QoE-Driven Mobile Edge Caching Placement for Adaptive Video Streaming

Transcription

1 1 QoE-Dren Moble Edge Cachng Placement for Adapte Vdeo Streamng Chengln L, Member, IEEE, Laura Ton, Member, IEEE, Junn Zou, Member, IEEE, Hongka Xong, Senor Member, IEEE, and Pascal Frossard, Senor Member, IEEE Abstract Cachng at moble edge serers can smooth temporal traffc arablty and reduce serce load of base statons n moble deo delery. Howeer, the assgnment of multple deo representatons to dstrbuted serers s stll a challengng queston n the context of adapte streamng, snce any two representatons from dfferent deos or een from the same deo wll compete for the lmted cachng storage. It s therefore mportant, yet challengng, to optmally select the cached representatons for each edge serer n order to effectely reduce the serce load of base staton whle mantanng a hgh qualty of experence (QoE) for users. To address ths, we study on a QoE-dren moble edge cachng placement optmzaton problem for dynamc adapte deo streamng that properly takes nto account the dfferent ratedstorton (R-D) characterstcs of deos and the coordnaton among dstrbuted edge serers. Then, by the optmal cachng placement of representatons for multple deos, we maxmze the aggregate aerage deo dstorton reducton of all users whle mnmzng the addtonal cost of representaton downloadng from the base staton, subject not only to the storage capacty constrants at the edge serers, but also to the transmsson and ntal startup delay constrants at the users. We formulate the proposed optmzaton problem as an nteger lnear program (ILP) to prode the performance upper bound, and as a submodular maxmzaton problem wth a set of knapsack constrants to deelop a practcally feasble cost beneft greedy algorthm. The proposed algorthm has polynomal computatonal complexty and a theoretcal lower bound on ts performance. Smulaton results further show that the proposed algorthm s able to achee a near-optmal performance wth ery low tme complexty. Therefore, the proposed optmzaton framework reeals the cachng performance upper bound for general adapte deo streamng systems, whle the proposed algorthm prodes some desgn gudelnes for the edge serers to select the cached representatons n practce based on both the deo popularty and content nformaton. Index Terms Moble edge cachng, adapte deo streamng, wreless deo delery, deo-on-demand, submodular functon maxmzaton. I. INTRODUCTION In the last decade, moble multmeda serces, such as streamng of moble deos, hae become the man reason for the exponental growth of global moble data traffc oer cellular networks [1]. For example, as reealed by [2] n 216, real-tme entertanment that conssts of streamng deo and audo has become the largest traffc category on rtually eery network, and ts contnued growth s expected to lead all the networks. Such a dramatc growth of moble deo data poses sgnfcant challenges to both the deo content proders and the network serce proders. One notceable consequence s the resultant acceleraton of busy-hour traffc n relaton to the aerage traffc growth. Unlke other data traffc (e.g., web usage) that occurs throughout the day, deo usage s more lkely to occur durng eenng hours and thus has a prme tme. Globally, moble busy-hour traffc s expected to be 88 percent hgher than aerage-hour traffc by 22, compared to 66 percent n 215 [1]. Therefore, the moble deo traffc presents a hgh temporal arablty, whch ncurs congeston durng peak traffc hours and under-utlzaton durng off-peak hours. To reduce the heay traffc load of the base staton and prode context-aware serces n close proxmty to the moble multmeda users, moble edge computng has been ntroduced to push moble computng, network control and storage to network edges [3]. In partcular, moble edge cachng (MEC) s able to utlze the storage space of edge serers across the network and to perform multmeda content placement durng off-peak hours, thereby smoothng out the temporal traffc arablty and reducng congeston and access latences [4]. Smultaneously, the growng heterogenety of user populaton n terms of demands for specalzed deo content, dsplay deces, and access network capacty, has made the moble deo streamng a much more complex task. Adapte streamng technque, such as the dynamc adapte streamng oer HTTP (DASH), has emerged as an effecte method for deo streamng oer heterogeneous networks, whch can mproe the oerall user satsfacton by offerng seeral representatons of the same deo content to dfferent clents [5]. Each representaton s encoded wth a pre-defned btrate and/or resoluton by the content proder. The users then select the representaton that better fts to ther requrements and the network condtons. Therefore, t s promsng to study the potental performance gan ntroduced by the dynamc adapte streamng n addton to the moble edge cachng, and to nestgate the proper moble edge cachng placement schemes for dynamc adapte streamng systems, n order to alleate the traffc load of the base staton and reduce the access latences of the users (.e., beneft of cachng), and to satsfy heterogenous users demands (.e., beneft of adapte streamng). The basc queston n ths context s how to place the local caches of the dstrbuted edge serers wth approprate deo representatons such that the oerall users QoE n terms of deo qualtes and latences s maxmzed, gen the cache storage capacty of these edge serers. Dfferent from the cachng schemes for tradtonal deo streamng, the number of deo representatons stored at the content serer (whch s managed by the content proder) may become extremely huge snce multple representatons are stored for each deo. Ths results n a much more dffcult problem formulaton wth a hgher computatonal complexty to sole t. Therefore, n adapte streamng based MEC systems, people are not only concerned about whch deo should be cached at whch edge serer, they also want to know whch representaton of that deo should be selected for cachng. Studes to date hae nestgated related work to deal wth the aforementoned cachng and adapte streamng from dfferent perspectes. For moble deo delery, cachng at dstrbuted edge serers s demonstrated to be capable of greatly reducng the serce load of base staton, and replacng the usually weak backhaul connectons from the base staton wth hgh-speed local lnks from the edge serers to guarantee the low delay requrement of users [6] 1. An effcent cachng placement strategy s desgned for twoter wreless content delery networks to reduce the system desgn complexty by usng separate channels for content dssemnaton and serce [7]. For adapte streamng, the work n [8] deres a logarthmc QoE model based on emprcal results and formulates the cache management problem as a conex optmzaton problem. In order to cope wth dynamc deo segment requests, an onlne pre-fetchng algorthm s proposed n [9] to adaptely pre-fetch adapte streamng deo segments whle consderng the lmted bottleneck bandwdth between the content serer and the edge serer. 1 Accordng to [6], snce the edge serers are much closer to the moble users, localzed hgh-bandwdth communcaton from the edge serers can be acheed through enablng hgh frequency reuse or hgh-densty spatal reuse of communcaton resources, whle the backhaul communcaton from the base staton fals to do so.

2 2 Howeer, the lmtaton of these state-of-the-art cachng schemes s that the deo content characterstcs are not taken nto account. They manly focus on the rate (btrate of encoded representatons) and delay (transmsson delay) perspectes, and thus deo sources wth dfferent R-D behaors are treated n the same way, whch s not the optmal soluton for the adapte streamng scenaro where dfferent representatons hae dfferent R-D behaors. We therefore propose n ths paper to deelop a noel moble edge cachng placement optmzaton framework for the adapte streamng based deo-on-demand (VoD) system wth proper consderaton of the R-D propertes of the representatons from dfferent deos. Specfcally, we formulate the cachng placement optmzaton problem as an ILP, and target at maxmzng aggregate aerage deo dstorton reducton of all users whle takng nto account the mposed constrants on the backhaul lnk, the edge serers storage capacty and the users transmsson and ntal startup delay. Ths s accomplshed by the optmal assgnment of adapte streamng representatons of multple deo sources to dstrbuted edge serers. Through solng the proposed ILP to obtan the optmal soluton, we are able to prode a performance upper bound for the cachng placement. Howeer, t s NP-hard and thus too tme-consumng to be a practcal soluton for delay-senste deo streamng. In order to reduce the executon tme of the cachng placement algorthm n practce, we conert the orgnal optmzaton problem to an equalent set functon optmzaton problem and show ts submodularty. By usng the dmnshng return property of the submodular functons, we deelop a cost-beneft greedy algorthm for the cachng placement, whch has polynomal computatonal complexty and offers close-tooptmal performance (approxmaton rato s theoretcally proed to hae a lower bound and practcally shown to be aboe 95% under dfferent smulaton settngs n Secton VI). We conduct extense smulatons under dfferent system settngs. The smulaton results show that the proposed algorthm can scale ery well wth the sze of the system. It also strkes the tradeoff between the algorthm executon tme and the performance n terms of both the aerage dstorton reducton per user and the base staton transmsson rate. Oerall, the contrbutons of ths paper can be summarzed as follows. 1) Through ntroducng adapte streamng to allow cachng multple representatons for the same deo, the proposed cachng placement optmzaton framework addresses the users heterogenety ssue and thus achees an addtonal cachng performance gan (n terms of hgher aerage dstorton reducton per user and lower base staton transmsson rate) oer the cachng schemes desgned for general deo fles (.e., sngle representaton for each deo). It optmally allocates the cachng resources of edge serers not only among dfferent deos, but also among multple representatons of the same deo. 2) In addton to deo content popularty and network condtons that are commonly consdered by exstng cachng schemes for adapte streamng, deo content characterstcs (.e., the R- D property) are further taken nto account, to assgn dfferent utltes to the representatons wth the same btrate but from dfferent deos. In ths way, the actual performance of the cachng system s properly ealuated n terms of the users ewng qualty. 3) To effcently sole the proposed cachng placement optmzaton, we conert t to an equalent submodular maxmzaton problem wth a set of knapsack constrants. We deelop a polynomal-tme greedy algorthm and prode a theoretcal proof on the lower bound of ts approxmaton rato. The rest of ths paper s organzed as follows. Secton II reews the related works n lterature. In Secton III, we ntroduce the moble edge cachng placement framework and related system models. In Secton IV, we formulate the cachng problem as an ILP by consderng the users QoE and edge serers cache capacty constrants. In Secton V, we transform the orgnal ILP to an equalent submodular maxmzaton problem, and deelop a practcal approxmaton algorthm to sole ths problem wth close-to-optmal performance. Secton VI presents expermental results, and ealuates the gans of the proposed algorthm compared to exstng algorthms. The concludng remarks are gen n Secton VII. II. RELATED WORK The dea of usng moble edge cachng to support the cellular leel communcaton has been recently explored n [6], [1] [18]. In [1], Lu et al. summarze the desgn aspects and challenges of moble edge cachng. They further reeal that cachng at the wreless edge for 5G cellular networks s stll an open problem snce the unque lmtatons n wreless networks due to the archtecture and channel (such as the network topology, lnk nterference, users moblty, and lmted battery) must be consdered when desgnng an approprate cachng placement strategy. In [11], the authors study a cachng scheme for the 5G edge cloud network where contents are stored wth a prce determned by the moble network operator. The noel FemtoCachng archtecture n [6], [12] proposes moble edge cachng at the small-cell access ponts, by compensatng the backhaul capacty wth the storage capacty at the moble edge to effcently handle some hghly predctable bulky traffc (e.g., VoD traffc). The moble deo cachng placement oer dstrbuted edge serers s essentally used to mnmze the aerage downloadng delay of users. The authors n [13] deelop a dstrbuted cachng optmzaton algorthm a belef propagaton for the heterogeneous cellular networks wth edge serers, n order to mnmze the oerall downloadng delay. Sengupta et al. [14] study the fundamental nformaton theoretc lmt of moble edge cachng, reealng the optmal tradeoff between the latences and cache szes. The work n [15] formulates a jont routng and cachng problem that targets at maxmzng the fracton of content requests sered locally by the deployed edge serers, under the consderaton of some mportant features such as the storage and bandwdth capactes of edge serers, and the content request patterns of users. By further ncorporatng the users lnk nterference ssue, a jont cachng, routng and channel assgnment problem s proposed n [16] to maxmze the throughput of the deo delery oer coordnated small-cell cellular systems.whle most of the aboe works assume a pror knowledge about the content popularty, the authors n [17], [18] propose a context/trend-aware cachng scheme to predct the popularty nformaton based on the users context (e.g., hs/her personal characterstcs, equpment, or external factors), whch explctly learns the context-specfc popularty of deo content through onlne learnng and uses t to determne the cachng replacement decson. The onlne learnng here ndcates that the context nformaton becomes aalable n a sequental order and s used to update the best predctor for the short-term popularty of content at each tme step, as opposed to the learnng technques that generate the best predctor by learnng on the entre tranng set at only one dedcated tranng phase. Howeer, all these aboe studes only focus on the cachng assgnment problems for general (deo) fles. Ths s howeer not suffcent n the context of adapte deo streamng [1], where approprate btrate representatons need to be carefully determned and pre-fetched n the edge serers. In another lne of research, some works hae been done to leerage cachng n the dynamc adapte deo streamng system [8], [9], [19] [25]. From the rate adaptaton perspecte, Lee et al. [19] nestgate the btrate oscllaton and sudden rate change problem occurrng through the nteracton between the clents and caches, and

3 3 TABLE I COMPARISON WITH THE MOST RELEVANT WORKS ON MOBILE EDGE CACHING FOR VIDEO STREAMING. Ths work [6], [12] [13], [15], [16] [2] [9] [25] [23], [24] Applcable to adapte streamng Yes No No Yes Yes Yes Yes Optmal performance upper bound Yes No Yes Yes Yes No No Approxmaton algorthm guarantee Yes Yes Yes N/A Yes No No Operatonal-cost/rate-cost aware Yes Yes Yes Yes Yes No Yes Vdeo content characterstcs aware Yes No No No No Yes No propose an approach that uses shapng to elmnate such oscllatons. Jn et al. [2] apply cachng to adapte streamng, and study the optmal transcodng and cachng allocaton scheme n meda cloud n order to mnmze the total operatonal cost of delerng on-demand adapte deo streamng, wth the assumpton that each moble user accesses one edge serer for deo downloadng. Gao et al. [21] nestgate the tradeoff between storage and transcodng computaton n the cloud, and propose a cost-effcent partal transcodng scheme for content management based on user ewng patterns. Zhao et al. [22] further deelop a deo segment-based cachng strategy for multple representaton VoD systems to mnmze the storage and transcodng costs. In order to cope wth dynamc requests, the work n [9] proposes an onlne pre-fetchng algorthm to adaptely pre-fetch adapte streamng deo segments whle respectng the lmted bottleneck bandwdth between the content serer and the edge serer. To mproe the users QoE, the authors n [8] dere a logarthmc QoE model based on emprcal results and formulate a cache management problem for adapte streamng as a conex optmzaton problem, thereby prodng an analytcal framework for ths engneerng problem. The work n [23] proposes an nnetwork deo cachng polcy for nformaton centrc networks to enhance users QoE n terms of aerage user throughput, based on the content popularty dstrbuton. A QoE-dren DASH deo cachng and adaptaton algorthm s proposed n [24] to make the cachng and replacement decson based on the content context (e.g., segment popularty) and the network context (e.g., downlnk bandwdth). Howeer, all these works only focus on the operatonal-cost/rate perspecte and thus neglect the deo content characterstcs of the representatons from dfferent deo contents. Here, the deo content means the dstnct foreground, background and moton n the deo, whch results n dfferent rate-dstorton (R-D) behaors (consdered as the deo content characterstcs) for dfferent deo sources after encodng. In other words, ths dfference of deo content (or R-D behaors) between dfferent deos s not consdered n the aboe works, where the multple representatons encoded from dfferent raw deos but wth the same btrate are assumed to hae the same system utlty. Therefore, ther cachng performance depends only on the deo content popularty and network condtons. Howeer, as wll be justfed by the expermental results n Secton VI, t s only by carefully consderng the deo content characterstcs (.e., the R-D behaor) that the actual performance of the cachng system can be properly ealuated n terms of user utlty. In our preous work [25], we hae partally addressed ths ssue by proposng a wreless deo cachng placement optmzaton problem for dynamc adapte deo streamng and a fast approxmaton algorthm to mnmze the aerage deo dstorton of all clents, under the edge serers storage capacty constrants. In ths work, we further prode a general optmzaton formulaton as an ILP along wth ts optmal soluton as a performance upper bound. In addton, we also take nto account other QoE metrcs, such as the ntal startup delay, n order to better reflect the actual utlty of each deo stream. Fnally, we study n detal the approxmaton algorthm for the cache allocaton, and prode a theoretcal lower bound on ts performance. In summary, Table I lsts the dfferences between ths work and the most releant papers n the lterature on moble edge cachng for deo streamng. Wthn these references, [6] and [12] are the most related model. Through the comparson n Table I, t can be seen that the work n [6] and [12] s a cachng scheme desgned for general deo fles (.e., sngle representaton for each deo) and only consders deo content popularty dstrbuton and network condtons, whle ths work addresses the cachng resource allocaton among dfferent deos and dfferent representatons of adapte streamng through the consderaton of deo content characterstcs (.e., the R-D property). In addton, the femto-cache algorthm proposed n [6] and [12] has been selected as a comparson algorthm n Secton VI, whch justfes that compared to the femto-cache algorthm, ths work can achee a hgher cachng performance gan n terms of hgher aerage dstorton reducton per user and lower base staton transmsson rate. III. FRAMEWORK AND SYSTEM MODELS In ths secton, we ntroduce the moble edge cachng placement framework for dynamc adapte deo streamng systems and related models. A. Framework Consder a wreless adapte streamng based VoD system as llustrated n Fg. 1. Suppose that the base staton stores F deo fles, each of whch s encoded nto M dfferent representatons. S edge serers wth certan capabltes of pre-fetchng deo content are determnstcally placed n the wreless coerage regon of the base staton, and are assumed to connect to the base staton through sngle hop transmsson. If the connecton between the base staton and edge serers n some cases s mult-hop, the mult-hop connecton characterstcs can be consdered as the end-to-end transmsson rate between them. These edge serers are geographcally closer to the moble users and enable hgh-densty spatal reuse of the wreless resources wth hgh-speed localzed communcaton, whch s usually assumed to be much faster than the backhaul lnks connected to the base staton [12]. For the VoD serce wth a pror knowledge of the deo popularty dstrbuton, some popular deo fles can be pre-fetched by the edge serers durng the off-peak hours to relee the serce load of the base staton and to replace the weak backhaul communcaton. The moble edge cachng placement crtera for adapte streamng are as follows. Wheneer a moble user sends a playback request for a specfc deo, t attempts to download the hghest possble qualty representaton from ts adjacent edge serers n accordance wth the content placement and the aalable download lnk capacty. If the same hgh qualty representaton s cached n multple edge serers, the user mght want to download t from the edge serer wth the hghest transmsson rate, n order to reduce the ntal startup delay. That s, the user wll frst determne whether there s a representaton wth the hghest btrate aalable at one of ts adjacent edge serers and the download of ths representaton can be supported by the lnk capacty wth an acceptable downloadng delay. If yes, the user could download and playback that representaton; otherwse, t would make a further selecton for the representaton wth the next lower

4 4 btrate. Ths determnaton wll contnue untl a representaton wth an affordable btrate s found at an edge serer or the representaton wth the smallest btrate s reached. When no representaton of the requested deo s aalable at any adjacent edge serer, the user has to turn to the base staton and download the representaton wth the hghest btrate that could be afforded by the backhaul lnk connected to the base staton. Howeer, downloadng from the base staton wll result n a much more expense transmsson cost snce the backhaul communcaton resource s typcally ery lmted compared to the hgh-speed lnks offered by the adjacent edge serers. BS Vdeo 1 R1,M R1,2 R1,1 F M Tme VdeoF RF,M RF,2 RF,1 Tme B. System Models We now descrbe n more detal the model that we consder n ths work, and ntroduce the notaton. Let frst F denote the set of F deo fles that are offered to the users. Any deo fle f F s encoded nto a set of M representatons Z f = {z f,m m = 1,2,...,M} wth the m-th representaton z f,m hang an encodng btrate beng R f,m. We further suppose that ths set s sorted n a decreasng order of the encodng btrate,.e., R f, > R f,j, 1 < j M. Therefore, the complete set ncludng all representatons for all the deo fles can be denoted as Z = f F Z f. For the sake of smplcty, and wthout loss of fundamental generalty, we adopt the assumpton from [2], that each deo fle has the same lengtht. Such assumpton s manly proposed for the notatonal conenence, and could be easly lfted by breakng a longer fle nto multple fles of the same length [12]. If n some scenaros the deo lengths are sgnfcantly heterogeneous and ths assumpton becomes no longer reasonable, we can use the notaton T f to represent the length of deo fle f n the cache capacty constrant of ILP n Eq. (8b) (or ts equalent submodular problem n Eq. (12b)), whch would not fundamentally change the correspondng analyss and algorthm desgn. To llustrate the connecton between the edge serers and the users, the wreless network s defned by a bpartte graph G su = (S,U,E su), where S represents the set of S edge serers, U denotes the set of U moble users, and a graph edge (s,u) E su ndcates that a wreless communcaton lnk exsts from the edge serer s S to the user u U. The download lnk transmsson rate of the wreless lnk (s,u) s denoted by c (s,u) 2. For each edge serer s S, the cache storage capablty s constraned by the capacty. Fnally, we denote by N(u) the neghborng edge serers of user u U. We assume that N(u) s sorted n a decreasng order of the download lnk capacty, such that () u N(u) represents the edge serer wth the -th largest capacty of the lnk to the user u. In ths paper, we study the cachng system wth the cachng placement decson to be made for a certan tme perod (e.g., seeral hours durng the peak hours, or een seeral days), durng whch the aerage demand for the set of F deo fles s assumed to be known n adance, as n [12], [2], [29]. In ths way, the backhaul s only used to refresh the caches at the rate at whch the user request dstrbuton eoles oer tme, whch s a much slower process than the tme scale at whch the users place ther requests [12]. Therefore, we adopt the assumpton from [12], [2], that users requests are statstcally ndependent and a probablty mass functon P u,f s used to represent the aerage 2 In ths paper, we assume that we hae detected and known the accurate channel state nformaton (CSI) for the upcomng transmsson frame and that the transmsson rate c (s,u) s known a pror. For the tme-ary wreless channel when c (s,u) s not perfectly known and may change oer tme, channel predcton technques [26] can be used to estmate the lnk transmsson rate. For example, the fnte state Marko channel model [27], [28] s wdely adopted as a good approxmaton n modelng and predctng the tme-aryng processes of wreless lnks. Howeer, the detaled descrpton of these channel predcton technques s beyond the scope of ths paper. (a) Fg. 1. (a) Example of the system layout, where moble users are randomly dstrbuted, whle edge serers are connected to the base staton wth backhaul lnks and can be determnstcally placed n the coerage regon. (b) The connectty bpartte graph ndcatng how moble users are connected to the edge serers. probablty that the deo fle f F s requested by the user u U wthn ths tme perod. Ths ndependent user request model s an acceptable approxmaton n an aerage sense or when the content popularty araton oer tme s relately slow. We further consder a cachng system where a representaton of a deo fle s ether cached fully (.e., the whole representaton of the length T ) or not cached at all n any edge serer 3, the representaton placement strategy can be represented by a bpartte graph G zf,m,s = (Z,S,E zf,m,s) between ertces representng edge serers n S, and ertces descrbng deo representatons n Z. An edge (z f,m,s) E zf,m,s s drawn when z f,m (.e., the m-th representaton of deo flef) s stored n the cache of edge serer s. To better understand the representaton placement strategy as shown by the bpartte graph, we can further denote A F M S as the F M S adjacency matrx of G zf,m,s, such that s S, a s f,m = 1 ndcates that an edge (z f,m,s) E zf,m,s exsts and a s f,m = denotes the absence of an edge between z f,m and s,.e., a s f,m = C. Qualty-of-Experence Models (b) 1, f the edge serer s caches the m-th representaton of deo f;, otherwse. Accordng to [32], both the ntal startup delay (the watng tme nteral between the clent s request and the begnnng of the playback) and the aerage deo qualty (the aerage deo dstorton) are the key factors that affect the qualty of experence (QoE) of deo streamng serces. For each user u U, the ntal startup delay constrant requres that the watng tme nteral between submttng a request and the actual deo playback should not exceed the maxmum tolerable 3 In some scenaros where the szes of deo fles are ery large (e.g., HD deos, or deo length T s too long) and the cachng storage resource becomes the crtcal concern, we can alternately adopt the partal cachng strategy that caches the frst porton of the same length T (T T ) for each representaton of each deo. The reason s as follows. Based on the studes on users behaor and ewng patterns n some practcal VoD systems, such as YouTube [3] and PPTV [31], t s obsered that usually users only watch a small porton of the full content of a deo. For example, statstcs n [3] show that 95% of the ews last shorter than 2 seconds. Therefore, the consumpton of cachng storage greatly decreases by only partally cachng the frst T seconds of each representaton (e.g., T = 2 s), and the system s stll effcent snce most of the tme (e.g., > 95%) the users are satsfed wth the partally cached content. (1)

5 5 watng tme of that user, whch s denoted as d u,max. Let us assume frst that the deo representaton z f,m Z s aalable n the cache of user u s adjacent edge serer s S. Let us further denote wth T the tme fracton wthn a deo fle that s requred to be buffered by the user before the actual playback starts on the user s screen. Then the ntal startup delay experenced by the user u to download the representaton z f,m from the edge serer s s: d s u,f,m = R f,m T, u U, z f,m Z, s S. (2) c(s, u) Here, we set the transmsson rate of lnks from the non-adjacent edge serers of a user to a small poste alue that s arbtrarly close to zero,.e., for all s / N(u) we hae c (s,u) = ε, where ε and accordngly d s u,f,m +. Smlarly, when the requested deo s not aalable n the edge serers, the ntal startup delay experenced by the user u to download z f,m from the base staton s: d u,f,m = R f,m T c(bs,u), u U, z f,m Z, (3) where c(bs, u) s the download lnk transmsson rate of the wreless lnk connectng the base staton and the user. Then, we use a general rate-dstorton functon D max D f (R f,m ) to denote the dstorton of the m-th representaton of the deo f wth the encodng btrate R f,m, where D max and D f (R f,m ) represent a constant maxmal dstorton when no deo s decoded and the dstorton reducton (or qualty mproement) after successfully decodng ths representaton, respectely.by utlzng the R-D model n [33], D f (R f,m ) can be expressed as: D f (R f,m ) = D max D θ R f,m R (4) where the arables, θ, R and D, are emprcal parameters that depend on the actual deo content; they can be estmated as the fttng parameters from the emprcal rate-dstorton cures of dfferent deos by usng regresson technques. IV. QOE-DRIVEN CACHING PLACEMENT OPTIMIZATION PROBLEM In ths secton, we descrbe the QoE-dren moble edge cachng placement optmzaton problem for adapte streamng, and formulate t as an ILP. A. Problem Descrpton and Challenges The QoE-dren moble edge cachng placement problem for adapte streamng can be summarzed as follows: gen the representaton set of source deo fles, the fle popularty dstrbuton, the edge serer storage capacty and the network topology, how to place the representatons of the deo fles n the dstrbuted edge serers such that the total system utlty (whch s defned by Eqs. (7) and (8a) n the next subsecton) s maxmzed subject to the cachng capacty constrant of each edge serer and the downloadng delay requrement of each user. If each deo fle has only one representaton and each user has only access to one edge serer, the optmal placement strategy becomes smple and straghtforward. That s, each edge serer should cache as many of the most popular deo fles as possble untl ts storage s full. Howeer, for the case of dense edge serer deployment where each user can hae access to more than one edge serers, the optmal content placement strategy becomes hghly nontral. Furthermore, f each deo fle s aalable n dfferent representatons wth dfferent btrates, the optmal placement problem becomes een more complcated. Compared to the cachng problem wth general fles, the fundamental techncal challenges ntroduced by the adapte deo streamng,.e., multple representatons of a deo fle need to be cached, can be explaned as follows. The general fle cachng problem usually addresses the cachng resource competton ssue among dfferent fles by placng approprate fles n the dstrbuted edge serers. It s also based on the assumpton that there s no dfference between dfferent fles n terms of the system utlty,.e., downloadng a dfferent fle would lead to the same utlty mproement (e.g., the ncrease of ht rato). When the adapte deo streamng s taken nto account, howeer, people are not only concerned wth whch deo fle should be cached at whch edge serer, they also want to know whch representaton(s) should be selected to cache n order to maxmze the oerall system utlty. Ths means that not only dfferent deo fles, but also the multple representatons of the same deo fle wll compete for the cachng resource at the edge serers. In addton, due to the dfference of deo content characterstcs, downloadng the same btrate representaton of dfferent deo fles would also result n dfferent utlty mproement (e.g., the dstorton reducton). Een for the same deo fle, the cachng resource allocaton problem becomes more complcated snce the relatonshp between the utlty mproement (e.g., the dstorton reducton) and the btrate of the dfferent representatons s nonlnear and presents the dmnshng return property. It should be noted that all of the aboe ssues ntroduced by the adapte streamng cannot be straghtforwardly addressed by the general fle cachng problem, whch motates us to study the followng cachng placement optmzaton problem for adapte streamng. B. System Utlty Functon Frst, we ntroduce two sets of auxlary bnary arables: 1, f user u gets the m-th representaton βu,f,m s = of deo f from edge serer s;, otherwse. 1, f user u gets the m-th representaton γ u,f,m = of deo f from the base staton;, otherwse. We then defne the followng utlty functon, based on both the aerage deo dstorton reducton experenced by the user u and the cost of the representaton downloadng ether from the edge serer or the base staton: Q u = M βu,f,m s P u,f [ D f (R f,m ) η R f,m ] f F m=1 s N(u) + M γ u,f,m P u,f [ D f (R f,m ) η R f,m ] (7a) f F m=1 M βu,f,m s P u,f D f (R f,m ) f F m=1 s N(u) + M γ u,f,m P u,f [ D f (R f,m ) η R f,m ]. (7b) f F m=1 As usually done n many rate-dstorton optmzaton problems [34], n the utlty functon defned n Eq. (7a), we mpose the bandwdth constrants (from the edge serers and the BS) as the cost penalty, rather than puttng them as hard constrants. It represents a typcal optmzaton objecte that trades bandwdth (resource cost) for deo qualty. Specfcally, [ D f (R f,m ) η R f,m ] n the frst term of Eq. (7a) ncludes the deo dstorton reducton D f (R f,m ) (5) (6)

6 6 of downloadng the representaton z f,m, and a transmsson cost penalty η R f,m where η s the unt prce parameter correspondng to the representaton downloadng of z f,m from the adjacent edge serers. As constraned by Eq. (8f), for any user u U and any deo fle f F, at most one βu,f,m, m s = 1,2,...,M, s N(u) equals to 1. Therefore, the weghted summaton (where the weght s the deo request probablty P u,f ) oer all F deo fles, f F M m=1 s N(u) βs u,f,m P u,f [ D f (R f,m ) η R f,m ], represents the aerage deo dstorton reducton plus the aerage transmsson cost penalty experenced by user u downloadng requested deo representatons from ts adjacent edge serers. Lkewse, the second term n Eq. (7a) represents the aerage deo dstorton reducton plus the aerage transmsson cost penalty experenced by user u downloadng requested deo representatons from the base staton. Due to the lmted bandwdth aalable n the backhaul channel, the unt prce for downloadng from the base staton s much hgher than the unt prce for accessng the adjacent edge serers (.e., η η ) 4. As a consequence, the oerall cachng system wll prefer to store representatons n the edge serers, snce downloadng the same representaton from an edge serer achees the same dstorton reducton gan whle the transmsson cost s much lower. Users wll only access the base staton for representaton downloadng n some rare cases when they are hghly rewarded. Ths happens ether when there s no representaton of the requested deo cached n ther adjacent edge serers, or when the cached content has a ery poor qualty and the dstorton reducton gan of a better qualty representaton s so hgh that downloadng t from the base staton wth a hgher transmsson cost s worthy for the oerall utlty mproement. For the sake of smplcty, herenafter, we assume that η and η s a poste constant, and thus defne the utlty functon as shown n Eq. (7b). C. Optmzaton Problem Formulaton Mathematcally, the QoE-dren moble edge cachng placement problem for adapte streamng can be formulated as an nteger lnear program (ILP), as follows: ILP: s.t. max A,β,γ f F m=1 u U Q u M a s f,m R f,m T, s S, (8a) (8b) β s u,f,m d s u,f,m d u,max, u U, z f,m Z, s S, (8c) γ u,f,m d u,f,m d u,max, u U, z f,m Z, (8d) βu,f,m s a s f,m, u U, z f,m Z, s S, (8e) M M γ u,f,m + βu,f,m s 1, u U, f F, (8f) m=1 m=1 s N(u) β s u,f,m {,1}, u U, z f,m Z, s S, γ u,f,m {,1}, u U, z f,m Z, a s f,m {,1}, z f,m Z, s S. (8g) (8h) (8) 4 For the sake of smplcty, we assume n ths paper that the unt downloadng prce η s the same for dfferent edge serers, snce the downloadng cost of the same representaton from dfferent edge serers dffers ery slghtly compared to the much larger downloadng cost from the base staton. Ths assumpton could be lfted by assgnng a dfferent unt downloadng prce η s n Eq. (7a) to an edge serer s. Then, the ILP n Eq. (8) can be smlarly soled by settng the optmzaton objecte accordng to Eq. (7a). For the equalent submodular maxmzaton problem and ts approxmaton algorthm, we only need to re-sort the set of neghborng edge serers N u for each user u, n such a way that () u N(u) represents the edge serer offerng the -th smallest unt downloadng prce. In the aboe ILP, the objecte s to maxmze the aggregate utlty defned n Eq. (7b), or equaelent to maxmze the aerage deo dstorton reducton of all users (whch s equalent to mnmzng the aggregate aerage deo dstorton) whle mnmzng the transmsson cost of the representaton downloadng from the base staton. The decson arables are the representaton placement strategy represented by the adjacency matrx A F M S {,1} F M S and the sets of auxlary bnary arables β and γ. The constrant n Eq. (8b) represents the cache capacty constrants of each edge serer, where T s the tme duraton of each deo fle. The startup delay constrants n Eqs. (8c) and (8d) specfy that the ntal startup delay experenced by the user u to download the representaton z f,m ether from the edge serer s or the base staton should not exceed the maxmum tolerant watng tme d u,max. The constrant n Eq. (8e) sets up a consstent relatonshp between the decson matrx A and auxlary arables β, ensurng that the representaton selected by a user s already cached and aalable at the edge serer s. The constrant n Eq. (8f) mposes that for any deo f, the user u can only download at most one representaton from at most one edge serer (or the base staton), to aod duplcated downloadng of multple representatons for the same deo or the same representaton from multple edge serers (or the base staton). Together wth the startup delay constrants n Eqs. (8c) and (8d), t ensures that only one representaton wll be downloaded by the user u for the deo f. Furthermore, ths representaton s the largest possble btrate representaton under the user s download lnk capacty and the startup delay constrants, snce otherwse the alue of the objecte functon n Eq. (8a) decreases, whch ndcates a non-optmal soluton. The constrants n Eqs. (8g)-(8) defne the bnary decson and auxlary arables, respectely. The optmal soluton of the ILP can be obtaned by the generc soler IBM ILOG CPLEX [35], usng a branch-and-cut search. The branch-and-cut procedure follows a search tree consstng of nodes, each of whch represents a relaxed LP subproblem to be soled. It then noles runnng a branch and bound algorthm to create two new nodes from a parent node, and addng addtonal cuttng planes to tghten the LP relaxatons and reduce the number of branches requred to sole the orgnal ILP. In general, the branch-and-cut search requres exponental computatonal complexty to achee the optmal soluton n the worst case [36], [37]. Therefore, the ILP problem n Eq. (8) s NP-hard. Specfcally, t can be obsered that the cardnalty of the decson arables A, β, and γ s FMS, UFMS, and UF M, respectely. By usng the branch and bound method for the bnary decson arables, n the worst case, the number of nodes obsered by the CPLEX soler would be upper bounded by 2 FMS 2 UFMS 2 UFM. At each node the soler needs to sole a relaxed LP problem wth the SIMPLEX method. Ths corresponds to an exponental computatonal complexty O(2 2U 3F 3M 2S ) and thus ncurs an ncredbly long executon tme when the problem scale becomes large. V. EQUIVALENT SUBMODULAR MAXIMIZATION PROBLEM AND ALGORITHM DESIGN In order to effcently cope wth the dffcultes of solng the ILP n Eq. (8), n ths secton, we conert t to an equalent set functon optmzaton problem. We proe that t s a submodular maxmzaton problem oer ndependence constrants. We fnally deelop new practcally effcent algorthms wth polynomal computatonal tme complexty and theoretcal approxmaton guarantees. A. Equalent Problem Formulaton as a Set Functon Optmzaton In accordance wth the adjacency matrx A F M S n the ILP n Eq. (8), the fnte ground set of the equalent set functon

7 7 optmzaton problem can be ewed as: V = {V 1,...,V s,...v S}, (9) V s = { s 1,1,..., s 1,M,..., s f,m,..., s F,1,..., s F,M}, s S, where the ground set s parttoned nto S dsjont subsets. Each subset V s denotes the full set of all representatons of all fles that may be cached on the edge serer s, and the element s f,m represents the placement of the m-th representaton of deo fle f (.e., z f,m ) on the cache of the edge serer s. For a gen adjacency matrx A F M S, the correspondng representaton placement set A V can be defned n such a way that s f,m A corresponds to the case a s f,m = 1 and ce ersa. When ntal startup delay constrants are taken nto account, the feasble set should be re-defned by elmnatng the elements that olate the maxmum tolerance of the ntal startup delay from the ground set V n Eq. (9). From the perspecte of users, for any u U, the ntal startup delay constrant ndcates that a representaton that could be downloaded from an edge seer wthn the maxmum delay bound s consdered feasble and mght contrbute to the aggregate expected dstorton reducton. In the ILP n Eq. (8), such a constrant s ndcated by Eq. (8c), whch corresponds to a feasble subset of the ground set V: { } Ω u = f,m s V ds u,f,m d u,max, s S, z f,m Z V, f,m (A Ωu)) u U. (1) It should be noted that for a gen representaton set F M and known transmsson rate for lnks between S and U, the feasble subset Ω u s also gen wth respect to the alue of d u,max. Accordngly, the utlty functon of user u n Eq. (7) can be rewrtten n terms of the set functon, by also consderng the ntal startup delay constrants, as: Q u(a) = N(u) M [ m 1 N(u) ] (1 1 (j)u (A Ωu)) (11) f,n f F m=1 =1 n=1 [ 1 ] (1 1 (j)u 1 ()u P f,m (A Ωu) u,f D f (R f,m ) + f F [ M N(u) (1 1 (j)u (A Ωu)) f,m m=1 ] P u,f [ D f (R f,m ) η R f,m ]. The defnton of Eq. (11) follows the dstrbuted cachng placement crteron n Secton III-A. In Eq. (11), 1 x X s an ndcator functon, whch s 1 f x X and otherwse; and the term [ m 1 n=1 1 ()u f,m N(u) (1 1 (j)u f,n (A Ωu))] [ 1 (1 1 (j)u f,m (A Ωu))] = 1 s the ndcator functon defned oer the feasble (A Ωu) placement set A Ω u for the case where the m-th representaton of deo fle f s the best representaton that user u could fnd n ts neghborng edge serers whle the ntal startup delay constrant s satsfed, and ths representaton s at the cache of edge serer () u. In N(u) partcular, [ m 1 n=1 (1 1 (j)u (A Ωu))] = 1 ndcates that f,n no representaton wth an ndex smaller than m s aalable at any of the adjacent edge serers; and [ 1 (1 1 (j)u (A Ωu))] = 1 f,m ndcates that the m-th representaton s not aalable at any of the edge serers wth a larger download lnk rate (shorter ntal startup N(u) delay) than the edge serer () u. The term [ M m=1 (1 1 (j)u (A Ωu))] = 1 ndcates that no representaton of deo fle f f,m can be found n any neghborng edge serer of user u, and the user u wll download from the base staton the representaton z f,m that has the hghest btrate whle stll respectng the ntal startup delay constrant, namely z f,m = argmax {zf,m Z, d u,f,m d u,max}r f,m. Therefore, the orgnal optmzaton problem ILP n Eq. (8) can be reformulated as a constraned set functon optmzaton problem that leads to the same soluton of the ILP based on the dstrbuted cachng placement crteron n Secton III-A, as follows: SUB: max Q u(a) A V u U (12a) s.t. A I, (12b) { M } I = A V 1 s f,m A R f,m T, s S. f F m=1 Comparng the orgnal problem ILP n Eq. (8) wth the equalent set functon optmzaton formulaton SUB n Eq. (12), t can be seen that the objecte functon and the frst constrant n the problem ILP n Eq. (8) are transformed to Eqs. (12a) and (12b) n problem SUB, respectely. The ntal startup delay constrant of each user u n Eq. (8c) s presered by the feasble subset Ω u appled n the objecte functon Q u(a) as defned n Eq. (11), whle the delay constrant n Eq. (8d) s ensured by the defnton of z f,m n Eq. (11). The constrants n Eqs. (8e) and (8f) are also guaranteed snce Q u(a) n Eq. (11) s dered accordng to the dstrbuted cachng placement crteron n Secton III-A.That s, for each deo, only one acheable representaton wth the hghest btrate wll be selected for each user wth ts coeffcent, ether [ m 1 n=1 N(u) (1 1 (j)u N(u) (1 f,n (A Ωu))] [ 1 (1 1 (j)u (A Ωu))] 1 ()u or [ M f,m f,m (A Ωu) m=1 (A Ωu))], n Eq. (11) beng one, whle the coeffcents of the 1 (j)u f,m other representatons are all zeros. B. Submodular Maxmzaton Problem Submodularty, often ewed as a dscrete analogue of conexty, plays a central role n dscrete optmzaton. Its characterzng property, dmnshng margnal returns, makes submodular maxmzaton an effcent approach for many real-world applcatons, ncludng approxmaton algorthms and many challengng problems n machne learnng. We show now that problem SUB n Eq. (12) s a submodular maxmzaton problem. We frst reew and nclude the defnton of submodular functons accordng to [38] [4]. Defnton 1. Submodularty: Let V be a fnte ground set, and a set functon g : 2 V R s submodular f and only f for any sets X Y V and for any element (Y \X), we hae or equalently g(x)+g(y) g(x Y)+g(X Y), (13) g(x {}) g(x) g(y {}) g(y), (14) whch captures the dmnshng margnal return characterstcs such that the beneft of addng a new element nto the set decreases as the set becomes larger. We now proe that the objecte functon of the problem SUB n Eq. (12) s monotone submodular. Proposton 1. The objecte functon n Eq. (12a) s a monotone submodular functon oer the ground set V as defned n Eq. (9). Proof: Ths proposton can be proed by usng the defnton of monotoncty and submodularty. We further obsere the cache storage constrant of edge serer s S n Eq. (12b), and note that each element s f,m A (correspondng to the casea s f,m = 1 na F M S) has a non-unform

8 8 cost of R f,m T and s has a storage budget of. Ths constrant can be ewed as a knapsack constrant on the subset V s V. Oerall, the dstrbuted cachng placement problem n Eq. (12) s a submodular maxmzaton problem subject to a set of knapsack constrants, whch stll s generally NP-hard and requres exponental computatonal complexty to reach the optmum by ether ILP or other optmzaton methods. It s expected that by explotng submodularty, the polynomal-tme greedy algorthm s able to prode an effecte approxmaton of the optmal soluton of ths NP-hard problem [41]. Howeer, accordng to [41], [42], the greedy algorthm can only effcently address the smplest case (.e., a submodular maxmzaton problem subject to one knapsack constrant) wth theoretcal approxmaton guarantee. When the number of knapsack constrants becomes greater than one, the greedy algorthm n general s no longer effcent, and n the worst case ts approxmaton rato wll be arbtrarly bad. An excepton exsts f the set of multple knapsack constrants form a matrod [38], such as the cache placement problem n [6] and [12] where the knapsack constrants are proed to be a partton matrod snce all deo fles hae the same sze. In comparson, the proof of matrod for the multple knapsack constrants n [6] and [12] no longer holds n our case because of the dfferent deo fle szes ntroduced by adapte streamng. Howeer, due to the specal structure of the knapsack constrants n Eq. (12) (.e., each knapsack constrant s mposed on the subset V s V, and the set of all knapsack constrants s mposed on the fnte ground set V), we deelop n the next subsecton a polynomal-tme greedy algorthm and prode a theoretcal proof on the approxmaton rato of the proposed greedy algorthm. C. Approxmaton Algorthm To effcently sole the submodular maxmzaton problem n Eq. (12) wth polynomal tme complexty and theoretcal approxmaton guarantees, we deelop a k-cost beneft (k-cb) greedy algorthm. The system parameter, k =,1,2,... specfes the sze of the ntal set. Specfcally, the proposed k-cb greedy algorthm consders all feasble ntal sets A V of cardnalty k. Startng from any ntal set A, at step t, the cost beneft greedy procedure terately searches oer the remanng set V t 1 \ A t 1 and nserts nto the partal solutona t 1 an element accordng to Eqs. (16) and (17), untl the remanng set reduces to an empty set. In other words, the cost beneft procedure adds at each teraton an element that maxmzes the rato between margnal beneft Q(A t 1 { s f,m}) Q(A t 1 ) and cost R f,m T among all elements stll affordable under the remanng storage budget untl no more elements can be added. The proposed k-cb greedy algorthm then enumerates all ntal sets A V of cardnalty k, augments each of them followng the cost beneft greedy procedure, and selects the ntal set acheng the largest alue of the objecte functon Q(A) = u U Qu(A) and fnds ts soluton set as the fnal placement set A k. For the specal case of k =, the algorthm reduces to a smple cost beneft greedy algorthm startng wth A =. On the other hand, f we remoe the cost term R f,m T n Eqs. (15) and (16) and only add at each teraton an element maxmzng the margnal beneft Q(A t 1 { s f,m}) Q(A t 1 ), the algorthm reduces to a k-smple greedy algorthm. The complete k-cost beneft greedy algorthm s descrbed n Algorthm 1. Snce the k-smple greedy algorthm s only slghtly dfferent from Algorthm 1, t s thus omtted due to the space lmt. In terms of computatonal complexty, the runnng tme of the proposed k-cb greedy algorthm s O((SFM) k+1 U), ndcatng a polynomal tme complexty and a ery short addtonal mplementaton delay that s ntroduced by runnng the algorthm to fnd Algorthm 1 k-cost beneft (k-cb) greedy algorthm Input: system parameter k; fnte ground set V; deo length T ; encodng btrate R f,m for any representaton z f,m Z; and cache storage capacty for any edge serer s S. Output: cachng placement set A k 1: d := 1 // the ndex of the ntal set 2: for any ntal set A V and A = k do 3: V := V and t := 1 // ntalzaton 4: for t = 1,2,3,... do 5: // greedy search teraton 6: 7: 8: f Q(A t 1 { s f,m θ t := max }) Q(At 1 ) f,m s Vt 1 \A t 1 R f,m T (15) s Q(A t 1 { s t f,m f t,m := arg max }) Q(At 1 ) t f,m s Vt 1 \A t 1 R f,m T (16) M 1 s t f,m (At 1 Vs t ) { s t } R f,m T t (17) f f F m=1 t,m t then 9: A t := A t 1 { st f t,m t } and V t := V t 1 1: else 11: A t := A t 1 and V t := V t 1 \{ st f t,m t } 12: end f 13: f V t \A t then 14: t := t+1 15: else 16: break 17: end f 18: end for 19: A d := A t and d := d+1 2: end for 21: A k := argmax {1,2,...,d 1} u U Qu(A ) the fnal cachng placement set. As the alue of k ncreases, the runnng tme of the proposed algorthm becomes longer whle the performance mproes. In Theorem 1, we proe that when k = 2, the theoretcal worst-case performance guarantee of the proposed algorthm s 1 (1 1/e),.e., ts soluton achees at least the rato 2 1 (1 1/e).316 of the optmal objecte alue. In practce, as t 2 wll be shown n the smulaton results n Secton VI, the algorthm performance approxmaton rato s much hgher than the theoretcal lower bound, whch s generally aboe.95. Theorem 1. The better cache placement result acheed by runnng separately and comparng the 2-cost beneft greedy algorthm gen n Algorthm 1 and the 2-smple greedy algorthm prodes a 1 (1 2 1/e) approxmaton. That s, n the worst case, t can achee a performance guarantee of rato 1 (1 1/e) to the optmum. 2 Proof: Ths theorem can be proed by usng the dmnshng return property of submodular functons. For the detals, please refer to Appendx A. VI. EXPERIMENTAL RESULTS In ths secton, we ealuate the performance of our moble edge cachng placement optmzaton algorthms, and dere smple gudelnes for effecte cache allocaton n wreless adapte streamng systems under dfferent smulaton settngs. We compare ther performance wth two schemes n the recent lterature: 1) Femto-Cache, the femto-cachng system and ts assocated greedy algorthm proposed n [12], whch ams at mnmzng the aerage downloadng delay