Redundant Trees for Preplanned Protection & Multipath Routing

1. Redundant Trees for Preplanned Protection & Multipath Routing By: Umang Patel CSC-772 Presentation

2. Redundant Trees (Blue & Red) source

3. Advantages • Suitable for preplanned protection • Dual benefits for multicast networks: Routing & Protection • Topological requirement are relaxed. • Requires any 2-edge-connected topology for protection against single-link failure & any 2-vertex-connected topology for protection against single-vertex failure. Figure -1 Figure -2 Figure -3

4. Problem Statement • To design two directed trees (Blue & Red) in such a fashion that the elimination of any vertex (edge) in the graph (other than the source) leaves each destination vertex connected to the source by at least one of the directed trees for any source, and destination vertices in any vertex (edge) redundant graph.

5. Algorithm-1 (For 2-vertex connected topology) • Step-1 Let TB and TR contains node{s}. Assign N(sM)=MAX, N(sM’)=0

6. Algorithm-1 • Step-2 - Find a cycle [s, v1,v2 ,..., vk,s]with k>=2. - Put s->v1->v2->…->vk on the blue chain - Put s->vk->vk-1->…->v1 on the red chain - Add blue Chain to TB,red chain to TR. - Assign. Numbers N(SM) > N(v1)>…>N(vk)>N(SM’) N(SM)>N(1)>N(6)>N(8)>N(5)>N(4)>N(3) >N(SM’)

7. Algorithm-1 • Step-3 - If TB contains all nodes in G, stop. 4. Step-4 - Find path [x, v1,v2 ,..., vk,y] s.t. x,y distinct nodes on blue chain. If x=s, N(x)=N(sM), if y=s N(y)=N(sM’). N(x)>N(y) , k>=1, vi not on blue chain. - Put x->v1->v2->…->vk on the blue chain - Put y->vk->vk-1->…->v1 on the red chain - Add blue Chain to TB,red chain to TR. - Assign. Numbers N(x) > N(v1)>…>N(vk)>N(y)>N(x’) N(x’) is max. of assigned values lower than N(x)

8. Algorithm-1 Interation-5 Interation-4 ,

9. Algorithm-2 (For 2-edge connected topology) • Step-4 modified condition Find path [x, v1,v2 ,..., vk,y] s.t. x,ydistinct nodes on blue chain. If x=s, N(x)=N(sM), if y=s N(y)=N(sM’). N(x)>=N(y), k>=1, vi not on blue chain.

10. Algorithm-3(Heuristic for low average delay on Blue tree) • Running time: O(n2(m+nlogn)) • For n nodes, 3n and nlogn link topologies. • For each topology, 100 2-connected graphs • Bandwidth, cost, delay of links are random integers uniformly distributed in range [1,10] • Results are average over 100 runs • Step-2 modified condition Find cycle such that delay of this Cycle minus the last edge is minimum among all such cycles. • Step-4 modified condition Find Path [x, v1,v2 ,..., vk,y] that has minimum delay from (s,x) plus (x,vk) among all such paths.

11. Algorithm-4(Heuristic for reducing total cost) • Running time: O(n2(m+n)) • For n nodes, 3n and nlogn link topologies. • For each topology, 100 2-connected graphs • Bandwidth, cost, delay of links are random integers uniformly distributed in range [1,10] • Results are average over 100 runs • Scaled cost For cycle [x, v1,v2 ,..., vk,x] scaled cost is ((c(x,v1)+c(v1,v2)+…+c(vk,x))/k For path [x, v1,v2 ,..., vk,y] scaled cost is ((c(x,v1)+c(v1,v2)+…+c(vk,y))/k • Steps 2 and 4 of the Algorithm 1 are modified to find low scaled cost cycle and path in each iteration.

12. Algorithm-5(Optimal algo. for maximizing bottleneck bandwidth) • For n nodes, 3n and nlogn link topologies. • For each topology, 100 2-connected graphs • Bandwidth, cost, delay of links are random integers uniformly distributed in range [1,10] • Results are average over 100 runs • Use bisection method on bandwidth values to find largest B such that G(B) is 2-connected. • Apply Algo. 1 or Algo. 2 for constructing Redundant Trees.

13. Algorithm-6(Algorithm for enhancing QoP) • Running time: O(n2m) • For n nodes, 3n and nlogn link topologies. • For each topology, 100 2-connected graphs • Bandwidth, cost, delay of links are random integers uniformly distributed in range [1,10] • Results are average over 100 runs • QoP of pair of single-link recovery trees equals total number of cycles and paths used in construction process. • Step 2 and step 4 of algorithm 2 is modified to find minimum hop cycle and paths in each iteration.

14. MPLS Domain protection using Redundant Trees • Assume MPLS domain represented by graph G(N,L). N is set of nodes and L is set of links between nodes. G is two-edge connected and therefore can protect against single link failure. • Protection paths for all working path terminating in an egress router are calculated simultaneously.

15. MPLS Domain protection using Redundant Trees • Initialization: - Find spanning tree of graph G rooted in egress router A. - Let P be set of nodes for which the protection paths have been established. Initially P={e}

16. MPLS Domain protection using Redundant Trees • Repeat until all nodes are protected (P=N) - Select one of branches of spanning tree attached to egress node and mark all nodes except egress node. - scan all marked nodes to find node i that has link to unmarked node j. - consider ring path consisting of links of spanning tree leading from A to i, the link between I and j and links of spanning tree between j and e.

17. MPLS Domain protection using Redundant Trees - Place two protection paths along the ring: one in clock- wise , the other in counterclockwise direction. The paths originate in two nodes of the ring that are adjacent to egress router and follow the ring all the way to egress node. Merge the created protection paths with the protection paths established in previous iterations. All nodes on the ring are now connected to both protection paths and added to P.

18. MPLS Domain protection using Redundant Trees - In the subsequent iteration of the algorithm consider new graph constructed by treating all nodes in P as a single node that will act as the egress node and removing all links that connect two protected nodes.

19. MPLS Domain protection using Redundant Trees

20. Redundant Multicast Trees for optical networks Multicast Session from S to {6,8,14,12} Edge-disjoint paths from S to node 6

21. Redundant Multicast Trees for optical networks Edge-disjoint paths from S to nodes 6,8 Edge-disjoint paths from S to nodes 6,8.11

22. Redundant Multicast Trees for optical networks Edge-disjoint paths from S to nodes 6,8,11,12

23. Redundant Multicast Trees for optical networks

24. Protecting multicast sessions in WDM Mesh Networks using Redundant Trees • Multicast Session (S->1,2) Protected • - Working Tree: S->1(λ1), 1->2(λ1) • - Protection Tree: S->2(λ1), 2->1(λ1) • Multicast Session (S->1,2,3), Not-protected • Working Tree: S->1(λ2), 1->2(λ2), 1->3(λ1)

25. ILP for finding Light Trees

26. Redundant Trees For Multipath Routing • Every Node in the network has two preferred neighbors to drain: red and blue. • Source marks packet with color and intermediate node forwards packet according to color of packet. • Path from any source to drain on blue and red trees are link/node disjoint. • Network can be viewed as two trees rooted at drain and the paths on these trees are directed towards drain.

27. Redundant Trees For Multipath Routing • Every Node in the network has two preferred neighbors to drain: red and blue. • Source marks packet with color and intermediate node forwards packet according to color of packet. • Path from any source to drain on blue and red trees are link/node disjoint. • Network can be viewed as two trees rooted at drain and the paths on these trees are directed towards drain.

30. Distributed construction of colored trees

31. Analysis of Multipath Routing • System Model - Source A and destination B - n disjoint routes between them - Each route can support m=1 connection - Connection arrival is Poisson Process with rate λ - When connection request arrives, no knowledge about availability of routes - One(or more) routes are selected randomly to attempt reservation - Overall period of reservation and connection duration time is exponentially distributed with mean 1/μ

32. Single-Path Reservation • For each connection arrival, one route out of n is randomly chosen to attempt bandwidth reservation on it. - No additional attempts are made upon failure to bandwidth reservation. • System can be modeled as n+1 state birth-death process with transition rates • Markov chain is depicted below

33. Greedy Multipath-Path Reservation • For each connection arrival, all n routes are chosen to attempt bandwidth reservation on it. - No additional attempts are made upon failure to bandwidth reservation on any route. • System can be modeled as n+1 state birth-death process with transition rates λs= λ and μs= μ • Markov chain is depicted below

34. Greedy Multipath-Path Reservation Flow-balance equation Steady-state probability of being in state s Summation of all probabilities Steady-state probability of being in state 0 Probability of successful connection

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