1 On the Fronthaul Statistical Multiplexing Gain Liumeng Wang and Sheng Zhou, Member, IEEE Abstract—Breakingthefronthaulcapacitylimitationsisvitalto makecloudradioaccessnetwork(C-RAN)scalableandpractical. BBU One promising way is aggregating several remote radio units (RRUs) as a cluster to share a fronthaul link, so as to enjoy the Fronthaul Segment II statisticalmultiplexinggainbroughtbythespatialrandomnessof Aggregator thetraffic.Inthisletter,atractablemodelisproposedtoanalyze Fronthaul Segment I 7 thefronthaulstatisticalmultiplexinggain.Wefirstderivetheuser 1 blocking probability caused by the limited fronthaul capacity, RRU 0 including its upper and lower bounds. We then obtain the 2 limits of fronthaul statistical multiplexing gain when the cluster size approaches infinity. Analytical results reveal that the user n blocking probability decreases exponentially with the average UE a fronthaul capacity per RRU, and the exponent is proportional J to the cluster size. Numerical results further show considerable 8 fronthaulstatisticalmultiplexinggainevenatasmalltomedium 2 cluster size. Fig.1. C-RANarchitecturewithfronthaullinksoftwo-segmentswithcluster ] Index Terms—Cloud-Radio Access Network (C-RAN), fron- sizeK=3. T thaul, statistical multiplexing, stochastic geometry. I . s for next generation fronthaul interface (NGFI) 1. Moreover, c I. INTRODUCTION with appropriate RRU-BBU functional split, data rates in [ Recent evolution of radio access network (RAN) features the fronthaul links can be traffic-dependent [5]. As a result, 1 thebasebandprocessingcentralization,whichenablesefficient the randomness of user traffic can be exploited to enjoy the v cooperative signal processing and can potentially reduce the statistical multiplexinggain, in the hope of reducingthe fron- 6 operation and deployment costs. In a typical cloud radio thaul capacity demand. Some initial numerical studies have 6 2 access network (C-RAN) [1], baseband processing functions validatedthefronthaulstatisticalmultiplexinggainbroughtby 8 arecentralizedinbasebandunits(BBUs)whileradiofunctions the packeterized fronthaul and functional split [8]. Teletraffic 0 are integratedin remote radio units(RRUs). BBUs and RRUs theory and event-driven simulations are adopted in [9] to . 1 are connected with fronthaul network, over which baseband analyze the fronthaul statistical multiplexing gain in C-RAN. 0 signals are transported. Despite many advantages, one major Inthispaper,weproposeatractablemodeltoquantitatively 7 design challenge of C-RAN is to meet its high fronthaul analyze the fronthaul statistical multiplexing gain. We derive 1 capacity demand. One typical interface between RRUs and the probability of user blocking due to the limited fronthaul : v BBUs is Common Public Radio Interface (CPRI) [2], which capacity, and obtain its upper and lower bounds. We then use i X can only support fixed-rate delivery of raw baseband signals. large-limit analysis to get the closed-form expression of the Forexample,1Gbpsfronthaulrateisrequiredevenbyasingle fronthaul statistical multiplexing gain, which enables quanti- r a 20MHz LTE antenna-carrier. fyingthe statistical multiplexinggain underlarge cluster size. To ease the severe burden on the fronthaul,many solutions Numericalresultsfurtherconfirmthatevenasmalltomedium have been proposed. On the link level, one direct way is to cluster size can obtain notable fronthaul rate reduction. increase the fronthaul capacity, such as using single fiber bi- direction, wavelength-division multiplexing, and etc [3]. The II. SYSTEMMODEL other is to reduce the required data rate on the fronthaul, Consider a fronthaul network with two segments shown by means of baseband signal compression [4], RRU-BBU in Fig.1. Fronthaul segment I represents the direct links functionality splitting [5], radio resource allocation [6], and between RRUs and the aggregators,treated as the ”last mile” etc. On the network level, packet switching can provide of the fronthaul network. It can be implemented by CPRI hierarchical and flexible fronthaul networking [7], allowing with acceptable costs due to the short transmission distances. multiple RRUs to form a cluster and share a fronthaul link, SegmentII is the connection between the aggregatorsand the which is adopted by the recent IEEE 1914 working group BBU,andeachlinkissharedbyK RRUsformedasacluster. Our analysis is focusing on the fronthaul links in segment II. ThisworkissponsoredinpartbytheNationalScienceFoundationofChina (NSFC)undergrantNo.61571265,No.61461136004,andNo.91638204,and The capacity of a fronthaul link in segment II is denoted IntelCollaborativeResearchInstituteforMobileNetworkingandComputing. by T, shared by an RRU cluster. If the RRU-BBU function is Liumeng Wang and Sheng Zhou are with Tsinghua National splitbetween layer1 andlayer 2 [5], or with properfronthaul Laboratory for Information Science and Technology, Dept. of Electronic Engineering, Tsinghua University, Beijing 100084, China (Email: [email protected]; [email protected]). 1http://sites.ieee.org/sagroups-1914/ 2 compression schemes [4], the required fronthaulrate depends TheexpressionofPK,T isthesumofaninfiniteseries,andit b onthenumberofusersinthecorrespondingRRUs.Weassume ishardtohaveintuitivederivations.Nevertheless,theelements that each user requires the same fronthaul rate, and unify in the summation of (4) have the following property: the fronthaul capacity T as the maximum number of users P {N =n+1}(n+1−T)/(n+1) whose baseband signals can be transported in the fronthaul K P {N =n}(n−T)/n link simultaneously. K The spatial distribution of RRUs is modeled by a homoge- (λu/λr)n(n+Ka)(n+1−T) = neous Poisson point process (PPP) with density λr, and the (λu/λr+b)(n+1)2(n−T) spatial distribution of users is also assumed a homogeneous n+1−T = aKλ, (5) PPPwithdensityλ [10].Tosimplifytheanalysis,weassume n−T n u thatthereisnocoverageoverlapbetweenRRUs,i.e.,eachuser where λ=(λ /λ)/(λ /λ +b), aK =n(n+Ka)/(n+1)2. is severed by its nearest RRU, and as a result the coverage u r u r n As n > T, according to the summation in (4), when T > area of each RRU is a Voronoi cell. We assume that each 2/(Ka−2), RRU’s location in the cluster is independent to each other [9], and thus the coverage sizes of different RRUs are also 1<aK ≤(T +1)(T +Ka+1)/(T +2)2, (6) n independent. Denote the coverage size of the ith RRU as x , i and the distribution of xi can be approximated by a gamma and only when n=T +1, aKn = (T+1()T(T++2)K2a+1). In fact, as distribution, i.e., x ∼Γ(a,1/(bλ)), K ≥1 and T ≥1, T >2/(Ka−2) will always hold unless i r Γ(a,1/(bλr))=(bλr)a(xi)a−1e(−bλrxi)/Γ(a), aKna=lys1isa.nWdeTca=n1th,uwshgiceht aisloawterirvbiaolusncdenoafriPoKig,Tnoarsed in our b wherea=3.5,b=3.5[11],andΓ(·)istheGammafunction. ∞ Aarsetthheesscaamlee,pdaernaomteettehrestboλ1tralocfovalelrathgeeKsizegaomfKmaRdRisUtrsibaustixon=s PbK,T > PK{NT=+T1 +1} (n−T)λn−T−1 n=XT+1 K x , x∼Γ Ka, 1 [12]. The probability distribution i=1 i (cid:16) bλr(cid:17) = PK{N =T +1} ,PK,T. (7) Pof x given K is then (T +1)(1−λ)2 b,LB (bλ)Ka f (x)= r (x)Ka−1e(−bλrx). (1) Similarly, we find an upper bound of PK,T as x|K Γ(Ka) b Giventhe RRU cluster coveragesize x, the totalnumberof P {N =T +1} users in the RRU cluster , denoted by N, obeys the Poisson PK,T < K b T +1 distribution with mean λ x, i.e., u ∞ n−T−1 PK{N =n|x}= (λux)ne−λux. (2) × (n−T)(cid:18)(T +1)((TT++2K)2a+1)λ(cid:19) n! n=XT+1 According to (1) and (2), the distribution of N is ∞ n−1 P {N =T +1} (T +1)(T +Ka+1)λ K ∞ = n T +1 (cid:18) (T +2)2 (cid:19) PK{N =n}= PK{N =n|x}fx|K(x)dx nX=1 Z 0 P {N =T +1} = ∞ (λux)neλux(bλr)Ka(x)Ka−1e(−bλrx)dx =(T +1) K1− (T+1)(T+Ka+1)λ 2 ,PbK,U,BT. (8) Z n! Γ(Ka) (T+2)2 0 (cid:16) (cid:17) = bKa(λu/λr)nΓ(n+Ka) . (3) Furthermore, when T approaches infinity, (λ /λ +b)Ka+nΓ(Ka)n! u r 2 III. USERBLOCKING PROBABILITY lim PbK,L,BT = lim (cid:16)1− (T+1)((TT++2K)2a+1)λ(cid:17) =1, (9) Due to the fronthaulcapacity limitation, the services of the T→+∞PK,T T→+∞ (1−λ)2 b,UB users may be blocked.If the total numberof users per cluster whichmeansthatwhenT approachesinfinity,theupperbound N is larger than the fronthaul capacity T, only the baseband tends to be equal to the lower bound. As a result, the user signals of T (randomly selected) users will be transmitted in blockingprobabilityPK,T canbewellapproximatedbyeither thefronthaul,whileservicesoftheremainingN−T usersare b the upper bound or the lower bound when T is large. blocked. The quality of service (QoS) requires the blocking probability be smaller than a threshold Pth. Given the RRU b cluster size K and the shared fronthaul capacity T, the user IV. MULTIPLEXINGGAIN ANALYSIS blocking probability is expressed as Denoted by T (Pth) the minimum fronthaul capacity to K b ∞ satisfyagivenuserblockingprobabilityrequirementPthwhen PbK,T = PK{N =n}(n−T)/n K RRUsaresharingthefronthaullink,thefronthaulstbatistical n=XT+1 multiplexing gain G (Pth) is accordingly defined as K b ∞ bKa(n−T)(λ /λ)nΓ(n+Ka) = u r . (4) T (Pth)−T (Pth)/K T (Pth) n=XT+1 n(λu/λr+b)Ka+nΓ(Ka)n! GK(Pbth)= 1 b T (PKth) b =1− KKT (Pbth), 1 b 1 b 3 which represents the relative saving of the equivalent fron- Thus when K approachesinfinity, lim PK,T =1−T, K→+∞ b µ thaul capacity per RRU compared with the scenario with no and the minimum required equivalent fronthaul capacity is fronthaul sharing, i.e., K =1. T (Pth) The closed-form expression of G (Pth) with general K is lim K b =(1−Pth)µ. (17) K b K→+∞ K b hard to derive. Nevertheless, we can still get the closed-form expression with large-K limit, as follows. According to (10) and (17), we get T (Pth) (1−Pth)µ Proposition 1. When the cluster size K approaches infinity, G (Pth)= lim 1− K b =1− b . (18) (1−Pth)µ ∞ b K→+∞ KT1(Pbth) T1(Pbth) G (Pth)=1− b , (10) ∞ b T (Pth) 1 b Thispropositionrevealstheultimatemultiplexinggainwith where µ = λ /λ is the average number of users in the RRU fronthaul sharing, which is obtained when the RRU u r coverage of an RRU. cluster size K approaches infinity, and via numerical results we will show that the majority of the gain can be achieved Proof: Given the RRU cluster size K, we have even with small to medium RRU cluster sizes. Furthermore, E(N|K) = Kµ as the coverage sizes of different RRUs are we characterizehow the RRU cluster size K can expeditethe independent.Accordingto the law of largenumbers,when K decreasingrateoftheuserblockingprobabilityPK,T w.r.t.the approaches infinity, for any positive real ǫ, we have b equivalent fronthaul capacity T in the following proposition. lim PK(|N/K−µ|>ǫ)=0. (11) Proposition 2. When T approaches infinity, PK,T decreases K→∞ b exponentially with the equivalent fronthaul capacity T, and We define the equivalent fronthaul capacity allocated to each the exponent is proportional to K. RRU as T =T/K. If T >µ, according to (11), ∞ Proof: For fronthaul capacity T +1, the user blocking n−KT lim PK,T = lim P (N =n) probability can be expressed as K→+∞ b K→+∞n=XKT+1 n K ∞ bKa(n−T −1)(λ /λ)nΓ(n+Ka) ∞ PK,T+1 = u r ≤ lim PK(N =n) b n=XT+2 n(λu/λr+b)Ka+nΓ(Ka)n! K→+∞n=XKT+1 ∞ bKa(n−T)(λ /λ)n+1Γ(n+1+Ka) u r =K→lim+∞PK(NK −µ>T −µ)=0. (12) =n=XT+1(n+1)(λu/λr+b)Ka+n+1Γ(Ka)(n+1)! ∞ bKa(n−T)(λ /λ)nΓ(n+Ka) If T <µ, for any positive real ǫ<(µ−T), = aKλ u r . ∞ n=XT+1 n n(λu/λr+b)Ka+nΓ(Ka)n! lim PK,T = lim n−KTP (N =n) (19) K→+∞ b K→+∞ X n K When T >2/(Ka−2), according to (6), we have n=KT+1 n=K(µ+ǫ)n−KT PK,T+1 (T +1)(T +Ka+1) = lim P (N =n) λ< b < λ. (20) K→+∞n=KX(µ−ǫ) n K PbK,T (T +2)2 K(µ−ǫ)−KT Taking the limit of (20) with T approaching infinity, leads to ≥ lim P (|N/K−µ|≤ǫ) K K(µ−ǫ) K→+∞ PK,T+1 lim b =λ, =1− T . (13) T→+∞ PbK,T µ−ǫ lim log(PK,T+1)−log(PK,T)=log(λ). (21) b b Similarly, we get T→+∞ Note that λ=(λ /λ)/(λ /λ +b). According to (21), n=K(µ+ǫ) u r u r n−KT T K→lim+∞n=KX(µ−ǫ) n PK(N =n)≤1− µ+ǫ. (14) T→lim+∞dlogd(PTbK,T) =Klog(λ). (22) According to (13) and (14), for any positive real δ, there always exists a positive real ǫ<min{ µ2δ ,µ−T}, s.t. T+µδ V. NUMERICAL RESULTS n=K(µ+ǫ) n−KT T The average number of users per RRU λ /λ is set to u r (cid:12)(cid:12)(cid:12)(cid:12)K→lim+∞n=KX(µ−ǫ) n PK(N =n)−(1− µ)(cid:12)(cid:12)(cid:12)(cid:12)<δ, 5totianllyou1r00n0umReRrUicsa,lasntuddeya.cFhoKr th=e1s,im3,u5l,a1ti0o,n2s0,,w50e,caonndsi1d0e0r (cid:12) (cid:12) (15) (cid:12) (cid:12) RRUsareselectedasacluster,respectively.Theuserblocking i.e., for any positive real δ, probabilityw.r.t. the equivalentfronthaulcapacity per RRU T T is presented in Fig. 2. With the same T, the larger the cluster lim PK,T −(1− ) <δ. (16) (cid:12)K→+∞ b µ (cid:12) sizeK,thesmallertheuserblockingprobabilitywillbe.With (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 4 100 12 Pth=0.1 K,TUser blocking probability Pb1111100000−−−−−54321 0.00.501 6KKKKKK======113355,,, SSSiiimmm...8 10 K increases thEquivalent fronthaul capacity per RRU T(P)/KKb110156789 Pbth increases PPPPPbbtbtbtbtbthhhhh=====00000.....10000,5511 S,, SSimiimm... 10−6 4 0 5 10 15 10 20 30 40 50 60 70 80 90 100 EquilaventfronthaulcapacityperRRUT Cluster size K Fig. 2. User blocking probability versus equivalent fronthaul capacity per Fig. 4. Required equivalent fronthaul capacity TK(Pbth)/K versus RRU RRUT underdifferent clustersizeK. clustersizeK.Dashed-lines denoteT∞(Pbth). 100 Blocking Probability derive the user blocking probability PK,T given the RRU 10−1 LUopwpeerr BBoouunndd clustersizeK andthesharedfronthaulcabpacityT.Wefurther Tangent Line K,TPb10−2 get the upper bound and the lower bound of PbK,T. We then User blocking probability 11110000−−−−6543 abfcfrrnlloouoannscltyttkehhziraaneuugslltiphzmcreeaou.pblstaNatiacpbutiliitemlsyixtt,ieyicnraaidgncle,dactmlhrteeuhraelerstseieupqeslluxteesipxrxoeipfnnduogeranntvhegtenearirtiasinagrlpeelayrvnfoerdwpoaonlifitrthhtntihadotahuntelathcltaaahvttproeoarucuatsighgteyheer perRRUcanbenotablyreduced,andevenasmalltomedium 10−7 clustersizecanobtainconsiderablefronthaulcapacitysavings. 10−8 5 10 15 EquilaventfronthaulcapacityperRRUT REFERENCES Fig.3. Theupperboundandlower boundoftheuserblocking probability [1] China Mobile, “C-RAN: the road towards green RAN,” White Paper, whenK=5. Version3.0,Dec2013. [2] CPRI Specification V6.0, “Common Public Radio Interface (CPRI),” Interface Specification, 2013. larger K, the user blocking probability also decreases more [3] I.Chih-Lin,J.Huang,R.Duan,C.Cui,J.X.Jiang,andL.Li,“Recent progress on C-RAN centralization and cloudification,” IEEE Access, rapidly. 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CONCLUSIONS [12] P.G.Moschopoulos,“Thedistributionofthesumofindependentgamma Inthispaper,wehaveproposedatractablemodeltoanalyze random variables,” Annals of the Institute of Statistical Mathematics, vol.37,no.1,pp.541–544, 1985. the fronthaul statistical multiplexing gain in C-RAN. We