# A Mathematical Model for Efficient and Fair Resource Assignment in Multipath Transport
## 1 Introduction
The analytical description of multipath transport requires mathematical formulation which avoids flow splitting \"on the way\"; it has to be restricted to the end nodes by respective constraints. The topic should not be confused with multipath _routing_ where data is sent from a source node to a destination node as a single flow from the end-to-end view, whereas the flow splitting occurs in routers along the way between sender and receiver. Multipath transport is run between end-user devices as previously said. In this way, both methods complement each other. Furthermore, it was previously mentioned that multipath transport fairness can be performed not only inside the resource of an entire network, but also in an individual bottleneck link, whereas multipath routing always has the network as the scope. Existing research about analytical modeling of multipath transport focuses on analytical modeling of the behaviour and stability of practical MPTCP congestion control [7; 8; 9; 10; 11]. Any practical transport-layer algorithm is however limited by the fact that it does not have knowledge about the internal states of the network; it takes estimation by observing the performance of the links. Furthermore, practical algorithms respond to changing link conditions, which results in a time-dependent behaviour. In contrast to this, the aim of our paper is looking at a network in \"god view\" in order to find the optimum resource assignment assuming perfect knowledge of the network topology in a static equilibrium. The considerations in our paper are independent from a particular algorithm; however, they consider fairness which is also inherent to the different practical algorithms as discussed in Sections 3.2.1-3.2.3. Another group of publications discusses the aforementioned fairness in multipath routing where the survey [12] gives an overview and some further examples including [13; 14; 15; 16]. Our paper differs, as already mentioned, by considering fairness between end-to-end connections instead of routes in a network. Further publications discuss legacy (single-path) TCP where various methods are summarized in three surveys [17; 18; 19]. In addition, for UDP transport, an application-layer congestion control scheme has been proposed to support fairness [20]. Finally, authors have investigated different practical implementations of MPTCP; these references are summarized in Sections 3.2.1-3.2.3 where they are given as examples of how different fairness methods were realized. The contribution of this paper is (1) presenting formal definitions of a network and the different transport-layer fairness methods based on selecting resources and participants; (2) based on these definitions, designing a novel mathematical method based on mixed-integer (non)linear programming(MINLP) for the optimum assignment of data flows to physical paths across the network; and (3) evaluating the developed mathematical methods using 30 example scenarios. The proposed method should consider a trade-off between high link utilization and fair share as discussed in the previous paragraphs. Since fairness is an important aspect of the resource assignment problem, well-known fairness methods from the literature are used to measure the fairness in a quantitative way. The methods can serve as a benchmark to evaluate practical resource assignment methods for multipath transport protocols. The remaining parts of this paper are organized as follows: Section 2 gives the definition of resources and participants in a network and specifies the different fairness methods. Based on these definitions, the mathematical model is developed in Section 3. Metrics for the performance evaluation are discussed in Section 4, whereas, for the evaluation itself, a number of example scenarios are investigated in Section 5. Finally, Section 6 gives a conclusion and an outlook.