2,  P(s=1) : decreases from to 0.11  E[S] : decreases from to  S 10 : decreases from 0.8 to 0.72  S * : unchanged Step 6. Calculate survivability function (2)

38 Step 6. Calculate survivability function (3) Although the worst-case survivability is 0.28, 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

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

40 OUTLINE 1. Introduction 2. Survivability of a Centralized Ring Network under Link Failures 3. General Procedure for Finding Survivability Function 4. Finding Survivability Function for a Network 5. Conclusions

41 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.

42 Future Work The framework can accommodate situations that involve dependency among failures.

43 The End Thank you for your listening~

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

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

46 Fig. 6. A network for survivability characterization. max l i min l i