# Multi-Objective Caching Optimization for Wireless Backhauled Fog Radio Access Network

^{*}

## Abstract

**:**

## 1. Introduction

- Closed-form expressions of the probabilities of a F-AP being a direct F-AP or a transit F-AP with respect to the requested content are derived, then the expressions are used to calculate the association probabilities.
- Using stochastic geometry tools, we derive expressions of the STP and average delay in the general signal-to-interference-plus-noise ratio (SINR) regime. To reduce the complexity, closed-form expressions of the asymptotic multi-objective STP and delay in the high signal-to-noise ratio (SNR) regime are derived.
- The multi-objective optimization problem is formulated to maximize the STP or minimize the average delay. Then, the asymptotic multi-objective optimization problem in the high SNR is considered to reduce the computational complexity.
- A novel projected multi-objective cuckoo search algorithm (PMOCSA) is proposed to compute the Pareto front of the optimal cache placement.
- The numerical results show that the developed PMOCSA outperforms the original multi-objective cuckoo search algorithm (MOCSA). Also, the proposed multi-objective caching scheme is shown to achieve higher performance than the benchmark caching schemes.

## 2. System Model

#### 2.1. Network Model

#### 2.2. Caching Model

#### 2.3. Association Model

- If there are F-APs caching content m within R, ${u}_{0}$ will be associated with the nearest F-AP caching m${F}_{m,0}$ to serve ${u}_{0}$ directly, e.g., user A in Figure 1. Therefore, ${F}_{m,0}$ is dubbed ’direct F-AP’. Denote $Pr[{X}_{m}={F}_{m,0}]$ as the probability of ${u}_{0}$ is being associated with a direct F-AP when it requests content m. As all the F-APs caching content m can serve as a direct F-AP with respect to content m, the point process of the direct F-APs is the thinned PPP ${\mathsf{\Phi}}_{m}\subseteq {\mathsf{\Phi}}_{F}$ with density ${p}_{m}{\lambda}_{F}$, i.e., ${\mathsf{\Phi}}_{m}$ is the point process of the F-APs caching content m. Then, using the null property of PPP, $Pr[{X}_{m}={F}_{m,0}]$ can be obtained as in the following lemma
**Lemma****1.**When ${u}_{0}$ requests content m, the probability of ${u}_{0}$ being associated to a direct F-AP within R is given by$$Pr[{X}_{m}={F}_{m,0}]=1-exp\left(-\pi {p}_{m}{\lambda}_{F}{R}^{2}\right)$$**Proof.**Please refer to Appendix A. □ - If content m in not cached within R, then ${u}_{0}$ will be associated with the nearest available F-AP ${F}_{a,0}$ within R to fetch m from the nearest C-AP ${C}_{0}$, e.g., user B in Figure 1. Here, ${F}_{a,0}$ is dubbed ’transit F-AP’ due to the 2-hop transmission. Moreover, the available F-AP is defined as the F-AP that caches inactive content (i.e., a content not requested by users within its association area). Let the random variable ${Y}_{\mu}\in \{0,1\}$ denote whether content $\mu $ cached by ${X}_{\mu}$ is being requested by users within its Voronoi cell, such that ${Y}_{\mu}=0$ stands for the event of content $\mu $ being not requested, and ${Y}_{\mu}=1$ otherwise. Then, the probability of the content $\mu $ being inactive can be obtained using proposition 1 of [20] as$${b}_{\mu}=Pr\left[{Y}_{\mu}=0\right]={\left(1+\frac{{a}_{\mu}\phantom{\rule{0.166667em}{0ex}}{\lambda}_{U}}{3.5\phantom{\rule{0.166667em}{0ex}}{p}_{\mu}\phantom{\rule{0.166667em}{0ex}}{\lambda}_{F}}\right)}^{-3.5}$$The probability that a F-AP can be an available F-AP ${\mathrm{\Lambda}}_{m}$ when content m is requested can be obtained as in the following lemma
**Lemma****2.**The probability of available F-APs with respect to content m is$${\mathrm{\Lambda}}_{m}=\sum _{\mu \in \mathcal{M}\backslash m}{p}_{\mu}{b}_{\mu}$$**Proof.**Please refer to Appendix B. □To proceed, we denote ${\mathsf{\Phi}}_{a,m}\subseteq {\mathsf{\Phi}}_{F}$ as the point process of available F-APs with respect to content m. As ${\mathsf{\Phi}}_{a,m}$ is a thinned PPP, its density can be obtained as ${\mathrm{\Lambda}}_{m}{\lambda}_{F}$. Then, the probability of the event that ${u}_{0}$ is associated with a transit F-AP within R when it requests content m can be obtained as in Lemma 3**Lemma****3.**When ${u}_{0}$ requests content m, the probability of ${u}_{0}$ is being associated with a transit F-AP within R is given by$$Pr\left[{X}_{m}={F}_{a,0}\right]=exp\left(-\pi {p}_{m}{\lambda}_{F}{R}^{2}\right)\left(1-exp\left(-\pi {\mathrm{\Lambda}}_{m}{\lambda}_{F}{R}^{2}\right)\right)$$**Proof.**Please refer to Appendix C. □ - If neither a direct nor a transit F-AP exists within R to be associated with, then ${u}_{0}$ will be associated with the nearest C-AP ${C}_{m,0}$ within R, e.g., user C in Figure 1. The probability of this event $Pr\left[{X}_{m}={C}_{m,0}\right]$ is given as in the following lemma
**Lemma****4.**When ${u}_{0}$ requests content m, the probability of ${u}_{0}$ is being associated with the nearest C-AP within R is given by$$Pr\left[{X}_{m}={C}_{m,0}\right]=exp\left(-\pi ({p}_{m}+{\mathrm{\Lambda}}_{m}){\lambda}_{F}{R}^{2}\right)\left(1-exp\left(-\pi {\lambda}_{C}{R}^{2}\right)\right)$$**Proof.**Please refer to Appendix D. □

## 3. Performance Analysis

#### 3.1. STP Analysis

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Corollary**

**1**

**.**When $\frac{P}{{N}_{0}}\to \infty $, we have

**Proof.**

#### 3.2. Delay Analysis

**Theorem**

**5.**

**Proof.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Theorem**

**8.**

**Proof.**

**Corollary**

**2**

**.**When $\frac{P}{{N}_{0}}\to \infty $, the average delay of ${u}_{0}$ can be expressed as

**Proof.**

## 4. Problem Formulation and Optimization

**Problem**

**1.**

**Problem**

**2.**

**Problem**

**3.**

Algorithm 1: Projected Multi-Objective Cuckoo Search Algorithm (PMOCSA). |

## 5. Numerical Results and Discussions

#### 5.1. Numerical Results

#### 5.2. Discussions

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Proof of Lemma 1

## Appendix B. Proof of Lemma 2

## Appendix C. Proof of Lemma 3

## Appendix D. Proof of Lemma 4

## Appendix E. Proof of Theorem 1

## Appendix F. Proof of Theorem 2

## Appendix G. Proof of Theorem 3

## References

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**Figure 2.**Euclidean distance versus iteration index at $M=10$, ${\lambda}_{U}=0.05$ users/m${}^{2}$, ${\lambda}_{F}=0.01$ fog access points (F-APs)/m${}^{2}$, ${\lambda}_{C}=0.001$ cloud access points (C-APs)/m${}^{2}$, $R=50$ m, $\gamma =0.8$, $\alpha =4$, ${W}_{F}=10$ MHz, ${W}_{C}=100$ MHz, $T=1$ ms, and $\tau =0.01$ Mbps.

**Figure 3.**Successful transmission probability (STP) and delay values of all visited points during the optimization at $M=10$, ${\lambda}_{U}=0.05$ users/m${}^{2}$, ${\lambda}_{F}=0.01$ F-APs/m${}^{2}$, ${\lambda}_{C}=0.001$ C-APs/m${}^{2}$, $R=50$ m, $\gamma =0.8$, $\alpha =4$, ${W}_{F}=10$ MHz, ${W}_{C}=100$ MHz, $T=1$ ms, and $\tau =0.01$ Mbps.

**Figure 4.**STP and delay versus discovery range at $M=20$, ${\lambda}_{U}=0.1$ users/m${}^{2}$, ${\lambda}_{F}=0.01$ F-APs/m${}^{2}$, ${\lambda}_{C}=0.001$ C-APs/m${}^{2}$, $\gamma =0.8$, $\alpha =4$, ${W}_{F}=10$ MHz, ${W}_{C}=50$ MHz, $T=1$ ms, and $\tau =0.01$ Mbps.

**Figure 5.**STP and delay versus Zipf exponent at $M=20$, ${\lambda}_{U}=0.1$ users/m${}^{2}$, ${\lambda}_{F}=0.01$ F-APs/m${}^{2}$, ${\lambda}_{C}=0.001$ C-APs/m${}^{2}$, $R=150$ m, $\alpha =4$, ${W}_{F}=10$ MHz, ${W}_{C}=50$ MHz, $T=1$ ms, and $\tau =0.01$ Mbps.

**Figure 6.**STP and delay versus F-APs’ bandwidth at $M=20$, ${\lambda}_{U}=0.1$ users/m${}^{2}$, ${\lambda}_{F}=0.01$ F-APs/m${}^{2}$, ${\lambda}_{C}=0.001$ C-APs/m${}^{2}$, $R=150$ m, $\gamma =0.8$, $\alpha =4$, ${W}_{C}=50$ MHz, $T=1$ ms, and $\tau =0.01$ Mbps.

**Figure 7.**STP and delay versus C-APs’ bandwidth at $M=20$, ${\lambda}_{U}=0.1$ users/m${}^{2}$, ${\lambda}_{F}=0.01$ F-APs/m${}^{2}$, ${\lambda}_{C}=0.001$ C-APs/m${}^{2}$, $R=150$ m, $\gamma =0.8$, $\alpha =4$, ${W}_{F}=10$ MHz, $T=1$ ms, and $\tau =0.01$ Mbps.

**Figure 8.**STP and delay versus total number of contents at ${\lambda}_{U}=0.1$ users/m${}^{2}$, ${\lambda}_{F}=0.01$ F-APs/m${}^{2}$, ${\lambda}_{C}=0.001$ C-APs/m${}^{2}$, $R=150$ m, $\gamma =0.8$, $\alpha =4$, ${W}_{F}=100$ MHz, ${W}_{C}=500$ MHz, $T=1$ ms, and $\tau =0.01$ Mbps.

**Figure 9.**STP and delay versus user density at $M=20$, ${\lambda}_{F}=0.01$ F-APs/m${}^{2}$, ${\lambda}_{C}=0.001$ C-APs/m${}^{2}$, $R=150$ m, $\gamma =0.8$, $\alpha =4$, ${W}_{F}=10$ MHz, ${W}_{C}=50$ MHz, $T=1$ ms, and $\tau =0.01$ Mbps.

**Figure 10.**STP and delay versus F-AP density at $M=20$, ${\lambda}_{U}=0.1$ users/m${}^{2}$, ${\lambda}_{C}=0.001$ C-APs/m${}^{2}$, $R=150$ m, $\gamma =0.8$, $\alpha =4$, ${W}_{F}=10$ MHz, ${W}_{C}=50$ MHz, $T=1$ ms, and $\tau =0.01$ Mbps.

**Figure 11.**STP and delay versus C-AP density at $M=20$, ${\lambda}_{U}=0.1$ users/m${}^{2}$, ${\lambda}_{F}=0.01$ F-APs/m${}^{2}$, $R=150$ m, $\gamma =0.8$, $\alpha =4$, ${W}_{F}=10$ MHz, ${W}_{C}=50$ MHz, $T=1$ ms, and $\tau =0.01$ Mbps.

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

$\mathcal{M}$ | Content library |

M | Total number of contents |

${\mathsf{\Phi}}_{F}$ | Point process of the F-APs |

${\mathsf{\Phi}}_{C}$ | Point process of the C-APs |

${\mathsf{\Phi}}_{U}$ | Point process of the users |

${\mathsf{\Phi}}_{m}$ | Point process of the F-APs that cache content m |

${\mathsf{\Phi}}_{-m}$ | Point process of the F-APs that do not cache content m |

${\mathsf{\Phi}}_{a,m}$ | Point process of the available F-APs with respect to content m |

${\mathsf{\Phi}}_{-a,m}$ | Point process of the unavailable F-APs with respect to content m |

${\lambda}_{F}$ | Density of ${\mathsf{\Phi}}_{F}$ |

${\lambda}_{C}$ | Density of ${\mathsf{\Phi}}_{C}$ |

${\lambda}_{U}$ | Density of ${\mathsf{\Phi}}_{U}$ |

${a}_{m}$ | Probability of content m being randomly requested by a user |

$\mathit{p}$ | Caching distribution of the contents |

${p}_{m}$ | Probability of content m being cached at each F-AP |

${b}_{\mu}$ | Probability of the content $\mu $ being inactive |

${\mathrm{\Lambda}}_{m}$ | Probability of available F-APs with respect to content m |

${F}_{m,0}$ | Direct F-AP with respect to content m |

${F}_{a,0}$ | Transit F-AP with respect to content m |

${C}_{m,0}$ | Nearest C-AP to ${u}_{0}$ |

${C}_{0}$ | Nearest C-AP to ${F}_{a,0}$ |

$Pr[{X}_{m}={F}_{m,0}]$ | Probability of association with ${F}_{m,0}$ when content m is requested |

$Pr\left[{X}_{m}={F}_{a,0}\right]$ | Probability of association with ${F}_{a,0}$ when content m is requested |

$Pr\left[{X}_{m}={C}_{m,0}\right]$ | Probability of association with ${C}_{m,0}$ when content m is requested |

${\mathcal{A}}_{m}$ | Total probability of association with an access point when content m is requested |

$SIN{R}_{m,0}$ | SINR at ${u}_{0}$ when it is associated with ${F}_{m,0}$ |

$SIN{R}_{a,0}$ | SINR at ${u}_{0}$ when it is associated with ${F}_{a,0}$ |

$SIN{R}_{C,a}$ | SINR at ${F}_{a,0}$ when ${u}_{0}$ is associated with ${F}_{a,0}$ |

$SIN{R}_{C,0}$ | SINR at ${u}_{0}$ when it is associated with ${C}_{m,0}$ |

${D}_{0,0}$ | Distance between ${F}_{m,0}$ and ${u}_{0}$ |

${D}_{\ell ,0}$ | Distance between access point ℓ and ${u}_{0}$ |

${D}_{a,0}$ | Distance between ${F}_{a,0}$ and ${u}_{0}$ |

${D}_{C,a}$ | Distance between ${C}_{0}$ and ${F}_{a,0}$ |

${D}_{\ell ,a}$ | Distance between access point ℓ and ${F}_{a,0}$ |

${D}_{C,0}$ | Distance between ${C}_{m,0}$ and ${u}_{0}$ |

${h}_{0,0}$ | Small-scale channel coefficient between ${F}_{m,0}$ and ${u}_{0}$ |

${h}_{\ell ,0}$ | Small-scale channel coefficient between access point ℓ and ${u}_{0}$ |

${h}_{a,0}$ | Small-scale channel coefficient between ${F}_{a,0}$ and ${u}_{0}$ |

${h}_{C,a}$ | Small-scale channel coefficient between ${C}_{0}$ and ${F}_{a,0}$ |

${h}_{\ell ,a}$ | Small-scale channel coefficient between access point ℓ and ${F}_{a,0}$ |

${h}_{C,0}$ | Small-scale channel coefficient between ${C}_{m,0}$ and ${u}_{0}$ |

${q}_{m,0}\left(\mathit{p}\right)$ | STP of content m when ${u}_{0}$ is associated with ${F}_{m,0}$ |

${q}_{m,0,{D}_{0,0}}(\mathit{p},d)$ | ${q}_{m,0}\left(\mathit{p}\right)$ conditioned on ${D}_{0,0}=d$ |

${q}_{C,a,0}\left(\mathit{p}\right)$ | STP of content m when ${u}_{0}$ is associated with ${F}_{a,0}$ |

${q}_{a,0}\left(\mathit{p}\right)$ | STP of content m over the link ${F}_{a,0}$ to ${u}_{0}$ |

${q}_{a,0,{D}_{a,0}}(\mathit{p},d)$ | ${q}_{a,0}\left(\mathit{p}\right)$ conditioned on ${D}_{a,0}=d$ |

${q}_{C,a}\left(\mathit{p}\right)$ | STP of content m over the link ${C}_{0}$ to ${F}_{a,0}$ |

${q}_{C,a,{D}_{c,a}}(\mathit{p},d)$ | ${q}_{C,a}\left(\mathit{p}\right)$ conditioned on ${D}_{C,a}=d$ |

${q}_{C,0}\left(\mathit{p}\right)$ | STP of content m when ${u}_{0}$ is associated with ${C}_{m,0}$ |

${q}_{C,0,{D}_{C,0}}(\mathit{p},d)$ | ${q}_{C,0}\left(\mathit{p}\right)$ conditioned on ${D}_{C,0}=d$ |

$q\left(\mathit{p}\right)$ | STP of ${u}_{0}$ |

${q}_{\infty}\left(\mathit{p}\right)$ | Asymptotic STP of ${u}_{0}$ when $\frac{P}{{N}_{0}}\to \infty $ |

${\tau}_{m,0}\left(\mathit{p}\right)$ | Average delay of content m when ${u}_{0}$ is associated with ${F}_{m,0}$ |

${\tau}_{C,a,0}\left(\mathit{p}\right)$ | Average delay of content m when ${u}_{0}$ is associated with ${F}_{a,0}$ |

${\tau}_{a,0}\left(\mathit{p}\right)$ | Average delay of content m over the links from ${F}_{a,0}$ to ${u}_{0}$ |

${\tau}_{C,a}\left(\mathit{p}\right)$ | Average delay of content m over the link from ${C}_{0}$ to ${F}_{a,0}$ |

${\tau}_{C,0}$ | Average delay of content m when ${u}_{0}$ is associated with ${C}_{m,0}$ |

$\tau \left(\mathit{p}\right)$ | Average delay of ${u}_{0}$ |

${\tau}_{\infty}\left(\mathit{p}\right)$ | Asymptotic delay of ${u}_{0}$ when $\frac{P}{{N}_{0}}\to \infty $ |

## Short Biography of Authors

Alaa Bani-Bakr received B.Sc. and M.Sc. degrees in electrical/telecommunication engineering from Mutah University, Karak, Jordan, in 2002 and 2006, respectively. He is currently pursuing a Ph.D. with the University of Malaya, Kuala Lumpur, Malaysia. He was a Lecturer at the Department of Electrical Engineering, AlBaha University, Saudi Arabia, from 2010 to 2012. His research interests are fog radio access networks, cache-enabled wireless networks, mmWave communication systems, stochastic analysis, and optimization. | |

MHD Nour Hindia received a Ph.D. from the Faculty of Engineering in Telecommunication, University of Malaya, Kuala Lumpur, Malaysia, in 2015. He is currently involved with research in the field of wireless communications, especially in channel sounding, network planning, converge estimation, handover, scheduling, and quality of service enhancement for 5G networks. He is currently a Post-Doctoral Fellow from the Faculty of Engineering in Telecommunication, University of Malaya. Besides that, he is involved with research with the Research Group in Modulation and Coding Scheme for Internet of Things for Future Network. He has authored or co-authored a number of science citation index journals and conference papers. Dr. Hindia has participated as a Reviewer and a committee member of a number of ISI journals and conferences. | |

Kaharudin Dimyati graduated from the University of Malaya, Malaysia, in 1992. He received a Ph.D. from the University of Wales Swansea, U.K., in 1996. He is currently a Professor at the Department of Electrical Engineering, Faculty of Engineering, University of Malaya. Since joining the university, he is actively involved in teaching, postgraduate supervision, research, and also administration. To date, he has supervised 15 Ph.D. students and 32 masters by research students. He has published over 100 journal articles. He is a member of IET and IEICE. He is a Professional Engineer and a Chartered Engineer. | |

Effariza Hanafi rreceived a B.Eng. in telecommunications (first class Hons.) from the University of Adelaide, Adelaide, S.A., Australia, and the Ph.D. degree in electrical and electronic engineering from the University of Canterbury, Christchurch, New Zealand, in 2010 and 2014, respectively. She joined the University of Malaya, Kuala Lumpur, Malaysia, where she is now a Senior Lecturer at the Faculty of Engineering. In 2015, she was the recipient of the University of Malaya Excellence Awards. She is currently a Senior Member for IEEE (Institute of Electrical and Electronics Engineers). Her main research interests include wireless communications, Internet of Things, cognitive radio, cooperative communications, 5G networks, and beyond. | |

Tengku Faiz Tengku Mohmed Noor Izam received a Ph.D. in electronic engineering from the University of Surrey, U.K., in 2016. He is currently a Lecturer with the Department of Electrical Engineering, University of Malaya, Malaysia. His research interests include, parasitic antenna, and MIMO system with antenna selection. |

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## Share and Cite

**MDPI and ACS Style**

Bani-Bakr, A.; Hindia, M.N.; Dimyati, K.; Hanafi, E.; Tengku Mohmed Noor Izam, T.F.
Multi-Objective Caching Optimization for Wireless Backhauled Fog Radio Access Network. *Symmetry* **2021**, *13*, 708.
https://doi.org/10.3390/sym13040708

**AMA Style**

Bani-Bakr A, Hindia MN, Dimyati K, Hanafi E, Tengku Mohmed Noor Izam TF.
Multi-Objective Caching Optimization for Wireless Backhauled Fog Radio Access Network. *Symmetry*. 2021; 13(4):708.
https://doi.org/10.3390/sym13040708

**Chicago/Turabian Style**

Bani-Bakr, Alaa, MHD Nour Hindia, Kaharudin Dimyati, Effariza Hanafi, and Tengku Faiz Tengku Mohmed Noor Izam.
2021. "Multi-Objective Caching Optimization for Wireless Backhauled Fog Radio Access Network" *Symmetry* 13, no. 4: 708.
https://doi.org/10.3390/sym13040708