Energy-Efficient Caching for Scalable Videos in Heterogeneous Networks

1 Ener gy-Ef ficient Caching for Scal able V ideos in Heterogeneous Netw orks Xuewei Zhang, Student Member , IEEE , T iejun Lv , Senior Member , IEEE , W ei N i, Senior Member , IEEE , J ohn M. Ciof fi, F e llow , IEEE , Norman C. Beaulieu, F ellow , IEEE , and Y . Jay Guo, F ellow , IEEE Abstract —By suppressing r epeated conten t deliv eries, wireless caching has the potential to substantially impr ove the energy efficiency (EE) of the fifth g en eration (5G) communication networks. In this paper , we propose two nov el energy-efficient caching schemes in heter ogeneous networks, namely , scalable video codin g (S VC)-based fractional caching and S VC-based random caching, which can p rovide on-demand video services with differ en t perceptual qualities. W e deriv e the expressions for successful transmission p robabilities and erg odi c service rates. Based on t h e derivations and the established power consumption models, the EE maximization problems are formulated for the two proposed caching schemes. By taking logarithmic approx- imations of the l 0 -norm, the p roblems are efficien tly solved by the standard gradient projection method. Numerical results validate the theor etical analysis and demonstrate the superiority of our proposed caching schemes, compar ed to t hree benchmark strategies. Index T erms —Energy efficiency (EE), heterog en eous networks, scalable video coding (SVC), standard gradient projection method, wi reless caching. I . I N T RO D U C T I O N Exposed to inf o rmation explosion and data tsunami, we have witnessed explo si ve traffic surges for socializing , workin g and entertainme n t. It is forecasted that the total am ount of data traffic is expected to achieve a t 100 exabytes in 20 23, and over 75% of this traffic is expected to b e ge n erated from ba ndwidth- demand in g multimedia video serv ices [1] . Notably , there a re a large number of repeated deliveries for po pular video files [2], which would ca u se h uge resource w a stes and aggravate traffic burdens ov er ba c k haul links. Therefor e, inn ovati ve techniq ues are desired to address the repeated deliv er ies of bandwidth - demand in g popular video s. In light of this, wireless caching has been pro posed as the appealing can d idate techniq ue in the fifth gen eration (5G) c ommun ic a tion networks [3]. Wireless caching can ef f ectiv ely relieve the severe tra ffic burden over backhau l lin ks and redu ce service delay [4]–[6]. Additio n ally , The financi al support of the Natio nal Natura l Sci ence Foundatio n of China (NSFC) (Grant No. 61671072) is gratefully ackno wledged. ( Corr esponding author: T iejun Lv . ) X. Z hang, T . L v and N. C. Bea ulieu are with the School of Informatio n and Communicati on Enginee ring, Beijing Uni versit y of Posts and T elecom- municati ons (BUPT), Beijing 10087 6, China (e-mail: {zhangxw , lvti ejun, nborm}@bup t.edu.cn). W . Ni is with Data61, Commonwealth Scientific and Industrial Research (e-mail: wei.ni@data61.csi ro. au). J. M. Ciof fi is with the Department of Electric al Engineering, Stanford Uni versity , Stanford, CA 94305 USA (e-mail: ciof fi@stanford.edu). Y . J. Guo is wit h the Globa l Big Data T echnologies Centre, Univ ersity of T echnol ogy Sydney , Australia (e- m ail: ja y . guo@uts.edu.au). it also exhibits stron g potential to reduce power con sumption and improve energy ef ficiency (EE) of wireless systems [7]. W ireless cach ing allows base station s ( BSs) to pre fetch video files f rom the co re network throug h capacity- limited backhau l links during off-peak hours. These v id eos can be placed in the local sto rage of the BSs [8], and delivered to users when requested. The r eby , it can relieve th e req uests of backhaul bandwidth during peak-h o urs. Dependin g on different content placement strategies, ca ching can be typically classified into two categories, inclu ding uncoded caching [4], [5] and coded cachin g [9]– [12]. Uncoded caching aims at storing complete video files in each of the BS, which is very suitable for p opular video s. In ord e r to effecti vely leverage th e cumulative cache size o f ne arby BSs, co ded caching e n ables each BS to store dif feren t fragmen ts or proportions o f the encoded contents, an d a recipien t of the vid e o can con struct the co ntent file b ased on a set of pre - defined decod ing rules. Recently , the network coding-based caching has been accepted as a pro mising content pla c e ment scheme, such a s the maxi- mum distance sep a r able (MDS) cod ing-based caching schem es [10]–[13]. Random caching schemes hav e also gaine d a lot of interest [14], [15], where video files o r their combinatio n s ar e random ly placed und er certain pr obability distributions to yield optimal successful transmission probability . No te that rando m caching can be regarded as a special case of uncoded caching, since complete video files are cached; howe ver , th e caching probab ilities have y et to be determ ined. T ill no w , wireless caching has b een extensively studied in cloud radio acce ss networks (C-RAN) [ 4], [16], heterogeneo u s networks [11], [14], device-to-d evice (D2D) communication s [17], [1 8], small cell networks (SCNs) [1 3], [19], [20] an d networks comb ined with D2D an d SCNs [2 1], etc. Consensus has b e e n reached that wireless cachin g is able to red uce power co nsumption , service de lay and b ackhaul- link traffic burden, as well as improve the req u est hit ra tio. Perceptual req uirements of v ideo s u bscribers can ha ve strong impact on the d esign of wireless caching system. For example, people ge n erally req u est standard viewing qua lity for ne ws reports and spo rts g ames, and high viewing qual- ity fo r movies and TV series. However , this issue has not been captured in th e aforementione d existing studies. Scalable video coding (SVC), as part of th e H. 2 65 stan dard [ 22], is able to flexibly rem ove part of th e video bit streams to adju st to various u ser requ irements and ne twork states while guaranteein g acceptable video quality . T o elabor ate a little furthe r, in SVC, each video file is divided into a base 2 layer (BL) an d multiple enhancement laye rs (ELs). V ideos with only BL can provide funda m ental v iewi ng quality , while EL con tents can complem ent to the BL to provide superior video quality . It is worth noting th at E L contents cannot be decoded withou t the corr e sponding BL [23]. Some research efforts have been dev oted to combining SVC with wireless caching. For example, W u et al. in [ 24] analy ze the successful transmission rate and backh aul lo ad in cellular networks by caching an d transm ittin g scalable vid eos, and confir m the effecti veness of tak ing different v iewing quality requiremen ts into c onsideration . Addition ally , th e authors in [25] propo se an SVC-based lay er placemen t strategy , wh ic h can sign ificantly reduce the average conten t download time. These works focus on a n alyzing the successful transmission rate, bac k haul load and a verage con tent download time. In the near futu re, mor e div erse and compr ehensive performan c e metrics, such a s EE, are a d vocated so that more perf ormanc e gain can be obtaine d throug h the com bination of wireless c aching and SVC. In a different yet r elev ant con text of 5G, EE is an impor tant design criterion [26], [27], where the energy saving should not be at the cost of th e quality of service (QoS) [7], [28]. The op timal EE design is critical, and can make p r eferable balance between total power consumptio n and sp ectrum e f - ficiency (SE) so as to improve resource u tilization [29]. It is obvious that this issue also merits consideratio n in cache- enabled networks. Specifically , the authors in [30] provide the closed-for m expression for EE and iden tify the conditio n s to obtain benefits from cachin g. Chen et al. in [8] an d Zhang et al. in [3 1] also der i ve the expre ssion s fo r EE by exp lo iting stochastic geo metry . Further more, some research e s fo cus on the de sig n of optimal conten t p lacement po licy to enhance EE. T o b e more specific, th e authors in [11] min imize the total power consu m ption to improve EE by designing the placement scheme of the coded packets under the MDS coding -based caching strategy . These works have b een fo cusing on the optimization of E E, especially the m inimization of total p ower consump tion to impr ove EE. They fail to apply to the mo re practical scenario, wher e different viewing requiremen ts need to be taken into conside r ation. I t is of great impo rtance to provide scalable video services when conc e ntrating on EE optimization s in cach e-enabled networks. T o fill this void, in this p aper, we in vestigate th e en ergy- efficient caching schemes to yield optimal EE performance in c ache-enab led heterogeneo us networks, where vid eo files to be requ e sted are encoded by SVC an d ar e divided into a BL an d an EL. The BL and E L conte n ts are locally cache d and coo peratively transmitted by two clu ster s of small-cell BSs (SBSs). W e propo se two cachin g schemes, n amely , SVC-based fractional caching and SVC-based r andom caching, to improve the EE of the cache-enabled networks. A compreh ensiv e perfor mance analysis is carried out to char acterize the EE of cache-ena b led networks supp o rting SVC. T o d e riv e the closed- form expressions fo r EE, we e stab lish the power consumptio n models and theoretically analyze the ergodic service rates. Afterwards, the optimization p roblems are formulated to maxi- mize EE of the two pro p osed cach ing schemes, which are then effecti vely solved by the standa rd g radient projection method after taking the approx imations of l 0 -norm . The m ain con tributions of this paper are summarized as follows: • W e propose two new SVC-based caching schem es to sup- port scalable video services. Both the content popular ity and the quality p referen c e of the videos are taken into account. In specific, standard definition video (SD V) an d high definition v ideo (HDV) can both be provided to the users, which are able to im p rove re so urce utilizatio n and and thereby enhance EE. • Relying on stocha stic geom etry , wh en the designated user is served by the neare st MBS a nd coop erative SBSs, the closed-fo rm expr e ssions f or successful transmission probab ilities and ergodic service rates are deriv e d , fr om which some useful insights are shed. • It is more challeng ing to deal with the p roposed EE optimization problems. After taking logarithm ic ap proxi- mations of the l 0 -norm , we transform the ob jectiv e func- tions into the con tinuous and d ifferential o n es. Finally , accordin g to their specific form s, the EE o ptimization problem s are e fficiently solved by the prop osed standard gradient projection method. The rest of this p a p er is organized as follows. Section II presents th e system mod el, including the network mo del, SVC-based ca c hing schem es, channel m odel a n d power con- sumption model of the cache-enabled heter ogeneo us n etworks. Closed-form expressions for successful transmission proba- bilities and ergo d ic service rates ar e derived in Section II I. Under the two proposed SVC-based cach ing schemes, the EE optimization pr oblems are fo rmulated and solved in Section IV and Section V , respectiv e ly . E x tensive simulation results are presented in Section VI, and this p aper concludes in Section VII. I I . S Y S T E M M O D E L In this sectio n, we p resent the system model of th e cache- enabled networks supp orting SVC, includin g the network model, SVC-based cachin g schem es, ch annel model and power consump tion mod el. A. Netw o rk Model Fig. 1 illustrates th e hetero geneou s network of in terest, including an MBS tier and an SBS tier . Multiple MBSs and SBSs are inde p endently and identically distributed in the n etwork o f in te r est, whose locatio ns fo llow homog e neous Poisson Point Proce sses (PPPs) Φ M and Φ S with den sities λ M and λ S , respectiv ely . Without loss of gen erality , our an alysis is carried ou t for a design ated user , which is located at the center of the network [8], [14], [31], [32]. All the BSs and the user are assumed to be equip ped with a single antenna. T o im plement different cach ing and transmission assignmen ts, the SBSs are group ed in to two c lusters. In specific, taking the user’ s position as the center , SBSs with in the circle with rad iu s a form clu ster N 1 and the n u mber of SBSs in th is clu ster is N 1 = |N 1 | . Th e SBSs located in the an nulus with rad ii a and b ( a < b ) form cluster N 2 , and the number of SBSs in this cluster is N 2 = |N 2 | . 3 MBS MBS MBS File Set MBS Serving MBS/SBS SBS SBS SBS SBS SBS SBS SBS SBS SBS SBS SBS SBS SBS SBS SBS SBS SBS 2 Clu ster 2 1 Clu ster 1 Fig. 1. In the proposed system model , the designated user is loc ated at the cente r of the observed netwo rk and requests the video file colo red in red. The serving SBSs are those who cache the BL and E L contents color ed in red and are loc ated in the circle and annul ar areas. If the user cann ot obt ain complete video layers, the nearest MBS will be trigger ed to retrie ve the remaining video content s. Employing SVC, each video file is encod ed into a BL and an EL 1 . The BL provides fund a mental video quality , while users with both BL an d EL contents can acquire superio r video qu ality . V ideo files with on ly BL con tents are defined as SD Vs, and videos with both BL and EL co ntents are defined as HD Vs [24]. The SBSs in clu ster N 1 are assigned to cach e BL conten ts, while ELs are cached in th e local stor a ge of SBSs belonging to cluster N 2 2 . When the design ated user requests the f -th video file, the SBSs located in cluster N 1 are respon sible fo r transmitting the BL con tent of this file; and the SBSs loc a ted in cluster N 2 will deliver the EL co ntent if HD V is req uired. Some SBSs in clusters N 1 and N 2 that do not possess the required BL and EL contents remain silent until the next transmission process begins. When cooperative SBSs fail to provide the comp lete v ideo layers, the nearest MBS is activ ated to deliver the remain ing vid eo contents from the core networks via backhaul links. B. SVC-Based Caching Schemes Assume that there are a total of F video files requested by th e user . The sizes o f each BL and EL are L B and L E , 1 In this paper , we consider two-layer video cac hing and transmission for illustra tion con ve nience . When the video files are di vided into more layers, there can be more clusters of SBSs to cooperati vel y serve the user , while the performanc e ana lysis and the problem formulati on can follo w the exac tly same steps proposed in the paper . T o thi s en d, the propo sed two-lay er video cachi ng and transmission model is instrumental , and can be readily extende d to the mult i-layer cases. 2 BL cont ent is the most fundamenta l part of a scalabl e video, since the decodin g of EL largely depends on the recei ved BL. Owing to the paramount importanc e of BLs, the SBSs loca ted in cl uster N 1 , whic h are closer to the user and are capable of providing stronger signal strength than those in cluster N 2 , are responsible for caching and deli veri ng BLs, while SBSs located in cluster N 2 are assigned to provi de ELs. respectively 3 . The total cach e size at each SBS is den oted by M . According to what mentioned above, the num ber of locally c a ched BL and EL contents can be giv e n by M B = ⌊ ( M /L B ) ⌋ and M E = ⌊ ( M /L E ) ⌋ . All videos are arranged in the descending order o f po pularity , where more p o pular videos are ran ked with smaller indice s. W e assume that the probab ility o f video requests follows the Zipf ’ s law , a s giv en by [33] p f = f − α P F n =1 n − α , f = 1 , 2 , ..., F , (1) where α is the skewness p arameter that ch a racterizes the concentr a tion o f video requ ests [34]. T he viewing qu ality preferen ce fo r SD V and HD V is also c o nsidered. Accor ding to [2 4], the SD V p erceptual prefe rence of th e f -th video file can be modeled as g SDV ( f ) = f − 1 F − 1 , f = 1 , 2 , ..., F . (2) The HDV p erceptual p referenc e can be a c cording ly denoted as g HDV ( f ) = 1 − g SDV ( f ) . Based on the above con tent tran smission p rotocol and user request model, we prop ose two caching sch emes, wh ich are described as follows: • Scheme I (SVC-based fractional cach ing). The SBSs located in cluster N 1 cache parts of the BL co ntents of th e v ideo files, where th e cached par ts are the same across the SBSs. F o r SBSs located in c luster N 2 , some parts of the EL contents are cached in their local stor a g e, where the cach ed parts ar e also the same across th ese SBSs. Let Q 1 ,f and Q 2 ,f denote the caching fractions of the BL and EL contents o f the f -th video file in SBSs belongin g to clusters N 1 and N 2 , respectiv ely . They can be stacked into vectors Q 1 = [ Q 1 , 1 , Q 1 , 2 , ..., Q 1 ,F ] and Q 2 = [ Q 2 , 1 , Q 2 , 2 , ..., Q 2 ,F ] . • Scheme I I (SVC-based random caching). The SBSs located in clusters N 1 and N 2 random ly cach e the complete BL and EL conten ts under certain probab ility distributions, respectively . Let T 1 ,f and T 2 ,f denote the probab ilities fo r cach ing BL and EL contents of the f - th video file in SBSs belo n ging to clusters N 1 and N 2 , respectively . They can be stacked into vectors T 1 = [ T 1 , 1 , T 1 , 2 , ..., T 1 ,F ] and T 2 = [ T 2 , 1 , T 2 , 2 , ..., T 2 ,F ] . In the following sections, we will deri ve th e op timal cac h ing distributions to yield o p timal EE of th e cache-enabled n etwork. C. Cha nnel Model Interfer e n ce typica lly d ominates over no ises in modern cellular networks. This pape r is therefore focusing on an interferen ce-limited case, wher e the backgro und additive white Gaussian noise is c omparatively negligible at the user, as compare d to the co-c h annel interfer ence [13], [1 4], [24]. When the cached BL content o f the f -th video file is rece ived, 3 W ithout loss of generalit y , equal video sizes are assumed in this paper . Of course, we can also consider diff erent content sizes in the proposed schemes, which has little ef fect on the per formance analysis. 4 the signal-to- interferen ce ratio (SIR) of the designated user is expressed as SIR S,B L =    P k ∈N 1 ,f h S,k √ P S r − α S 2 S,k    2 P n ∈ Φ S \N 1 | h S,n | 2 P S r − α S S,n + P m ∈ Φ M | h M ,m | 2 P M r − α M M ,m , (3) where N 1 ,f denotes the set of SBSs located in clu ster N 1 that cache the BL co ntent of the f -th vide o file. h S,k and h M ,m are the chan nel gains from the k -th SBS and the m -th MBS, f ollowing th e com plex Gaussian distribution with zero mean and unit variance, i.e., C N ∼ (0 , 1 ) . P S and P M are the tran smit p owers of SBS and MBS, respectively . r S,k is the distance between the k -th SBS and the user . Like wise, r M ,m is the distance b etween the m -th MBS and the user . Add itionally , α S and α M are the path lo ss expon ents from the SBSs and MBSs to the u ser . In (3), the fir st term in the deno minator of the right han d side accou nts for the interferen ce fro m other SBSs, e x cept th o se lo cated in cluster N 1 , and the second term represents the cross-tier interferen ce fr om a ll the MBSs. When the cached EL content of the f - th video file is delivered to the user, the received SIR is giv en by SIR S,E L =    P k ∈N 2 ,f h S,k √ P S r − α S 2 S,k    2 P n ∈ Φ S \N 2 | h S,n | 2 P S r − α S S,n + P m ∈ Φ M | h M ,m | 2 P M r − α M M ,m , (4) where N 2 ,f denotes the set of SBSs located in clu ster N 2 that cache EL con tent of the f -th video file. As mentioned, the required video layers may n o t be completely provided by cooper a tive SBSs. In this case, th e remaining video c ontents can proceed to be deli vered by the nearest MBS. As a result, the received SIR is denoted as SIR M = | h M ,m 0 | 2 P M r − α M M ,m 0 P n ∈ Φ S | h S,n | 2 P S r − α S S,n + P m ∈ Φ M \ m 0 | h M ,m | 2 P M r − α M M ,m , (5) where m 0 refers to th e nearest MBS th at provide s the user with the required video content. D. P o wer Consumption Model The total power consumption of the cache-enabled network can be modeled as P T otal = P TR + P CA + P BH + P Fix , (6) where P TR , P CA , P BH and P Fix are th e power consum p tions for data transmission , content cachin g, b ackhau l deliv ery and other fixed b udg ets, r espectively . De tails of these kinds of power consu m ptions are illustrated as follows. After p lacing par ts of the video layers in the local storage of SBSs, the cach ing power co nsumptio n exists. As describ e d in [11], [3 5], the caching power consumptio n is proportio n al to the number o f the data bits stored in the BSs. Theref ore, the caching power con sumption is ca lculated as P CA = c ca N ca , (7) where c ca is the caching coefficient in W / bit and N ca is the total number of data bits cac h ed in the local stor age of SBSs belongin g to clusters N 1 and N 2 . Due to the limited stora g e of SBSs, all req uired v ideo layers cannot be locally cach e d. Parts of them need to b e retrie ved from the nearest MBS via b ackhaul links. This leads to the backhau l power consumptio n. The back haul consum ption is propo rtional to the to tal number o f the da ta b its tran smitted via backhaul links, which is given by P BH = c bh N bh , (8) where c bh is the coefficient of backhau l power consumption in W / bit and N bh is the total number o f data bits delivered by b ackhaul lin k s. In p r actical imp lementations, for video files with the sam e sizes, caching them typically consumes less power con su mption than deliv er ing th em via microwa ve backhau l links. In th is sen se, c ca is typically less than c bh [30]. As for the fixed power co nsumption P Fix , it is mainly caused by site-c ooling, contr olling and run ning circuit compo - nents and the oscillator . In the proposed network model, the fixed power consumptio n is calcu lated a s P Fix = ( N 1 + N 2 ) P S , Fix + P M , Fix , (9) where P S , Fix and P M , Fix are the fixed po wer consum ption constants for the SBSs and MBSs, respectively . I I I . T H E E R G O D I C S E RV I C E R AT E In this section , we de r iv e the e x p ressions for suc c e ssful transmission probabilities and ergodic service rates when the user is served by the nearest MBS and cooperative SBSs , which provides the c orner stone to derive the expression s for EE. A. The Ergodic S ervice Rate F r om the Near est MBS Giv en limited stor age capacity o f SBSs, they can har dly cache all of the BL a n d EL contents locally . When the complete content layers of the req u ired video s can not be delivered to the user’ s side, the n earest MBS proc eeds to provide th e remaining video contents. In th e following, we derive the ergodic service rate in the case where the user is served b y its nearest MBS. Definition 1. When the nearest MBS pr ovides th e user with the r equ ir ed vid eo layers, the er godic service r a te is written as R M ( γ ) , W E { log 2 (1 + SIR M ) | SIR M ≥ γ } , (10) wher e W is th e sp ectrum ba ndwidth an d γ is set as the minimum QoS r equ ir ement. E {·} takes the expectation with r espect to small-scale fa ding, and the loc a tions of PPP- distributed MBS s a nd SBSs. Note that in (10), the min imum QoS req uiremen t is gu ar- anteed with the given R M ( γ ) . T o d eriv e the exp r ession for 5 R M ( γ ) , we have to first deri ve the successful transmission probab ility , i.e., P (SIR M ≥ γ ) , which can be provide d by Lemma 1. Lemma 1. When the user is served by the near est MBS under the minimum Qo S r equirement γ , th e successful transmission pr obab ility is g iven in (11), wher e G α ( x ) = R ∞ x 1 1+ t α 2 d t . Pr oof: See Appen dix A. From ( 11), we can fin d that the d eriv e d expre ssion for P (SIR M ≥ γ ) can be comp licated. F o rtunately , when α M = α S = 4 , ( 11) can be expr essed in a much simpler form , as shown in the fo llowing coro llary . Corollary 1. When α M = α S = 4 , the expr ession for P (SIR M ≥ γ ) can be simplified as P (SIR M ≥ γ ) = (1 + λ − 1 M γ 1 2 ( π 2 λ S ( P S P M ) 1 2 + λ M arccot( γ − 1 2 ))) − 1 . (14) Pr oof: See Appen dix B. In Corollary 1, th e simplified fo rm of P (SIR M ≥ γ ) is obtained, from which som e u seful insights are she d , as follows. • W ith the increasing ratio between the tr a n smit powers o f SBS and MBS, i.e., P S P M , P (SIR M ≥ γ ) degrades. This is due to the fact that SBSs with larger signa l strength can produ ce stronger cross-tier interferen ce while the useful signal strength remains the same. • W ith th e increasing ratio between th e PPP d ensities of SBS and MBS, i.e., λ S λ M , P (SIR M ≥ γ ) deterio r ates. This is because ther e are m ore SBSs, leading to sev er er cross-tier interferen ce. • As the minimum Qo S requiremen t γ increases, the user can imp ose strict video q uality re quiremen t and P (SIR M ≥ γ ) decre a ses. The rea so n is b ecause the co-exiting intra-tier and in ter-tier interf erence restricts further im provement of the succ e ssfu l tr ansmission p rob- ability . Based o n th e expression fo r P (SIR M ≥ γ ) in Lemma 1, R M ( γ ) c an be calculated by employing the following theor em. Theorem 1. When the user is served by the nea r est MBS, the er god ic service r a te is expr essed as R M ( γ ) = W log 2 (1 + γ ) + 2 π λ M W ln 2 Z ∞ 0 xe − λ M π x 2 d x Z ∞ γ P (SIR M ≥ t | r M ,m 0 = x ) P (SIR M ≥ γ | r M ,m 0 = x )(1 + t ) d t, (15) wher e the e xpress ions for P (SIR M ≥ t | r M ,m 0 = x ) and P (SIR M ≥ γ | r M ,m 0 = x ) can be found in (43), as g ive n in Appen dix A. Pr oof: See Appen dix C. Remark 1. F r om Theorem 1, it is o bvious th at a many o f factors have impa ct on the ergodic service rate, such as the minimum QoS requir ement, the P PP densities o f SBSs a n d MBSs, and the system bandwidth. Among them, the minimum QoS requir ement plays the most impo rtant r ole in the er g odic service rate, since the two terms sho wn in (15) a r e both r ela ted to the minimum QoS r equir emen t. As th e QoS r equir e ment gr ows, the first term of (15) in cr eases. In the mea nwhile, the successful transmission p r obability decreases, leading to a r educ tio n of the second term of (15). When the incr ease of the first term can compen sate for th e loss of the second term, the er god ic service rate is e n hanced . Otherwise , the performance deteriorates. Thu s, it is not always th e case that high QoS r equirement wou ld lead to good er godic service rate. B. The Ergodic S ervice Rates F r om Cooperative S BSs W e proceed to derive the expre ssion s for ergodic service rates when th e required BL and EL contents a r e deli vered by cooper a tive SBSs, wh ich are denoted by R S,B L and R S,E L , respectively . The definitions of R S,B L and R S,E L are gi ven in the following. Definition 2. Wh en cooperative SBSs p r ovide the user with the r equir ed BL and E L contents, th e ergodic service rates ar e defin e d a s R S,B L ( γ B L , n 1 ) , W E { log 2 (1 + SIR S,B L ) | SIR S,B L ≥ γ B L } , (16) R S,E L ( γ E L , n 2 ) , W E { lo g 2 (1 + SIR S,E L ) | SIR S,E L ≥ γ E L } , (17) wher e γ B L and γ E L ar e the minimum QoS r eq uir emen ts fo r delivering BL and EL contents, respectively . n 1 = |N 1 ,f | ≤ N 1 and n 2 = |N 2 ,f | ≤ N 2 ar e the n umbers of the serving SBSs that cache the r equir ed BL an d E L contents. Remark 2. Accor ding to Definitions 1 and 2, it can be seen that the er god ic service rates ar e obtained un der the co nstraint of minimum QoS r equ ir ements. Thus, in the EE o ptimization pr oblems formulated later , the QoS constraints are inhe rently satisfied in the devised EE e xpressi ons, and we on ly foc u s on the cache size restri c tions. In order to obtain the expressions devised in Definition 2, we need to acq uire the successful tran smission probab ilities when the d esignated user receives BL and E L contents from cooper a tive SBSs, i.e., P (SIR S,B L ≥ γ B L ) and P (SIR S,E L ≥ γ E L ) . The successful tr ansmission p robabilities are g i ven in the following lemma. Lemma 2 . When th er e ar e n 1 and n 2 SBSs to pr ovid e BL and EL contents, r espectively , the successful transmission p r oba- bilities ar e shown in (12) an d (13), wher e c = γ BL P n 1 k =1 x − α S BL ,k , d = γ E L P n 2 k =1 x − α S E L,k , x B L = [ x B L, 1 , ..., x B L,n 1 ] and x E L = [ x E L, 1 , ..., x E L,n 2 ] . Pr oof: See Appen dix D. Considering the special c ase of α M = α S = 4 , we are able to achieve the simplified for ms of P (SIR S,B L ≥ γ B L ) and P (SIR S,E L ≥ γ E L ) , as dictated in the following co rollary . Corollary 2. When α M = α S = 4 , P (SIR S,B L ≥ γ B L ) and 6 P (SIR M ≥ γ ) = 2 π λ M Z ∞ 0 x exp( − π ( λ S ( γ P S P M x α M ) 2 α S G α S (0) − λ M x 2 ( γ 2 α M G α M ( γ − 2 α M ) + 1)))d x, (11) P (SIR S,B L ≥ γ B L ) = Z a 0 Z a 0 ... Z a 0 n 1 Y k =1 2 x B L,k a 2 exp( − π ( λ S c 2 α S G α S ( a 2 c − 2 α S ) − λ M ( c P M P S ) 2 α M G α M (0)))d x B L , (12) P (SIR S,E L ≥ γ E L ) = Z b a Z b a ... Z b a n 2 Y k =1 2 x E L,k b 2 − a 2 exp( − π ( λ M ( d P M P S ) 2 α M G α M (0) − λ S d 2 α S ( Z a 2 d − 2 α S 0 1 1 + t α S 2 d t + G α S ( b 2 d − 2 α S ))))d x E L , (13) P (SIR S,E L ≥ γ E L ) can be simplified as P (SIR S,B L ≥ γ B L ) = Z a 0 ... Z a 0 n 1 Y k =1 2 x B L,k a 2 exp( − π uc 1 2 )d x B L , (18) P (SIR S,E L ≥ γ E L ) = Z b a ... Z b a n 2 Y k =1 2 x E L,k b 2 − a 2 exp( − π v d 1 2 )d x E L , ( 19) wher e u = λ S arccot( a 2 c − 1 2 ) + π 2 λ M ( P M P S ) 1 2 , (20) v = λ S (arctan( a 2 d − 1 2 ) + ar ccot( b 2 d − 1 2 )) + π 2 λ M ( P M P S ) 1 2 . (21) Pr oof: Corollary 2 can be proved by following the step s shown in Appe n dix B, and therefore suppressed. Based on Lem ma 2 , R S,B L ( γ B L , n 1 ) and R S,E L ( γ E L , n 2 ) can be derived, as presented in Theorem 2. Theorem 2 . Let vectors r S BL = [ r B L, 1 , r B L, 2 , ..., r B L,n 1 ] an d r S E L = [ r E L, 1 , r E L, 2 , ..., r E L,n 2 ] den o te the positions of the serving SBSs be lo nging to clusters N 1 and N 2 , res p ectively . When the d esignated user is served by coop erative SB Ss to obtain BL a nd EL con tents, the er g odic service rates can be calculated as R S,B L ( γ B L, n 1 ) = W log 2 (1 + γ B L ) + W ln 2 Z a 0 Z a 0 ... Z a 0 n 1 Y k =1 2 x B L,k a 2 d x B L Z ∞ γ BL P (SIR S,B L ≥ t | r S BL = x B L ) P (SIR S,B L ≥ γ B L | r S BL = x B L )(1 + t ) d t, (22) R S,E L ( γ E L, n 2 ) = W log 2 (1 + γ E L ) + W ln 2 Z b a Z b a ... Z b a n 2 Y k =1 2 x E L,k ( b 2 − a 2 ) d x E L Z ∞ γ E L P (SIR S,E L ≥ t | r S E L = x E L ) P (SIR S,E L ≥ γ E L | r S E L = x E L )(1 + t ) d t, (23) wher e the expr essions for P (SIR S,B L ≥ t | r S BL = x B L ) and P (SIR S,B L ≥ γ B L | r S BL = x B L ) can b e found in (51), and the expr essions fo r P (SIR S,E L ≥ t | r S E L = x E L ) and P (SIR S,E L ≥ γ E L | r S E L = x E L ) can be found in (57). Pr oof: This theor e m can be proved by following the steps shown in Appe n dix C, and therefore omitted for brevity . From Th e orem 2, some usefu l insights can also be obser ved, which are similar to those shown in Remark 1. For av oidin g repetition, they are omitted her e. Fro m the above deriv ations, we have successfully obtained th e expressions for ergodic service r a tes, which are the key in termediate steps to gain the EE expressions. I V . T H E E E O P T I M I Z A T I O N P RO B L E M F O R S C H E M E I In this section, we deri ve the expre ssion s for total power consump tion and th e sum rate, based on which the EE max- imization problem is fo rmulated. The prop osed optim ization problem is appro x imated, an d then efficiently solved by the propo sed standard gr adient projection m e thod. A. EE Optimizatio n Pr oblem F ormula tion Under Sche m e I, the total power co n sumption of the net- work of interest can be quantified as P T otal , 1 = P TR , 1 + P CA , 1 + P BH , 1 + P Fix , (24) where P TR , 1 = F X f =1 p f ( ζ S ( N 1 k Q 1 ,f k 0 + g HDV ( f ) N 2 k Q 2 ,f k 0 ) P S + ζ M ( k 1 − Q 1 ,f k 0 + g HDV ( f ) k 1 − Q 2 ,f k 0 ) P M ) , (25) P CA , 1 = c ca F X f =1 ( Q 1 ,f L B N 1 + Q 2 ,f L E N 2 ) , (26) and P BH , 1 = c bh F X f =1 p f ((1 − Q 1 ,f ) L B + g HDV (1 − Q 2 ,f ) L E ) . (27) 7 In (2 5), ζ S and ζ M are the power ef ficien cy coefficients of the po w e r amplifiers of the SBSs and MBSs , resp ectiv ely , and k·k 0 stands for l 0 -norm . Based on the proposed SVC-ba sed factional caching and co- operative tr a n smission schemes, the sum rate of the d e signated user is given by R Sum , 1 = F X f =1 p f ((1 − Q 1 ,f ) R M ( γ B L ) + g HDV ( f )(1 − Q 2 ,f ) R M ( γ E L ) | {z } served by the nearest MBS to obtain BL and EL contents + Q 1 ,f R S,B L ( γ B L , N 1 ) + g HDV ( f ) Q 2 ,f R S,E L ( γ E L , N 2 ) | {z } served by co op erati ve SBSs t o obtai n BL and E L c onten ts ) . (28) From ( 2 8), in tuitions can be found that the per forman ce of R Sum , 1 can pred ominately d epend on the cachin g fra ctions, the number of serving SBSs and the minimum Q o S r equireme n ts. Moreover , if HDV service is r equired by th e user, th e SBSs located in cluster N 2 are activ ated to deliver the EL contents. This can im prove the sum r ate performance. Accord ing to the expressions f or total power co nsumption and sum rate presented ab ove, the EE maximization p roblem w ith BL and EL caching fraction design can be for m ulated as max Q 1 , Q 2 E E 1 = R Sum , 1 P T otal , 1 (29a) s . t . F X f =1 Q 1 ,f = M B , (29b) F X f =1 Q 2 ,f = M E , (29c) 0 ≤ Q 1 ,f ≤ 1 , 0 ≤ Q 2 ,f ≤ 1 , ∀ f , (29d) where (29 b) and (29c) are th e cache size restrictions of e a ch SBS when caching BL a nd EL contents, respecti vely; (29d) indicates the fea sib le solu tion regions of th e o ptimization variables Q 1 ,f and Q 2 ,f . B. The Pr oposed Algorithm In (29), constraints (29b) to (29d) for m a conv ex variable set, while the objective function has a complicated expression. Moreover , the l 0 -norm in P T otal , 1 makes the problem mor e challengin g to de a l with. Note that the l 0 -norm can be ap- proxim a ted by a lo garithmic fu nction, an expo nential function or an arctange nt fu nction [4]. Without loss of gen erality , the logarithm ic func tion is ad opted in this paper as th e smo oth function . Then, the l 0 -norm in P TR , 1 can be appr oximated as ˆ P TR , 1 = F X f =1 p f ( ζ S ( N 1 f θ ( Q 1 ,f ) + g HDV ( f ) N 2 f θ ( Q 2 ,f )) P S + ζ M ( f θ (1 − Q 1 ,f ) + g HDV ( f ) f θ (1 − Q 2 ,f )) P M ) , (30) where f θ ( x ) = log( x θ + 1) / log( 1 θ + 1) . (31) Algorithm 1 The pr oposed standard g radient projection method for solving the EE optimization problem (29). 1) Initialization: Set t = 1 , ǫ (1) = 1 , and find Q 1 ,f and Q 2 ,f that are feasible for constraints (29b)-(29 d). 2) For f ∈ { 1 , ..., F } , calcu late ∂ E E 1 Q 1 ,f , and then obtain ˆ Q 1 ,f ( t + 1) = Q 1 ,f ( t ) + ǫ ( t ) ∂ E E 1 Q 1 ,f | Q 1 ,f = Q 1 ,f ( t ) ,Q 2 ,f = Q 2 ,f ( t ) . 3) For f ∈ { 1 , ..., F } , calculate Q 1 ,f ( t + 1) = min { [ ˆ Q 1 ,f ( t + 1) − u ′ ] + , 1 } , where u ′ satisfies P F f =1 min { [ ˆ Q 1 ,f ( t + 1) − u ′ ] + , 1 } = M B . 4) For f ∈ { 1 , ..., F } , calcu late ∂ E E 1 Q 2 ,f , and then obtain ˆ Q 2 ,f ( t + 1) = Q 2 ,f ( t ) + ǫ ( t ) ∂ E E 1 Q 2 ,f | Q 1 ,f = Q 1 ,f ( t ) ,Q 2 ,f = Q 2 ,f ( t ) . 5) For f ∈ { 1 , ..., F } , calculate Q 2 ,f ( t + 1) = min { [ ˆ Q 2 ,f ( t + 1 ) − v ′ ] + , 1 } , where v ′ satisfies P F f =1 min { [ ˆ Q 2 ,f ( t + 1) − v ′ ] + , 1 } = M E . 6) If conv ergen ce, the algorithm terminates. Otherwise, set t = t + 1 a n d ǫ ( t ) = 1 t , then go back to Step 2). In (31), θ is a constant parameter to reflect the smoothness of f θ ( x ) . A larger value for θ lead s to a smoother function but less accu r ate appro ximation. After the appr oximation s, the objective function (29a) is a co ntinuou s an d differentiable function with respe c t to Q 1 ,f and Q 2 ,f . Due to the specific form of th e transformed EE optim ization pro blem, the stan- dard gr adient pro jection algorithm is emp loyed [14], [1 5], [36], an d the suboptimal solution of (29) can be obtain ed, as sum marized in Algorithm 1. In the pro p osed algorithm , ǫ ( t ) is the iteration step size, wh ich satisfi e s lim t → 0 ǫ ( t ) = 0 , lim T →∞ P T t =1 ǫ ( t ) = ∞ and lim T →∞ P T t =1 ǫ 2 ( t ) < ∞ . W e set ǫ ( t ) = 1 t in the t -th iteration. Note that Step s 3 ) and 5) g iv e the projection s of ˆ Q 1 ,f ( t + 1) and ˆ Q 2 ,f ( t + 1) onto the o ptimization variable set so that constraints (2 9b) and (29c) are satisfied, where [ x ] + , max[x , 0] . V . T H E E E O P T I M I Z AT I O N P RO B L E M F O R S C H E M E I I For Scheme II, acco rding to the power con sumption model established p reviously , the total po w e r consump tion is gi ven by P T otal , 2 = P TR , 2 + P CA , 2 + P BH , 2 + P Fix , (32) where P TR , 2 = F X f =1 p f ( ζ S ( N 1 T 1 ,f + g HDV ( f ) N 2 T 2 ,f ) P S + ζ M ((1 − T 1 ,f ) + g HDV ( f )(1 − T 2 ,f )) P M ) , (3 3) P CA , 2 = c ca F X f =1 ( T 1 ,f L B N 1 + T 2 ,f L B N 2 ) , (34) 8 and P BH , 2 = c bh F X f =1 p f ((1 − T 1 ,f ) N 1 L B + g HDV (1 − T 2 ,f ) N 2 L E ) . (35) T o d e r iv e the expr e ssion for EE, the sum rate expression needs to be first derived. T o this end, the numbers of the serving SBSs in clu ster s N 1 and N 2 remain to be determined. Under Scheme II, each SBS randomly s elects BL and EL contents to cach e in its local storage under probability dis- tributions T 1 and T 2 , respectively . As a r esult, the nu mber of the serving SBSs fo llows the binom ial distribution. T o be more sp e cific, the numb er of serving SBSs in cluster N 1 which cache BL content of the f -th file, i.e., |N 1 ,f | , follows the binomial distribution with parameter s N 1 and T 1 ,f , while |N 2 ,f | follows the b inomial distribution with parameters N 2 and T 2 ,f . Therefore, the sum rate of the designated user can be written as R Sum , 2 = F X f =1 p f ((1 − T 1 ,f ) N 1 R M ( γ B L ) + g HDV ( f )(1 − T 2 ,f ) N 2 R M ( γ E L ) | {z } served by the n earest MBS to obtain BL and EL contents + N 1 X n 1 =1 C n 1 N 1 ( T 1 ,f ) n 1 (1 − T 1 ,f ) N 1 − n 1 R S,B L ( γ B L , n 1 ) | {z } served by co op erati ve SBSs t o obtai n BL contents + g HDV ( f ) N 2 X n 2 =1 C n 2 N 2 ( T 2 ,f ) n 2 (1 − T 2 ,f ) N 2 − n 2 R S,E L ( γ E L , n 2 ) | {z } served by co op erati ve SBSs to obtai n EL contents ) . (36) From (36), it can be conclu d ed that R sum , 2 depend s on the caching probab ilities, the number of serv in g SBS s, and the minimum QoS re q uiremen ts. Particularly , wh en th ere a r e more serving SBSs, the requested video layers are more likely to be obtained lo cally . This can lead to less power consumptio n an d relieve traffic cong estion in back haul. Acco rding to (32) an d (36), the EE max im ization problem with BL and EL caching probab ility design can b e formulated a s max T 1 , T 2 E E 2 = R Sum , 2 P T otal , 2 (37a) s . t . F X f =1 T 1 ,f = M B , (37b) F X f =1 T 2 ,f = M E , (37c) 0 ≤ T 1 ,f ≤ 1 , 0 ≤ T 2 ,f ≤ 1 , ∀ f , (37d) where (37b) and (37c) are th e cache size con stra ints of each SBS when cachin g the BL and EL c o ntents, r espectively . (37d) s p ecifies th e feasible solution region s o f the caching probab ilities T 1 ,f and T 2 ,f . It is obvious tha t ( 37) has th e same form as (29). Ther efore, in the same manner, this pro b lem can also b e effectiv e ly solved by using the standard grad ient T able I V A L U E S O F S I M U L A T I O N P A R A M E T E R S Parameter V alue N 1 , N 2 4 P S , P M 23 dBm (default), 43 dBm λ S , λ M 1 / (100 2 π ) , 1 / (250 2 π ) a, b 50 m, 100 m α S , α M 4 M 500 Mbits (default) F 20 L B , L E 100 Mbits, 200 Mbits α 1 (default) W 10 Mbits γ B L , γ E L 10 dB (default), 5 dB (default) ζ S , ζ M 4.7 c ca 6 . 25 × 1 0 − 12 W/bit [30] c bh 5 × 1 0 − 7 W/bit [30] P S , Fix 6 . 8 W [37] P M , Fix 130 W [37] projection method, and more details can r efer to Algorithm 1. Remark 3. F or Scheme II, th e SBSs a r e cap able o f caching the complete video la y e rs, while Scheme I aims at ca ching parts of the BL an d EL contents. In S cheme II , video layers of th e popu - lar files ar e mo r e likely to be comp letely cached an d deliveries fr om the b ackhaul links ar e pr evented when these videos ar e r eque sted . In contrast, for Scheme I, each SBS caches parts of the popula r video layers, and the remaining co ntents need to be r etrieved fr o m ba ckhauls, consuming more po wer than caching. As a con sequence, it can b e con cluded that rando mly caching the co mplete video lay e rs pr ovides better EE than fractionally caching pa rts of them. The superiority o f Scheme II is demonstrated in terms of EE , especially in the pr esence of high b ackhaul power co nsumption. These co n clusions can be co llaborated by extensive simulations as will be shown in Section VI. V I . S I M U L A T I O N R E S U L T S In this section, we show the simulation results of the derived exp r essions for successful tran smission pro babilities and ergodic servic e rates, as well as the EE perfor mance u n der the pr oposed SVC-based caching sch emes. Th e simulation parameters are listed in T able I . For comparison p u rpose, three benchm a rks are simulated, which are de scribed as follows: • Most Popu la r Content Placemen t (MPCP): For SBSs located in clusters N 1 and N 2 , M B BL co ntents an d M E EL co n tents of the most popu lar video files are cached in their local storage, respectively . • Uniform Content Placement (UCP): Re g ardless of the video po pularity , for SBSs located in clusters N 1 and N 2 , the same fractions of the BL and EL contents are cached, respectively . • Indepen dent Con te n t Placement (ICP): The SBSs belo ng- ing to clusters N 1 and N 2 random ly select M B and 9 -10 -6 -2 2 6 10 (dB) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 P(SIR M >= ) P S =23dBm, Analysis P S =23dBm, Simulation P S =33dBm, Analysis P S =33dBm, Simulation (a) P (SIR M ≥ γ ) versus va rying γ unde r dif ferent P S . 10 12 14 16 18 20 BL (dB) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 P(SIR S,BL >= BL ) n 1 =1,Analysis n 1 =1,Simulation n 1 =2,Analysis n 1 =2,Simulation n 1 =3,Analysis n 1 =3,Simulation n 1 =4,Analysis n 1 =4,Simulation (b) P (SIR S,BL ≥ γ BL ) versus varying γ BL under diff erent n 1 . -5 -3 -1 1 3 5 EL (dB) 0 0.1 0.2 0.3 0.4 0.5 0.6 P(SIR S,EL >= EL ) n 2 =1,Analysis n 2 =1,Simulation n 2 =2,Analysis n 2 =2,Simulation n 2 =3,Analysis n 2 =3,Simulation n 2 =4,Analysis n 2 =4,Simulation (c) P (SIR S,E L ≥ γ E L ) versus vary ing γ E L under differe nt n 2 . Fig. 2. The successful transmission probabili ties when the user is serve d by the nearest MBS and cooperati ve SBSs. -10 -6 -2 2 6 (dB) 10 15 20 25 30 35 R M ( ) (Mbit/s) P S =23dBm P S =33dBm (a) R M ( γ ) v ersus v arying γ BL under differe nt P S . 0 2 4 6 8 10 35 40 45 50 55 60 65 70 75 γ BL (dB) R S,BL ( γ BL ,n 1 ) (Mbit/s) n 1 =4 n 1 =3 n 1 =2 n 1 =1 (b) R S,BL ( γ BL, n 1 ) versus varying γ BL under diff erent n 1 . -5 -3 -1 1 3 5 EL (dB) 10 15 20 25 30 35 R S,EL ( EL ,n 2 ) (Mbit/s) n 2 =4 n 2 =3 n 2 =2 n 2 =1 (c) R S,E L ( γ E L, n 2 ) versus varying γ E L under differe nt n 2 . Fig. 3. The ergodic s ervice rate s when the user is served by the nearest MBS and cooperati ve SBSs. M E different BL and EL contents to c ache in their local storage. In Fig. 2 , we show the successful transmission p robab ilities derived from theoretical analy sis and M onte Carlo simulations. The plots of theoretical analysis and Monte Carlo simula tio ns are overlapped, confirming the accuracy of successful trans- mission probabilities gi ven in Lem mas 1 and 2. A commo n trend is also revealed that h igher QoS requiremen ts lead to lower successful transmission probabilities. From Fig. 2 (a), it can be seen th at a larger value of P S results in lower P (SIR M ≥ γ ) . The reason for this is because, with the increase o f P S , the co-existing SBSs can produce stronger interferen ce towards the serv in g MBS, and hence redu c e P (SIR M ≥ γ ) . Moreover , Figs. 2 (b) an d (c) lead to the conclusion th at with more c ooperative SBSs, the successful transmission proba bility can b e improved. In Fig. 3, we show th e ergodic service r ates, i.e., R M ( γ ) , R S,B L ( γ B L , n 1 ) and R S,E L ( γ E L , n 2 ) , with varying number s of cooperative SBSs and QoS requir e ments. Under o ur pa- rameter settings, the ergodic service rates increase as the QoS requirem ents grow , since the impr ovement of the first term in (15), (22) and (2 3) can make up for the loss o f the secon d term. Howe ver , h igher P S can lead to lower R M ( γ ) due to a reduction of the suc cessful transmission prob ability , which coincides with the conclusion d rawn from Fig. 2 (a). Moreover , in Figs. 3 (b) an d 3 (c) , the ergodic service rates are improved 20 22 24 26 28 30 P T,S (dBm) 0.17 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.27 Energy Efficiency (Mbit/J) Scheme II Scheme I without Approximation Scheme I MPCP ICP UCP Fig. 4. The EE performan ce versus P S under diff erent caching s chemes. as n 1 and n 2 grow , since there are more coo p erative SBSs to enhance the suc c essful tran smission p r obabilities. Th erefore, it can be conclu d ed that, in large scale h eterogene ous networks with multiple SBSs, th e perfor mance o f ergodic service rates can be furthe r enhan ced. Fig. 4 presents the relationsh ip be tween th e EE perf ormance and the tran smit p ower of the SBSs u n der different caching schemes. The superio r ity of ou r propo sed SVC-based cach ing 10 0 2 4 6 8 10 0.16 0.18 0.2 0.22 0.24 0.26 γ BL (dB) Energy Efficiency (Mbit/J) Scheme II Scheme I MPCP ICP UCP Fig. 5. The EE performan ce versus γ BL under diff erent caching schemes. 0 500 1000 1500 2000 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 Cache Size M (Mbit) Energy Efficiency (Mbit/J) Scheme II Scheme I MPCP ICP UCP 500 0.24 0.25 0.26 0.27 Fig. 6. T he EE per formance ve rsus cache size under di fferent cachi ng schemes. schemes is v a lid ated. W ith the increase o f P S , the EE grows and the growth slo ws down. When P S grows further, thou gh the sum rate increases, the improvement o f the sum rate cannot scale up with the increase of the total power c onsumptio n. This results in the degraded EE performance . When we o b tain the optimal caching fractions u nder Scheme I, the op timal caching f ractions a re used to validate th e accur acy of the l 0 - norm appr oximation . As shown in Fig. 4, the approximation is accurate, and the perfor mance loss resulting from the approx- imation is m arginal. It can be concluded that wh e n selecting the proper smoo th par ameter θ , the l 0 -norm of the caching fractions can be accura tely estimated. For the ICP scheme, the file selection process is pe rformed under each chan nel realization. When ther e are sufficient channel realizations, the EE of ICP is comparab le to that of UCP . Fig. 5 shows the E E p erforma nce of different cachin g schemes u nder various values o f γ B L . It is o bvious that the propo sed c a ching schemes are sup erior to the three benc h - marks, especially achieving m u ch more perfor mance gain than the UCP a n d ICP schem es. Furtherm o re, it can be seen that Scheme II can achieve higher EE than Scheme I. This indicates that rando mly caching comp lete v ideo layers p rovides b etter 0 0.5 1 1.5 2 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 Skewness Parameter α Energy Efficiency (Mbit/J) Scheme II Scheme I MPCP ICP UCP Fig. 7. The EE performance v ersus skewn ess paramet er under diff erent cachi ng s chemes. EE than storing only parts of them, even und e r the op timal caching fractions. In Fig. 6, we presen t the EE perfor mance of different caching schemes with varying ca c he size M . When a larger cache size is eq uipped at each SBS, more vid eo contents can be locally cached. Therefo re, the demand fo r backhau l links can b e r eliev ed substantially , which in turn can significantly reduce serv ice delay and back haul power consumption . In practice, fo r video files with the same sizes, caching them can consume less power con sumption than retrieving them from micr owa ve backha u l links. As a re su lt, larger c ache sizes can lead to higher EE. Particularly , when M = 0 , all caching schemes have the same EE. This is equiv alen t to the case with no caching. The p roposed cach ing schemes pr ovid e better EE in small c a che size region, a n d reach the maximum perfor mance gap at abou t M = 600 Mbits. Howe ver, when the cach e size gr ows further, th e per forman c e gaps gra d ually diminish, since all v ideo files will be cach ed when the cach e size of each SBS is large e nough . In Fig. 7, the relation ship between the EE perfo rmance an d ske wne ss p arameter α is p lotted. Note that larger α means that fewer video files can mee t the majo rity of user req uests. Therefo re, video files with sma ller ind ices ar e much mo re likely to b e stored in the loc al cache of SBSs . Wh e n α is small, the UCP and ICP schemes pr ovide better EE. I n this case, v id eo popular ities are unif o rmly distributed, and the UCP and ICP sch emes are able to increase file d iv er sity , and then increase request hit ratio. In Fig. 8, we show the co nvergence property o f the proposed algorithm . It can be seen that the pr oposed algor ith m is able to converge after a small num b er of iteratio ns, which v alida tes the effectiveness of the standard g radient pr ojection method in practical implemen tatio ns. V I I . C O N C L U S I O N This paper p roposed two ene rgy-efficient SVC-ba sed caching scheme s to boost EE in cache-en abled heterogeneou s networks. Based o n the p roposed caching sch emes, we es- tablished the power con sumption m odel, and deriv e d the ex- 11 0 1 2 3 4 5 6 7 0 0.05 0.1 0.15 0.2 0.25 Iteration Number Energy Efficiency (Mbit/J) Scheme II Scheme I Fig. 8. The con vergen ce prope rty of th e proposed algorithm. pressions f or successful transmission pro babilities and ergodic service rates. W e further formulated tw o EE m aximization problem s, which are sub ject to the cache size constraint of each SBS. After taking app roximatio ns of the l 0 -norm , the EE o ptimization p roblems c a n be efficiently solved. Num erical results confir med th e a ccuracy o f our analy sis a s well as the super iority o f th e pro p osed caching sch emes to three benchm a rks. A P P E N D I X A P R O O F O F L E M M A 1 In the pr o posed model, the designated user is loc a te d at the origin of the observed n etwork and the distance b etween the near est MBS an d the user is den oted by r M ,m 0 , whose probab ility density fun c tion ( PDF) is shown as [8] f m 0 ( r M ,m 0 ) = 2 πλ M r M ,m 0 e − λ M π r 2 M,m 0 . (38) Then, P (SIR M ≥ γ ) is calculated as P (SIR M ≥ γ ) = Z ∞ 0 f m 0 ( x ) P (SIR M ≥ γ | r M ,m 0 = x )d x. (39) For simplicity of notations, the interfe rence from all SBSs and other non- serving MBSs are deno ted b y I S 1 = X n ∈ Φ S \N 2 | h S,n | 2 P S r − α S S,n , I M 1 = X m ∈ Φ M \ m 0 | h M ,m | 2 P M r − α M M ,m . Relying on stochastic geome try , P (SIR M ≥ γ | r M ,m 0 = x ) can be calculated as P (SIR M ≥ γ | r M ,m 0 = x ) = P ( | h M ,m 0 | 2 P M r − α M M ,m 0 I S 1 + I M 1 ≥ γ | r M ,m 0 = x ) = P ( | h M ,m 0 | 2 ≥ γ P − 1 M x α M ( I S 1 + I M 1 )) ( a ) = E I S 1 ,I M 1 [exp( − ( I S 1 + I M 1 ) γ P − 1 M x α M )] = L I S 1 ( γ P − 1 M x α M ) L I M 1 ( γ P − 1 M x α M ) , (40 ) where ( a ) f ollows the fact that | h M ,m 0 | 2 ∼ exp(1) a nd exp( µ ) de notes the expon ential d istribution with mean µ . Additionally , L I S 1 ( γ P − 1 M x α M ) and L I M 1 ( γ P − 1 M x α M ) are the Laplace transform s of in terferenc e I S 1 and I M 1 , respectively . Let k 1 = γ P − 1 M x α M . Then , L I S 1 ( k 1 ) can be obtain ed as follows L I S 1 ( k 1 ) = E I S 1 [exp( − X n ∈ Φ S k 1 I S 1 ,n )] = E I S 1 [ Y n ∈ Φ S exp( − k 1 | h S,n | 2 P S r − α S S,n )] = E I S 1 [ Y n ∈ Φ S 1 1 + k 1 P S r − α S S,n ] = exp( − 2 π λ s Z ∞ 0 (1 − 1 1 + k 1 P S ρ − α S ) ρ d ρ = exp( − π λ s ( k 1 P S ) 2 α S Z ∞ 0 1 1 + t α S 2 d t ) . (41) W e denote G α ( x ) = R ∞ x 1 1+ t α 2 d t , then L I S 1 ( k 1 ) = exp( − π λ s ( k 1 P S ) 2 α S G α S (0)) . In the similar manner, L I M 1 ( k 1 ) can be calculated as L I M 1 ( k 1 ) = E I M 1 [exp( − X m ∈ Φ M \ m 0 k 1 I M 1 ,m )] = E I M 1 [ Y m ∈ Φ M \ m 0 exp( − k 1 | h M ,m | 2 P M r − α M M ,m )] = E I M 1 [ Y m ∈ Φ M \ m 0 1 1 + k 1 P M r − α M M ,m ) = exp( − 2 π λ M Z ∞ x (1 − 1 1 + k 1 P M ρ − α M ) ρ d ρ = exp( − π λ M ( k 1 P M ) 2 α M G α M ( x 2 ( k 1 P M ) − 2 α M )) . (42) Therefo re, we can obtain that P (SIR M ≥ γ | r M ,m 0 = x ) = exp( − π λ M x 2 γ 2 α M G α M ( γ − 2 α M )) exp( − π λ S ( γ P S P M x α M ) 2 α S G α S (0)) . (43) Finally , substituting (4 3) and (38) into (39), P (SIR M ≥ γ ) is obtained. A P P E N D I X B P R O O F O F C O RO L L A RY 1 When α M = α S = 4 , it is intuitio nal to obtain that G α S (0) = π 2 and G α M ( γ − 2 α M ) = π 2 − arcta n( γ − 2 α M ) = arccot( γ − 2 α M ) . Next, th e simplified form of P (SIR M ≥ γ ) can be obtained as follows P (SIR M ≥ γ ) = Z ∞ 0 f m 0 ( x ) ex p( − π x 2 γ 1 2 ( π 2 λ S ( P S P M ) 1 2 + λ M arccot( γ − 1 2 )))d x = π λ M Z ∞ 0 exp( − π x 2 ( λ M + γ 1 2 ( π 2 λ S ( P S P M ) 1 2 + λ M arccot( γ − 1 2 ))))d x 2 12 = λ M ( λ M + γ 1 2 ( π 2 λ S ( P S P M ) 1 2 + λ M arccot( γ − 1 2 ))) − 1 = (1 + λ − 1 M γ 1 2 ( π 2 λ S ( P S P M ) 1 2 + λ M arccot( γ − 1 2 ))) − 1 . A P P E N D I X C P R O O F O F T H E O R E M 1 For n o tational sim p licity , let y M = SIR M and Ω( y M ) denotes the suc c e ssful transmission event SIR M ≥ γ . The condition al PDF of y M is the n denoted as g ( y M | Ω( y M )) . Therefo re, we have th e following deriv ations E [log 2 (1 + y M ) | Ω( y M )] = Z ∞ 0 f m 0 ( x )d x Z ∞ 0 log 2 (1 + y M ) g ( y M | Ω( y M ))d y M = 1 ln 2 Z ∞ 0 f m 0 ( x )d x Z ∞ 0 ln(1 + y M ) g ( y M | Ω( y M ))d y M = 1 ln 2 Z ∞ 0 f m 0 ( x )d x Z ∞ 0 ( Z y M 0 1 1 + t dt ) g ( y M | Ω( y M )d y M ( b ) = 1 ln 2 Z ∞ 0 f m 0 ( x )d x Z ∞ 0 1 1 + t d t Z ∞ t g ( y M | Ω( y M ))d y M ( c ) = 1 ln 2 Z ∞ 0 f m 0 ( x )d x Z ∞ 0 1 1 + t P ( y M ≥ t | Ω( y M ))d t = 1 ln 2 Z ∞ 0 f m 0 ( x )d x Z ∞ 0 1 1 + t P (SIR M ≥ t, SIR M ≥ γ | r M ,m 0 = x ) P (SIR M ≥ γ | r M ,m 0 = x ) d t = 1 ln 2 Z ∞ 0 f m 0 ( x )d x Z ∞ 0 1 1 + t P (SIR M ≥ max(t , γ ) | r M , m 0 = x) P (SIR M ≥ γ | r M ,m 0 = x ) d t = 1 ln 2 Z ∞ 0 f m 0 ( x )d x  Z γ 0 1 1 + t d t + Z ∞ γ P (SIR M ≥ t | r M ,m 0 = x ) P (SIR M ≥ γ | r M ,m 0 = x )(1 + t ) d t  = log 2 (1 + γ ) + 1 ln 2 Z ∞ 0 f m 0 ( x )d x Z ∞ γ P (SIR M ≥ t | r M ,m 0 = x ) P (SIR M ≥ γ | r M ,m 0 = x )(1 + t ) d t, (44) where ( b ) and ( c ) follow the facts th a t Z ∞ 0 ( Z y M 0 1 1 + t dt ) g ( y M | Ω( y M )d y M = Z ∞ 0 1 1 + t d t Z ∞ t g ( y M | Ω( y M ))d y M , Z ∞ t g ( y M | Ω( y M ))d y M = P ( y M ≥ t | Ω( y M )) . Finally , substituting (44) an d (3 8) into ( 10), the expression for R M ( γ ) shown in (15) is obtained. A P P E N D I X D P R O O F O F L E M M A 2 Let vectors r S BL = [ r B L, 1 , r B L, 2 , ..., r B L,n 1 ] and r S E L = [ r E L, 1 , r E L, 2 , ..., r E L,n 2 ] represent the positions of the serving SBSs located in two clusters. T he serving SBSs in clusters N 1 and N 2 are independ ently and un iformly distrib u ted with in the circle with radius a an d an nulus with radii a and b . T herefor e, the joint PDF of r S BL and r S E L are given by [8] f ( r B L, 1 , ..., r B L,n 1 ) = n 1 Y k =1 2 r B L,k a 2 , (45) f ( r E L, 1 , ..., r E L,n 2 ) = n 2 Y k =1 2 r E L,k b 2 − a 2 . (46) A. The deriva tio n of P (SIR S,B L ≥ γ B L ) From the definition of succ essful transmission pr obability , P (SIR S,B L ≥ γ B L ) can be expressed as P (SIR S,B L ≥ γ B L ) = Z a 0 ... Z a 0 f ( x B L, 1 , ..., x B L,n 1 ) P (SIR S,B L ≥ γ B L | r S BL = x B L )d x B L . (47 ) For n otational simplicity , we let I S 2 = X n ∈ Φ S \N 1 | h S,n | 2 P S r − α S S,n , I M 2 = X m ∈ Φ M | h M ,m | 2 P M r − α M M ,m . Next, P (SIR S,B L ≥ γ B L | r S BL = x B L ) is calculated as P (SIR S,B L ≥ γ B L | r S BL = x B L ) = P (      n 1 X k =1 h S,k p P S x − α S 2 B L,k      2 ≥ γ B L ( I S 2 + I M 2 )) ( d ) = E I S 2 ,I M 2 [exp( − 1 P n 1 k =1 x − α S B L,k γ B L ( I S 2 + I M 2 ) P − 1 S )] = L I S 2 ( γ B L P − 1 S P n 1 k =1 x − α S B L,k ) L I M 2 ( γ B L P − 1 S P n 1 k =1 x − α S B L,k ) , ( 4 8) where (d) follows the fact th at    P n 1 k =1 h S,k √ P S r − α S 2 B L,k    2 ∼ P − 1 S exp( 1 P n 1 k =1 x − α S BL ,k ) . Let k 2 = γ BL P − 1 S P n 1 k =1 x − α S BL ,k . Following the similar steps shown before, L I S 2 ( k 2 ) an d L I M 2 ( k 2 ) ar e cal- culated as L I S 2 ( k 2 ) = exp( − πλ S ( k 2 P S ) 2 α S G α S ( a 2 ( k 2 P S ) − 2 α S )) , (49) L I M 2 ( k 2 ) = exp( − πλ M ( k 2 P M ) 2 α M G α M (0)) , (50) respectively . Substitutin g formulas (4 9) a n d (50) into ( 48), we can obtain that P (SIR S,B L ≥ γ B L | r S BL = x B L ) = exp( − π λ S c 2 α S G α S ( a 2 c − 2 α S ) − π λ M ( c P M P S ) 2 α M G α M (0)) . (51) Finally , sub stituting (51) and (45) into (47), we can ob tain the e x pression f or P (SIR S,B L ≥ γ B L ) . Thus, the proof of P (SIR S,B L ≥ γ B L ) is complete d . 13 B. The derivation o f P (SIR S,E L ≥ γ E L ) Follo win g the similar meth ods, P (SIR S,E L ≥ γ E L ) can be expressed a s P (SIR S,E L ≥ γ E L ) = Z b a ... Z b a f ( x E L, 1 , ..., x E L,n 2 ) P (SIR S,E L ≥ γ E L | r S E L = x E L )d x E L . (52 ) In order to simplify notation s, we let I S 3 = X n ∈N 1 | h S,n | 2 P S r − α S S,n , I S 4 = X n ∈ Φ S \{N 1 ∪N 2 } | h S,n | 2 P S r − α S S,n . Afterwards, the expression f o r P (SIR S,E L ≥ γ E L | r S E L = x E L ) can be calculated as P (SIR S,E L ≥ γ E L | r S E L = x E L ) = L I S 3 ( k 3 ) L I S 4 ( k 3 ) L I M 2 ( k 3 ) , (53) where k 3 = γ E L P − 1 S P n 2 k =1 x − α S E L,k , (54) L I S 3 ( k 3 ) = exp( − πλ S ( k 3 P S ) 2 α S Z a 2 ( k 3 P S ) − 2 α S 0 1 1 + t α S 2 d t ) , (55) L I S 4 ( k 3 ) = exp( − πλ S ( k 3 P S ) 2 α S G α S ( b 2 ( k 3 P S ) − 2 α S ) . (5 6) Thus, (53) can be rewritten a s P (SIR S,E L ≥ γ E L | r S E L = x E L ) = exp( − π ( λ M ( d P M P S ) 2 α M G α M (0) − λ S d 2 α S ( Z a 2 d − 2 α S 0 1 1 + t α S 2 d t ) + G α S ( b 2 d − 2 α S ))) . (57) Substituting (57) and (4 6) in to (5 2), we can obtain the th eo- retical expression f o r P (SIR S,E L ≥ γ E L ) . 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