## Ramy Amer , Mohamed Baza , Tara Salman , M. Majid Butt , Ahmad Alhindi and Nicola Marchetti## |

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

Displacement standard deviation | [TeX:] $$\sigma$$ | 10m |

Popularity index | [TeX:] $$\beta$$ | \beta |

Path loss exponent | [TeX:] $$\alpha$$ | 4 |

Library size and cache size per device | [TeX:] $$N_{f}, M$$ | 100, 8 files |

Average number of devices per cluster | [TeX:] $$\bar{n}$$ | 4 |

Density of clusters | [TeX:] $$\lambda_{p}$$ | [TeX:] $$10 \text { clusters } / \mathrm{km}^{2}$$ |

SIR threshold | [TeX:] $$\vartheta$$ | 0dB |

the caching probability b in the optimization problem below.

**Lemma 3:** For fixed [TeX:] $$q^{*}, \mathbb{P}_{\rtimes}\left(\|^{*}, b\right)$$ is a concave function w.r.t. b and the optimal caching probability b* that maximizes the offloading gain is given by

[TeX:] $$b_{i}^{*}= \begin{cases}1 & , v^{*}<p_{i}-p_{i}\left(1-e^{-\bar{n}}\right) \Upsilon \\ 0 & , v^{*}>p_{i}+\bar{n} p_{i} \Upsilon \\ \psi\left(v^{*}\right) & , \text { otherwise }\end{cases}$$

where [TeX:] $$\psi\left(v^{*}\right)$$ is the solution of [TeX:] $$v^{*}=p_{i}+p_{i}\left(\bar{n}\left(1-b_{i}^{*}\right) e^{-\bar{n} b_{i}^{*}}-\right. \left.\left(1-e^{-\bar{n} b_{i}^{*}}\right)\right) \Upsilon,$$ that satisfies [TeX:] $$\sum_{i=1}^{N_{f}} b_{i}^{*}=M.$$

Proof: Please see Appendix B.

Clearly, the optimal caching solution b* depends on the scheduling of devices through channel access probability q* from [TeX:] $$\Upsilon,$$ while q* is independent of b*. [9] shows that a PPP network exhibits the same property, i.e., the caching scheme is scheduling-dependent. To gain some insights, it is useful to consider a simple case when only one D2D link per cluster is allowed. In this case, the rate coverage probability of the proposed clustered model with one active D2D link within a cluster will be [20, Lemma 2]:

Substituting in (10) for [TeX:] $$\Upsilon,$$ we get the offloading gain as

**Remark 1:** From (17), it is clear that the offloading gain increases as [TeX:] $$\sigma \text { and } \lambda_{p}$$ decrease. Particularly, the offloading gain is inversely proportional to the density of clusters [TeX:] $$\lambda_{p}$$ and the variance of the displacement [TeX:] $$\sigma^{2}.$$ This is because smaller [TeX:] $$\sigma$$ results in higher levels of the desired signal, while lower [TeX:] $$\lambda_{p}$$ leads to smaller encountered interference at the typical device.

We first validate the developed mathematical model via Monte Carlo simulations. Then we benchmark the proposed caching scheme against conventional caching schemes. Unless otherwise stated, the network parameters are selected as shown in Table 1.

In Fig. 2, we plot the rate coverage probability [TeX:] $$\Upsilon$$ against the channel access probability q. The theoretical and simulated results are plotted together, and they are consistent. Clearly, there is an optimal q*; before it, [TeX:] $$\Upsilon$$ tends to increase as the probability of accessing the channel increases, and beyond it, [TeX:] $$\Upsilon$$ tends to decrease due to the effect of aggressive interference. It is intuitive to observe that the optimal access probability q*, which maximizes [TeX:] $$\Upsilon$$, decreases as [TeX:] $$\vartheta$$ increases. This reflects the fact the system becomes more sensitive to the effect of interference when a higher SIR threshold is required.

Fig. 3 manifests the effect of the access probability q on the offloading gain. The offloading gain is plotted against q for different caching schemes, namely, the proposed probabilistic caching (PC), Zipf caching (Zipf), and uniform random caching (RC). Fig. 3 is plotted for an SIR threshold [TeX:] $$\vartheta=0 \mathrm{~dB},$$ hence, the optimal access probability q* is near one from Fig. 2. Clearly, the offloading gain for the different caching schemes improves as q approaches its optimal value, which reveals the crucial impact of the device scheduling on the content placement and ac

cordingly, on the offloading gain. Moreover, the proposed PC is shown to attain the best performance as compared to other benchmark schemes.

To show the effect of q on the caching probability, in Fig. 4, we plot the histogram of the optimal caching probability at different q values. Specifically, [TeX:] $$p=q^{*}$$ in Fig. 4(a) and [TeX:] $$q<q^{*}$$ in Fig. 4(b). It is clear from the histograms that the optimal caching probability b* tends to be more skewed when [TeX:] $$q<q^{*} \text {, }$$ i.e., when [TeX:] $$\Upsilon$$ decreases. This shows that file sharing is more difficult when q is not optimized. Broadly speaking, for [TeX:] $$q<q^{*},$$ the system is too conservative, while for [TeX:] $$q>q^{*} \text {, }$$ the outage probability is high due to the aggressive interference. In such regimes, each device tends to cache the most popular files leading to fewer opportunities of content transfer.

Fig. 5 illustrates the prominent effect of the content popularity on the offloading gain, and compares the achievable gain of three different caching schemes. Clearly, the offloading gain of the proposed PC attains the best performance as compared to other schemes. Particularly, 10% improvement in the offloading gain is observed compared to the Zipf caching when [TeX:] $$\beta=1.$$ Moreover, we note that all caching schemes encompass the same offloading gain when [TeX:] $$\beta=0$$ owing to the uniformity of content popularity.

To show the effect of network geometry, in Fig. 6, we plot the closed-form offloading gain in (17) against [TeX:] $$\sigma$$ at different [TeX:] $$\lambda_{p}.$$ Fig. 6 shows that the offloading gain monotonically decreases with both [TeX:] $$\sigma \text { and } \lambda_{p}.$$ This is because content sharing between

devices turns out to be less successful when the distance between devices is large, i.e., larger [TeX:] $$\sigma.$$ Analogously, file sharing among the cluster devices is accompanied with higher interference when [TeX:] $$\lambda_{p} \text { and } \sigma$$ are higher. Accordingly, this expected degradation prohibits successful content delivery via D2D communication.

Last, in Fig. 7, we plot the offloading gain versus the popularity index [TeX:] $$\beta$$ at different densities of cluster devices [TeX:] $$\bar{n}.$$ Fig. 7 first shows that the proposed optimized probabilistic caching scheme achieves the best performance as compared to caching popular files (CPF) and Zipf caching. In addition, Fig. 7 shows that the attained offloading gain increases as the number of devices per cluster increases. This is attributed to the fact that the probability of having requested contents cached at a neighbor device within the same cluster increases when the number of cluster members is higher.

In this paper, we have proposed a joint communication and caching optimization framework for clustered D2D networks. In particular, we have conducted joint optimization of channel access probability and content placement in order to maximize the offloading gain. We have characterized the optimal content caching scheme as a function of the system parameters, namely, density of clusters, average number of devices per cluster, content caching, placement and access probabilities. A bisection search method is also proposed to calculate the optimal channel access probability. We have demonstrated that deviating from the optimal access probability makes file sharing more difficult, i.e., the system is too conservative for small access probabilities, while the interference is too aggressive for larger access probabilities. Results showed up to 10% enhancement in offloading gain compared to the Zipf caching technique.

Laplace transform of the inter-cluster aggregate interference [TeX:] $$I_{\Phi_{P}^{1}}$$ can be evaluated as

[TeX:] $$\mathscr{L}_{I_{\Phi} !}(s)=\mathbb{E}\left[e^{-s \sum_{\Phi_{p}^{!}}}\right. \left.\sum_{y \in \mathcal{B}^{\mathrm{II}}} g_{y_{x}}\|x+y\|^{-\alpha}\right] \\ \stackrel{(a)}{=} \mathbb{E}_{\Phi_{p}}\left[\prod_{\Phi_{p}} \mathbb{E}_{\Phi_{c q}} \prod_{y \in \mathcal{B}^{\mathrm{II}}} \frac{1}{1+s\|x+y\|^{-\alpha}}\right] \\ \stackrel{(b)}{=} \mathbb{E}_{\Phi_{p}} \prod_{\Phi_{p}^{\prime}} e -q \bar{n} \int_{\mathbb{R}^{2}}\left(1-\frac{1}{1+s\|x+y\|-\alpha}\right) f_{Y}(y) \mathrm{dy} \\ \stackrel{(c)}{=} e^{-\lambda_{p} \int_{\mathbb{R}^{2}}\left(1-e^{-q \bar{n} \int_{\mathbb{R}}^{2}\left(1-\frac{1}{1+s\|x+y\|-\alpha}\right) f_{Y}(y) \mathrm{dy}} \mathrm{dx}\right.},$$

where (a) follows from the Rayleigh fading assumption, (b) follows from the probability generating functional (PGFL) of Gaussian PPP [TeX:] $$\Phi_{c q},$$ and (c) follows from the PGFL of the parent PPP [TeX:] $$\Phi_{p^{*}}.$$ By using change of variables [TeX:] $$z=x+y \text { with } \mathrm{dz}=\mathrm{dy},$$ we proceed as

where (d) follows from converting the cartesian coordinates to the polar coordinates with [TeX:] $$u=\|z\|.$$ To clarify how in (d) the normal distribution [TeX:] $$f_{Y}(z-x)$$ is converted to the Rice distribution [TeX:] $$f_{U}(u \mid v),$$ consider a remote cluster centered at [TeX:] $$x \in \Phi_{p}^{!},$$ with a distance [TeX:] $$v=\|x\|$$ from the origin. Every interfering device belonging to the cluster centered at x has its coordinates in [TeX:] $$\mathbb{R}^{2}$$ chosen independently from a Gaussian distribution with standard deviation . Then, by definition, the distance from such an interfering device to the origin, denoted as u, has a Rice distribution, denoted as [TeX:] $$f_{U}(u \mid v)=u / \sigma^{2} \exp \left(-\left(\mathrm{u}^{2}+\mathrm{v}^{2}\right) / 2 \sigma^{2}\right) \mathrm{I}_{0}\left(\mathrm{uv} / \sigma^{2}\right),,$$ where [TeX:] $$I_{0}$$ is the modified Bessel function of the first kind with oRrder zero and[TeX:] $$\sigma$$ is the scale parameter. Letting [TeX:] $$\varphi(s, v)= \ \int_{u=0}^{\infty} s /\left(s+u^{\alpha}\right) f_{U}(u \mid v)$$ du, the proof is completed.

First, to prove concavity, we proceed as follows.

It is clear that the second derivative [TeX:] $$\frac{\partial^{2} \mathbb{P}_{o}}{\partial b_{i} \partial b_{j}}$$ is negative. Hence, the Hessian matrix [TeX:] $$\mathbf{H}_{i, j} \text { of } \mathbb{P}_{o}\left(p^{*}, b_{i}\right)$$ w.r.t. [TeX:] $$b_{i}$$ is negative semidefinite, and the function [TeX:] $$\mathbb{P}_{o}\left(p^{*}, b_{i}\right)$$ is concave with respect to bi. Also, the constraints are linear, which implies that the necessity and sufficiency conditions for optimality exist. The dual Lagrangian function and the KKT conditions are then employed to solve P2 [23]. The KKT Lagrangian function of the energy minimization problem is given by

where [TeX:] $$v, w_{i}, \text { and } \mu_{i}$$ are the dual equality and two inequality constraints, respectively. Now, the optimality conditions are written

The optimality conditions imply that:

BPy combining (28), (29), and (30), with the fact that [TeX:] $$\sum_{i=1}^{N_{f}} b_{i}^{*}=M,$$ Lemma 3 is proven.

Ramy Amer received his Ph.D. in Electrical and Communications Engineering from Trinity College Dublin in November 2020. He was a Visiting Scholar at Virginia Tech, USA from September 2018 to March 2019. He holds one best paper award, IEEE student travel grant, and IEEE exemplary reviewer award. His research interests include edge caching and Intelligence, edge computing and IoT, machine learning, and UA Vs. Ramy has been first author in one book chapter and six journals and eight IEEE conference papers.

Mohamed Baza is currently an Assistant Professor at the Department of Computer Science at College of Charleston, SC, USA. He received his Ph.D. degree in Electrical and Computer Engineering from Tennessee Tech University, Cookeville, Tennessee, United States in Dec. 2020 and B.S. and M.S. degrees in Electrical Computer Engineering from Benha University, Egypt in 2012 and 2017 respectively. From August 2020 to May 2021, he worked as a Visiting Assistant Professor at the Department of Computer Science at Sam Houston State University, TX, USA. He also has more than two years of industry experience in information security in Apachekhalda petroleum company, Egypt. Dr. Baza is the Author of numerous papers published in major IEEE conferences and journals, such as IEEE Wireless Communications and Networking Conference (IEEE WCNC), IEEE International Conference on Communications (IEEE ICC), IEEE Vehicular Technology Conference (IEEE VTC), IEEE Transactions on Dependable and Secure Computing, IEEE Transactions of Vehicular Technology (TVT), IEEE Transactions on Network Science and Engineering (TNSE), and IEEE Systems Journal. He served as a Reviewer for several journals and conferences such as IEEE Transactions on Vehicular Technology, IEEE IoT Journal, and the journal of Peer-to-Peer Networking. His research interests include blockchains, cyber-security, machine learning, smart-grid, and vehicular ad-hoc networks. He also a recipient of best IEEE paper award in the International Conference on Smart Applications, Communications and Networking (SmartNets 2020).

Tara Salman is finishing a graduate Research Assistant at Washington University in St. Louis. She received her B.S. in Computer Engineering and M.S. degrees in Computer Networking from Qatar University Doha, Qatar in 2012 and 2015, respectively. She is currently pursuing a Ph.D. in Computer Science Engineering at Washington University in St Louis, Missouri, USA. From 2012 -2015, she worked as a Research Assistant with Qatar University on an NPRP (National Priorities Research Program) funded project targeting physical layer security. Since 2015, she has been working as a Graduate Research Assistant at Washington University in St. Louis. Her research interests span network security, distributed systems, the Internet of things, and financial technology. She is an Author of 1 book chapter and numerous papers published at major IEEE conferences and journals. Tara Salman is an EECS Rising Star in UC Berkeley 2020, is a Recipient of the Cisco Certified Network Associate (CCNA) certification in 2012, and has completed all four levels of CCNA at Cisco academy-Qatar university branch.

M. Majid Butt received the M.Sc. degree in Digital Communications from Christian Albrechts University, Kiel, Germany, in 2005, and the Ph.D. degree in Telecommunications from the Norwegian University of Science and Technology, Trondheim, Norway, in 2011. He is currently a Senior Research Specialist 5G+ at Nokia Bell Labs, Paris-Saclay, France, and also an adjunct Research Professor at Trinity College Dublin, Dublin, Ireland. Prior to that, he has held various positions at the University of Glasgow, U.K., Trinity College Dublin, Ireland, Fraunhofer HHI, Germany, and the University of Luxembourg. His current research interests include communication techniques for wireless networks with a focus on radio resource allocation, scheduling algorithms, energy efficiency, and machine learning for RAN. He has Authored more than 70 peer-reviewed conference and journal publications, and filed some 10 patents in these areas. He frequently gives invited talks, as well as technical tutorial talks on various topics in IEEE conferences, including ICC, Globecom, PIMRC, VTC, ISWCS, etc. He was a Recipient of the Marie Curie Alain Bensoussan Post-Doctoral Fellowship from the European Research Consortium for Informatics and Mathematics. He has served as the Organizer/chair for technical workshops on various aspects of communication systems in conjunction with major IEEE conferences. He has been an Associate Editor of the IEEE ACCESS and the IEEE Communication Magazine since 2016.

Ahmad Alhindi received the B.Sc. degree in Computer Science from Umm Al-Qura University (UQU), Makkah, Saudi Arabia, in 2006, and the M.Sc. degree in Computer Science and the Ph.D. degree in Computing and Electronic Systems from the University of Essex, Colchester, U.K., in 2010 and 2015, respectively. He is currently an Assistant Professor in Artificial Intelligence (AI) with Computer Science Department, UQU. His current research interests include evolutionary multi-objective optimization and machine learning techniques. He is currently involved in AI algorithms, focusing particularly on machine learning and optimization with a willingness to implement them in a context of decision making and solving combinatorial problems in real-world projects.

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