**4.4 Decomposition evaluation**

With the purpose of evaluating the decomposition strategy, in this simulation set we consider randomly generated networks, flows and paths, and use the decomposition algorithm to partition the network in sub-networks. The networks were generated starting from a grid of nodes; in particular, the considered network width is 10 nodes. Each column of the grid can be assigned a number of nodes; in the considered network, the number of nodes per column is [18, 18, 18, 16, 10, 10, 16, 18, 18, 18]. 30 flows were considered, starting from a random node of the first column of the network and directed to a random node of the last column. Similarly, each network path is directed from a node of the first column of the network and directed to a node of the last column Fig. 4 a) shows an example of randomly generated network, whereas Fig. 4 a) shows an example of sub-network. The results were obtained by averaging 20 simulations. The average number of variables of the original problem (i.e., the non-decomposed one) is 1984.8, whereas the decomposition manages to decompose the network in 10.2 sub-network (in the average): each sub-network optimization problem has therefore 194.6 variables, i.e., each sub-network problem is reduced by about one order of magnitude.

Optimal Control Strategies for Multipath Routing:

modifications suggested in Bruni *et al.*, 2010).

flow and the possible presence of ballast links.

according to the present contest.

**5. Conclusion** 

higher traffic flow.

**6. References** 

From Load Balancing to Bottleneck Link Management 419

In this work we formulate the multipath routing problem as an optimal control problem considering various performance indices. In particular, the scenario includes the load balancing problem already dealt with in a previous work Bruni *et al.*, 2010, as well as the bottleneck minimax control problem, in which the traffic load of the bottleneck (raised to a given power *m*) is minimized. The mathematical structure of the problem might easily suggest some issues which are evidentiated by the results of Section 4, simply intended to provide a numerical example of more general behaviours. On one side, the load balancing performance index obviously allows to achieve a higher uniformity in the loading of the various links, but it cannot prevent overloading of possible ballast links (apart from *ad hoc*

On the other side, the minimax (bottleneck) approach succeeds in keeping the bottleneck loads (including the ones of the ballast links), as low as possible, with an effort which happens to be more successful the higher the value of *m* is. This allows accommodating for a

Moreover, we stress the fact that the choice of the proper performance index is a matter left to the network manager in charge of the routing control problem, who will have to take into account at the same time the network structure and capacity, as well as the admitted traffic

As a final conclusion, we have considered several cost functions for the multipath routing which are suitable for a certain network load situation. Those cost functions can be properly switched during the operations according to the network needs. In that way our approach is strongly oriented with the most innovative vision of the Future Internet perspective (see Delli Priscoli, 2010), in which the core idea is to take consistent and coordinated decisions

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Fig. 4. a) example of a network (width=10, height=18), b) one of the sub-networks resulting from the decomposition of the network in Fig. 4 a).
