#
Diffusion Model of a Non-Integer Order PI^{γ} Controller with TCP/UDP Streams

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## Abstract

**:**

## 1. Introduction

## 2. Background and Related Work

#### 2.1. AQMs Based on the Control Theory Approach

#### 2.2. Diffusion Approximation

## 3. RED, NLRED and a Non-Integer Order ${\mathit{PI}}^{\mathit{\gamma}}$ Controllers

## 4. Diffusion Approximation of the TCP and UDP Network Streams

## 5. Numerical Results

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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PI (Simulation) [18] | Study of the TCP/AQM mechanisms based on PI controllers |

PID (Simulation) [19] | Evaluation of the AQM based on non-integer order PID controller |

$P{I}^{\gamma}$ (Fluid-Flow) [20] | First application of non-integer order $P{I}^{\gamma}$ controller to an AQM strategy |

$P{I}^{\gamma}$ (Fluid-Flow/Simulation) [21] | Fluid flow approximation and discrete-event simulation to investigate the influence of the AQM policy based on non-integer order $P{I}^{\gamma}$ controller on the packet loss probability, the queue length and its variability |

$P{I}^{\gamma}{D}^{\omega}$ (Simulation) [22] | Model of AQM mechanism based on non-integer order $P{I}^{\gamma}{D}^{\omega}$ controller |

$P{I}^{\gamma}$ (Simulation) [23] | Finding optimal parameters of the non-integer order $P{I}^{\gamma}$ controller used as AQM mechanisms. The optimization was made by using the well-known Hooke and Jeeves direct search method applied for minimization of a multivariate score function |

Adapted $P{I}^{\gamma}$ (Simulation) [24] | Choice of non-integer order $P{I}^{\gamma}$ controller parameters based on machine learning algorithms. The controller parameters automatically adjust to network traffic parameters (traffic intensity and self-similarity) |

TCP $P{I}^{\gamma}$ (Diffusion) [17] | The diffusion approximation model of the simple TCP traffic. Evaluation (in close loop scenario) of the effectiveness of active queue management (AQM) mechanisms based non-integer order $P{I}^{\gamma}$ controller |

TCP $P{I}^{\gamma}$ (Combined Diffusion and Simulation) [25] | Combined diffusion approximation and simulation model based on non-integer order $P{I}^{\gamma}$ controller |

${K}_{P}$ | Proportional term |

${K}_{I}$ | Integral term |

$\gamma $ | Integral order |

${e}_{i}$ | Error in current slot |

${q}_{i}$ | Actual queue length |

q | Desired queue length |

$\lambda $ | Intensity of the input traffic |

$\mu $ | Intensity of packet processing and dispatching |

${\sigma}_{A}^{2}$ | Variance of interarrival time distribution |

${\sigma}_{B}^{2}$ | Variance of service time distribution |

${C}_{A}^{2}$ | Squared coefficient of variation of interarrival time distribution |

${C}_{B}^{2}$ | Squared coefficient of variation of service time distribution |

$X\left(t\right)$ | Diffusion process |

$\beta $ | Diffusion parameter; $\beta dt$ is the mean value of changes of $X\left(t\right)$ during $dt$ |

$\alpha $ | Diffusion parameter; $\alpha dt$ is the variance of changes of $X\left(t\right)$ during $dt$ |

$f(x,t,{x}_{0})$ | Probability density that the process will be in state x at time t, for initial conditions ${x}_{0}$ |

${\mathit{K}}_{\mathit{p}}$ | ${\mathit{K}}_{\mathit{i}}$ | $\mathit{\gamma}$ | Setpoint | Type of Controller | |
---|---|---|---|---|---|

1 | 0.0001 | 0.005 | −0.4 | 10 | non integer order controller |

2 | 0.0001 | 0.005 | −1.0 | 10 | classical controller |

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**MDPI and ACS Style**

Marek, D.; Domański, A.; Domańska, J.; Szyguła, J.; Czachórski, T.; Klamka, J. Diffusion Model of a Non-Integer Order *PI ^{γ}* Controller with TCP/UDP Streams.

*Entropy*

**2021**,

*23*, 619. https://doi.org/10.3390/e23050619

**AMA Style**

Marek D, Domański A, Domańska J, Szyguła J, Czachórski T, Klamka J. Diffusion Model of a Non-Integer Order *PI ^{γ}* Controller with TCP/UDP Streams.

*Entropy*. 2021; 23(5):619. https://doi.org/10.3390/e23050619

**Chicago/Turabian Style**

Marek, Dariusz, Adam Domański, Joanna Domańska, Jakub Szyguła, Tadeusz Czachórski, and Jerzy Klamka. 2021. "Diffusion Model of a Non-Integer Order *PI ^{γ}* Controller with TCP/UDP Streams"

*Entropy*23, no. 5: 619. https://doi.org/10.3390/e23050619