Distortion-Memory Tradeoffs in Cache-Aided Wireless Video Delivery

Transcription

1 Dstorton-Memory Tradeoffs n Cache-Aded Wreless Vdeo Delvery P. Hassanzadeh, E. Erkp, J. Llorca, A. Tulno arxv: v1 [cs.it] 12 Nov 2015 Abstract Moble network operators are consderng cachng as one of the strateges to keep up wth the ncreasng demand for hgh-defnton wreless vdeo streamng. By prefetchng popular content nto memory at wreless access ponts or end user devces, requests can be served locally, relevng stran on expensve backhaul. In addton, usng network codng allows the smultaneous servng of dstnct cache msses va common coded multcast transmssons, resultng n sgnfcantly larger load reductons compared to those acheved wth conventonal delvery schemes. However, pror work does not explot the propertes of vdeo and smply treats content as fxed-sze fles that users would lke to fully download. Our work s motvated by the fact that vdeo can be coded n a scalable fashon and that the decoded vdeo qualty depends on the number of layers a user s able to receve. Usng a Gaussan source model, cachng and coded delvery methods are desgned to mnmze the squared error dstorton at end user devces. Our work s general enough to consder heterogeneous cache szes and vdeo popularty dstrbutons. I. INTRODUCTION Wth the recent explosve growth n cellular vdeo traffc, wreless operators are heavly nvestng n makng nfrastructural mprovements such as ncreasng base staton densty and offloadng traffc to W-F. Cachng s a technque to reduce traffc load by explotng the hgh degree of asynchronous content reuse and the fact that storage s cheap and ubqutous n today s wreless devces [1] [3]. Durng offpeak perods when network resources are abundant, popular content can be stored at the wreless edge (e.g., access ponts or end user devces, so that peak hour demands can be met wth reduced access latences and bandwdth requrements. The smplest form of cachng s to store the most popular vdeo fles at every edge cache [4]. Requests for popular cached fles can then be served locally, whle cache msses need to be served by the base staton, achevng what s referred to as a local cachng gan. However, replcatng the same content on many devces can result n an neffcent use of the aggregate cache capacty [5]. In fact, recent studes [6] [10] have shown that makng users store dfferent portons of the vdeo fles creates coded multcast opportuntes that enable a global cachng gan. In [7], t s shown that unform random cachng, n whch users cache portons of every fle unformly at random, n combnaton wth lnear ndex codng, acheves a worst-case rate that s wthn a P. Hassanzadeh and E. Erkp are wth the ECE Department of New York Unversty, Brooklyn, NY. Emal: {ph990, elza}@nyu.edu J. Llorca and A. Tulno are wth Bell Labs, Alcatel-Lucent, Holmdel, NJ, USA. Emal: {jame.llorca, a.tulno}@alcatel-lucent.com A. Tulno s wth the DIETI, Unversty of Naples Federco II, Italy. Emal: {antonamara.tulno}@unna.t constant factor of an nformaton theoretc lower bound; and hence, s order-optmal. The case of random demands accordng to a Zpf popularty dstrbuton s analyzed n [8], [9]. The authors characterze the optmal average rate as a functon of all system parameters and provde an orderoptmal cachng and coded multcast scheme desgned to balance the gans from local cache hts and coded multcast opportuntes. Whle exstng work on wreless cachng s motvated by vdeo applcatons, specfc propertes of vdeo are not exploted n the cachng and delvery phases. In scalable vdeo codng (SVC [11], vdeo fles are encoded nto layers such that the base layer contans the lowest qualty level and addtonal enhancement layers allow successve mprovement of the vdeo streamng qualty. In ths work, we analyze the use of cachng as a method to enhance vdeo qualty at users streamng devces. We consder a scenaro n whch users store vdeos at dfferent encodng rates (e.g., vdeo layers n SVC. Upon vdeo streamng requests, dependng on the avalable network resources, users receve addtonal layers that successvely refne the vdeo playback qualty. To formulate the problem mathematcally, we assume that the lbrary conssts of Gaussan sources wth dfferent varances. We allow for users to have dfferent cache szes and probablty dstrbutons for accessng these sources. The goal s to desgn cachng and delvery schemes that, for a gven broadcast rate, mnmze the average dstorton experenced at user devces. We frst show that under uncast delvery, the optmal cachng polcy admts a reverse water-fllng type soluton whch can be mplemented locally and ndependently across users, wthout the need of global coordnaton. Each vdeo streamng request just needs to specfy the maxmum qualty level avalable at the correspondng local cache, such that the sender (e.g., base staton can effectvely compute the optmal delvery rates allocated to each user. We then show how usng coded multcast offers notable performance mprovements n terms of average vdeo dstorton. In ths case, the optmal polcy requres jont optmzaton of the rates at whch vdeos are cached by each user. After users place ther requests, the sender, wth knowledge of the users cache contents, computes a common multcast codeword that smultaneously delvers addtonal enhancement layers to each user. Our smulaton results confrm the sgnfcant gans achevable va coordnated cachng and coded multcast, wth more than 10 reducton n average dstorton observed n wreless cachng networks wth 20 user caches and 100 vdeos.

2 Fg. 1: System Model. Cachng s used for mprovng vdeo playback qualtes. The remander of ths paper s organzed as follows. The problem settng s ntroduced n Sec. II. The use of local cachng and uncast transmsson s analyzed n Sec. III. Sec. IV descrbes the proposed achevable scheme based on cooperatve cachng and coded multcast transmssons. Fnally, numercal results that llustrate the achevable dstortonmemory tradeoffs are presented n Sec. VI. II. PROBLEM SETTING Consder the system n Fg. 1 where one sender (e.g., base staton s connected through an error-free shared lnk to n recevers (e.g., access ponts or user devces wth rate (capacty R bts/source-sample. The sender has access to a content lbrary, F = {1,..., m}, contanng m vdeo fles (sources each composed of F source samples. Recever {1,..., n} has a cache of sze M bts/source-sample, or equvalently, M F bts. Recevers place requests for vdeos n the lbrary accordng to a demand dstrbuton Q = [q,j ], = 1,, n, j = 1,, m, assumed to be known at the sender, where q,j [0, 1] and m j=1 q,j = 1, [n] 1. The demand dstrbuton s defned such that recever requests vdeo fle j wth probablty q,j. We use d to denote the random request at recever, wth d F beng a realzaton of d. We consder a vdeo streamng applcaton, n whch each fle f F represents a vdeo segment, compressed usng SVC [11]. In SVC, the base layer contans the lowest level of detal spatally, temporally, and from a qualty perspectve. Addtonal layers, named enhancement layers, can mprove the qualty of the stream. Note that an enhancement layer s useless, unless the recever has access to the base layer and all precedng enhancement layers. The decoded vdeo qualty depends on the total number of layers receved n sequence. The vdeo delvery system operates n two phases, a cachng (or placement phase followed by a transmsson phase: Cachng Phase: The cachng phase occurs durng a perod of low network traffc. In ths phase, all recevers have access to the entre lbrary for fllng ther caches. In the followng, wthout loss of generalty, and n lne wth [8] [10], we assume that the cachng phase s 1 Throughout the rest of ths paper, [n] denotes the dscrete set of ntegers from 1 to n.e. [n] = {1,..., n} descrbed by a set of n vectors, p = [p,1,..., p,m ], [n], referred to as the cachng dstrbutons, wth m j=1 p,j = 1, [n]. Element p,j represents the cache porton assgned to vdeo j at recever, and M,j = p,j M s the number of bts/source-sample of vdeo j cached at recever. We refer to the m- dmensonal vector, M = [M,1,..., M,m ], as the cache placement of user. Desgnng the cachng phase conssts of desgnng cachng dstrbutons, p [n], and the correspondng cache content. Ths can be done locally by the recevers, based on ther local nformaton, or globally (n a cooperatve manner ether drectly by the sender, or by the recever tself based on nformaton from the overall network. As n [6] [10], we assume that lbrary fles and ther popularty change at a much slower tme-scale than the vdeo delvery tme-scale, and neglect the resource requrements assocated wth the cache-update process. Hence, the cachng phase s sometmes referred to as the placement or prefetchng phase. Transmsson Phase: The transmsson phase takes place after completon of the cachng phase. Durng ths phase, only the sender has access to the lbrary. The network s repeatedly used n a tme slotted fashon. At the begnnng of each tme slot, each recever requests a vdeo fle d F. Havng been nformed of the demand realzaton d = [d 1, d 2,..., d n ], the sender decdes on the playback qualtes of the requested vdeos. The sender computes the rate, R,d (bts/source-sample, that wll be transmtted to recever [n], based on the demand, cached content, and the channel capacty constrant. We refer to R,d as the per-user rate for demand realzaton d. The sender encodes the chosen vdeo layers nto a codeword, X d, whch s then transmtted to the recevers over the shared lnk. Recever decodes ts vdeo of nterest, d, (at the correspondng qualty level usng the receved codeword, X d, and ts cache content. Recever s requested vdeo playback qualty depends on both M,d and R,d. For ease of exposure and analytcal tractablty, we assume that vdeo fle j conssts of F..d. Gaussan samples wth varance σj 2 and dstorton-rate functon D j(r = σj 22 2r [12]. In ths settng, based on the fact that Gaussan sources wth squared error dstorton are successvely refnable wth a multple-stage descrpton that s optmal at each stage [13], scalable codng does not have any codng overhead. The goal s to desgn the cachng and transmsson schemes that mnmze the expected dstorton (over the demand dstrbuton, defned as E(D = ( 1 Π d σd 2 n 2 2(M,d +R,d. (1 d D =1 In (1, D s the set of possble demands, d D represents the specfc demand realzaton, and Π d s the probablty of demand d. We assume that recevers request vdeo fles ndependently; hence, Π d = n =1 q,d.

3 Note that, n ths paper, the goal s to mnmze the expected dstorton when recevers are connected to the sender through a shared lnk of fnte capacty, R bts/sourcesample. Ths s n contrast to pror work, [6] [10], [14], [15], n whch the goal s to mnmze the total rate transmtted over the shared lnk, n order to recover all requested fxed-sze (and hence fxed-qualty vdeo fles. In the followng, we focus on two scenaros for desgnng the cachng and transmsson schemes, that dffer n performance, computatonal complexty and requred codng overhead. Specfcally, n the frst scenaro, n order to lmt the computatonal complexty and reduce the communcaton overhead, we assume that the sender compresses the requested vdeos ndependently for each recever, merely based on ther local cached content. We refer to such an encodng strategy (cachng & transmsson scheme, as the Local Cachng-aded Uncast (LC-U scheme. On the other hand, n the second scenaro, we assume that the sender compresses the requested vdeos jontly across all recevers based on global network knowledge (cache contents and demand dstrbutons. We refer to ths second encodng strategy as the Cooperatve Cachng-aded Coded Multcast (CC-CM scheme. Optmzaton of the cachng and transmsson phases for these two schemes are conducted dfferently and requre dfferent levels of complexty. Note that CC-CM uses global network knowledge to construct codes that fully explot the multcast nature of a wreless system, whle LC-U elmnates addtonal codng complexty, but the total wreless resource (rate must be orthogonally dvded among recevers. III. LOCAL CACHING-AIDED UNICAST (LC-U In LC-U, the encoder s equvalent to n ndependent fxedto-varable source encoders, each dependng only on the local cache of the correspondng recever. The resultng codeword hence corresponds to n uncast transmssons. Each recever s cachng and transmsson rates, M,j and R,d (, j [n] [j], are computed as follows: 1 In the cachng phase, each recever computes the optmal cache allocaton that mnmzes the expected dstorton, assumng that the sender doesn t transmt further vdeo layers (R,d = 0. Snce users are not expectng to receve further layers durng the transmsson phase, each recever caches vdeo layers ndependently, based on ts own demand dstrbuton. Recever [n] solves the followng convex optmzaton problem: mn E(D = s.t q,j σj 2 2 2M,j j=1 M,j M j=1 M,j 0, j [m] The resultng cache allocatons are gven by (2 Fg. 2: Illustraton of the local cachng scheme for 6 fles at user 1, assumng ndependent Gaussan sources (vdeo fles when R = 0. M,j = log 2 2 ln 2q,j σ 2 j λ +, (3 wth λ such that j M,j = M. The soluton admts a reverse water-fllng form [12]. User only stores portons of vdeo fles wth q,j σj 2 less than λ 2 ln 2 ; hence, q,j M,j = mn{ λ 2 ln 2, q,jσj 2 }, as llustrated n Fg Durng the transmsson phase, the optmal transmsson rates for demand d D, R,d, are derved at the sender by solvng: mn s.t. 1 σd 2 n 2 2(M,d +R,d =1 R,d R =1 R,d 0, The optmal rates are then gven by R,d = log 2 2 ln 2σ 2 d γ d [n] M,d + (4, (5 wth γd chosen such that R,d = R. Note that recevers cache based on ther own preferences, q,j, and fle characterstcs, σj 2. The cachng process s decentralzed and does not requre any coordnaton from the sender. On the other hand, delvery of the requested vdeo fles s done n a centralzed manner. The sender gathers nformaton of the cached vdeo layers at the recevers, and jontly optmzes the transmsson rates. A. Implementaton of LC-U va SVC LC-U s a scalable vdeo codng scheme descrbed by two layers, one base layer and one enhancement layer. Durng the cachng phase recever stores a base layer of vdeo fle j at rate M,j (bts/source-sample j [m]. In the transmsson phase, the sender uncasts the enhancement layers of the requested vdeo fles to the correspondng recevers, at rates R,d [n] whch have been jontly optmzed across all

4 recevers. Hence, there are n dsjont encoders at the sender for transmttng the enhancement layers. IV. COOPERATIVE CACHING-AIDED CODED MULTICAST (CC-CM In CC-CM, dfferently from LC-U, the sender encodes the requested vdeos jontly across all recevers. Equvalently, there s one fxed-to-varable encoder at the sender, whch jontly compresses the nformaton that needs to be delvered to each recever based on the cached content dstrbuted among all of them. The sender multcasts the compressed nformaton over the shared lnk; therefore, the total avalable rate, R, s no longer smply dvded among the recevers. Ths results n more effcent transmssons over the shared lnk, whch, n turn, ncreases the decoded vdeo qualtes. As mentoned n Sec. I, t has been shown that the jont desgn of cachng and coded multcastng enables multplcatve cachng gans n terms of the aggregate rate (or load requred on the shared lnk to delver the desred per-user rate;.e., the aggregate load decreases lnearly wth the local cache sze. [6] [10], [14] [16]. However, all coded multcast schemes avalable n lterature are based on fxed-to-varable source encodng, desgned to mnmze the aggregate load on the shared lnk so that the requested fles are recovered n whole and wth zero dstorton. In other words, the per-user rates are gven as constrants, and the goal s to mnmze the requred aggregate rate over the shared lnk. In contrast, n ths work, we constran the overall rate that can be transmtted over the shared lnk, and optmze the per-user rates, whch n turn determne the recevers vdeo playback qualtes. The per-user rate (n bts/source-sample delvered to recever [n], s denoted by R,d. We splt R,d nto two portons: a porton,,d delvered va coded multcast, and a porton, R,d delvered va uncast, so that R,d =,d + R,d [n]. As descrbed later, the per-user coded multcast rate,d s computed ndependently from the entre demand vector realzaton d D. In the followng, we use aggregate coded multcast rate to refer to the total load over the shared lnk assocated to the multcast transmsson used to delver the per-user coded multcast rates,d, [n], and aggregate uncast rate to refer to the total load over the shared lnk assocated to the uncast transmsson used to delver the per-user uncast rates R,d, [n]. Note that whle the aggregate uncast rate s equal to n =1 R,d, the aggregate coded multcast rate s, n general, much smaller than n =1,d (due to multcastng gans, and depends on the specfc scheme adopted for the multcast transmsson. The cachng-aded coded multcast scheme used n CC-CM s descrbed n the followng. A. Random Popularty-based Cachng wth Greedy Index Codng The scheme adopted n CC-CM for the multcast transmsson s the achevable scheme proposed n [9], referred to as RAndom Popularty-based (RAP cachng wth Greedy Constraned Colorng (GCC, or RAP-GCC. 1 Durng the cachng phase, the sender computes the optmal per-user cachng and coded multcast transmsson rates, M,j and R,j (, j [n] [j], as descrbed n Sec. IV-C as a functon of the lnk capacty R, the cache sze M [n], and the demand dstrbuton Q = [q,j ] (, j [n] [j], usng (15a-(15d. Next, recever caches F M,j bts unformly at random from the frst F (M,j +,j bts of fle j [m]. 2 Durng the transmsson phase, the sender delvers the remanng F,d bts, wth [n], va coded multcast transmssons usng the GCC scheme [9]. Specfcally, gven a demand realzaton d, the sender computes the multcast codeword as a functon of recever s caches and demand realzaton d usng a (centralzed ndex codng based transmsson scheme mplemented va GCC. Next, the sender uses ts remanng avalable rate to uncast further layers wth rate R,d wth [n]. Let Rd GCC denote the aggregate coded multcast rate when usng RAP-GCC. In the followng, we frst provde a closedform expresson for Rd GCC as a functon of M j and,j (see Sec, IV-B. Ths expresson s then used n the optmzaton problem descrbed n Sec. IV-C to evaluate the optmal M j,,j and R,d that mnmze the total average dstorton. A detaled descrpton of the mplementaton of CC-CM n the context of SVC s gven n Sec. IV-D. B. Achevable Rate In the next two theorems we provde closed form expressons for the asymptotc rate (as F goes to nfnty of the RAP-GCC scheme. Specfcally, Theorem 1, quantfes the rate acheved wth RAP-GCC for a demand realzaton d, Rd GCC, whle Theorem 2 gves the average rate over all demand realzatons, R GCC. These rate expressons are, then, used n the optmzaton problems gven n Sec. IV-C for fndng the optmal (n the sense of mnmum total average dstorton values of M j,,j and R,d. For convenence, n the followng, we let M p c,j = p,j M,j +. (6,j Theorem 1. For a shared lnk network wth n recevers, lbrary sze m, and cache capacty M [n], fx a cachng dstrbuton p (or equvalently a cachng placement M [n] and a demand realzaton d. Then, the aggregate coded multcast rate (n bts/source-sample requred to delver,d (bts/source-sample to each recever [n], as F, s gven by: where ψ d (p = R GCC d ({p } n =1 mn {ψ d (p, m d }, (7 l=1 U l U max U l {λ(, d (M,d +,d }, (8 and where m d s the rate (n bts/source-sample sent va uncoded (nave multcast gven by m d = 1 {f d} max (M,f +,f, (9 f=1

5 and fnally λ(, d = (1 p c,d (p c u,d (1 p c u,d u U l \{} u U\U l (10 denotes the probablty that a packet from fle d requested by recever has been cached by exactly l 1 recevers, where U = {1,..., n} s the set of all recevers and U l denotes a gven set of l recevers. Averagng over all possble demand realzatons we obtan the followng result: Theorem 2. For the shared lnk network wth n recevers, lbrary sze m, cache capacty M and demand dstrbuton q [n], fx a cachng dstrbuton p [n]. Then, for all ɛ > 0, the average aggregate coded multcast rate (n bts/source-sample requred to delver,j (bts/sourcesample to each recever [n] requestng fle j, as F, satsfes: lm P ( R GCC mn {ψ ({q, p } n =1, m} + ɛ = 1, F (11 where ψ ({q, p } n =1 = and m = f=1 l=1 U l U f=1 =1 u U l γ f,u,u l λ(u, f(m u,f + u,f, (12 ( n 1 (1 q,f max(m,f +,f, (13 wth λ(u, f gven as n (10, and where γ f,u,u l = P(f = arg max f u f(u l λ(u, f u (M u,fu + u,fu (14 denotes the probablty that f s the fle that maxmzes the term λ(u, f among f(u l, whch s the set of fles requested by recevers U l. We remark that, the proofs for Theorems 1 and 2 omtted n ths paper due to space lmtatons, are based on an extenson of the procedure n Appendx A n [9] for heterogeneous cache szes, popularty dstrbutons, and fle szes. C. CC-CM cachng and transmsson rate optmzaton Gven the achevable scheme from Sec. IV-A, the objectve s to desgn the cachng dstrbutons p, the set of per-user coded multcast rates,j, and the set of per-user uncast rates R,d (, j, d [n] [m] D, such that the average dstorton s mnmzed and the channel capacty constrant s not volated. Usng Rd GCC gven n Eq. (7-(10 of Theorem 1, the optmal per-user cachng, coded multcast and uncast rates, M,j,,j, R,d are derved va the followng optmzaton: mn ( 1 Π d σd 2 n 2 2(M,d +,d + R,d (15a d D =1 s.t. mn {ψ d (p, m d } + R,d R, d D (15b =1 M,j M, [n] (15c =1 M,j,,j, R,d 0, (, j, d [n] [m] D (15d Eq. (15b corresponds to the rate constrant, wth ts frst term beng the aggregate average coded multcast rate acheved wth RAP-GCC. The second term n (15b s the aggregate uncast rate for demand d. The optmzaton problem n (15 s hghly non-convex and has an exponental number of constrants due to (15b, whch depends on the cardnalty of D. In order to reduce complexty, we allow satsfyng the rate constrant on average over all demands rather than for each demand realzaton, and replace (15b wth the followng expresson: mn {ψ ({q, p } n =1, m} + d D n Π d =1 R,d R, (16 where ψ ({q, p } n =1 and m are gven by Eq.(12-(14 n Theorem 2. D. Implementaton of CC-CM va SVC In order to mplement the CC-CM scheme va SVC, the followng steps are executed: 1 The optmal cachng and transmsson rates, M,j,,j, R,d, are computed by solvng the optmzaton problem (15a-(15d wth (15b replaced by (16. 2 The vdeos are splt nto multple layers and each layer s encoded at the same rate, b bts/source-sample, represented as a bnary vector of length (entropy bf bts. The value of b s chosen as follows: Let M,j,j µ,j =, ρ,j = b b, ρ,d = R,d b, then, the common layer rate b (bts/source-sample s such that µ,j, ρ,j, and ρ,d are ntegers (, j, d [n] [m] D. Note that µ,j denotes the number of layers of vdeo fle j cached by recever, ρ,j represents the number of layers delvered va coded multcast, whle ρ,d denotes the number of layers that are uncast to recever. Fnally, the sum ω,j = µ,j + ρ,j, referred to as the storng range of recever for vdeo fle j, ndcates the hghest layer 2 of vdeo fle j that recever s allowed to cache. In other words, ω,j b represents the average rate (n bts/source-sample guaranteed to recever after the coded multcast transmsson. 3 Each recever confgures ts cache based on the mplementaton of RAP va SVC descrbed n Sec. IV-D.I. 2 We ndex vdeo layers accordng to ther sequental order n SVC, where layer k s only useful f layers {1,..., k 1} are present.

6 Fg. 3: Layer-packet dvson of a fle where µ = 3 and ω = 6. Fles are encoded at multple layers (represented wth dfferent colors. Each layer has rate b (bts/sample and conssts of B = 5 packets. The recever caches 15 packets from a total of 30 packets of the frst 6 layers. The above steps are all part of the cachng phase. Durng the delvery phase, at each use of the network, after a demand realzaton d s generated, the sender uses the mplementaton of GCC va SVC descrbed n Sec. IV-D.II. D.I RAP va SVC: As n [6], [8], [9], n order to maxmze coded multcast opportuntes, fle layers are dvded nto B equal sze packets each wth bf/b bts. Recever [n] caches µ j B packets unformly at random from the ω,j B packets of the frst ω,j layers of vdeo fle j [m]. In ths context, p c,j admts the nterpretaton of the probablty that a packet from the frst ω,j layers of vdeo fle j s cached at user,.e.: p c,j = p,j M M,j +,j = µ,j ω,j = ( ω,jb 1 µ (,jb 1 ω,jb µ,jb. (17 Fg. 3 llustrates the layer-packet dvson of a vdeo fle stored at a recever. The layers are encoded at the same rate b bts/sample and each layer of length bf bts s dvded nto 5 packets. The number of cached layers s µ = 3 and the storng range for ths vdeo fle s ω = 6. The recever has cached 15 packets from the 30 packets formng the frst 6 layers. We remark that, n contrast to LC-U, n whch the cached content can be confgured locally by each recever, the cachng scheme n CC-CM requres coordnaton from the sender, whch computes the cachng rates jontly across all recevers, as descrbed n Sec. IV-C. D.II GCC plus Uncast Transmsson va SVC: For a gven cache placement and demand realzaton d, the sender dentfes the set of packets that need to be delvered va coded multcast so that each recever [n] s guaranteed average rate ω,d b. Specfcally, the sender has to delver ρ,d B = (ω,d µ,d B packets, or equvalently,,d = ρ,d b bts/sample. In CC-CM, the sender uses the GCC coded multcast transmsson scheme, whch s based on a greedy soluton for the mnmum vertex colorng of the correspondng ndex codng conflct graph [17], whose detaled descrpton can be found n [8], [9]. Havng exploted all coded multcast opportuntes for demand d, recevers wll have receved the frst ω,d layers of ther requested vdeo fles wth rate,d. The sender, then, uses ts remanng avalable rate to uncast further layers wth rate R,b. It s mportant to remark that n SVC, a layer s used as part of the decodng process only f all ts precedng layers are avalable. Therefore, one needs to ensure that after completon of the transmsson phase, recever [n] has up to the (µ,d + ρ,d + ρ,d th layer of fle d. However, snce for reducng complexty, the coded multcast rates,,d = ρ,d b, are computed usng the average rate constrant (16, there may be specfc demand realzatons for whch the lnk capacty R s volated. Consequently the sender s not able to delver all the ρ,d B packets that ensure reconstructon of the ω,d layers by recever. In such cases, CC-CM employs a greedy approach, n whch the per-user coded multcast rates,,d, are reduced n decreasng order of σj 2 untl the lnk capacty constrant s satsfed. V. CC-CM OPTIMIZATION: SIMPLIFICATIONS In ths secton, we focus on optmzaton problem (15 wth rate constrant (15b replaced by (16, and assume key network symmetres whch allows us to quantfy the solutons. A. Symmetry Across Users We assume that recevers have the same cache sze and they request vdeo fles accordng to the same dstrbuton,.e M = M and q,j = q j, (, j [n] [m]. Hence, wthout loss of optmalty, we can assume that the cachng dstrbutons p, and the correspondng cachng placements M are constant across all recevers (.e., M = M. Analogously, we can assume that all recevers cache µ j B packets of vdeo fle j up to the same storng range ω j, or equvalently, the optmal rate delvered to each recever va coded multcast for d = j s,j = j [n]. Therefore, the probablty of a packet from fle j beng cached at any recever s p c j = µj ω j = Mj M j+. The average aggregate rate j requred for delverng the per-user coded multcast rate j (bts/source-sample to any recever requestng fle j, gven by (11, s smplfed as n [9] to: where ψ (q, p = and ( γ j,l P R GCC = mn {ψ (q, p, m}, (18 m = (1 (1 q j n (M j + j, (19 j=1 ( n m γ j,l (p c l j l 1 (1 p c j n l+1 (M j + j, l=1 j=1 (20 j = arg max(p c f l 1 (1 p c f n l+1 (M f + f, f D (21 wth D beng the random subset of l fles chosen n an..d manner from the lbrary (wth replacement.

7 We refer to the scheme resultng from the soluton of the followng optmzaton problem as CC-CM usng RAP-GCC: mn s.t. d D Π d ( 1 n =1 mn {ψ (q, p, m} + d D M j M j=1 M j, j, R,d 0, σd 2 2 2(M d + d + R,d n Π d =1 R,d R, (22a (22b (22c (, j, d [n] [m] D (22d As proposed n [9], the cachng placement can be smplfed accordng to the followng truncated unform cachng dstrbuton: { p j = 1 m, j m 0, j m + 1, (23 where the cut-off ndex m M s a functon of system parameters. The resultng cachng scheme s referred to as the Random LFU (RLFU cachng scheme [9]. RLFU cachng s equvalent to all recevers unformly cachng the most m popular vdeos fles. Therefore: { { M = M M j = m j m 0 j m + 1, j m j = 0 j m + 1. A packet of fle j s cached at any recever wth probablty: p c j = { M M+ j m 0 j m + 1, (24 whch results n the followng optmzaton problem: mn ( 1 Π d σd 2 n 2 2(M d + d + R,d (25a d D =1 ( ( ng m s.t. M + 1 M + ( M + + n n(1 G m + Π d R,d R, (25b d D =1 M,, R,d 0, (, d [n] D (25c where G m = m j=1 q j, and the coded multcast rate expresson n Eq. (25b, s derved n Eq. (17 of [9]. We refer to the resultng scheme as CC-CM usng RLUF-GCC. B. Symmetry Across Users and Fles (Sources Fnally, the smplest network settng would be for all users to have equal-sze caches, to request vdeo fles unformly and for all sources to have the same dstrbuton,.e. M = M, q,j = 1/m, σj 2 = σ2, (, j [n] [m]. Consequently, wth no loss of optmalty, we can assume that,j = and M,j = M (, j [n] [m] whch results n p c = M. It s mmedate to see that n ths settng the optmal M+ soluton assgns R,d = 0 (, d [n] D, and only the coded multcast rates need to be accounted for. Optmal values of and M n terms of mnmum average dstorton are gven by mn s.t. σ 2 2( M+ 2 ( ( n 1 M M + ( M + R M M M, 0 VI. SIMULATION RESULTS (26 In ths secton, we numercally compare the performance of the uncast and multcast cachng-aded transmsson schemes ntroduced n Sec. III and IV for the smplfed RAP- GCC settngs analyzed n Sec.V. We consder a network composed of n = 20 recevers and a lbrary wth m = 100 vdeo fles. We assume that vdeos are requested accordng to a Zpf dstrbuton wth parameter α: j α q j = m f=1 f j [m]. α In order to reduce the complexty of the RAP-GCC optmzaton, we assume R,d s ndependent of the demand and depends only on the fle dentty. Note that ths assumpton may lead to suboptmal solutons and hence the results n ths secton represent an upper bound on the optmal performance (n terms of dstorton. Fg. 4, dsplays the expected average dstorton acheved wth the LC-U and CC-CM schemes usng RAP-GCC. It s assumed that all recevers have the same cache sze, α = 0.6 and σj 2 s unformly dstrbuted n [0.7, 1.6]. The dstortons have been plotted (on a logarthmc scale for lnk capacty values of R = 2, 5, 8 bts/sample as recever cache szes vary from 5 to 100 bts/sample. As expected, CC-CM sgnfcantly outperforms LC-U n terms of average dstorton. Ths means that for a gven shared lnk capacty constrant, R, CC-CM s able to delver more vdeo layers to the recevers, reducng ther experenced dstortons, and equvalently, ncreasng ther vdeo playback qualty. Specfcally, for capacty R = 2 and cache sze M = 50, CC-CM acheves a 2.1 reducton n expected dstorton compared to LC-U. Observe that when the rate goes up to R = 8, for the same cache sze M = 50, the gan of CC-CM ncreases to 5.4. In Fg. 5, we smulate a homogeneous network scenaro wth α = 0 (fles are requested unformly and σj 2 = 1.5 j [100]. The expected dstortons acheved for LC-U and CC-CM (usng RAP-GCC are plotted for the values of R = 2, 5, 10 bts/sample as recever cache szes vary from 5 to 100 bts/sample. Observe how the gans of CC-CM are even hgher n ths scenaro. Ths s due to the ncreased coded multcast opportuntes that arse when fles have unform popularty [9]. Note that n ths case, for R = 10 and M = 50, the average dstorton wth CC-CM s 9.5 tmes lower

8 Expected Dstorton, log 2 (E(D LC U, R = 2 CC CM, R = 2 LC U, R = 5 CC CM, R = 5 LC U, R = 8 CC CM, R = Cache Sze, M (bts/sample Fg. 4: n = 20 recevers, m = 100 vdeos and Zpf parameter α = 0.6. For the multcast transmsson scenaro, cachng and transmsson schemes are based on RLUF and GCC respectvely. Expected Dstorton, log 2 (E(D LC U, R = 2 CC CM, R = 2 LC U, R = 5 CC CM, R = 5 LC U, R = 10 CC CM, R = Cache Sze, M (bts/sample Fg. 5: n = 20 recevers, m = 100 vdeos and Zpf parameter α = 0. For the multcast transmsson scenaro, cachng and transmsson schemes are based on RAP-GCC, analyzed n Sec. V-B. than wth LC-U. The mprovement factor goes up to 14 wth M = 70. In practce, recevers cache content based on the solutons derved from the optmzaton problems, and the sender transmts further layers through coded multcast and nave multcast (nstead of uncast whch results n dstortons lower than those ntally computed. We also remark that, when mplementng the schemes (B, the nteger constrant on µ,j, ρ,j and ˆρ,d can be releved to µ,j + ρ,j +ˆρ,d beng an nteger (, j, d [n] [m] [D]. For example, µ 1,1 = 2.5 means user 1 caches the two most sgnfcant layers fully and only caches half of the packets from the thrd layer. VII. CONCLUSION In ths paper, we have nvestgated the use of cachng for enhancng vdeo streamng qualty, or n a more abstract sense, reducng source dstorton. Recevers cache low rate versons of the vdeo fles and durng the transmsson phase further vdeo layers are delvered to enhance the vdeo playback qualty. We show that whle local cachng and uncast transmsson results n acceptable dstorton wthout the need of global coordnaton, the use of cooperatve cachng and coded multcast transmsson s able to provde order mprovements n average achevable dstorton by delverng more enhancement vdeo layers wth the same avalable broadcast rate. We remark that whle parttonng vdeos nto multple equal-sze layers s key to fully explotng multcast opportuntes, the performance of SVC scheme degrades as the number of layers ncreases due to codng overhead. Ths tradeoff between multcastng gans and codng overhead s the subject of ongong work. REFERENCES [1] [Onlne]. Avalable: vdeo-aware-wreless-networks [2] N. Golrezae, A. G. Dmaks, A. F. Molsch, and G. Care, Wreless vdeo content delvery through dstrbuted cachng and peer-to-peer gosspng, n 2011 Conference Record of the Forty Ffth Aslomar Conference on Sgnals, Systems and Computers (ASILOMAR, 2011, pp [3] A. F. Molsch, G. Care, D. Ott, J. R. Foerster, D. Bethanabhotla, and M. J, Cachng elmnates the wreless bottleneck n vdeo-aware wreless networks, arxv preprnt arxv: , [4] X. Wang, M. Chen, T. Taleb, A. Ksentn, and V. Leung, Cache n the ar: explotng content cachng and delvery technques for 5g systems, IEEE Communcatons Magazne, vol. 52, no. 2, pp , [5] N. Golrezae, K. Shanmugam, A. G. Dmaks, A. F. Molsch, and G. Care, Femtocachng: Wreless vdeo content delvery through dstrbuted cachng helpers, n Proceedngs IEEE INFOCOM, 2012, pp [6] M. A. Maddah-Al and U. Nesen, Fundamental lmts of cachng, n Proceedngs IEEE Internatonal Symposum on Informaton Theory (ISIT, 2013, pp [7] M. Maddah-Al and U. Nesen, Decentralzed coded cachng attans order-optmal memory-rate tradeoff, IEEE/ACM Transactons on Networkng,, [8] M. J, A. Tulno, J. Llorca, and G. Care, On the average performance of cachng and coded multcastng wth random demands, n 11th Internatonal Symposum on Wreless Communcatons Systems (ISWCS, 2014, pp [9], Order-optmal rate of cachng and coded multcastng wth random demands, arxv: , [10], Cachng and coded multcastng: Multple groupcast ndex codng, n IEEE Global Conference on Sgnal and Informaton Processng (GlobalSIP, Dec 2014, pp [11] H. Schwarz, D. Marpe, and T. Wegand, Overvew of the scalable vdeo codng extenson of the h. 264/avc standard, IEEE Transactons on Crcuts and Systems for Vdeo Technology, vol. 17, no. 9, pp , [12] T. M. Cover and J. A. Thomas, Elements of Informaton Theory. John Wley & Sons, [13] W. H. Equtz and T. M. Cover, Successve refnement of nformaton, IEEE Transactons on Informaton Theory, vol. 37, no. 2, pp , [14] M. J, K. Shanmugam, G. Vettgl, J. Llorca, A. Tulno, and G. Care, An effcent multple-groupcast coded multcastng scheme for fnte fractonal cachng, n Proc. IEEE Internatonal Conference on Communcatons (ICC, [15] G. Vettgl, M. J, A. Tulno, J. Llorca, and P. Festa, An effcent coded multcastng scheme preservng the multplcatve cachng gan, n Proceedngs IEEE INFOCOM, [16] K. Shanmugam, M. J, A. Tulno, J. Llorca, and A. Dmaks, Fnte length analyss of cachng-aded coded multcastng, n Proceedngs of Annual Allerton Conference on Communcaton, Control, and Computng, Oct [17] Z. Bar-Yossef, Y. Brk, T. Jayram, and T. Kol, Index codng wth sde nformaton, IEEE Transactons on Informaton Theory, vol. 57, no. 3, pp , 2011.