# Energy Efficiency Maximization of Two-Time-Slot and Three-Time-Slot Two-Way Relay-Assisted Device-to-Device Underlaying Cellular Networks

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## Abstract

**:**

## 1. Introduction

#### 1.1. Related Work

#### 1.2. Main Contributions

- Firstly, the closed-form expressions for the STP, the TATR, and the TAEE for the cellular users and D2D users in 2TS and 3TS mode of the two-way D2D communication network are derived.
- Secondly, there is a proposed optimization problem aiming at maximizing the EE of D2D users while ensuring the QoS of the cellular and D2D users. This problem is solved thanks to a DB algorithm formulated from the fact that the multi-hop D2D users are subject to transmission power and OP constraints. This problem is proven to be non-convex and separated into two sub-problems. Solving the first sub-problem provides the maximized EE of the D2D users along with the optimum transmission power of the direct (one-hop) D2D user in 2TS mode. In the second sub-problem, an objective function is formulated to calculate the maximized EE and the optimum transmission power of the D2D users in 3TS mode with a two-way relay assisting the communication (multi-hop). It is noteworthy that the objective function is a sum of a number of sub-functions. When all of the sub-functions are maximized, by summing them up, it is possible to obtain the optimal result for the second sub-problem.
- Finally, a comparison between this study and the study in [31,32] given similar system model and assumptions is done. It can be observed from the simulation results that the DB algorithm provides a near-optimal solution in 2TS and an outstanding performance in 3TS comparing to the conventional Branch and Bound (BB) algorithm [33].

**Notations**: $\mathrm{E}\left\{.\right\}$ denotes expectation operation, $\mathrm{Pr}\left\{.\right\}$ is OP function. $\mathcal{L}\left(x\right)$ is Laplace transform of x. $\mathrm{\Gamma}\left(.\right)$ is the gamma function with $\mathrm{\Gamma}\left(z\right)={\int}_{0}^{\infty}{t}^{z-1}{e}^{-t}dt$.

## 2. System Model

**A. Scenario Description:**In this study, a D2D communication underlaying a general cellular network is considered. It should be noted that the cellular network’s uplink frequency resources are shared with the D2D communication. The resource allocation of the whole network is handled by a base station (BS). Moreover, there are network and channel model depicted. Dissimilar to [34] where the authors conducted a power allocation problem study on a single band, the power allocation on multiple bands is considered in this paper. As its name suggests, the cellular network operates in a spectrum that is split into K number of bands, whereas each K has a subscript i indicating the ${i}_{th}$ band, given that $i=1,2,\u2025,K$. Figure 1 depicts the two transmission modes focused in this study. The two are based on a cellular network with cellular links between cellular users and the BS receiver. As D2D users exchange information with one another in multiple bands scenario, they operate in two modes:

- Two-time-slot (2TS) mode: direct, one-hop communication.
- Three-time-slot (3TS) mode: indirect, via an in-between D2D user working as a two-way relay to assist the signal transmission, multi-hop communication.

**B. Network Models:**There are four assumptions that are made based on the stochastic geometry theory in this study:

**Assumption**

**A1.**

**Assumption**

**A2.**

**Assumption**

**A3.**

**Assumption**

**A4.**

**C. Channel Models:**The whole network is a combination of a cellular system and D2D communication. In this context, from [36], the path loss and Rayleigh fading are placed in a propagation channel model and their effects on the network are studied. Accordingly, a parameter namely received power for either cellular users or D2D users, ${\mathrm{P}}_{rx}$, is proposed and its expression is given as follows

**Remark**

**1.**

## 3. Problem Formulation

#### 3.1. The Signal to Noise Plus Interference Ratio

**A. Two Time Slot Communications Mode**

**B. Three Time Slot Communications Mode**

#### 3.2. The Successful Transmission Probability of Typical Receivers

**A. Two Time Slot Communications Mode**

**Proposition**

**1.**

**Proof.**

**Remark**

**2.**

**Proposition**

**2.**

**Proof.**

**Remark**

**3.**

**B. Three Time Slot Communication Mode**

**Proposition**

**3.**

**Proof.**

## 4. Performance Energy Efficiency

#### Total Average Transmissions Rate and Total Average Energy Efficiency

**A. Two Time Slot Communication**

**B. Three Time Slot Communication**

## 5. Optimization Problem

#### 5.1. Optimization Total Average Energy Efficiency in 2TS

#### 5.2. Optimization Total Average Energy Efficiency in 3TS

#### 5.3. Derivative-Based Algorithm

**A. Two Time Slot Communication**

**Theorem**

**1.**

**Proof.**

- Provided that ${P}_{d,i,up}\le {\left(2{B}_{i}^{2TS}/\phantom{2{B}_{i}^{2TS}m}\phantom{\rule{0.0pt}{0ex}}m\right)}^{\frac{m}{2}}$, then there is a monotonic increase of ${f}_{i}\left({P}_{d,i}\right)$ on the feasible region. Hence, ${f}_{i}\left({P}_{d,i}\right)$ reaches its maximum at ${P}_{d,i}={P}_{d,i,up}$ leading to ${P}_{d,i,max}={P}_{d,i,up}$.
- Provided that ${P}_{d,i,down}<{\left(2{B}_{i}^{2TS}/\phantom{2{B}_{i}^{2TS}m}\phantom{\rule{0.0pt}{0ex}}m\right)}^{\frac{m}{2}}<{P}_{d,i,up}$, then the global maximum of ${f}_{i}\left({P}_{d,i}\right)$ is located within the feasible region leading to ${P}_{d,i,max}={\left(2{B}_{i}^{2TS}/\phantom{2{B}_{i}^{2TS}m}\phantom{\rule{0.0pt}{0ex}}m\right)}^{\frac{m}{2}}$.
- Provided that ${P}_{d,i,down}\ge {\left(2{B}_{i}^{2TS}/\phantom{2{B}_{i}^{2TS}m}\phantom{\rule{0.0pt}{0ex}}m\right)}^{\frac{m}{2}}$, then there is a monotonic decrease of ${f}_{i}\left({P}_{d,i}\right)$ on the feasible region. Hence, ${f}_{i}\left({P}_{d,i}\right)$ reaches its maximum at ${P}_{d,i,max}={P}_{d,i,down}$.

Algorithm 1: The DB algorithm for 2TS |

**B. Three Time Slot Communication**

Algorithm 2: The DB algorithm for 3TS |

## 6. Numerical Results

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Primary Parameters | Description | Values |
---|---|---|

K | Number of bands | 5 |

${W}_{i}$ | Bandwidth ith | 20 MHz |

${P}_{c,i}$ | Cellular transmission power | 125 mW |

${P}_{d,i}$ | D2D user transmission power | 60 mW |

${P}_{d,i,up}$ | D2D user transmission power threshold | 20 mW |

m | Path loss coefficient | 4 |

$\alpha $ | Distances fraction between relay and D2D user | 0.5 |

$\beta $ | Power allocation at relay D2D user | 0.5 |

${\epsilon}_{c,i}$ | Cellular OP threshold | 0.05 |

${\epsilon}_{d,i}$ | D2D user OP threshold | 0.05 |

${\zeta}_{c,i}$ | Cellular SIR threshold | 0 dB |

${\zeta}_{d,i}$ | D2D user SIR threshold | 0 dB |

$\left[{R}_{c,00,1},{R}_{c,00,2},\cdots ,{R}_{c,00,K}\right]$ | Cellular link distance | $\left[50,60,70,80,90\right]$ |

$\left[{R}_{d,00,1},{R}_{d,00,2},\cdots ,{R}_{d,00,K}\right]$ | D2D link distance | $\left[10,20,30,20,10\right]$ |

$\left[{\delta}_{c,1},{\delta}_{c,2},\cdots ,{\delta}_{c,K}\right]$ | Cellular user density | $\left[10,1,10,10,10\right]\times {10}^{-5}$ |

$\left[{\delta}_{d,1},{\delta}_{d,2},\cdots ,{\delta}_{d,K}\right]$ | D2D user density | $\left[10,1,10,10,10\right]\times {10}^{-4}$ |

$\left[{\delta}_{r,1},{\delta}_{r,2},\cdots ,{\delta}_{d,K}\right]$ | Potential two-way relay D2D user density | $\left[10,1,10,10,10\right]\times {10}^{-4}$ |

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**MDPI and ACS Style**

Huynh, V.-V.; Tan-Loc, N.; Quoc-Phu, M.; Sevcik, L.; Nguyen, H.-S.; Voznak, M.
Energy Efficiency Maximization of Two-Time-Slot and Three-Time-Slot Two-Way Relay-Assisted Device-to-Device Underlaying Cellular Networks. *Energies* **2020**, *13*, 3422.
https://doi.org/10.3390/en13133422

**AMA Style**

Huynh V-V, Tan-Loc N, Quoc-Phu M, Sevcik L, Nguyen H-S, Voznak M.
Energy Efficiency Maximization of Two-Time-Slot and Three-Time-Slot Two-Way Relay-Assisted Device-to-Device Underlaying Cellular Networks. *Energies*. 2020; 13(13):3422.
https://doi.org/10.3390/en13133422

**Chicago/Turabian Style**

Huynh, Van-Van, Nguyen Tan-Loc, Ma Quoc-Phu, Lukas Sevcik, Hoang-Sy Nguyen, and Miroslav Voznak.
2020. "Energy Efficiency Maximization of Two-Time-Slot and Three-Time-Slot Two-Way Relay-Assisted Device-to-Device Underlaying Cellular Networks" *Energies* 13, no. 13: 3422.
https://doi.org/10.3390/en13133422