Topology Design for Service Overlay Networks with Bandwidth Guarantees

1. Topology Design for Service Overlay Networks with Bandwidth Guarantees Sibelius Vieira* Jorg Liebeherr** *Department of Computer ScienceCatholic 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 Transport link Access 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 • ESiEndsystem i • PNjProvider node j • αij Cost of reserving one Mbps from ESi to PNj, through the ISP which provides the minimal cost (access cost) • lij Cost of reserving one Mbps between PNitoPNj through the ISP that provides the minimal cost of connecting the two provider nodes (transport cost) • ωij Required bandwidth fromESi toESj • ΩjTotal bandwidth for traffic generated at ESj (Ωj = j ωij).

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 rnm denote the least-cost path between PNnto PNm • Cost of the least-cost path per unit of reserved bandwidth from PNnto PNm is bnm = (ij)rnm lij .

11. 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 Optimization problem • Let yij be a 0-1 decision variable that indicates if ESiis assigned to PNj • Solving the topology design problem requires: Minimize i k Ωi αik yik + i j kl yij ykl ωik bjl + jl Ωj αjl yjl subject toj yij = 1 for i = 1,..,M

12. Complexity Minimize i k Ωi αik yik + i j kl yij ykl ωik bjl + jl Ωj αjl yjl subject toj yij = 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. Finding simpler solutionsSpecial case • The optimization problem can be expressed as an equivalent matrix-combination problem • Define: u(i) = j, iff yij = 1. • Then: u = (u(1),u(2), ..,u(M)) is assignment of endsystems to provider nodes. • We can write optimization as: MinimizeZ(u) = i j ωij (αiu(i) + bu(i)u(j) + αju(i)) • Side conditions of the original problem are implicitly given via the definition of the u(i)´s.

14. Finding simpler solutions: Special case • Choose v(i) such that αiv(i) = minj{αij}. • Consider the following conditions: (C1) bij≤ bik + bkjfor all i,j,k ≤ N. (C2)αij ≥ αiv(i) + bv(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)

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., yiv(i)=1 iff. αiv(i) = minj{αij} • 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 (ri,si) and traffic load Ωi of each endsystem ESi 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 Rkis the set of endsystems assigned to the kth centroid, the centroid position is given by: • rk = i: ESi є Rk ri. Ωi/ i: ESi є Rk Ωi • sk = i: ESi є Sk si. Ωi/ i: ESi є Sk Ωi • 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 pα·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 (Repmax) controls the number of solutions evaluated by simulated annealing • Conclusion: Simulated annealing seems to work well

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 = 0.1 P = 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 pa = 0.5 pa = 0.9

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