**4.3 Comparisons between load balancing and bottleneck link management**

In this simulation set we consider that all the traffic sources transmit with an increasing trend from 4.5 Mbps to 8.5 Mbps. Tables 2-5 show the network load as the sources bit rate increase, and compares the optimal bottleneck control solutions for *m* = 1, 2, 3 with the load balancing optimal solution.

In Tables 2-5, we denote by bold characters the normalized link loads exceeding 1; hereinafter, the corresponding links will be denoted as overloaded links.

The bottleneck control for *m* ≥ 2 manages a higher network load than the load balancing approach. In fact, the tables show that the solutions of the bottleneck control problem are such that no link is overloaded until the flow rates exceed 5 Mbps, 6.5 Mbps and 6.5 Mbps for *m* = 1,2,3, respectively; on the other hand, the load balancing solutions are such that no link is overloaded until the flow rates exceed 5 Mbps. Similar results are obtained for subnetwork 2.


Table 2. Sub-network 1: Optimal Solutions under bottleneck control, *m* =1.

Optimal Control Strategies for Multipath Routing:

*u*(*f*1,*v*1) 0,60 *u*(*f*1, *v*2) 0,40 *u*(*f*4, *v*7) 0,45 *u*(*f*4, *v*8) 0,55

Table 5. Sub-network 1: Optimal Solutions under load balancing control.

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

**4.4 Decomposition evaluation** 

reduced by about one order of magnitude.

From Load Balancing to Bottleneck Link Management 417

Rate [Mbps] 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5

*g*(*v*1) 0,24 0,27 0,30 0,33 0,36 0,39 0,42 0,45 0,48 0,51 *g*(*v*2) 0,16 0,18 0,20 0,22 0,24 0,26 0,28 0,30 0,32 0,34 *g*(*v*7) 0,33 0,37 0,41 0,45 0,50 0,54 0,58 0,62 0,66 0,70 *g*(*v*8) 0,41 0,46 0,51 0,56 0,62 0,67 0,72 0,77 0,82 0,87 *l1* 0,74 0,83 0,92 **1,01 1,11 1,20 1,29 1,38 1,48 1,57**  *l2* 0,45 0,50 0,56 0,61 0,67 0,72 0,78 0,84 0,89 0,95 *l5* 0,30 0,33 0,37 0,41 0,44 0,48 0,52 0,55 0,59 0,63 *l6* 0,33 0,37 0,41 0,45 0,50 0,54 0,58 0,62 0,66 0,70 *l7* 0,41 0,46 0,51 0,56 0,62 0,67 0,72 0,77 0,82 0,87 *l10* 0,41 0,46 0,51 0,56 0,62 0,67 0,72 0,77 0,82 0,87 *l11* 0,24 0,27 0,30 0,33 0,36 0,39 0,42 0,45 0,48 0,51 *l14* 0,16 0,18 0,20 0,22 0,24 0,26 0,28 0,30 0,32 0,34 *l16* 0,33 0,37 0,41 0,45 0,50 0,54 0,58 0,62 0,66 0,70 *l19* 0,41 0,46 0,51 0,56 0,62 0,67 0,72 0,77 0,82 0,87




Table 4. Sub-network 1: Optimal Solutions under bottleneck control, *m* =3.

*g*(*v*1) 0,60 0,67 0,75 0,82 0,90 0,97 1,05 1,12 1,20 1,27 *g*(*v*2) 0,55 0,62 0,69 0,76 0,83 0,90 0,97 1,04 1,11 1,18 *g*(*v*7) 0,55 0,62 0,69 0,76 0,83 0,90 0,97 1,04 1,11 1,18 *g*(*v*8) 0,60 0,67 0,75 0,82 0,90 0,97 1,05 1,12 1,20 1,27 *l1* 0,32 0,36 0,40 0,44 0,48 0,52 0,57 0,61 0,65 0,69 *l2* 0,08 0,09 0,10 0,11 0,12 0,13 0,13 0,14 0,15 0,16 *l5* 0,14 0,16 0,18 0,20 0,21 0,23 0,25 0,27 0,29 0,30 *l6* 0,60 0,67 0,75 0,82 0,90 0,97 **1,05 1,12 1,20 1,27**  *l7* 0,55 0,62 0,69 0,76 0,83 0,90 0,97 **1,04 1,11 1,18**  *l10* 0,60 0,67 0,75 0,82 0,90 0,97 **1,05 1,12 1,20 1,27**  *l11* 0,14 0,16 0,18 0,20 0,21 0,23 0,25 0,27 0,29 0,30 *l14* 0,14 0,16 0,18 0,20 0,21 0,23 0,25 0,27 0,29 0,30 *l16* 0,60 0,67 0,75 0,82 0,90 0,97 **1,05 1,12 1,20 1,27**  *l19* 0,60 0,67 0,75 0,82 0,90 0,97 **1,05 1,12 1,20 1,27** 

Rate [Mbps] 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5

*u*(*f*1,*v*1) 0,81 *u*(*f*1, *v*2) 0,19 *u*(*f*4, *v*7) 0,19 *u*(*f*4, *v*8) 0,81

Table 3. Sub-network 1: Optimal Solutions under bottleneck control, *m* =2.

Table 4. Sub-network 1: Optimal Solutions under bottleneck control, *m* =3.

*u*(*f*1,*v*1) 0,79 *u*(*f*1, *v*2) 0,21 *u*(*f*4, *v*7) 0,21 *u*(*f*4, *v*8) 0,79

Rate [Mbps] 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5

*g*(*v*1) 0,59 0,66 0,73 0,81 0,88 0,95 1,03 1,10 1,18 1,25 *g*(*v*2) 0,57 0,64 0,71 0,78 0,85 0,92 0,99 1,06 1,13 1,20 *g*(*v*7) 0,57 0,64 0,71 0,78 0,85 0,92 0,99 1,06 1,13 1,20 *g*(*v*8) 0,59 0,66 0,73 0,81 0,88 0,95 1,03 1,10 1,18 1,25 *l1* 0,32 0,36 0,40 0,44 0,48 0,52 0,56 0,59 0,63 0,67 *l2* 0,08 0,09 0,10 0,11 0,12 0,13 0,14 0,16 0,17 0,18 *l5* 0,15 0,17 0,19 0,21 0,23 0,25 0,27 0,29 0,31 0,33 *l6* 0,59 0,66 0,73 0,81 0,88 0,95 **1,03 1,10 1,18 1,25**  *l7* 0,57 0,64 0,71 0,78 0,85 0,92 0,99 **1,06 1,13 1,20**  *l10* 0,59 0,66 0,73 0,81 0,88 0,95 **1,03 1,10 1,18 1,25**  *l11* 0,15 0,17 0,19 0,21 0,23 0,25 0,27 0,29 0,31 0,33 *l14* 0,15 0,17 0,19 0,21 0,23 0,25 0,27 0,29 0,31 0,33 *l16* 0,59 0,66 0,73 0,81 0,88 0,95 **1,03 1,10 1,18 1,25**  *l19* 0,59 0,66 0,73 0,81 0,88 0,95 **1,03 1,10 1,18 1,25** 


Table 5. Sub-network 1: Optimal Solutions under load balancing control.
