Statistical Multiplexing Gain Analysis of Heterogeneous Virtual Base Station Pools in Cloud Radio Access Networks
aa r X i v : . [ c s . I T ] M a y Statistical Multiplexing Gain Analysis ofHeterogeneous Virtual Base Station Pools in CloudRadio Access Networks
Jingchu Liu,
Student Member, IEEE , Sheng Zhou,
Member, IEEE ,Jie Gong,
Member, IEEE , Zhisheng Niu,
Fellow, IEEE ,and Shugong Xu,
Fellow, IEEE
Abstract —Cloud radio access network (C-RAN) is pro-posed recently to reduce network cost, enable cooperativecommunications, and increase system flexibility throughcentralized baseband processing. By pooling multiple vir-tual base stations (VBSs) and consolidating their stochasticcomputational tasks, the overall computational resourcecan be reduced, achieving the so-called statistical mul-tiplexing gain. In this paper, we evaluate the statisticalmultiplexing gain of VBS pools using a multi-dimensionalMarkov model, which captures the session-level dynamicsand the constraints imposed by both radio and computa-tional resources. Based on this model, we derive a recursiveformula for the blocking probability and also a closed-formapproximation for it in large pools. These formulas arethen used to derive the session-level statistical multiplexinggain of both real-time and delay-tolerant traffic. Numericalresults show that VBS pools can achieve more than of the maximum pooling gain with VBSs, but furtherconvergence to the upper bound (large-pool limit) is slowbecause of the quickly diminishing marginal pooling gain,which is inversely proportional to a factor between the one-half and three-fourth power of the pool size. We also findthat the pooling gain is more evident under light trafficload and stringent Quality of Service (QoS) requirement.
Index Terms —C-RAN, VBS pooling, statistical multi-
J. Liu, S. Zhou, and Z. Niu are with Department of Electronic En-gineering and Tsinghua National Laboratory for Information Scienceand Technology, Tsinghua University, Beijing 100084, China. Email:[email protected], [email protected], [email protected]. Gong was with Department of Electronic Engineering and Ts-inghua National Laboratory for Information Science and Technology,Tsinghua University, Beijing 100084, China. He is now with Schoolof Data and Computer Science, Sun Yat-sen University, Guangzhou510006, Guangdong, China. Email: [email protected]. Xu is with Intel Cooperation. Email: [email protected] of this work has been presented at IEEE Globecom 2014. Thiswork is sponsored in part by the National Basic Research Program ofChina (973 Program: 2012CB316001), the National Science Founda-tion of China (NSFC) under grant No. 61201191, No. 61321061, No.61401250, and No. 61461136004, and Intel Collaborative ResearchInstitute for Mobile Networking and Computing. plexing.
I. I
NTRODUCTION I N recent years, the proliferation of mobile de-vices such as smart phones and tablets, togetherwith the diverse applications enabled by mobileInternet, has triggered the exponential growth ofmobile data traffic [1]. To accommodate the rapidtraffic growth, cellular networks have been con-tinuously evolving toward smaller cell size, widerbandwidth, and more advanced transmission tech-nologies. However, the problems that arise, suchas the increased interference and operational cost,are difficult to be solved via the traditional radioaccess network (RAN) architecture, in which theprocessing functionalities are packed into stand-alone base stations (BSs) and the cooperation be-tween BSs is restricted by the limited inter-BSbackhaul bandwidth.To overcome the shortcomings of the traditionalRAN architecture, cloud RAN (C-RAN) [2] isproposed with centralized baseband processing. C-RAN can facilitate the adoption of cooperative sig-nal processing and potentially reduce the operationalcost. A similar idea is also proposed under with thename wireless network cloud (WNC) [3]. This kindof novel architecture has attracted substantial atten-tions recently: the key building blocks of C-RANare investigated and its major use cases are identified[4]–[7]. Centralized processing is combined withdynamical fronthaul switching to address the mo-bility and energy efficiency issues of small cells in[8], [9]. Concerning realization-related issues, it isdemonstrated in [10]–[14] that BBU functionalitiescan be implemented as software, i.e. virtual base station (VBS), which runs on general purpose plat-forms (GPP). Compared with specialized-platform-based implementations, GPP-based implementationis more flexible in terms of the implementation ofnew functionalities and the management of com-putational resource. Further more, a VBS pool canbe constructed by consolidating multiple VBSs toshare the same set of computational resource. Inthis way, the computational resource can be utilizedmore efficiently and related cost can be reduced.Despite the evident advantages of C-RAN, themassive bandwidth requirement of its fronthaul net-work poses a serious challenge: transmitting thebaseband sample of a single MHz LTE antenna-carrier (AxC) requires around 1Gbps link bandwidth[15], [16]. Large-scale centralization will thus in-cur enormous fronthaul expenditure and potentiallycancel out the gains. Fortunately, it is observed in[11], [17] that substantial statistical multiplexinggain can be obtained even with small-scale central-ization, justifying the deployment of small clustersof C-RAN. Yet, these results are obtained fromsimulations that are based on short-term small-scaletraffic logs, and a generalized analytical model is inneed to derive the optimal VBS cluster size. To thisend, a session-level VBS pool model is proposed in[18] under the assumption of unconstrained radioresource and dynamic resource management. Hereuser sessions represent the time period in whichusers occupy computational resource in the pool.Nevertheless, this model does not reflect two realis-tic factors. Firstly, radio resource are often the mainperformance bottleneck for real networks, and thusthe influence of radio resource should be reflectedin the VBS pool model. Secondly, dynamic resourcemanagement, which re-assigns resources at the ar-rival and departure of each user session, may incurtoo much control overhead and overload the system[11]. Hence, semi-dynamic resource management,which assigns resources on much larger time scales(e.g. hours to days) than the arrival and departureof user sessions (e.g. seconds to minutes), is morerealistic. This assumption is also reasonable becausethe traffic statistics also vary in similarly large timescales, therefore the management plan designed forsome traffic statistics can be useful for a fairly longperiod, and only need to be occasionally adjustedin the long run.In our previous work [19], we analyze the statis-tical multiplexing gain of homogeneous VBS pools, in which each VBS has the same traffic arrival,resource configuration, and service strategy. Wederive a product-form expression for the stationarydistribution of user sessions in each VBS and givea recursive method to compute the session blockingprobability of the VBS pool. In this paper, weextend these results to heterogeneous VBS pools, inwhich there are multiple classes of VBSs with dif-ferent session arrival, resource configurations, andservice strategies. The computational complexity ofthe recursive method is also analyzed. Under theassumption of large VBS pools, we also derive aclosed-form approximation for the blocking proba-bility. We show through simulation that the approxi-mation is precise even for medium-sized pools witharound VBSs. We then use this approximation toquantitatively investigate the statistical multiplexinggain of VBS pools under the influence of differentdifferent factors, including pool sizes, VBS hetero-geneity, traffic load, and the desired levels of QoS.The main contributions of this paper are as fol-lows: • We propose a realistic session-level modelfor heterogeneous VBS pools with both ra-dio and computational resource constraints andsemi-dynamic resource management. We showthat this model constitutes a continuous-timemulti-dimensional Markov chain and derive itsproduct-form stationary distribution. We alsoillustrate how this model can be used to analyzereal-time and delay-tolerant traffics. • We give a recursive method to compute theblocking probability for the proposed model.This method has quadratic computational com-plexity, much lower than brute-force evaluationwhich has exponential complexity. For largeVBS pools, we also derive a closed-form for-mula to approximate the blocking probability. • We provide an in-depth analysis on the statisti-cal multiplexing gain of VBS pools. We shownumerically the influence of various factorsincluding the pool size, traffic load, and QoSrequirements. We also prove that the statisti-cal multiplexing gain increases slowly as thepool size grows large, with the residual gaindiminishing at a speed between | M | − / and | M | − / . Here | M | denotes the pool size.The rest of the paper is organized as follows. Sec-tion II introduces the proposed model and presents the product-form stationary distribution of user ses-sions. Section III derives the recursive formula forthe blocking probability and gives a closed-formapproximation for large VBS pools. In Section IV,we derive the expression of statistical multiplexinggains and apply it to both real-time and delay-tolerant traffics in Section V. Section VI presentsthe numerical results and discusses the implicationson realistic system design. Finally the paper isconcluded in section VII.II. M ODEL F ORMULATION
In this section, we introduce the Markov modelfor VBS pools and derive its stationary distribution.The model captures the session-level dynamics in aVBS pool. To endow our model with enough gener-ality, we assume V different classes of VBSs. Thetotal number of VBSs in class v ( v = 1 , , · · · , V ) is M v . Each VBS in class v is assigned with K v unitsof radio resource. To perform baseband signal pro-cessing on user sessions, all VBSs share a total of N units of computational resource. The overall settingis illustrated in Fig. 1. We assume homogeneousresource demands: every active session is assumedto occupy one unit of radio resource and one unitof computational resource. Note that computationalworkloads that are independent of user dynamicsdo exist in cellular systems. However, the dominantconsumers for computational resources are mostlyper-user functions and thus the overall workloadroughly follows a linear relationship with the num-ber of users [11]. For simplicity, hereafter we denoteradio and computational resource by r-servers andc-servers, respectively. A. Session Arrival, Service Discipline, and Admis-sion Control
User sessions are initiated following independentPoisson processes in the coverage area of theirserving VBSs. Obviously, the overall session arrivalrate in a VBS is proportional to its coverage area.We allow VBSs in different classes to have differentsizes of coverage areas, and consequently, differentaggregated session arrival rates. We denote the ar-rival rate for class- v VBSs by λ v .Each user session demands an exponentially dis-tributed amount of service capacity before it leaves.Note the notion of service capacity can be flexiblyinterpreted according to the specific scenario this K K K K v K v K V K V K V K V ... ... Fronthaul linksFronthaul linksFronthaul links
Fig. 1. An example of heterogeneous VBS pool. There are V classesof VBSs, each class may have different number of VBSs and amountof radio resource. These VBSs are consolidated in a data center andshares N units of computational resource. model is applied to. For example, service capacitycan be considered as time duration for voice callsessions or the amount of information bits forcellular data sessions. For statistical QoS scenariossuch as soft-real-time video, service capacity canstill be interpreted as duration or information bits.The physical resources that enables such capacityis defined as the minimum amount of resourcethat can constantly satisfy a session. This meansif the instantaneous requirement of a session islower than provided, the remaining resources willbecome under-utilized. We assume that a VBS poolscheduler manages the service capacity so that theservice capacity assigned to a class- v VBS is afunction of the total number of sessions in this VBS,and the assigned capacity is equally divided amongthese sessions for fairness Denote the number ofsessions in the m -th VBS of class- v at time t as U v,m ( t ) . Then the above service strategy canbe translated into a session departure rate function f v ( U v,m ( t )) for sessions in the m -th VBS of class- v at time t . The above Poisson assumptions havebeen widely used in existing literature to evaluatethe impact of randomness on system performance[20], [21].To guarantee that active sessions always haveenough r-servers and c-servers, the VBS pool has toenforce admission control on the arriving sessions:whenever a new session arrives, an admission con-trol agent in the VBS pool will decide whether ornot to accept this session according to the currentresource usage. For class- v sessions, the new sessionis accepted only if the number of r-servers in itsserving VBS is less than K v and the number of c- servers in the pool is less than N ; otherwise thesession is blocked. B. State Transitions
Recall that we denote the number of sessions inthe m -th VBS of class- v by U v,m ( t ) , we can furtherdescribe the session dynamics in the VBS pool witha continuous-time stochastic process U ( t ) = ( U , ( t ) , · · · , U ,M , · · · , U V, , · · · , U V,M V ( t )) T . Given the Markovian property of the arrival andservice of processes, it is obvious that U ( t ) is aMarkov chain. Taking the admission control policyinto consideration, we can get the set of possiblesystem states U ( t ) ∈ U = { u | ≤ u v,m ≤ K v , ≤ V X v =1 M v X m =1 u v,m ≤ N,u v,m ∈ N } , (1)where u = ( u , , · · · , u ,M , · · · , u V, , · · · , u V,M V ) T is the state vector. Because the session arrivalsand departures are Markovian, U ( t ) is a multi-dimensional birth-and-death process. The transitionrate of U ( t ) from an arbitrary state u ( i ) to anotherstate u ( j ) is: q u ( i ) u ( j ) = λ v , if u ( j ) − u ( i ) = e v,m f v ( u ( i ) v,m ) , if u ( j ) − u ( i ) = − e v,m , otherwise , (2)where u ( i ) v,m is the ( v − P w =1 M w + m ) -th entry of u ( i ) ,and e v,m = (0 , · · · , , |{z} ( v − P w =1 M w + m ) -th , , · · · , T is a column vector of length V P v =1 M v . For the easeof understanding, we illustrate the state transitiongraph of a simple example with V = 1 , M = 2 , K = 3 , N = 4 and f ( n ) = nµ in Fig. 2.A similar problem has been formulated in thecontext of Stochastic Knapsack Problem [22], whichis a stochastic extension of the traditional knapsackproblem. Specifically, the items which occupy acertain amount of space come into and leave a λμ λ λ μ μλ λ λμ μ μλ λμ μλμλλλ λλλ λλ λμ μ μ μ μ μ μ μ μ (0,0) k k Fig. 2. Transition graph of a VBS pool with two VBSs. The k and k axes indicate the number of active sessions in these twoVBSs, respectively. Each yellow point represents a possible poolstate, and the states in the gray region are prohibited because ofthe computational and radio resource constraints. ( K = 3 , N = 4 ,and f ( n ) = nµ ) knapsack randomly. The model we formulate ismathematically equivalent to stochastic knapsacksunder coordinate convex [23] admission controlpolicies. However, the focus of these previous workwas to find the policy that maximizes the rewardof storing items, and the analysis was limited toproblems with small dimensionality because thecomplexity increases dramatically as the number ofitem classes grows. In contrast, we aim at evaluatingthe blocking probability instead of reward, and wehave to address the large-dimensionality problemsdue to the large sizes of VBS pools. C. Stationary Distribution
To perform further analysis, we need to derivethe stationary distribution of U ( t ) . Fortunately, themodel we formulated guarantees the reversibilityof U ( t ) as in the following theorem, which inturn results in a product-form expression for thestationary distribution. Theorem 1 (Reversibility) . A continuous-timeMarkov chain with a state set as in (1) and transi-tion rates as in (2) is reversible.
The reversibility of U ( t ) has been proved in [24]for more general cases. We provide an alternativeproof in the Appendix using Kolmogorov’s Criterionof Reversibility. Since U ( t ) is reversible, the local balance equation holds in the statistical equilibrium Pr (cid:8) U ( ∞ ) = u ( i ) (cid:9) q u ( i ) u ( j ) = Pr (cid:8) U ( ∞ ) = u ( j ) (cid:9) q u ( j ) u ( i ) . (3)For simplicity, Pr { U ( ∞ ) = u } ishereafter denoted by Pr { u } or Pr { u , , · · · , u ,M , · · · , u V, , · · · , u V,M V } . Withoutloss of generality, let u ( i ) and u ( j ) be two arbitraryneighboring states: u ( i ) = ( u , · · · , u v,m , · · · , u V,M V ) T u ( j ) = ( u , · · · , u v,m + 1 , · · · , u V,M V ) T , and substituting (2) into (3) yields: Pr { u , · · · , u v,m , · · · , u V,M V } λ v = Pr { u , · · · , u v,m + 1 , · · · , u V,M V } f v ( u v,m + 1) . (4)Clearly, this implies a recursive equation for com-puting the stationary distribution. Continuing therecursion down to for the ( v − P w =1 M w + m ) -th entry: Pr { u , · · · , u v,m + 1 , · · · , u V,M V } = Pr { u , · · · , , · · · , u V,M V } · λ u m +1 v Q u v,m +1 i =1 f v ( i ) . (5)Repeating the same process for other entries yields: Pr { u } = P V Y v =1 M v Y m =1 λ u v,m v Q u v,m i =1 f v ( i ) , (6)in which P = Pr { , · · · , , · · · , } = X u ∈ U V Y v =1 M v Y m =1 λ u v,m v Q u v,m i =1 f v ( i ) ! − (7)is the probability of zero state and can be derived di-rectly from the unity of probability distribution. Ascan be seen in (6) and (7), the stationary distributionof any state u is proportional to the product of termswhich can be solely determined by the entry valuesof u . Remark 1.
Although we formulate our problemwith the assumption of exponentially distributedservice demand, it is worthy of noting that the aboveproduct-form stationary distribution also applies toother non-exponential service demand distributions.It is proved in [24] that the product form distribu-tion is valid for any service time distributions withrational Laplace transforms.
III. B
LOCKING P ROBABILITY
A. Brute-force Evaluation
The admission control policy we have enforcedon the VBS pool will cause session blockings.These blocking events can be classified into twoclasses: radio blockings (denoted by B r ) andcomputational blockings (denoted by B c ). Ra-dio blocking is defined as the blocking events solely due to insufficient r-servers, i.e. U v,m ( t − ) = K, and P Vv =1 P M v m =1 U v,m ( t − ) < N, while compu-tational blocking is defined as the blocking eventsdue to insufficient c-servers regardless of r-servers ,i.e. P Vv =1 P M v m =1 U v,m ( t − ) = N. Here t − means theepoch just prior to a session arrival. Because wedefine radio blocking as the events that are solelydue to insufficient r-servers, the blocking events thatare due to simultaneously insufficient r-servers andc-servers are explicitly classified as computationalblocking. It is worth noting that these doubly block-ing events can be instead classified as radio blockingwithout affecting the overall blocking probability.But doing so will nevertheless result in less concisemathematical definition for both classes of events.Therefore, radio and computational blockings eventsare mutually exclusive, i.e. B r ∩ B c = ∅ . The unionset of radio and computational blocking is furtherdefined as the overall blocking B = B r ∪ B c .Next we derive the expression for the probabilityof radio and computational blockings. Since Poissonarrivals see time-averages (PASTA) [25], the block-ing probability can be evaluated from the stationarydistribution we have just derived. Concretely, theradio blocking probability for sessions in class- v VBSs is: P br v = M v X m =1 M v X u ∈ U v,m br Pr { u } (8) = P X u ∈ U v, V Y w =1 M w Y m =1 λ u w,m w Q u w,m i =1 f w ( i ) (9) = P λ K v v Q K v i =1 f v ( i ) · (10) X u ∈ U v, " ( Y w = v M w Y m =1 λ u w,m w Q u w,m i =1 f w ( i ) ) · ( M v Y m =2 λ u v,m v Q u v,m i =1 f v ( i ) ) , (11) where U v,m br = { u | u v,m = K v , u , + · · · + u ,M + · · · + u V, + · · · + u V,M V < N } and (9) holds because(6) is symmetric for entries with same values forindex v , i.e. Pr {· · · , u v,i , · · · , u v,j , · · · } = Pr {· · · , u v,j , · · · , u v,i , · · · } . (12)Similarly, the computational blocking probability is: P bc = X u ∈ U N bc Pr { u } = P X u ∈ U N bc V Y v =1 M v Y m =1 λ u v,m v Q u v,m i =1 f v ( i ) , (13)where U N bc = { u | u , + · · · + u ,M + · · · + u V, + · · · + u V,M V = N, u v,m ≤ K v } . The overall blockingprobability for class- v VBSs can then be brute-forceevaluated by summing up radio and computationalblocking probability P b v = Pr { B } = P br v + P bc . (14) B. Recursive Evaluation
Theoretically, the blocking probability under arbi-trary system parameter can be calculated with brute-force evaluation. However the calculation processis exponentially hard and can become intractablewhen the pool size is extremely large. To reduce thecomputational complexity, we next give a recursivemethod for calculating the blocking probability. Wewill first introduce two auxiliary functions and re-express the blocking probability with respect tothese functions. Then we will establish a recursiverelationship to evaluate those two auxiliary func-tions and provide an analysis on the computationalcomplexity of the proposed recursive evaluationmethod. These two auxiliary functions are definedas follows: C ( N, M ) = X u ∈ U N bc V Y w =1 M w Y m =1 λ u w,m w Q u w,m i =1 f w ( i ) , (15) R ( N, M ) = X u ∈ ( U N bc ) C V Y w =1 M w Y m =1 λ u w,m w Q u w,m i =1 f w ( i ) , (16)where M = ( M , · · · , M v , · · · , M V ) T and (cid:0) U N bc (cid:1) C = { u | u , + · · · + u ,M + · · · + u V, + · · · + u V,M V < N, u v,m ≤ K v } is the complement set of set U N bc in set U . Clearly, C ( N, M ) and R ( N, M ) are proportional to the sum of probabilityterms over U N bc and (cid:0) U N bc (cid:1) C , respectively. Therefore,the blocking probability (11) and (13) can be re-expressed as, P br v = P · λ K v v Q K v i =1 f v ( i ) R ( N − K v , M − ˆ e v ) ,P bc = P · C ( N, M ) ,P = R − ( N + 1 , M ) , (17)where ˆ e v = (0 , · · · , , v -th z}|{ , , · · · , T is a columnvector of length V . From the definition of C ( N, M ) and R ( N, M ) , the following recursive relationshipsexist: C ( N, M ) = λ N v ) v Q N v ) i =1 f v ( i ) , M = ˆ e vN ( v ) X n = N ( v ) λ nv Q ni =1 f v ( i ) C ( N − n, M − ˆ e v ) , M > ˆ e v , (18) R ( N, M )= , N = 1 R ( N + 1 , M ) − C ( N, M ) , < N < M T K + 1 Q Vw =1 (cid:16)P K w n =1 λ nw Q ni =1 f w ( i ) (cid:17) M w , N = M T K + 1 , (19)where N ( v ) = max " , N − X w = v M w K w − ( M v − K v ,N ( v ) = min ( K v , N ) , and M T K = P Vw =1 M w K w . Following these re-cursive relationship, we can calculate the value C ( N, M ) and R ( N, M ) for arbitrary input throughiterative calculation. Concretely, we can iterate for C ( N, M ) from any ˆ e v following (18). After that,we can reuse the calculated C ( N, M ) values tocalculate for R ( N, M ) by iterating from either N = 0 or N = M T K + 1 according to (19). Notethat the comparison between M and ˆ e in (18) iselement-wise, and in this sense M is always greateror equal to ˆ e in non-empty pools. With the aboverecursive relationship, the computational complexityof blocking probability can be reduced to at most the second power of the pool size, as stated in thefollowing theorem. Theorem 2.
The upper bound for the overall com-putational complexity of the proposed recursivemethod is: C ≤ h (max v K v ) + max v K v i · | M | . (20) Proof:
See Appendix B for proof.
C. Large Pool Approximation
The above quadratic computational complexitycan also become intractable for very large pools.To overcome this inconvenience, next we present aclosed-form approximation for the blocking proba-bility with large VBS pools. Although the aboverecursive expression cannot lead us to a directapproximation, the product-form stationary distribu-tion of U does facilitate an indirect one.First define some auxiliary variables. Let ˜ U w,m be the number of sessions in the m -th class- w VBSwhen N ≥ M T K ,µ w = E h ˜ U w,m i ,σ w = Var h ˜ U w,m i . Also, let ˜ S M = | M | P Vw =1 P M w m =1 ˜ U w,m , and ˜ S M w = M w P M w m =1 ˜ U w,m . Using these notations, the large-pool approximation is stated in the following The-orem. Theorem 3 (Large Pool Blocking Probability) . For
N > | M | µ , the session blocking probability forclass- v VBSs is: lim | M |→∞ P b v = 1 p π | M | σ e α / − P br v , (21) where µ = P Vw =1 β w µ w , σ = P Vw =1 σ w , and β w =lim | M |→∞ M w | M | ; α = N −| M | µ √ | M | σ is the normalized numberof c-servers; ˜ P br v is the overall blocking probabilityin class- v VBSs when
N > M T K .Proof: See Appendix C for proof.
Remark 2.
With the approximation in (21), theblocking probability can be calculated in one shot It is obvious that ˜ U w,m are i.i.d random variables for m =1 , , · · · , M w . as long as the first-order and second-order statisticsof ˜ U v,m are known. We will see in Section V thatthese statistics are rather easy to obtain in somevery practical scenarios. Remark 3.
Under the large-pool assumption, theblocking probability in (21) is decomposed into twoterms. The first term √ π | M | σ e α / − reflects theportion of blockings that are solely due to insuf-ficient computational resource, while the secondterm ˜ P brv reflects the portion that are solely dueto insufficient radio resource. This result revealsthe decoupling feature between radio and compu-tational blockings in large VBS pools. Remark 4.
Although we assume K v < ∞ in ourderivation, (21) is still true when K → ∞ . In thiscase, the approximation used in (59) may not holdanymore. But this will not cause any problem sincethe radio blocking probability ˜ P b v will be whenwe have infinite radio resource. This will force thesecond term of (21) to become zero, canceling outthe inconsistency in the above approximation. IV. S
TATISTICAL M ULTIPLEXING G AIN
Since stochastic user sessions from differentVBSs are consolidated, it is natural to expect areduction in the required amount of computationalresource compared with non-pooling schemes dueto statistical multiplexing. Next we provide a theo-retical analysis for the statistical multiplexing gain.We first derive the asymptotic utilization ratio ofcomputational resource in large VBS pools.
Theorem 4 (Large Pool Utilization Ratio) . When c-servers are sufficiently provisioned (i.e. the numberof c-servers is greater or equal to the number ofr-servers, or N ≥ M T K ), the utilization ratio ofcomputational resource converges almost surely toa constant number that is smaller than 1 as | M | →∞ : lim | M |→∞ η , P Vw =1 P M w m =1 ˜ U w,m N a . s . −−→ | M | µN < . (22) Proof:
See Appendix D for proof.This theorem implies that there exist some ( − η )redundant computational resource when the VBSpool is large enough. Thus this limit can be seenas an the maximum portion of c-servers that onecan turn down to save computational resource. The potential to further turn down c-servers can inturn be defined as the difference between currentutilization ratio of c-servers and the large-pool limit η : Definition 1 (Residual Pooling Gain) . The residualpooling gain of a VBS pool is: g r , N M T K − η. (23)Although some c-servers can be turned down dueto the statistical multiplexing effect, the negativeeffect is that the overall blocking probability P b will increase following (21). Hence we have totrade QoS for the statistical multiplexing gain. Thistradeoff will be favorable as long as the degrada-tion in the QoS is not very significant. Using theresults in Theorem 3, we can directly derive thefollowing corollary to quantify such “significance”and approximate the gain of VBS pools of differentsizes. Corollary 1 (Critical Tradeoff Point) . When | M | →∞ , the minimum computational resource α ∗ re-quired to keep the overall blocking probability P b v ≤ ˜ P br v + δ ( δ ≈ ) for all v is α ∗ = s p π | M | σ δ + 1) . (24)We will show later in Fig. 3, that this criticaltradeoff point is essentially the point where theblocking probability start to increase at a signifi-cantly higher speed. This type of points are oftenreferred to “knee points”.The residual pooling gain at this critical tradeoffpoint is bounded as follows: g ∗ r = N − | M | µ M T K = σ α ∗ p | M | M T K ∈ α ∗ p | M | · (cid:20) σ max v K v , σ min v K v (cid:21) , (25)by which g ∗ r is roughly proportional to α ∗ / p | M | .Note α ∗ is also a function of the pool size | M | ,so g ∗ r is not necessarily proportional to / p | M | .Investigating two extreme cases will help to revealthe true relationship between g ∗ r and the pool size | M | . Extreme Case 1: If | M | is not very large suchthat p π | M | σ δ ≪ and p | M | ≪ /δ , then α ∗ is approximately constant because: α ∗ ≈ s p π | M | σ δ )= s ln( 12 πσ ) + ln( 1 δ ) + ln( 1 | M | ) ≈ r ln( 12 πσ δ ) . (26)In this case g ∗ r ∝ | M | − / , which decreases slowlywith | M | . Even so, considering the fact that theresidual pooling gain is at most 1, we can stillget considerable pooling gain with a small valueof | M | . Extreme Case 2: , if | M | is very large such that p π | M | σ δ ≫ , notice lim x → ln(1 + x ) ≈ x : α ∗ ≈ s p π | M | σ δ ∝ | M | − / . (27)In this case g ∗ r ∝ | M | − / , which means that thedecrease in the residual pooling gain will speed-upas | M | grows large. Remark 5.
As can be seen in (24), the critical pointis invariant of the VBS class index v . This impliesthat the VBS heterogeneity is decomposed in largeVBS pools. The reason for this phenomenon may bethat in large VBS pools, the absolute number of c-servers is large. Therefore, different class of VBSsmay tend to interfere less with each other. V. E
XAMPLE S CENARIOS
In this section, we will apply the results derivedso far to two specific scenarios: real-time and delay-tolerant traffic. For each scenario, we will firstexplain how they can be mapped to our model andthen we will perform necessary derivations. Note inreal-life systems, both type of traffic may exist at thesame time. In such a case, the following analysis canstill be applied if the available resources are dividedto serve these two type of traffic separately.
A. Real-time Traffic
For real-time traffic such as voice calls, activesessions will constantly bring in signal process-ing workload. Therefore, dedicated r-servers andc-servers need to be provisioned upon admission to guarantee the QoS of active sessions. The ser-vice capacity in this scenario equals the temporalduration of sessions, which is not affected by thescheduling policy of VBS pools once the sessionsare accepted. As a result, the departure rate functioncan be simplified as f v ( i ) = iµ . The QoS targetin this case is to keep the overall session blockingprobability for class- v VBSs under a certain smallthreshold P bth v ≈ . Obviously, the session dy-namics in different VBSs are mutually independentwhen computational resource are sufficiently provi-sioned ( N > M T K ). Therefore the radio blockingprobability ˜ P br v can be calculated as ˜ P br v = a K v v K v ! K v X i =0 a iv i ! ! − ≤ P bth v ≈ , (28)where a v = λ v /µ v . Then the first-order and second-order statistics of ˜ U v,m can be approximated asfollows: E h ˜ U v,m i = K v P i =0 i a iv i ! K v P i =0 a iv i ! = a K v − P i =0 a iv i ! K v P i =0 a iv i ! = a v − a K v v K v ! K v X i =0 a iv i ! ! − ≈ a v , (29) E h ˜ U v,m i = K v P i =0 i a iv i ! K v P i =0 a iv i ! = a v K v − P i =0 ( i + 1) a iv i ! K v P i =0 a iv i ! = a v ( K v − P i =0 a iv i ! + a v K v − P i =0 a iv i ! ) K v P i =0 a iv i ! ≈ a v + a v . (30)Then µ v ≈ a v , (31) σ v ≈ a v . (32) B. Delay-tolerant traffic
For delay-tolerant traffic such as packet data, thepool scheduler can opportunistically divide the totalservice capacity among active sessions. Here we assume constant service capacity rate f v ( i ) = µ v forclass- v VBSs and
Proportional Fairness schedulingalgorithm which effectively divides the total ser-vice capacity equally among active sessions. Al-though this assumption manifests a processor shar-ing model, it is essentially equivalent to a Marko-vian queueing model with the same λ v and µ v . Notebecause sessions would require certain amount ofradio resources for signaling, new sessions would berejected if there’re no more signaling radio channelsregardless of the data channels left. Therefore eventhe sessions can wait they still cannot be admittedinto the system. If the rejected session decides towait and retry, it will be considered as a newsession. Whats more, many delay-tolerant traffic orelastic traffic still have a minimum rate requirement,which also limits the number of sessions that canbe simultaneously served by the system.To derive the statistics in this scenario, first let a v = λ v /µ v be the traffic load of the VBSs anddefine the following auxiliary function A ( a, K ) : A ( a, K ) = K X i =0 a i = 1 − a K +1 − a ,A ′ a ( a, K ) = K X i =0 a i ! ′ a = K X i =1 ia i − = 1 − ( K + 1) a K + Ka K +1 (1 − a ) ,A ′′ a ( a, K ) = K X i =0 a i ! ′′ a = K X i =2 i ( i − a i − . (33)With these definitions, the average and covarianceof ˜ U v,m can be expressed as: E h ˜ U v,m i = K v P i =1 ia ivK v P i =0 a iv = a v A ′ a ( a v , K v ) A ( a v , K v ) , (34) E h ˜ U v,m i = K v P i =1 i a ivK v P i =0 a iv = K v P i =1 ia iv + K v P i =2 i ( i − a ivK v P i =0 a iv = a v A ′ a ( a v , K v ) + a v A ′′ a ( a v , K v ) A ( a v , K v ) . (35) Again when computational resource are sufficientlyprovisioned (
N > M T K ), we have ˜ P br v = a kvK v P i =0 a iv = a kv A ( a v , K v ) . (36)Although these formulas are already enough forus to evaluate the performance of a VBS pool,the evaluation is nevertheless quite cumbersome. Tosimplify these formulas, we further assume that K v for all v are large enough such that K v a K v v → . In this regime, A ( a, K ) ≈ − a ,A ′ a ( a, K ) ≈ − a ) ,A ′′ a ( a, K ) ≈ − a ) . (37)Using (37), (34) and (35) can be simplified as E h ˜ U v,m i ≈ a v − a v , (38) E h ˜ U v,m i ≈ a v − a v + 2 a v (1 − a v ) . (39)Thus µ v ≈ a v − a v , (40) σ v ≈ a v − a v + a v (1 − a v ) . (41)VI. N UMERICAL R ESULTS
In this section, we will use the recursive methodto numerically evaluate the blocking probability andcompare them with the large-pool approximations.
A. Basic Characteristics
Fig. 3 shows the exact and large-pool-approximated blocking probability of a VBSpool under real-time traffic and different number ofc-servers, and Fig. 4 shows the same metrics for aVBS pool under delay-tolerant traffic. Note x-axisis normalized by the number of c-servers requiredwithout pooling to show the relative pooling gain.As can be seen, the trend in both figures are similar. This assumption is realistic because a K v v will diminish exponen-tially when a v < . Thus K v a K v v will be driven to for large enough K v . This coincide with our large-pool approximationresults that the blocking probability are affectedonly by the first- and second-order statistics ofthe number of sessions in the VBS pool. For thisreason, we will only present results for real-timetraffic from now on, and the conclusions we drawshould apply to the delay-tolerant case as well.We can observe some basic blocking characteris-tic from these two figures: 1) when the number ofc-servers is sufficient, the computational blockingprobability P bc is very small and below the scopeof this figure; the overall blocking probability isdominated by radio blocking probability P br , whichis around the desired threshold P bth . 2) As thenumber of c-servers decreases from its largest value M T K , the computational blocking probability in-creases rapidly while the radio blocking probabilitystart to decrease slightly; the net result of these twotrends is a plateau before the critical tradeoff point(“knee point”) and a significant increase after it. 3)If the number of c-servers is to further decrease, theoverall blocking probability will be dominated bythe computational blocking probability and saturatesat probability . The radio blocking probability willdecrease rapidly and its influence on overall block-ing probability will diminish. Fig. 3 and Fig. 4 alsoshow the approximated blocking probability. Thefact that the “knee point” configuration saved morethan computational resources with a penaltyof only − increase in the blocking probabilitydemonstrates the benefit of statistical multiplexing.the As expected, these approximations are coherentwith the exact values. B. Heterogeneous VBSs
In Fig. 5, we show the blocking probability of aVBS pool with two class of VBSs. The two classeshave the same number of VBSs and the same trafficload, but the QoS, and thus the number of provi-sioned r-servers, is different. From the curves wecan observe similar basic blocking characteristicsas in the single class case. Also, we can see thatthe overall blocking probability for the two classesare different: they are respectively close to theirthreshold blocking probability when c-servers aresufficient since the overall blocking probability aredominated by the radio blocking, and converges tothe same curve when c-servers become insufficientbecause the computational blocking probability be-gins to overwhelm. −3 −2 −1 B l o c k i n g p r o b a b ili t y Normalized number of c-servers N/ ( M T K ) Overall blocking probability P b Radio blocking probability P br Computational blocking prob. P bc Approx. overall blocking prob.Approx. computational blocking prob.
Fig. 3. Blocking probability of a homogeneous VBS pool asa function of normalized number of c-servers ( N/ M T K ) underreal-time traffic. Black box indicates the knee point. Simulationparameters: M = 40 , a = 20 , P bth1 = 10 − , K = 30 . −4 −3 −2 −1 B l o c k i n g p r o b a b ili t y Normalized number of c-servers N/ ( M T K ) Overall blocking probability P b Radio blocking probability P br Computational blocking prob. P bc Approx. overall blocking prob.Approx. computational blocking prob.
Fig. 4. Blocking probability of a homogeneous VBS pool as afunction of normalized number of c-servers ( N/ M T K ) under delay-tolerant traffic. Simulation parameters: M = 100 , a = 0 . , P bth1 =5 × − , K = 10 . C. Influence of Traffic Load and QoS Target
In Fig. 6, we illustrate the influence of differenttraffic load and QoS target. As can be seen, the de-sired level of QoS ( P bth ) determines the minimumblocking probability(i.e. height of the “plateau” tothe right of the figure); while the traffic load a determines the position of the “knee point” and howfast the blocking probability saturates to . D. Statistical Multiplexing Gain
Most importantly, we can quantify the statisticalmultiplexing gain of the simulated VBS pool withthe equations derived previously. In Fig. 7, we −2 −1 Normalized number of c-servers N/ ( M T K ) B l o c k i n g p r o b a b ili t y Overall blocking prob. of class−1 VBSs P Overall blocking prob. of class−2 VBSs P Radio blocking prob. of class−1 VBSs P
Radio blocking prob. of class−1 VBSs P
Computational blocking prob. P bc Fig. 5. Blocking probability of a heterogeneous VBS pool as afunction of normalized number of c-servers ( N/ M T K ) under real-time traffic. Simulation parameters: M = [20 , T , a = [20 , T , P bth = [1 , T × − , K = [30 , T . −3 −2 −1 Normalized number of c-servers N/ ( M T K ) B l o ck i ng p r obab ili t y a=20, K=30, P bth =0.01a=20, K=28, P bth =0.02a=32, K=44, P bth =0.01 Fig. 6. Blocking probability of a homogeneous VBS pool as afunction of normalized number of c-servers ( N/ M T K ) under real-time traffic with different traffic load and QoS guarantees. Pool size M = 40 . compare the overall blocking probability of threeVBS pools under varying pool size. The “kneepoint” position and large-pool limit are also markedout with vertical lines. As the pool size increases,the blocking probability curve (and so does the“knee point”) is pushed to the left. But the closerthe “knee point” is to the large-pool limit, the slowerthe remaining distance decreases with the pool size | M | . This indicates a decreasing marginal statisticalmultiplexing gain. Comparing the curves, we canfind that the traffic load and the desired level ofQoS have influence on the blocking probability andthe statistical multiplexing gain.To better investigate this influence, we show theknee point position versus varying pool size in Fig. 8. Firstly we can find that a medium sizedVBS pool can readily obtain considerable statisticalmultiplexing gain and the marginal gain diminishesfast. Thus a huge number of VBSs is needed so thatthe pooling gain can approach the large-pool limit.These observations imply that a C-RAN formedwith multiple medium sized VBS pools can obtainalmost the same pooling gain as the one formedwith a single huge pool. If we further take theexpenditure of fronthaul network into consideration,the former choice may be far more economical thanthe latter one.By contrasting the left and middle curves, wecan see that stricter QoS requirements can increasethe pooling gain. This is because on one hand,we need to increase the number of r-servers K in order to reduce the blocking probability, whichwill in turn increase M T K ; on the other hand,the average number of c-servers occupied is alwaysaround | M | µ . Therefore the stricter the QoS, themore idle c-servers there will be in the VBS pooland consequently the more the pooling gain. Also,we can see that the increase in traffic load willreduce the pooling gain by pushing the “knee point”to larger values. This observation indicates that, wemay need to dynamically adjust the size of VBSpools in order to get a satisfactory pooling gainunder fluctuating traffic load.VII. C ONCLUSION
In this article, we proposed a multi-dimensionalMarkov model for VBS pools to analyze theirstatistical multiplexing gain. We showed that theproposed model have a product-form expression forthe stationary distribution. We derived a recursivemethod for calculating the blocking probability ofa VBS pool, and gave closed-form approximationwhen the pool is large enough. Based on theseresults, we derived the expressions for the statisticalmultiplexing gains and applied them to both real-time and delay-tolerant traffic. Numerical resultsreveal that 1) the pooling gain reaches a significantlevel even with medium pool size (more than of the pooling gain can be achieved with around VBSs); 2) the marginal gain of larger pool size tendto be negligible; 3) lighter traffic load and tighterQoS level can increase the pooling gain. This can be achieved by dynamically changing the switchingconfiguration of fronthaul so that the traffic of VBSs can be sentto a data center of the desired size.
Our model can be extended in several aspects toaccommodate for more general scenarios. Firstly,we assume that user sessions occupy equal andfixed amounts of radio and computational resources.Yet realistic resource scheduling algorithm mayallocate different amount of resources for each in-dividual session. To accommodate such cases, ourmodel need to be further relaxed to allow statetransitions among non-neighboring states. Secondly,we assume sessions are only attached to one cellof the system. Nevertheless, coordinated-multipoint(CoMP) transmission/reception may introduce ses-sions that simultaneously consume radio resourcesfrom multiple cells. This means the admission con-trol of CoMP session is hinged upon the availableradio resources in all its serving cells, which can beaccounted for by introducing more comprehensiveadmission controls in the model. Thirdly, we assumesession arrival and service to be Poission. However,many emerging types of multi-media traffic is foundto exhibit certain burstiness. The influence of bursti-ness can be investigated by assuming more generalstochastic traffic and service models, e.g InteruptedPoisson Process [26]. Fourthly, we investigated real-time and delay-tolerant traffic separately whereasthey are likely to coexist in real system. To evaluatesuch heterogeneous traffic, our model need to beextended to account for the different resource usagepatterns of the two session types. Last but not theleast, our resource reservation model may not bethe most efficient possible. For example, unusedservice capacity can be further shared to increasestatistical multiplexing gain. Regarding this, ourmodel need to be further refined to support moregeneral admission control and service strategies.A
PPENDIX AR EVERSIBILITY OF P ROPOSED M ODEL
To proof that U ( t ) is reversible, we first give theKolmogorov’s Criterion of Reversibility. Theorem 5 (Kolmogorov’s Criterion) . Acontinuous-time Markov chain is reversible ifand only if its transition rates satisfy q u (1) u (2) q u (2) u (3) · · · q u ( n − u ( n ) q u ( n ) u (1) (42) = q u (1) u ( n ) q u ( n ) u ( n − · · · q u (3) u (2) q u (2) u (1) (43) for all finite sequences of states u (1) , u (2) , · · · , u ( n ) ∈ U . −3 −2 −1 B l o c k i n g p r o b a b ili t y (a) a = 20 ,K = 28 ,P bth = 0 . −3 −2 −1 (b) a = 20 ,K = 30 ,P bth = 0 . −3 −2 −1 (c) a = 32 ,K = 44 ,P bth = 0 . Overall blocking prob.Knee point indicatorLarge pool limit
Normalized number of c-servers
N/M T K pool size increases Fig. 7. Blocking probability in homogeneous VBS pools as a function of normalized number of c-servers under real-time traffic and differentpool size. Red vertical line indicates the knee point position. Red dashed line indicates the large-pool limit. Curves to the left correspond tolarger pool size. (a) a = 20 ,K = 28 ,P bth = 0 . K nee po i n t i n t e r m s o f no r m a li z ed nu m be r o f c − s e r v e r s (b) a = 20 ,K = 30 ,P bth = 0 . (c) a = 32 ,K = 44 ,P bth = 0 . ApproximationNumerical searchLarge pool limit
Po ol size | M | Fig. 8. Knee point position as a function of pool size. Knee point position is measured in terms of the number of c-servers required normalizedby the number of c-servers needed without pooling. Red dotted line shows the knee point values derived from large-pool approximation,whereas blue line shows the values directly searched from numerical results. Red dashed lines represent the large-pool limit η as pool sizeapproaches infinity. The percentage of the pooling gain at VBSs with respect to the maximum possible gain ( − η ) is also noted in thefigure. We next verify that the Kolmogorov’s Criterion issatisfied by U ( t ) for any finite sequences of states u (1) , u (2) , · · · , u ( n ) ∈ U . Case 1 . If (42) (or (43)) equals to , then theremust be at least one product term being in (42)(or (43)). Assuming this term to be q u ( i ) u ( i +1) , thenaccording to (2), the term q u ( i +1) u ( i ) in (43) (or (42))must also be . Thus (43) (or (42)) must also equalto . Case 2 . If neither (42) nor (43) is , then noneof the terms in (42) and (43) equals . Accordingto (2), the transition resulting in the term q u ( i ) u ( i +1) must be between neighboring states, i.e. u ( i +1) − u ( i ) = (0 , · · · , , ± , , · · · , T . Notice that a transition such that the m -th entry ofa state is changed from u m to u m + 1 will produce aproduct term λ in (42) and a product term f ( u m +1) in (43). And reversely, a transition such that the m -th entry is changed from u m + 1 to u m will producea product term f ( u m +1) in (42) and a product term λ in (43).If we denote the number of state transitions inthe transition loop ( u (1) , u (2) , · · · , u ( n ) , u (1) ) (44)such that the ( v − P w =1 M w + m ) -th state entry is changedfrom u v,m to u v,m + 1 by n + v,m,u v,m times, and thenumber of transitions such that the ( v − P w =1 M w + m ) -thentry is changed from u v,m + 1 to u v,m by n − v,m,u v,m times, thenEq. (42)= V Y w =1 M w Y m =1 K Y u w,m =1 λ n + w,m,uw,m f n − w,m,uw,m w ( u w,m + 1) , (45)Eq. (43)= V Y w =1 M w Y m =1 K Y u w,m =1 λ n − w,m,uw,m f n + w,m,uw,m w ( u w,m + 1) . (46)To make (44) a closed loop, we must have n + v,m,u v,m = n − v,m,u v,m (47)for all v = 1 , , · · · , V , m = 1 , , · · · , M v and u v,m = 1 , , · · · , K . Substituting (47) into (45) and(46) and we get that (42) equals to (43). Thus U ( t ) is reversible. A PPENDIX BP ROOF OF T HEOREM C ( N, M ) term in-volves at most K v summations. Also, there areat most N ≤ M T K such terms for some spe-cific M . Hence the computational complexity forall C ( N, M ) terms for a VBS pool sized M isbounded as C ≤ (max v K v ) M T K | M | ≤ (max v K v ) | M | , (48)where | M | = P Vw =1 M w . At the same time, thecalculation of R ( N, M ) involves only a singlesubtraction (or summation). Again, there are at most M T K such terms for some specific M . Thereforethe computational complexity of R ( N, M ) for aVBS pool with size vector M is bounded as C ≤ M T K | M |≤ (max v K v ) | M | . (49)All together, the overall computational complexityis bounded as C = C + C ≤ h (max v K v ) + max v K v i | M | . (50)This bound is essentially quadratic in the pool size | M | . Therefore, the computational complexity ofblocking probability should also be quadratic in thepool size. A PPENDIX CP ROOF OF T HEOREM ˜ S M we know lim | M |→∞ ˜ S M = lim | M |→∞ | M | V X w =1 M w X m =1 ˜ U w,m = lim | M |→∞ V X w =1 M w | M | P M w m =1 ˜ U w,m M w = lim | M |→∞ V X w =1 β w ˜ S M w . (51)According to the Central Limit Theorem, ˜ S M w con-verges in distribution to a normal random variablesas | M | → ∞ : lim M v →∞ ˜ S M w ∼ N ( µ w , σ w M w ) . (52) Since ˜ U v,m are independent random variables for all v and m , ˜ S M v are also independent. Therefore ˜ S M will also converge to a normal distributed randomvariable: lim | M |→∞ ˜ S M = lim | M |→∞ V X w =1 β w ˜ S w ∼ N ( V X w =1 β w µ w , V X w =1 β w σ w M w ) ∼ N ( µ, σ | M | ) . (53)To express the blocking probability in terms ofthis normal distribution, we next establish a rela-tionship between the stationary distributions of U and ˜ U . Let ˜ P be the probability of zero state for ˜ U ,then from the product-form stationary distributionof U and ˜ U we can get the following scalingrelationship between the stationary distribution of U and ˜ U : Pr { U = u } P = Pr n ˜ U = u o ˜ P . (54)From the definition of P and ˜ P , the followingrelationship exists: P ˜ P = Pr ( V X w =1 M w X m =1 ˜ U w,m ≤ N ) − = Pr ( | M | V X w =1 M w X m =1 ˜ U w,m ≤ N | M | ) − = Pr (cid:26) ˜ S M ≤ N | M | (cid:27) − . (55)Notice in the above relationship, P / ˜ P is deter-mined by the probability distribution of ˜ S M . There-fore we can use the large-pool limit of ˜ S M to getthe following approximation lim | M |→∞ P ˜ P = (cid:20) − Q ( N | M | ) (cid:21) − , (56)where q ( x ) and Q ( x ) are respectively the proba-bility density function (PDF) and cumulative taildistribution of N ( µ, σ | M | ) . With these relationships, we can now approximate the blocking probability: lim | M |→∞ P bc = lim | M |→∞ Pr ( V X w =1 M w X m =1 U w,m = N ) = lim | M |→∞ P ˜ P Pr ( V X w =1 M w X m =1 ˜ U w,m = N ) = lim | M |→∞ P ˜ P Pr (cid:26) ˜ S M = N | M | (cid:27) = P ˜ P | M | q (cid:18) N | M | (cid:19) , (57) lim | M |→∞ P brv = lim | M |→∞ Pr ( U v, = K v , V X w =1 M w X m =1 U w,m < N ) = lim | M |→∞ P ˜ P Pr ( ˜ U v, = K v , V X w =1 M w X m =1 ˜ U w,m < N ) = lim | M |→∞ P ˜ P Pr n ˜ U v, = K v o · Pr ( M v X m =2 ˜ U v,m + X w = v M w X m =1 ˜ U w,m < N − K v ) = lim | M |→∞ P ˜ P ˜ P brv Pr ( ˜ S M − ˜ U v, | M | − < N − K v | M | − ) = P ˜ P ˜ P brv (cid:20) − Q ( N | M | ) (cid:21) . (58)Notice the fifth equality of (58) holds because, as | M | → ∞ , N should also approach infinity as N > | M | µ . Hence lim | M |→∞ Q ( N − K v | M | ) = Q ( N | M | ) . (59)Also, lim | M |→∞ Q ( N | M | ) = e − α / . Therefore, the ap-proximation for the overall session blocking proba- bility of class- v VBSs is lim | M |→∞ P bv = lim | M |→∞ ( P bc + P brv )= (cid:20) − Q ( N | M | ) (cid:21) − · (cid:26) | M | q (cid:18) N | M | (cid:19) + ˜ P brv (cid:20) − Q ( N | M | ) (cid:21)(cid:27) = p | M || M |√ πσ e − α / − e − α / + ˜ P brv = 1 p π | M | σ e α / − P brv . (60)A PPENDIX DP ROOF OF T HEOREM ˜ S M will also converge to a normal dis-tributed random variable N ( µ, σ | M | ) as | M | → ∞ ,according to the strong law of large numbers: Pr (cid:26) lim | M |→∞ η = | M | µN (cid:27) = Pr ( lim | M |→∞ P Vw =1 P M w m =1 ˜ U w,m N = | M | µN ) = Pr (cid:26) lim | M |→∞ ˜ S M = µ (cid:27) = 1 . (61)Hence η a . s . −−→ | M | µN . 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IEEE Transactions on Wireless Communi-cations , vol. 12, no. 8, pp. 4196–4209, August 2013.PLACEPHOTOHERE
Jingchu Liu (S’14) received his B.S. degreefrom Department of Electronic Engineering ofTsinghua University, China, in 2012. He iscurrently a PhD student at the Departmentof Electronic Engineering, Tsinghua Univer-sity. From October 2015 to April 2016, hevisited the Autonomous Networks ResearchGroup, Ming Hsieh Department of ElectricalEngineering, University of Southern Califor-nia, CA, USA. His research interests include cloud-based wirelessnetworking, data-driven network management, network data analyt-ics, and green wireless communications.PLACEPHOTOHERE
Sheng Zhou (S’06-M’12) received the B.E.and Ph.D. degrees in electronic engineeringfrom Tsinghua University, Beijing, China, in2005 and 2011, respectively. From January toJune 2010, he was a visiting student at theWireless System Lab, Department of Electri-cal Engineering, Stanford University, Stanford,CA, USA. He is currently an Assistant Pro-fessor with the Department of Electronic En-gineering, Tsinghua University. His research interests include cross-layer design for multiple antenna systems, cooperative transmissionin cellular systems, and green wireless communications. Dr. Zhoucoreceived the Best Paper Award at the Asia-Pacific Conferenceon Communication in 2009 and 2013, the 23th IEEE InternationalConference on Communication Technology in 2011, and the 25thInternational Tele-traffic Congress in 2013. PLACEPHOTOHERE
Jie Gong (S’09-M’13) received his B.S. andPh.D. degrees in Department of ElectronicEngineering in Tsinghua University, Beijing,China, in 2008 and 2013, respectively. FromJuly 2012 to January 2013, he visited In-stitute of Digital Communications, Universityof Edinburgh, Edinburgh, UK. During 2013-2015, he worked as a postdoctorial scholarin Department of Electronic Engineering inTsinghua University, Beijing, China. He is currently an associateresearch fellow in School of Data and Computer Science, Sun Yat-sen University, Guangzhou, Guangdong Province, China. He was aco-recipient of the Best Paper Award from IEEE CommunicationsSociety Asia-Pacific Board in 2013. His research interests includeCloud RAN, energy harvesting and green wireless communications.PLACEPHOTOHERE
Zhisheng Niu (M’98-SM’99-F’12) graduatedfrom Beijing Jiaotong University, China, in1985, and got his M.E. and D.E. degrees fromToyohashi University of Technology, Japan, in1989 and 1992, respectively. During 1992-94,he worked for Fujitsu Laboratories Ltd., Japan,and in 1994 joined with Tsinghua University,Beijing, China, where he is now a professor atthe Department of Electronic Engineering. Heis also a guest chair professor of Shandong University, China. Hismajor research interests include queueing theory, traffic engineering,mobile Internet, radio resource management of wireless networks,and green communication and networks.Dr. Niu has been an active volunteer for various academic soci-eties, including Director for Conference Publications (2010-11) andDirector for Asia-Pacific Board (2008-09) of IEEE CommunicationSociety, Membership Development Coordinator (2009-10) of IEEERegion 10, Councilor of IEICE-Japan (2009-11), and council memberof Chinese Institute of Electronics (2006-11). He is now a distin-guished lecturer (2012-15) and Chair of Emerging Technology Com-mittee (2014-15) of IEEE Communication Society, a distinguishedlecturer (2014-16) of IEEE Vehicular Technologies Society, a memberof the Fellow Nomination Committee of IEICE CommunicationSociety (2013-14), standing committee member of Chinese Instituteof Communications (CIC, 2012-16), and associate editor-in-chief ofIEEE/CIC joint publication China Communications.Dr. Niu received the Outstanding Young Researcher Award fromNatural Science Foundation of China in 2009 and the Best PaperAward from IEEE Communication Society Asia-Pacific Board in2013. He also co-received the Best Paper Awards from the 13th,15th and 19th Asia-Pacific Conference on Communication (APCC)in 2007, 2009, and 2013, respectively, International Conference onWireless Communications and Signal Processing (WCSP’13), andthe Best Student Paper Award from the 25th International TeletrafficCongress (ITC25). He is now the Chief Scientist of the NationalBasic Research Program (so called ”973 Project”) of China on”Fundamental Research on the Energy and Resource OptimizedHyper-Cellular Mobile Communication System” (2012-2016), whichis the first national project on green communications in China. He isa fellow of both IEEE and IEICE. PLACEPHOTOHERE