# A Mathematical Model for Efficient and Fair Resource Assignment in Multipath Transport
## 5 Performance Evaluation  
This section includes validation and performance comparisons of different allocation models. In order to validate the developed fair allocation methods, eight network topologies are considered which represent different topology types such as a linear network, a network with two alternative paths, a full-meshed network, etc. The details about the topologies are given in Figures 13-20. In the figures, the blue circles identify the nodes, the connectors represent the links and the arrow in each link specifies the direction of the data flow if the link is used. The numbers besides the connectors specify the respective link speeds. For each of the topologies, different scenarios are created by using different nodes as senders and receivers for data flows and in some cases the capacities of individual links are changed, resulting in 31 scenarios. The link speeds are given in the respective topology figures, whereas the data flows injected into the topologies are given in Tables 2-9. The same tables also show the end-to-end throughput which each flow can achieve within a given scenario considering the different fairness methods discussed in Section 3.2. The NFF column in the tables shows the results for the NFF-objective-VI function specified in Equations (25) and (26). The computation of the linear programming solutions is performed with the AIMMS (non-)linear programming software system [33]. The formulations for BSF and BFF are specified as a mixed-integer linear programming (MILP) problem (Source code available on [34]) which is processed by the CPLEX solver included in the AIMMS system. The NFF solution contains nonlinear components and is computed using AIMMS's Outer Approximation (AOA) module. The AIMMS software does, however, not support batch jobs, the problem description had to be entered manually into a GUI, so that an automated evaluation of different network scenarios could not be performed. For this reason, the evaluation is performed with a selected number of scenarios. Figure 14: Topology of scenarios 5 to 8: all stations in a line. The numbers specify the link speeds in Mbps. Scenarios 5 and 6: A = 10; Scenarios 7 and 8: A = 20. Figure 15: Topology of scenarios 9 to 14: Two-path connection between a and h where one of the paths is again split into two sub-paths. The numbers specify the link speeds in Mbps. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Scenario 1**} \\ \hline Flow & BSF I & BSF II & BFF I & BFF II & NFF \\
[a,f] & 25.0 & 25.0 & 25.0 & 25.0 & 25.0 \\
[m,n] & 10.0 & 10.0 & 10.0 & 10.0 & 10.0 \\ \hline \multicolumn{5}{c}{**Scenario 2**} \\ \hline Flow & BSF I & BSF II & BFF I & BFF II & NFF \\
[a,f] & 18.3 & 18.3 & 10.0 & 18.3 & 12.5 \\
[a,c] & 8.3 & 8.3 & 15.0 & 8.3 & 12.5 \\
[m,n] & 8.3 & 8.3 & 10.0 & 8.3 & 10.0 \\ \hline \multicolumn{5}{c}{**Scenario 3**} \\ \hline Flow & BSF I & BSF II & BFF I & BFF II & NFF \\
[n,h] & 7.5 & 15.8 & 7.5 & 15.8 & 10.0 \\
[m,h] & 15.0 & 8.3 & 15.0 & 8.3 & 10.0 \\
[a,h] & 7.5 & 7.5 & 7.5 & 7.5 & 10.0 \\
[a,d] & 10.0 & 8.3 & 10.0 & 8.3 & 10.0 \\ \hline \multicolumn{5}{c}{**Scenario 4**} \\ \hline Flow & BSF I & BSF II & BFF I & BFF II & NFF \\
[a,f] & 8.3 & 13.3 & 5.0 & 13.3 & 8.8 \\
[a,c] & 8.3 & 8.3 & 15.0 & 8.3 & 8.8 \\
[a,e] & 10.0 & 5.0 & 5.0 & 5.0 & 8.8 \\
[m,n] & 8.3 & 8.3 & 10.0 & 8.3 & 8.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Flow set and linear programming (LP) results for the flows’ end-to-end throughput for scenarios 1 to 4 described in Figure 13. All values in Mbps. Figure 16: Topology of scenarios 15 to 18: 2 two-path connections i-l and a-f with shared bottleneck links d-e and b-c. The numbers specify the link speeds in Mbps. Scenarios 15 and 16: A = B = C = 10, D = 15; scenarios 17 and 18: A = 30, B = 40, C = 10, D = 35.