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1 Riikka Susitaival and Samuli Aalto. Adaptive load balancing with OSPF. In Proceedings of the Second International Working Conference on Performance Modelling and Evaluation of Heterogeneous Networks (HET NETs 2004), pages P9/1 P9/10, Networks UK Publishers Reprinted with permission.

5 traffic loads ỹ l serve as new capacity constraints: Minimize x k l subject to the constraints k K l L x k l 0, for each k K and l L, x k l ỹ l, for each l L, k K Ax k = R k, for each k K, (2) The dual of the problem above is: Maximize k K d k U k t k l L ỹ l W l subject to the constraints W l 0 for each l L, U k s k = 0, for each k K (3) U k j U k i W (i,j) + 1, for each k K and (i, j) L. The required link weights are then given by w l = W l + 1, where the variables W l are determined as the solution to the dual problem. In addition, optimal destination based traffic splitting ratios φ t ij are determined from the link loads xk l of the solution of the primal problem. These splitting ratios are calculated as follows [6]: φ t ij = 2.4 Optimization of the splitting ratios k:t k =t x k (i,j) j :(i,j ) L k:t k =t x k (i,j ) The traffic control reacting to changes in traffic demands by changing the link weights is often too time consuming or impractical. In such a case with fixed link weights, we can still affect the traffic distribution by optimizing the traffic splitting ratios used in the routers. We present a procedure to determine the splitting ratios that minimize the maximum link utilization with the given link weights. As before, let Pk SP denote the set of shortest paths for IE-pair k with respect to these link weights. Let φ p denote the fraction of traffic demand d k that uses path p Pk SP. We start by solving these splitting ratios for each IE-pair k from the following LP-problem: Minimize α + r k K α 0, φ p 0, k K l L p P SP k :l p d k φ p subject to the constraints for each p p P SP k :l p d k φ p αb l, for each l L, p P k φ p = 1, for each k K. Let φ p be the optimal traffic share on path p. This induces the following link loads: x k l = p P SP k :l p d k φ p. The destination based splitting ratios for each node i can then be calculated as in (4). k K P SP k, (4) (5) P9/4

7 Algorithm At time t n, after receiving the information ŷ(n) concerning all the measured loads, node i adjusts the traffic splitting ratios for all destination nodes t as follows: 1. Calculate the cost D p (ŷ(n)) for each path p P SP (i,t). 2. Find the path q P(i,t) SP with maximum cost, i.e. D q(ŷ(n)) = max p P SP D p (ŷ(n)), and decrease the (i,t) splitting ratio of the first link (i, j) of that path as follows: φ t ij(n) = φ t ij(n 1) 1 g φt ij(n 1). 3. Choose another path r P(i,t) SP randomly and increase the splitting ratio of its first link (i, k) as follows: φ t ik(n) = φ t ik(n 1) + 1 g φt ij(n 1). 4. For all other admissible next hops j, keep the old splitting ratio, φ t ij (n) = φt ij (n 1). 4 Numerical performance evaluation In this section we evaluate numerically the performance of the proposed adaptive load balancing algorithm. First, in subsection 4.1, a simple but efficient numerical evaluation method is described. The method is similar to that developed in [7]. Thereafter, in subsection 4.2, the results of this evaluation method applied to two different test networks are presented. 4.1 Evaluation method The evaluation method is iterative and runs as follows. The test network (including the nodes n, links l, IE-pairs k, and paths p), the link weights w l and the traffic demands d k are first fixed. 4 Traffic of each IE-pair is initially allocated to the shortest paths with respect to the fixed link weights w l. If multiple shortest paths exist, traffic is initially split equally in each node (). At each iteration n, the measured link loads ŷ l (n) induced by the splitting ratios φ t ij (n 1) are calculated as follows. First we calculate, for each IE-pair k, the induced traffic splitting ratios φ p (n 1) for each path p Pk SP by φ p (n 1) = φ t k ij (n 1). Then the measured link loads ŷ l (n) are determined by (i,j) p ŷ l (n) = k K(d k + ɛ k (n)) p P SP k :l p φ p (n 1), where the ɛ k (n) are independent Gaussian random variables with mean 0 and variance δ 2 d 2 k describing the random fluctuations of traffic during measurement period n around the fixed demands d k. The coefficient of variation, δ, for this random variable is assumed to be the same for all IE-pairs k. After this, the new traffic splitting ratios φ t ij (n) are determined from the measured loads ŷ l(n) as presented in subsection Numerical results Two different test networks (see Figure 2) are used with the following characteristics: nodes, 52 links, and 72 IE-pairs; nodes, 102 links, and 380 IE-pairs. P9/6

8 Figure 2: Left-hand-side: 10-node network. Right-hand-side: 20-node network. The test networks are random networks generated by the mechanism described in [8]. Three different traffic scenarios are used. In the first one the random traffic fluctuations in the time-scale of measurements are ignored by setting the fluctuation parameter δ to 0. In the second one these random fluctuations are taken into account by setting the fluctuation parameter δ to 0.1. In the third scenario, we consider the traffic fluctuations in a longer time-scales by letting the traffic demands to change drastically three times during the evaluation period (after 500, 1000 and 1500 iterations, correspondingly). The results of the adaptive algorithm are compared with 1. : the standard policy where the traffic is splitted equally to the shortest paths with the unit link weights, 2. Sub-optimal : the optimal value of the restricted optimization problem (5) with link weights fixed to 1, and 3. : the optimal value of the static load balancing problem (1). No traffic fluctuations In this scenario δ = 0. The shortest paths needed for the adaptive algorithm are calculated using link weights w l = 1 for all links l. Figure 3 shows the resulting maximum link utilization for the 10-node and 20-node networks as a function of the number of iterations for granularity parameters g = 20 and g = 50. We can see that the performance of the adaptive algorithm approaches the sub-optimal value and improves the performance remarkably as compared to the standard equal splitting. A small step size in the algorithm ensures that oscillations are insignificant. The convergence times are only two times greater in the 20-node network (approx. 200 iterations) than in the 10-node network (approx. 100 iterations) in spite of the huge growth in the complexity of the network. Random traffic fluctuations in a shorter time-scale In this scenario δ = 0.1. The shortest paths needed for the adaptive algorithm are again calculated using link weights w l = 1 for all links l. Figure 4 shows the resulting maximum link utilization for the 10-node and 20-node networks as a function of the number of iterations for granularity parameters g = 20 and g = 50. We find that the random traffic fluctuations in a time-scale of the measurement period induce oscillations to the maximum link utilization. However, even with granularity parameter g = 20, oscillations are tolerable and the algorithm converges close to the sub-optimal value. Traffic fluctuations in a longer time-scale In this scenario the traffic demands change drastically three times. The shortest paths needed for the adaptive algorithm are first calculated using link weights w l = 1 for all links l. Figure 5 shows the resulting maximum link utilization for the 10-node and 20-node networks as a function of the number of iterations for granularity parameters g = 20 and g = 50. The adaptive algorithm reacts to the changes and provides a result close to the sub-optimal value in the 10-node network and close to 4 Note that the traffic demands are used only for the evaluation purposes. The algorithm itself does not use any information on these demands. P9/7

9 Figure 3: The maximum link utilization as a function of the number of iterations, when there are no traffic fluctuations, δ = 0. Left-hand-side: 10-node network. Right-hand-side: 20-node network Figure 4: The maximum link utilization as a function of the number of iterations, when there are random traffic fluctuations in a shorter time-scale, δ = 0.1. Left-hand-side: 10-node network. Right-hand-side: 20-node network. the optimal value in the 20-node network. Note that in the 10-node network in the iteration rounds from 500 to 1000 the result of heuristics and the sub-optimal value are equal. Then we assumed that the link weights and the splitting ratios of each router can be optimized in a longer time scale (hours to days). The shortest paths needed for the adaptive algorithm are thus first calculated using the optimal link weights corresponding to the original traffic demands d k and determined from the dual problem (3) and the splitting ratios from the solution of the primal problem (2) and formula (4). After that the link weights remain unchanged. Figure 6 shows the resulting maximum link utilization for the 10-node and 20-node networks as a function of the number of iterations for granularity parameters g = 20 and g = 50. Now the sub-optimal values in the 10-node network are close to the optimal values and thus heuristics can yield to the results close to the optimal solution. In the 20-node network the results are similar to the previous case in Figure 5, except the first iterations where the results are now immediately close to the optimal one. As a conclusion, the optimization of the link weights in the longer time-scale improves the performance of the network. In the 20-node network also the unit weights provide a good result. An explanation is that, in the 10-node network, the number of shortest paths related to the unit link weights is 116 whereas it is 153 in the case of the optimal link weights. Thus the number of φ-parameters is greater in the latter case and also the results are better. In the 20-node network, the corresponding numbers of the shortest paths are 782 and 785. Also the results are quite similar. P9/8

10 Figure 5: The maximum link utilization as a function of the number of iterations, when there are traffic fluctuations in a longer time-scale. Left-hand-side: 10-node network. Right-hand-side: 20-node network Figure 6: The maximum link utilization as a function of the number of iterations. Left-hand-side: 10-node network. Righthand-side: 20-node network. 5 Conclusion In this paper we studied how load can be balanced adaptively in OSPF-networks using a distributed approach. We also considered the procedure of optimizing the OSPF-weights by primal-dual methods and how this can be combined to adaptive heuristics. The results show that the optimization of traffic splitting ratios improves the performance of the network when compared to equal splitting. When the set of shortest paths is small also the changing of OSPF-weights is worthwhile. In the future the approach which combines the shorter and longer time-scale optimization has to be developed further. Also the actual converge time of the adaptive algorithm has to be studied carefully in realistic traffic scenarios. In addition, we have to study if the disorder of packets is really a problem and how this problem can be solved. Acknowledgements. This work was financially supported by the Academy of Finland (grant n:o 74524). P9/9

11 References [1] J. MOY, OSPF Version 2, IETF RFC2328, April [2] E. ROSEN, A. VISWANATHAN AND R. CALLON, Multiprotocol Label Switching Architecture IETF RFC3031, January [3] B. FORTZ, M. THORUP, Internet Traffic Engineering by Optimizing OSPF Weights, in Proc. of IEEE Infocom [4] B. FORTZ AND M. THORUP Optimizing OSPF/IS-IS Weights in a Changing World, IEEE Journal on Selected Areas in Communications, Vol. 20, No. 4, pp , [5] Y. WANG, Z. WANG AND L. ZHANG Internet Traffic Engineering without Full Mesh Overlaying, Proc. of IEEE Infocom [6] A. SRIDHARAN, R. GUÉRIN AND C. DIOT, Achieving Near- Traffic Engineering Solutions for Current OSPF-IS-IS Networks, in Proc. of IEEE Infocom [7] R. SUSITAIVAL, S. AALTO AND J. VIRTAMO, Adaptive load balancing using MPLS, accepted for publication in Proc. of MMB&PGTS, [8] C. VILLAMIZAR, OSPF Optimized Multipath (OSPF-OMP), IETF Internet-draft, draft-ietf-ospf-omp-02, February P9/10

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