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System cost minimization in cloud RAN withlimited fronthaul capacity

Tang, Jianhua; Tay, Wee Peng; Quek, Tony Q. S.; Liang, Ben

2017

Tang, J., Tay, W. P., Quek, T. Q. S., & Liang, B. (2017). System cost minimization in cloud RANwith limited fronthaul capacity. IEEE Transactions on Wireless Communications, 16(5),3371‑3384. doi:10.1109/TWC.2017.2682079

https://hdl.handle.net/10356/102695

https://doi.org/10.1109/TWC.2017.2682079

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1

System Cost Minimization in Cloud RAN withLimited Fronthaul Capacity

Jianhua Tang,Member, IEEE,Wee Peng Tay,Senior Member, IEEE,Tony Q.S. Quek,Senior Member, IEEEandBen Liang,Senior Member, IEEE

Abstract—Cloud radio access network (C-RAN) is emerging asa potential alternative for the next generation RAN by mergingRAN and cloud computing together. In this paper, we considerthe baseband unit (BBU) pool of C-RAN as a collection of virtualmachines (VMs). We allow each user equipment (UE) to associatewith multiple VMs in the BBU pool, and each remote radiohead (RRH) can only serve a limited number of UEs. Under thismodel, we jointly optimize the VM activation in the BBU pooland sparse beamforming in the coordinated RRH cluster, which isconstrained by limited fronthaul capacity, to minimize the systemcost of C-RAN. We formulate this problem as a mixed-integernonlinear programming (MINLP) problem, and then proposeefficient methods to optimize the number of active VMs, as wellas the sparse beamforming vectors. Moreover, we derive closed-form solution for the beamforming vectors. Simulation resultssuggest that our proposed algorithms have better performancethan the benchmark algorithms in terms of both system cost androbustness.

Index Terms—C-RAN, VM activation, limited fronthaul capac-ity, computation capacity

I. I NTRODUCTION

The evolution of radio access network (RAN) over thepast decade has been driven by fast data proliferation. CiscoSystems predicts that mobile data traffic will increase 8-foldfrom 2015 to 2020, and the number of mobile devices percapita will reach 1.5 by the year 2020 [1]. To maintain ahigh quality-of-service (QoS), the principal solution forRANservice providers is enhancing RANs’ capacity and coverage.However, they are facing many challenges when adopting thissolution approach [2]: Firstly, the explosive increase in net-work capacity demand (especially busy-hour demand) triggersan exponential increase in the number of base stations (BSs),which leads to a significantly higher power consumption.

Part of this paper was presented by the Asilomar Conference on Signals,Systems, and Computers (ASILOMAR), Pacific Grove, CA, USA, November2015.

This work was supported in part by the Startup Funds of Chongqing Uni-versity of Posts and Telecommunications under Grant A2016-114, SingaporeMOE ARF Tier 2 under grant MOE2014-T2-1-028, the SUTD-ZJU ResearchCollaboration under Grant SUTD-ZJU/RES/01/2014, and the MOE ARF Tier2 under Grant MOE2014-T2-2-002.

J. Tang is with the School of Communication and Information Engineering,Chongqing University of Posts and Telecommunications, China. He was withthe School of Electrical and Electronic Engineering, Nanyang TechnologicalUniversity, Singapore. e-mail: jtang4@e.ntu.edu.sg

W. P. Tay is with the School of Electrical and Electronic Engineering,Nanyang Technological University, Singapore. e-mail: wptay@ntu.edu.sg.

T. Q. S. Quek is with Singapore University of Technology and Design. Heis also with Institute for Infocomm Research, A*STAR, Singapore. e-mail:tonyquek@sutd.edu.sg.

B. Liang is with Department of Electrical and Computer Engineering,University of Toronto, Canada. e-mail: liang@comm.utoronto.ca.

BBU pool RRHsFronthaul

links

Dispatcher

UEs

Assembler

Fig. 1. A typical structure of C-RAN.

Secondly, costly capital and operating expenditure leads tofalling average revenue per user. Moreover, with the dynamicnature of mobile traffic, the utilization of some BSs is actuallyquite low during non-peak hours.

Along with RANs’ evolution, cloud computing has emergedas a popular computing paradigm, since cloud computinghas its attractive characteristics like resource pooling andrapid elasticity. By introducing the merits of cloud comput-ing into RANs, cloud radio access network (C-RAN) hasbeen proposed as a prospective architecture to overcome theaforementioned challenges [3]. A typical C-RAN consists ofthree components (cf. Figure 1): baseband unit (BBU) pool,fronthaul links and remote radio heads (RRHs). The mostsignificant innovation of C-RAN is utilizing a centralizedcloud-based BBU pool instead of the conventional distributedbaseband processing devices co-located with the BSs. Thatmeans, in C-RAN, baseband signal processing functionalitiesare decoupled from the RRHs, and RRHs just need to keepbasic signal transmission and reception functionalities.

C-RAN possesses several advantages compared to the con-ventional RAN: Firstly, utilizing centralized signal process-ing in the BBU pool instead of the distributed BSs in theconventional RAN can significantly save the capital and op-erating expenditure. Secondly, joint processing in the BBUpool and cooperative radio techniques over RRHs, whichare interconnected via the BBU pool, improves the spectrumefficiency, link reliability and the communication qualityofthe cell edge users. Thirdly, BBU pool consists of manygeneral purpose servers. Applying cloud computing as thecomputing paradigm of the centralized BBU pool can reducethe power consumption and improve hardware utilization,through resource sharing and virtualization, i.e., a server canbe further virtualized into many virtual machines (VMs). How-ever, several challenges in C-RAN remain to be addressed, and

2

I summarize these challenges as a “limited versus unlimited”problem:

• Due to the high amount of data transfers (especiallywhen joint transmission techniques are adopted in theC-RAN downlink) in the fronthaul, whose capacity isactually limited, efficient data transfer algorithms needto be developed.

• As all the computation resources are migrated fromthe BSs into a centralized BBU pool, the amount ofcomputation resources in BBU pool is relativelyunlimited(compared to these in BSs). We need to effectively man-age and dispatch those computation resources in C-RAN.In particular, with the ability to elastically scale servicecapacities in the cloud-based BBU pool, many problem-s well studied in the conventional RANs have to berelooked at in C-RAN. For example, resource allocationschemes for conventional RANs are typically obliviousto computation capatities/costs since they are fixed. InC-RAN, however, the computation capatities/costs at thecloud BBU pool can be dynamically scaled, e.g., turningon or off VMs, according to system demands.

In this paper, we jointly optimize the VM activation inthe BBU pool and sparse beamforming vector in the RRHs,which have limited fronthaul capacity, to minimize the systemcost of C-RAN, including cloud processing cost and wirelesstransmission cost.

A. Related Work

C-RAN [4]–[6] has attracted increasing research interestover the past three years. C-RAN provides a centralized BBUpool to improve resource utilization, such as the hardware andenergy utilization, and enables centralized processing ofthereceive and transmit signals at the RRHs. However, the mainconcern for this centralized processing structure is the highamount of data transfer in the capacity-limited fronthaul [7]–[20]. In fact, there are two main different definitions for thefronthaul capacity in the literature:

1) Fronthaul capacity was defined as the maximum sum datarate transmitted on each fronthaul, such as in [11] and[18].

2) Fronthaul capacity was defined as the maximum numberof user equipments (UEs) can be served on each fron-thaul, such as in [9], [10] and [21].

The authors always implicitly assume that each fronthaul canserve unlimited number of users when they adopt the firstdefinition. However, due to the signaling and coordinationoverhead, in the real system, this assumption can not hold.Thus, we adopt the second definition in this work (See thedetailed mathematical definition in Section II-C). Moreover,the second definition is also applied in the simulation part of[22].

With respect to the problem formulations, the works [7],[8] aim to minimize the number of active UE-RRH pairs tomitigate the amount of data transfer in the fronthaul, whilein [9], [11], [13], [23], the authors consider the fronthaulcapacity as a constraint in their optimization formulation. The

references [14]–[17], [24] develop efficient signal compres-sion/quantization algorithms to downsize the load in the C-RAN fronthaul. Admission control in heterogeneous networkswith wireless backhaul is studied in [25]. In addition, to reducethe amount of data transfer in C-RAN, caching at accesspoints is a promising approach [26]–[28], and users devicelevel caching is also applicable [29].

Instead of focusing on the wireless transmission part of C-RAN, several works also investigate the problems introducedby the BBU pool. Computationally aware strategies are pro-posed to reduce computational outage in [30], and to maximizesum-rate in [31]. The reference [32] uses task assignment tominimize the power consumption in the BBU pool of C-RAN.However, most of these works fail to consider the interactionbetween cloud processing and wireless transmission in C-RAN. In [33], we jointly optimize the elastic service scalingin the BBU pool, RRH selection, and the beamformer designat the RRHs. However, the system model studied in [33] issomewhat idealized. This work considers a more practicalsystem, which differs from our previous work in the followingaspects:

1) In this paper, to fully utilize the computation capacityof VMs, we consider the case where each UE can beassociated with multiple VMs in the BBU pool, and weneed to optimize the number of active VMs in the BBUpool. However, in [33], we assume each UE can onlyassociate with one VM, and one VM can only serverone UE. Hence, the number of active VMs is just simplyequal to the number of UEs, while this is not practical.

2) We consider the limited fronthaul capacity in this paper,which is assumed to be unlimited in [33]. As a conse-quence, in this paper, each UE can only access a limitednumber of RRHs in the active RRH cluster. In [33], weassume each UE can access every RRH in the active RRHcluster, which is not practical.

B. Main Contributions

In this paper, we jointly optimize the VM activation andsparse beamforming in order to minimize the overall cost fora C-RAN with limited fronthaul capacity. Our main contribu-tions are as follows:

• We formulate the “unlimited versus limited” problemas a mixed-integer nonlinear programming (MINLP) byminimizing the overall cost, which consists of two parts:the cloud processing cost (in the BBU pool) with respectto (w.r.t.) the number of active VMs, and the wirelesstransmission cost (in the fronthaul and RRHs) w.r.t. thetransmit beamformers.

• To avoid the feasibility problem caused by relaxing thel2,0-norm constraint directly, we reformulate the originalMINLP into an equivalent problem, which introduces aprice vector. This equivalent problem can then be solvedby adjusting the value of the price vector, and reducingto a subproblem. We propose two different approachesto solve this subproblem: an integer search (IS) approachand a joint optimization with integer recovery (JR) ap-proach. Moreover, we derive the closed-form solution forthe JR approach.

3

• We provide simulation results that suggest that our pro-posed approach can provide better feasibility guaran-tee and obtain lower system cost than the benchmarkalgorithms, for example, the recently proposed staticclustering algorithm in [11].

The remainder of this paper is organized as follows. Wepresent the C-RAN system model in Section II, and proposethe problem formulation and its equivalent formulation inSection III. In Section IV and V, we propose approachesto solve the problem step-by-step. And in Section VI, thenumerical results are presented. We conclude the paper inSection VII.

Notations

We use calligraphy letters to represent sets, boldface lowercase letters to denote vectors, and boldface upper case lettersto denote matrices. The notation‖·‖2 stands for the Euclideannorm, while(·)T and(·)H are the transpose and the conjugatetranspose, respectively. We useN, C andR+ to represent thenatural numbers, complex numbers and non-negative real num-bers, respectively. The notationA\B denotes the setA with itssubsetB removed. We also use the notationx+ = max(0, x).⌊x⌋ stands for the largest integer smaller than or equalsx and⌈x⌉ stands for the smallest integer larger than or equalsx.

II. SYSTEM MODEL

In this section, we present our C-RAN system model andits practical constraints.

A. System description

Suppose that there areN single-antenna UEs andL RRHs,each withK antennas, in a C-RAN cluster. We denote theset of all UEs and all RRHs asN = {1, · · · , N} andL ={1, · · · , L}, respectively. There areM homogenous VMs inthe BBU pool. Each has computation capacityµ and incurs aVM costϕ > 0 when it is active. We denote the number ofactive VMs asm ∈ N, wherem ≤ M . This model reflectsthe popular commercial cloud service models, e.g., AmazonElastic Compute Cloud (EC2). In EC2, there are thousands ofinstances (VMs) in the data center, and each instance has afixed computation capacity. Cloud users just need to decidehow many instances they need to rent.

In the downlink of a C-RAN (cf. Figure 1), all UEs’incoming traffic is first processed by a dispatcher. We assumethat the mean arrival data rate of UEi to the dispatcher isλi,∀i ∈ N , and letα =

∑

i∈N λi. Then, each transport block[34] (or even a code block within each transport block) in thedata flow to UEi can be routed to one ofm active VMs forprocessing (e.g., turbo coding) with probability1/m by thedispatcher. Therefore, the mean incoming traffic rate routed toeach active VM isα/m.

In the wireless transmission part, we consider joint trans-mission as the CoMP technique in C-RAN, i.e., each UE’sdata can be shared among all the coordinated/associated R-RHs, while the RRHs have limited fronthaul link capacity(the fronthaul links between the BBU pool and the RRHs

Assem

bler

Dispatch

er

Cloud

Processing

Wireless

Transmission

...

...

...

λ1

λN

c1

cNµ

αm

αm

αm

µ

µα

Fig. 2. Queueing network model representation of a C-RAN cloud processingand wireless transmission.

are heterogeneous, and can be fiber links, copper cables orwireless channels). After processing by the VMs, each UE’sdata is forwarded to the UE via at mostL RRHs (since thedata is shared among the limited fronthaul RRHs). Let theachievable wireless transmission rate to UEi be ci.

B. Queueing system model

Each active VM in the BBU pool can be modeled as aqueue. Specifically, for each queue, the mean arrival rate isα/m and the mean service rate isµ. Throughout the paper,we assume the tasks within each queue is served in a firstin first out (FIFO) manner and the buffer length is infinite.We note that the use of queuing models, where the wirelesstransmission rate is the queue’s service rate, is not new, andhas been widely used to characterize wireless communicationsystems [35].

We consider a double-layer queueing network to representeach UE’s data processing and transmitting behavior in the C-RAN downlink (cf. Figure 2). Specifically, in the BBU pool,the transport blocks to each UE is processed (e.g., encoded)by m parallel active VMs, each of which is abstracted as aqueue with mean service rateµ. Then, the processed data istransmitted to UEi via RRHs over wireless channels, whichare modeled by a wireless transmission queue with meanservice rateci.

We denote the mean processing delay for data to UEi inthe BBU pool asbi. Let di be the mean transmission delayof the data to UEi in the wireless transmission queue (i.e.,the expected delay incurred at the queue before the data iscompletely transmitted). We assume that UEi’s packet arrivalprocess to the dispatcher is a Poisson process with meanrate λi. Hence, the arrival process to each VM also formsPoisson process with mean arrival rateα/m. Suppose that theservice time of each data packet in each VM queue follows anexponential distribution with mean1/µ, for µ > α/m. Then,for each UE’s data, the arrival rate to the wireless transmissionqueue is the same as the one to the dispatcher [36], [37]1. Weassume that the service time of each data packet in the wirelesstransmission queue follows an exponential distribution with

1Note that, we do not consider the impact, introduced by baseband process-ing, on the arrival data rate to the wireless transmission queue. Specifically,we assume that, after being processed by the VM, the size of the transportblocks and the inter-arrival times to the assembler still remain the same asthose to the dispatcher.

4

mean1/ci. Therefore, the data processing and transmissionin our C-RAN model can be treated as two layers of M/M/1queues in tandem. In addition, from queueing theory [38], wehave

bi =m

mµ− α, (1)

di =1

ci − λi(2)

whereµ > α/m, ci > λi, ∀i ∈ N .Let ui denote the data symbol for UEi with E[|ui|2] = 1,

and wij ∈ CK×1 denote the transmit beamformer for UEifrom RRH j. We assume block fading wireless transmissionchannels from the RRHs to the UEs. We define thel2,0-norm andl2,1-norm of vectorwij as ‖wij‖2,0 ,

∥∥wH

ijwij

∥∥0

and ‖wij‖2,1 , wHijwij respectively. Then, the associations

between UEs and RRHs can be represented by‖wij‖2,0, i.e.,‖wij‖2,0 = 1 if and only if UE i is connected toj.

The block fading channel from RRHj to UE i is denotedashH

ij , wherehij ∈ CK×1, for i ∈ N andj ∈ L. The received

signal at UEi is then given by

ui =∑

j∈L

hHijwijui +

N∑

k 6=i

∑

j∈L

hHijwkjuk + δi,

where the first term is the useful signal for UEi, the secondterm is the interference to UEi, and δi ∼ CN (0, σ2

i ) is theadditive white Gaussian noise (AWGN) at UEi. The signal-to-interference-plus-noise ratio (SINR) at UEi is

SINRi =|∑

j∈L hHijwij |2

σ2i +

∑Nk 6=i |

∑

j∈L hHijwkj |2

. (3)

The downlink achievable rateci to UE i satisfies

ci ≤ Bi log(1 + SINRi),

whereBi is the wireless transmission bandwidth for UEi.Each RRHj’s maximum transmit power is denoted asEj ,i.e.,

N∑

i=1

‖wij‖2,1 ≤ Ej , for j ∈ L.

C. Practical constraints

In addition to the basic system model above, we include thefollowing two practical constraints to capture the features ofC-RAN:

1) System delay constraint: To couple cloud processing andwireless transmission in C-RAN, we propose the cross-layer system delay constraint:

bi + di ≤ τi, ∀i ∈ N ,

whereτi is a predefined maximum system delay for UEi, which can be treated as its QoS requirement.

2) Fronthaul capacity constraint: We denoteSj ∈ N asthe fronthaul linkj’s capacity, i.e., the maximum numberof UEs that can be connected with this fronthaul link. In

other words, at mostSj UEs can be associated with RRHj. We can cast this fronthaul capacity constraint as

∑

i∈N

‖wij‖2,0 ≤ Sj , ∀j ∈ L,

where‖wij‖2,0 = 1 if and only if RRH j is associatedwith UE i.

III. PROBLEM FORMULATION

The system cost incurred by a C-RAN is the total VMcost in the BBU pool and the power consumption incurredby the RRHs. Our aim is to minimize the system cost byoptimizing the number of active VMs and the beamformersat the RRHs. Specifically, (i) the cost for cloud process-ing is mϕ; (ii) the cost incurred by wireless transmissionis η

∑Ni=1

∑Lj=1 ‖wij‖2,1 +

∑Ni=1

∑Lj=1 ‖wij‖2,0 Pj , where

η > 0 is a weight andPj is a static cost when RRHj (and itsfronthaul) is active. Our optimization problem is formulatedas:

(P0) minm,ci,wij

mϕ+ η

N∑

i=1

L∑

j=1

‖wij‖2,1 +N∑

i=1

L∑

j=1

Pj ‖wij‖2,0

s.t.m

mµ− α+

1

ci − λi≤ τi, ∀i ∈ N , (4)

α < mµ, λi < ci, ∀i ∈ N , (5)

0 < m ≤ M, m ∈ N, (6)

ci ≤ Bi log (1 + SINRi) , ∀i ∈ N , (7)N∑

i=1

‖wij‖2,1 ≤ Ej , ∀j ∈ L, (8)

N∑

i=1

‖wij‖2,0 ≤ Sj , ∀j ∈ L, (9)

where “s.t.” stands for “subject to” and SINRi is given by (3).We assume the feasible region of problem (P0) is nonempty.Let the optimal solution for problem (P0) be{(m∗, c∗i ,w

∗ij) :

i ∈ N , j ∈ L}.Remark 1:Actually, the system cost in this paper is a wide

concept, and its physical meaning can be varied. For instance,

1) It can be the monetary cost per unit time, ifφ stands forthe price of renting/turning on a VM per unit time,Pj

denotes the unit time electricity price of turning on RRHj andη captures the electricity price per Watts per unittime.

2) It also can be the power consumption, ifφ stands for thestatic power consumption of turning on a VM,Pj denotesthe static power consumption of turning on RRHj andη captures the inefficiency coefficient of the amplifier ofeach RRH.

For problem (P0), we first note that there is no loss inoptimality if we restrict the constraints (4) to be equalities.

Proposition 1: There exists an optimal solution{(m∗, c∗i ,w

∗ij) : i ∈ N , j ∈ L} such that constraints

5

(4) are active, i.e.,(m∗, c∗i ) for problem (P0) satisfies thefollowing equation

m∗

m∗µ− α+

1

c∗i − λi= τi, ∀i ∈ N . (4′)

Proof: For any optimal solution, if there exists ani ∈ Nsuch that (4) is a strict inequality, then we can decreasec∗iuntil equality in (4) holds. This is still a feasible solution for(P0), and the proposition is proved.

Proposition 1 shows the interaction between cloud process-ing and wireless communication. For example, if we need alower processing delaybi, which means more VMs should beadded, then this will result in a higher cloud processing cost.However, on the other hand, based on Proposition 1, a lowerprocessing delaybi will lead to a higher transmission delaydiin return. That means we can save some wireless transmissionpower and active RRHs. Then a lower wireless transmissioncost can be achieved. This interaction reveals C-RAN as acoupled system.

Problem (P0) is difficult to solve, due to the followingreasons: (i) it is an MINLP with two integer constraints (6)and (9); and (ii) the problem is nonconvex even if we assumem ∈ R+ and the fronthaul capacity constraint (9) is removed.In the following section, we propose a reformulation and somerelaxation techniques to make problem (P0) tractable.

A. An equivalent formulation for problem (P0)

In problem (P0), one of the solution challenges is thel2,0-norm constraint (9). Two commonly used approaches to dealwith the l2,0-norm are: smoothing function approximation[8], [39] and reweightedl2,1-norm approximation [11], [40].However, if we just relax the left hand side of constraint (9)with a smoothing function orl2,1-norm approximations, andsolve the relaxed problem directly, then we have no guaranteethat the optimal solution derived from the relaxed problem isalso feasible for the original problem (P0).

Let’s consider the following problem, which considers thetrade-off between the system cost and fronthaul capacity:

(P1) minm,ci,wij

mϕ+ η

N∑

i=1

L∑

j=1

‖wij‖2,1

+

N∑

i=1

L∑

j=1

(Pj + γj) ‖wij‖2,0

s.t. (4′), (5), (6), (7) and (8),

whereγj ≥ 0 is the price for RRHj. Let Γ , [γ1, · · · , γL]T .We denote{m(Γ),wij(Γ)} as the optimal solution for

problem (P1) for a given price vectorΓ. Define βj(Γ) =∑N

i=1 ‖wij(Γ)‖2,0. The following theorem shows the relation-ship between problem (P0) and problem (P1).

Theorem 1:For problem (P1), if

βj(Γ) = Sj , ∀j ∈ L, (10)

then the optimal solution to problem (P1) is also optimal forproblem (P0).

Proof: See Appendix A.

P0P0 P1P1

Price

Adjd usting

Price

Adjusting

P2P2

RelaxationRelaxation

P2-1P2-1

P2-2P2-2

Integer

Search

Integer

Search

Joint

Optimization

Joint

Optimization

{m, ci,wij}

{m∗, c∗i ,w∗ij}

Fig. 3. An iterative approach to solve problem (P0).

Based on Theorem 1, if a price vectorΓ can be found so that(10) holds, then we can solve (P1) instead of (P0). However,such a price vector may not exist, and in general, the solutionof (P1) is sub-optimal for (P0) if

βj(Γ) ≤ Sj , ∀j ∈ L. (11)

Instead of solving problem (P0) directly, in what follows, wepropose a step-by-step relaxation and reformulation approachto simplify the problem, and obtain a reasonable but sub-optimal solution:

1) In Section IV, we introduce some properties of theprice vector Γ in problem (P1), and we propose aprice adjusting algorithm to find a price vectorΓ thatsatisfies equation (11). For a fixed price vectorΓ, weapply reweightedl1-norm relaxation on problem (P1) tosimplify it into another problem (P2) in subsection IV-B.

2) In Section V, we propose two different approaches tosolve problem (P2).

We show the logic flow for solving problem (P0) in Fig-ure 3.

IV. A PPROXIMATION FOR PROBLEM(P1)

In this section, we first present some properties that theprice vectorΓ satisfies. Then, we propose aprice adjustingalgorithm to obtain a sub-optimal solution for problem (P0).

A. Bisection search for price vectorΓ

In the following proposition, we present results that allowus to iteratively adjust the price vector in problem (P1) suchthat equation (11) holds. The proposition is inspired by [9].

Proposition 2: Fix eachγk as a constantγk for all k ∈L\j, and let Γj = [γ1, · · · , γj−1, γj , γj+1, · · · , γL]T . Thenthe following holds.(i) βj(Γj) is a non-increasing function w.r.t.γj .(ii) There is a threshold price for RRHj, θj = ϕM +

∑

j∈L(ηEj +PjSj) +∑

k∈L\j γkSk, such that forγj ≥θj , βj(Γj) ≤ Sj .Proof: The proof is similar with the one in [9]. We

provide it for completeness in Appendix B.Recall that the feasible region of problem (P0) is nonempty,

therefore, we can always satisfy equation (11) by iterativelysearching overγj ∈ [0, θj ]. We elaborate the algorithm tosolve problem (P1) by iteratively adjusting the price vector inAlgorithm 1, in whichγ

(l)j is the j-th component ofΓ(l) in

6

the l-th iteration, andθ(l)j

= ϕM +∑

j∈L(ηEj + PjSj) +∑

k∈L\j γ(l−1)k Sk.

Algorithm 1 Price adjusting algorithm for problem (P1)

1: Initialize: Let Γ(0) = [0, · · · , 0]T .2: Iterationl: Solve problem (P1) with givenΓ(l−1), obtain-

ing βj(Γ(l−1)), for j ∈ L.

3: if βj(Γ(l−1)) ≤ Sj , ∀j ∈ L, then

4: break;5: else6: for thosej ∈ A(l),{j : βj(Γ

(l−1)) > Sj , ∀j ∈ L}, setγ(l)

j= θ

(l)

j. Fix γ

(l)k = γ

(l−1)k , ∀k ∈ L\A(l).

7: end if8: Let l = l + 1, go to step 2.

In Algorithm 1, the main iteration in Step 2 involves analgorithm to solve problem (P1). Although we avoid thefeasibility problem by reformulating problem (P0) into (P1),the l2,0-norm still remains unsolved in the objective function.In the next subsection, we introduce reweightedl2,1-normapproximation for problem (P1).

B. Reweightedl2,1-norm relaxation

In compressive sensing [41], reweightedl1-norm is regardedas an effective way to deal with thel0-norm in the objectivefunction, sincel1-norm is the convex relaxation forl0-norm[40]. In the same spirit, we adopt an iterative procedure tosolve problem (P1), and the terms involvingl2,0-norm in theobjective function of problem (P1) as:

‖wij‖2,0 ≈ ρ(p)ij ‖wij‖2,1 , (12)

whereρ(p)ij =

(∥∥∥w(p−1)

ij

∥∥∥

2

2+ φ

)−1

is the weight in thep-th

iteration,w(p−1)ij is a constant vector obtained from previous

iteration andφ is a small positive constant to guarantee thenumerical stability. The intuition behind the weightρ

(p)ij is that

the beamformer vector that has smaller norm in iterationp−1is allocated a larger weightρ(p)ij in iterationp, and hence, thenorm is further reduced after solving the problem in iterationp.

With the l2,1-norm relaxation, problem (P1) can be approx-imated as the following problem in thep-th iteration:

(P2) minm,ci,wij

mϕ+

N∑

i=1

L∑

j=1

z(p)ij ‖wij‖2,1

s.t. (4′), (5), (6), (7) and (8),

wherez(p)ij = η + (Pj + γj)ρ(p)ij .

After the reweightedl2,1-norm relaxation, a sub-optimalsolution can be obtained for problem (P1). However, in eachiteration of the reweightedl2,1-norm relaxation, problem (P2)is required to be solved, which is still an MINLP. In the nextsection, we discuss two different approaches to solve problem(P2).

V. TWO OPTIMIZATION APPROACHES FOR PROBLEM(P2)

In this section, we propose two different approaches tosolve problem (P2), which obtain its global and local optimalsolution respectively.

First of all, constraints (4′) and (5) imply that

m = α

(τini

+1

ni(ni(ci − λi)− µ)

)

≥ α

(τini

+1

ni(ni(ci − λi)− µ)

)

, mi, ∀i ∈ N , (13)

and

ci = λi +µi

ni+

α

ni(nim− ατi), gi(m), ∀i ∈ N , (14)

whereni = τiµ−1 > 0, andci is an upper bound ofci, whichcan be derived by applying the Cauchy-Schwarz inequality on(7) as follows:

ci ≤ Bi log

1 +1

σ2i

L∑

j=1

‖hij‖22

L∑

j=1

‖wij‖22

≤ Bi log

1 +1

σ2i

L∑

j=1

‖hij‖22 Ej

, ci, ∀i ∈ N . (15)

Based on (13) and (14), in what follows, we discuss twodifferent approaches to solve problem (P2).

A. Integer search (IS) approach

Oncem is fixed as an integerm, problem (P2) is reducedinto the following weighted sum-power minimization (WSPM)problem:

(P2-1) minwij

N∑

i=1

L∑

j=1

z(p)ij ‖wij‖2,1

s.t. ci ≤ Bi log (1 + SINRi) , ∀i ∈ N , (16)N∑

i=1

‖wij‖2,1 ≤ Ej , ∀j ∈ L,

where ci = gi(m) is a constant. Since phase rotation ofwij

does not affect problem (P2-1), we can recast constraint (16)as the following second-order cone (SOC) [42], [43]:

‖r i‖2 ≤

√

1 + 1/(2ciBi − 1)Re[Rii], ∀i ∈ N , (17)

whereRik =∑

j∈L hHijwkj , r i = [Ri1, · · · , RiN , σi]

T andRe(·) stands for the real part of a complex number. Thus,problem (P2-1) can be reformulated as a second-order coneprogramming (SOCP), which can be easily solved by interiorpoint method with standard optimization tool boxes like CVX[44].

Sinceci = gi(m) is decreasing inm, the optimal objectivefunction value in (P2-1) is non-increasing inm. Therefore,a straightforward approach to obtain an optimal solution forproblem (P2) is to first perform a search for the optimalm∗ ∈N in the following interval

[

maxi∈N

⌈mi⌉ , M

]

, (18)

7

which minimizes the optimal objective function value inproblem (P2). This IS approach can obtain the global optimalsolution for problem (P2).

B. Joint optimization with integer recovery (JR) approach

If the number of available VMsM is very large, theaforedescribed integer search algorithm may not be applicable.For the case with largeM , we can relaxm from naturenumbers to non-negative real numbers, i.e.,m ∈ R+.

For the received signalui at UE i, let yi ∈ C be the receivebeamformer. Then, the mean square error (MSE)ei is definedas [11]:

ei , E

[∥∥yHi ui − ui

∥∥2

2

]

=

∣∣∣∣∣∣

yHi∑

j∈L

hHijwij − 1

∣∣∣∣∣∣

2

+

N∑

l 6=i

∣∣∣∣∣∣

yHi∑

j∈L

hHijwlj

∣∣∣∣∣∣

2

+ σ2i |yi|

2

=

N∑

l=1

∣∣∣∣∣∣

yHi∑

j∈L

hHijwlj

∣∣∣∣∣∣

2

− 2Re

yHi∑

j∈L

hHijwij

+ σ2i |yi|

2

+ 1. (19)

Hence, for a given transmit beamformerwij , the minimummean square error (MMSE) is

emmsei = 1−

∑

j∈L

wHijhijy

mmsei , (20)

where ymmsei is the well-known MMSE receive beamformer

given by

ymmsei =

∑

j∈L hHijwij

∑

k∈N

(∑

j∈L hHijwkj

)(∑

j∈L wHkjhij

)

+ σ2i

.

(21)

Lemma 1:Each UEi’s achievable rateBi log (1 + SINRi)satisfies the following equation [45], [46]:

Bi log (1 + SINRi) = maxxi,yi

(log xi − xiei + 1)Bi, (22)

wherexi ∈ R+ is theMSE weight.With Lemma 1, we have the following Proposition.Proposition 3: For m ∈ R

+, problem (P2) can be repre-sented as:

(P2-2) minxi,yi,m,wij

mϕ+N∑

i=1

L∑

j=1

z(p)ij ‖wij‖2,1

s.t. gi(m) ≤ (log xi − xiei + 1)Bi,

∀i ∈ N , (23)

maxi∈N

⌈mi⌉ ≤ m ≤ M, m ∈ R+, (24)

N∑

i=1

‖wij‖2,1 ≤ Ej , ∀j ∈ L,

wheregi(m) andei are given by (14) and (19) respectively.Let the optimal solution for problem (P2-2) be{(m, ci, wij) :i ∈ N , j ∈ L}.

We partition the variables in problem (P2-2) into threegroups, i.e.,xi, yi and{m,wij}. The reasons that we introducetwo new groups of variablesxi andyi in problem (P2-2) are:

• If we fix xi and yi, then problem (P2-2) can be easilyrecast as a convex optimization problem w.r.t.{m,wij},sincegi(m) is a convex function.

• For the right hand side of (23), by checking the first orderoptimality condition, optimal receive beamformersyi canbe obtained using (21).

• The optimal MSE weightxi in the right hand side of (23)is given by

xi = (emmsei )−1, (25)

for fixed {m,wij} andyi.

Problem (P2-2) is convex w.r.t. each variable group whilekeeping other variable groups fixed. Therefore problem (P2-2) is much easier to solve using an alternating optimizationapproach than problem (P2). Specifically, in problem (P2-2), the optimalyi and xi can be obtained with closed-formsolutions if we fix the other variable groups as constants, and{m,wij} can be obtained by solving the following convexoptimization problem, for fixedxi andyi:

(P2-2.1) minm,wij

mϕ+

N∑

i=1

L∑

j=1

z(p)ij ‖wij‖2,1

s.t. gi(m) +Bixiei ≤ Bi(log xi + 1),

∀i ∈ N , (26)

(24) and (8).

A local optimal solution for problem (P2-2) can be achievedby this alternating optimization procedure.

Problem (P2-2.1) can be solved by applying the interiorpoint method. However, to reduce the complexity further, wepropose the following dual decomposition approach, whichobtains the closed-form solution for Problem (P2-2.1).

1) Dual decomposition approach: The Lagrangian associ-ated with problem (P2-2.1) is (we drop the superscript(p) inthis subsubsection)

L (m,wij , εi, νj) = mϕ+N∑

i=1

L∑

j=1

zij ‖wij‖2,1

+

N∑

i=1

εi (gi(m) +Bixiei −Bi(log xi + 1))

+

L∑

j=1

νj

(N∑

i=1

‖wij‖2,1 − Ej

)

, (27)

where εi ≥ 0, ∀i ∈ N and νj ≥ 0, ∀j ∈ L are theLagrange multipliers associated with constraint (26) and (8)respectively2. Then, the Lagrange dual function can be writtenas

f(εi, νj) = minm,wij

L (m,wij , εi, νj)

2Constraint (24) can be simply omitted in the Lagrangian, since it is justa constant bound form.

8

= minm,wij

mϕ+

N∑

i=1

εigi(m)

+

N∑

i=1

L∑

j=1

(zij + νj) ‖wij‖2,1 +N∑

i=1

Biεixiei

−N∑

i=1

εiBi(log xi + 1)−L∑

j=1

νjEj , (28)

wherem should satisfy (24).The dual problem is then formulated as

maxεi,νj

f(εi, νj) (29)

s.t. εi ≥ 0, ∀i ∈ N

νj ≥ 0, ∀j ∈ L.

To solve the dual problem, we first observe that the La-grangian (27) is separable overm and wij . Hence, in theLagrange dual function, the minimization can be achieved bythe following two subproblems:

minm

mϕ+

N∑

i=1

εigi(m) (30)

s.t. maxi∈N

⌈mi⌉ ≤ m ≤ M, m ∈ R+,

and

minwij

N∑

i=1

L∑

j=1

(νj + zij) ‖wij‖2,1 +N∑

i=1

Biεixiei, (31)

whereei is given by (19).Problem (30) can be easily solved numerically. And the

closed-form solution for problem (31) can be derived asfollows.

Let wi = [wi1,wi2, · · · ,wiL] ∈ CKL×1 and hi =[hi1, hi2, · · · , hiL] ∈ C

KL×1 be the network-wide beam-former and channel for UEi respectively. DefineAj ={0K , · · · ,0K︸ ︷︷ ︸

j−1

, IK ,0K , · · · ,0K︸ ︷︷ ︸

L−j

} ∈ RK×KL, where0K is the

K × K zero matrix andIK is the K × K identity matrix.Then, we have

wij = Ajwi. (32)

Thus, problem (31) is equivalent to

minwi

N∑

i=1

wHi Qiwi −

N∑

i=1

2BiεixiRe[yHi wH

i hi

], (33)

where Qi =∑

j∈L(νj + zij)AHj Aj +

(∑

l∈N yHl hlhHl yl)Biεixi. And Qi can be easily proven as

a positive definite matrix. Then, the closed-form solution forproblem (33) can be derived as

w∗i = Qi

†Re [yihi]Biεixi, (34)

whereQi† is the pseudo-inverse ofQi.

Therefore, the dual problem (29) can be solve via thefollowing gradient projection algorithm:

εi(t+ 1) =[εi(t) + π1(t)(gi(m(t))

+Bixiei(t)−Bi(log xi + 1))]+, (35)

and

νj(t+ 1) =

[

νj(t) + π2(t)

(N∑

i=1

‖wij(t)‖2,1 − Ej

)]+

,

(36)

whereπ1(t) > 0 andπ2(t) > 0 are step sizes,m(t) denotesthe number of active VMs calculated by solving problem (30)in the t-th iteration,wij(t) is the beamformer derived from(32) and (34) in thet-th iteration, and

ei(t) =

N∑

l=1

∣∣∣∣∣∣

yHi∑

j∈L

hHijwlj(t)

∣∣∣∣∣∣

2

− 2Re

yHi∑

j∈L

hHijwij(t)

+ σ2i |yi|

2+ 1. (37)

2) Implementation: With dual decomposition and gradientprojection, problem (P2-2.1) is ready to be tackled in parallel.Specifically, optimalm(t) and dual variableεi(t) can becalculated by one certain computation resource block in BBUpool, and the optimalwij(t) and dual variableνj(t) can be cal-culated by another computation resource block. Moreover, theparallel computing property of the cloud BBU pool provides anature environment to implement this parallel computing, andthe enormous computation resource in the cloud BBU poolcan help solve the problem quickly.

From subsection IV-B and subsection V-B, we see that oneapproach to solve problem (P2) includes two nested loops:an outer loop to update the weightsz(p)ij and an inner loopto solve problem (P2-2) by the iteratively weighted minimummean square error (WMMSE) method [47], [48]. To reducethe complexity, we can combine the two nested loops togetherwith in a single loop [11], as elaborated in Algorithm 2, inwhich

O(p) = m(p)ϕ+

N∑

i=1

L∑

j=1

z(p)ij

∥∥∥w(p)

ij

∥∥∥

2

2.

VI. N UMERICAL RESULTS

In this section, we conduct simulations to verify the per-formance of our proposed algorithms, and compare them withcurrent benchmark algorithms in the literature.

A. Simulation setup

In our simulation, we define the VM cost as its powerconsumption (in Watts), i.e.,ϕ = kµ3, which is measuredby [49], and adopted by [50] and [51]. Herek is a parameterdetermined by the processor structure, andµ (cycles/s) is thecomputation capacity of the VM. We choosek = 10−26

andµ = 109 in our simulation, which is consistent with themeasurements in [52]. Moreover, we assume that, for each databyte arrives at the BBU pool, 1900 processor cycles are neededto finish its baseband (cloud) processing [53], and mean packetsize is 1000 bytes.

We consider a C-RAN system of 3 RRHs, where RRH 1to 3 are located on a circle with radius 0.5 km. The 3 RRHs

9

Algorithm 2 Joint reweightedl2,1-norm relaxation and itera-tively WMMSE approach for problem (P2)

1: Initialize: w(0)ij andp = 1.

2: while∣∣O(p) −O(p−1)

∣∣ > ξ do

3: Givenw(p−1)ij , obtain receive beamformery(p)i by (21);

4: Fix w(p−1)ij andy(p)i , obtain the MSE weightx(p)

i from(20) and (25);

5: Given x(p)i , y

(p)i and z

(p)ij , utilize the proposed low-

complexity dual decomposition approach to solve theconvex optimization problem (P2-2.1), obtaining thenumber of active VMsm(p) and transmit beamformerw(p)

ij ;

6: Updatez(p)ij ;7: Let p = p+ 1.8: end while9: Integer recovery: Setm = m(p). Therefore,m∗ is chosen

from {⌊m⌋, ⌈m⌉} to minimize the optimal objective func-tion value of (P2). Thenw∗

ij can be obtained by solvingproblem (P2-1) withci = gi(m

∗).10: Output:{(m∗, c∗i ,w

∗ij) : i ∈ N , j ∈ L}.

0.5 km

RRH 1RRH 3

RRH 2

Fig. 4. Simulation topology.

are placed at equal distances apart, as shown in Figure 4. UEsare randomly, uniformly and independently distributed withinthis disk. The wireless transmission bandwidth is 10 MHz foreach UE. We adopt the path loss model used by the 3GPPspecification for Evolved Universal Terrestrial Radio Accessin [54], where the received power at a UEd km from a RRHis given by

p (dB) = 128.1 + 37.6 log10 d.

The transmit antenna gain at each RRH isϑ. The lognormalshadowing standard deviation iss.

In our simulations, we consider homogeneous RRHs withE1 = E2 = · · · = EL = E, P1 = P2 = · · · = PL = P , andS1 = S2 = · · · = SL = S. We also consider homogeneousUEs with σ1 = σ2 = · · · = σN = σ, λ1 = · · · = λN = λ,and τ1 = τ2 = · · · = τN = τ . We summarize our defaultsimulation parameters in Table I, if not specified.

B. The optimality of price adjusting

It is noted that once the sparsity of the beamforming vectoris obtained, the association relationship between the RRHsandthe UEs is also determined. Specifically,

∥∥w∗

ij

∥∥2> 0 if and

only if UE i is associated with RRHj. In Section IV, weobtain the number of active VMs and the sparse beamformingvectors by solving problem (P1). Hence, the sparsity of thebeamforming vectors, or the RRH-UE associations, can beachieved by our proposed price adjusting (PA) algorithm. Toshow the optimality of our proposed PA algorithm, we utilizethe following benchmark algorithms:

• Static Clustering (SC). This is proposed by the authorsin [11], which obtains the sparse beamforming vectors intwo steps:

1) The heuristic Algorithm 3 in [11] is used to obtainthe UE setNj that associates with RRHj, such that∥∥w∗

ij

∥∥2> 0 for i ∈ Nj and

∥∥w∗

ij

∥∥2= 0 for i ∈ Nj

c,whereNj

c is the complementary set ofNj .2) Based on the given user association setNj , a solution

to the following problem is found:

(P0-S) minm,ci,wij

mϕ+ η

N∑

i=1

L∑

j=1

‖wij‖2,1

+N∑

i=1

L∑

j=1

Pj ‖wij‖2,0

s.t. (4′), (5), (6), (7) and (8),

‖wij‖2 = 0, ∀i ∈ Njc, j ∈ L.

Problem (P0-S) is very similar with problem (P2), and,hence, can be solved by either IS or JR approach.

• Exhaustive Search (ES). This algorithm aims to find outthe best association by searching all possible RRH-UEassociations. It has an extremely high complexity. In theworst case, the number of possible RRH-UE associationrelationship can be

∏Lj=1

(∑Sj

i=0

(Ni

))

, where(Ni

)stands

for the number ofi-combinations of a set withN ele-ments. For each possible RRH-UE association case, wedenoteLi as the RRH set that serves UEi, and useLi

c

as the complementary set ofLi. The following problem(similar with problem (P0-S)) is needed to be solved (byIS or JR):

(P0-E) minm,ci,wij

mϕ+ ηN∑

i=1

L∑

j=1

‖wij‖2,1

+

N∑

i=1

L∑

j=1

Pj ‖wij‖2,0

s.t. (4′), (5), (6), (7) and (8),

‖wij‖2 = 0, ∀j ∈ Lic, i ∈ N .

• Closest Clustering (CC). This algorithm simply assumeseach UE only associates with one RRH which providesthe best channel gain. Specifically, the RRH associateswith UE i is determined by

Li = argmaxj

‖hij‖2 . (38)

10

TABLE ISIMULATION PARAMETERS

Parameter Value Parameter ValueComputation capacityµ 1× 109 cycles/s Weight η 5× 104

Static costP 5 Number of UEsN 4Maximum number of VMsM 25 VM cost ϕ 10Fronthaul capacityS 2 Number of antennasK 2Maximum transmitting powerE 1 W QoS requirementτ 50 msWhite noise power densityσ2 -174 dBm/Hz Mean arrival rateλ 10 Mb/sTransmit antenna gainϑ 15 dBi Lognormal shadowings 10 dB

However, this algorithm may easily obtaininfeasibleassociations, i.e., the number of UEs associated withcertain RRH is more than its fronthaul capacityS. Forany feasible RRH-UE association case, problem (P0-E)is also needed to be solved, whereLi is calculated by(38).

As we can learn from Section V, IS always performs betterthan JR (since IS obtains the global optimal for problem (P2),while JR only obtains the local optimal). In this subsection, toidentify the performance gap between PA and its benchmarkalgorithms above, we only utilize IS to solve problem (P2)(and its similar problems (P0-S) and (P0-E)).

We show the number of feasible associations in Figure 5under 500 channel realizations. Specifically, we fixτ = 50(ms) in Figure 5(a) and increaseλ gradually. While in Figure5(b), λ is set as10 Mb/s andτ is varied. We can concludethat CC and SC algorithms are unable to guarantee feasibilitywhen λ becomes high andτ becomes low because that theuser association sets obtained from CC and SC algorithmsare oblivious to UEs’ incoming traffic rates and their QoSrequirements.

In Figure 6, we present the system cost with different trafficrate, different QoS requirements, and differentη values respec-tively, under different algorithms. We observe that, firstly, PAalgorithm outperforms the CC and SC algorithms. Secondly,PA algorithm has much lower complexity than ES algorithm,but still has close performance with ES algorithm.

C. Allocations by PAIS

With the “cross-layer” resource allocation in both BBUpool and RRHs, an interesting question is how the allocationworks in the system by applying our proposed algorithms. InFigure 7, we present the cost and delay allocations under theQoS requirement as 20 ms, 50 ms and 80 ms respectively(with fixed λ = 10 (Mb/s)). Leveraging our proposed PAISalgorithm, we can obtain the number of active VMs underthese three different QoS requirements as 11, 10 and 10respectively. And the optimal achievable rate for the UE are11.33, 10.67 and10.19 (Mb/s) respectively.

In Figure 7(a), we show the cost of cloud processing (in theBBU pool, i.e., the first term in problem (P0)) and the costof wireless transmission (in the fronthauls and RRHs, i.e.,thesecond and third terms in problem (P0)). The first interestingobservation is, whenτ increases from 50 to 80 (ms), thecost of cloud processing remains the same. That becausethe optimal numbers of VMs forτ = 50 and τ = 80 are

5 10 15 20

350

400

450

500

λ (Mb/s)N

o. o

f fea

sibl

e as

soci

atio

ns

CCISSCISPAISESIS

(a) Different arrival rates.

20 50 80

350

400

450

500

τ (ms)

No.

of f

easi

ble

asso

ciat

ions

CCISSCISPAISESIS

(b) Different system delay constraints.

Fig. 5. Feasibility under different user association algorithms.

identical. However, the cost of wireless transmission decreasesstrictly with τ . Secondly, the cost of wireless transmissionis much lower than the cost of cloud processing (under theparameters in Table I). In Figure 7(b), we show the delay incloud processing and the delay in wireless transmission. Itcanbe learnt that, interestingly, the delay in wireless transmissionstrictly increases withτ , while the delay in cloud processingmay not vary during some increments ofτ .

D. Performance gain by cross-layer design

Most of the previous work in C-RAN just optimizes the costin wireless transmission, without any considerations of the costin cloud processing, for instance, [11] and [42]. We call thosealgorithms which consider the cost in wireless transmissionand cloud processing independently as decoupled-layer (DL)

11

5 10 15 20

100

150

200

250

λ (Mb/s)

Sys

tem

Cos

t

CCISSCISPAISESIS

(a) Different arrival rates.

20 30 40 50 60 70 80115

120

125

130

135

140

145

τ (ms)

Sys

tem

Cos

t

CCISSCISPAISESIS

(b) Different QoS requirements.

2 3 4 5 6 7 8115

120

125

130

135

140

η (104)

Sys

tem

Cos

t

CCISSCISPAISESIS

(c) Different values of the weight.

Fig. 6. System cost under different user association algorithms.

algorithms. We assume that, for the DL algorithm in oursimulations, the delay in the cloud processing queuebi and thedelay in the wireless transmission queuedi satisfy bi ≤ τi/2and di ≤ τi/2, respectively. Specifically, in our simulations,we use the following DL algorithm:

1) we obtain the optimal number of VMsm∗ =

maxi∈N

⌈τiα

τiµ−2

⌉

from (1);2) we obtain the optimal beamformersw∗

ij by solving prob-lem (P2-1) withci = λi +

2τi

.

This DL algorithm can be used in tandem with the PAalgorithm, in place of the IS and JR algorithms. Hence, in

20 50 8020

40

60

80

100

τ (ms)

Cos

t

Cloud ProcessingWireless Transmission

(a) Cost allocation.

20 50 800

10

20

30

40

50

τ (ms)

Del

ay

Cloud ProcessingWireless Transmission

(b) Delay allocation.

Fig. 7. Allocations in the system.

this subsection, we have three algorithms: PADL, PAJR andPAIS.

In Figure 8, we present the system cost with differenttraffic rate and number of UEs under different algorithms.In particular, we show the relationship between UEs’ meanarrival rate and system cost forN = 4 in Figure 8(a), andthe performance of system cost versus the number of UEsis depicted in Figure 8(b) whenλ = 10 Mb/s. We observethat, firstly, JR algorithm have very close performance withISalgorithm. In addition, IS and JR algorithms have lower systemcost than DL algorithm, since the optimal delay allocation forcloud processing and wireless transmission is not trivial (as wecan learn from Section VI-C). However, DL algorithm alwaystrivially allocates this two delays.

VII. C ONCLUSION

In this paper, we considered the joint VM activation andsparse beamforming problem in C-RAN, which has limitedfronthaul capacity. We aim to minimize the system cost of C-RAN, including VM cost (w.r.t. the number of active VMs) inthe BBU pool and RRH cost (w.r.t. the beamformer vectors).To tackle the limited fronthaul capacity constraint, we proposea price adjusting algorithm. To find out the the optimal numberof VMs, we proposed two different algorithms: integer searchand joint optimization. Simulation results suggest that our

12

5 10 15 2050

100

150

200

250

300

λ (Mb/s)

Sys

tem

Cos

t

PADLPAJRPAIS

(a) Different arrival rates.

3 4 590

100

110

120

130

140

150

160

N

Sys

tem

Cos

t

PADLPAJRPAIS

(b) Different number of UEs.

Fig. 8. System cost under different algorithms.

proposed algorithms have more robust performance and lowersystem cost than the benchmark algorithms.

With dense RRH placement in C-RAN, the huge amountof channel state information (CSI) exchange will lead toadditional overhead and even cause potential problems for C-RAN. Therefore, in future work, it would be of interest tostudy effective techniques to reduce CSI overhead. Besides,more practically, we will examine both maximum sum datarate and maximum number of associated UEs as the fronthualcapacity constraint.

APPENDIX APROOF OFTHEOREM 1

Firstly, if βj(Γ) = Sj , which implies that{m(Γ),wij(Γ)}is also a feasible solution for problem (P0).

Then, based on{m∗,w∗ij} and {m(Γ),wij(Γ)} are the

optimal solutions for problem (P0) and (P1) respectively, wehave

m(Γ)ϕ+ η

N∑

i=1

L∑

j=1

‖wij(Γ)‖22 +

L∑

j=1

(Pj + γj)βj(Γ)

≤ m∗ϕ+ η

N∑

i=1

L∑

j=1

∥∥w∗

ij

∥∥2

2+

N∑

i=1

L∑

j=1

(Pj + γj)∥∥w∗

ij

∥∥2,0

≤ m∗ϕ+ η

N∑

i=1

L∑

j=1

∥∥w∗

ij

∥∥2

2+

N∑

i=1

L∑

j=1

Pj

∥∥w∗

ij

∥∥2,0

+

L∑

j=1

γjSj

≤ m(Γ)ϕ+ ηN∑

i=1

L∑

j=1

‖wij(Γ)‖22 +

N∑

i=1

L∑

j=1

Pj ‖wij(Γ)‖2,0

+

L∑

j=1

γjSj ,

where the first inequality is based on that{m(Γ),wij(Γ)} isthe optimal solution for problem (P1), the second inequalityin based on constraint (9) in problem (P0), the third inequalityis based on that{c∗i ,w

∗ij} is the optimal solution for problem

(P0) and{m(Γ),wij(Γ)} is a feasible solution for problem(P0). Then, let’s substitute the equationβj(Γ) = Sj into theright hand side of the third inequality above, we can have

m(Γ)ϕ+ η

N∑

i=1

L∑

j=1

‖wij(Γ)‖22 +

N∑

i=1

L∑

j=1

Pj ‖wij(Γ)‖2,0

= m∗ϕ+ ηN∑

i=1

L∑

j=1

∥∥w∗

ij

∥∥2+

N∑

i=1

L∑

j=1

Pj

∥∥w∗

ij

∥∥2,0

(39)

Therefore, the theorem is now proved.

APPENDIX BPROOF OFPROPOSITION2

Let Γ′j , [γ′

1, · · · , γ′j , · · · , γ

′L]

T be a different price vectorfrom Γj , such thatγ′

j > γj and γ′k = γk, for k ∈ L\j. We

have

m(Γj)ϕ+ η

N∑

i=1

L∑

j=1

∥∥wij(Γj)

∥∥2

2+

N∑

i=1

L∑

j=1

Pj

∥∥wij(Γj)

∥∥2,0

+ γjβj(Γj) +∑

k∈L\j

γkβk(Γk)

≤ m(Γ′j)ϕ+ η

N∑

i=1

L∑

j=1

∥∥wij(Γ

′j)∥∥2

2+

N∑

i=1

L∑

j=1

Pj

∥∥wij(Γ

′j)∥∥2,0

+ γjβj(Γ′j) +

∑

k∈L\j

γkβk(Γ′k),

and

m(Γ′j)ϕ+ η

N∑

i=1

L∑

j=1

∥∥wij(Γ

′j)∥∥2

2+

N∑

i=1

L∑

j=1

Pj

∥∥wij(Γ

′j)∥∥2,0

+ γ′jβj(Γ

′j) +

∑

k∈L\j

γ′kβk(Γ

′k)

≤ m(Γj)ϕ+ η

N∑

i=1

L∑

j=1

∥∥wij(Γj)

∥∥2

2+

N∑

i=1

L∑

j=1

Pj

∥∥wij(Γj)

∥∥2,0

+ γ′jβj(Γj) +

∑

k∈L\j

γ′kβk(Γk),

where the first inequality is based on the assumption that{m(Γj),wij(Γj)} is the optimal solution for problem (P1),and the second inequality is based on the assumption that{m(Γ′

j),wij(Γ′j)} is the optimal solution for problem (P1).

13

Adding up both sides of the two inequalities above andsimplifying it, we have

(γ′j − γj)βj(Γ

′j) ≤ (γ′

j − γj)βj(Γj).

Hence, the first statement is now proved.We denote{m, wij} as a feasible solution for problem (P0),

whose feasible region is nonempty. Then, we have

m(Γj)ϕ+ η

N∑

i=1

L∑

j=1

∥∥wij(Γj)

∥∥2

2+

N∑

i=1

L∑

j=1

Pj

∥∥wij(Γj)

∥∥2,0

+ γjβj(Γj) +∑

k∈L/j

γkβk(Γj)

≤ mϕ+ η

N∑

i=1

L∑

j=1

‖wij‖22 +

N∑

i=1

L∑

j=1

Pj ‖wij‖2,0

+ γj

N∑

i=1

‖wij‖2,0 +∑

k∈L/j

γk

N∑

i=1

‖wik‖2,0 ,

Then, we obtain

βj(Γj)−N∑

i=1

‖wij‖2,0

< (mϕ+ ηN∑

i=1

L∑

j=1

‖wij‖22 +

N∑

i=1

L∑

j=1

Pj ‖wij‖2,0

+∑

k∈L/j

γk

N∑

i=1

‖wik‖2,0)/γj

≤ (Mϕ+ η

L∑

j=1

Ej +

L∑

j=1

PjSj +∑

k∈L/j

γkSk)/γj .

Therefore, if γj ≥ θj , then βj(Γj) −∑N

i=1 ‖wij‖2,0 < 1.

Sinceβj(Γj) and∑N

i=1 ‖wij‖2,0 are both integers, then we

haveβj(Γj) =∑N

i=1 ‖wij‖2,0 ≤ Sj .This completes the proof.

REFERENCES

[1] Cisco Systems, “Cisco visual networking index: Global mobile datatraffic forecast update, 2015-2020,” Cisco Systems, Technical Report,Feb. 2016.

[2] China Mobile Research Institute, “C-RAN: The road towards greenRAN,” China Mobile Research Institute, White Paper, V2.5, Oct. 2011.

[3] M. Peng, Y. Li, J. Jiang, J. Li, and C. Wang, “Heterogeneous cloud radioaccess networks: a new perspective for enhancing spectral and energyefficiencies,”IEEE Wireless Commun., vol. 21, no. 6, pp. 126–135, Dec.2014.

[4] A. Checko, H. Christiansen, Y. Yan, L. Scolari, G. Kardaras, M. Berger,and L. Dittmann, “Cloud RAN for mobile networks - a technologyoverview,” IEEE Commun. Surveys Tuts., vol. 17, no. 1, pp. 405–426,First Quarter 2015.

[5] M. Peng, Y. Sun, X. Li, Z. Mao, and C. Wang, “Recent advances incloud radio access networks: System architectures, key techniques, andopen issues,”IEEE Commun. Surveys Tuts., vol. 18, no. 3, pp. 2282–2308, Third Quarter 2016.

[6] T. Q. S. Quek, M. Peng, O. Simeone, and W. Yu,Cloud Radio AccessNetworks: Principles, Technologies, and Applications. Cambridge,U.K.: Cambridge University Press, 2016.

[7] J. Zhao, T. Q. S. Quek, and Z. Lei, “Coordinated multipoint transmissionwith limited backhaul data transfer,”IEEE Trans. Wireless Commun.,vol. 12, no. 6, pp. 2762–2775, Jun. 2013.

[8] F. Zhuang and V. K. N. Lau, “Backhaul limited asymmetric cooperationfor MIMO cellular networks via semidefinite relaxation,”IEEE Trans.Signal Process., vol. 62, no. 3, pp. 684–693, Feb. 2014.

[9] V. N. Ha, L. B. Le, and N.-D. Dao, “Energy-efficient coordinatedtransmission for Cloud-RANs: Algorithm design and trade-off,” in Proc.48th Annual Conference on Information Sciences and Systems(CISS),Princeton, NJ, USA, Mar. 2014, pp. 1–6.

[10] ——, “Cooperative transmission in cloud RAN considering fronthaulcapacity and cloud processing constraints,” inProc. IEEE WCNC,Istanbul, Turkey, Apr. 2014, pp. 1862–1867.

[11] B. Dai and W. Yu, “Sparse beamforming and user-centric clustering fordownlink cloud radio access network,”IEEE Access, vol. 2, pp. 1326–1339, Oct. 2014.

[12] M. Peng, C. Wang, V. K. N. Lau, and H. V. Poor, “Fronthaul-constrainedcloud radio access networks: insights and challenges,”IEEE WirelessCommun., vol. 22, no. 2, pp. 152–160, Apr. 2015.

[13] B. Dai and W. Yu, “Backhaul-aware multicell beamforming for downlinkcloud radio access network,” inProc. IEEE ICC Workshop, London,U.K., Jun. 2015, pp. 2689–2694.

[14] L. Liu, S. Bi, and R. Zhang, “Joint power control and fronthaul rateallocation for throughput maximization in OFDMA-based cloud radioaccess network,”IEEE Trans. Commun., vol. 63, no. 11, pp. 4097–4110,Nov. 2015.

[15] Y. Zhou and W. Yu, “Optimized backhaul compression for uplink cloudradio access network,”IEEE J. Sel. Areas Commun., vol. 32, no. 6, pp.1295–1307, Jun. 2014.

[16] S.-H. Park, O. Simeone, O. Sahin, and S. Shamai, “Joint precodingand multivariate backhaul compression for the downlink of cloud radioaccess networks,”IEEE Trans. Signal Process., vol. 61, no. 22, pp.5646–5658, Nov. 2013.

[17] T. Vu, H. Nguyen, and T. Q. S. Quek, “Adaptive compression and jointdetection for fronthaul uplinks in cloud radio access networks,” IEEETrans. Wireless Commun., vol. 63, no. 11, pp. 4565–4575, Nov. 2015.

[18] D. Liu, S. Han, C. Yang, and Q. Zhang, “Semi-dynamic user-specificclustering for downlink cloud radio access network,”IEEE Trans. Veh.Technol., vol. 65, no. 4, pp. 2063–2077, Apr. 2016.

[19] V. N. Ha, L. B. Le, and N. D. Dao, “Coordinated multipointtransmissiondesign for cloud-RANs with limited fronthaul capacity constraints,”IEEE Trans. Veh. Technol., vol. 65, no. 9, pp. 7432–7447, Sep. 2016.

[20] D. Liu, S. Han, C. Yang, and Q. Zhang, “Semi-dynamic user-specificclustering for downlink cloud radio access network,”IEEE Trans. Veh.Technol., vol. 65, no. 4, pp. 2063–2077, Apr. 2016.

[21] K. Guo, M. Sheng, J. Tang, T. Q. S. Quek, and Z. Qiu, “Exploitinghybrid clustering and computation provisioning for green C-RAN,” IEEEJ. Sel. Areas Commun., vol. 34, no. 12, pp. 4063–4076, Dec. 2016.

[22] M. A. Marotta, N. Kaminski, I. Gomez-Miguelez, L. Z. Granville, J. Ro-chol, L. DaSilva, and C. B. Both, “Resource sharing in heterogeneouscloud radio access networks,”IEEE Wireless Commun., vol. 22, no. 3,pp. 74–82, 2015.

[23] V. N. Ha and L. B. Le, “Joint coordinated beamforming andadmissioncontrol for fronthaul constrained cloud-rans,” inProc. IEEE GLOBE-COM, Austin, TX, USA, Dec. 2014, pp. 4397–4402.

[24] B. Dai and W. Yu, “Energy efficiency of downlink transmission strategiesfor cloud radio access networks,”IEEE J. Sel. Areas Commun., vol. 34,no. 4, pp. 1037–1050, Apr. 2016.

[25] J. Zhao, T. Q. S. Quek, and Z. Lei, “Heterogeneous cellular networksusing wireless backhaul: Fast admission control and large system analy-sis,” IEEE J. Sel. Areas Commun., vol. 33, no. 10, pp. 2128–2143, Oct.2015.

[26] X. Peng, J.-C. Shen, J. Zhang, and K. B. Letaief, “Joint data assignmentand beamforming for backhaul limited caching networks,” inProc. IEEEPIMRC, Washington, DC, USA, Sep. 2014, pp. 1370–1374.

[27] D. Liu and C. Yang, “Energy efficiency of downlink networks withcaching at base stations,”IEEE J. Sel. Areas Commun., vol. 34, no. 4,p. 907922, Apr. 2016.

[28] M. Tao, E. Chen, H. Zhou, and W. Yu, “Content-centric sparse multicastbeamforming for cache-enabled cloud RAN,”IEEE Trans. WirelessCommun., vol. 15, no. 9, pp. 6118–6131, Sep. 2016.

[29] J. Tang and T. Q. S. Quek, “The role of cloud computing in content-centric mobile networking,”IEEE Commun. Mag., vol. 54, no. 8, pp.52–59, Aug. 2016.

[30] P. Rost, S. Talarico, and M. C. Valenti, “The complexity-rate tradeoffof centralized radio access networks,”IEEE Trans. Wireless Commun.,vol. 14, no. 11, pp. 6164–6176, Nov. 2015.

[31] P. Rost, A. Maeder, M. C. Valenti, and S. Talarico, “Computationallyaware sum-rate optimal scheduling for centralized radio access network-s,” in Proc. IEEE GLOBECOM, San Diego, CA, USA, Dec. 2015.

14

[32] M. Qian, W. Hardjawana, J. Shi, and B. Vucetic, “Baseband processingunits virtualization for cloud radio access networks,”IEEE WirelessCommun. Lett., vol. 4, no. 2, pp. 189–192, Apr. 2015.

[33] J. Tang, W. P. Tay, and T. Q. S. Quek, “Cross-layer resource allocationwith elastic service scaling in cloud radio access network,” IEEE Trans.Wireless Commun., vol. 14, no. 9, pp. 5068–5081, Sep. 2015.

[34] M. C. Valenti, S. Talarico, and P. Rost, “The role of computational outagein dense cloud-based centralized radio access networks,” in Proc. IEEEGLOBECOM, Austin, TX, USA, Dec. 2014, pp. 1489–1495.

[35] D. Wu and R. Negi, “Effective capacity: a wireless link model forsupport of quality of service,”IEEE Trans. Wireless Commun., vol. 2,no. 4, pp. 630–643, Jul. 2003.

[36] P. J. Burke, “The output of a queuing system,”Operations Research,vol. 4, no. 6, pp. 699–704, Dec. 1956.

[37] E. Reich, “Waiting times when queues are in tandem,”The Annals ofMathematical Statistics, vol. 28, no. 3, pp. 768–773, Sep. 1957.

[38] D. Bertsekas and R. Gallager,Data Networks, 2nd ed. New Jersey,U.S.: Prentice Hall, 1992.

[39] H. Mohimani, M. Babaie-Zadeh, and C. Jutten, “A fast approach forovercomplete sparse decomposition based on smoothedl

0 norm,” IEEETrans. Signal Process., vol. 57, no. 1, pp. 289–301, Jan. 2009.

[40] E. J. Candes, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity byreweightedl1 minimization,” Journal of Fourier Analysis and Applica-tions, vol. 14, no. 5, pp. 877–905, 2008.

[41] D. Donoho, “Compressed sensing,”IEEE Trans. Inf. Theory, vol. 52,no. 4, pp. 1289–1306, Apr. 2006.

[42] Y. Shi, J. Zhang, and K. B. Letaief, “Group sparse beamforming forgreen cloud-RAN,”IEEE Trans. Wireless Commun., vol. 13, no. 5, pp.2809–2823, May 2014.

[43] A. Wiesel, Y. C. Eldar, and S. S. (Shitz), “Linear precoding via conicoptimization for fixed MIMO receivers,”IEEE Trans. Signal Process.,vol. 54, no. 1, pp. 161–176, Jan. 2006.

[44] CVX Research, Inc., “CVX: Matlab software for disciplined convexprogramming, version 2.0,” http://cvxr.com/cvx, Aug. 2012.

[45] M. Razaviyayn, M. Hong, and Z.-Q. Luo, “Linear transceiver designfor a MIMO interfering broadcast channel achieving max-minfairness,”in Proc. Asilomar Conference on Signals, Systems and Computers(ASILOMAR), Pacific Grove, CA, Nov. 2011, pp. 1309–1313.

[46] W.-C. Liao, M. Hong, H. Farmanbar, X. Li, Z.-Q. Luo, and H. Zhang,“Min flow rate maximization for software defined radio accessnetwork-s,” IEEE J. Sel. Areas Commun., vol. 32, no. 6, pp. 1282–1294, Jun.2014.

[47] Q. Shi, M. Razaviyayn, Z.-Q. Luo, and C. He, “An iteratively weightedMMSE approach to distributed sum-utility maximization fora MIMOinterfering broadcast channel,”IEEE Trans. Signal Process., vol. 59,no. 9, pp. 4331–4340, Sep. 2011.

[48] J. Tang, W. P. Tay, T. Q. S. Quek, and B. Liang, “Towards systemcost minimization in cloud radio access network,” inProc. AsilomarConference on Signals, Systems and Computers (ASILOMAR), PacificGrove, CA, USA, Nov. 2015, pp. 1460–1464.

[49] S. Kaxiras and M. Martonosi,Computer Architecture Techniques forPower-Efficiency. Morgan & Claypool, 2008.

[50] J. Tang, W. P. Tay, and Y. Wen, “Dynamic request redirection andelastic service scaling in cloud-centric media networks,”IEEE Trans.Multimedia, vol. 16, no. 5, pp. 1434–1445, Aug. 2014.

[51] M. H. Chen, B. Liang, and M. Dong, “A semidefinite relaxation approachto mobile cloud offloading with computing access point,” inProc. IEEESPAWC, Stockholm, Sweden, Jun. 2015, pp. 186–190.

[52] A. P. Miettinen and J. K. Nurminen, “Energy efficiency ofmobile clientsin cloud computing,” inProc. USENIX HotCloud, Boston, MA, USA,Jun. 2010, pp. 1–7.

[53] L. Chen and N. Li, “On the interaction between load balancing and speedscaling,” IEEE J. Sel. Areas Commun., vol. 33, no. 12, pp. 2567–2578,Dec. 2015.

[54] 3GPP, “LTE; Evolved universal terrestrial radio access (E-UTRA); Radiofrequency (RF) requirements for LTE Pico Node B (release 9),” 3rdGeneration Partnership Project (3GPP), TS 36.931, May 2011, v9.0.0.

Jianhua Tang (S’11-M’15) received the B.E. degreein communication engineering from NortheasternUniversity, China, in 2010, and the Ph.D. degree inelectrical and electronic engineering from NanyangTechnological University, Singapore, in 2015. Hewas a Post-Doctoral Research Fellow with the Sin-gapore University of Technology and Design from2015 to 2016. He is with the School of Commu-nication and Information Engineering, ChongqingUniversity of Posts and Telecommunications, China.Currently, he is a Research Assistant Professor with

the Department of Electrical and Computer Engineering, Seoul NationalUniversity, Korea. His research interests include cloud computing, content-centric network, and cloud radio access network.

Wee Peng Tay (S’06-M’08-SM’14) received theB.S. degree in Electrical Engineering and Mathemat-ics, and the M.S. degree in Electrical Engineeringfrom Stanford University, Stanford, CA, USA, in2002. He received the Ph.D. degree in ElectricalEngineering and Computer Science from the Mas-sachusetts Institute of Technology, Cambridge, MA,USA, in 2008. He is currently an Assistant Professorin the School of Electrical and Electronic Engineer-ing at Nanyang Technological University, Singapore.His research interests include distributed inference

and signal processing, sensor networks, social networks, information theory,and applied probability.

Dr. Tay received the Singapore Technologies Scholarship in1998, theStanford University President’s Award in 1999, the Frederick Emmons TermanEngineering Scholastic Award in 2002, and the Tan Chin Tuan ExchangeFellowship in 2015. He is the coauthor of the best student paper award atthe Asilomar conference on Signals, Systems, and Computersin 2012, andcoauthor for the IEEE Signal Processing Society Young Author Best PaperAward in 2016. He is currently an Associate Editor for the IEEE Transactionson Signal Processing, serves on the MLSP TC of the IEEE SignalProcessingSociety, and is the chair of DSNIG in IEEE MMTC. He has also served as atechnical program committee member for various international conferences.

15

Tony Q.S. Quek (S’98-M’08-SM’12) received theB.E. and M.E. degrees in Electrical and ElectronicsEngineering from Tokyo Institute of Technology, re-spectively. At MIT, he earned the Ph.D. in ElectricalEngineering and Computer Science. Currently, he isa tenured Associate Professor with the SingaporeUniversity of Technology and Design (SUTD). Healso serves as the Associate Head of ISTD Pillarand the Deputy Director of the SUTD-ZJU IDEA.His main research interests are the application ofmathematical, optimization, and statistical theories

to communication, networking, signal processing, and resource allocationproblems. Specific current research topics include heterogeneous networks,wireless security, internet-of-things, and big data processing.

Dr. Quek has been actively involved in organizing and chairing sessions,and has served as a member of the Technical Program Committeeas wellas symposium chairs in a number of international conferences. He is servingas the Workshop Chair for IEEE Globecom in 2017, the TutorialChair forthe IEEE ICCC in 2017, and the Special Session Chair for IEEE SPAWC in2017. He is currently an elected member of IEEE Signal Processing SocietySPCOM Technical Committee. He was an Executive Editorial CommitteeMember for the IEEE TRANSACTIONS ON WIRELESSCOMMUNICATIONS,an Editor for the IEEE TRANSACTIONS ONCOMMUNICATIONS and an Editorfor the IEEE WIRELESSCOMMUNICATIONS LETTERS. He is a co-author ofthe book “Small Cell Networks: Deployment, PHY Techniques,and ResourceAllocation” published by Cambridge University Press in 2013 and the book“Cloud Radio Access Networks: Principles, Technologies, and Applications”by Cambridge University Press in 2017.

Dr. Quek was honored with the 2008 Philip Yeo Prize for OutstandingAchievement in Research, the IEEE Globecom 2010 Best Paper Award, the2012 IEEE William R. Bennett Prize, the IEEE SPAWC 2013 Best StudentPaper Award, the IEEE WCSP 2014 Best Paper Award, the 2015 SUTDOutstanding Education Awards – Excellence in Research, the2016 ThomsonReuters Highly Cited Researcher, and the 2016 IEEE Signal ProcessingSociety Young Author Best Paper Award.

Ben Liang (S’94-M’01-SM’06) received honors-simultaneous B.Sc. (valedictorian) and M.Sc. de-grees in Electrical Engineering from PolytechnicUniversity in Brooklyn, New York, in 1997 andthe Ph.D. degree in Electrical Engineering with aminor in Computer Science from Cornell Universityin Ithaca, New York, in 2001. In the 2001 - 2002academic year, he was a visiting lecturer and post-doctoral research associate with Cornell University.He joined the Department of Electrical and Comput-er Engineering at the University of Toronto in 2002,

where he is now a Professor. His current research interests are in networkedsystems and mobile communications. He has served as an editor for the IEEETransactions on Communications since 2014, and he was an editor for theIEEE Transactions on Wireless Communications from 2008 to 2013 and anassociate editor for Wiley Security and Communication Networks from 2007to 2016. He regularly serves on the organizational and technical committeesof a number of conferences. He is a senior member of IEEE and a memberof ACM and Tau Beta Pi.