# A Mathematical Model for Efficient and Fair Resource Assignment in Multipath Transport
## Appendix A Mathematical Annex
### Allocation-Specific Constraints
#### a.2.1 Bottleneck Subflow Fair Allocation (BSF)
Subflow allocation \(sf\_alloc^{st}_{ohij}=sf\_link\_share_{ij}\) when \(sf\_bneck^{st}_{ohij}=1\):  
\[sf\_alloc^{st}_{ohij}\leq sf\_link\_share_{ij}\qquad\forall(o,h)\in A,(i,j) \in A,[s,t]\in K,\] (A40)  
\[sf\_alloc^{st}_{ohij}\geq sf\_link\_share_{ij}-cap_{ij}(1-sf\_bneck^{st}_{ ohij})\qquad\forall(o,h)\in A,(i,j)\in A,[s,t]\in K.\] (A41)  
In bottleneck subflow fairness, single-path flows and individual subflows of mulitpath flows are equivalent and should in the ideal case get the same share of the link capacity. Equation (A42) and Equation (A43) ensure the fair share for a single path flow on the bottleneck link. A single path flow \([s,t]\in K\) gets the maximum share of the capacity on its bottleneck link \((i,j)\in A\) i.e., \(f^{st}_{ij}=sf\_link\_share_{ij}\) when \(flow\_type^{st}=0\) and \(y^{st}_{ij}=1\). For a multipath flow, Equation (A42) and Equation (A43) do not have any effect on the allocation:  
\[m^{st}\leq\varphi^{st}\qquad\forall[s,t]\in K, \tag{108}\]0 (A42)  
\[m^{st}\leq\varphi^{st}\qquad\forall[s,t]\in K, \tag{108}\]1 (A43)  
**Bottleneck flow fair allocation constraints**: Let \(flow\_link\_share_{ij}\) is the maximum amount of the capacity a flow can be allocated on the link \((i,j)\in A\). Equations (A44) and (A45) ensure equal share to all flows which are bottlenecked on the link \((i,j)\in A\). Equation (A44) expresses \(flow\_link\_share_{ij}\) as the maximum share of the capacity to a flow on the link \((i,j)\in A\). If the link \((i,j)\in A\) is bottlenecked for the flow \([s,t]\in K\) i.e., \(y^{st}_{ij}=1\), Equation (A45) ensures maximum share of the capacity \(flow\_link\_share_{ij}\) to the flow \([s,t]\in K\) on that bottleneck link. If \(y^{st}_{ij}=1\), then \(flow\_link\_share_{ij}=f^{st}_{ij}\) which means equal share of the bottleneck capacity to all the flows bottlenecked on the link \((i,j)\in A\):  
\[m^{st}\leq\varphi^{st}\qquad\forall[s,t]\in K, \tag{108}\]2 (A44)  
\[m^{st}\leq\varphi^{st}\qquad\forall[s,t]\in K, \tag{108}\]3 (A45)  
Besides assigning equal share to the flows on a bottlenecked link, BFF optionally takes a second step if there are flows which are represented by more than one subflow on the link. For each of these flows, BFF tries to share the capacity assigned to the respective flow equally among its subflows. Equation (A46) and Equation (A47) make sure that all the subflows of a flow get equal share from the flow capacity on that bottleneck link. Let \(sf\_link\_share^{st}_{ij}\) be the maximum allocation that a subflow of the flow \([s,t]\in K\) can get on the link \((o,h)\in A\). The difference between \(sf\_link\_share^{st}_{ij}\) and previously used \(sf\_link\_share_{ij}\) in Appendix A.2.2 is that \(sf\_link\_share_{ij}\) is the generic value irrespective of any flow where as \(sf\_link\_share^{st}_{ij}\) is the flow specific value. Equation (A46) means that \(sf\_link\_share^{st}_{ij}\) is the maximum amount of the capacity the subflow \([o,h]\in A\) of the flow \([s,t]\in K\) can get on the link \((o,h)\in A\). Equation (A47) ensures that the subflow \([o,h]\in A\) of the flow \([s,t]\in K\) gets the maximum share of the flow capacity on the link \((i,j)\in A\) if the subflow is bottlenecked on that link. In other words, subflows of the flow \([s,t]\in K\) which are bottlenecked on the link \((i,j)\in A\) get equal allocation from that link:  
\[m^{st}\leq\varphi^{st}\qquad\forall[s,t]\in K, \tag{108}\]4 (A46)  
\[m^{st}\leq\varphi^{st}\qquad\forall[s,t]\in K, \tag{108}\]5 (A47)  
Appendix A.2.3. Network Flow Fair Allocation (NFF)  
In this paragraph, the constraints to compute the allocation gain for the objective function NFF-objective-III are derived. Let \(\varphi_{single-path}^{st}\geq 0\) be the allocation of the single path flow \((s,t)\in K\). Equation (A48) refers to if the flow \((s,t)\in K\) is a multipath path flow, then \(\varphi_{single-path}^{st}=0\). Equations (A49) and (A50) make sure that \(\varphi_{single-path}^{st}\) is the allocation of the single path flow \((s,t)\in K\).