# The Relation between the Probability of Collision-Free Broadcast Transmission in a Wireless Network and the Stirling Number of the Second Kind

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## Abstract

**:**

## 1. Introduction

- We establish a closed-relation between the number of simultaneously contending nodes and the packet collision probability mathematically through the Stirling number of Second kind. Packet collision probability is then used to evaluate the channel quality. Our results are not only applicable to VANETs, but also to 802.11x wireless protocols in general. Currently, expensive hardware (or a side-channel, or both) is used on every vehicle to measure the channel quality. Using our results, channel quality can be estimated with the least expensive hardware available.
- In the same vein as other research papers available in the literature, we also simulated the VANET environment to obtain the results. The simulated results are compared with the mathematical results for relative-error analysis. Our simulation results show an accuracy of 99.9% compared with the analytical model. Even on a smaller sample size, our simulation results show an accuracy of 95% and above.
- We compared our analytical results with the Markov model proposed by Bianchi [4]. The Markovian model exhibits an error margin of up to 10%. For a smaller contention window size, error margin is less and it increases as the contention window size increases. Also, the marginal error follows a normal distribution curve, for the number of nodes between 1 and CNWD.

## 2. The 802.11 $\mathit{DCF}$ Overview

## 3. Estimation of Channel Quality in VANETs

- Adjusting the transmission power
- Adapting the frequency of the beacon transmission
- Adjusting the transmission data rate
- Adjusting the contention window of the backoff mechanism
- Modifying the carrier sense threshold
- Transmit frames with some associated probability (e.g., Cristhian et al. [9])

## 4. The Probability of Collision-Free Broadcast Transmission

**Lemma**

**1.**

**Proof.**

**Case 1:**$n<w$. In this case, at least $w-n$ slots are empty. Now E can be partitioned into exactly k non-empty parts and $(w-k)$ empty parts, where $1\le k\le n$

**Case 2:**$n\ge w$.

**Lemma**

**2.**

**Proof.**

**Theorem**

**1.**

## 5. Simulation and Comparison of Results

## 6. Comparison of Our Analytical Model with Biachi’s Model

- Markov model results in probability of successful transmission (($1-{P}_{C}$)), that is less than the exact probability. It exhibits an error margin of up to 10%. For a smaller contention window size like 8, error margin is less and it increases gradually as the contention window size increases to 16, 24, 32 and 64.
- It can be observed from the graphs that if n is small (typically $<CNWD$), Markovian values are closer to the actual probability. It then increases gradually and peaks around $n=2\ast CNWD$ for CNWD 8 and 16 and $1.5\ast CNDW$ for CNWD 24, 32 and 64. It then equals to the exact probability around $n=4\ast CNWD$ for all values of CNWD we simulated.
- It can be seen from the graphs (Figure 7) that marginal error follows a normal distribution curve for the number of nodes between 1 and CNWD. Whenever n is less than $1,5\ast CNWD$, there is more probability that some slots may not be taken by any nodes. If the initial slots are empty, they are counted towards $1-{P}_{s}$. Thus, the system has more marginal error for $1\le n\le 1.5\ast CNWD$. Whenever $n>1.5\ast CNWD$, the probability of having an empty slot is close to zero. Thus the Markovian values are close to the exact probability.

## 7. Conclusions and Future Direction

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

$BC$ | Backoff Counter |

$CCA$ | Cooperative Collision Avoidance |

$CNWD$ | Contention Window |

$CSMA$ | Carrier Sense Multiple Access |

$CSMA/CA$ | Carrier Sense Multiple Access with Collision Avoidance |

$DCF$ | Distributed Coordinated Function |

$DIFS$ | Distributed Inter Frame Space |

$MAC$ | media access control |

$MANET$ | mobile ad hoc network |

$PDR$ | Packet Delivery Ratio |

$VANET$ | vehicular ad hoc network |

$V2V$ | Vehicle-to-Vehicle |

$V2I$ | Vehicle-to-Infrastructure |

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**MDPI and ACS Style**

Veeraraghavan, P.; Khomami, G.; Fontan, F.P. The Relation between the Probability of Collision-Free Broadcast Transmission in a Wireless Network and the Stirling Number of the Second Kind. *Mathematics* **2018**, *6*, 127.
https://doi.org/10.3390/math6070127

**AMA Style**

Veeraraghavan P, Khomami G, Fontan FP. The Relation between the Probability of Collision-Free Broadcast Transmission in a Wireless Network and the Stirling Number of the Second Kind. *Mathematics*. 2018; 6(7):127.
https://doi.org/10.3390/math6070127

**Chicago/Turabian Style**

Veeraraghavan, Prakash, Golnar Khomami, and Fernando Perez Fontan. 2018. "The Relation between the Probability of Collision-Free Broadcast Transmission in a Wireless Network and the Stirling Number of the Second Kind" *Mathematics* 6, no. 7: 127.
https://doi.org/10.3390/math6070127