# New RED-Type TCP-AQM Algorithms Based on Beta Distribution Drop Functions
## 4 The Beta RED Scheme
### Beta RED Algorithm  
In [11; 17] the authors generalize the RED scheme by replacing the linear packet drop probability function \(p_{\max}\cdot z(q_{\mathrm{avg}})\) in (1) by the nonlinear function \(p_{\max}\cdot I_{z(q_{\mathrm{avg}})}(\alpha,\beta)\), where \(0\leq z\leq 1\) and \(\alpha,\beta>0\). Since \(I_{z}(1,1)=z\), we recover the classical RED scheme for \(\alpha=\beta=1\). The purpose of this generalization was to improve the stability properties by introducing the additional control parameters \(\alpha\) and \(\beta\). Following the analysis of [PERSON] [38], in [12; 18] the authors perform a detailed theoretical study of chaos, bifurcation diagrams, Lyapunov exponents and global stability robustness for different control parameters and fixed system parameters. Bifurcation diagrams are discussed for specific values of \(\alpha\) and \(\beta\) in different scenarios, as well as biparametric sweeps of these parameters in which it was found that there are parameter regions in which the system performs very successfully in terms of stability and robustness. However, for the design of our algorithms it is much more convenient to consider the biparametric family of packet drop probability functions with respect to the parameters \(\mu\) and \(\sigma\), since they have an intuitive meaning in statistical terms, namely, the mean and the standard deviation. In this case the packet drop probability function takes the expression  
\[p(q_{\mathrm{avg}})= \begin{cases}0&\text{if }q_{\mathrm{avg}}<q_{\min},\\ 1&\text{if }q_{\mathrm{avg}}>q_{\max},\\ p_{\max}\cdot\tilde{I}_{z(q_{\mathrm{avg}})}(\mu,\sigma)&\text{otherwise}.\end{cases}  \tag{7}\]  
where \(\mu\in(0,1)\) and \(\sigma<\sqrt{\mu(1-\mu)}\). From Equation (4), the classical RED (\(\alpha=\beta=1\)) is recovered when \(\mu=\frac{1}{2}=0.5\) and \(\sigma=\frac{1}{2\sqrt{3}}\approx 0.2886\). As in the RED algorithm, the basic idea of BetaRED is to maintain the average queue length \(q_{\text{avg}}\) (calculated through the EWMA algorithm) within minimum and maximum thresholds \(q_{\text{min}}\) and \(q_{\text{max}}\), but as close as possible to a predetermined target queue length that we denote by \(q_{\text{target}}\). With this objective in mind, the next step is to provide concrete values of \(\mu\) and \(\sigma\) for which a good balance between stability and performance of the algorithm is obtained. From this point of view, our selection of the mean is given as a function of the target queue length \(q_{\text{target}}\) and will be completely determined by means of the expression:  
\[\mu=\frac{q_{\text{target}}-q_{\text{min}}}{q_{\text{max}}-q_{\text{min}}}. \tag{8}\]  
With this choice, we match the mean of the packet drop probability function to the \(q_{\text{target}}\) value. Moreover, the standard deviation \(\sigma\) is a spread measure of the values of the distribution around \(\mu\). This means that the smaller \(\sigma\) is, the more concentrated the packet drop probability mass is around its mean \(\mu\) (see Figure 2). Therefore, when the buffer starts to fill up and the average length of the buffer queue \(q_{\text{avg}}\) approaches \(q_{\text{target}}\), the probability of packet drop will increase faster the smaller \(\sigma\) is. Moreover, the smaller \(\sigma\) is, the closer to \(q_{\text{target}}\) the system will stabilize. Consequently, the value of the standard deviation is a parameter that has to be adjusted in our BetaRED scheme. However, the fact that the value of the standard deviation verifies \(0<\sigma<\sqrt{\mu(1-\mu)}=\sigma_{\text{max}}\) causes a slight inconvenience since the user has to calculate the maximum value \(\sigma_{\text{max}}\) before selecting the value of \(\sigma\). To avoid this inconvenience, the \(\sigma\) value is selected by mean of a scale factor \(\theta\) that verifies \(0<\theta<1\), being \(\sigma=\theta\cdot\sqrt{\mu(1-\mu)}\), which facilitates the task to the user.