Optimized Base-Station Cache Allocation for Cloud Radio Access Network with Multicast Backhaul

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1 Optimized Base-Station Cache Aocation for Coud Radio Access Network with Muticast Backhau Binbin Dai, Student Member, IEEE, Ya-Feng Liu, Member, IEEE, and Wei Yu, Feow, IEEE arxiv: v [cs.it] 28 Apr 208 Abstract The performance of coud radio access network C-RAN) is imited by the finite capacities of the backhau inks connecting the centraized processor CP) with the basestations BSs), especiay when the backhau is impemented in a wireess medium. This paper proposes the use of wireess muticast together with BS caching, where the BSs pre-store contents of popuar fies, to augment the backhau of C-RAN. For a downink C-RAN consisting of a singe custer of BSs and wireess backhau, this paper studies the optima cache size aocation strategy among the BSs and the optima muticast beamforming transmission strategy at the CP such that the user s requested messages are deivered from the CP to the BSs in the most efficient way. We first state a muticast backhau rate expression based on a joint cache-channe coding scheme, which impies that arger cache sizes shoud be aocated to the BSs with weaker channes. We then formuate a two-timescae joint cache size aocation and beamforming design probem, where the cache is optimized offine based on the ong-term channe statistica information, whie the beamformer is designed during the fie deivery phase based on the instantaneous channe state information. By everaging the sampe approximation method and the aternating direction method of mutipiers ADMM), we deveop efficient agorithms for optimizing cache size aocation among the BSs, and quantify how much more cache shoud be aocated to the weaker BSs. We further consider the case with mutipe fies having different popuarities and show that it is in genera not optima to entirey cache the most popuar fies first. Numerica resuts show considerabe performance improvement of the optimized cache size aocation scheme over the uniform aocation and other heuristic schemes. Index Terms Aternating direction method of mutipiers ADMM), base-station BS) caching, coud radio access network C-RAN), data-sharing strategy, muticasting, wireess backhau I. INTRODUCTION Coud radio access network C-RAN) has been recognized as one of the enabing technoogies to meet the ever-increasing demand for higher data rates for the next generation 5G) wireess networks [2] [4]. In C-RAN, the base-stations BSs) Manuscript submitted on December 0, 207; revised on Apri, 208; accepted on Apri 8, 208. The materias in this paper have been presented in part at the IEEE Internationa Conference on Acoustics, Speech and Signa Processing ICASSP), Cagary, Canada, 208 []. This work was supported in part by the Natura Sciences and Engineering Research Counci NSERC) of Canada and in part by the Nationa Natura Science Foundation of China NSFC) grants 6749 and B. Dai and W. Yu are with The Edward S. Rogers Sr. Department of Eectrica and Computer Engineering, University of Toronto, Toronto, ON M5S 3G4, Canada e-mais: {bdai, weiyu}@comm.utoronto.ca). Y.-F. Liu is with the State Key Laboratory of Scientific and Engineering Computing, Institute of Computationa Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 0090, China e-mai: yafiu@sec.cc.ac.cn). are connected to a centraized processor CP) through highspeed fronthau/backhau inks, which provide opportunities for cooperation among the BSs for inter-ce interference canceation. The performance of C-RAN depends cruciay on the capacity of the fronthau/backhau inks. The objective of this paper is to expore the benefit of utiizing caching at the BSs to augment the fronthau/backhau inks. There are two fundamentay different fronthauing strategies that enabe the cooperation of the BSs in C-RAN. In the data-sharing strategy [5] [8], the CP directy shares the user s messages with a custer of BSs, which subsequenty serve the user through cooperative beamforming. In the compression strategy [5], [9], the CP performs the beamforming operation and sends the compressed version of the anaog beamformed signa to the BSs. The reative advantage of the data-sharing strategy versus the compression strategy depends highy on the fronthau/backhau channe capacity [0], []. In genera, the compression strategy outperforms data-sharing when the fronthau/backhau capacity is moderatey high, in part because the data-sharing strategy reies on the backhau to carry each user s data mutipe times to mutipe cooperating BSs. Thus, the finite backhau capacity significanty imits the BS cooperation size. The capacity imitation in fronthau/backhau is especiay pertinent for sma-ce depoyment where high-speed fiber optic connections from the CP to the BSs may not be avaiabe and wireess backhauing may be the most feasibe engineering option. The purpose of this paper is to point out that under this scenario, the data-sharing strategy has a distinct edge in that it can take advantage of: i) the abiity of the CP to muticast user messages wireessy to mutipe BSs at the same time; and ii) the abiity of the BSs to cache user messages to further aeviate the backhau requirement. Note that the muticast opportunity in the wireess backhau and the caching opportunity at the BSs are ony avaiabe to faciitate the datasharing strategy in C-RAN, but not the compression strategy, as the atter invoves sending anaog compressed beamformed signas from the CP to the BSs, which are different for different BSs and are aso constanty changing according to the channe conditions, so are impossibe to cache. This paper considers a downink C-RAN in which the CP utiizes mutipe antennas to muticast user messages to a singe custer of BSs using the data-sharing strategy, whie the BSs pre-store fractions of popuar contents during the offpeak time and request the rest of the fies from the CP using coded deivery via the noisy wireess backhau channe. Given a tota cache constraint, we investigate the optima cache size aocation strategy across the BSs and the optima muticast

2 2 beamforming transmission strategy at the CP so that the fie requests can be deivered most efficienty from the CP to the BSs. It is important to emphasize that the optimizations of the BS cache size aocation and the beamforming strategy at the CP occur in different timescaes. Whie the beamformer can dynamicay adapt to the instantaneous channe reaization, the cache size is optimized ony at the cache aocation phase and can ony adapt to the ong-term statistics of the backhau channe. This paper proposes a sampe approximation approach to sove the above two-timescae optimization probem. The optima cache size aocation considers the ong-term channe statistics in aocating arger cache sizes to the BSs with weaker backhau channes, whie accounting for the potentia effect of beamforming. It aso considers the difference in fie popuarities in caching arger portions of more popuar fies. A. Reated Work Whie caching has been extensivey used at the edge of Internet, the idea of coded caching that takes advantage of muticast opportunity has recenty attracted extensive research interests due to the pioneering work of [2], which uses the network coding method to simutaneousy deiver mutipe fies through a common noiseess channe to mutipe receivers, each caching different parts of the fies. This paper studies a different scenario in which the same content is requested by mutipe receivers BSs), hence no network coding is needed and the coded muticasting in this paper refers to channe coding across mutipe wireess backhau channes with different channe conditions between the CP transmitter and the BS receivers. C-RAN with BS caching has been previousy considered in [3] [6], but most previous works assume fixed cache aocation among the BSs. More specificay, [3] and [4] examine how BS caching heps in reducing both backhau capacity consumption and BS power consumption under given users quaity-of-service constraints; [5] studies how BS caching changes the way that backhau is utiized and proposes a simiar scheme as in [7] that combines the data-sharing strategy and the compression strategy to improve the spectra efficiency of the downink C-RAN. This paper differs from the above works in focusing on how to optimay aocate the cache sizes among the BSs and design muticast beamformers at the CP to improve the efficiency of sharing user s requested fies via the wireess backhau channe. Previous works on caching strategy optimization rey on the assumptions of either simpified networks [8], [9] or Poisson distributed networks [20] [22] that are reasonabe in a network with a arge number of BSs and users, and focus on anayzing how BS caching heps in improving the performance of the BS-to-user ayer. This paper instead considers a C-RAN with a singe custer of BSs and investigates how BS caching improves the efficiency of fie deivery between the coud and the BSs ayer. This paper is motivated by [23] which shows from an information-theoretica perspective the advantage of aocating different cache sizes to different BSs depending on their channe conditions. In addition, [23] proposes a joint cachechanne coding scheme that optimay utiizes the caches at the BSs in a broadcast erasure channe, which is further generaized to the degraded broadcast channe in [24]. We take the findings in [23] one step further by considering the effect of mutipe-antenna beamforming in a downink C- RAN backhau network. We aso extend [23] to the case of mutipe fies with different popuarities and demonstrate that the optima caching strategy aso highy depends on the fie popuarities. B. Main Contributions This paper considers the joint optimization of BS cache size aocation and muticast beamformer at the CP in two timescaes for a downink C-RAN with a singe BS custer under the data-sharing strategy. The main contributions of this paper are summarized as foows: Probem Formuation: We derive a new muticast backhau rate expression with BS caching based on the joint cache-channe coding scheme of [23]. We then formuate two new cache size aocation probems of minimizing the expected fie downoading time and maximizing the expected fie downoading rate subject to the tota cache size constraint. The cache size aocation is optimized offine and is fixed during the fie deivery phase, whie the transmit beamformers are adapted to the rea-time channe reaization. Agorithms: We propose efficient agorithms for soving the formuated cache size aocation probems. More specificay, to dea with the intractabiity of taking expectation over the channe reaizations in the objective functions, we approximate the expectation via samping [25]. Note that the sampe size generay needs to be arge in order to guarantee the approximation accuracy. We further propose to sove the sampe approximation probem using the successive inear approximation technique and the aternating direction method of mutipiers ADMM) agorithm [26], which decomposes the potentiay argescae probem due to the arge sampe size) into many sma-scae probems on each sampe. Engineering Insight: We quantify how much cache shoud be aocated among the BSs in a practica C-RAN setup, and show that, as compared to the uniform and proportiona cache size aocation schemes, the proposed scheme aocates aggressivey arger cache sizes to the fies with higher popuarities, and for each fie the proposed scheme aocates aggressivey arger cache sizes to the BSs with weaker backhau channes. C. Paper Organization and Notations The remainder of this paper is organized as foows. Section II introduces the considered system mode for C-RAN. We derive the backhau muticast rate with BS caching in Section III and state the probem setup in Section IV. Sections V and VI focus on the proposed cache size aocation schemes for the singe fie case and the mutipe fies case,

3 3 repacements Centra Processor Wireess Backhau Cache Cache Cache BS BS 2 BS 3 User Fig.. Downink C-RAN with BS caching, where each BS is equipped with a oca storage unit that caches some contents of user s requested fies. respectivey. Simuation resuts are provided in Section VII. Concusions are drawn in Section VIII. Throughout this paper, ower-case etters e.g. x) and owercase bod etters e.g. x) denote scaars and coumn vectors, respectivey. We use C to denote the compex domain. The transpose and conjugate transpose of a vector are denoted as ) T and ) H, respectivey. The expectation of a random variabe is denoted as E[ ]. Caigraphy etters are used to denote sets. II. SYSTEM MODEL Consider a downink C-RAN mode in Fig. consisting of BSs connected to a coud-based CP through shared wireess backhau. The coud empoys a data-sharing strategy which deivers each user s intended message to a predefined custer of BSs and the BS custer subsequenty serves the user through cooperative beamforming. The capacities of the backhau is a significant imiting factor to the performance of the C-RAN [5], [8]. To aeviate the backhau requirement, this paper considers the scenario where each BS is equipped with a oca cache, as shown in Fig., that can pre-store a subset of the fies during off-peak traffic time in order to reduce the peak time backhau traffic. For simpicity, we consider a network consisting of a singe custer of L cooperative BSs, i.e., a the BSs in the network beamform cooperativey to serve each user. In this case, the user s intended message needs to be made avaiabe at a BSs in order to aow for cooperation. We assume that the CP has access to a the fies and deivers the user s requested fie to the BSs through muticast beamforming over the wireess backhau channe. The backhau connecting the CP with the BSs is impemented in a shared wireess medium, assumed to foow a bock-fading channe mode. We assume that the CP is equipped with M transmit antennas, whie a the BSs are equipped with a singe antenna. We denote the channe vector between the CP and the BS L := {,2,...,L}, as h C M, which remains constant within a coherent bock but changes independenty and identicay according to some distribution in different coherent bocks. The received signa at BS can be written as y = h H x+z, ) where x C M is the transmit signa of the CP transmitter, y C is the received signa at BS, and z CN 0, ) is the background noise at BS obeying the compex Gaussian distribution with zero mean and variance. Each BS is equipped with a oca cache of size C that can pre-store some contents of the fie. This paper addresses two questions: Given fixed oca cache sizes and fixed cached contents, at a fast timescae, how shoud the transmit beamforming strategy be designed as function of the instantaneous reaization of the wireess channe in order to most efficienty deivery a common user message to a the BSs? At a sow timescae, how shoud the contents be cached and how shoud the cache sizes be aocated across the BSs so that the expected deivery performance across many channe reaizations is optimized? The answers to the above two questions woud be trivia if the cache size at each BS is arge enough to store the entire fie ibrary in the network, in which case no backhau transmission is needed. This paper considers a more reaistic scenario where the network operator has a fixed budget to depoy ony a imited amount of tota cache size C. Because of the imited cache size, each BS can ony cache a subset of the fies. In the next section, we define the fie deivery performance in the backhau ink in terms of both the deivery rate and the downoading time, which are expressed as functions of cache sizes at the BSs. III. BROADCAST CHANNEL WITH RECEIVER CACHING In this section, we investigate the optima caching strategy for the backhau network with given cache size at each BS. We then formuate the two-stage joint cache and beamforming design probem considered in this paper in the next section. A. Separate Cache-Channe Coding Without BS caching, the downink C-RAN wireess backhau network with a singe custer of BSs as shown in Fig. can be modeed as a broadcast channe BC) with common message ony, whose capacity is given as R 0 Ix;y ), L, 2) where R 0 denotes the muticast rate, Ix;y ) is the mutua information between the transmit signa x at the CP and the received signa y at BS. It can be seen from 2) that the common information rate is imited by the worst channe across the BSs. To dea with the channe disparity issue in 2), this paper considers the use of BS caching to smooth out the difference

4 4 in channe quaity across the BSs. Assuming that BS has cache size C bits, fied up by caching the first C bits of a fie with a tota size of F bits, a simpe caching strategy is to et the CP deiver ony the rest F C bits of the fie to BS. However, since the BSs are served through muticasting, the CP has to send the maximum of the rest of the requested fie, i.e., max {F C }, to make sure that the BS with east cache size can get the entire fie. Assuming that the channe coherent bock is arge enough so that the fie can be competey downoaded within one coherent bock, then the amount of time needed to finish the fie downoading is T 0 = max {F C } min {Ix;y )} and the effective fie downoading rate is 3) D 0 = F T 0 = min {Ix;y )} max { C /F}. 4) As we can see from 3) or 4), with this naive caching strategy, it is optima to aocate the cache size uniformy among the BSs, i.e., C = C/L, L. B. Joint Cache-Channe Coding It is possibe to significanty improve the naive separate cache-channe coding strategy by considering cached content as side information for the broadcast channe. The achievabe rate of this strategy, named as joint cache-channe coding in [23], can be characterized as beow. Lemma [23]): Consider a BC with common message, if receiver L cachesα 0 α ) fraction of the message, then the muticast common message rateris achievabe if and ony if the foowing set of inequaities are satisfied: R α ) Ix;y ), L, 5) where x is the input and y s are the output of the broadcast channe. Proof: We outine an information-theoretic proof as foows. Consider that a message w is chosen uniformy from the index set {,2,...,W} and is to be transmitted to a set of receivers L over n channe uses at a rate of R = ogw n bits per channe use. A codebook C of size [ 2 nr,n ] is first generated by drawing a symbos x i j),i =,2,...,n, and j =,2,...,2 nr, independenty and identicay according to the channe input distribution, where each row of C corresponds to a codeword. To send the message w, the w-th row of C, denoted as X n w) = [x w) x 2 w)... x n w)], is transmitted over the channe. Note that the codebook C is reveaed to both the transmitter and the L receivers. After receiving Y n, the receiver tries to decode the index w by ooking for a codeword in the codebook C that is jointy typica with Y n and the cached content. Suppose that each receiver caches a fraction of the message specificay caches the first α ogw bits of w. Then, receiver ony needs to search among those codewords whose indices start with the same α ogw bits as the cached In a simiar vein, a reated probem formuation of using secondary backhau inks to compensate for channe disparity is investigated in [27]. bits. Since there exist a tota number of 2 nr α ogw such codewords, by the packing emma [28], receiver woud be abe to find the correct codeword with diminishing error probabiity in the imit n as ong as the inequaity nr α ogw nix;y ) is satisfied, or equivaenty R α R Ix;y ), where x is the input channe symbo. This inequaity needs to be satisfied by a L to ensure that the common message is recovered by a the receivers, which eads to the proof of the achievabiity of 5) in Lemma. For the proof of converse, we refer to [23] for the detais. In the setup of this paper, given cache size aocation C and the fie size F, each BS can cache C /F fraction of the fie. Hence, by Lemma, the fie deivery rate D c with the joint cache-channe coding strategy can be formuated as D c = min { Ix;y ) C /F }, 6) and the downoading time T c can be written as T c = F { } F C = max. 7) D c Ix;y ) Ceary, the above fie downoading time and deivery rate are stricty better than the ones in 3) and 4) except when a Ix;y ) are equa to each other. Instead of aocating the cache size C uniformy, 6) and 7) suggest that it is advantageous to aocate more cache to the BSs with weaker channes to achieve an overa higher muticast rate or shorter downoading time. The difficuty, however, ies in the fact that in practice the channe condition changes over time whie the cache size aocations among the BSs can ony be optimized ahead of time at the cache depoyment phase. In the next section, we formuate a two-stage optimization probem that jointy optimizes the cache size aocation strategy based on the ong-term channe statistics and the beamforming strategy based on the short-term channe reaization. IV. TWO-STAGE CACHING AND BEAMFORMING DESIGN We are now ready to formuate the two-stage joint cache size aocation and beamforming design probem. At a sow timescae, cache size aocation is done at the cache depoyment phase, so they can ony adapt to the channe statistics. At a fast timescae, the beamforming vector can be designed to adapt to each channe reaization during the fie deivery phase. First, we fix cache size aocation and content pacement and focus on the beamforming design in the fast timescae. Assuming that the BS uses a singe-datastream muticast beamforming strategy for the mutipe-antenna BC ), the transmit signa is given by x = ws, where w C M is the beamformer vector and s C is the user message, which can be assumed to be compex Gaussian distributedcn 0, ). Then, the mutua information in the previous section becomes Ix;y ) = og + TrH W) ), 8) where H = h h H, and W = ww H is the beamforming covariance matrix of the transmit signa x restricted within the constraint set W = {W 0 TrW) P, rankw) = } 9)

5 5 with P being the transmit power budget at the CP. The above set is nonconvex due to the rank-one constraint. To obtain a numerica soution, a common practice is to drop the rank-one constraint to enabe convex optimization, then to recover a feasibe rank-one beamformer from the resuting soution [29]. Whie the soution so obtained is not necessariy goba optimum, this strategy often works very we in practice, when compared to the gobay optima branch-and-bound agorithm [30]. Under fixed channe reaization H and cache size C, the optima beamformer design probem, after dropping the rankone constraint, can now be formuated in terms of maximizing the deivery rate or equivaenty minimizing the downoading time): maximize {W} D c subject to TrW) P, W 0, 0a) 0b) which can be reformuated as the foowing convex optimization probem: maximize ξ a) {W, ξ} subject to og + TrH ) W) ξf C ), L, b) TrW) P, W 0. c) This probem can be soved efficienty using standard optimization toobox such as CVX [3]. To obtain a rank-one muticast beamforming vector afterwards, we can adopt a strategy of using the eigenvector corresponding to the argest eigenvaue of soution W. The simuation section of this paper ater examines the performance oss due to such a reaxation of the rank-one constraint. Next, we consider the aocation of cache sizes in the sow timescae. The chaenge is now to find the optima aocation C that s the expected fie downoading time or maximizes the expected fie downoading rate over the channe distribution. Intuitivey, the roe of caching at the BSs is to even out the channe capacity disparity in the CP-to-BS inks so as to improve the muticast rate, which is the minimum capacity across the BSs. At the fast timescae, transmit beamforming aready does so to some extent. BS caching aims to further improve the minimum. The chaenge here is to optimize the cache size aocation, which is done in the sow timescae, whie accounting for the effect of beamforming, which is done in the fast timescae as a function of the instantaneous channe. We note that the caching strategy outined in Lemma is universa in the sense that it depends ony on C and not on H. In the next two sections, we devise efficient agorithms that optimize the cache size aocation at the BSs based on the ong-term channe statistics using a sampe approximation technique. V. CACHE ALLOCATION OPTIMIZATION ACROSS THE BSS In this section, we formuate the cache size aocation probem for deivering a singe fie case of fixed size F bits in order to iustrate a sampe approximation technique that aows us to quantify how much cache shoud be aocated to the BSs with different average channe strengths. The mutipe fies case is treated in Section VI. A. Minimizing Expected Downoading Time For given cache size C, the optima fie downoading time 7) can be written as Tc = minmax F C W og + TrH W). 2) ) Note that the fie downoading time has aso been considered in [6], [32] as the objective function. Differenty, in this paper, we take the expectation oftc over the channe distribution and aim to find an optima cache size aocation that s the ong-term expected fie downoading time. The cache optimization probem under a tota cache size constraint C across the BSs is formuated as: {C } subject to E {H }[Tc] C C, 0 C F, L. L 3a) 3b) Finding a cosed-form expression for the objective function in 3a) is difficut. This paper proposes to repace the objective function in 3a) with its sampe approximation [25] and to reformuate the probem as: {C, W n } subject to N N max F C og + TrHn Wn ) ) C C, 0 C F, L, TrW n ) P, W n 0, n N, 4a) 4b) 4c) where N is the sampe size, N := {,2,...,N}, {H n } n N are the channe sampes drawn according to the distribution of H, and W n is the beamforming covariance matrix adapted to the sampes{h n } L. Note that we do not assume any specific channe distribution here. In fact, the above sampe approximation scheme works for any genera channe distribution. Furthermore, even if in practice when the channe distribution is unknown, we can sti use the historica channe reaizations as the channe sampes, as ong as they are samped from the same distribution. Probem 4) is sti not easy to sove mainy due to the foowing two reasons. First, the objective function of probem 4) is nonsmooth and nonconvex, abeit a of its constraints are convex. Second, the sampe size N generay needs to be sufficienty arge such that the sampe average is a good approximation to the origina expected downoading time, eading to a high compexity for soving probem 4) directy. In the foowing, we first reformuate probem 4) as a smooth probem and inearize the nonconvex term, then everage the ADMM approach to decoupe the probem inton

6 6 ow-compexity convex subprobems to improve the efficiency of soving the probem. First, drop the constant /N in 4a) and introduce the auxiiary variabe {ξ n }, and reformuate probem 4) as {C, W n, ξ n } subject to N ξ n og + TrHn Wn ) 4b) and 4c). ) ξ n F C ), 5a) L, n N, 5b) The above probem 5) is smooth but sti nonconvex due to constraint 5b). To dea with this nonconvex constraint, we approximate the nonconvex term ξ n F C ) in 5b) by its first-order Tayor expansion at some appropriate point ξ n, C ), i.e., ξ n F C ) ξ n F C )+ [ F C, ξ n] [ ξ n ξ n, C C ] T 6) = ξ n F C )+ F C ) ξ n ξ n). 7) Based on 7), an iterative first-order approximation is proposed in Agorithm for soving probem 4). More specificay, et {ξ n t), C t)} be the iterates at the t-th iteration, the agorithm soves {C, W n, ξ n } N subject to og ξ n + TrHn Wn ) ) ξ n t)f C ) 8a) +F C t))ξ n ξ n t)), L, n N, 8b) ξ n ξ n t) rt), n N, 8c) C C t) rt), L, 8d) 4b) and 4c), with fixed {ξ n t),c t)}, where 8c) and 8d) are the trust region constraints [33], within which we trust that the inear approximation in 8b) is accurate, and rt) is the trust region radius at the t-th iteration, which is chosen in a way such that the foowing condition is satisfied: N N ξ n t) N max F C t) og + TrHn Wn t)) N ξ n t) ξ n t) ) τ, 9) where W n t), C t), ξn t) are soutions to probem 8) and τ 0,) is a constant. Notice that the numerator in 9) is the actua reduction in the objective of probem 4) and the denominator is the predicted reduction. The condition in 9) basicay says that the trust region radius is accepted ony if the ratio of the actua reduction and the predicted reduction Agorithm Optimized Cache Aocation with Singe Fie Initiaization: Initiaize C ) = C/L, L, and ξ n ) as the soution to probem 5) with C = C ); set t = ; Repeat: ) Initiaize the trust region radius rt) = ; Repeat: a) Use the ADMM approach in Appendix B to sove probem 8); b) Update rt) = rt)/2; Unti condition 9) is satisfied. 2) Update {ξ n t+),c t+)} according to 20) and 2), respectivey; 3) Set t = t+; Unti convergence is greater than or equa to a constant, in which case probem 8) is a good approximation of the origina probem 4). After soving probem 8), the agorithm updates the parameters for the next iteration by substituting the soution obtained from the ineary approximated probem 8) to the origina probem 4): ξ n t+) = min L og ) + TrHn Wn t)) F C t), n N, 20) C t+) = C t), L. 2) For the initia point, we can set C ) to be C/L for a L, then probem 5) can be decouped into N convex optimization subprobems to sove for ξ n ) for a n N. It remains to sove probem 8). Note that probem 8) is a convex probem but with a potentiay arge number of variabes due to the arge sampe size. We propose an ADMM approach [26] to sove probem 8), which decoupes the high-dimensiona probem into N decouped smadimensiona subprobems. The detais of soving probem 8) using the ADMM approach can be found in Appendix B. It can be shown that the ADMM approach is guaranteed to converge to the goba optimum soution of the convex optimization probem 8). Once probem 3) is soved using Agorithm, we fix the obtained optimized cache size aocation and evauate its effectiveness under a different set of independenty generated channes and cacuate the fie downoading time 2 for each channe by soving the convex probem ). Agorithm is guaranteed to converge to a stationary point of the optimization probem 4). In the rest of this section, we prove the convergence of Agorithm. First, we define the stationary point of probem 4) as in [34]. Definition : Consider a more genera probem x X Fx) 22) 2 Note that the optima downoading time for each given channe is the inverse of the optima objective vaue of probem ).

7 7 where X is the feasibe set and Fx) is defined as Fx) := N N max{f n x)}. 23) Here, {f n x)} is a set of continuousy differentiabe functions. Given any feasibe point x, define { Φ x) = max { d, x + d X} N N F x) { max fn x)+ f n x) T d }}. 24) A point x X is caed a stationary point of probem 22) if Φ x) = 0. Two remarks on the above definition of the stationary point are in order. First, it is simpe to see that Φ x) is aways nonnegative as d = 0 is a feasibe point of 24). If Φ x) = 0, it means that there does not exist any feasibe and decreasing direction at point x in the first-order approximation sense. Second, probem 4) is in the form of probem 22) if we set ) x = {C,W n }, f n x) = F C )/og + TrHn Wn ) σ, 2 and X to be the feasibe set of probem 4), which is convex and bounded. Based on the above stationary point definition, we now state the convergence resut of Agorithm in the foowing theorem. Theorem : Agorithm is guaranteed to converge. Any accumuation point of the sequence generated by Agorithm is a stationary point of probem 4), or equivaenty probem 5). Proof: Agorithm is a specia case of the genera nonsmooth trust region agorithm discussed in [35, Chapter ], which can be proved to converge to a stationary point of the genera probem 22). For competeness of this paper, we provide a proof outine in Appendix A. B. Maximizing Expected Downoading Rate In this subsection, we consider maximizing the expected fie downoading rate as the objective function to optimize the BS cache size aocation, which can be formuated as [ ] F maximize E {H } {C } Tc 25a) subject to C C, 0 C F, L, 25b) L where Tc is the optima fie downoading time defined in 2) under given channe reaization and cache size aocation. Note that the expected vaue of the inverse of a random variabe X, E [ X], is in genera different from the inverse of the expected vaue ofx, E[X]. Thus, the cache size aocation obtained from soving probem 25) is aso different from the one obtained from soving probem 3). We use the same idea as in the previous subsection to sove probem 25). First, we repace the objective function 25a) with its sampe approximation and reformuate the probem as N maximize {C, W n, ξ n } ξ n 26a) subject to 4b), 4c), and 5b), in which we have dropped the constants N and F from the objective function. Then, we repace the nonconvex term in constraint 5b) by its inear approximation 7) and sove probem 26) via optimizing a sequence of ineary approximated probems simiar to probem 8). The approximated probem at each iteration is soved via an ADMM approach simiar to the one described in Appendix B with the ony difference being that the first term ξ in the subprobem 37) n needs to be repaced by ξ n. Same as in the previous subsection, once the optimized cache size aocation is obtained from soving probem 26), we evauate its effectiveness on different sets of channes and sove the muticast rate 3 by optimizing probem ). VI. CACHE ALLOCATION OPTIMIZATION ACROSS FILES In this section, we consider the cache size aocation probem for the genera case with mutipe fies having different popuarities. Due to the minima difference between the downoading rate and the downoading time as described in the previous section, we ony focus on minimizing the expected fie downoading time as the objective function in this section. We assume that each fie k of equa size F bits is requested from the user with given probabiity p k,k K := k p k =, and that BS caches C k /F {,2,...,K}, fraction of fie k with a tota cache size constraint given by,k C k C. Given that fie k is requested, according to Lemma, the optima downoading time for fie k, denoted as Tk, can be written as Tk = min max W k W F C k og + TrH W k ) ). 27) Different from the downoading time 2) in the singe fie case, the above optima downoading time Tk is a random variabe depending on not ony the channe reaization but aso the index of the requested fie. We take the expected vaue of Tk on both the channe reaization H and the fie index k as the objective function and formuate the muti-fie cache size aocation probem as p k E {H }[Tk] 28a) {C k } k subject to C k C, 0 C k F, L, k K.,k 28b) Athough intuitivey the more popuar fie shoud be aocated arger cache size, the question of how much cache shoud be aocated to each fie is nontrivia. In particuar, it is in 3 The optima muticast rate for given channe is the optima objective vaue of probem ).

8 8 genera not true that one shoud aocate the most popuar fie in its entirety first, then the second most popuar fie, etc., unti the cache size is exhausted. This is because the gain in term of the objective function of the optimization probem 28) due to aocating progressivey more cache size to one fie diminishes as more cache is aocated. At some point, it is better to aocate some cache to the ess popuar fies, even when the most popuar fie has not been entirey cached. The optima aocation needs to be found by soving probem 28). To sove probem 28), we use the same sampe approximation idea as in the singe fie case. With an additiona set of axiary variabes ξk n, probem 28) after the sampe approximation can be formuated as {C k, W n k, ξn k} subject to K N p k k= og ξ n k + TrHn k Wn k ) 29a) ) ξk n F C k), L, n N, k K, 29b) C k C, 0 C k F,,k TrW n k ) P, Wn k 0, L, k K, 29c) n N, k K. 29d) Probem 29) is then soved in an iterative fashion. At each iteration the nonconvex term on the right hand side of 29b) is repaced by its first-order approximation and the resuting convex probem to be soved at the t-th iteration is given by: {C k, W n k, ξn k} K N p k k= subject to og ξ n k + TrHn k Wn k ) 30a) ) ξk n t)f C k) +F C k t))ξ n k ξ n kt)), L, n N, k K, 30b) ξ n k ξ n kt) rt), n N, k K, 30c) C k C k t) rt), L, k K, 30d) 29c) and 29d), where ξ n k t) and C kt) are fixed parameters obtained from the previous iteration and are updated for the next iteration according to ξ n kt+) = min L og ) + TrHn k Wn k t)) F C k t), n N, k K, 3) C k t+) = C kt), L, k K, 32) where Wk n t) and C k t) are soutions to probem 30). Simiar to 9), the trust region radius rt) in 30c) and 30d) Agorithm 2 Optimized Cache Aocation with Mutipe Fies Initiaization: Initiaize C k ) = C/LK, L, k K and ξ n k ) as the soution to probem 29) with C k = C k ); set t = ; Repeat: ) Initiaize the trust region radius rt) = ; Repeat: a) Use the ADMM approach in Appendix C to sove probem 30); b) Update rt) = rt)/2; Unti condition 33) is satisfied. 2) Update {ξ n k t+),c kt+)} according to 3) and 32), respectivey; 3) Set t = t+; Unti convergence is picked to satisfy the foowing condition: K N p k F C ξ n max k t) k= k t) og k= + TrHn k Wn t)) ) k K N p k ξk nt) ) ξk n t) τ 33) for some constant τ 0,). Note that probem 30) can aso be soved by using an ADMM approach as expained in Appendix C, which decoupes probem 30) into NK subprobems and each subprobem corresponds to a pair of sampe channe and fie request. The overa proposed agorithm for soving the cache size aocation probem with mutipe fies is summarized in Agorithm 2. Once the cache size aocation for the mutipe fies case is optimized through Agorithm 2, we cacuate the downoading time for fie k by soving probem 27) with fixed C k, which can be formuated as a convex optimization probem simiar to ). We then compute the average downoading time by averaging under different sets of channe reaizations. VII. SIMULATION RESULTS This section evauates the performance of our proposed caching schemes through simuations. Consider a downink C-RAN mode with L = 5 BSs randomy paced on the haf pane beow the CP with the reative distances between the CP and the 5 BSs shown in Fig. 2. We generate 000 sets of channe reaizations from the CP to the BSs according to h = K /2 v, where K modes the correation between the CP transmit antennas to BS and is generated mainy according to the ange-of-arriva and the antenna pattern, as described in [36], with the path-oss component modeed as og 0 d) db and d is the distance between the coud and the BS in kiometers;v is a Gaussian random vector with each eement independenty and identicay distributed as CN 0,). The first N = 00 sets of channe reaizations are

9 Coud Uniform Proportiona Optimized - Downoading Time Optimized - Downoading Rate -00 meter BS 5 BS 4 BS 3 BS 2 BS Cache Size meter 0 BS BS 2 BS 3 BS 4 BS 5 Fig. 2. A downink C-RAN setup with 5 BSs. The distances from the CP to the 5 BSs are 398, 278, 473, 286, 267) meters, respectivey. Fig. 3. Cache aocation for different schemes under tota cache size C = 00, normaized with respect to fie size F = 00. TABLE I SIMULATION PARAMETERS. Parameters Vaues Number of BSs 5 Backhau channe bandwidth 20 MHz Number of antennas at CP 0 Number of antennas at each BS Maximum transmit power P at CP 40 Watts Antenna gain 7 dbi Background noise 50 dbm/hz Path oss from CP to BS og 0 d) Rayeigh sma scae fading 0 db Normaized fie size 00 Training sampe size N 00 Test sampe size 900 used in the sampe approximation to optimize the cache aocation whie the rest 900 are used to evauate the performance under the obtained cache size aocation. The detais of the simuation parameters are isted in Tabe I. A. Cache Aocation for BSs with Varying Channe Strengths In this subsection, we evauate the performances of the proposed schemes for caching a singe fie across mutipe BSs with different channe strengths as discussed in Section V. We compare the optimized cache size aocations obtained from minimizing the expected fie downoading time 3) and maximizing the expected fie downoading rate 25) with the foowing set of schemes: No Cache: Cache sizes C = 0 for a BSs; Uniform Cache Aocation: Cache sizes among the BSs are uniformy distributed as C = C/L, which serves as a baseine scheme; Proportiona Cache Aocation: Cache sizes among the BSs are proportionay ) aocated such that F C )/og + P TrK ) Lσ for a are equaized, if 2 possibe, which serves as another baseine scheme; Lower/Upper Bound: Cache sizes among the BSs are dynamicay and optimay aocated by soving probem ) for each channe reaization by treating {C } as the optimization variabes, which is impractica in reaity but serves as a ower bound for minimizing the expected fie downoading time and an upper bound for maximizing the expected fie downoading rate; Rank-One Muticast Beamformer: Cache sizes among the BSs are the same as the optimized caching schemes, but the muticast beamformer is restricted to be rank-one and is set to be the eigenvector corresponding to the argest eigenvaue of the optimized beamforming matrix W n in each test sampe channe. In Fig. 3, we compare the aocated BS cache sizes between the proposed schemes trained on the first 00 channes and the baseine schemes under normaized fie size F = 00 and tota cache size C = 00. As we can see, both of the proposed caching schemes are more aggressive in aocating arger cache sizes to the weaker BS 3 as compared to the uniform and proportiona caching schemes. We then evauate the performances of different cache size aocation schemes on the rest 900 sampe channes and report the fie downoading time and downoading rate or spectra efficiency) in Tabe II and III, respectivey, under two different settings of tota cache size C = 00 and C = 200, normaized with respect to fie size F = 00. As we can see, the proposed caching scheme improves over the uniform and proportiona caching schemes by 0% 5% on average, but the gains are more significant for the 90th-percentie downoading time and the 0th-percentie downoading rate, which are around 20% 27% and 26% 36%, respectivey. We note here that without caching, the average and 90thpercentie fie downoading time are.45 ms/mb and 4.76 ms/mb, respectivey, in this setting. The average and 0thpercentie fie downoading rate are 4.63 bps/hz and 3.39 bps/hz. Thus, the optimized BS caching schemes with C = 00 and C = 200 normaized with respect to F = 00) improve the average downoading time by about 33% and 50% respectivey, and improve the average downoading rate

10 0 TABLE II FILE DOWNLOADING TIME MS/MB) COMPARISON FOR DIFFERENT TOTAL CACHE SIZES, NORMALIZED WITH RESPECT TO FILE SIZE F = 00. Cache Scheme Tota Cache C = 00 Tota Cache C = 200 Average 90th-Percentie Average 90th-Percentie Uniform Proportiona Optimized Rank-One Lower Bound TABLE III FILE DOWNLOADING RATE BPS/HZ) COMPARISON FOR DIFFERENT TOTAL CACHE SIZES, NORMALIZED WITH RESPECT TO FILE SIZE F = 00. Cache Scheme Tota Cache C = 00 Tota Cache C = 200 Average 0th-Percentie Average 0th-Percentie Uniform Proportiona Optimized Rank-One Upper Bound Cumuative Distribution Function No Cache, C = 0 Uniform, C = 00 Proportiona, C = 00 Optimized, C = 00 Lower Bound, C = 00 Rank-One, C = 00 Uniform, C = 200 Proportiona, C = 200 Optimized, C = 200 Lower Bound, C = 200 Rank-One, C = 200 Cumuative Distribution Function No Cache, C = 0 Uniform, C = 00 Proportiona, C = 00 Optimized, C = 00 Upper Bound, C = 00 Rank-One, C = 00 Uniform, C = 200 Proportiona, C = 200 Optimized, C = 200 Upper Bound, C = 200 Rank-One, C = Downoading Time in ms/mb Fig. 4. CDF of downoading time under different caching schemes with tota cache size C = 00 and C = 200, respectivey, normaized with respect to fie size F = Muticast Rate in bps/hz Fig. 5. CDF of downoading rates under different caching schemes with tota cache size C = 00 and C = 200, respectivey, normaized with respect to fie size F = 00. by about 43% and 9% respectivey. In Figs. 4 and 5, we compare the cumuative distribution functions CDFs) of the downoading time and the downoading rates evauated on the 900 test channes with different caching schemes. Simiar to what we have seen in Tabes II and III, the proposed caching scheme shows significant gain on the high downoading time regime in Fig. 4 and on the ow downoading rate regime in Fig. 5 as compared to the baseine schemes. From Figs. 4 and 5, we can aso see that the rankone muticast beamformer shows negigibe performance oss as compared to the genera-rank muticast beamformer matrix W n obtained by soving ). It is aso worth remarking that the ower bound scheme in Fig. 4 and the upper bound scheme in Fig. 5 sove the cache size aocation probem dynamicay for each channe reaization, which is impractica, and ony serve as benchmark schemes in this paper. To summarize the insight from the simuation resuts in this subsection for the singe fie case: First, athough both the uniform and the proportiona caching schemes perform fairy we in terms of the average fie downoading time and downoading rate, the proposed caching scheme shows significant gains in improving the high downoading time regime and the ow downoading rate regime. This is due to the fact that BSs farther away from the coud are more aggressivey aocated arger amount of cache under the optimized scheme. Second, the rank-one beamformer derived from the generarank covariance matrix does not degrade the performance much at a. Hence, we ony focus on the performance of the proposed caching schemes without the rank- constraint on the covariance matrix in the next subsection for the mutipe fies case. B. Cache Aocation for Fies of Varying Popuarities In this subsection, we present simuation resuts for the caching schemes with mutipe fies having different popuarities and focus on the expected fie downoading time as the performance metric. We first consider ony two fies with different pairs of request probabiities p,p 2 ) isted on

11 TABLE IV OPTIMIZED CACHE ALLOCATIONC,C 2 ) FOR A 2-FILE CASE WITH DIFFERENT FILE POPULARITIES UNDERC = 00 AND F = 00. Fie Popuarity p,p 2 ) = 0.5,0.5) p,p 2 ) = 0.6,0.4) p,p 2 ) = 0.7,0.3) p,p 2 ) = 0.8,0.2) p,p 2 ) = 0.9,0.) BS 8.2, 8.2) 3.2, 2.7) 6.8, 0) 20.2, 0) 22.2, 0) BS2 0, 0) 0, 0) 0, 0) 4.6, 0) 7., 0) BS3 4.8, 4.8) 48.2, 35.9) 53.6, 27) 56.8, 0.9) 58.8, 0) BS4 0, 0) 0, 0) 2.6, 0) 7.5, 0) 0., 0) BS5 0, 0) 0, 0) 0, 0) 0, 0).8, 0) Tota 50, 50) 6.4, 38.6) 73, 27) 89., 0.9) 00, 0) the first row of Tabe IV, where each coumn denotes the cache size aocation among the 5 BSs under the specific fie popuarity given in the first row and each ce gives the cache size aocation between the two fies within each BS. The cache sizes in each coumn add up to the tota cache size C = 00, normaized with respect to fie size F = 00. From Tabe IV we see that for each coumn with given fie popuarity, the weakest BS 3 aways gets the most cache size as in the singe fie case shown in Fig. 3. Moreover, as the difference between the popuarities of the two fies increases across the coumns, more cache is aocated to the first fie. For exampe, the proposed caching scheme decides to aocate a the cache to ony the more popuar fie when p,p 2 ) = 0.9,0.). In Fig. 6, we compare the average fie downoading time between the optimized cache scheme and the foowing baseine schemes: No Cache: Cache size C k = 0 for a BSs and fies; Uniform Cache Aocation: Cache size for fie k at each BS is set to be as C k = C/LK for a k and ; Proportiona Cache Aocation: We first set the tota cache size aocated for fie k as p k C, then distribute p k C among the BSs according to the rue descried in the Proportiona Cache Aocation scheme in Section VII-A; Caching the Most Popuar Fie: We cache the most popuar fie in its entirety first, then the second most popuar fie, etc. When a fie cannot be cache entirey, we distribute the remaining cache among the BSs according to the Proportiona Cache Aocation scheme described in Section VII-A. In Fig. 6, we fix the number of fies to be K = 4 and generate the fie popuarity according to the Zipf distribution [37] given by p k = k α K, k, with different settings of α. As the i= Zipf distribution i α exponent α increases, the difference among the fie popuarities aso increases. As we can see from Fig. 6, the average downoading time for a schemes, except for the uniform caching scheme, decreases as α increases. This is because in uniform cache aocation the cache size is the same for a fies, hence the downoading time is the same no matter which fie is requested. In contrast, a other three schemes tend to aocate more cache to the more popuar fies. In particuar, the proposed caching scheme converges to the scheme of caching the most popuar fie when α =.5, whie it consistenty outperforms the proportiona caching scheme. From Fig. 6 we concude that first, the uniform cache size aocation scheme performs poory when the fies have different popuarities and especiay when the difference is Average Downoading Time in ms/mb No Cache Uniform Proportiona Most Popuar Optimized Zipf Distribution Exponent α Fig. 6. Average downoading time for different Zipf fie distributions under the same number of fies K = 4 and tota cache size C = 400, normaized with respect to fie size F = 00. arge. Second, it is advantageous to aocate arger cache size to the more popuar fie, however, it is not trivia to decide how much more cache is needed for the more popuar fie. Our proposed caching scheme provides a better cache size aocation soution as compared to the heuristic proportiona caching scheme and the most popuar fie caching scheme. VIII. CONCLUSION This paper points out that caching can be used to even out the channe disparity in a muticast scenario. We study the optima BS cache size aocation probem in the downink C- RAN with wireess backhau to iustrate the advantage of muticast and caching for the data-sharing strategy. We first derive the optima muticast rate with BS caching, then formuate the cache size optimization probem under two objective functions, minimizing the expected fie downoading time and maximizing the expected fie downoading rate, subject to the tota cache size constraint. By everaging the sampe approximation method and ADMM, we propose efficient cache size aocation agorithms that consideraby outperform the heuristic schemes. APPENDIX A PROOF OF THEOREM We use the notations introduced in Definition in the foowing convergence proof. First of a, it is simpe to show that the objective sequence {Fxt))} generated by Agorithm monotonicay decreases and is ower bounded by zero.