# New RED-Type TCP-AQM Algorithms Based on Beta Distribution Drop Functions

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## Abstract

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## 1. Introduction

## 2. Description of the Problem and Related Works

## 3. Scenarios of Simulation and Metrics

#### 3.1. Scenarios of Simulation

- Scenario 1:
- Constant congestion level. In this case the number N of active flows is constant throughout the simulation time, which is of 250 s.
- Scenario 2:
- Changing congestion level. In this case the number N of active flows varies as a function of time according to the following distribution:
- Between 0 and 50 s, the number of active flows is $N=100$.
- Between 50 and 100 s, the number of active flows is $N=200$.
- Between 100 and 150 s, the number of active flows is $N={N}_{max}$.
- Between 150 and 200 s, the number of active flows is $N=200$.
- Between 200 and 250 s, the number of active flows is $N=100$.

#### 3.2. Performance Metrics

- Average queue length (AQL). The queue size indicates the number of packets pending transmission in the buffer queue. An unstable system is usually characterized by a synchronization in the TCP queues, accompanied by strong oscillations.Under suitable settings, the average queue length tends to stabilize at a value that we refer to as the equilibrium point. One of the main objectives of AQM algorithms is to stabilize the buffer queue size and, in this sense, we shall refer to the stability of an AQM algorithm as the ability to maintain the average queue length (or the average queuing delay) around a certain target value. The stability is an important performance characteristic of TCP/IP networks.
- The packet drop rate. It measures the ratio of the number of packets dropped by an AQM to the total number of packets in queue. In this count, packets dropped by the link or channel at the physical layer are not counted, considering only the drop rate at the network layer. The main objective of an AQM is to maintain a stable queue size with as low packet drop rate as possible. This increases the performance, since dropped packets are an early signal of congestion to the TCP, causing a decrease in its send rate.
- End-to-end throughput. This is a performance measure obtained between two interlocutors (server–host). It measures the actual transmission of the total data propagated with respect to the simulation time (from the time the data is sent to the time it is received). It is defined as the number of bits received correctly per unit of time. Specifically, the calculation of this metric is obtained from the ratio between the number of bits received by the server/host and the time elapsed between the reception of the first segment and the last one. To calculate the throughput in the dumbbell type topology, we have averaged the ratio between the number of bits received by the left and right hosts, and the sum of elapsed time between the reception of the first segment and the last one at each of these nodes.
- End-to-end delay (latency). It is one of the most significant metrics in a communication system and, in general terms, it is the time required to transmit a segment along its entire length, end-to-end. Specifically, it is calculated using the equation:$$\mathrm{Latency}=\mathrm{propagation}\phantom{\rule{4.pt}{0ex}}\mathrm{time}+\mathrm{transmission}\phantom{\rule{4.pt}{0ex}}\mathrm{time}+\mathrm{queuing}\phantom{\rule{4.pt}{0ex}}\mathrm{time}+\mathrm{processing}\phantom{\rule{4.pt}{0ex}}\mathrm{delay}$$For its calculation in the dumbbell topology, the end-to-end delay times of all the segments sent between the left and right hosts are summed, divided by the number of segments received on both hosts.
- Jitter. Jitter in a flow is defined as the variation in delay of arriving packets over time. A very high jitter can cause packet loss due to buffer overflow. In the dumbbell topology, an average value is calculated by summing the time variations between the correlative packets of all flows, and dividing by the number of variations.

## 4. The Beta RED Scheme

#### 4.1. Normalized Incomplete Beta Function

#### 4.2. Beta RED Algorithm

Algorithm 1 Pseudo-code outline for the Beta RED algorithm |

#### 4.3. Simulations for the BetaRED Algorithm

## 5. Dynamic Algorithms Based on BetaRED

#### 5.1. The Adaptative Beta RED Algorithm (ABetaRED)

Algorithm 2 Pseudo-code outline for the Adaptative Beta RED algorithm. By default, $\mathrm{Alpha}=\mathrm{Beta}=1$ and ${T}_{\mathrm{update}}=0.5\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$, were set in all simulations |

#### 5.2. The Dynamic Beta RED Algorithm (DBetaRED)

Algorithm 3 Pseudo-code outline for the Dynamic Beta RED algorithm. By default, $\mathrm{Alpha}=\mathrm{Beta}=1$ and ${T}_{\mathrm{update}}=0.5\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$, were set in all simulations |

#### 5.3. Simulations

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**The lower the standard deviation $\sigma $, the higher the concentration of the packet drop probability mass around ${q}_{\mathrm{target}}$.

**Figure 3.**Comparison of the BetaRED algorithm according to the number N of FTP active flows and different values of the maximum probability threshold ${p}_{max}$. Different thresholds ${q}_{min}$ and ${q}_{max}$ are also considered. Other tunable parameters: ${q}_{\mathrm{target}}=250\phantom{\rule{0.166667em}{0ex}}\mathrm{packets}$, $w=0.1$, $\theta =0.1$. Scenario 1 on the dumbbell topology (described in Section 3.1) is used. As ${p}_{max}$ and/or the level of congestion increase, the equilibrium point of the average queue length gets closer to the prefixed ${q}_{\mathrm{target}}$. On the other hand, if minimum and maximum thresholds ${q}_{min}$ and ${q}_{max}$ are closer to the target queue length ${q}_{\mathrm{target}}$, this also results in the average queue length equilibrium point being closer to ${q}_{\mathrm{target}}$.

**Figure 4.**Comparison of the BetaRED algorithm according to the number N of FTP active flows and different values of $\theta $. Different thresholds ${q}_{min}$ and ${q}_{max}$ are also considered. Other tunable parameters: ${q}_{\mathrm{target}}=250\phantom{\rule{0.166667em}{0ex}}\mathrm{packets}$, $w=0.1$, ${p}_{max}=1$. Scenario 1 on the dumbbell topology (described in Section 3.1) is used. As the standard deviation $\sigma $ decreases and the level of congestion increases, the equilibrium point of the average queue length gets closer to the prefixed ${q}_{\mathrm{target}}$. Similarly, if the minimum and maximum thresholds ${q}_{min}$ and ${q}_{max}$ are closer to the target queue length ${q}_{\mathrm{target}}$, then the average queue length equilibrium point is closer to ${q}_{\mathrm{target}}$.

**Figure 5.**Comparison between ABetaRED and DBetaRED algorithms according to the standard deviation $\sigma $. Different levels of congestion are also considered by means of a constant number N of active flows. Other tuning parameters: ${T}_{\mathrm{target}}=40\phantom{\rule{0.166667em}{0ex}}\mathrm{ms}$ (${q}_{\mathrm{target}}=250\phantom{\rule{0.166667em}{0ex}}\mathrm{packets}$) and $w=0.1$. Scenario 1 on the dumbbell topology (described in Section 3.1) is used. Overall, the DBetaRED algorithm exhibits better performance, especially for the end-to-end jitter metric.

**Figure 6.**Comparison between the ABetaRED and DBetaRED algorithms according to the parameter $\theta $. Different levels of congestion are also considered, but, unlike in Figure 5, the number N of active flows varies dynamically with time. Other tuning parameters: ${T}_{\mathrm{target}}=40\phantom{\rule{0.166667em}{0ex}}\mathrm{ms}$ (${q}_{\mathrm{target}}=250\phantom{\rule{0.166667em}{0ex}}\mathrm{packets}$) and $w=0.1$. Scenario 2 on the dumbbell topology (described in Section 3.1) is used. As in the non-dynamic scenario (Figure 5), the DBetaRED algorithm performs better than the ABetaRED algorithm.

**Figure 7.**Performance comparison of the Dynamic Beta RED algorithm according to a number N of active flows that vary dynamically over time and different values of the weight parameter w. Other tunable parameters: ${T}_{\mathrm{target}}=40\phantom{\rule{0.166667em}{0ex}}\mathrm{ms}$ (${q}_{\mathrm{target}}=250\phantom{\rule{0.166667em}{0ex}}\mathrm{packets}$) and $\theta =0.1$. Scenario 2 on the dumbbell topology (described in Section 3.1) is used. We observe that the trend is for the performance of the DBetaRED algorithm to increase as the value of w increases.

**Figure 8.**Performance comparison between related AQM algorithms when considering different levels of congestion by varying dynamically the number N of active flows. Other tuning parameters: ${T}_{\mathrm{target}}=40\phantom{\rule{0.166667em}{0ex}}\mathrm{ms}$ for all AQM algorithms; $\theta =0.1$ and $w=0.1$ for ABetaRED and DBetaRED. Scenario 2 on the dumbbell topology (described in Section 3.1) is used. The performance of the DBetaRED algorithm outperforms the other algorithms as the congestion level increases.

**Figure 9.**Stability comparison between related AQM algorithms when considering different levels of congestion by varying dynamically the number N of active flows with ${N}_{max}=800$. Other tuning parameters: ${T}_{\mathrm{target}}=40\phantom{\rule{0.166667em}{0ex}}\mathrm{ms}$ for all AQM algorithms; $\theta =0.1$ and $w=0.1$ for ABetaRED and DBetaRED. Scenario 2 on the dumbbell topology (described in Section 3.1) is used. The bottom right panel shows a comparison of the average queue length (for the total simulation time) of all selected AQM algorithms. The other panels illustrate the instantaneous queue length pattern, together with its moving average queue length (of the last 25 s) for each of the AQM algorithms. Among all the algorithms, DBetaRED exhibits the highest stability.

AQM Algorithm | Name | Description | |
---|---|---|---|

Tunable parameters | ABetaRED, DBetaRED, ARED, CoDel, PIE | ${T}_{\mathrm{target}}$ | Target delay |

${T}_{\mathrm{update}}$ | Update interval time | ||

Alpha, Beta | Control parameters with different objectives according to the AQM. | ||

BetaRED, ABetaRED, DBetaRED | $\theta $ | Scale factor determining the standard deviation of the drop probability function | |

w | Averaging weight | ||

BetaRED | ${q}_{\mathrm{target}}$ | Target queue length | |

${q}_{min}$ | Lower threshold | ||

${q}_{max}$ | Upper threshold | ||

${p}_{max}$ | Maximum packet drop probability | ||

System parameters | All | B | Buffer size (maximum number of packets that the buffer of Router 1 can store) |

C | Capacity of the channel (the maximum amount of error-free information that can be transmitted over the channel per unit time) | ||

N | Number of flows in the dumbbell topology | ||

M | Packet size | ||

Variables | All | p | Drop probability |

${q}_{\mathrm{cur}}$ | Current queue length at Router 1 | ||

${q}_{\mathrm{avg}}$ | Average queue length at Router 1 |

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**MDPI and ACS Style**

Giménez, A.; Murcia, M.A.; Amigó, J.M.; Martínez-Bonastre, O.; Valero, J.
New RED-Type TCP-AQM Algorithms Based on Beta Distribution Drop Functions. *Appl. Sci.* **2022**, *12*, 11176.
https://doi.org/10.3390/app122111176

**AMA Style**

Giménez A, Murcia MA, Amigó JM, Martínez-Bonastre O, Valero J.
New RED-Type TCP-AQM Algorithms Based on Beta Distribution Drop Functions. *Applied Sciences*. 2022; 12(21):11176.
https://doi.org/10.3390/app122111176

**Chicago/Turabian Style**

Giménez, Angel, Miguel A. Murcia, José M. Amigó, Oscar Martínez-Bonastre, and José Valero.
2022. "New RED-Type TCP-AQM Algorithms Based on Beta Distribution Drop Functions" *Applied Sciences* 12, no. 21: 11176.
https://doi.org/10.3390/app122111176