Cache-Aided Content Delivery in Fog-RAN Systems with Topological Information and no CSI

Transcription

1 Cache-Aie Content Delivery in Fog-RAN Systems with Topological Information an no CSI Wei-Ting Chang, Ravi Tanon, Osvalo Simeone Abstract In this work, we consier a Fog Raio Access Network (F-RAN) system with a partially connecte wireless topology an no channel state information available at the clou an Ege Noes (ENs). An F-RAN consists of ENs that are equippe with caches an connecte to the clou through fronthaul links. We first focus on the case where clou connectivity is isable, an hence the ENs have to satisfy the user emans base only on their cache contents. For a class of partially connecte regular networks, we present a elivery scheme which combines intra-file MDS coe caching at the ENs an blin interference avoiance on the wireless ege channel. This scheme requires minimum storage an leas to an achievable Normalize Delivery Time (NDT) that is within a constant multiplicative factor of the best known NDT with full storage. We next provie achievable schemes for the case where clou connectivity is enable, an we provie new insights on the interplay between clou connectivity an ege caching when only topological information is available. Keywors Fog Networking, ege caching, latency. I. INTRODUCTION The growth in the eman of multimeia contents is increasing the nee to reuce content elivery latency in wireless networks, which is one of the goals of 5G []. To tackle this challenge, ege caching is emerging as one of the main caniate solutions. Ege caching enables the Ege Noes (ENs) in a wireless system, such as base stations or access points, to locally store popular content. By prefetching popular content uring off-peak hours or online, the ENs can eliver these content without retrieving them from core networks, thus reucing latency. Caching for wireless networks is an active area of recent research. Cache-aie interference channels with full Channel State Information (CSI) in the case of three ENs an three users were stuie in [2]. A performance upper boun in terms of the inverse of sum Degrees of Freeom (DoF) for this setting was obtaine in [2]. Information-theoretic lower bouns for cache-aie wireless networks with full CSI for any number of ENs an W.-T. Chang an R. Tanon are with the Department of Electrical an Computer Engineering at the University of Arizona, Tucson, AZ, USA ( {wchang, tanonr}@ .arizona.eu). O. Simeone is with the Department of Informatics at King s College Lonon, Lonon, UK ( osvalo.simeone@kcl.ac.uk). O. Simeone has receive funing from the European Research Council (ERC) uner the European Union s Horizon 2020 Research an Innovation Programme (Grant Agreement No ). His work was also partially supporte by the U.S. NSF through grant

2 2 users were foun in [3]. Ege caching was investigate in the presence of caches at the receivers in [4] [7]. A novel approach that achieves approximately optimal DoF by separating physical an network layers was presente in [4]. Upper an lower bouns on the minimum Normalize Delivery Time (NDT), which is relate to the inverse of the DoF, were provie in [5], an their optimality was shown for certain cache storage regimes. The scheme of [6] was shown to be within a constant multiplicative factor of 2 of the optimal in the presence of full CSI uner linear precoing. Ege caching was then stuie in combination with clou-aie transmission to yiel the Fog Raio Access Network (F-RAN) architecture in [8]. There, the optimal NDT trae-off was characterize for two ENs an two users. More general upper an lower bouns on the NDT were erive in [9] for any number of ENs an users, characterizing the minimum NDT within a multiplicative factor of 2. Online caching in F-RANs was stuie in [0] an scenario with heterogeneous contents was consiere in []. The prior works summarize above on F-RAN assume that the wireless network is fully connecte an that all the ENs an the clou have full CSI. The assumption of full connectivity an CSI may not be vali in practice. In fact, links between ENs an users may be too weak ue to large geographic separation or severe faing. As an approximation, noes with such weak links can be viewe as isconnecte from each other. Uner such an approximation, one obtains a partially connecte topology. Furthermore, full CSI requires significant overhea, particularly for large networks. To alleviate these assumptions, we stuy the extreme case in which clou an ENs have no CSI but only knowlege of wireless network topology. Interference management techniques for partially connecte network with only topology knowlege have been stuie uner the rubric of Topological Interference Management (TIM), TIM has interesting connection to inex coing [2]. Cooperative TIM (or TIM-CoMP) was stuie in [3], by allowing ENs to cooperate with each other. Achievable schemes an upper bouns on the NDT uner TIM-CoMP setting were provie in [3] (also see [4] for another variation of cooperative TIM). The work of [5] is the closest to the problem consier in this paper. In [5], the ENs can only store a subset of files in the library. This reference focuse on blin zero-forcing base interference management an stuies the minimization of the elivery latency from an algorithmic stanpoint. Main Contributions: In this paper, we stuy a class of partially connecte networks, referre to as (K, ) regular network, with no CSI at the clou an ENs. In this network, as seen in Fig., there are K ENs an K users, an each EN is connecte to the corresponing user an the subsequent users in a cyclic fashion in the wireless channel. During the offline pre-fetching phase, the ENs can cache a function of the files in the library of popular contents subject to storage constraints. In aition, the ENs are connecte to the clou with finite capacity fronthaul links. Users can request any file in the library, an the requeste files must be recovere through the transmission over the wireless channel. This paper aresses the following questions: What is the minimum necessary cache storage at each EN so that the users can reliably ecoe their esire files when there is no clou connectivity? What caching strategies allow the ENs to leverage the topological knowlege of a network? How to evise elivery schemes uner cache storage constraint at each EN an fronthaul capacity constraint? The specific main contributions are as follows. We first stuy the case with µ = / an no clou connectivity. We propose a Maximum Distance Separable

3 3 (MDS) coe caching scheme that ensures that each user can recover any arbitrary requeste file by means of a blin interference avoiance-base elivery scheme. We characterize the corresponing achievable NDT as a function of connectivity egree an cache storage size µ. We show that the resulting NDT is within a constant multiplicative factor of 4 of the best known NDT with full caching propose in [3]. For the case when clou connectivity is enable, we present ifferent caching an elivery strategies as a function of cache sizes an fronthaul capacity. In the low an meium cache size regimes, if the fronthaul capacity is low, the propose scheme is shown to obtain a lower NDT than the scheme in [3] by means of clou transmission espite the fronthaul overhea. The remainer of the paper is organize as follows. We escribe the K-user F-RAN moel uner stuy in Section II. In Section III, we present the propose scheme in the absence of clou connectivity. Section IV iscusses the full F-RAN moel where clou processing is enable. We conclue the paper in Section V. II. SYSTEM MODEL AND PRELIMINARIES C F = r log P C F EN µnf EN 2 µnf T 3 =2 UE UE 2 Clou N Files C F C F EN 3 µnf UE 3 C F EN 4 µnf R 4 UE 4 EN 5 µnf UE 5 Fig. : A (5, 2) regular F-RAN moel. We consier a partially connecte K-user F-RAN network with one antenna at each EN an user an connectivity egree, as illustrate in Fig.. G represents the network topology, where G is efine by the sets (T, T 2,..., T K, R, R 2,..., R K ). Specifically, the set of ENs that are connecte to user UE j is enote as T j, an the set of users that are connecte to EN i is enote as R i. If each EN i transmits a symbol X i (t), the receive signal at user UE j at time instant t is Y j (t) = i T j H j,i (t)x i (t) + U j (t), () where H j,i (t) is the channel coefficient between EN i an UE j at time instant t. The channel coefficients are assume to be rawn from a continuous istribution, an are inepenent an ientically istribute (i.i..) across all t an k.

4 4 The aitive white Gaussian noise U j (t) is assume to have zero mean an unit variance an inepenent to all variables H j,i (t) an X i (t). Each EN is connecte to a clou processor through a fronthaul link of capacity C F bits per symbol of the wireless channel. Each EN has a local cache of size µnf bits, where µ [0, ] is the fractional cache size. We enote Z k as the cache content store at EN k uring the pre-fetching phase, an we focus on cases in which only intra-file coing is allowe. Accoringly, the cache content after the pre-fetching phase is given as Z k = (Z k,, Z k,2,..., Z k,n ), where Z k,n is a function of the n th file W n an each Z k,n is of size H(Z k,n ) µf, n =,..., N, k =,..., K. (2) At the start of the elivery phase, each user requests one file from the library, an the eman vector is enote as D (, 2,.., K ). We note that the cache content oes not epen on the user s emans. We focus on F-RANs that operate in serial fashion: once the eman vector is reveale, the clou first sens the ata through iniviual fronthaul links to the ENs, an then the ENs transmit signals through wireless channel (). The latency is then the sum of latency of fronthaul transmission an latency of wireless transmission. The block length of the channel coe is enote by T E an the receive block signal at UE j is Y T E j = H T E j,i XT E i + U T E j, (3) i T j where we impose the average power constraint T E the following worst-case constraint on the probability of error for any ɛ > 0, i.e., where Ŵ k max D E( XT E 2 ) P. Any feasible sequence of policies must satisfy i max Pr(W k k Ŵ k ) ɛ, (4) is the ecoe file at UE k. Conition (4) imposes that the probability of ecoing error across all users an over all possible eman vectors D be mae arbitrarily small as T E. Throughout the paper, we focus on a class of partially connecte regular ege channels as efine next. Definition. (Regular Network) In regular (K, ) symmetric partially connecte ege channel, the set of users that are connecte to EN i is efine as R i {i, i +,..., i + ( )}, for all i [K], an the set of ENs that are connecte to UE j is efine as T j {j +, j + 2,..., j}, for all j [K], where all inices are moulo K. To be specific, every EN i is connecte to its corresponing user an subsequent users in cyclic manner, where K. We note that, for a (K, ) regular network without clou connectivity, the fractional cache size µ must be at least / for reliable ecoing, i.e. / µ. In fact, when µ < / an clou connectivity is isable, a user can receive at most µf bits of a file from its connecte ENs. Therefore, since µf < F, some bits of a file cannot be receive to a user. In this paper, we use NDT as the performance metric, which is efine as follows [9].

5 5 Definition 2. (Delivery time per bit). For a given sequence of feasible policies, an achievable elivery time per bit (µ, C F,, P ) is given as (µ, C F,, P ) = lim F T F + T E, (5) F T with fronthaul an ege contributions given as E (µ, C F,, P ) = lim E T F F an F (µ, C F,, P ) = lim F F F, where T F is the uration of the fronthaul transmission an T E is the uration of the wireless transmission. Letting the fronthaul capacity C F scale as r log P with the power P, the parameter r can be viewe as pre-log of the fronthaul capacity. We then efine the NDT as follows. Definition 3. (Normalize Delivery Time). For any achievable (µ, C F,, P ), an given connectivity egree, the NDT is efine as δ (µ, r) = lim P (µ, r log P,, P ). (6) / log(p ) In aition, for any given pair of (µ, r) an a fixe, the minimum NDT is efine as δ (µ, r) = inf{δ (µ, r) : δ (µ, r) is achievable}. (7) Furthermore, we enote as δ E, (µ, r) = lim E (µ,r log P,,P ) P / log(p ) an similarly as δ F, (µ, r) = lim F (µ,r log P,,P ) P / log(p ) the achievable NDTs of wireless an fronthaul transmissions, respectively. The corresponing values achieve by an optimal scheme are efine as δe, (µ, r) an δ F, (µ, r). As for the efinition above, the NDT compares the elivery time per bit of the scheme of interest to that of an ieal interference-free system, which has the elivery time per bit of / log(p ) [9]. Hence, the NDT satisfies the inequality δ (µ, r). The minimum NDT δ (µ, r) is convex in µ for any fixe value of C F, as it can be prove by means of file-splitting an cache-sharing [9, Lemma ]. Notation: Throughout the paper, [K] {,..., K} an [N] {,..., N}. i, j, k an n represent the inex of the i th, j th, k th EN/user an the n th file. W n represents the n th message an W [N]\n represents all messages in the library but the n th message. III. CACHE-AIDED BLIND INTERFERENCE AVOIDANCE In this section, we present the propose achievable NDT for the F-RAN system uner stuy in the absence of fronthaul connections. Proposition. (Upper boun on minimum NDT) For a (K, ) regular network with no fronthauling an minimum storage, i.e., with r = 0 an µ = /, an upper boun on the minimum NDT is given as δe, (µ = ) 2(+) 2, 0 δe, ach, 2 =, =. (8) The NDT escribe in Proposition is achieve by means of an MDS coe caching scheme an a carefully esigne elivery scheme. For the caching scheme, each file is split into subfiles, an a (K, ) MDS coe is

6 6 applie to create K coe subfiles. Each EN stores one coe subfile for each file. This is one so that any user nees to obtain one subfile from each of its connecte ENs, to recover its requeste file. The elivery scheme is base on a novel blin interference avoiance scheme. The etails of the propose scheme are escribe in Appenix A, along with the proof of Proposition. As a general remark, the role of intra-file coe caching in obtaining the NDT in (8) shoul be emphasize. In fact, as iscusse in [9], in the presence of full connectivity an CSI, intra-file coe caching can only bring minor improvements in the NDT, which are limite to a factor of 2 for any F-RAN parameters. In contrast, with partial connectivity an no CSI, coe caching is instrumental in reucing the elivery latency. In fact, it can be seen that with µ = /, it is not possible to obtain a finite NDT with uncoe caching as long as the inequality K > hols, even with full CSI. As a benchmark, we now compare the NDT of the propose scheme to the NDT in [3]. This is the best known scheme for a (K, ) regular network with no CSI [3, Theorem 4]. For a (K, ) regular network with full storage, i.e., with µ =, the scheme achieves the NDT δ full E, = + 2, 2, =. The achievable scheme in [3] uses blin interference alignment with full EN cooperation. In orer to achieve the NDT in (9), the coherence time of the channel must be greater than +. Uner such an assumption, the ENs break each esire file into two uncoe subfiles, an esign precoing vectors jointly so that the interference only affects imensions out of + imensions at each user. The remaining two imensions are then reserve for the two uncoe esire subfiles. While the scheme in [3] assumes full caching, whereby each EN stores all files, we note here that, when using intra-file coe caching, the NDT in (9) can be achieve with a smaller cache storage. Specifically, consier a (K, 2) MDS coe caching scheme, where each file is split into two uncoe subfiles, so that K coe subfiles are create. Instea of storing the entire file, each EN stores only one coe subfile for each file. This reuces the require cache size to µ = /2, while still ensuring that the scheme in [3] can be implemente. Therefore, the scheme in [3] can, in fact, be applie for any value of µ between /2 an. Corollary. The propose scheme achieves an NDT that is within a multiplicative factor of 4 of the best known scheme with full caching [3], i.e., where inequality (a) hols for 2. δ ach E, δ full E, = 4 2 4( 2 + ) We next emonstrate the propose scheme through an example. = (9) (a) 4, (0) Example. Let us consier an (8, 2) regular network with µ = /2 (see Fig. 2). For the caching scheme, each file is split into two subfiles, so that eight coe subfiles are create. We now escribe the propose blin interference avoiance scheme for this example. Note that each user can ecoe the esire file with any two coe subfiles.

7 7 Fig. 2: Illustration of the propose scheme for a (8, 2) regular network. The goal is to eliver two coe subfiles to every user using as few time slots as possible. Denote the requeste file vector as D = (A, B, C, D, E, F, G, H) an with subscript i the inex of the coe subfile, so that EN i has available (A i, B i, C i, D i, E i, F i, G i, H i ). With reference to Fig. 2 for an illustration, for t =, EN transmits A to UE. File A is seen at UE 2 as interference. In orer to maximize the amount of users being serve in the same time slot, EN 2 transmits C 2 to UE 3. For UE 3 to ecoe C 2, EN 3 nees to be silent to avoi interference. Similarly, EN 4 sens D 4 to UE 4, an EN 5 sens F 5 to UE 6 while EN 6 stays silent. For t = 2, we shift the scheule ENs ownwar by one EN serving UE 2, UE 4, UE 5 an UE 7. The ENs are scheule in a similar way for t = 3 serving UE 3, UE 5, UE 6 an UE 8. However, for t = 4, EN 4 an EN 5 o not have new subfiles for UE 4 an UE 6. Therefore, they will stay silent, an only EN 7 an EN 8 are scheule. Similarly, for t = 5, only EN 8 an EN are scheule. A total of 6 subfiles were elivere in 5 time slots, yieling a sum DoF of 6/5 an an NDT of 5/2. IV. CACHE AND CLOUD-AIDED TOPOLOGICAL INTERFERENCE MANAGEMENT In this section, we present an achievable NDT in the presence of both fronthaul connections an EN caches. We consier either the propose scheme that achieves δe, ach in Proposition or the scheme in [3] achieving the NDT δ full E, in (9). The propose scheme requires a fractional cache capacity µ /, for values µ < /, we let the clou sen the remaining file fraction (/ µ) to the ENs in orer to enable the propose scheme. This yiels the achievable NDT δ ach (µ, r) = µ + δ ach r E,, () where the first term represents the fronthaul NDT, since each EN must receive a fraction (/ µ) for files. As for the scheme in [3], which requires µ /2, we similarly obtain for µ < /2 δ full (µ, r) = 2 since each EN must receive a fraction (/2 µ) for two files from the clou. 2 µ + δ full E, r, (2) Note that, for values / µ < /2, the cache size is large enough to implement the propose scheme without the ai of the clou. Hence, the achievable NDT is simply δ ach (µ, r) = δ ach. To apply the scheme in [3], we still nee the clou to sen the remaining file fraction to the ENs, an the achievable NDT δ full is given by (2). E,

8 8 Furthermore, for values µ /2, both schemes can be implemente without the ai of fronthaul transmission. The achievable NDTs are δ ach (µ, r) = δe, ach the following. an δfull Proposition 2. For an F-RAN with (K, ) regular ege topology, we have δ full r r = or r r 2 = (µ, r) = δ full E,, respectively. The main result of this section is (µ, r) δ ach (µ, r) when 2( 2)µ ( + )(4 2 ), an 0 < µ <, (3) 2( 2µ) ( + )(4 2 ), an µ < 2. (4) This result inicates that the propose scheme is particularly useful when the availability of fronthaul capacity is limite. Intuitively, this is the case since the propose scheme requires smaller values of the cache capacity µ. The proof of Proposition 2 can be foun in Appenix C. V. CONCLUSIONS In this paper, we focuse on cache an clou enable regular networks with partial wireless connections an no CSI. The main contribution of this work is the proposal of a novel cache-aie blin interference avoiance scheme. We showe that through intra-file coe caching, it is possible to obtain a finite NDT when the ege network is partially connecte, with no CSI an minimum require cache size at the ENs. We further showe that the propose scheme outperforms the state of the art when the network resources are limite, i.e., with low fronthaul capacity an/or low cache size. There are several interesting irections for future work. A first irection woul be to obtain lower bouns on the minimum NDT an subsequently characterizing the optimal NDT in the presence of partial connectivity an no CSI. A secon interesting irection woul be to generalize the ieas presente herein for arbitrary (irregular) network topologies. A. Proof of Proposition APPENDIX In orer to prove the achievability of the propose scheme, we first present the MDS coe caching scheme, followe by the interference avoiance-base elivery scheme. For the (K, ) MDS coe caching strategy, each file W n is split into subfiles, i.e., W n = (W n (), W n (2),..., W n () ), where W (i) n inicates the i th subfile of file W n. We create K coe subfiles out of these subfiles. Each coe subfile is of size F/ bits, an each EN stores one coe subfile per file to meet the storage constraint NF/ bits. Note that each UE can recover the esire file from any coe subfiles. We exploit this fact in the elivery scheme escribe next. For elivery, we propose an interference avoiance-base strategy. Note that each user is connecte to ENs, an each one of the ENs has an unique coe subfile for that particular user. We efine these unique subfiles as types.

9 9 Fig. 3: Subfile types. Fig. 4: A (7, 2) regular network with two blocks. Definition 4. (Subfile Type). For any (K, ) regular network with the propose caching scheme, there are types of possible subfiles τ {, 2,..., }. We efine the subfile for any UE k from EN k as type, the subfile for any UE k from EN k as type 2, an so on. Consequently, the subfile for UE k from EN k + is type. Each type represents a istinct coe subfile. We then exploit the topology consiere in this paper to simultaneously scheule as many ENs an users as possible, an eventually ensuring that all subfile types are receive at all users. We briefly illustrate the notion of subfile type an blin interference avoiance through an example. Example 2. For a (7, 2) regular network (see Fig. 4), each file is split into two subfiles, so that seven coe subfiles are create. Each EN stores one istinct coe subfile per file. To avoi interference, EN is scheule to sen type subfile to UE, an EN 2 is scheule to sen type 2 subfile to UE 3 while EN 3 is scheule to be silent. Similarly, EN 4 is scheule to sen type subfile to UE 4, an EN 5 is scheule to sen type 2 subfile to UE 6 while EN 6 is scheule to be silent. EN 7 has to be silent to avoi interference at UE (complete examples will be shown later in this section). Clearly, in this example, three consecutive ENs nee to be scheule jointly to maximize the amount of users being serve simultaneously. We enote these ENs as forming a block as efine next. Definition 5. (Block). A block refers to a group of ENs an users that are scheule together to avoi interference at the intene users. Each block consists of + EN an user pairs. There are at most K + blocks. As shown in the previous example, two subfile types are elivere within each block to two ifferent users. We efine a series of transmissions that ensures every user to get those two subfile types as a stage. Definition 6. (Stage) A series of transmissions at each Stage s, where s {, 2,..., 2 }, focuses on sening subfile types s an s + to all users.

10 0 Fig. 5: A general view of stages an phases, an their respective urations. To eliver type s an type s + subfiles to all the user, we can shift blocks in a cyclic manner. However, if we simply shift the blocks + number of times, we will eventually arrive at a point where ENs have no new subfiles for most of users an sen reunant subfiles. Thus, we break each stage into two phases to avoi such a scenario (see Fig. 5). (a) Phase : At t-th time slot of Phase of Stage s, we scheule EN i to eliver type s subfile to user UE i+(s ), an scheule EN i+s to eliver type s+ subfile to user UE i+, where i = {t, t+(+),..., t+( K + )(+)}. A total of + time slots are neee for Phase, i.e. t {,..., ( + )}, because once it reaches the ( + 2)-th time slot, most of ENs o not have new coe subfiles about the esire files for intene users. (b) Phase 2: When K is not ivisible by +, at the en of Phase of any Stage s, UE i, where i = {,,..., (K mo ( + )) + }, will only receive type s subfiles. Last (K mo ( + )) users will only receive type s + subfiles, i.e. UE i, where i = {K (K mo ( + )) +,..., K}. Therefore, at t-th time slot of Stage s, where t > +, we scheule EN i to eliver type s subfiles to UE i+(s ), an scheule EN i+s to eliver type s + subfiles to user UE i+, where i = {t + ( K + )( + )}. The uration of Phase 2 is (K mo ( + )) time slots, which oes not excee + time slots. Hence, any stage will take at most 2( + ) time slots. The number of stages can be easily obtaine by counting how many subfile types can be paire, i.e., 2. In sum, K subfiles can be sent in 2 stages, an each stage occupies at most 2( + ) time slots. This gives a lower boun on the maximum sum DoF of K/2( + ) 2 an yiel an upper boun on the minimum NDT of 2( + ) 2 /. This completes the proof of Proposition. One might notice that after Stage 2, the range of users suffers from interference wien. Users will start seeing interference from the previous block. Lemma shows that this effect oes not cause interference at intene users. Lemma. (Impact of interference on intene users). As long as the subfile types elivere at the same time sum up to +, the intene users will not suffer from the interference coming from the preceing block. We next present an example with two stages to show that Lemma is inee true an efer the proof of Lemma to Appenix B.

11 Fig. 6: A block of ENs an users, an its interference to the next block. Soli lines represent esire link an ash lines represent interference, with inices on the sie an i = j. Example 3. For a (, 4) regular network, there are two blocks of size + = 5 an 2 stages. At Stage, type an type 4 subfiles will be serve to all users. During Stage Phase at t = (see Fig. 7), EN sens A (type subfile) to UE, EN 2 sens E 2 (type 4 subfile) to UE 5, EN 6 sens F 6 (type subfile) to UE 6, an EN 7 sens I 7 (type 4 subfile) to UE 0. Notice that at the en of t =, UE an UE 6 receive type subfiles, UE 5 an UE 0 receive type 4 subfiles. At t = 2, EN 2, EN 3, EN 7, EN 8 transmit type an type 4 subfiles to user UE 2, UE 6, UE 7, UE in a similar fashion. Same form of transmissions are repeate for 5 time slots in orer to cover type an type 4 subfiles for most of users. When t = 5, UE 4 has not receive its type 4 subfile an UE has not receive its type subfile. Therefore, we enter Stage Phase 2, where EN, EN sen D an K to UE 4 an UE, respectively. Phase 2 takes time slot. Stage takes a total of 6 time slots. We then move onto Stage 2 Phase (see Fig. 8), in which we sen type 2 an type 3 subfiles together. Note that we no longer scheule two ajacent ENs now. At t = 7, EN transmits B (type 2 subfile) to UE 2, EN 3 transmits E 3 (type 3 subfile) to UE 5. Same form of transmissions is again repeate for 5 time slots. In Phase 2, D 2 an A are sent by EN 2, EN to UE 4 an UE, respectively. Therefore, we sen a total of 44 subfiles in 2 time slots an achieve a sum DoF of /3 an an NDT of 3. B. Proof of Lemma Since every time slot is just a shifte version of the previous time slot an ENs are scheule in the same way in each block, we focus on just one single time slot an any two consecutive blocks. During Phase of any Stage s, we sen type s an type s + subfiles, where the sum of them is +. ENs that will be scheule are EN i, EN i+s, EN i+(+) an EN i+s+(+). However, since we are intereste in the impact of interference between any two consecutive blocks, we focus on EN i+s an EN i+(+) only. The receiver sets of EN i+s, EN i+(+) are R i+s = {j + s,..., j + s + ( )} an R i+(+) = {j + ( + ),..., j }. Their respective intene users are UE j+ an UE j+s+.

12 2 Fig. 7: Ege transmission in Stage for a (, 4) regular network with cache size µ = /4. Fig. 8: Ege transmission in Stage 2 for a (, 4) regular network with cache size µ = /4. As shown in Fig. 6, when s 2, R i+s overlaps with R i+(+). Users who see signals from both ENs are R i+s R i+(+) = {j + ( + ),..., j + s + ( )}. It can be seen that the intene user of EN i+(+), UE j+s+, is not in the set. Thus, we conclue that the intene user in the following block will not see interference.

13 3 Fig. 9: Choice of caching an elivery schemes for any (K, ) regular network. (Left) µ [0, /). (Right) µ [/, /2). C. Proof of Proposition 2 Recall that in Section IV, we have the following results. For values µ < /, the achievable NDTs of the propose scheme an the scheme in [3], both with fronthaul connections, are δ ach (µ, r) = µ + δ ach r E,, δ full (µ, r) = 2 2 µ + δ full E, r, (5) respectively. To show that the propose scheme is a better choice when the fronthaul capacity is low, we obtain a threshol r on the fronthaul capacity by comparing δ ach µ + 2( + ) 2 r 2( + ) ( + )(4 2 ) 2 (µ, r) to δ full (µ, r) as follows. δ ach (µ, r) δ full (µ, r) (6) 2 2 µ + + r 2 2µ µ r r ( 2)µ (7) (8) r (9) 2( 2)µ r ( + )(4 2 ) = r. (20) We conclue that the propose scheme yiels a lower NDT when the fronthaul capacity is lower or equal to r for µ < /. Similarly, for values / µ < /2, the achievable NDTs of the propse scheme an the scheme in [3] are δ ach (µ, r) = δe,, ach δ full (µ, r) = 2 2 µ + δ full E, r, (2) respectively. We obtain a threshol r 2 on the fronthaul capacity for this cache size regime by comparing δ ach (µ, r) to δ full (µ, r). δ ach (µ, r) δ full (µ, r) (22) 2( + ) r 2 (23) 2( + ) 2 + 2µ 2 r (24) ( + )(4 2 ) 2µ 2 r (25) 2( 2µ) r ( + )(4 2 ) = r 2. (26) It can be seen that a lower NDT can be achieve with the propose scheme when r r 2 for / µ < /2.

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