Power Minimization for Wireless Backhaul Based Ultra-Dense Cache-enabled C-RAN
aa r X i v : . [ ee ss . SP ] S e p Power Minimization for Wireless Backhaul Based Ultra-DenseCache-enabled C-RAN
Jun Xu, Pengcheng Zhu, Jiamin Li, and Xiaohu You,
Fellow, IEEE
Abstract —This correspondence paper investigates joint designof small base station (SBS) clustering, multicast beamformingfor access and backhaul links, as well as frequency allocation inbackhaul transmission to minimize the total power consumptionfor wireless backhaul based ultra-dense cache-enabled cloudradio access network (C-RAN). To solve this nontrivial problem,we develop a low-complexity algorithm, which is a combination ofsmoothed ℓ -norm approximation and convex-concave procedure.Simulation results show that the proposed algorithm convergesfast and greatly reduces the backhaul traffic. Index Terms —Wireless backhaul, cache, multicast beamform-ing, ultra-dense, C-RAN.
I. I
NTRODUCTION
To meet the explosive data demand of the fifth-generation(5G) wireless system, ultra-dense (UD) networking have beenwidely recognized as one of the most promising technolo-gies [1]. The significant reduction of average access distancebetween users and UD base stations (BSs) increases thenetwork capacity and spectrum efficiency [2]. However, it alsoleads to drastic interference, which constitutes the performancebottleneck for UD networking. Cloud radio access network (C-RAN) has been regarded as a promising network architectureto address this issue [3], where interference mitigation canbe effectively realized in the powerful baseband unit (BBU)pool by applying coordinated multi-point (CoMP). In UD C-RAN, macro base stations (MBSs) are connected to BBU withwired links and transmit data signal and control signal tosmall base stations (SBSs), which then provide high qualityof service to users. To enable UD C-RAN, a reliable and cost-effective backhaul link (BL) connecting the MBS and SBSs isprerequisite. The expensive expenditure of wired links makesit unaffordable to be densely deployed. As an alternative,wireless backhaul (WB) emerges to present a viable solutionto solve the expensive backhaul installation in UD C-RAN [4].Due to limited bandwidth and large channel fading, a criticalissue in WB based UD C-RAN is to offload the peak trafficof wireless backhaul. Two enabling techniques to alleviate thebackhaul are caching and SBS clustering. Motivated by thefact only a small portion of the files is highly popular withthe majority of users while the rest is rarely requested, thepopular files can be proactively cached in SBSs during theoff-peak time, which can bring files closer to users and greatlyreduce the backhaul congestion. With SBS clustering, the dataof each user only needs to be delivered to its serving SBSs
This work was supported by Natural Science Foundation of JiangsuProvince (Grant No. BK20180011), and by Natural Science Foundation ofChina (Grant Nos. 61571120, 61871122 and 61871465).J. Xu, P. Zhu, J. Li, and X. You are all with National Mobile Communi-cations Research Laboratory, Southeast University, Nanjing, China ( emails: { xujunseu, p.zhu, lijiamin, xhyu } @seu.edu.cn). from the MBS rather than all the SBSs, which greatly reducesthe overall backhaul load. A joint design of content-centricclustering and sparse multicast beamforming to minimize thetotal network cost for wired backhaul based cache-enabled C-RAN was investigated in [5]. Without considering cache, [6]jointly designed WB multicast beamforming and user-centricclustering to maximize the weighted sum rate in C-RAN.This paper considers a joint design of content-centric SBSclustering, multicast beamforming for access links (ALs) andBLs, as well as frequency allocation in backhaul transmissionto minimize the total power consumption for WB based UDcache-enabled C-RAN, which is a NP-hard problem. We firstanalyze the nature of the intractable constraints, and thenpropose a smoothed ℓ -norm approximation based convex-concave procedure (CCP) algorithm to efficiently solve theconsidered problem with low complexity. Simulation resultsshow that the proposed algorithm converges fast and greatlyreduces the backhaul traffic.II. S YSTEM D ESCRIPTIONS
Consider a cache-enabled UD C-RAN, which consists ofa MBS with N m antennas, M SBSs with N s antennas and K single-antenna users. Let M ∆ = { , . . . , M } and K ∆ = { , , . . . , K } represent the set of SBSs and users respectively.The MBS is connected to BBU through a wired link andcan access a library of F files with normalized size, indexedby F ∆ = { , . . . , F } . All SBSs are connected to the MBSwith WB. It is assumed that the WB links between theMBS and SBSs use out-of-band spectrum, so there is nointerference between ALs and BLs. Users requesting the samefile are grouped together and served by SBSs using multicasttransmission. Let L ∆ = { , , . . . , L } represent the set ofgroups. The set of users belong to group l is denoted by U l ,and the index of group that user k belongs to is denoted by l k .This paper considers SBS clustering and defines a binary SBSclustering matrix C ∈ { , } M × L , where c m,l = 1 indicatesthat SBS m belongs to the serving cluster for the group l and0 otherwise. Hence, we let S A l = { m ∈ M| c m,l = 1 } denotethe cluster of SBSs serving group l . We define a binary cacheplacement matrix S ∈ { , } M × F , where s m,f = 1 indicatesthat the f -th file is cached in the m -th SBS and 0 otherwise.For each SBS m ∈ S A l , if the file f l requested by group l has been cached in its local cache, it can directly deliver thefile to all users in U l via ALs. Otherwise, it needs to fetchthis file from the MBS via the BLs. The cluster of un-cachedSBSs belong to S A l that do not cache file f l can be denoted as S B l = (cid:8) m ∈ S A l | s m,f l = 0 (cid:9) . Note that S B l will be an emptyset if all SBSs in S A l have cached the file f l , which meansthey can access the file f l directly without costing backhaul. We let L u ∈ L denote the set of groups with cardinality L u that needs backhaul transmission. A. Downlink Data Transmission Models
The downlink data transmission consists of BL transmissionand AL transmission. The details of them will be analyzedthereinafter.
1) BL Transmission:
To improve the efficiency of BL trans-mission, the MBS adopts multicast transmission to transmitrequested files to corresponding SBS clusters. Note that oneSBS may belong to multiple un-cached clusters and need tofetch multiple files from the MBS through BLs. To eliminatethe interference during the decoding process at SBSs, thispaper adopts a frequency division multiple access (FDMA)scheme to schedule the BL transmissions for different files,where ≤ b l ≤ is the faction of bandawidth allocated fortransmitting file f l . Let v l ∈ C N m × denote the multicastbeamforming vector at the MBS for group l . The receivedsignal at SBS m is written as y lm = H H m v l x l + z m , ∀ m ∈ M , ∀ l ∈ L , (1)where H m ∈ C N m × N s denotes the channel between theMBS and SBS m ; x l is the normalized signal symbol offile f l ; z m ∼ N c (cid:0) , z m (cid:1) is the noise at SBS m , where N c ( µ, σ ) denotes the circularly symmetric complex Gaussiandistribution with mean µ and variance σ . Then the BLcapacity R B m,l of group l at SBS m is given by R B m,l = b l log (cid:16) (cid:12)(cid:12) H H m v l (cid:12)(cid:12) /z m (cid:17) , ∀ m ∈ M , ∀ l ∈ L .
2) AL Transmission:
We consider the decode-and-forwardscheme at SBSs. All SBSs in S A l adopt multicast transmissionto simultaneously forward file f l to each user in U l after allun-cached SBSs in S B l perfectly decoding file f l . Let w m,l ∈ C N s × denote the beamforming vector at SBS m for group l ,then w l = h w H1 ,l , · · · , w H M,l i H is the aggregate beamformer forgroup l . The received signal at user k can be written as y k = h H k w l k x l k + X Lj = l k h H k w j x j + n k , ∀ k ∈ K , (2)where h k ∈ C MN s × denote the channel between all SBSs touser k ; n k ∼ N c (cid:0) , σ k (cid:1) is the additive white Gaussian noiseat user k . Therefore, the signal-to-interference-plus-noise ratio(SINR) of user k is ξ A k = (cid:12)(cid:12) h H k w l k (cid:12)(cid:12) / (cid:16)P Lj = l k (cid:12)(cid:12) h H k w j (cid:12)(cid:12) + σ k (cid:17) .According to [7], the common AL rate R A l of group l iscomputed by R A l = log (cid:0) k ∈U l ξ A k (cid:1) , ∀ l ∈ L .III. P ROBLEM S TATEMENT
In this section, we first present a power consumption model.Then we formulate a joint optimization problem to minimizethe total power consumption.
A. Power Consumption
The total power consumption of a downlink cached-enabledC-RAN with WB mainly roots in transmit power at SBSs andthe MBS, signal processing power at SBSs and circuit powerat all BSs. Note that the signal processing at each SBS refers to the decoding process of the received signal from the MBS.If SBS m has cached file f l or it is not selected to serve group l , it does not consume power to decode file f l . Hence, the totalpower consumption can be calculated as P tot = X l,m η m | w m,l | + X Ll =1 η | v l | + X l,m c m,l (1 − s m,f l ) P sp m + X Mm =0 P c m , (3)where η m and P c m are respectively the inverse of the transmitamplifier efficiency and the circuit power consumption at BS m , where m = 0 refers to the MBS. P sp m denotes the signalprocessing power consumption used to decode at SBS m . Notethat both P sp m and P c m are given constants. B. Problem Formulation and Analysis
With the given S , this paper aims to formulate a joint de-sign of content-centric SBS clustering, multicast beamformingvectors in ALs and BLs, as well as frequency allocation inbackhaul transmission to minimize the total power consump-tion. All the channel state information (CSI) and user requestsare assumed to be available at the BBU for joint design. Theconsidered problem can be stated as P : min P tot (4a) s . t . ξ A k ≥ γ l k , ∀ k ∈ K , (4b) R A l ≤ R B m,l , ∀ m ∈ S B l , ∀ l ∈ L u , (4c) X Ll =1 | w m,l | ≤ P m , ∀ m ∈ M , (4d) X Ll =1 | v l | ≤ P , (4e) ≤ b l ≤ , ∀ l ∈ L , X Ll =1 b l ≤ , (4f) c m,l = { , } , ∀ m ∈ M , ∀ l ∈ L , (4g)where (4b) requires that the SINR of each user should beabove certain threshold; (4c) denotes that the AL rate R A l forgroup l should be upper bounded by the BL rate R B m,l of SBS m ∈ S B l ; (4d) and (4e) represent transmit power constraints ateach SBS and the MBS respectively; (4f) denotes the backhaulfrequency allocation constraint for each group. Remark 1:
Different from the considered problem in [5]that jointly design content-centric BS clustering and multicastbeamforming for cached-enabled C-RAN with wired backhaul,this paper considers a cached-enabled UD C-RAN with WB.WB induces different power consumption model, more impor-tantly, leads to an intractable backhaul rate constraints in (4c).In addition, the consideration of backhaul frequency allocationmakes the problem more complex. P is a mixed integer nonlinear program (MINLP) withmultiple optimization variables, which is NP-hard and can notbe globally solved in polynominal complexity. In next section,we propose a low-complexity algorithm to solve P .IV. S MOOTHED ℓ - NORM A PPROXIMATION B ASED
CCPA
LGORITHM
In this section, we first simplify the intractable constraints(4c). Then smoothed ℓ -norm is used to approximate discreteSBS clustering with continuous beamforming vetors. At last, aCCP algorithm is proposed to solve the approximated problem. A. Equivalent Form
To simplify (4c), we first provide the following theorem,and the proof is given in Appendix A.
Theorem 1:
Assuming W ∗ = [ w ∗ , · · · , w ∗ L ] is the optimalsolution to P , we have min k ∈U l ξ A k ( W ∗ ) = γ l , ∀ l ∈ L at theoptimal point.Based on Theorem 1, the expression of R A l in (4c) can bereplaced by its threshold r l = log (1 + γ l ) , and we have r l ≤ R B m,l , ∀ m ∈ S B l , ∀ l ∈ L u . (5)Note that S B l and L u are unavailable to BBU because ofunknown SBS clustering. To tackle this difficulty, we refor-mulate (5) into an equivalent form as shown in the followingproposition, and the proof is omitted due to limited space. Proposition 1:
Constraint (5) can be equivalently trans-formed to the following constraint c m,l (1 − s m,f l ) r l ≤ R B m,l , ∀ m ∈ M , ∀ l ∈ L . (6)However, P is still hard to be solved duo to the discrete C . Note that c m,l depends on w m,l as c m,l = (cid:13)(cid:13)(cid:13) | w m,l | (cid:13)(cid:13)(cid:13) .However, the ℓ -norm is not convex. In next subsection, weapproximate it with a smooth logarithmic function. B. Smoothed ℓ -norm Approximation According to [8], we have c m,l = (cid:13)(cid:13)(cid:13) | w m,l | (cid:13)(cid:13)(cid:13) =ln (cid:16) | w m,l | ς − (cid:17) / ln (cid:0) ς − (cid:1) , where ς is a sufficientlysmall constant. Using the right-hand side to replace c m,l inthe objective and (6), we obtain the following problem P : min P tx + X l,m ν m,l ln (cid:16) | w m,l | ς − (cid:17) (7a) s . t . (4 b ) , (4 d ) − (4 f ) , (7b) ln (cid:16) | w m,l | ς − (cid:17) ζ m,l ≤ R B m,l , ∀ m, l, (7c)where the last constant term in (3) is omitted, and P tx denotes the sum of first two terms in (3). In addi-tion, ν m,l = (1 − s m,f l ) P sp m / ln (cid:0) ς − (cid:1) and ζ m,l =(1 − s m,f l ) r l / ln (cid:0) ς − (cid:1) . However, the nonconvex objec-tive (7a) and constraints (4b) and (7c) make it still NP-hard tofind the global optimum to P . In next subsection, we convert P into a form of difference of convex (DC) program, andthen find a suboptimal solution to it by CCP. C. CCP Algorithm to Solve P Because ln (cid:16) | w m,l | ς − (cid:17) is concave in | w m,l | ,it is upper bounded by its first-order expansion as ln (cid:16) | w m,l | ς − (cid:17) ≤ θ ( n ) m,l | w m,l | + q ( n ) m,l , where θ ( n ) m,l =1 / (cid:18)(cid:12)(cid:12)(cid:12) w ( n ) m,l (cid:12)(cid:12)(cid:12) + ς (cid:19) with the solution w ( n ) m,l at n -th iteration and q ( n ) m,l = ln (cid:18) (cid:12)(cid:12)(cid:12) w ( n ) m,l (cid:12)(cid:12)(cid:12) ς − (cid:19) − θ ( n ) m,l (cid:12)(cid:12)(cid:12) w ( n ) m,l (cid:12)(cid:12)(cid:12) . Using this upperbound, P can be approximated as P : min X l,m π ( n ) m,l | w m,l | + X Ll =1 η | v l | (8a) s . t . (4 b ) , (4 d ) − (4 f ) , (8b) ζ m,l (cid:16) θ ( n ) m,l | w m,l | + q ( n ) m,l (cid:17) ≤ R B m,l , ∀ m, l, (8c) where π ( n ) m,l = η m + ν m,l θ ( n ) m,l and the constant term in theobjective is omitted. At this time, the objective is convex andwe turn to nonconvex constraints (4b) and (8c). By introducingauxiliary variables ψ m,l to replace the term (cid:12)(cid:12) H H m v l (cid:12)(cid:12) /z m , (8c)can be equivalently replaced by the following two constraints ζ m,l θ ( n ) m,l | w m,l | b l + q ( n ) m,l b l ! ≤ log (1 + ψ m,l ) , ∀ m, l, (9a) ψ m,l − (cid:12)(cid:12) H H m v l (cid:12)(cid:12) /z m ≤ , ∀ m, l. (9b)Here, the constraint (9a) is convex, and the left-hand side(LHS) of the introduced constraint (9b) is a DC function. Inaddition, the nonconvex constraints (4b) can be rewritten as (cid:18)X Lj = l (cid:12)(cid:12) h H k w j (cid:12)(cid:12) + σ k (cid:19) γ l − (cid:12)(cid:12) h H k w l (cid:12)(cid:12) ≤ , ∀ k ∈ U l , ∀ l ∈ L , (10)where the LHS is also a DC function. Hence, we can applythe CCP algorithm to obtain a suboptimal solution. The sub-problem in each iteration is a convex quadratically constrainedquadratic program (QCQP) and can be efficiently solved byCVX. The details are omitted.V. S IMULATION R ESULTS
We consider a C-RAN network deployed in a circle areawith radius e = 250 m , such as stadiums, where the MBS with N m = 50 antennas is located at the center of this cell. Thereare M = 14 SBSs with N s = 2 antennas and K = 8 single-antenna users randomly distributed in the cell. The densitiesof SBS in the considered scenario comply with the 5G UDnetwork, where the density of 5G BS is highly anticipated tocome up to 40-50 BS / km [1]. The channels of both BLs andALs are modeled as h = p d − β χ g , where d is the distance; β is the path loss exponent; χ is a log-normal shadow fadingcoefficient; g denotes the small scale fading following Raleighdistribution CN (0 , I ) . Here, the path loss exponents for BLsand ALs are 3 and 3.2, and the variances of shadow fading forthem are 3dB and 4dB. The noise power at SBSs and users are-90dBW and -65dBW. In addition, we set P m = 10 W, ∀ m ∈M , P = 50 W , P sp = 1 W and r l = 1 . . We assume thatthere are F = 100 files and each SBS can cache at most Z = 5 files. The user requests follow a Zipf distribution withshewness parameter α . We consider three heuristic cachingstrategies: PopC [5]: each SBS caches the top Z popular files;RanC [5]: each SBS randomly caches Z files; MosC [9]: eachSBS m caches files [1 , , · · · , Z ] + ( m − Z to cache themost files.In Fig.1, we show the convergence of the proposed al-gorithm with three cache strategies. It can be seen clearlythat the proposed algorithm converge within less than 10iterations for all considered cases. Fig.2 shows the cumu-lative distribution functions (CDFs) of power consumptionfor three methods under PopC strategy with α = 1 , where“No SC” does not consider SBS clustering, and “No C&SC”does not consider caching and SBS clustering simultaneously.They can be achieved by slightly modifying Algorithm 1with setting c m,l = 1 , ∀ m, l and c m,l = 1 , ∀ m, l ; Z = 0 respectively. The CDF curves are from 1000 independent Iteration index P o w e r c on s u m p t i on ( W ) PopC, =0RanC, =0MosC, =0PopC, =2RanC, =2MosC, =213 14 15 16 17051015
Fig. 1. Convergence behavior of the proposedalgorithm
Power consumption (W) C u m u l a t i v e d i s t r i bu t i on f un c t i on Empirical CDF
ProposedNo SCNo C&SC
Fig. 2. Performance comparison between the pro-posed algorithm and other methods T r an s m i t po w e r a t SBS s -1 T r an s m i t po w e r a t t he M BS ProposedNo SCNo C&SC S i gna l p r o c e ss i ng po w e r a t SBS s Fig. 3. Power consumption comparison of eachterm in P tot realizations of channels for a random user request profile [1 , , , , , , , , where the k -th element denotes theindex of the file requested by user k . Note that we set P =200 W for “No C&SC” to make it feasible. The compara-tive results show that considering caching and SBS clusteringgreatly reduces the power consumption. Furthermore, the sec-ond subfigure in Fig.3 demonstrates that backhual traffic canbe greatly reduced by considering caching and SBS clustering.VI. C ONCLUSION
In this correspondence paper, we proposed a low-complexityalgorithm to jointly optimize SBS clustering, multicast beam-forming for ALs and BLs, as well as frequency allocation inbackhaul transmission for WB based UD C-RAN. Simulationresults show that the proposed algorithm converges fast andgreatly alleviates the backhaul traffic.A
PPENDIX
A. Proof of Theorem 1
Theorem 1 can be proved by using contradiction. Specifi-cally, we assume that one of the following two cases occurs:1) The minimum SINR of each multicast group is strictlylarger than its threshold, i.e., min k ∈U l ξ A k ( W ∗ ) >γ l , ∀ l ∈ L ;2) The minimum SINR of several multicast groups arestrictly larger than their thresholds, while those of therest are equal to their thresholds.First, we prove that the first case cannot happen. Supposingit happens, we can find another solution c W = √ a W ∗ , where a = max k ∈U l ,l ∈L σ k / (cid:16)(cid:12)(cid:12) h H k w ∗ l (cid:12)(cid:12) /γ l − P Lj = l (cid:12)(cid:12) h H k w ∗ j (cid:12)(cid:12) (cid:17) . Be-cause min k ∈U l ξ A k ( W ∗ ) > γ l , ∀ l ∈ L , we have a < .Hence, we have ξ A k ( W ∗ ) > ξ A k (cid:16) c W (cid:17) ≥ γ l , ∀ k ∈ U l , ∀ l ∈ L .Furthermore, it can be easily verified that c W must be a feasiblesolution to P because a < . For the same reason, c W yields a lower objective value compared to that of W ∗ , whichcontradicts the assumption that W ∗ is the optimal solution.Thus the first case cannot happen.For the second case, we divide L multicast groups intotwo categories, i.e., L = (cid:8) l | min k ∈U l ξ A k ( W ∗ ) > γ l , ∀ l ∈ L (cid:9) and L = (cid:8) l | min k ∈U l ξ A k ( W ∗ ) = γ l , ∀ l ∈ L (cid:9) . Similar to the first case, we can also find another constant a satisfy a =max k ∈U l ,l ∈L σ k / (cid:16)(cid:12)(cid:12) h H k w ∗ l (cid:12)(cid:12) /γ l − P Lj = l (cid:12)(cid:12) h H k w ∗ j (cid:12)(cid:12) (cid:17) . Defining c W = [ √ a w ∗ l , ∀ l ∈ L , w ∗ l , ∀ l ∈ L ] , we have ξ A k ( W ∗ ) >ξ A k (cid:16) c W (cid:17) ≥ γ l , ∀ k ∈ U l , ∀ l ∈ L and ξ A k (cid:16) c W (cid:17) > ξ A k ( W ∗ ) ≥ γ l , ∀ k ∈ U l , ∀ l ∈ L . Because the minimum SINR of group l ∈ L increase, constraints (4c) may not hold. To resolvethis problem, we iteratively update group sets L and L .In each iteration, the new constructed c W yields a lowerobjective value. Because the objective value is lower bounded,the convergence of this procedure is guaranteed. When theprocedure converges, the minimum SINRs of all multicastgroups are equal to their thresholds, and the final c W isa feasible solution. However, the final c W yields a lowerobjective value than that with W ∗ , which contradicts theassumption that W ∗ is the optimal solution. Hence, the secondcase cannot happen either. The proof is completed.R EFERENCES[1] X. Ge, S. Tu, G. Mao, C. Wang, and T. Han, “5g ultra-dense cellularnetworks,”
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