# A Fuzzy Method for Joint Resource Allocation and Stable Pairing in D2D Communications

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## Abstract

**:**

## 1. Introduction and Related Works

#### 1.1. Pairing in D2D Communications

#### 1.2. Fuzzy Mathematics

**Definition**

**1**

**Definition**

**2**

- ${T}_{L}(x,y)=max(x+y-1,0)$,
- ${T}_{M}(x,y)=min(x,y)$,
- ${T}_{P}(x,y)=x.y$.

**Definition**

**3**

- ${S}_{L}(x,y)=min(x+y,1)$, (Lukasiewicz t-conorm);
- ${S}_{M}(x,y)=max(x,y)$, (Maximum t-conorm);
- ${S}_{P}(x,y)=x+y-x.y$, (Probabilistic sum).

#### 1.3. Contributions

- A joint resource allocation and pairing scenario in D2D communications is investigated. A number of D2D receivers are considered for the reuse of cellular network resources. For each D2D receiver, there exists a number of potential D2D transmitters, and a stable fuzzy pairing criteria is proposed for the selection of the best transmitter.
- The set of transmitters is considered as a fuzzy set in such a way that a fuzzy degree is assigned to each node, concerning the data rates and battery levels in connection with their potential receiver. We claim that this fuzzy membership function can be considered as a measure of the stability of the connections.
- To examine the proposed method, this method is compared with three other pairing methods, so that the three parameters of stability, fairness, and sum rate in different modes such as changing the D2D search radius and increasing the number of D2D transmitters and receivers for all four methods are investigated.
- The proposed method for D2D pairing, in addition to reducing the traffic load of the cellular network, leads to more stable connections with a higher quality of service. One of the significant applications of this method is that bulky content can be transferred more reliably.

## 2. System Model

#### 2.1. Communication Models

- ${h}_{c,B}$ and ${d}_{c,B}$ are the direct channel and distance between the cellular user and evolved node base station (eNB), respectively.
- ${h}_{{t}_{ki},{r}_{j}}$ and ${d}_{{t}_{ki},{r}_{j}}$ are the channel and distance between the D2D transmitter ${t}_{ki}$ and the D2D receiver ${r}_{j}$, respectively where ${t}_{ki}\in {\mathcal{D}}_{{t}_{i}}$, ${r}_{j}\in {\mathcal{D}}_{r}$, $k\in \{1,\cdots ,K\}$ and $i,j\in \{1,\cdots ,N\}$.
- ${h}_{{t}_{ki},B}$ and ${d}_{{t}_{ki},B}$ are the interference channel and distance between the D2D transmitter ${t}_{ki}$ and eNB, respectively.
- ${h}_{c,{r}_{i}}$ and ${d}_{c,{r}_{i}}$ are the interference channel and distance between the cellular user and D2D receiver ${r}_{i}$, respectively.

#### 2.2. Construing the Fuzzy Membership Function

## 3. Simulation Results

- Fuzzy-based pairing (FP) method;
- Max-sum-rate pairing (MSP) method;
- Random pairing (RP) method;
- Constant pairing (CP) method.

## 4. Applications

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**The stability as a function of (

**a**) the number of potential D2D transmitters, (

**b**) the number of co-channel D2D receivers, and (

**c**) the D2D search radius.

**Figure 3.**Jain’s fairness is computed and compared according to (

**a**) the number of potential D2D transmitters, (

**b**) the number of co-channel D2D receivers, and (

**c**) the D2D search radius.

**Figure 4.**The sum rate as a function of (

**a**) the number of potential D2D transmitters, (

**b**) the number of co-channel D2D receivers, and (

**c**) the D2D search radius.

Notation | Description |
---|---|

${r}_{i}$ | ${i}^{th}$ D2D receiver |

${t}_{ki}$ | ${k}^{th}$ Potential D2D transmitter for ${r}_{i}$ D2D receiver |

N | The maximum number of co-channel D2D receivers |

K | The number of D2D transmitters around each receiver |

${R}_{max}$ | The maximum search radius of D2D receivers |

${h}_{c,B}$ | The direct channel between the cellular user and eNB |

${d}_{c,B}$ | The distance between the cellular user and eNB |

${h}_{{t}_{ki},{r}_{j}}$ | The channel between the D2D transmitter ${t}_{ki}$ and the D2D receiver ${r}_{j}$ |

${d}_{{t}_{ki},{r}_{j}}$ | The distance between the D2D transmitter ${t}_{ki}$ and the D2D receiver ${r}_{j}$ |

${h}_{{t}_{ki},B}$ | The interference channel between the D2D transmitter ${t}_{ki}$ and eNB |

${d}_{{t}_{ki},B}$ | The distance between the D2D transmitter ${t}_{ki}$ and eNB |

${h}_{c,{r}_{i}}$ | The interference channel between the cellular user and D2D receiver ${r}_{i}$ |

${d}_{c,{r}_{i}}$ | The distance between the cellular user and D2D receiver ${r}_{i}$ |

${\mathrm{SIN}\mathrm{R}}_{B}$ | The cellular link SINR |

${R}_{B}$ | The cellular link data rate |

${\mathrm{SIN}\mathrm{R}}_{ki}$ | The D2D link (consists of transmitter ${t}_{ki}$ and receiver ${r}_{i}$) SINR |

${R}_{ki}$ | The D2D link (consists of transmitter ${t}_{ki}$ and receiver ${r}_{i}$) data rate |

${P}^{c}$ | Cellular user’s power |

${P}^{d}$ | D2D transmitters’ power |

$\alpha $ | Path loss exponent |

${N}_{0}$ | The received noise power |

${E}_{ki}$ | The battery level of transmitter ${t}_{ki}$ |

${E}_{0}$ | The battery level threshold |

${R}_{0}$ | The date-rate threshold |

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

${R}_{c}$ (Cellular radius) | 500 m |

${R}_{max}$ (D2D search radius) | 200 m |

N (The number of D2D receivers for each cellular users) | 1–4 |

K (The number of D2D transmitters for each receiver) | 1–4 |

${P}^{c}$ (Cellular user’s power) | 23 dBm |

${P}^{d}$ (D2D transmitters’ power) | 15 dBm |

$\alpha $ (Path loss exponent) | 3.5 |

${R}_{0}$ (Data-rate threshold) | 0.3 |

${E}_{0}$ (Battery-level threshold) | 0.25 |

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**MDPI and ACS Style**

Pourmoslemi, A.; Rajabi, S.; Salimi, M.; Ferrara, M.
A Fuzzy Method for Joint Resource Allocation and Stable Pairing in D2D Communications. *Appl. Sci.* **2022**, *12*, 1343.
https://doi.org/10.3390/app12031343

**AMA Style**

Pourmoslemi A, Rajabi S, Salimi M, Ferrara M.
A Fuzzy Method for Joint Resource Allocation and Stable Pairing in D2D Communications. *Applied Sciences*. 2022; 12(3):1343.
https://doi.org/10.3390/app12031343

**Chicago/Turabian Style**

Pourmoslemi, Alireza, Siavash Rajabi, Mehdi Salimi, and Massimiliano Ferrara.
2022. "A Fuzzy Method for Joint Resource Allocation and Stable Pairing in D2D Communications" *Applied Sciences* 12, no. 3: 1343.
https://doi.org/10.3390/app12031343