Content Delivery in Erasure Broadcast. Channels with Cache and Feedback

Transcription

1 Content Delivery in Erasure Broadcast Channels with Cache and Feedack Asma Ghorel, Mari Koayashi, and Sheng Yang LSS, CentraleSupélec Gif-sur-Yvette, France arxiv: v [cs.t] 5 Fe 206 {asma.ghorel, mari.koayashi, sheng.yang}@centralesupelec.fr Astract We study a content delivery prolem in a K-user erasure roadcast channel such that a content providing server wishes to deliver requested files to users, each equipped with a cache of a finite memory. Assuming that the transmitter has state feedack and user caches can e filled during off-peak hours relialy y the decentralized content placement, we characterize the achievale rate region as a function of the memory sizes and the erasure proailities. The proposed delivery scheme, ased on the roadcasting scheme y Wang and Gatzianas et al., exploits the receiver side information estalished during the placement phase. Our results can e extended to centralized content placement as well as multi-antenna roadcast channels with state feedack.. NTRODUCTON Today s exponentially growing moile data traffic is mainly due to video applications such as content-ased video streaming. The skewness of the video traffic together with the ever-growing cheap on-oard storage memory suggests that the quality of experience can e oosted y caching popular contents at (or close to the end-users in wireless networks. A numer of recent works have studied such concept under different models and assumptions (see [], [2] and references therein. Most of existing works assume that caching is performed in two phases: placement phase to prefetch users caches under their memory constraints (typically during off-peak hours prior to the actual demands; delivery phase to transmit codewords such that each user, ased on the received signal and the contents of its cache, is ale to decode the requested file. n this work, we study the delivery phase ased on a coded caching model where a server is connected to many users, each equipped with a cache of finite memory []. By carefully choosing the su-files to e distriuted across users, coded caching exploits opportunistic multicasting such that a common signal is simultaneously useful for all users even with distinct file requests. A numer of extensions of coded caching have een developed (see e.g. [, Section V]. These include the decentralized content placement [5], online coded caching [7], non-uniform popularities [4], [6], more general networks such as device-to-device (D2D enaled network [3], hierarchical networks [22], heterogeneous networks [0], as well as the performance analysis in different regimes [9], [20]. Further, very recent works have attempted to relax the unrealistic assumption

2 2 of a perfect shared link y replacing it y wireless channels (e.g. [8], [9], [8], [25], [26]. f wireless channels are used only to multicast a common signal, naturally the performance of coded caching (delivery phase is limited y the user in the worst condition of fading channels as oserved in [8]. This is due to the information theoretic limit, that is, the multicasting rate is determined y the worst user [24, Chapter 7.2]. However, if the underlying wireless channels enoy some degrees of freedom to convey simultaneously oth private messages and common messages, the delivery phase of coded caching can e further enhanced. n the context of multi-antenna roadcast channel and erasure roadcast channel, the potential gain of coded caching in the presence of channel state feedack has een demonstrated [8], [25], [26]. The key oservation ehind [8], [26] is that opportunistic multicasting can e performed ased on either the receiver side information estalished during the placement phase or the channel state information acquired via feedack. n this work, we model the ottleneck link etween the server with N files and K users equipped with a cache of a finite memory as an erasure roadcast channel (EBC. The simple EBC captures the essential features of wireless channels such as random failure or disconnection of any server-user link that a packet transmission may experience during high-traffic hours, i.e. during the delivery phase. n this work, we consider a memoryless EBC in which erasure is independent across users with proailities {δ k } and each user k can cache up to M k files. Moreover, the server is assumed to acquire the channel states causally via feedack sent y the users. Assuming that users fill the caches randomly and independently according to the decentralized content placement scheme as proposed in [5], we study the achievale rate region of the EBC with cache and state feedack. Our main contriution is the characterization of the rate region in the cache-enaled EBC with state feedack for the case of the decentralized content placement (Theorem. The converse proof uilds on the genie-aided ounds exploiting two key lemmas, i.e. a generalized form of the entropy inequalities (Lemma as well as the reduced entropy of messages in the presence of receiver side information (Lemma 2. For the achievaility, we present a multi-phase delivery scheme extending the algorithm proposed independently y Wang [5] and y Gatzianas et al. [6] to the case with receiver side information and prove that it achieves the optimal rate region for special cases of interest. We provide, as a yproduct of the achievaility proof for the symmetric network, an alternative proof for the sum capacity of the EBC with state feedack and without cache. More specifically, we characterize the order- capacity defined as the maximum transmission rate of a message intended to users and express the sum capacity in a convenient manner along the line of [4]. This allows us to characterize the rate region of the symmetric cache-enaled EBC with state feedack easily, since as such all we need is to incorporate the packets generated during the placement phase [8]. However, such proof exploits the specific structure of the rate region of symmetric networks, and unfortunately cannot e applied to a general network setting considered here. Our current work provides a non-trivial extension of [8] to such networks. Furthermore, we show that our results can e extended in a straightforward manner to the centralized content placement [] as well as the multi-antenna roadcast channel with state feedack. Finally, we provide some numerical examples to quantify the enefit of state feedack, the relative merit of the centralized caching to the decentralized counterpart, as well as the gain due to the optimization of memory sizes, as a function of other system parameters.

3 3 Fig.. A cached-enaled erasure roadcast channel with K 3. The rest of the paper is organized as follows. n section, we descrie the system model together with some definitions and then summarize the main results. Section gives the converse proof of the achievale rate region of the cache-enaled EBC with state feedack. After a high-level description of the well-known algorithm y Wang and Gatzianas et al. in section V, section V presents our proposed delivery scheme and provides the achievaility proof for some special cases of interest. Section V provides the extensions of the previous results and section V shows some numerical examples. Throughout the paper, we use the following notational conventions. The superscript notation X n represents a sequence (X,..., X n of variales. X is used to denote the set of variales {X i } i. The entropy of X is denoted y H(X. We let [k] {,..., k}. We let ɛ n denote a constant which vanishes as n, i.e. lim n ɛ n 0.. SYSTEM MODEL AND MAN RESULTS A. System model and definitions We consider a cache-enaled network depicted in Fig. where a server is connected to K users through an erasure roadcast channel (EBC. The server has an access to N files W,..., W N where the i-th file W i consists of F i packets of L its each (F i L its. Each user k has a cache memory Z k of M k F packets for M k [0, N], where F N N i F i is the average size of the files. Under such a setting, consider a discrete time communication system where a packet is sent in each slot over the K-user EBC. The channel input X k F q elongs to the input alphaet of size L log 2 q its. The erasure is assumed to e memoryless and independently distriuted across users so that in a given slot we have K Pr(Y, Y 2,..., Y K X Pr(Y k X ( δ k, Y k X, Pr(Y k X δ k, Y k E where Y k denotes the channel output of receiver k, E stands for an erased output, δ k denotes the erasure proaility of user k. We let S l S 2 {,...,K} denote the state of the channel in slot l and indicate the set of users who (2

4 4 received correctly the packet. We assume that all the receivers know instantaneously S l, and that through feedack the transmitter only knows the past states S l during slot l. The caching network is operated in two phases: the placement phase and the delivery phase. n the content placement phase, the server fills the caches of all users, Z,..., Z K, up to the memory constraint. As in most works in the literature, we assume that the placement phase incurs no error and no cost, since it takes place usually during off-peak traffic hours. Once each user k makes a request d k, the server sends the codewords so that each user can decode its requested file as a function of its cache contents and received signals during the delivery phase. We provide a more formal definition elow. A (M,..., M K, F d,..., F dk, n caching scheme consists of the following components. N message files W,..., W N independently and uniformly distriuted over W W N with W i F Fi q for all i. K caching functions defined y φ k : F N i Fi q F F M k q contents that map the files W,..., W N into user k s cache Z k φ k (W,..., W N, k [K]. (3 A sequence of encoding functions which transmit at slot l a symol X l f l (W d,..., W dk, S l F q, ased on the requested files and the state feedack up to slot l for l,..., n, where W dk message file requested y user k for d k {,..., N}. K decoding functions defined y ψ k : F n q F F M k q denotes the S n F F d k q, k [K], that decode the file Ŵd k ψ k (Y n k, Z k, S n as a function of the received signals Y n k, the cache content Z k, as well as the state information S n. A rate tuple (R,..., R K is said to e achievale if, for every ɛ > 0, there exists a (M,..., M K, F d,..., F dk, n caching strategy that satisfies the reliaility condition as well as the rate condition max max Pr(ψ k (Y (d,...,d K {,...,N} K k n, Z k, S n W dk < ɛ k R k < F d k n k [K]. (4 Throughout the paper, we express for revity the entropy and the rate in terms of packets in oder to avoid the constant factor L log 2 q. B. Decentralized content placement We mainly focus on the decentralized content placement proposed in [5]. Under the memory constraint of M k F packets, each user k independently caches a suset of p k F i packets of file i, chosen uniformly at random for i,..., N, where p k M k N. By letting L (W i denote the su-file of W i stored exclusively y the users in, the cache memory of user k after the decentralized placement is given y Z k {L (W i : [K], k, i,..., N}. (5

5 5 The size of each su-file is given y L (W i p [K]\ ( p F i + ɛ Fi (6 as F i. t can e easily verified that the memory constraint of each user is fulfilled, namely, Z k N i :k N N L (W i (F i p k + ɛ Fi M k F + ɛ Fi (7 as F i for all i. Throughout the paper, we assume that F and meanwhile Fi F F i > 0. Thus, we identify all ɛ Fi with a single ɛ F. i i converges to some constant To illustrate the placement strategy, let us consider an example of K 3 users. After the placement phase, each file will e partitioned into 8 su-files: W i {L (W i, L (W i, L 2 (W i, L 3 (W i, L 2 (W i, L 3 (W i, L 23 (W i, L 23 (W i }. (8 Oviously, the su-files received y the destination, e.g. L (W, L 2 (W, L 3 (W, L 23 (W for user requesting W, need not e transmitted in the delivery phase. C. Main results n order to present the main results, we specify two special cases. Definition. The cache-enaled EBC (or the network is symmetric if the erasure proailities as well as the memory sizes are the same for all users, i.e. δ δ K δ, M M K M, p p K p. Definition 2. The rate vector is said to e one-sided fair in the cache-enaled EBC if δ k δ and for k implies p k R k p R, and (9 p k p δ k R k δ R. (0 For the special case where p k 0, k [K], it is reduced to δ k R k δ R which coincides with the one-sided fairness originally defined in [5]. Focusing on the case of most interest with N K and K distinct demands, we present the following main results of this work. Theorem. For K 3, or for the symmetric network with K 3, or for the one-sided fair rate vector with K > 3, the achievale rate region of the cached-enaled EBC with the state feedack under the decentralized content placement is given y for any permutation π of {,..., K}. K k ( p π k δ π R πk (

6 6 Fig. 2. A two-user rate region with (p, p 2 ( 3, 2 3, (δ, δ 2 ( 4, 2. The aove region has a polyhedron structure determined y K! inequalities in general. t should e remarked that Theorem covers some existing results. For the symmetric network, the aove region simplifies to [8] K ( p k δ k R π k, π. (2 For the case without cache memory, i.e. p k 0 for all k, Theorem oils down to the capacity region of the EBC with state feedack [5], [6] given y K k δ π R πk, π (3 which is achievale for K 3 or the symmetric network or the one-sided fair rate vector where δ k δ implies δ k R k δ R for any k. Comparing ( and (3, we immediately see that the presence of cache memories decreases the weights in the weighted rate sum and thus enlarges the rate region. n order to gain some further insight, Fig. 2 illustrates a toy example of two users with (p, p 2 ( 3, 2 3 and (δ, δ 2 ( 4, 2. According to Theorem, the rate region is given y which is characterized y three vertices ( R R R R 2 (4 63, 0 (0.78,.20, and (0, 6. The vertex (0.78,.20, achieving the sum rate of.98, corresponds to the case when the requested files satisfy the ratio F d2 /F d 20/3. On the other

7 7 hand, the region of the EBC without cache is given y 4 3 R R R + 2R 2 (5 which is characterized y three vertices ( 3 4, 0, (0.63, 0.4, (0, 2. The sum capacity of 0.77 is achievale for the ratio R 2 /R 2/9. The gain due to the cache is highlighted even in this toy example. Theorem yields the following corollary. Corollary. For K 3, or for the symmetric network with K 3, or for the one-sided fair rate vector with K > 3, the transmission length to deliver requested files to useres in the cached-enaled EBC under the decentralized content placement is given y as F. T tot max π { K k ( p π k δ π F dπk } + ɛ F, (6 The corollary covers some existing results in the literature. For the symmetric network with files of equal size (F i F, i, the transmission length simplifies to T tot K ( p k δ k F + ɛ F, (7 as F [8]. For the case with files of equal size and without erasure, the transmission length in Corollary normalized y F coincides with the rate-memory tradeoff under the decentralized content placement for asymmetric memory sizes [7] given y T tot F K k ( p, (8 where the maximum over all permutations is chosen to e identity y assuming p p K. f additionally we restrict ourselves to the case with caches of equal size, we recover the rate-memory tradeoff given in [5] T tot F N ( M { ( M } K. (9 M N N n fact, the aove expression readily follows y applying the geometric series to the RHS of (8.. CONVERSE n this section, we prove the converse of Theorem. First we provide two useful lemmas. The first one is a generalized form of the entropy inequality, while the second one is a simple relation of the message entropy in the presence of receiver side information. Although the former has een proved in [8], we restate it for the sake of completeness. here. n [5] and all follow-up works, the rate is defined as the numer of files to deliver over the shared link, which corresponds to our T tot

8 8 Lemma. [2, Lemma 5] For the erasure roadcast channel, if U is such that X l UY l S l (S l+,..., S n,, δ for any sets, such that {,..., K}. Proof: We have, for, H(Y n U, S n δ H(Y n U, S n (20 H(Y n U, S n (2 n H(Y,l Y l, U, S n (22 l n l n l l H(Y,l Y l, U, S l, S l (23 Pr{S l } H(X l Y l, U, S l, S l (24 n ( δ i H(Xl Y l, U, S l (25 ( i i δ i n l H(X l Y l, U, S l (26 where the first equality is from the chain rule; the second equality is ecause the current input does not depend on future states conditioned on the past outputs/states and U; the third one holds since Y,l is deterministic and has entropy 0 when all outputs in are erased (S l ; the fourth equality is from the independence etween X l and S l ; and we get the last inequality y removing the terms Y l \ same steps, we have from which and (26, we otain (20. H(Y n U, S n ( i δ i n l in the condition of the entropy. Following the H(X l Y l, U, S l, (27 Lemma 2. Under the decentralized content placement [5], the following inequality hold for any i and K [K] H(W i {Z k } k K k K ( p k H(W i.

9 9 Proof: Under the decentralized content placement, we have H(W i {Z k } k K H(W i {L (W l } K, l,...,n (28 H(W i {L (W i } K (29 H({L (W i } K (30 H(L (W i (3 : [K]\K : [K]\K H(L (W i L (32 where the first equality follows from (5; the second equality follows due to the independence etween messages W,, W N ; the third equality follows y identifying the unknown parts of W i given the cache memories of K and using the independence of all su-files; (3 is again from the independence of the su-files. Note that L is a random variale indicating which suset of packets of file W i are shared y the users in. The size of the random ( suset L follows thus the inomial distriution B H(W i, p k [K]\ ( p k. t is readily shown that H(L (W i L E{ L }. This implies that H(W i {Z k } k K : [K]\K k K( p k p k [K]\ : [K]\K ( p k H(W i (33 p k [K]\K\ ( p k H(W i (34 k K ( p k H(W i (35 where the last inequality is otained from the asic property that we have M p k M\ ( p k for a suset M [K] \ K. We apply genie-aided ounds to create a degraded erasure roadcast channel y providing the messages, the channel outputs, as well as the receiver side information (contents of cache memories to the enhanced receivers. Without loss of generality, we focus on the case without permutation and the demand (d,..., d K (,..., K. n k ( p R k k ( p H(W k (36 H(W k Z k S n (37 (W k ; Y n [k] Zk S n + nɛ n,k (38 (W k ; Y n [k], W k Z k S n + nɛ n,k (39 (W k ; Y n [k] W k Z k S n + nɛ n,k (40 where the second inequality is y applying Lemma 2 and noting that S n is independent of others; (38 is from Fano s inequality; the last equality is from (W k ; W k Z k S n 0 since the caches Z k only store disoint pieces of individual files y the decentralized content placement [5]. Putting all the rate constraints together, and defining

10 0 ɛ n,k ɛ n,k / k ( p, we have n n( p (R ɛ n, H(Y n Z S n H(Y n W Z S n. K ( p (R K ɛ n,k H(Y[K] n W K Z K S n H(Y[K] n W K Z K S n. (4 We now sum up the aove inequalities with different weights, and apply K times Lemma, namely, for k,..., K, H(Y n [k+] W k Z k+ S n [k+] δ H(Y n [k+] W k Z k S n [k+] δ (42 H(Y [k] n W k Z k S n [k] δ, (43 where the first inequality follows ecause removing conditioning increases entropy. Finally, we have K [k] ( p [k] δ (R k ɛ n which estalishes the converse proof. H(Y n Z S n n( δ H(Y n [K] W K Z K S n n( [k] δ (44 H(Y n n( δ (45 V. BROADCASTNG WTHOUT RECEVER SDE NFORMATON n this section, we first revisit the algorithm proposed in [5], [6] achieving the capacity region of the EBC with state feedack for some cases of interest, as an important uilding lock of our proposed scheme. Then, we provide an alternative achievaility proof for the symmetric channel with uniform erasure proailities across users. A. Revisiting the algorithm y Wang and Gatzianas et al. We recall the capacity region of the EBC with state feedack as elow. Theorem 2 ( [5], [6]. For K 3, or for the symmetric channel with K 3, or for the one-sided fair rate vector 2 with K > 3, the capacity region of the erasure roadcast channel with state feedack is given y K k δ R πk, π. (46 π We provide a high-level description of the roadcasting scheme [5], [6] which is optimal under the special cases as specified in the aove theorem. We recall that the numer of private packets {F k } is assumed to e aritrarily large so that the length of each phase ecomes deterministic. Thus, we drop the ɛ F term wherever confusion is not 2 δ k δ implies δ k R k δ R for any k.

11 proale. The roadcasting algorithm has two main roles: roadcast new information packets and 2 multicast side information or overheard packets ased on state feedack. Therefore, we can call phase roadcasting phase and phases 2 to K multicasting phase. Phase consists of ( K su-phases in each of which the transmitter sends packets intended to a suset of users for. Similarly to the receiver side information otained after the placement phase, we let L (V K denote the part of packet V K received y users in and erased at users in [K] \. Here is a high-level description of the roadcasting algorithm: Broadcasting phase (phase : send each message V k W k of F k packets sequentially for k,..., K. This phase generates overheard symols {L (V k } to e transmitted via linear comination in multicasting phase, where [K] \ k for all k. 2 Multicasting phase (phases 2 K: for a suset of users, generate V as a linear comination of overheard packets such that V F ( {L\ (V } :, (47 where F denotes a linear function. Send V sequentially for all [K] of the cardinality 2,..., K. The achievaility result of Theorem 2 implies the following corollary. Corollary 2. For K 3, or for the symmetric channel with K > 3, or for the one-sided fair rate vector with K > 3, the total transmission length to convey W,..., W K to users,..., K, respectively, is given y K F πk T tot k δ + ɛ F. π The proof is omitted ecause the proof in section V-B covers the case without user memories. TABLE NOTATONS FOR THE ERASURE BROADCAST CHANNEL. R k t t {k} V K L (V K N {k} Message rate for user k Length of su-phase Length needed y user k for su-phase Packets intended to users in K Part of packets V K received y users in and erased at users in [K] \ Numer of packets useful for user k generated in su-phase and to e sent in su-phase n order to calculate the total transmission length of the algorithm, we need to introduce further some notations and parameters (Tale which are explained as follows. A packet intended to is consumed for a given user k, if this user or at least one user in [K] \ receives it. The proaility of such event is equal to [K]\ {k} δ. A packet intended to ecomes a packet intended to and useful for user k [K], if erased at user k and all users in [K] \ ut received y \. The numer of packets useful for user k generated in

12 2 Fig. 3. Phase organization for K 3 and packet evolution viewed y user. su-phase and to e sent in su-phase, denoted y N {k}, is then given y N {k} t{k} δ ( δ (48 [K]\ {k} \ where t {k} N {k} as denotes the length of su-phase viewed y user k to e defined shortly. We can also express N {k} \k L \ (V {k}, (49 where we let V {k} denotes the part of V required for user k. The duration t of su-phase is given y where t max k t{k}, (50 k t {k} N {k} [K]\ {k} δ. (5 The total transmission length is given y summing up all su-phases, i.e. T tot [K] t. Fig. 3 illustrates the phase organization for K 3 and the packet evolution viewed y user. The packets intended to {, 2, 3} are created from oth phases and 2. More precisely, su-phase {} creates L 23 (V to e sent in phase 3 if erased at user and received y others (ERR. The numer of such packets is N {} 23. Su-phase {, 2} creates L 3 (V 2, L 23 (V 2 if erased at user ut received y user 3 (EXR, while su-phase {, 3} creates L 2 (V 3, L 23 (V 3 if erased at user and received y user 2 (ERX. The total numer of packets intended to {, 2, 3} generated in phase 2 and required y user is N {} N {} B. Achievaility in the symmetric channel We focus now on the special case of the symmetric channel with uniform erasure proailities, i.e. δ k δ for all k. n this case, the capacity region of the EBC with state feedack in (46 simplifies to K δ k R π k, π. (52

13 3 t readily follows that the capacity region yields the symmetric capacity, i.e. R R K R sym (K, given y R sym (K K. (53 δ k n the following, we provide an alternative proof of the achievaility of the symmetric capacity. Notice that other vertices of the capacity region can e characterized similarly as proved in susection V-C. Our proof follows the footsteps of [4] and uses the notion of order- packets. Let us define message set {W } independently and uniformly distriuted over {W } for all [K]. For with the cardinality, the message set {W } are called order- messages. We define R an achievale rate of the message W and define the sum rate of order- messages as R (K : R ( K R. (54 The supremum of R (K is called the sum capacity of order- messages. We characterize the sum capacity of order- messages, in the erasure roadcast channel with state feedack in the following theorem. Theorem 3. n the K-user erasure roadcast channel with state feedack, the sum capacity of order- packets is upper ounded y R (K ( K K + The algorithms in [5], [6] achieve the RHS with equality. ( K k δ k,,..., K. (55 Proof: We first provide the converse proof. Similarly to section, we uild on genie-aided ounds together with Lemma. Let us assume that the transmitter wishes to convey the message W to a suset of users {,..., K}, and receiver k wishes to decode all messages W k {W } : k for,..., K. n order to create a degraded roadcast channel, we assume that receiver k provides the message set Wk and the channel output Y n k to receivers k + to K for k,..., K, Under this setting and using Fano s inequality, we have for receiver : n R ɛ n, H(Y n S n H(Y n W S n. (56 [K] For receiver k 2,..., K, we have: n R ɛ n,k H(Y n... Yk n W k S n H(Y n... Yk n W k S n, (57 k {k,...,k} where we used W k \ W k {W } :\{k,...,k} in the LHS. Summing up the aove inequalities and applying Lemma K times, we readily otain: K k {k,...,k} (R ɛ n,k δ k H(Y n S n n( δ (58. (59

14 4 TABLE NOTATONS FOR THE SYMMETRC CHANNEL. W R R (K t t t i N i N i N i i Message intended to users in Rate of W Sum rate of order- messages Length of any su-phase in phase Length of any su-phase when starting from phase i Numer of packets created in su-phase and to e sent in su-phase for any of cardinality i < N i when starting from phase i for i i We further impose the symmetric rate condition such that R R for any with the same cardinality. By focusing on of the same cardinality in (58 and noticing that there are ( K k such suset, R is upper ounded y This estalishes the converse part. R K + ( K k δ k,,. (60 n order to prove the achievaility of R i (K in Theorem 3, we apply the roadcasting algorithm of [5], [6] from phase i > y sending N i packets to each suset [K] with i. First, we redefine some parameters y taking into account the symmetry across users as summarized in Tale. Due to the symmetry, we drop the user index k in t {k}, N {k} and replace them y t, N i, respectively for [K] with i,. Now, we introduce variants of these notations to reflect the fact that the algorithm starts from phase i >, rather than from phase. The length of any su-phase in phase when starting the algorithm from phase i, denoted y t i, is given y where t i δ K + li ( N i l l, > i, (6 N i l t i lδ K + ( δ l (62 denotes the numer of order- packets generated during a given su-phase in phase i, again starting from phase i. For i, we have t i i N i. (63 δk i+ By counting the total numer of order-i packets and the transmission length from phase i to phase K, the sum rate of order-i messages achieved y the algorithm [5], [6] is given y R i (K ( K i Ni K i ( K t i, i. (64

15 5 t remains to prove that Ri (K coincides with the RHS expression of (55. We notice that the transmission length from phase to K can e expressed in the following different way, i.e. K ( K K t i U i, (65 where we let U i i li i ( t i l l, i. (66 By following similar steps as [6, Appendix C], we otain the recursive equations given y U i i ( δ K + ( l+ ( δ K +l+ U l i (67 l for > i. Since we have U i i ti i N i δ K i+ l and using the equality ( c ( c i ( i ( i c and the inomial theorem n ( n k0 k x k y n k (x + y n, it readily follows that we have ( U i N i δ K +, i. (68 i By plugging the last expression into (64 using (65, we have R i (K ( K i Ni K N i ( i δ K + i ( K i K i+ (69 ( K k i δ k (70 which coincides the RHS of (55 for i,..., K. This estalishes the achievaility proof. As a corollary of Theorem 3, we provide an alternative expression for the sum capacity. Corollary 3. The sum capacity of the K-user symmetric roadcast erasure channel with state feedack can e expressed as a function of R 2 (K,..., R K (K y where KN δ K phase. R (K KN KN + K ( K i N i δ K i2 R i (K, (7 is the duration of phase, ( K N corresponds to the total numer of order- packets generated in Proof: By letting f denote the RHS of (7, we wish to prove the equality f R (K K K δ k proving f R (K. f it is true, from the achievaility proof of Theorem 3 that proves R i R i for all i, the proof is complete. n the RHS of (7, we replace R i y the expression R i in (64 y letting N i N i for i 2. Then, we have f KN KN + K K ( K δ K i2 i t i (72 KN KN + K ( K. δ K 2 i2 ti (73 y

16 6 Comparing the desired equality f R (K KN K ( K t we immediately see that it remains to prove the following equality. t with the aove expression and noticing that KN δ K Kt, t i 2. (74 We prove this relation recursively. For 2, the aove equality follows from (6 and (63. δ K + l t 2 i2 N 2 δ K t2 2. (75 Now suppose that (74 holds for l 2,..., and we prove it for. From (6 we have ( t l δ K + δ K + δ K + δ K + [ N + N l (76 l2 [ N + l2 [ N + l2 ( ] t l δ K + ( δ l l ( l ] t i l lδ K + ( δ l i2 l ] ( l [ N + i2 li i2 ( l N i l N i l t + t i, (8 i2 where (77 follows from (62; (78 follows from our hypothesis (74; (79 follows from (62; (80 is due to the equality l l2 i2 i2 li ; the last equality is due to (6. Therefore, the desired equality holds also for. This completes the proof of Corollary 3. ] (77 (78 (79 (80 V. ACHEVABLTY We provide the achievaility proof of Theorem for the case of one-sided fair rate vector as well as the symmetric network. The proof for the case of K 3 is omitted, since it is a straightforward extension of [5, Section V]. A. Proposed delivery scheme for K > 3 We descrie the proposed delivery scheme for the case of K > 3 assuming that user k requests file W k of size F k packets for k,..., K without loss of generality. Compared to the algorithm [5], [6] revisited previously, our scheme must convey packets created during the placement phase as well as all previous phases in each phase. Here is a high-level description of our proposed delivery scheme. Placement phase (phase 0: fill the caches Z,..., Z K according to the decentralized content placement (see susection -B. This phase creates overheard packets {L \k (W k } for [K] and all k to e delivered during phases to K.

17 7 2 Broadcasting phase (phase : the transmitter sends V,..., V K sequentially until at least one user receives it, where V k L (W k corresponds to the order- packets. 3 Multicasting phase (phases 2-K: for a suset of users, generate V as a linear comination of overheard packets during the placement phase as well as during phases to. Send V sequentially for [K], ( V F {L\ (V } :, L \{k} (W k. (82 The proposed delivery scheme achieves the optimal rate region only in two special cases. We provide the proof separately in upcoming susections. B. Proof of Theorem for the case of one-sided fair rate vector We assume without loss of generality δ δ K, δ R δ K R K, and p p R p2 p 2 R K. Under this setting, we wish to prove the achievaility of the following equality. K k ( p k δ R k. (83 By replacing R k F d k T tot equivalent to and further assuming d k k for all k without loss of generality, the aove equality is T tot K k ( p k δ F k. (84 The rest of the susection is dedicated to the proof of the total transmission length (84. We start y rewriting t {k} in (5 y incorporating the packets generated during the placement phase. Namely we have for k [K] :k t {k} N {k} + L \{k}(w k [K]\ {k} δ. (85 We recall that the length of su-phase is given y t max k t {k}. Our proof consists of four steps. a Step : We express t {k} as a function of key parameters {δ k }, {p k }, {F k } in two different ways. By following similar steps as in [6, Appendix C], the aggregate length of su-phases required y user k for a fixed [K] is given y :k We have an alternative expression for t {k} y user k such that k [K] is equal to t {k} The proof is provided in Appendix A. H:H \{k} t {k} [K]\ {k} ( p [K]\ {k} δ F k. (86 which is useful as will e seen shortly. The length of su-phase needed ( H [K]\ {k} H ( p [K]\ {k} H δ F k. (87

18 8 Step 2 : The length of su-phase is determined y the worst user which requires the maximum length, i.e. arg max k t {k}. For the special case of one-sided fair rate vector, y using (87 it is possile to prove that the worst user is given y arg max k t{k} min{}, [K], (88 where min{} is the smallest index in the set of users that corresponds to the user with the largest erasure proaility. The proof is provided in Appendix B. This means that the user permutation (which determines the su-phase length is preserved in all su-phases for the one-sided fair rate vector. c Step 3 : By comining the two previous steps, the total transmission length can e derived as follows. T tot : [K] : [K] K max k t{k} (89 {min } t (90 :k {k,...,k} K t {k} (9 k F ( p k k δ, (92 where (90 is otained from (88; the last equality follows from (86. Then, we otain the desired equality (84. d Step 4 : The final step is to prove that under the one-sided fair rate vector (83 implies all the other K! inequalities of the rate region (. This is proved in Appendix C. Hence, the achievaility proof for the one-sided rate vector is completed. C. Proof of Theorem for the symmetric network First we recall the rate region of the symmetric network with uniform channel statistics and memory sizes given in (2, K ( p k δ k R π k, π. (93 Exploiting the polyhedron structure and following the same footsteps as [4, Section V], we can prove that the vertices of the aove rate region are characterized as: R sym ( K, k K R k 0, k / K for K [K], where the symmetric rate R sym (K is given y R sym (K K (94 ( p k δ k. (95 This means that when only K users are active in the system, each of these users achieves the same symmetric rate as the reduced system of dimension K. Then, it suffices to prove the achievaility of the symmetric rate for a given

19 9 dimension K. As explained in susection V-A, the placement phase generates overheard packets {L \k (W k } for [K] and all k. We let N 0 L \k (W k denote the numer of order- packets created during the placement phase. Then, we can express the sum rate of the cached-enaled EBC y incorporating the packets generated from the placement phase into (7 as follows, KR sym (K KN 0 β KF + K 2 ( K (N 0 +N R (K. (96 By repeating the same steps as the proof of Corollary 3, it readily follows that the aove expression oils down to K K ( p k δ k. This estalishes the achievaility proof for the symmetric network. V. EXTENSONS n this section, we provide rather straightforward extensions of our previous results to other scenarios such as the centralized content placement and the multi-antenna roadcast channel with the state feedack. A. Centralized content placement So far, we have focused on the decentralized content placement. We shall show in this susection that the rate region under the decentralized content placement can e easily modified to the case of the centralized content placement proposed in []. We restrict ourselves to the symmetric memory size M k {0, N/K, 2N/K,..., N} so that the parameter MK N M such that M is an integer. Each file is split into ( K disoint equal size su-files. Each su-file is cached at a suset of users, [K] with cardinality. Namely, the size of any su-file of file i is given y which satisfies the memory constraint for user k Z k N i :k ; L (W i ( K F i, (97 L (W i N i ( K Fi ( K N i K F i MF. (98 n analogy to Lemma 2 for the decentralized content placement, we can characterize the message entropy given the receiver side information. Lemma 3. For the centralized content placement [], the following equalities hold for any i and K [K] H(W i {Z k } k K ( K K ( K H(W i.

20 20 Proof: Under the centralized content placement H(W i {Z k } k K [K]\K [K]\K; [K]\K; ( K K H(L (W i (99 H(L (W i (00 ( K H(W i (0 ( K H(W i, (02 where the first equality follows y repeating the same steps from (28 to (3; (00 and (0 follows from the definition of the centralized content placement (97. Then, we present the rate region of the cache-enaled EBC under the centralized content placement. Theorem 4. For the symmetric network, the rate region of the cached-enaled EBC with the state feedack under the centralized content placement is given y for any permutation π of {,..., K}. K ( K k ( / K δ k R πk (03 Proof: Following the same steps as in section and replacing Lemma 2 with Lemma 3, the converse proof follows immediately. For achievaility, as explained in susection V-C, it is sufficient to consider the case of symmetric rate for a given dimension. By focusing without loss of generality on the dimension K, we fix the numer of packets per user to e F and prove that our proposed scheme can deliver requested files to users within the total transmission length given y T tot F K ( K k ( / K δ k + ɛ F, (04 as F. We proceed our proposed delivery scheme from phase + y sending packets of order +. More precisely, in phase + we generate and send the packets intended to y the following linear comination V F ( L\k (W k, (05 for [K] with +. n susequent phases + 2 to K, we repeat ( V F {L\ (V } : (06 for [K] with + 2,..., K. n order to calculate the total transmission length required y our delivery algorithm, we follow the same footsteps as in susection V-C and exploit Theorem 3 on the sum capacity of order-i messages that we recall here for the sake of clarity. R i (K ( K i K i+ ( K k i δ k. (07

21 2 Noticing that there are ( K + su-phases in phase + and in each su-phase we send a linear comination whose F size is ( K, the total transmission length is given y ( K ( T tot + / K R + F (08 K ( K k ( / K δ k F, (09 where the last equality follows y plugging the expression R +. For the case without erasure, Theorem 4, in particular, the expression of the transmission length in (04, ecomes the rate-memory tradeoff under the centralized content placement [] given y B. MSO-BC T tot F K ( M/N + KM/N. (0 We consider the multi-input single-output roadcast channel (MSO-BC etween a N t -antennas transmitter and K single-antenna receivers. The channel state S l in slot l is given y the N t K matrix and we restrict ourselves to the i.i.d. channels across time and users. Here, we are interested in the capacity scaling in the high signal-to-noise ratio (SNR regime and define the degree of freedom (DoF of user k as DoF k We define the sum DoF of order- messages given y DoF lim snr lim snr : R k log 2 snr. R log 2 snr. ( First we recall the mains results on the MSO-BC with state feedack y Maddah-Ali and Tse [4]. n [4, Theorem 3], the DoF region of the MSO-BC with state feedack has een characterized as K DoF πk, π. (2 k The sum DoF of order- messages has een characterized in [4, Theorem 2] and is given y ( K DoF K + ( K k k. (3 t is worth comparing the DoF region of the MSO-BC in (2 and the capacity region of the EBC in (52. n fact, as remarked in [2], oth regions have exactly the same structure and can e unified through a parameter α k k for the MSO-BC and α k δ k for the EBC. The same holds for the sum DoF of order- messages in the MSO-BC in (3 and the sum capacity of order- packets in the EBC characterized in Theorem 3. By exploiting this duality and replacing δ k with k in the rate region of the symmetric EBC (2, we can easily characterize the DoF region of the cache-enaled MSO-BC with state feedack. Namely, under the decentralized content placement, the DoF region is given y K ( p k DoF πk, π (4 k

22 22 for N t K, while under the centralized content placement, the DoF region is given y ( / K K ( K k k DoF πk, π (5 for N t K. The converse follows exactly in the same manner except that we use the entropy inequality for the MSO-BC given in [2, Lemma 4] y replacing the entropy y the differential entropy and again δ k y k. The achievaility can e proved y modifying the scheme in [4] to the case of receiver side information along the line of [3]. As a final remark, for the case of the centralized content placement, our DoF region in (5 yields the following transmission length which coincides with [26, Corollary 2]. T tot K ( K k ( / K F, (6 k V. NUMERCAL EXAMPLES n this section, we provide some numerical examples to show the performance of our proposed delivery scheme. Fig. 4 illustrates the tradeoff etween the erasure proaility and the memory size for the symmetric network with K 3 for the case of the decentralized content placement Each curve corresponds to a different symmetric rate R sym (3 3. The arrow shows the increasing symmetric rate from /3, corresponding to case with no ( p k δ memory and no erasure, k to infinity. The memory size increases the rate performance even in the presence of erasure and the enefit of caching is significant for smaller erasure proailities as expected from the analytical expression. Fig. 5 compares the transmission length T tot, normalized y the file size F, achieved y our delivery scheme with feedack and the scheme without feedack for the case of the decentralized content placement. We consider the system with N 00, K 0 and the erasure proailities of δ 0 (perfect link, 0.2, and 0.6. We oserve that state feedack can e useful especially when the memory size is small and the erasure proaility is large. n fact, it can e easily shown that the rate region of the cached-enaled EBC without feedack under the decentralized content placement is given y K ( M k N R πk (7 δ where the denominator in the LHS reflects the fact that each packet must e received y all K users. This yields the transmission length given y T tot nofb K ( M k N F + ɛ F. (8 δ Under the centralized content placement, the rate region of the cached-enaled EBC without feedack is given y ( / K K ( K k δ R πk (9

23 23 Fig. 4. The tradeoff etween the memory and the erasure for K 3. Fig. 5. The transmission length T tot as a function of memory size M for N 00, K 0. yielding T tot nofb K ( M/N +KM/N F + ɛ F. (20 δ Without state feedack, the transmission length in (8, (20 corresponds to the transmission length over the perfect link expanded y a factor δ >, ecause each packet must e received y all users. The merit of feedack ecomes significant if the packets of lower-order dominate the order-k packets. The case of small p M N and large erasure proaility corresponds to such a situation. Fig. 6 plots the normalized transmission length T tot /F versus the memory size M in the symmetric network with N 00, K 0. We compare the performance with and without feedack under the decentralized and the centralized caching for δ 0 and δ 0.6. size. The relative merit of the centralized content placement compared to the decentralized the counterpart can e oserved. Fig. 7 plots the normalized transmission length T tot /F versus average memory size M in the asymmetric

24 24 Fig. 6. The transmission length T tot as a function of memory size M for N 00, K 0. Fig. 7. δ i i 5, N 20, K 4 and F i. network with N 20 and K 4 under the decentralized content placement. We let erasure proailities δ k k 5 for k,..., 4 and consider files of equal size. We compare symmetric memory (M k M, k, asymmetric memory otained y optimizing over all possile sets of {M k } using our delivery scheme, as well as lower ound otained y optimizing over all possile of {M k } ased on (6. This result shows the advantage (in terms of delivery time of optimally allocating cache sizes across users, whenever possile, according to the condition of the delivery channels. V. CONCLUSON n this paper, we investigated the content delivery prolem in the erasure roadcast channel (EBC with state feedack, assuming that the content placement phase is performed with existing methods proposed in the literature. Our main contriution was the characterization of the optimal rate region of the channel under these conditions, ased on a scheme that optimally exploits the receiver side information acquired during the placement phase. This appears as a non-trivial extension of the work y Wang and Gatzianas et al. [6], [7] which have characterized the capacity region of the EBC with state feedack for some cases of interest. We provided an intuitive interpretation

25 25 of the algorithm proposed in these works and revealed an explicit connection etween the capacity in the symmetric EBC and the degree of freedom (DoF in the MSO-BC. More specifically, we showed that there exists a duality in terms of the order- multicast capacity/dof. Such a connection was fully exploited to generalize our results to the cache-enaled MSO-BC. Our work demonstrated the enefits of coded caching comined with state feedack in the presence of random erasure. An interesting future direction is to include some more practical constraints, such as the popularity profile of contents and the non-asymptotic file size, into the current system model. APPENDX n the appendix, we repeatedly use the following weight expression. w ( p δ p (2 δ where we let p p and use a short-hand notation δ δ. A. Length of su-phase n this section, we prove (87 given y t {k} H:H \{k} To this end, we first introduce a new variale g {k} We first need to prove the following lemma ( H [K]\ {k} H ( p [K]\ {k} H δ F k. (22 :k t{k} F k Lemma 4. For any nonempty set [K] and [K]. t holds : H:H for k [K]. Using (86 we otain g {k} w [K]\ {k}. (23 ( H w [K]\ H w [K]\ (24

26 26 Proof: ( H w [K]\ H ( H w [K]\(\H (25 : H:H : H:H : H :H H :H :H H :H H :H w [K]\H w [K]\H w [K]\ + ( \H w [K]\H (26 H :H ( \H w [K]\H (27 :H ( \H : \H ( w [K]\H : \H ( (28 (29 (30 w [K]\. (3 We set H \ H and \ H to otain (26 and (29, respectively. The last equality follows from : ( 0 for all. We prove (87 y induction on. For {i} we have :i g{i} w [K]\ {i}. By apply (23 for {i}, we otain the proof for. g {i} and H:H \{i} ( H w [K]\ {i} H Now suppose (87 holds for any [K] such that < and we prove in the following that it holds for too. We have Thus, we otain :i g {i} w [K]\ {i} (32 g {i} + :i g {i}. (33 g {i} w [K]\ {i} :i w [K]\ {i} w [K]\ {i} w [K]\ {i} g {i} (34 :i H:H \{i} :i H:H \{i} : \{i} H:H w [K]\ {i} w [K]\(\{i} + H:H \{i} ( H w [K]\ {i} H (35 ( H w [K]\ {i} H + ( H w [K]\ H + H:H \{i} H:H \{i} H:H \{i} ( H w [K]\ {i} H (36 ( H w [K]\ {i} H (37 ( H w [K]\ {i} H (38 ( H w [K]\ {i} H, (39

27 27 where (38 is from Lemma 4. B. Existence of the permutation n this section, we prove that the worst user under the one-sided fair rate vector is determined y (88, namely arg max k t{k} min{}, [K]. (40 We set m min( for any suset [K] such that 2. Proving (88 is equivalent to prove R m g {m} R i g {i} i. (4 Recall that from our one-sided rate vector assumption we have for i, δ m δ i ; δ m R m δ i R i and pm p m R m p i p i R i. Plugging (6 and (48 into (85, we otain and g {i} g {m} δ [K]\ {i} δ [K]\ {m} We prove y induction on that R m g {m} following and [ :i [ :m g {i} [ δ [K]\ {i} ] g {i} δ \ δ [K]\ {i} + p \{i} p [K]\ {i}, (42 ] g {m} δ \ δ [K]\ {m} + p \{m} p [K]\ {m}. (43 R i g {i} : For 2, {m, i} hence (42 and (43 imply the g {i} i δ m δ [K]\ {i} + p m p [K]\ {i} ], (44 g {m} [ g {m} δ δ ] m i δ [K]\ {m} + p i p [K]\ {m}. (45 [K]\ {m} Since δ m δ i, it holds δ [K]\ {m} δ [K]\ {i} and δ i δ m. Since pm p m Rm pi p i Ri, then it holds p i p [K]\ {m} R m p m p [K]\ {i} R i. n addition we have from (87 : g {m} m thus we otain R m g {m} R i g {i} for 2. g {i} i p [K] δ [K] and δ m R m δ i R i, Suppose that (4 holds for any [K] such that < and we prove that it holds also for in the following. Since δ m δ i, it holds δ [K]\ {m} δ [K]\ {i}. Since pm p m R m pi p i R i, it holds p \{m} p [K]\ {m} R m p \{i} p [K]\ {i} R i. By oserving (43 and (42, it remains to prove that We have for user m :m R m :m g {m} δ \ δ [K]\ {m} g {m} δ \ δ [K]\ {m} R i :{m,i} :{m,i} :i g {m} δ \ δ [K]\ {m} + g {m} δ \ δ [K]\ {m} + g {i} δ \ δ [K]\ {i}. (46 m \{i} : \{i,m} g {m} δ \ δ [K]\ {m} (47 g {m} {m} δ \\{m} δ [K]\ {m}, (48

28 28 and similarly for user i :i g {i} δ \ δ [K]\ {i} :{m,i} :{m,i} g {i} δ \ δ [K]\ {i} + g {i} δ \ δ [K]\ {i} + :i \{m} : \{m,i} g {i} δ \ δ [K]\ {i} (49 g {i} {i} δ \\{i} δ [K]\ {i}. (50 For any satisfying {m, i} we have <, min( m and i so y the hypothesis we have g {m} R m g {i} R i. n addition we have δ m δ i thus :{m,i} g {m} δ \ δ [K]\ {m} R m :{m,i} For any satisfying \ {m, i} we have from (87 g {m} {m} g {i} δ \ δ [K]\ {i} R i. (5 g{i} {i}. n addition we have δ i R m δ m R i δ i, then δ \\{m} δ [K]\ {m} R m δ \\{i} δ [K]\ {i} R i. As a result we otain R m \{i,m} Hence the proof is completed. g {m} {m} δ \\{m} δ [K]\ {m} R i \{m,i} δ m and g {i} {i} δ \\{i} δ [K]\ {i}. (52 C. The outer-ound under the one-sided fair rate vector Suppose that there exists π such that K R π (w π(..π ( and that π (i π (i + holds for some i [K ]. We prove that for any permutation π 2 that satisfies π 2 (i + π (i k, π 2 (i π (i + k and π ( π 2 ( [K] \ {i, i + }, it holds K R π 2(w π2(..π 2(. t suffices to show that equivalent to w π(..π (ir π(i + w π(..π (i+r π(i+ w π2(..π 2(iR π2(i + w π2(..π 2(i+R π2(i+ (w π(..π (i w π2(..π 2(i+R π(i (w π2(..π 2(i w π(..π (i+r π(i+ equivalent to (w k w kk R k (w k w kk R k, (53 where π (..π (i. By replacing the weight y its expression (2 we otain and similarly p kk w k w kk p k δ k δ kk [ p k δ k δ kk ] p k + δ kk [ ] ( δkk ( δ k p k p k + ( δ k ( δ kk δ kk [ p k δk ( δ k + p k δ kk ( δ k (54 (55 (56 ], (57