# Accurate Analytical Model and Evaluation of Wi-Fi Halow Based IoT Networks under a Rayleigh-Fading Channel with Capture

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## Abstract

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## 1. Introduction

- We model the channel access within a RAW slot under a Rayleigh fading channel with capture, considering the geometric distribution of stations around the AP.
- We develop a renewal theory based framework to model the contention within the RAW slot by deploying a counting process to track transmissions within the RAW slot period. We then evaluate the RAW slot throughput representing the channel usage ratio in delivering data frames during the RAW slot period.
- We derive the throughput of a RAW consisting of several RAW slots, which represents the channel usage ratio to deliver data frames during the overall RAW duration. We also evaluate the RAW performance in a no-capture channel scenario and derive the capture ratio in a given RAW configuration.
- We present a meticulous validation of the analytical findings through extensive simulations obtained via a discrete-event simulator developed with the MATLAB software. We henceforth carry out a discussion and analysis of several parameters affecting the performance of a RAW slot and a RAW consisting of several RAW slots.

## 2. Background

#### 2.1. The IEEE 802.11ah RAW Mechanism

#### 2.2. Related Work

## 3. System Model

#### 3.1. Scenario

#### 3.2. Channel Model

- ${\omega}_{T}$ is the transmitted signal power.
- $A\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{r}_{k}^{-\alpha}$ represents the deterministic path-loss law, where ${r}_{k}$ is the distance between the station k and the AP.
- $\alpha $ is the path-loss exponent, and it depends on the propagation environment. For the convenience of analysis, we assume $\alpha =4$ for the rest of this paper.
- A is a dimensionless constant in the path-loss law.

#### 3.3. Capture Aware Channel Access

#### 3.4. Channel Slot State

- Idle: When there are no ongoing transmissions in the slot. That is, all stations are listening to the medium while counting down their backoff counter. A randomly chosen slot is idle with probability ${P}_{i}$.
- Success: When only one station transmits and its packet is delivered successfully. Given that this state is a subset of busy slots, a randomly chosen slot contains a successful single transmission with probability $(1-{P}_{i}){P}_{s}$.
- Capture: When two or more stations transmit and the packet of one station is captured successfully from the collision. This type of slot is a subset of slots containing collisions, which are a subset of busy slots. Thus, a randomly chosen slot contains a collision with a capture with probability $(1-{P}_{i})(1-{P}_{s}){P}_{cap}$.
- Failure: When two or more stations transmit and no packet is captured. Given that this state is a subset of slots containing collisions, the probability that a randomly chosen slot contains a collision and no packet is captured is given by $(1-{P}_{i})(1-{P}_{s})(1-{P}_{cap})$.

## 4. Analytical Model and Performance Evaluation

#### 4.1. Stochastic Model

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

#### 4.2. Evaluating a Single RAW Slot

#### 4.3. Evaluating Several RAW Slots

## 5. Results and Discussion

#### 5.1. RAW Slot

#### 5.1.1. RAW Slot Period

#### 5.1.2. Contending Stations

#### 5.1.3. Capture Threshold

#### 5.2. RAW Performance

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 6.**Average number of busy and idle slots, and the holding period usage ratio within the RAW slot interval, with channel capture threshold $z=8\phantom{\rule{0.166667em}{0ex}}\mathrm{dB}$.

**Figure 7.**Average number of slots with successful single transmission, captured packet, and failure, within the RAW slot interval, with channel capture threshold $z=8\phantom{\rule{0.166667em}{0ex}}\mathrm{dB}$.

**Figure 9.**Average number of slots with successful single transmission, captured packet, and failure, within a RAW slot interval of length ${T}_{S}=20\phantom{\rule{0.166667em}{0ex}}\mathrm{m}\mathrm{s}$ and channel capture threshold $z=8\phantom{\rule{0.166667em}{0ex}}\mathrm{dB}$.

**Figure 10.**Channel usage ratio by different events occupying the RAW slot interval with capture threshold $z=8\phantom{\rule{0.166667em}{0ex}}\mathrm{dB}$, in terms of contending stations.

**Figure 12.**Average number of slots with with successful single transmission, captured packet, and failure, in terms of the capture threshold, within a RAW slot of duration ${T}_{S}=20\phantom{\rule{0.166667em}{0ex}}\mathrm{m}\mathrm{s}$ and $N=10$ assigned stations.

**Figure 15.**Capture gain in terms of allocated RAW slots for different sets of assigned stations to the RAW.

Parameter | Value |
---|---|

Data rate | 1.95 Mbps |

MacHeader | 272 bits |

${T}_{\mathrm{ACK}}$ | 1000 μs |

${T}_{\mathrm{PLCP}}$ | 80 μs |

$\sigma $ | 52 μs |

SIFS | 160 μs |

DIFS | 264 μs |

$C{W}_{min}$ | 8 |

$C{W}_{max}$ | 16 |

m | 1 |

Payload | 160 bytes |

$\rho $ | 100 $\mathrm{m}$ |

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**MDPI and ACS Style**

Taramit, H.; Camacho-Escoto, J.J.; Gomez, J.; Orozco-Barbosa, L.; Haqiq, A.
Accurate Analytical Model and Evaluation of Wi-Fi Halow Based IoT Networks under a Rayleigh-Fading Channel with Capture. *Mathematics* **2022**, *10*, 952.
https://doi.org/10.3390/math10060952

**AMA Style**

Taramit H, Camacho-Escoto JJ, Gomez J, Orozco-Barbosa L, Haqiq A.
Accurate Analytical Model and Evaluation of Wi-Fi Halow Based IoT Networks under a Rayleigh-Fading Channel with Capture. *Mathematics*. 2022; 10(6):952.
https://doi.org/10.3390/math10060952

**Chicago/Turabian Style**

Taramit, Hamid, José Jaime Camacho-Escoto, Javier Gomez, Luis Orozco-Barbosa, and Abdelkrim Haqiq.
2022. "Accurate Analytical Model and Evaluation of Wi-Fi Halow Based IoT Networks under a Rayleigh-Fading Channel with Capture" *Mathematics* 10, no. 6: 952.
https://doi.org/10.3390/math10060952