# Intelligent Reflecting Surfaces Enhanced Mobile Edge Computing: Minimizing the Maximum Computational Time

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## Abstract

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## 1. Introduction

#### 1.1. Related Work

#### 1.2. Motivation and Contribution

- We proposed a task’s computational time minimization problem with joint optimization of computational and communication resources, such as edge computational resource, phase shift control, transmission power, and task segmentation, under energy and system constraints;
- To address the non-linearity and non-convexity that characterize the original optimization problem, we decoupled the original optimization problem into sub-problems and iteratively solved them. Furthermore, we also drive a closed-form solution for task segmentation and computational resource allocation at MEC;
- Numerical results are compared to the exhaustive search and other benchmark schemes to demonstrate the efficacy of the proposed scheme. By taking into account, task computational time and energy consumption as performance matrices, numerical results show that the proposed scheme performs epsilon equally to the exhaustive search and outperforms all other schemes.

## 2. System Model

#### 2.1. Communication Model

#### 2.2. Task Computational Model

#### 2.2.1. Local Computation

#### 2.2.2. Edge Computing

#### 2.3. Problem Formulation

## 3. Proposed Solution

#### 3.1. Task Segmentation

#### 3.2. Edge Computational Resource Allocation

**Lemma**

**1.**

**Proof.**

**H**were zeros except for the digital elements, i.e.,

**H**have a positive value. Therefore, in accordance with the theorem [44], the objective function mentioned in (11) is convex in nature. As a results, the optimization problem mentioned in (10) is a convex optimization problem. □

**Lemma 2.**

**Proof.**

**Lemma 1**, the Lagrange function of optimization problem (10) can be mathematically represented as:

#### 3.3. Transmission Power and Phase Shift Control

Algorithm 1: Framework for Optimal Resource Allocation. |

#### 3.4. Worst Case Per Iteration Complexity Analysis

## 4. Result and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Name | Symbol | Value |
---|---|---|

Transmission Bandwidth | B | 20 MHz |

Maximum Transmission Power | ${P}^{max}$ | $0.1$ W |

Noise Power | ${N}_{o}$ | $-173$ dBm |

Static Circuit Power | $\mathsf{\Gamma}$ | 90 W/Gcycle |

Maximum battery Capacity | ${E}^{max}$ | ${10}^{3}$ J |

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**MDPI and ACS Style**

Sarfraz, M.; Alshahrani, H.M.; Tarmissi, K.; Alshahrani, H.; Elfaki, M.A.; Hamza, M.A.; Nauman, A.; Khurshaid, T.
Intelligent Reflecting Surfaces Enhanced Mobile Edge Computing: Minimizing the Maximum Computational Time. *Sensors* **2022**, *22*, 8719.
https://doi.org/10.3390/s22228719

**AMA Style**

Sarfraz M, Alshahrani HM, Tarmissi K, Alshahrani H, Elfaki MA, Hamza MA, Nauman A, Khurshaid T.
Intelligent Reflecting Surfaces Enhanced Mobile Edge Computing: Minimizing the Maximum Computational Time. *Sensors*. 2022; 22(22):8719.
https://doi.org/10.3390/s22228719

**Chicago/Turabian Style**

Sarfraz, Mubashar, Haya Mesfer Alshahrani, Khaled Tarmissi, Hussain Alshahrani, Mohamed Ahmed Elfaki, Manar Ahmed Hamza, Ali Nauman, and Tahir Khurshaid.
2022. "Intelligent Reflecting Surfaces Enhanced Mobile Edge Computing: Minimizing the Maximum Computational Time" *Sensors* 22, no. 22: 8719.
https://doi.org/10.3390/s22228719