systematic resource allocation in cloud ran with caching...

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 11, NOVEMBER 2019 7755

Systematic Resource Allocation in Cloud RANWith Caching as a Service Under

Two TimescalesJianhua Tang , Member, IEEE, Tony Q. S. Quek , Fellow, IEEE, Tsung-Hui Chang , Senior Member, IEEE,

and Byonghyo Shim , Senior Member, IEEE

Abstract— Recently, cloud radio access network (C-RAN) withcaching as a service (CaaS) was proposed to merge the func-tionalities of communication, computing, and caching (CC&C)together. In this paper, we dissect the interactions of CC&Cin C-RAN with CaaS from two dimensions: physical resourcedimension and time dimension. In the physical resource dimen-sion, we identify how to segment the baseband unit (BBU) poolresources (i.e. computation and storage) into different types ofvirtual machines (VMs). In the time dimension, we addresshow the long-term resource segmentation in the BBU poolimpacts on the short-term transmit beamforming at the remoteradio heads. We formulate the problem as a stochastic mixed-integer nonlinear programming (SMINLP) to minimize thesystem cost, including the server cost, VM cost and wireless

Manuscript received December 12, 2018; revised April 16, 2019 andJune 20, 2019; accepted August 4, 2019. Date of publication August 13, 2019;date of current version November 19, 2019. This work was supported in partby the Korea Research Fellowship Program through the National ResearchFoundation of Korea (NRF) funded by the Ministry of Science and ICT(Grant No. 2016H1D3A1938245), in part by the Hunan Provincial Science& Technology Project Foundation (2018TP1018, 2018RS3065), in part bythe SUTD-ZJU Research Collaboration under Grant SUTD-ZJU/RES/01/2016, in part by the SUTD-ZJU Research Collaboration under Grant SUTD-ZJU/RES/05/2016, in part by the NSFC, China, under Grant 61571385 andGrant 61731018, in part by the Shenzhen Fundamental Research Fund underGrant No. ZDSYS201707251409055 and No. KQTD2015033114415450, andin part by the MSIT, Korea, under the ITRC support program (IITP-2019-2017-0-01637) supervised by the IITP and the Samsung Research Funding& Incubation Center for Future Technology of Samsung Electronics underGrant SRFC-IT1901-17. This article was presented in part at the 17thIEEE International Workshop on Signal Processing Advances in WirelessCommunications (SPAWC), Edinburgh, U.K., July 2016 [1], and in part atthe 18th IEEE SPAWC, Sapporo, Japan, July 2017 [2]. The associate editorcoordinating the review of this article and approving it for publication wasS. Bhashyam. (Corresponding author: Byonghyo Shim.)

J. Tang is with the Hunan Provincial Key Laboratory of Intelligent Com-puting and Language Information Processing, College of Information Scienceand Engineering, Hunan Normal University, Changsha 410081, China. Hewas with the Institute of New Media and Communications, Department ofElectrical and Computer Engineering, Seoul National University, Seoul 08826,South Korea (e-mail: [email protected]).

T. Q. S. Quek is with Information Systems Technology and Design Pillar,Singapore University of Technology and Design, Singapore 487372, and alsowith the Department of Electronic Engineering, Kyung Hee University, Yongin17104, South Korea (e-mail: [email protected]).

T.-H. Chang is with the School of Science and Engineering, The ChineseUniversity of Hong Kong, Shenzhen, Shenzhen 518172, China, and alsowith the Shenzhen Research Institute of Big Data, Shenzhen 518172, China(e-mail: [email protected]).

B. Shim is with the Institute of New Media and Communications, Depart-ment of Electrical and Computer Engineering, Seoul National University,Seoul 08826, South Korea (e-mail: [email protected]).

Color versions of one or more of the figures in this article are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TCOMM.2019.2934854

transmission cost. After a series of approximation, includingsample average approximation, successive convex approximation,and semidefinite relaxation, the SMINLP is approximated asa global consensus problem. The alternating direction methodof multipliers (ADMM) is utilized to obtain the solution in aparallel fashion. Simulation results verify the convergence of ourproposed algorithm, and also confirm that the proposed schemeis more cost-saving than that without considering the integrationof CC&C.

Index Terms— C-RAN, caching as a service, two timescales,semidefinite relaxation (SDR), alternating direction method ofmultipliers (ADMM).

I. INTRODUCTION

AN approach to merge the radio access network (RAN)and cloud computing together, often referred to as

cloud RAN (C-RAN), has been received much attention asa prospective architecture for the 5th generation wirelesssystems (5G). C-RAN consists of three components: basebandunit (BBU) pool, fronthaul links, and remote radio heads(RRHs). The key feature of C-RAN is to decouple the base-band processing functionalities from RRHs and migrate thosefunctionalities into the centralized cloud based BBU pool,thereby achieving reduction of the capital expense (CAPEX)and improvement of the system energy-efficiency [3], [4].

There are two main logical functionalities in originalC-RAN: baseband processing and signal transceiving. Thebaseband processing functionality is executed in the BBU pooland the signal transceiving functionality is performed at theRRHs. Due to the cloud-based software-defined environmentof BBU pool, new functionality can be easily added to C-RAN.For example, in [5] and [6], we proposed that the contentcaching can be incorporated in the BBU pool as an extendedfunctionality of C-RAN. We call this scheme as C-RAN withcaching as a service (CaaS).

Under C-RAN with CaaS, popular contents, e.g., popularvideos and latest news, are placed in the BBU pool thatcontains a large number of general purpose servers. C-RANwith CaaS can greatly extend the capability of the conventionalFemtoCaching [7], [8] whose primary purpose is to storesome popular contents to the base stations (BSs). In particular,the most attractive benefit of CaaS is that the storage volumecan be adjusted dynamically and elastically, thanks to thesoftware-defined cloud environment. This is in contrast toFemtoCaching where the storage volume in the each BS is

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7756 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 67, NO. 11, NOVEMBER 2019

always low and fixed. Owing to the high storage volume inBBU pool, we can avoid the complicated content placementproblem. In FemtoCaching, the operator has to decide whichcontent to store into which BS [9]. While in C-RAN withCaaS, the operator may just simply cache the popular filesas many as possible into the allocated storage. In addition,C-RAN with CaaS can also help reduce the latency inC-RAN [5], since, 1) the content (cached in the BBU pool) isnow close to the end users; 2) the traffic that travels into thebackhaul is now decreased.

Due to the introduction of CaaS, C-RAN extends its func-tionalities from communication and computing to commu-nication, computing, and caching (CC&C). To be specific,communication refers to the transmission and reception ofwireless signal at the RRHs side, computing means the base-band processing in the BBU pool, and caching indicates theproposed CaaS in the BBU pool. In the C-RAN architecture,these three functionalities are not independent and, in fact,they are tightly coupled and intertwined with each other.We elaborate these two types of interactions in what follows.

The first type of interaction occurs inside the BBU pool.In the BBU pool, there are two main resources: computa-tion resource (e.g., CPU and memory) and storage resource(e.g., hard drives). In each general purpose server, thesetwo types of resources are limited. To support computingand caching functionalities in the BBU pool, the operatorneeds to segment and virtualize resources into two dif-ferent types of virtual machines (VMs): the computation-intensive VM (CIVM) to conduct the baseband processing tasksand the storage-intensive VM (SIVM) dedicated for contentcaching [10], [11]. While CIVM contains a large amountof computation resources and a small amount of storageresources, SIVM contains a small amount of computationresources, together with a large amount of storage resources.

The second type of interaction occurs between the BBUpool and RRHs. This interaction causes two issues: theresource allocation issue and the timescale issue. The resourceallocation issue states how resource segmentation in the BBUpool affects the beamforming at RRHs. Such issue has beeninvestigated in [1]. The second issue which has not beenconsidered in [1] is the timescale issue. In particular, resourcesegmentation in the BBU pool is usually executed oncein a large/slow timescale span (e.g, a half hour),1 whilethe beamforming is performed in every small/fast timescalespan (e.g, several milliseconds), i.e., channel coherence time,owing to the nature of wireless channels. Therefore, it is ofimportance to investigate the interactions of CC&C under twotimescales.

A. Main Contributions

The main purpose of this paper is to jointly address theresource segmentation in the BBU pool and beamforming atthe RRHs under two timescales. Our main contributions areas follows:

1Because that, firstly, the BBU pool supports many users, and the statisticalcharacterization of these users, e.g., the number of total users, changes slowly;Secondly, the management of VMs has to be in a large timescale span to savethe management overhead.

• We introduce CaaS in the BBU pool. By this setting,we identify and quantify the resource segmentation andbeamforming under two timescales to minimize the sys-tem cost, which covers the server cost, VM cost, andwireless transmission cost. The server cost and VM costare with respect to (w.r.t.) the resource segmentationvariables, and wireless transmission cost is w.r.t. thebeamforming vectors. The benefit (i.e., system cost reduc-tion) of introducing CaaS in the BBU pool is verified bysimulation.

• We formulate the two timescales problem as a sto-chastic mixed-integer nonlinear programming (SMINLP).To make the problem tractable, we first leverage thesample average approximation (SAA) [12], [13] to trans-fer the SMINLP problem to a mixed-integer nonlinearprogramming (MINLP) problem. We then employ thesuccessive convex approximation (SCA) [14]–[16] andsemidefinite relaxation (SDR) [17], [18] approaches totackle the MINLP problem. In this approach, instead ofsolving the MINLP directly, we solve a series of large-scale convex optimization problems iteratively. Fromanalysis, we prove that the SDR is asymptotically tight.

• To further reduce the complexity of solving thelarge-scale convex optimization problem, we convertthe large-scale convex optimization problem into aglobal consensus problem. In solving the problem,we exploit the alternating direction method of multipliers(ADMM) [19], [20].

• From simulation results, we empirically verify the conver-gence of our proposed algorithm. We also show that ourproposed algorithm outperforms benchmark algorithms,e.g., our previous work [2], under a variety of scenarios.

B. Prior Arts

By introducing cloud computing in the BBU pool, C-RANacquires a big leap over the conventional RAN. Recently, therehave been some efforts to analyze the benefits of cloud com-puting in C-RAN. For instance, the authors in [21] and [22]studied the elastic computation provisioning in C-RAN BBUpool, under the assumption that the computation resourcein the BBU pool is unlimited. In [23], the virtual machineallocation problem in the BBU pool was investigated byjointly considering the limited fronthaul capacity. In [24],the computational requirement of the central processed uplinksignals was quantified. Recently, problems under the limitedcomputation resource constraint in the BBU pool have beenexplored in [25] and [26], which make the research worksmore practical.

While these works concern mainly on the computing capa-bility in C-RAN BBU pool, approaches utilizing the software-defined environment of BBU pool to make C-RAN moreflexible have been proposed as well. On one hand, based onthe software-defined environment, C-RAN operator is able todecide each functionality module to be realized in whetherBBU pool or RRHs, based on different application scenar-ios. For example, the authors in [27] leveraged graph-basedframework to solve the functionality splitting and placement

TANG et al.: SYSTEMATIC RESOURCE ALLOCATION IN CLOUD RAN 7757

problem, aiming at reducing the fronthauling cost. On the otherhand, with the software-defined environment, the functionalityof BBU pool can be extended as well. For instance, the mobilecloud computing service was embedded in the BBU poolin [28] and [29]. Also in [1], we proposed CaaS as a possiblefunctionality extension, to place the popular contents in theBBU pool.

As mentioned, in order to jointly optimize the resources inthe BBU pool and wireless transmission, there is always a twotimescales problem. This means that the resource allocationsin the BBU pool and wireless transmission are not executedin the same timescale. Recently, the two timescales problemunder many different scenarios has received decent amountof attention. For instance, the joint problem of long-termuser association and short-term beamforming was studiedin [30] and the long-term resource block control and short-term user scheduling control was considered in [31]. Also,the two timescales resource management problem in hyper-dense small cell networks was investigated in [32]. In [33],the two timescales optimization framework was examinedto maximize the profit of transcoding service in the cloud.However, the two timescales problems in C-RAN have notbeen considered in previous works [1], [27].

Recently, we studied the two timescales problem underC-RAN with CaaS in [2]. However, the solution in [2] isbased on a complicated algorithm involving three loops ofiterative updates, i.e., an outer loop for the Difference oftwo Convex (DC) algorithm, a middle loop for ADMM,and an inner loop for the weighted minimum mean squareerror (WMMSE) algorithm. In this paper, we propose a newapproach based on SCA, SDR and ADMM. The proposedapproach yields an efficient parallel algorithm with only twoiterative loops, i.e., an outer loop for SCA + SDR algorithmand an inner loop for ADMM algorithm. We compare our newapproach with our previous work [2] and demonstrate that ournew approach not only reduces the time to solve our problem,but also produces a better solution (i.e., a lower systemcost).

C. Notations

We use calligraphy letters to represent the sets, boldfacelower case letters to denote the vectors, and boldface uppercase letters to denote the matrices. ‖x‖2, ‖x‖0, and ‖X‖F

stand for the Euclidean norm, l0-norm and Frobenius normrespectively. (·)H represents for the conjugate transpose. E[·]stands for the statistical expectation. X � 0 means that matrixX is Hermitian positive semidefinite. N, C and R+ stand forthe natural numbers, complex numbers and non-negative realnumbers respectively. The notation A\B denotes the set Awith its subset B removed. |A| represents the cardinality ofset A. �x� stands for the largest integer smaller than or equalsx and �x� stands for the smallest integer larger than orequals x. The log(·) function is the logarithm function withbase 2.

The remainder of this paper is organized as follows.We present the system model and problem formulation inSection II, and then introduce the SAA method in Section III.

Fig. 1. The two timescales system.

In Section IV, we leverage SCA and SDR to solve ourproblem in a centralized form, and followed by an ADMMbased parallel fashion solution in Section V. In Section VI,we discuss the approach to recover the integers. We presentsimulation results in Section VII, and conclude the paperin Section VIII.

II. SYSTEM MODEL AND PROBLEM FORMULATION

We consider a setup that partitions the time into twotimescales, including slow timescale and fast timescale. Eachslow timescale contains T fast timescale slots, as shownin Fig. 1. At the beginning of each slow timescale, the BBUallocates CIVM and SIVM in each server. The allocationof VMs remains unchanged within the slow timescale slot.In each fast timescale slot, the BBU pool generates theoptimal beamforming vectors for the downlink transmission.We assume that the channel is static within each fast timescaleslot and varies from one slot to another. Similarly, we assumethat the VM allocation can change from one slow timescaleslot to another. Therefore, without loss of generality, we onlyconsider one slow timescale.

In what follows, we introduce the system model and theproblem formulation on the basis of two different timescale.

A. Slow Timescale: BBU Resource Segmentation

The collection of general purpose servers in the BBU poolis denoted as L = {1, · · · , L}. We list the configuration ofservers and VMs as follows:

• Server configuration: We denote the amount of totalcomputation and storage resource in server l as Cl andSl respectively. In particular, Cl captures the number ofCPU cycles that can be finished in a unit time in server land Sl is the number of data bits can be stored in server l.

• CIVM configuration: In each CIVM, the amount ofcomputation resources and storage resources are config-ured as Cc and Sc respectively. Note that Cc < Cl andSc < Sl, ∀l ∈ L.


Fig. 2. An intuitive description for resource segmentation in each server.

• SIVM configuration: In each SIVM, the amount of com-putation resources and storage resources are representedby Cs and Ss respectively, where Cs < Cc < Cl andSc < Ss < Sl, ∀l ∈ L.

All of these configurations, for both servers and VMs, arepredefined and fixed. At the beginning of each slow timescale,we allocate nl ∈ N CIVMs and ml ∈ N SIVMs at serverl respectively. Therefore, we have the following resourceallocation constraints in the cloud BBU pool:

(nlCc + mlC

s) ≤ Cl, ∀l ∈ L, (1a)

(nlSc + mlS

s) ≤ Sl, ∀l ∈ L, (1b)

nl ∈ N, ml ∈ N, ∀l ∈ L. (1c)

Constraint (1a) means that the total computation resourcesconfigured in all VMs of server l is limited by the underlyingcomputation resource of server l. Similarly, constraint (1b)implies that the total storage resources configured in all VMsof server l is limited by the underlying storage resource ofserver l. The resource segmentation problem is illustratedin Fig. 2 intuitively.

Remark 1: The reasons that we need multiple type andmultiple number of VMs, instead of a single resource pool,are as follows. Firstly, we envision that there will be manyservices/ functionalities (not only CaaS) incorporated in theBBU pool [6], e.g., mobile edge computing as a service. In thiscase, it is beneficial to use different types of VMs to supportdifferent services accordingly. Secondly, from the granularityperspective, VMs are easier to be maintained and managed,compared to the single resource pool.

The main objective of CaaS is to cache popular contents inthe BBU pool of C-RAN. We assume all users are in the sameinterest group, which means that the file popularity distributionis identical for all users. This file popularity distributionremains unchanged within each slow timescale span, e.g., onehour, but may vary between slow timescale slots. The expectedfile length to be transmitted to each user within one slowtimescale span is denoted as F .

We assume that the baseband processing can be done inany active CIVM. Therefore, the total computation capacityover all active CIVMs should be no less than the number ofexpected CPU cycles needed for the baseband processing in

a slow timescale span. That is,

FI

G≤∑l∈L

nlCc, (2)

where I is the number of users, and G is the number of bitsthat can be processed per CPU cycle.

B. Fast Timescale: Beamforming

We consider the joint transmission technique in C-RANdownlink [34], that means each user equipment’s (UE’s) datacan be shared among all the coordinated RRHs. Suppose thatthere are I single-antenna UEs and J coordinated RRHs, eachwith K antennas, in the C-RAN cluster. We denote the set ofall UEs and all coordinated RRHs as I = {1, · · · , I} andJ = {1, · · · , J} respectively.

Let ui(t) be the data symbol for the i-th UE during thet-th fast timescale slot with E[|ui(t)|2] = 1, ∀t ∈ T �{1, · · · , T }, and wij(t) ∈ C

K denote the transmit beamformerfor UE i from RRH j during the t-th fast timescale slot. Theblock fading channel from RRH j to UE i during the t-thfast timescale slot is denoted as hij(t), where hij(t) ∈ CK ,for i ∈ I and j ∈ J . Suppose that hij(t) is drawn fromcertain random distribution, and the distribution is known inadvance by the C-RAN operator. Denote h as the collection ofall random channels hij(t), i.e., h = {hij(t)}. In this setup,the received signal at UE i during the t-th fast timescale slot is

ui(t) =∑j∈J

hij(t)Hwij(t)ui(t)

+I∑

k �=i

∑j∈J

hij(t)Hwkj(t)uk(t) + δi(t),

where the first term is the desired signal for UE i, the secondterm is the interference to UE i, and δi(t) ∼ CN (0, σ2

i )is the additive white Gaussian noise (AWGN) at UE i. Thecorresponding signal-to-interference-plus-noise ratio (SINR) atUE i over the t-th fast timescale slot is

SINRi(t) =|∑

j∈J hij(t)Hwij(t)|2

σ2i +

∑Ik �=i |

∑j∈J hij(t)Hwkj(t)|2

, (3)

and RRH j has its maximum transmitting power Ej constraint:∑i∈I

wij(t)Hwij(t) ≤ Ej , ∀j ∈ J , ∀t ∈ T . (4)

Further, we define the following cache-based adaptive rate(CBAR) requirement to link up communication and caching:

max[riη, ri

F∑l∈L mlSs

]≤ log(1 + SINRi(t)),

∀i ∈ I, ∀t ∈ T , (5)

where riη > 0 is the minimum transmission rate (MTR)requirement, 0 < η ≤ 1 is the scaling factor. The intuitionbehind the CBAR requirement is that (let’s assume η = 1for description simplicity), if the sum of storage resources inall active SIVMs is smaller than the expected file length tobe transmitted, i.e.,

∑l∈L mlS

s < F , then the system shouldincrease the wireless transmission rate to compensate for the


potential file retrieval delay in the core network; if the sum ofstorage resources is lager than the expected file length to betransmitted, the system can just keep the MTR ri to save thetransmit power.

It is worth mentioning that due to the CBAR constraint (5),the BBU resource segmentation and RRH beamforming arecoupled. In other words, the slow timescale VM allocationcan directly affect the fast timescale beamforming design.

However, the CBAR constraint may not be always satisfied,for example, when channel is in deep fade. Considering thefact that many wireless applications can tolerate a small outageprobability, we introduce the following probabilistic CBARconstraint

Pr

{max

[riη, ri

F∑l∈L mlSs

]≤ log(1 + SINRi(t))

}≥ε,

(6)

where ε ∈ [0, 1) is the maximum tolerable outage probability.The probability in (6) is taken w.r.t. the random variables h.

C. Problem Formulation

In this subsection, we describe the problem formulation bydefining the system cost first. We denote wi(t) = [wi1(t);wi2(t); · · · ; wiJ (t)] ∈ CJK×1, w(t) = [w1(t); w2(t); · · · ;wI(t)] ∈ CIJK×1, hi(t) = [hi1(t); hi2(t); · · · ; hiJ (t)] ∈CJK×1, h(t) = [h1(t); h2(t); · · · ; hI(t)] ∈ CIJK×1, m =[m1 m2 · · · mL]T ∈ NL, and n = [n1 n2 · · · nL]T ∈ NL,where wi(t) defines all beamforming vectors to UE i, w(t)defines all beamforming vectors in the system, hi(t) reprenstsall channels to UE i, h(t) defines all channels in the system,and m and n are systemwide VM allocation vectors.

There are three types of costs in the system:• Server cost: A server has two modes, i.e., sleep and

active. The condition to transit from sleep mode to activemode is that at least one VM is turned on in thisserver. To capture the mode transition at server l, we use‖nl + ml‖0, which means that server l is in sleep modewhen nl +ml = 0, or in active mode when nl +ml ≥ 1.A static cost Pl is incurred once server l is activated.Then, the total server cost is

∑l∈L ‖nl + ml‖0 Pl.

• VM cost: A static cost associated with each active VM.Let P c and P s be the cost of a CIVM and SIVM respec-tively, then the total VM cost is

∑l∈L(nlP

c + mlPs).

• Wireless transmission cost: We first define the FastTimescale wireless transmission Cost (FTC) in time slott ∈ T as U(w(t)). Note that U(w(t)) is a continuousdifferentiable, non-negative, convex and increasing func-tion w.r.t.

∑i∈I

∑j∈J ‖wij(t)‖2

2. In the fast timescale,we aim to minimize the FTC in each fast timescale slot t.That is,

minw(t)

U(w(t)). (7)

Then, the Slow Timescale wireless transmission Cost(STC) is defined as the time average of the minimumFTC [30]. That is,

1T

∑t∈T

minw(t)

U(w(t)). (8)

At the beginning of each slow timescale, we minimize thesystem cost, including server cost, VM cost and STC. Thecorresponding optimization problem can be formulated as

(P0) minn,m

{∑l∈L

‖nl + ml‖0 Pl +∑l∈L

(nlPc + mlP

s)

+1T

∑t∈T

minw(t)

U(w(t))

}

s.t. (1a), (1b), (1c), (2), (4), and (6).

Since we assume that the system design of any two con-secutive slow timescale slots are independent, we need tofind a solution of problem (P0) at the beginning of eachslow timescale slot. However, problem (P0) cannot be solvedat the beginning of a slow timescale slot directly, becausechannels in the future, i.e., h(t), for t ∈ {2, · · · , T }, are notavailable at that moment. To deal with this difficulty and hencemake problem (P0) tractable, we propose an approximationtechnique in the next section.

III. SAMPLE AVERAGE APPROXIMATION

First of all, if we consider (7) as a random variable, then theSTC in (8) is just the sample average of T random variables.Thus, we approximately have

1T

∑t∈T

minw(t)

U(w(t)) ≈ Eh[minw

U(w)], (9)

where h includes all random channels (drawn from a cer-tain distribution) and w includes all beamformers based onthe channel. Then, problem (P0) becomes a two-stage sto-chastic optimization problem with the following objectivefunction:∑l∈L

‖nl+ml‖0 Pl+∑l∈L

(nlPc+mlP

s)+Eh[minw

U(w)]. (10)

To solve the two-stage stochastic optimization problem,we exploit the sample average approximation (SAA). Themain idea of SAA is to approximate the expectation of a ran-dom variable by its sample average. To handle the probabilisticCBAR constraint (6) by SAA, the number of samples V shouldsatisfy [35], [36]

V ≥⌈

1ε

(IJK − 1+ln

1θ

+

√2(IJK − 1) ln

1θ

+ ln2 1θ

)⌉,

(11)

for any θ ∈ (0, 1). In particular, any solution to

max[riη, ri

F∑l∈L mlSs

]≤ log(1 + SINRi(v)),

∀i ∈ I, ∀v ∈ V , (12)

will satisfy the probabilistic CBAR constraint (6) with proba-bility at least 1 − θ, where V = {1, · · · , V }.


Then, problem (P0) can be approximated as

(P0-A) minn,m

{∑l∈L


(nlPc + mlP

s)

+1V

∑v∈V

minw(v)

U(w(v))

}

s.t.∑i∈I

wij(v)Hwij(v) ≤ Ej , ∀j ∈ J , ∀v ∈ V , (13)

(1a), (1b), (1c), (2), and (12),

where h(v) is the v-th sample generated by the C-RAN oper-ator (from a distribution known by the operator). To be morespecific, h(t) in problem (P0) is the real channel information,while h(v) in problem (P0-A) is the system generated sampleto approximate h(t).

By combining two minimizations in problem (P0-A),we obtain the simplified formulation

(P0-C) minn,m,w

∑l∈L


(nlPc + mlP

s)

+1V

∑v∈V

U(w(v))

s.t. (nlCc + mlC

s) ≤ Cl, ∀l ∈ L,

(nlSc + mlS

s) ≤ Sl, ∀l ∈ L,

nl ∈ N, ml ∈ N, ∀l ∈ L,F I

G≤∑l∈L

nlCc,

∑i∈I

wij(v)Hwij(v) ≤ Ej , ∀j ∈ J , ∀v ∈ V ,

max[riη, ri

F∑l∈L mlSs

]≤ log(1 + SINRi(v)), ∀i ∈ I, ∀v ∈ V , (14)

where w = [w(1) · · ·w(V )]. Problem (P0-C) is an MINLP andwe denote {n∗, m∗, w∗} as its optimal solution.

Remark 2: When compared to problem (P0), problem(P0-C) contains all known channel samples at the beginningof a slow timescale slot. On this basis, the outline to solveproblem (P0) is as follows:

1) We solve problem (P0-C) to obtain {n∗, m∗}, as anapproximated solution for the optimal {n, m} inproblem (P0).

2) With the obtained {n∗, m∗}, the optimal w(t) for prob-lem (P0) in each fast timescale slot can be obtainedby solving the following beamforming problem at thebeginning of each fast timescale slot:

(P0-F) minw(t)

U(w(t))

s.t.∑i∈I

wij(t)Hwij(t) ≤ Ej , ∀j ∈ J ,

max[riη, ri

F∑l∈L m∗

l Ss

]≤ log(1 + SINRi(t)), ∀i ∈ I.

Note that channels in (P0-F) are actual channels (not systemgenerated samples) at fast timescale slot t. It can be seen that(P0-F) is a convex optimization problem, since, for given m∗

l ,the second constraint in problem (P0-F) can be recasted asa second-order cone (SOC) [1]. Then problem (P0-F) can beeasily solved by an interior point method with some standardoptimization tool boxes, e.g. CVX [37].

Whereas, problem (P0-C) is difficult to solve because 1) ithas integer variables nl and ml, and non-smooth and non-convex l0-norm in the objective function and 2) because ofconstraint (14), the problem is still nonconvex even if theinteger variables and l0-norm are approximated to their corre-sponding convex relaxations. In the next section, we proposethe tractable solution approach for problem (P0-C).

IV. APPROACH TO SOLVE PROBLEM (P0-C)

To make problem (P0-C) tractable, we firstly relax integervariables nl and ml into positive real numbers, i.e., nl,ml ∈ R+. Then, in what follows, we propose a step-by-step approach to solve problem (P0-C). Specifically, the firststep is to approximate the nonconvex l0-norm in the objectivefunction by a smooth concave function. Then approximatingthe concave term in the objective function and the non-convex constraint (12) simultaneously by applying the SCAapproach, and at the same time, the SDR method is alsoleveraged. Then, a series of convex optimization subproblemsare obtained. However, each convex optimization subprob-lem may contain very high dimensional variables since itinvolves the beamforming vectors for all samples. Hence,in the following step, we transfer the subproblem as a globalconsensus problem and then apply ADMM to solve it. Lastly,we propose efficient algorithms to recover the real numbervariables nl and ml back to integers. We show the logical flowin Fig. 3.

A. Smooth Function Approximation for l0-norm

Two commonly used approaches to deal with the l0-normare the smoothing function approximation [38] and thereweighted l1-norm approximation [39]. It has been shownin [40] that the reweighted l1-norm approximation can beregarded as a special case of smoothing function approx-imation. In contrast to previous works exploiting specificsmoothing functions, we apply a general smooth functionapproximation and propose an algorithm based on this generalsmooth function. Our general smooth function approximationapproach can extend the results of those who utilize thespecific smoothing functions.

A general smooth function g(z) used to approximatel0-norm ‖z‖0 needs to satisfy the following three properties:

• g(z) is concave and non-decreasing w.r.t. z ≥ 0;• g(z) is continuously differentiable;• limz→0+ g(z) = 0.

Most of the commonly used smooth functions, such as g(z) =1− e−αz , g(z) = (z + α)p and g(z) = z/(z + α) (α > 0 and0 < p < 1), satisfy these three properties.


Fig. 3. The logical flow to solve problem (P0).

Based on the general smooth function approximation andthe relaxation for integer variables, problem (P0-C) can bereformulated as

(P1) minn,m,w

∑l∈L

g(nl + ml)Pl +∑l∈L

(nlPc + mlP

s)

+1V

∑v∈V

U(w(v))

s.t. (1a), (1b), (2), (12), and (13),

nl ∈ R+, ml ∈ R

+, ∀l ∈ L. (15)

In the sequel, we denote the optimal solution of problem (P1)as {n, m, w}.

It is worth mentioning that problem (P1) is nonconvex, sincethe objective function is in the form of the sum of a concavefunction and a convex function, and also its constraint (12)is nonconvex. In the next subsection, we handle these twoobstacles by employing the SCA method.

B. Successive Convex Approximation

SCA is an efficient way to solve various types of non-convex optimization problems [14]–[16]. The main idea ofSCA is that, a locally tight approximation of the originalproblem is performed at each iteration to produce a tightconvex objective function and constraint sets. In other words,instead of solving a nonconvex optimization problem directly,a series of convex optimization problem is solved iterativelyto obtain an approximate solution. In this paper, we utilize theapproximation functions to locally approximate the nonconvexfunctions (in both the objective function and the constraints)based on the following assumption [15].

Assumption 1: A function h(x, y) is called as the approx-imation function for the nonconvex function h(x), when thefollowing conditions hold:

• h(x, y) is continuous in (x, y).• h(x, y) is convex in x.• The function value of h(x, x) and h(x) are consistent,

i.e, h(x, x) = h(x), ∀x.

• The gradient ∂h(x,y)∂x |x=y and ∇h(x)|x=y are consistent,

i.e., ∂h(x,y)∂x |x=y = ∇h(x)|x=y , ∀x.

• h(x, y) is an upper-bound of h(x), i.e., h(x, y) ≥h(x), ∀x, y.

In the rest of this subsection, we apply SCA for both theobjective function and the constraints respectively, based onapproximation functions satisfying Assumption 1.

1) SCA for the Objective Function of Problem (P1): Asmentioned, the objective function of problem (P1) is non-convex since it contains a concave term. We can linearizethe concave term by its first order Taylor series. To bespecific, let zl = nl + ml in problem (P1), then g(zl) canbe approximated as

g(z(q)l , z

(q−1)l

)= g

(z(q−1)l

)+ ∇g

(z(q−1)l

)(z(q)l − z

(q−1)l

), (16)

where q ≥ 1 is the iteration number and ∇g(z(q−1)l

)is the

gradient of g(·) at point z(q−1)l . It can be easily verified that

g(z(q)l , z

(q−1)l

)satisfies Assumption 1.

Therefore, at the q-th iteration, it is enough to solve aoptimization problem with the following objective function:

∑l∈L

(nl+ml)Γ(q−1)l +

∑l∈L

(nlPc + mlP

s)+1V

∑v∈V

U(w(v)),

where Γ(q−1)l = Pl∇g

((nl + ml)(q−1)

)is a constant. Note

that this objective function is convex.2) SCA for the Constraint of (P1): Let Wi(v) = wi(v)

wi(v)H ∈ RJK×JK and Hi(v) = hi(v)hi(v)H ∈ RJK×JK ,then the achievable rate of UE i over the v-th channel samplecan be re-expressed as:

log(1 + SINRi(v)) = log

(∑k∈I hi(v)HWk(v)hi(v) + σ2

i∑Ik �=i hi(v)HWk(v)hi(v) + σ2

i

)

= log

(∑k∈I

tr (Hi(v)Wk(v)) + σ2i

)

− log

⎛⎝ I∑

k �=i


⎞⎠ .

If we use the following property,

Wi(v) = wi(v)wi(v)H ⇔ Wi(v) � 0, rank(Wi(v)) ≤ 1,


then constraints (13) and (12) can be rewritten as

max[riη, ri

F∑l∈L mlSs

]

≤ log

(∑k∈I


)

− log

⎛⎝ I∑

k �=i


⎞⎠ ,

∀i ∈ I, ∀v ∈ V , (17)

rank(Wi(v)) ≤ 1, ∀i ∈ I, ∀v ∈ V , (18)

Wi(v) � 0, ∀i ∈ I, ∀v ∈ V , and (19)∑i∈I

tr (GjWi(v)) ≤ Ej , ∀j ∈ J , ∀v ∈ V , (20)

where Gj is a square matrix with J×J blocks, and each blockin Gj is a K×K matrix. In Gj , the block in the j-th row andj-th column is a K × K identity matrix, and all other blocksare zero matrices.

Since constraints (17) and (18) are still nonconvex, we can-not solve the problem directly. Fortunately, for (17), we canmake use of the SCA approach once again. Specifically,we have

log

⎛⎝ I∑

k �=i


⎞⎠

≤ log xi(v)(q−1)

+

(∑Ik �=i tr (Hi(v)Wk(v)) + σ2

i − xi(v)(q−1))

xi(v)(q−1) ln 2, (21)

where q ≥ 1 is the iteration number, and xi(v)(q−1) =∑Ik �=i tr

(Hi(v)Wk(v)(q−1)

)+ σ2

i is a constant derived fromthe (q − 1)-th iteration. We can verify that the approxi-mation function in the right hand side of (21) aligns withAssumption 1 as well. Then, at the q-th SCA iteration, (17)can be relaxed to

max[riη, ri

F∑l∈L mlSs

]−log

(∑k∈I

tr (Hi(v)Wk(v))+σ2i

)

≤ −


i − xi(v)(q−1))

xi(v)(q−1) ln 2− log xi(v)(q−1), ∀i ∈ I, ∀v ∈ V . (22)

Now, one can observe that (22) is a convex constraint.

C. Semidefinite Relaxation

So far, we have converted the nonconvex constraint (17) intoa convex one in (22) by leveraging SCA. Note, however, thatthe rank constraint in (18) is still nonconvex. In this subsection,we resort to the semidefinite relaxation (SDR) approach toresolve this nonconvex rank constraint.

In the SDR approach, we drop the rank constraint first, andthen solving the problem without the rank constraint. If theresulting Wi(v) is of rank one or zero, we conclude thatthe SDR is tight and no more manipulation is needed [41].

On the other hand, if the rank of resulting Wi(v) is largerthan one, we must use a method to extract an approximatesolution, e.g., the randomization method [17].

Recalling from Section II-C that U(w(v)) is w.r.t.∑

i∈I∑j∈J ‖wij(v)‖2

2, then we can equivalently represent U(w(v))as U(

∑i∈I tr (Wi(v))). Therefore, combining SCA for both

the objective function and constraints, and also applyingSDR at the same time (i.e., removing the nonconvex rankconstraint (18)), what we need to solve at the q-th iteration is

(Q1) minn,m,W(v)

∑l∈L

(nl + ml)Γ(q−1)l +

∑l∈L

(nlPc + mlP

s)

+1V

∑v∈V

U

(∑i∈I

tr (Wi(v))

)

s.t. (1a), (1b), (2), (15), (19), (20), and (22),

where W(v) = [W1(v) · · ·WN (v)]. Problem (Q1) is a convexoptimization problem.

Let {n(q−1), m(q−1), W(q)i (v)} be the optimal solution for

problem (Q1) at the q-th iteration. We elaborate the SCAalgorithm for problem (P1) in Algorithm 1, in which O(q)

is the optimal objective function value of problem (P1) atiteration q and � > 0 is a small constant.

Algorithm 1 SCA Algorithm to Solve Problem (P1)

1: Initialize: n(0), m(0), and W(0)i (v), for i ∈ I, v ∈ V .

2: Iteration q ≥ 1: Solve problem (Q1) withgiven n(q−1), m(q−1), W(q−1)

i (v), and obtainn(q), m(q), W(q)

i (v) and O(q).3: if |O(q) − O(q−1)| < � then4: Problem (P1) achieves the approximated solution, stop

iteration;5: else6: Let q = q + 1, go to step 2.7: end if8: Output: n(q), m(q), and W(q)

i (v), for i ∈ I, v ∈ V .

The following theorem unravels the convergence ofAlgorithm 1 and the tightness of SDR for problem (P1).

Theorem 1: Every limit point W(∞)i (v) generated by

Algorithm 1 is a stationary point of problem (P1). That is,

limq→∞

∥∥∥W(q)i (v) − W(q−1)

i (v)∥∥∥

F= 0, ∀i ∈ I, ∀v ∈ V .

Furthermore, if the Slater condition holds at the limit pointW(∞)

i (v), then

1) W(∞)i (v) is a KKT point.

2) The SDR for problem (P1) is asymptotically tight. Thatis,

limq→∞

rank(W(q)i (v)) ≤ 1, ∀i ∈ I, ∀v ∈ V .

Proof: See Appendix A. �So far, we have obtained an approximation solution of

problem (P1) using Algorithm 1. In Algorithm 1, the mainiteration Step 2 involves solving a convex problem (Q1),whose scale can be very large when the number of samples V


is large. Solving a large-scale nonlinear convex optimizationin a centralized manner is still computationally demanding andtime consuming. In this case, a parallel algorithm is desirableto reduce the computational complexity. In the next section,we develop a low-complexity parallel algorithm to solve thislarge-scale convex optimization problem.

V. A PARALLEL ALGORITHM TO SOLVE PROBLEM (Q1)

In this section, to handle the large-scale convex optimiza-tion problem (Q1), we reformulate the problem as a globalconsensus problem [42]. A global consensus problem widelyexists in the distributed computing and multi-agent systems,where all agents have to reach an agreement on a commonvalue that is needed during computing. In this paper, we cantreat each fast timescale slot as an agent, and the BBU poolresource segmentation variable {n, m} as the common valueneeded at each agent (to generate the beamforming vectors).

We introduce V auxiliary variables (since the number ofsamples is V ) as the copies of nl and ml respectively, namely,nlv = nl, and mlv = ml, for v = 1, · · · , V . First, let

fv(nlv, mlv, W(v))=∑l∈L

(nlv +mlv)Γl+∑l∈L

(nlvP c+mlvPs)

+ U

(∑i∈I

tr (Wi(v))

),

then problem (Q1) can be equivalently expressed as

(Q2) minnlv,mlv,

nl,ml,W(v)

1V

∑v∈V

fv(nlv, mlv, W(v))

s.t. nlv = nl, mlv = ml, ∀l ∈ L, ∀v ∈ V , (23)

(nlvCc + mlvCs) ≤ Cl, ∀l ∈ L, ∀v ∈ V , (24)

(nlvSc + mlvSs) ≤ Sl, ∀l ∈ L, ∀v ∈ V , (25)

nlv ∈ R+, mlv ∈ R

+, ∀l ∈ L, ∀v ∈ V , (26)FI

G≤∑l∈L

nlvCc, ∀v ∈ V , (27)

max[riη, ri

F∑l∈L mlvSs

]

− log

(∑k∈I


)

≤ −


i − xi(v)(q−1))

xi(v)(q−1) ln 2− log xi(v)(q−1), ∀i ∈ I, ∀v ∈ V , (28)

(19), and (20).

One can observe that problem (Q2) is a global consensusproblem, with global variables {nl, ml} and local variables{nlv, mlv, W(v)}. In what follows, we make use of theADMM [20] to obtain a parallel algorithms for the globalconsensus problem (Q2). For brevity, we henceforth drop thecoefficient 1

V in the objective function of problem (Q2). Theaugmented Lagrangian for problem (Q2) is2

Lρ =∑v∈V

(fv(nlv, mlv, W(v)) +

∑l∈L

[λlv(nlv−nl)

+φlv(mlv − ml)+(‖nlv−nl‖22+‖mlv − ml‖2

2)ρ

2

] ).

(29)

Then, the corresponding ADMM algorithm for problem (Q2)at iteration k is shown from (30) to (34), as shown at thebottom of this page, where Fv stands for the feasible regionof {nlv, mlv, W(v)}. The feasible region Fv is defined as

Fv =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(nlvCc + mlvCs) ≤ Cl, (nlvSc + mlvSs) ≤ Sl,

nlv ∈ R+, mlv ∈ R+, ∀l ∈ L,F I

G≤∑

l∈L nlvCc,

Wi(v) � 0, ∀i ∈ I,∑i∈I tr (GjWi(v)) ≤ Ej , ∀j ∈ J ,

max[riη, ri

F∑l∈L mlvSs

]− log

(∑k∈I tr (Hi(v)Wk(v)) + σ2

i

)≤ −


i − xi(v)(q−1))

xi(v)(q−1) ln 2− log xi(v)(q−1), ∀i ∈ I.

2We do not include all constraints in the Lagrangian since, when applyingADMM algorithm, the excluded constraints are just local constraints in theconsensus problem.

{n

(k+1)lv , m

(k+1)lv , W(v)(k+1)

}= arg min

〈nlv ,mlv,W(v)〉∈Fv

(fv(nlv, mlv, W(v)) +

∑l∈L

[λ

(k)lv (nlv − n

(k)l )

+ φ(k)lv (mlv − m

(k)l ) + (ρ/2)(

∥∥∥nlv − n(k)l

∥∥∥2

2+∥∥∥mlv − m

(k)l

∥∥∥2

2)])

, (30)

n(k+1)l =

1V

∑v∈V

(n

(k+1)lv +

1ρλ

(k)lv

), (31)

m(k+1)l =

1V

∑v∈V

(m

(k+1)lv +

1ρφ

(k)lv

), (32)

λ(k+1)lv = λ

(k)lv + ρ

(n

(k+1)lv − n

(k+1)l

), (33)

φ(k+1)lv = φ

(k)lv + ρ

(m

(k+1)lv − m

(k+1)l

), (34)


It can be verified that problem (30) is convex, which can besolved by the standard optimization tool boxes, e.g. CVX [37].C-RAN operator can implement this parallel ADMM algo-rithm easily by exploiting the parallel computing environmentin the BBU pool.

VI. INTEGER RECOVERY

With Algorithm 1 and ADMM, we can solve problem (P1)successfully and efficiently. However, the optimal ml and nl,∀l ∈ L obtained from solving problem (P1) are real numbers,which are infeasible for problem (P0). In this subsection,we propose algorithms to recover {m∗

l , n∗l } from {ml, nl},

i.e., from real numbers to natural numbers.

A. The Rules

One might consider using flooring or ceiling function torecover {m∗

l , n∗l } from {ml, nl}, ∀l ∈ L. Note, however,

that these simple and identical operation for all {ml, nl} isnot a right choice, since there are some constraints related to{m∗

l , n∗l } in the original problem (P0), e.g., (1a), (1b), (2),

and (5). In particular,

• if we simply ceiling all {ml, nl}, for l ∈ L, the resulted{m∗

l , n∗l } may not be a feasible solution for problem

(P0), since constraints (1a) and (1b) may not be satisfied.• if we simply flooring all {ml, nl}, for l ∈ L, the resulted

{m∗l , n∗

l } may not be feasible for problem (P0) as well,since constraint (2) may not be satisfied. In addition,with this resulted {m∗

l }, we may not be able to obtain afeasible w(t) for problem (P0), due to constraint (5).

Therefore, the integer recovery manipulation is nontrivial.In other words, we cannot find a simple and identical integerrecovery manipulation which is suitable for all {ml, nl}, forl ∈ L. Instead, we should design algorithms to decide theinteger recovery manipulation for each single {ml, nl}. Thatis, we need to decide ml should recover to �ml� or �ml�, foreach l ∈ L case by case (as well as nl).

Based on these observations above, on can deduce that thefollowing rules should be satisfied in the integer recoveryalgorithms. The principal rule to recover {m∗

l , n∗l } is to

guarantee its feasibility for problem (P0). In particular,1) Since the objective function of problem (P1) is increas-

ing w.r.t. nl, ∀l ∈ L, we can easily obtain∑

l∈L nl =FI/TCc. Hence,∑

l∈Ln∗

l =⌈

FI

TCc

⌉� C. (35)

2) To recover the integer m∗l , ∀l ∈ L, we use the following

rule ∑l∈L

m∗l =

⌈∑l∈L

ml

⌉� S, (36)

since we may not find the feasible wij(t) for constraint(5), if we take the floor function of

∑l∈L ml.

The second rule is that, after integer recovery, the incre-ment of the “server cost + VM cost” from problem (P1) toproblem (P0) should be as small as possible.

B. The Algorithms

We start from recovering m∗l , and then recover n∗

l based onthe resulted m∗

l .3 We first define the initial active set as

A � {l | ml > 0, l ∈ L}, (37)

which indicates that, there is at least one SIVM activated ineach server of this set.

Then, we partition the initial active set A into two sets,ceillable set AI and floor-only set AD, where AI = {l |�ml�Ss ≤ Sl, �ml�Cs ≤ Cl, l ∈ L} and AD = A\AI .Based on the definition, for any ml, l in the floor-only setAD, the optimal m∗

l is just �ml�. While for any ml, l in theceillable set AI , the optimal m∗

l can be either �ml� or �ml�.In addition, we define the truncation summation for the initialactive set as

d =

⌈∑l∈A

(ml − �ml�)⌉

.

The maximum number of SIVMs that can be allocated atserver l is denoted as,

ml = min{⌊

Sl

Ss

⌋,

⌊Cl

Cs

⌋}, ∀l ∈ L.

The following Lemma provides a guideline to design theinteger recovery algorithm for m∗

l .Lemma 1: If |AI | ≥ d, then the initial active set A is

enough to recover the feasible m∗l , ∀l ∈ L, which satisfies

(1a), (1b), (5) and (36). Otherwise, more servers should beadded to the initial active set.

Proof: See Appendix B. �We summarize the approach to recover the integer m∗

l ,∀l ∈ L, in Algorithm 2. Based on Lemma 1, there are twopossible cases in Algorithm 2:

• The first case is that, if we ceil all the ml in the ceillableset and floor all the ml in the floor-only set, the resultingm∗

l can satisfy∑

l∈A m∗l ≥ S. This corresponds to Step

3 to Step 8.• Otherwise, we should add more servers into the initial

active set to satisfy∑

l∈A m∗l ≥ S, where A is the

updated active set. This corresponds to Step 10 to Step 15.Note that, Step 5 and Step 12 follow the second rule.

Then, once m∗l , ∀l ∈ L, and A are generated from

Algorithm 2, we can apply Algorithm 3 to obtain n∗l , ∀l ∈ L,

in which nl is defined as the maximum number of CIVMsthat can be allocated to server l, i.e.,

nl = min{⌊

Sl

Sc

⌋,

⌊Cl

Cc

⌋}, ∀l ∈ L.

Similarly, there are also two possible cases in Algorithm 3:• The first case is that, if we allocate the maximum number

of CIVM in each server that belongs to the updated activeset A, the resulting n∗

l satisfies∑

l∈A n∗l ≥ C. This

corresponds to Step 4 to Step 10.• Otherwise, we should add more servers into the updated

active set to make∑

l∈A n∗l ≥ C satisfied, where A is

3If we start from recovering n∗l , and then recover m∗

l based on the resultedn∗

l , the similar algorithms can be also derived, we omit them for brevity.


Algorithm 2 Integer Recovery Algorithm for ml

1: Input: A, AI and AD.2: Initialize: Set m∗

l = �ml�, ∀l ∈ AD .3: if |AI | ≥ d then4: for i = 1 : d do5: Find out l = arg minl∈AI (�ml�−ml). Let m∗

l=⌈ml

⌉,

and AI = AI\l.6: Set m∗

l = �ml�, ∀l ∈ AI .7: end for8: Set m∗

l = �ml�, ∀l ∈ AI . Let A = A. Stop iteration andgo to Step 17.

9: else10: Set m∗

l = �ml�, ∀l ∈ AI . Let A = A.11: while

∑l∈A m∗

l < S do12: Find out l = arg minl∈L\A(Pl/ml);13: Calculate m∗

l= min

{(S −

∑l∈A m∗

l

), ml

};

14: Update A = A⋃

l;15: end while16: end if17: Output: m∗

l , ∀l ∈ L, and A.

Algorithm 3 Integer Recovery Algorithm for nl

1: Input: m∗l , ∀l ∈ L, and A (obtained from Algorithm 2).

2: Initialize: n∗l = min

{⌊Sl−m∗

l Ss

Sc

⌋,⌊

Cl−m∗l Cs

Cc

⌋}, for l ∈

A. Set n∗l = 0, for l ∈ L\A, and C = C.

3: Sort n∗l , ∀l ∈ A, in ascending order, i.e., n∗

π1≤ · · · ≤

n∗πi

≤ n∗πi+1

≤ · · · ≤ n∗π|A| .

4: for i = 1 : |A| do5: if C ≤ n∗

πithen

6: Let n∗πi

= C , and n∗πj

= 0, for j = i + 1, · · · , |A|.Stop this loop and go to Step 17.

7: else8: Set C = C − n∗

πi.

9: end if10: end for11: Let A = A.12: while

∑l∈A n∗

l < C do13: Find out l = arg minl∈L\A(Pl/nl);14: Calculate n∗

l= min

{(C −

∑l∈A n∗

l

), nl

};

15: Update A = A⋃

l;16: end while17: Output: n∗

l , ∀l ∈ L.

the re-updated active set. This corresponds to Step 12 toStep 17.

Note that, Step 13 in Algorithm 3 follows the second rule.

VII. SIMULATION RESULTS

We perform the simulation under the heterogeneous RRHsscenario, there are four normal RRHs, one macro RRH,and 10 UEs. The four normal RRHs are placed at equaldistances apart on a circle with radius 0.5 km and the macroRRH is located at the center of the circle. The maximumtransmitting power of a macro RRH is much higher than

TABLE I

SIMULATION PARAMETERS

the one of a normal RRH. UEs are randomly, uniformlyand independently distributed within the disk. We adopt thepath loss model used by the 3GPP specification for EvolvedUniversal Terrestrial Radio Access in [43]. The received powerat a UE located d km away from a RRH is given by p (dB) =128.1 + 37.6 log10 d. The transmit antenna gain at each RRHis 5 dB. The lognormal shadowing parameter s is 10 dB. Thescaling factor η is set to 1. In our simulations, we considerhomogeneous UEs with σ2

i = σ2 dBm, and ri = r bit/s/Hz,∀i ∈ I. The maximum transmitting power of all normal RRHsare identical as E, and the maximum transmitting power of themacro RRH is Emacro. We summarize our default simulationparameters in Table I.

In addition, we use g(z) = 1− e−10 z as a general smoothfunction, and

U(w(v)) = β∑i∈I

∑j∈J

wij(v)Hwij(v) = β∑i∈I

tr (Wi(v)) ,

where β is a constant.

A. Convergence

We present the convergence of ADMM and SCA in Fig. 4.In Fig. 4(a), Gap_ADMM =

∥∥m(k+1) − m(k)∥∥

2, where k is

the iteration number of ADMM. In Fig. 4(b), Gap_SCA =∥∥m(q+1) − m(q)∥∥

2, where q is the iteration number of SCA.

We can observe from Fig. 4 that both ADMM and SCAconverge to a stationary point very quickly, i.e., within severaliterations.

It can be analyzed from problem (P0-C) that the optimalresource segmentation strategy may not be unique. That is,there are many possible {n∗, m∗} who can produce the sameoptimal objective function value for problem (P0-C), since{n∗, m∗} are natural numbers. We show this non-uniquenessproperty in Fig. 5. In Fig. 5, there are two possible results gen-erated by applying our proposed algorithm, i.e., “case1” and“case2”. In case1, m∗ = [3 3 4 4 0 0]T , n∗ = [3 3 4 4 2 0]T ,and the active server set is {1, 2, 3, 4, 5}. While in case2,m∗ = [2 4 4 2 2 0]T , n∗ = [3 3 4 5 1 0]T , and theactive server set is still {1, 2, 3, 4, 5}. If we substitute thesetwo different {n∗, m∗} into the {n, m} in problem (P0-C),the same objective function value for problem (P0-C) canbe obtained. In addition, it can be verified from Fig. 5 thatthe integer recovery scheme in this paper is not as simpleas just trivially ceiling or flooring, i.e., ceiling or flooringis determined based on the algorithms in Subsection VI-Brespectively. Lastly, in “case1” of Fig. 5(b), we add one moreserver, i.e., server 5, to satisfy

∑l∈A n∗

l ≥ C, which verifiesStep 12 to Step 17 in Algorithm 3.


Fig. 4. Convergence upon the iterations.

Fig. 5. Effectiveness of the proposed integer recovery algorithms.

B. The Performance

To show the effectiveness, in this subsection, we comparethe performance of our proposed algorithm against the follow-ing benchmark algorithms:

1) Exhaustive Search (ES). The main efforts in this paper isto handle the integer variables {n, m} and also use SCAto to obtain proper approximations for the nonconvexterms. In this subsection, we try to use numerical method

Fig. 6. The performance of our proposed algorithm.

to exhaustively search the optimal solution, although thecomplexity is prohibiting. To exhaustively search theoptimal solution for problem (P0), the brief procedure is:

a) Choosing proper integer {n, m} and substitutingthem to problem (P0);

b) Solving the remaining convex problem by theoptimization toolbox;

c) Calculating and recording the objective functionvalue.

We repeat this procedure exhaustively to obtain theoptimal solution. In this ES algorithm, we assume thatthe operator can perfectly predict the channels in thefuture (hence SAA is not required). This algorithm isunrealistic and just acts as the performance benchmark.

2) The algorithm in [2], which uses three loops of iterativeupdates, i.e., an outer loop for the DC algorithm, a mid-dle loop for ADMM, and an inner loop for the WMMSEalgorithm. Since both DC and WMMSE can only pro-duce local optimal solutions, the result generated fromthis algorithm might be far from the global optimal.

To distinguish this work from [2], we call the proposedapproach in this work as the two loops (TL) algorithm,i.e., an outer loop for the SCA algorithm and an inner loopfor the ADMM algorithm. In Fig. 6, we show the system


TABLE II

SIMULATION RUNNING TIME IN SECONDS

cost generated by these algorithms under different values ofF and β. It can be observed that, firstly, TL algorithm performsclose to the unrealistic ES algorithm, which means that theapproximations we used, e.g., SAA and SCA, do not havemuch loss. Although our problem (P0) is MINLP, however,it is a special one. For our problem (P0), in both the VM costof the objective function and constraints (2) and (6), the inte-ger terms that make differences are

∑l∈L nl and

∑l∈L ml,

instead of nl and ml, for l ∈ L, individually. That meansonce the results of

∑l∈L nl and

∑l∈L ml obtained from our

proposed TL algorithm are close to the optimal ones, we donot have much loss. The individual nl and ml, for l ∈ L, onlyaffect the server cost. And secondly, TL algorithm outperformsthe algorithm in [2], since, in TL algorithm, only the outer loopproduces the local optimal solution (while in [2], both outerand inner loops produce the local optimal solutions).

In addition, we show the simulation running time in secondsin Table II. For the 3 RRHs case, three normal RRHs areplaced at equal distances apart on a circle with radius 0.5 km.The running time is based on 4 channel samples, since oursimulation tool only supports 4 parallel workers. We canconclude that TL algorithm saves about 50% running time,compared with the algorithm in [2].

C. The Interaction of CC&C

In this paper, we leverage CBAR, i.e., inequality (5),to capture the connections between caching and wirelesstransmission. To verify the benefits of CBAR, we performthe simulation against the minimum transmission rate (MTR)scheme. In particular, under the MTR scheme, the followingconstraints are utilized, instead of CBAR:⎧⎪⎨

⎪⎩F ≤ η

∑l∈L

mlSs,

riη ≤ log(1 + SINRi(t)), ∀i ∈ I, ∀t ∈ T .

(38a)

(38b)

This means that caching and wireless transmission aredecoupled. Under the MTR scheme, the BBU pool resourcesegmentation is obtained in a greedy way as follows:

1) Choosing one server in the server set with the minimumserver price.

2) Assigning SIVM and CIVM to this server one by one,until no more VMs can be assigned in this server, i.e.both of the computation and storage resources are usedup, then removing this server from the server set.

3) Repeating steps 1) and 2) until the number of CIVMsand SIVMs satisfy (35) and (38a) respectively.

In Fig. 7, we can learn that the system cost of CBAR schemeis lower than that of MTR scheme when β and r are bothsmall. When β and r are both large, MTR scheme and CBAR

Fig. 7. Comparison against the benchmark.

Fig. 8. Limited storage volume.

scheme are converging, since in this case, the optimal cache-based adaptive rate achieves its lower bound r. The steep slopein Fig. 7(a) is because that one more server is required whenF goes from 550 to 600.

Finally, to identify the benefits of integrating CaaS in theBBU pool, we reduce the storage volume in each server. Thatmeans, if the storage volume is reduced to 0, then CaaS isentirely removed from the BBU pool. In Fig. 8, we limit thestorage volume in each server to 35% of its original value.4

4In this paper, we cannot set the storage volume to 0. Otherwise, the CBARconstraint will become infeasible.


It can be learned that the system cost becomes much higherwhen the storage volume is limited, which reveals the benefitsof integrating CaaS in the BBU pool in return.

VIII. CONCLUSION

In this paper, we studied the interactions of CC&C undertwo timescales. In the slow timescale, we segmented thecomputation and storage resources into different types of VMsin the BBU pool, and in the fast timescale, we performedbeamforming at the RRHs side. We formulated a SMINLPto minimize the system cost. By leveraging SAA, SCA, andSDR approaches, we transferred the original SMINLP probleminto a global consensus problem, then we take advantage ofADMM algorithm to handle the global consensus problemefficiently, i.e., in a parallel way. From simulation, we demon-strate that our proposed algorithm converges fast, and for theoperator, the system cost is able to be reduced significantly byexploring the interactions of CC&C.

In the future, we will introduce network slicing techniquein C-RAN with CaaS and develop the two timescales resourceallocation schemes for multiple network slices, and also ana-lyze the throughput or delay [44], [45] for each slice.

APPENDIX APROOF OF THEOREM 1

Since the approximation functions we utilized for both theobjective function and the constraints satisfy Assumption 1,the similar proof for the convergence of limit point can befound in [46], and the limit point is a KKT point was similarlyproven in [15] as well. We omit those proof for brevity. In thefollowing, we prove the SDR tightness.

The Lagrangian for problem (Q1) is,5

L =1V

∑v∈V

U

(∑i∈I

tr (Wi(v))

)−∑i∈I

∑v∈V

ΛivWi(v)

+∑j∈J

∑v∈V

μjv

∑i∈I

tr (GjWi(v))

+∑i∈I

∑v∈V

ξiv

⎧⎨⎩ 1

xi(v)(q−1) ln 2

⎛⎝ I∑

k �=i

tr (Hi(v)Wk(v))

⎞⎠

− log

(∑l∈I

tr (Hi(v)Wl(v)) + σ2i

)}, (39)

where Λiv � 0, ∀i ∈ I, ∀v ∈ V , μjv ≥ 0, ∀j ∈ J , ∀v ∈ V ,and ξiv ≥ 0, ∀i ∈ I, ∀v ∈ V , are the Lagrange multipliers forconstraints (19), (20), and (22) respectively. Λiv is a JK×JKmatrix.

On the one hand, since W(∞)i (v) is a KKT point, we

have,

∂L

∂W(∞)i (v)

= 0, and (40)

Λ(∞)iv W(∞)

i (v) = 0. (41)

5We only include the terms that relevant to this proof.

On the other hand, at the q-th iteration, we have

∂L

∂W(q)i (v)

=1V

U ′

(∑i∈I

tr(

W(q)i (v)

))I − Λ(q)

iv

+∑j∈J

μ(q)jv Gj +

I∑k �=i

ξ(q)kv

xk(v)(q−1) ln 2Hk(v)

−∑i∈I

ξ(q)iv

ln 2

(∑l∈I

tr(

Hi(v)W(q)l (v)

)+ σ2

i

)−1

Hi(v)

=1V

U ′

(∑i∈I

tr(

W(q)i (v)

))I − Λ(q)

iv +∑j∈J

μ(q)jv Gj

− ξ(q)iv

ln 2

(∑l∈I

tr(

Hi(v)W(q)l (v)

)+ σ2

i

)−1

Hi(v)

+I∑

k �=i

ξ(q)kv

ln 2

(1

xk(v)(q−1)

−(∑

l∈Itr(

Hk(v)W(q)l (v)

)+ σ2

k

)−1⎞⎠Hk(v), (42)

where I is a JK × JK identity matrix. Combining (40) and(42), which yields

Λ(q)iv =

∑j∈J

μ(q)jv Gj +

I∑k �=i

ξ(q)kv

ln 2

(1

xk(v)(q−1)

−(∑

l∈Itr(

Hk(v)W(q)l (v)

)+σ2

k

)−1⎞⎠Hk(v)

+1V

U ′

(∑i∈I

tr(

W(q)i (v)

))I

− ξ(q)iv

ln 2

(∑l∈I

tr(

Hi(v)W(q)l (v)

)+σ2

i

)−1

Hi(v). (43)

In addition, based on limq→∞

∥∥∥W(q)l (v) − W(q−1)

l (v)∥∥∥

2,1=

0, ∀l ∈ I, ∀v ∈ V , then

xk(v)(q−1) ≈ xk(v)(q) =I∑

l �=k

tr(

Hk(v)W(q)l (v)

)+ σ2

k

≤∑l∈I

tr(

Hk(v)W(q)l (v)

)+ σ2

k, for q → ∞.

Furthermore, since the Slater condition holds at thelimit point W(∞)

i (v), the limit of the Lagrange multipliersexists [47], i.e., 0 ≤ μ

(∞)jv < +∞, ∀j ∈ J , v ∈ V and

0 ≤ ξ(∞)iv < +∞, ∀i ∈ I, ∀v ∈ V .

Therefore, in the right hand side of (43), the first threeterms construct a matrix with full rank, i.e., rank = JK .In addition, the coefficient of the last term Hi(v) in (43) isnegative. Recall that Λ(q)

iv � 0, for q = 1, 2, · · · ,∞, andrank(Hi(v)) ≤ 1, we can conclude rank(Λ(∞)

iv ) ≥ JK − 1.


Further, combining the rank result on Λ(∞)iv and (41), we can

obtain rank(W(∞)i (v)) ≤ 1. This completes the proof.

APPENDIX BPROOF OF LEMMA 1

From (36), we have

∑l∈L

m∗l =

⌈∑l∈L

ml

⌉=

⌈∑l∈A

ml

⌉

=∑l∈A

�ml� +

⌈∑l∈A

(ml − �ml�)⌉

=∑

l∈AD

�ml� +∑l∈AI

�ml� + d

=∑

l∈AD

�ml� +∑l∈AI

�ml� − |AI | + d.

Hence, if |AI | ≥ d, we have∑l∈L

m∗l ≤

∑l∈AD

�ml� +∑l∈AI

�ml� . (44)

This means that, if we take the best effort action, i.e., ceilingall the ml in the ceillable set AI and flooring all the ml inthe floor-only set AD, the resulted m∗

l is able to satisfy (36).Otherwise, i.e., if |AI | < d, the resulted m∗

l still cannotsatisfy (36), in spite of taking the best effort action. Thenmore servers should be added to the initial active set A. Thiscompletes the proof.

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Jianhua Tang (S’11–M’15) received the B.E.degree in communications engineering from North-eastern University, China, in 2010, and the Ph.D.degree in electrical and electronic engineeringfrom Nanyang Technological University, Singapore,in 2015. He was a Post-Doctoral Research Fellowwith the Singapore University of Technology andDesign from 2015 to 2016 and a Research AssistantProfessor with the Department of Electrical andComputer Engineering, Seoul National University,from 2016 to 2018. He is currently with the College

of Information Science and Engineering, Hunan Normal University, China.His research interests include cloud computing, cloud radio access network,and network slicing.

Tony Q. S. Quek (S’98–M’08–SM’12–F’18)received the B.E. and M.E. degrees in electrical andelectronics engineering from the Tokyo Institute ofTechnology, Tokyo, Japan, in 1998 and 2000, respec-tively, and the Ph.D. degree in electrical engineeringand computer science from the Massachusetts Insti-tute of Technology, Cambridge, MA, USA, in 2008.He is currently the Cheng Tsang Man Chair Pro-fessor with the Singapore University of Technologyand Design (SUTD). He also serves as the ActingHead of ISTD Pillar, a Sector Lead of the SUTD AI

Program, and the Deputy Director of SUTD-ZJU IDEA. His current researchtopics include wireless communications and networking, network intelligence,the Internet of Things, URLLC, and big data processing.

He is serving as an Elected Member of the IEEE Signal ProcessingSociety SPCOM Technical Committee. He has been actively involved inorganizing and chairing sessions, and has served as a member for the technicalprogram committee and symposium chairs in a number of internationalconferences. He was honored with the 2008 Philip Yeo Prize for Outstand-ing Achievement in Research, the 2012 IEEE William R. Bennett Prize,the 2015 SUTD Outstanding Education Awards—Excellence in Research, the2016 IEEE Signal Processing Society Young Author Best Paper Award,the 2017 CTTC Early Achievement Award, and the 2017 IEEE ComSocAP Outstanding Paper Award. He is serving as the Chair for the IEEEVTS Technical Committee on Deep Learning for Wireless Communications.He is currently serving as an Editor for IEEE TRANSACTIONS ON WIRELESS

COMMUNICATIONS. He was an Executive Editorial Committee Memberfor IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, an Editorfor IEEE TRANSACTIONS ON COMMUNICATIONS, and an Editor for IEEEWIRELESS COMMUNICATIONS LETTERS. He was the 2016–2018 ClarivateAnalytics Highly Cited Researcher. He is a Distinguished Lecturer of theIEEE Communications Society.

Tsung-Hui Chang (S’07–M’08–SM’17) receivedthe B.S. degree in electrical engineering and thePh.D. degree in communications engineering fromNational Tsing Hua University (NTHU), Hsinchu,Taiwan, in 2003 and 2008, respectively. From2012 to 2015, he was an Assistant Professorwith the Department of Electronic and ComputerEngineering, National Taiwan University of Sci-ence and Technology (NTUST), Taipei, Taiwan.In August 2015, he joined the School of Science andEngineering, The Chinese University of Hong Kong,

Shenzhen, China, as an Assistant Professor, where he has been an AssociateProfessor since August 2018. He was a Visiting Ph.D. Student with theUniversity of Minnesota, Minneapolis, MN, USA, from 2006 to 2008, anda Post-Doctoral Researcher with NTHU from 2008 to 2011 and with theUniversity of California at Davis, Davis, CA, USA, from 2011 to 2012. Hisresearch interests include signal processing and optimization problems in datacommunications, machine learning, and big data analysis.

Dr. Chang was a recipient of the Young Scholar Research Award of NTUSTin 2014, the IEEE ComSoc Asian–Pacific Outstanding Young ResearcherAward in 2015, and the IEEE Signal Processing Society Best Paper Awardin 2018. He served as an Associate Editor for IEEE TRANSACTIONS ON

SIGNAL PROCESSING and IEEE TRANSACTIONS ON SIGNAL AND INFOR-MATION PROCESSING OVER NETWORKS from 2015 to 2018.

Byonghyo Shim (S’95–M’97–SM’09) received theB.S. and M.S. degrees in control and instrumentationengineering from Seoul National University, Seoul,South Korea, in 1995 and 1997, respectively, and theM.S. degree in mathematics and the Ph.D. degreein electrical and computer engineering from theUniversity of Illinois at Urbana–Champaign (UIUC),Champaign, IL, USA, in 2004 and 2005, respec-tively. From 1997 to 2000, he was with the Depart-ment of Electronics Engineering, Korean Air ForceAcademy, as an Officer (First Lieutenant) and an

Academic Full-time Instructor. From 2005 to 2007, he was with QualcommInc., San Diego, CA, USA, as a Staff Engineer. From 2007 to 2014, he waswith the School of Information and Communication, Korea University, Seoul,as an Associate Professor. Since 2014, he has been with Seoul NationalUniversity (SNU), where he is currently a Professor with the Departmentof Electrical and Computer Engineering. His research interests include wire-less communications, statistical signal processing, compressed sensing, andmachine learning. He is an Elected Member of the Signal Processing forCommunications and Networking (SPCOM) Technical Committee of the IEEESignal Processing Society. He was a recipient of the M. E. Van ValkenburgResearch Award from the ECE Department, University of Illinois, in 2005,the Hadong Young Engineer Award from IEIE in 2010, the Irwin JacobsAward from Qualcomm and KICS in 2016, and the Shinyang ResearchAward from the Engineering College of SNU in 2017. He has served as anAssociate Editor for IEEE TRANSACTIONS ON SIGNAL PROCESSING, IEEETRANSACTIONS ON COMMUNICATIONS, IEEE WIRELESS COMMUNICA-TIONS LETTERS, and Journal of Communications and Networks, and a GuestEditor for IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS.

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