# New RED-Type TCP-AQM Algorithms Based on Beta Distribution Drop Functions
## 2 Description of the Problem and Related Works
The large amount of research on the design and study of AQM algorithms has resulted in significant improvement in performance metrics: latency, link utilization, throughput, jitter, etc., reducing burst losses and solving the global synchronization problem. In the RED scheme, the probability function \(p\) of packet dropping at an instant is dependent on the average size of the queue length \(q_{\text{avg}}\) and its expression is given by  
\[p(q_{\text{avg}})= \begin{cases}0&\text{if }q_{\text{avg}}<q_{\text{min}},\\ 1&\text{if }q_{\text{avg}}>q_{\text{max}},\\ p_{\text{max}}\cdot z(q_{\text{avg}})&\text{otherwise}.\end{cases} \quad\text{ where}\quad z(q_{\text{avg}})=\frac{q_{\text{avg}}-q_{\text{min}}}{q_{\text{max}}-q_{ \text{min}}} \tag{1}\]  
The values \(q_{\text{min}}\) and \(q_{\text{max}}\) are the lower and upper thresholds for the average queue length \(q_{\text{avg}}\) where below \(q_{\text{min}}\) all packets are accepted, and above \(q_{\text{max}}\) all packets are rejected. The value \(p_{\text{max}}\) is the maximum value for the packet drop probability function when the average queue length is between \(q_{\text{min}}\) and \(q_{\text{max}}\), which is reached at the point \(q_{\text{avg}}=q_{\text{max}}\). The average queue length is updated upon packet arrival according to the Exponential Weighted Moving Average (EWMA),  
\[q_{\text{avg}}^{\text{new}}=(1-w)\cdot q_{\text{avg}}^{\text{old}}+w\cdot q_{ \text{cur}} \tag{2}\]  
between the previous average queue length \(q_{\text{avg}}^{\text{old}}\) and the current queue length \(q_{\text{cur}}\), where \(0<w<1\) is the averaging weight. The higher \(w\), the faster the RED mechanism reacts to the actual buffer occupancy. Although RED was able to eliminate some shortcomings of Drop Tail such as blocking, full queues and global synchronization, other shortcomings remained. One of the main problems of RED is that its parameters must be adjusted according to different Internet traffic load states. Experiments and numerical simulations have been the main tool used to adjust the parameters of many of the algorithms studied. In the literature we find many conclusions based solely on simulation results, but they lack a theoretical foundation that guarantees the results. The combination of end-to-end TCP congestion control and active RED queue management can be modeled as a discrete-time dynamic system and this system exhibits a variety of irregular behaviors, such as bifurcation and chaos. Recently, in [11; 17], a generalized RED-based model was proposed in which two new control parameters are introduced by means of a nonlinear packet dropping probability function, namely, the normalized incomplete beta function. In addition, in [12; 18] a theoretical analysis of the global stability was performed, and the results were used to find robust ranges of the new control parameters. This theoretical study has helped to design the new AQM algorithms proposed here for queue management, since they showed the existence of a wide range of parameters for which a higher stability than RED is achieved. One of the advantages of the algorithms proposed here is that the parameters are easily and intuitively adjusted with guarantees of achieving a good balance between stability and performance, and supported by the theoretical analysis made in [12; 18]. The idea of using a nonlinear packet drop probability function also appears in several papers. In [19], the authors designed an algorithm called Three-section Random Early Detection (TRED), where the probability function is divided into three sections in order to achieve a trade-off between delay and throughput by distinguishing between light,moderate and high loads. Following a similar idea, an AQM scheme called Flexible Random Early Detection (FXRED) is proposed in [20] where different variations of the packet drop probability function are applied according to the state of the network traffic load.