## 2 Network Scenarios and Problems
### Packet Scheduling over 802.11 ay from TRA Layer  
MPTCP at the TRA layer has the potential to harness bandwidths and handle handoffs seamlessly; however, the MPTCP Scheduling design is a delicate balancing act. Multipath scheduler design over IEEE 802.11 ay is essentially a decision problem, namely deciding which packets to transmit down which link at which time for given beamforming and device allocation. However, deriving an optimal ultra-low delay scheduler is intractable due to the large state space (the state is combinatorial in the number of packets in flight, which is often several hundred) combined with uncertain knowledge of the link delays and channel losses. To this end, we have the following observations on the capabilities of an existing MPTCP modeling approach, protocol designs and their inherent problems. Analytical modeling of multipath data transport (see [7] and references therein) focuses on using fluid models to enhance linked scheduler designs. It is generally known that a fluid-based model averages out short-term stochastic fluctuations in parameters of interest, such as windows. Naturally, it leads to coupled partial differential equations as the system dynamics description [7]. Such a pragmatic approach dramatically simplifies the difficulty in modeling; for example, the number of bytes transferred over a period of length \(\mathcal{T}\) seconds in the core assumption of a fluid flow model, with \(w(t)\) the congestion window and \(RTT(t)\) the RTT at time \(t\),  
\[\int_{t=0}^{\mathcal{T}}w(t)/RTT(t)\mathrm{d}t. \tag{1}\]  
where the period is the time taken for \(w(t)\) to start its increments from \(w_{\mathrm{max}}/2\) packets to \(w_{\mathrm{max}}\); \(w_{\mathrm{max}}\) denotes the maximum window size in packets. Within each round, the window growth is roughly linear. It will always reach \(w_{\mathrm{max}}\) (according to this assumption), which \begin{table}
\begin{tabular}{c c c} \hline \hline
**MCS** & **EDMG SC** & **EDMG OFDM** \\ \hline Maximum TCP Throughput & 28.96 Gbps & 30.95 Gbps \\ Mobility TCP Throughput & 9.6 Gbps & 9.25 Gbps \\ Mobility MPTCP Throughput & 16.36 Gbps & 16.15 Gbps \\ Maximum MPTCP Throughput & 41.32 Gbps & 42.13 Gbps \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Throughputs of various cases with four bonded channels. is difficult to realize over 802.11_ay_ links. Therefore, we need to _relax this approximation while designing a new multipath scheduler_, and, eventually, harness PHY/MAC/TRA layers advance over ergodic characteristics of multiple 802.11_ay_ links intelligently. On the other hand, the renowned _utility maximization_ idea for the dynamical system of single-path TCP users associates a utility function \(U_{s}(w_{s})\) with each device \(s\) and understand the source and the network by executing the distributed algorithm so as to maximize the aggregate utilities perceived by the devices. The fundamental notion of this idea is to join the equilibria of the dynamical systems with the solution of an optimization problem. Therefore, the nonlinear dynamical system can be observed as one that is solving the utility maximization problem in a distributed manner. For example, (\(w_{s}^{*}\), \(p_{t}^{*}\)) is an equilibrium of the dynamical system of single-path TCP users iff \(w_{s}^{*}\) is the optimal congestion window for  
\[\text{Maximize}\sum_{s\in\mathcal{S}}U_{s}(w_{s})\ s.t.\ \lambda_{l}\leq \theta_{l}\]  
where \(p_{t}^{*}\) is the Lagrange multiplier or price (path loss) associated with the link at the optimal congestion window \(w_{s}^{*}\). For the case of MPTCP, \(\lambda_{l}\leq\theta_{l}\) means that the aggregate traffic (\(\lambda_{l}\)) over each link is less than the capacity (\(\theta_{l}\)) of the link. In fact, almost all of the currently deployed utility functions of single-path TCP algorithms presented in the literature are concave, reflecting a single stable equilibrium [7].