# New RED-Type TCP-AQM Algorithms Based on Beta Distribution Drop Functions
## 4 The Beta RED Scheme
### Simulations for the BetaRED Algorithm  
In this section we present some numerical simulations with the purpose of analyzing the average queue length and the performance of the BetaRED algorithm for some values of the control parameters. However, space limitation conditions us to present only a selection of a larger set of the simulations that have been carried out. The analysis of the results obtained will help us to design two new dynamic algorithms based on BetaRED, which will be described in Section 5. In the first set of simulations (see Figure 3) we vary the parameter \(p_{\max}\) for different numbers of nodes in the dumbbell topology shown in Figure 1, and analyze the average queue length in the last 125 s (half of the total simulation time). We contemplate only the second half of the total simulation considering the first half as a transition period, and thus obtain a better estimate of the equilibrium point. Two different scenarios are setting according to the parameters \(q_{\min}\) y \(q_{\max}\). The first one is when \(q_{\min}=0\) and \(q_{\max}=1000\) (maximum buffer capacity). In the second scenario, we consider the minimum and maximum thresholds close to the target queue length \(q_{\text{target}}=250\), namely, \(q_{\min}=200\) and \(q_{\max}=400\). We note that, to achieve a higher robustness in terms of the stability of the average queue length, it is convenient to select the value \(p_{\max}=1\). This way, the average queue length stays closer to \(q_{\text{target}}\). However, the performance may decrease for increasing values of the parameter \(p_{\max}\), which means that we should look for the lowest possible \(p_{\max}\) parameter value that guarantees stability and increases performance. It is well known that the ARED algorithm [28] is based precisely on adjusting \(p_{\max}\) to obtain a predetermined target average queue length. In Section 5 we will follow the same idea of ARED adapted to BetaRED. Figure 3: _Cont_. Figure 2: The lower the standard deviation \(\sigma\), the higher the concentration of the packet drop probability mass around \(q_{\text{target}}\). In a second set of simulations (see Figure 4), we vary the \(\theta\) parameter for different numbers of nodes in the dumbbell topology presented in Figure 1. Again, to estimate the equilibrium point, we calculate the average queue length in the last 125 s, i.e., in the second half of the total simulation time. As in the first set of simulations, we also consider the same two scenarios according to the minimum and maximum thresholds \(q_{\text{min}}\) and \(q_{\text{max}}\). As expected (taking into account the interpretation of the standard deviation \(\sigma\)), better stability results are obtained as the value of \(\theta\) (and hence \(\sigma\)) decreases. In the following, the default value of \(\theta=0.1\) will be used for simulations. The discussion of the results when we vary the value of the weight parameter \(w\) is more intricate. Although we do not show concrete numerical simulations since the behavior is less predictable, it was observed that, as a general rule, increasing its value is accompanied by better performance, but stability gets worse. Nevertheless, we shall show that for the DBetaRED algorithm (described in Section 5.3) both high performance and good stability are obtained for a wide range of \(w\) values. Figure 4: _Cont._  
Figure 3: Comparison of the BetaRED algorithm according to the number \(N\) of FTP active flows and different values of the maximum probability threshold \(p_{\text{max}}\). Different thresholds \(q_{\text{min}}\) and \(q_{\text{max}}\) are also considered. Other tunable parameters: \(q_{\text{target}}=250\,\text{packets}\), \(w=0.1\), \(\theta=0.1\). Scenario 1 on the dumbbell topology (described in Section 3.1) is used. As \(p_{\text{max}}\) and/or the level of congestion increase, the equilibrium point of the average queue length gets closer to the prefixed \(q_{\text{target}}\).