Optimal Caching Policy for D2D Assisted Cellular Networks With Different Cache Size Devices

This paper studies the problem of optimal cache placement to maximize the offloading probability in a device-to-device (D2D) enabled cellular network with small base stations (SBSs). Different from most existing works, we consider unequal users’ equipment (UE) cache memory sizes and all wireless links are modeled as Nakagami- $m$ fading. User preference for each UE and global popularity for SBS, as well as the higher priority of content request from neighboring UEs vs. SBS, are the main factors that make the problem formulation of our work different from that of existing works. It is assumed that each UE caches its desired content with the order of searching its cache, neighboring UEs’ cache via D2D communications, and its serving SBS’ cache. A close to optimal low complexity heuristic cache placement policy is proposed and it is shown that its performance reaches the optimal caching strategy.

spectrum utilization. One of the main challenges of D2D 23 communications are interference reduction and management 24 among devices due to their proximity [2]. 25 While edge caching is considered to be a promising solu-26 tion to enhance offloading, caching is still surrounded by 27 challenges, i.e., cache placement and content delivery strat-28 egy [1], [3], where and which content should be cached 29 due to limited cache size of UEs and SBSs, and how to 30 The associate editor coordinating the review of this manuscript and approving it for publication was Omer Chughtai.
reliably deliver cached content from its stored location to a 31 requesting UE. 32 Heterogeneous user activity level, different content request 33 of users for the same content and unequal user cache size 34 make the aforementioned challenges even more complicated. 35 Most of the existing works, assume the cache size of devices 36 to be equal, while according to the device model and other 37 factors, this is an unrealistic assumption for real-world cellu-38 lar networks [4]. 39 Recently, extensive research has been conducted to address 40 the aforementioned challenges. To optimize caching policy 41 and increase the likelihood of offloading, a prior knowledge-42 based learning algorithm has been proposed to obtain user 43 preference in [5]. In that work, caching at the edge of the 44 wireless network, D2D communications, equal user cache 45 size, and Rayleigh fading have been assumed. In [6], [7], and 46 [8] the authors have maximized cache hit rate and through-47 put by optimizing cache policy at the edge of the wireless 48 network. Furthermore, the aforementioned works ignored the 49 user's preference and the cache size difference, while the 50 telecommunication channel has been assumed to be Rayleigh 51 fading.
downlink network to accommodate heterogeneous user pref-104 erences and spatial locality by allowing each UE and SBS to 105 cache files with a different probability distribution. Location 106 of UEs and SBSs are modeled by two independent homo-107 geneous Poisson Point Processes (PPP) u and b with 108 intensities λ u and λ b , respectively. Three pre-defined cache 109 hit scenarios are: 110 1) Self request: occurs when the requested file is cached 111 in its device.

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2) D2D cache hit: when the requested file is not cached in 113 a device but exists in nearby devices.

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3) SBS cache hit: when the requested file is not cached in 115 a device and its nearby devices, the file is fetched from 116 devices associated with the SBS cache.

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The table below presents variable notations used through-118 out the paper. In D2D communications, to increase the cache-hit prob-120 ability, each user is allowed to associate with one of the 121 N -nearest UEs to download the requested file. For the sake 122 of lower latency and required transmission power, the nearest 123 user that has the requested file is selected. For a given user, 124 if the requested file is not found in any of its N th nearest users, 125 that user requests its desired file from SBS in cellular mode. 126 We consider Nakagami-m fading channel model for D2D 127 and cellular connections. The Nakagami-m probability den-128 sity function (PDF) is given by v u q f ,u ;

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In practice, user preference q u and activity level v can be 156 learned through machine learning techniques [5].

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In this section, we evaluate the offloading probability or 159 equivalently cache hit probability, which is the probability 160 that a user finds its requested file in the local cache. A user 161 chooses the first nearest user that has the requested file (due 162 to lower latency and required transmission power); it searches 163 up to the ith nearest user in priority of lower distance to 164 itself. If the requested file is not found in any of the ith 165 nearest users, it requests its desired file mode from SBS in 166 cellular.

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A. CACHING POLICY 168 We consider self-request, D2D cache hit, and SBS cache hit 169 cases for offloading. 170 1) Self request: In this case, p self is the average proba-171 bility that a user finds its requested file in its cache, given 172 by 2) D2D cache hit: In this case, the probability of suc-175 cess is defined as the probability that a user retrieves the 176 requested file from the cache of a neighboring device. This 177 is possible when the received signal-to-interference-plus-178 noise ratio (SINR) at the user that acquires the requested 179 file, γ , is greater than a threshold denoted by γ 0 . The uth 180 user's received SINR downloading from the n u (i)th user is 181 VOLUME 10, 2022

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where n u (i) denotes the index of the ith nearest user to the uth 184 user, P n u (i) is the transmit power of the n u (i)th user, h n u (i) is 185 the channel gain between uth user and n u (i)th user, r n u (i) is σ 2 is the noise variance, and u is the user set that shares the 188 same frequency with the uth user. For evaluating the power 189 of the interference term in our analysis, we have assumed that 190 only the nearest user causes interference in D2D mode, which 191 generates the strongest interference.

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The D2D cache hit probability is given by (5) loading from the n u (i)th user is given by where G n u (i) (r n u (i) , γ 0 ) is given in (7), as shown at the bottom 207 of the next page.

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For a real valued µ, we have [13]: For the case where h n u (i) has Nakagami-m distribution, . The characteristic function of r −α u,û ζ is given Then Im{φ(t)e −jtx } is given in (12), as shown at the bottom 224 of the next page, and by substituting (12) and (8) in (9) we 225 reach (13), as shown at the bottom of the next page.

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To achieve the success probability in (6), the PDF of the 227 distance of a UE from its ith nearest UE, r n u (i) , according to 228 the PPP is given by [14]: 3) SBS cache hit: when a user's requested file has not been 231 cached in its cache or via D2D mode, then it can download 232 from the SBS cache. The received SINR of the uth user 233 downloading from the SBS is given by where P b is the transmit power of SBS, h u,b is the channel 236 gain between the uth user and the SBS, r u,b is the distance 237 between the uth user and the bth SBS, and b is the SBS set 238 that shares the same frequency with bth SBS. The SBS cache 239 hit probability is then given by Similar to Lemma 1, it can be shown that the success 242 probability when the uth user downloads from the SBS is 243 given by (17) 247 where G u,b (r u,b , γ 0 ) is given in (18), as shown at the bottom 248 of the next page. Also, f R (r u,b , λ b ) in (17) is calculated similar 249 to (14).

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For the general case of N (assuming the requested file 251 is stored in the ith nearest user), (6) and (17) are difficult 252 to solve. Furthermore, for the analysis to be tractable we 253 have assumed that only the nearest user (strongest interferer) 254 causes interference in D2D mode. Therefore, we have closed 255 form expressions for the case of N = 1, m = 2, α = 2, and 256 K = 2, where S n u (i) is given in (19), as shown at the bottom 257 of the next page. The expression for S u,b (γ 0 ) is similar to (19) 258 and it can be obtained by replacing λ u and P u with λ b and P b , 259 respectively.

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Finally, the total cache hit probability is given by Based on (20), the optimization problem is formulated as C2 :

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C3 : In problem P1, constraints C2 and C3 ensure that the cache 275 content of each user device and SBS do not exceed their 276 cache size. Note that (21) is a multivariate polynomial in 277 terms of c f ,u s andĉs, and the determinant of its Hessian 278 matrix is found to be infinite. Therefore, it is a non-convex 279 optimization problem and we suggest to solve it using meta-280 heuristic methods. Accordingly, its numerical solution using 281 particle swarm optimization (PSO) algorithm with adjustable 282 search steps [15] is considered. In this algorithm, an initial 283 velocity is assigned to each particle. These particles move 284 in the problem space, and the results are calculated based 285 on a predefined competency function, after each movement. 286 As the algorithm progresses, these new locations determine 287 the direction of congestion according to, In (22) and (23), the best position the particle has ever 292 achieved, also called the best particle nostalgia or its best solo 293 experience, is denoted by p best . Another parameter used by 294 S n u (i) (γ 0 ) = 1 λ u P u π(−1 + µ) 6 µ 4 ((−1 + µ)µ 4 (4λσ 2 µ(1 + µ(19 + 10µ)) + λ u P u π(−1 + µ)(1 + µ(−4 + µ(23 + 4µ)))) − 6µ 6 (λ u P u π(−1 + µ)(1 + 3µ) + 2λσ 2 (3 + µ(6 + µ)))Log[µ]) VOLUME 10, 2022 the algorithm is the best position ever obtained by the particle 295 mass, called g best . Also, pos in (23) and V in (22) It is noted that in (26)

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In the next section, we present numerical results for the 326 proposed solution.

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In this section, we present numerical results of the proposed cache capacity, N c u , is randomly selected and the SBS cache 341 capacity is set to N c b = 10.

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In Figures 2 and 3, we compare four caching strategies for 349 the D2D assisted cellular network: 350 1) Optimal strategy, which is found by solving the opti-351 mization problem of Section III.B.

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2) Proposed strategy, where the files with least popularity 353 are discarded for each user.  are not equal, while the optimization is performed assuming 362 they are equal; i.e., average cache size of UEs has been used 363 as the mistaken common cache size of UE's. The results for 364 this case are obtained as follows: 365 (i) All devices are assumed to have similar cache sizes 366 which is the average of the randomly generated unequal cache 367 sizes used for the results in the case of ''Proposed scheme' '. 368 (ii) Using the similar cache sizes, we computed the sys-369 tem's caching strategy. Since in practical systems, the cache 370 sizes of different devices are not equal then that caching 371 strategy has to fit the actual cache sizes of the devices.  of sight component, the Nakagami-m fading channel has a 409 much higher success probability in transferring files among 410 devices compared to that of Rayleigh fading and accordingly 411 higher probability of cache-hit. This is most beneficial in our 412 network as it represents the case of small cell networks in 413 5G, where mm-Wave frequencies are expected to be used 414 and with smaller cells the probability of having LoS is high. 415 However, since cache placement is done with the knowledge 416 of success probability, the cache-hit probability curves for 417 Rayleigh fading in prior figures are very close to that of 418 Nakagami-m fading. 420 We mathematically formulated the optimal cache place-421 ment problem to maximize offloading probability in a 422 D2D-enabled cellular network with SBSs. To the best of our 423 knowledge, no prior work has considered unequal UE cache 424 size in solving the cache placement problem, while this is a 425 realistic assumption in many practical scenarios. Our results 426 showed that solving the cache placement problem with an 427 equal cache size assumption, in a network where cache sizes 428 are actually unequal, leads to cache hit probability reduction. 429