# A Mathematical Model for Efficient and Fair Resource Assignment in Multipath Transport
## Appendix A Mathematical Annex
### General Constraints
Thus, the information through which links the subflow is routed can not be identified from \(z_{oh}^{st}\) which can be found through subflow identifier \(sf\_id_{ohij}^{st}\):  
\[\sum_{[s,t]\in K}f_{ij}^{st}\leq cap_{ij}\qquad\forall(i,j)\in A. \tag{11}\]2  
Equation (114) sets the variable \(sf\_id_{ihih}^{st}\) to zero if the flow starting at node \(i\) does not split into subflows, i.e., the flow only uses one single path to the destination. Equation (115) signifies that, if there is no subflow \([o,h]\in A\) for the flow \([s,t]\in K\), i.e., the \(z_{oh}^{st}=0\), the subflow identifier \(sf\_id_{ohij}^{st}\) should be set to zero for any link \((i,j)\in A\). Once a subflow is initialized from a certain node i.e., \(sf\_id_{ohij}^{st}\) is set to 1, Equation (116) extends the subflow identification over the path it takes to reach the destination. Equation (117) ensures that, if there is an incoming subflow \([o,h]\in A\) on link \((i,j)\in A\) to any node \(j\) which is not a source or destination node, the subflow \([o,h]\in A\) also exists on the link \((j,m)\in A\) to complete the subflow path:  
\[\sum_{[s,t]\in K}f_{ij}^{st}\leq cap_{ij}\qquad\forall(i,j)\in A. \tag{11}\]3 \[\sum_{[s,t]\in K}f_{ij}^{st}\leq cap_{ij}\qquad\forall(i,j)\in A. \tag{11}\]4 (115) \[\sum_{[s,t]\in K}f_{ij}^{st}\leq cap_{ij}\qquad\forall(i,j)\in A. \tag{11}\]5  
The previously introduced equations ensure that subflows are not created erroneously but do not limit the number of subflows which a single flow can be split into. In order to enforce this limit, a variable \(splitting\_node^{st}\) which counts the total number of paths used by the flow \([s,t]\in K\) is defined:  
\[\sum_{[s,t]\in K}f_{ij}^{st}\leq cap_{ij}\qquad\forall(i,j)\in A. \tag{11}\]6  
Equation (118) limits the number of paths resp. subflows for the flow \([s,t]\in K\) to the product of the sender and receiver interfaces. For each interface pair, maximum one subflow is possible:  
\[\sum_{[s,t]\in K}f_{ij}^{st}\leq cap_{ij}\qquad\forall(i,j)\in A. \tag{11}\]7  
A helping variable \(n\_sf^{st}\) is introduced to express that the number of subflows is identical to the number of paths used for a multipath flow. \(n\_sf^{st}\) counts the total number of subflows for the flow \([s,t]\in K\). Equations (119)-(112) result in \(splitting\_node^{st}=\sum_{(i,j)\in A}sf\_id_{ijij}^{st}=n\_sf^{st}\) when \(flow\_type^{st}=1\):  
\[\sum_{[s,t]\in K}f_{ij}^{st}\leq cap_{ij}\qquad\forall(i,j)\in A. \tag{11}\]8  
\[\sum_{[s,t]\in K}f_{ij}^{st}\leq cap_{ij}\qquad\forall(i,j)\in A. \tag{11}\]9  
\[f_{ij}^{st}\leq cap_{ij}x_{ij}^{st}\qquad\forall(i,j)\in A,[s,t]\in K. \tag{12}\]0  
Equations (112) to (114) prevent subflow creation if the source as well as the destination have only one interface. The equations also ensure that only one subflow can be created for any source-destinationinterface pair. Equation (A22) means that, for the flow \([s,t]\in K\), both subflows \([m,n]\in A\) and \([m,p]\in A\) can not be created from the node \(m\) using the same source interface \((s,j)\in A\) and destination interface \((i,t)\in A\) pair. Equation (A23) implies that there can not be two incoming subflows \((m,p)\in A\) and \((o,p)\in A\) on the link \(p\) for the flow \([s,t]\in K\) using the same source interface \((s,j)\in A\) and destination interface \((i,t)\in A\) pair. Equation (A24) refers both subflows \([m,p]\in A\) and \([o,n]\in A\) can not be created from two different nodes using the same source interface \((s,j)\in A\) and destination interface \((i,t)\in A\) pair for the flow \([s,t]\in K\):  
\[f_{ij}^{st}\leq cap_{ij}x_{ij}^{st}\qquad\forall(i,j)\in A,[s,t]\in K. \tag{12}\]1 (A22)  
\[f_{ij}^{st}\leq cap_{ij}x_{ij}^{st}\qquad\forall(i,j)\in A,[s,t]\in K. \tag{12}\]2 (A23)  
\[f_{ij}^{st}\leq cap_{ij}x_{ij}^{st}\qquad\forall(i,j)\in A,[s,t]\in K. \tag{12}\]3 (A24)  
**Congested links identifier constraints for flows:** Equation (A25) implies that there exists at least one congested link for each flow.