1. Topology Design for Service Overlay Networks with Bandwidth Guarantees Sibelius Vieira* Jorg Liebeherr** *Department of Computer Science Catholic University of Goias, Brazil **Department Computer Science University of Virginia

2. Service Overlay Networks Provisioning of end-to-end QoS across multiple autonomous systems (ASs) requires a level a cooperation that is difficult to achieve in the current architecture. Service Overlay Networks can avoid these difficulties We define a Provider Network as a value-added overlay network that supports end-to-end bandwidth guarantees to a collection of subscribers Problem studied in this paper: Building a topology for a provider network

3. Endsystems and Provider Nodes Provider network = Provider nodes + Endsystems Provider nodes and endsystems gain access to the Internet through ISPs Provider network buys bandwidth from ISPs and sells bandwidth guarantees to endsystems

4. Provider nodes, endsystems and ISPs Two provider nodes and/or endsystems can establish a link between themselves if they have a common ISP Access link Transport link

5. Topology Design Problem Given the connectivity of endsystems, provider nodes, and ISPs Given the bandwidth requests between endsystems How to construct a “good” topology ?

6. Solution to the topology of the provider network For each endsystem, select an ISP to connect endsystem to a provider node Connect provider nodes, so that there are end-to-end paths for traffic between endsystems

7. Resulting topology:

8. Formal problem statement M Number of endsystems N Number of provider nodes ES i Endsystem i PN j Provider node j α ij Cost of reserving one Mbps from ES i to PN j, through the ISP which provides the minimal cost (access cost) l ij Cost of reserving one Mbps between PN i to PN j through the ISP that provides the minimal cost of connecting the two provider nodes (transport cost) ω ij Required bandwidth from ES i to ES j Ω j Total bandwidth for traffic generated at ES j (Ω j =  j ω ij ).

9. Formal problem statement Each endsystem must be assigned to one provider node via an access link Provider nodes must be connected by transport links Cost of a link is weighted by the traffic sent over the link Total cost of network = Costs of the access links + transport links Goal: Minimize total cost of network

10. Irrespective of the amount of traffic, traffic between two provider nodes is sent at lowest cost if it is sent on the least- cost path between the two provider nodes Let r nm denote the least-cost path between PN n to PN m Cost of the least-cost path per unit of reserved bandwidth from PN n to PN m is b nm =  (ij)  r nm l ij.

11. Optimization problem Let y ij be a 0-1 decision variable that indicates if ES i is assigned to PN j Solving the topology design problem requires: Minimize  i  k Ω i α ik y ik +  i  j  k  l y ij y kl ω ik b jl +  j  l Ω j α jl y jl subject to  j y ij = 1 for i = 1,..,M Ingress access costs (from endsystems to provider nodes) Transport cost Egress access costs (from provider nodes to endsystems) Each endsystem is connected to one provider node

12. Complexity Minimize  i  k Ω i α ik y ik +  i  j  k  l y ij y kl ω ik b jl +  j  l Ω j α jl y jl subject to  j y ij = 1 for i = 1,..,M Bad news: The optimization is a variant of the NP-hard quadratic assignment problem (QAP) Good news: In some special cases, the problem can be much simplified Heuristics optimizations (e.g., simulated annealing) seem to work well for this problem

13. The optimization problem can be expressed as an equivalent matrix-combination problem Define: u(i) = j, iff y ij = 1. Then: u = (u(1),u(2),..,u(M)) is assignment of endsystems to provider nodes. We can write optimization as: Minimize Z(u) =  i  j ω ij (α iu(i) + b u(i)u(j) + α ju(i) ) Side conditions of the original problem are implicitly given via the definition of the u(i)´s. Finding simpler solutions Special case

14. Choose v(i) such that α iv(i) = min j {α ij }. Consider the following conditions: (C1) b ij ≤ b ik + b kj for all i,j,k ≤ N. (C2) α ij ≥ α iv(i) + b v(i)j for all i ≤ M and j, v(i) ≤ N. Note: (C1) always holds by construction of the least-cost paths, and (C2) is satisfied if the cost structure is such that access costs outweigh transport costs. Lemma 1. Under (C1) and (C2), Z(u) is minimized for the mapping u(i)=v(i) Finding simpler solutions: Special case

15. Finding simpler solutions: Heuristic solutions Without (C2), exact solutions can be obtained only for problems up to 30 endsystems and provider nodes Here, heuristic optimizations are necessary Simulated annealing has been shown to provide good results for QAP type problems. See paper for details of the simulated annealing algorithm

16. Finding simpler solutions: Greedy Algorithm Greedy assignment: assign endsystems to provider nodes with lowest access cost, i.e., y iv(i) =1 iff. α iv(i) = min j {α ij } When (C2) holds, greedy assignment yields the optimal solution The algorithm performs well when access costs dominate transport costs

17. Finding simpler solutions: Clustering Cluster endsystems into groups (regions) and assign complete regions to a provider node Rules for clustering: Endsystems that are geographically close are likely to be assigned to the same region Endsystems with higher traffic load are given more consideration when regions are being formed Use the k-means clustering algorithm to assign endsystems into regions: Input – M endsystems with position (r i,s i ) and traffic load Ω i of each endsystem ES i and number of desired regions, R. Output – R cluster centers (centroids) and assignment of each endsystem to each centroid.

18. Clustering Algorithm for Endsystems If R k is the set of endsystems assigned to the kth centroid, the centroid position is given by: r k =  i: ESi є Rk r i. Ω i /  i: ESi є Rk Ω i s k =  i: ESi є Sk s i. Ω i /  i: ESi є Sk Ω i After establishing the new centroid position, re-associate each endsystem with a region by reassigning each endsystem to the closest centroid, until the algorithm converges.

19. Numerical Evaluation Questions How well do the heuristic algorithms perform? How does cost change with the number of provider nodes? What is the impact of the clustering algorithm? Evaluation with random graphs Connectivity of provider nodes is determined by random graph (using the GT-ITM, ‘Pure Random’ model) Each endsystem can access a randomly subset of p α ·100% of provider nodes Access costs = Uniform[5,50] Transport costs = Uniform[5,50] Traffic matrix = Uniform[10,20] Mbps

20. Evaluation of Simulated Annealing Comparson with optimum solution for a small network (M = 9, N = 9) Repetition factor (Rep max ) controls the number of solutions evaluated by simulated annealing Conclusion: Simulated annealing seems to work well Repetition factor (Rep max ) Average deviation from minimum (%) Number of optimal solutions found (from 100) 106.59 %1 204.44%3 301.41%4 400.02%7 500.02%9

21. Evaluation of Simulated Annealing Enforce condition (C2)  optimum solution can be computed for large networks Here: Simulated annealing always gets close to optimum solution Set: M = N Value of “Repetition Factor” (Repmax) needed to get simualted annealing within 1% of optimal solution

22. Evaluation of Heuristic Algorithms General network (i.e.,do not assume (C2)) Number of endsystems and provider nodes: 10 to 100 Prob. of transport link between provider nodes: P = 0.1, 0.5, 0.9. Comparison of: simulated annealing greedy algorithm random assignment

23. Evaluation of Heuristic Algorithms Plots show cost of network relative to “Greedy algorithm” P = 0.1P = 0.5

24. Impact of the Number of Provider Nodes Network with M = 100 endsystems and N= 10-100 provider nodes Solution method: Simulated annealing Costs normalized to network with N=10 p  = 0.9 p  = 0.5

25. Impact of Clustering Network of M=100 endsystems and N= 10 provider nodes. Number of regions is 10 – 100 Solution method: Simulated annealing

26. Conclusions Formaluated network topology design problem for a service overlay network with QoS guarantees We showed that the general problem is NP- hard But when the underlying network satisfies certain conditions, the problem has only linear complexit Developed and evaluated several heuristic methods Caveat: Different cost structure may give different results and may require a different solution approach