# A Mathematical Model for Efficient and Fair Resource Assignment in Multipath Transport
## 3 Mathematical Model
### Extension of the Objective Function for Fairness
#### 3.2.3 Network Flow Fair Allocation (NFF)
\(max\_sf\_val^{st}\) is the equivalent single path allocation for the multipath flow \([s,t]\in K\) because, in single path allocation, a flow tries to take the path from which it can get maximum capacity:  
\[max\_sf\_val^{st}=\max_{(o,h)\in K,(i,j)\in K}sf\_alloc_{ohij}^{st}\quad \forall[s,t]\in K. \tag{17}\]  
The variable \(sf\_alloc_{ohij}^{st}\) specifies the capacity allocation to the subflow between interface \(o\) to \(h\), where the subflow is part of the flow from \(s\) to \(t\) and uses the link between nodes \(i\) and \(j\). The overall allocation gain due to multipath \(M_{\mathrm{gain}}\) is the difference between the aggregated allocation when flows inside the network utilize multipath and when flows would use single path only. Let, \(Alloc_{\mathrm{single-path-flow}}\) be the aggregated allocation when each flow would use single path  
\[Alloc_{\mathrm{single-path-flow}}=\sum_{[s,t]\in K}\varphi_{\mathrm{single- path}}^{st}+\sum_{[s,t]\in K}max\_sf\_val^{st}, \tag{18}\]  
where \(\varphi_{\mathrm{single-path}}^{st}\) is the allocation of the single path flow \((s,t)\in K\) which is computed by the constraints given in Equations (48) to (A50) in the Appendix A.2.3. The overall allocation gain due to multipath is then:  
\[M_{\mathrm{gain}}=Alloc-Alloc_{\mathrm{single-path-flow}}. \tag{19}\]  
Though the objective function NFF-objective-III (Equation (16)) overcomes the limitation of the objective function NFF-objective-II for the scenario in Figure 8, it may not always provide the optimum allocation. For example, consider the scenario in Figure 9. The allocation of the flow \([\mathrm{m},\mathrm{n}]\) is limited by the links \((\mathrm{c},\mathrm{d})\) and \((\mathrm{g},\mathrm{h})\). Flow \([a,d]\) is limited by the link \((\mathrm{a},\mathrm{b})\) and the flow \([\mathrm{e},\mathrm{h}]\) is limited by the link \((\mathrm{g},\mathrm{h})\). This leads to the ideal network flow fair allocation of 20 Mbps to each flow by utilizing the whole network capacity. The corresponding value of the objective function NFF-objective-III = \((20+20+20)-0+(3-1)\times 0=60\). However, if the allocation is considered as in Figure 10 where the allocation of the flows \([\mathrm{a},\mathrm{f}]\), \([\mathrm{e},\mathrm{h}]\) and \([\mathrm{m},\mathrm{n}]\) are 30, 30 and 10 Mbps, respectively, flows \([\mathrm{a},\mathrm{f}]\) and \([\mathrm{e},\mathrm{h}]\) limit the allocation of the flow \([\mathrm{m},\mathrm{n}]\) by utilizing multipath. In this case, the multipath gain of each individual multipath flow is 10 Mbps. This results in a value of the objective function NFF-objective-III = \((30+30+10)-((30-30)+(30-10)+(30-10))+(3-1)\times(10+10)=70\), which is higher than the previous allocation. Thus, the objective function NFF-objective-III would allocate 30 Mbps to the flows \([\mathrm{a},\mathrm{f}]\) and \([\mathrm{e},\mathrm{h}]\), whereas flow \([\mathrm{m},\mathrm{n}]\) would be allocated 10 Mbps, which is not an optimum network flow fair solution. Going back to the objective function NFF-objective-II, the reason behind the problem in Figure 8 was comparing all flows for the network flow fair solution even if they do not share any congested links, which means they are disjoint. Comparing all flows might create higher allocation differences which can not be compensated by the \(M\) value. Introducing \(M_{\text{gain}}\) with the multiplication factor \(\beta\) is more biased to multipath allocation, which has a negative effect on fairness. This results in the idea of comparing flows only when they share a congested link. To compare flows according to the shared congested link, let \(cong\_flow\_diff_{ij}^{st}\) be the sum of the allocation differences between the flow \([s,t]\in K\) and all other flows sharing the congested link \((i,j)\in A\) with the flow \([s,t]\in K\). In addition, consider a binary variable \(cong\_group\_id_{ij}^{st}\) which identifies the flow \([s,t]\in K\) sharing the congested link \((i,j)\in A\).