# A Modified TCP BBR to Enable High Fairness in High-Speed Wireless Networks
## 3 Proposed Work
### BBR Fluid Model  
By establishing a fluid model, we simplify the operational mechanism of BBR. We assume that n flows with different RTTs passing through a bottleneck link with a bandwidth of C. Let \(flow_{i}\) (\(i\in[1;n]\)) represent \(flow_{i}\) and \(d_{i}\)(t) represent the delivery rate at time \(t\). In the ideal state, \(d_{1}+d_{2}+\ldots+d_{n}=\) C. Let \(T_{i}\)(t) represents the RTT of \(flow_{i}\) at time t, which is calculated as follows:  
\[T_{i}(t)=T_{D_{i}}(t)+T_{p_{i}}=\frac{q_{i}(t)}{\text{C}}+T_{p_{i}} \tag{1}\]  
Figure 1: The state machine and network path model. where \(T_{D}\) denotes queuing delay, \(q_{i}\) represents queue length, and \(T_{p}\) represents RTprop. Let \(I_{i}\) represent the inflight data of _flow_, namely, the upper bound of the bottleneck link, which is calculated as follows:  
\[I_{i}(t)=d_{i}(t)\times T_{i}(t)=d_{i}(t)\times(\frac{q_{i}(t)}{C}+T_{p_{i}}) \tag{2}\]  
The estimated BtlBw of the _flow\({}_{i}\)_ at time t can be obtained with  
\[BtlBw_{i}(\text{t})=max(d_{i}(\text{t}))(T\in[t-10 RTT;\ t]) \tag{3}\]  
Combined with Equation (2), _flow\({}_{i}\)_ can obtain a 1.25-fold gain in the first probe-up cycle, so the maximum delivery rate at time t is shown as  
\[\max(d_{i}(t))=\frac{\max(I_{i}(t))}{T_{i}}=\frac{1.25\times T_{p_{i}}\times Btl Bw_{i}(t-\Delta t)}{T_{i}} \tag{4}\]  
The detection cycle of BBR flow is 8 RTprop, so the estimated BtlBw of _flow\({}_{i}\)_ in the new round is updated as  
\[BtlBw_{i}(t)=\frac{1.25\times T_{p_{i}}\times BtlBw_{i}(t-8T_{p_{i}})}{T_{i}} \tag{5}\]  
Assuming that the total bandwidth is 1, the relationship between pacing gain and bandwidth is shown in Figure 2, where A and B represent two different flows, respectively. The total bandwidth and the initial bandwidths of the A flow and the B flow are assumed to be 1, \(a\), and \(1-a\), respectively. As show in Figure 2, if A and B are, respectively, probe-up once, the bandwidth increment expressions of A and B can be obtained as follows:  
\[\Delta A=\frac{1-a}{0.25a+1}-\text{a}=\frac{0.25a\ (1-a)}{0.25a+1} \tag{6}\]  
\[\Delta B=\frac{1.25(1-a)}{1.25-0.25a}-(1-a)\ =\frac{0.25a(1-a)}{1.25-0.25a} \tag{7}\]  
It can be seen from Equations (6) and (7) that the two flows intersect at \(a=0.5\) and are symmetric at about \(a=0.5\). The RTT of different flows on the same bottleneck link will not be the same, and with its initial bandwidth, it is difficult to achieve fair sharing. Figure 2: The relationship between pacing gain and bandwidth. The influence of RTT difference on bandwidth occupancy was further deduced. The RTTs of _flow\({}_{1}\)_ and _flow\({}_{2}\)_ are set to \(T_{1}\) and \(T_{2}\), respectively, and \(T_{2}\) = \(\lambda T_{1}\) (\(\lambda\geq 1\)). The total inflight data in the time interval [_t_, \(t\) + \(\lambda\)] of the two flows can be calculated by Equations (8) and (9), where \(\alpha=\lfloor(t-1)\times\lambda\rfloor\). \[I_{1}(t)=[d_{1}(\alpha)\times\lambda t+d_{1}(\alpha-1)\times(1-\lambda t)] \times T_{1} \tag{8}\]  
Figure 1:0  
The bandwidth occupation _oBw_ of different flows can be determined by \(I_{i}\):  
Figure 1:1  
In substituting Equations (8) and (9) into Equation (10),  
Figure 1:2  
From Equation (11), the bandwidth occupation of _flow\({}_{2}\)_ is related to ratio \(\lambda\). When \(d_{i}\) increases, _oBw\({}_{2}\)_ also increases, which means that _flow\({}_{2}\)_ preempts the bandwidth of _flow\({}_{1}\)_. As a result, the BBR's throughput is affected by the RTT ratio. From the above derivation, it can be seen that the long RTT flow has a larger pacing gain in the ProbeBW phase, resulting in a higher sending rate. If BBR misjudgment or other flows result in a queuing delay where \(T_{D}\) is greater than 0.25\(T_{p}\) but less than \(T_{p}\), the pacing gain will become negative, and the sending rate will decrease instead of increasing. This is also the reason for bandwidth unfairness when BBR and other CCAs are deployed together.