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51

Distributed Virtual Resource Allocation in SmallCell Networks with Full Duplex Self-backhauls and

VirtualizationLei Chen, F. Richard Yu,Senior Member, IEEE, Hong Ji,Senior Member, IEEE, Gang Liu, and Victor C.M.

Leung,Fellow, IEEE

Abstract—Wireless network virtualization has attracted greatattentions from both academia and industry. Another emergingtechnology for next generation wireless networks is in-band fullduplex (FD) communications. Due to its promising performance,FD communication has been considered as an effective wayto achieve self-backhauls for small cells. In this paper, weintroduce wireless virtualization into small cell networks, andpropose a virtualized small cell network architecture with FDself-backhauls. We formulate the virtual resource allocationproblem in virtualized small cell networks with FD self-backhaulsas an optimization problem. Since the formulated problem isa mixed combinatorial and non-convex optimization problem,its computational complexity is high. Moreover, the centralizedscheme may suffer from signaling overhead, outdated dynamicsinformation, and scalability issues. To solve it efficiently, wedivide the original problem into two subproblems. For the firstsubproblem, we transfer it to a convex optimization problem,and then solve it by an efficient alternating direction methodof multipliers (ADMM)-based distributed algorithm. The secondsubproblem is a convex problem, which can be solved by eachinfrastructure provider. Extensive simulations are conducted withdifferent system configurations to show the effectiveness of theproposed scheme.

Index Terms—Virtualization networks, small cell networks,self-backhauls, in-band full duplex communications.

I. I NTRODUCTION

By abstracting and sharing resources among different par-ties, virtualization can significantly reduce the cost of equip-ment and management in networks [1]. With the tremendousgrowth in wireless traffic and service, it is inevitable toextend virtualization to wireless networks [2], [3]. In wirelessvirtualization, physical wireless network infrastructure andphysical radio resources are abstracted and sliced into virtualwireless resources, which can be shared by multiple parties.After virtualization, the wireless network infrastructure owned

Copyright (c) 2015 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to pubs-permissions@ieee.org.

This work is jointly supported by Project 61271182 and 61302080 of theNational Natural Science Foundation of China.

L. Chen, H. Ji and G. Liu are with the Key Laboratoryof Universal Wireless Communications, Ministry of Education,Beijing University of Posts and Telecommunications, Beijing, P.R.China (e-mail:chenlei1989bupt@gmail.com, jihong@bupt.edu.cn andbuptgangliu@gmail.com).

F. R. Yu is with the Dept. of Systems and Computer Eng., CarletonUniversity, Ottawa, ON, Canada (e-mail: Richard.yu@carleton.ca).

V. C. M. Leung is with the Department of Electrical and ComputerEngineering, the University of British Columbia, Vancouver, BC V6T 1Z4Canada (e-mail: vleung@ece.ubc.ca).

by an infrastructure provider (InP) can be decoupled from theservices that it provides. At the same time, mobile virtualnetwork operators (MVNOs) provide services to users. Sincethe physical resources are abstracted and sliced into virtualresources, it is possible that different MVNOs coexist onthe same InP to share the infrastructure and radio spectrumresources, which enables the reducing of capital expenses(CapEx) and operation expenses (OpEx) [2].

The authors of [4] discussed the control and managementframeworks of wireless network virtualization. Virtual re-source sharing mechanisms were investigated in [5], where thedynamic interactions among MVNOs and InPs are modeled asa stochastic game. In addition, there are some other worksfocusing on the virtualization of certain specific wirelessnetworks. For example, in the context of cellular networks,Zaki et al. [6] proposed a virtualization framework for long-term evolution (LTE) systems, in which a supervisor is usedto virtualize the eNB and manage the physical resources.For wireless local area networks (WLANs) virtualization, aSplitAP architecture was proposed in [7], and a resourcesharing algorithm based on control theory was designed in [8].Moreover, virtualization techniques for WiMAX networks [9]and wireless-optical heterogeneous networks [10] were alsostudied.

Although some excellent researches have been done forwireless virtualization, most existing works do not considersmall cell networks with self-backhauls. Recently, small cellnetworks have been regarded as one of the key componentsof next generation cellular networks to improve spectrumefficiency and energy efficiency [11], [12]. Traditionally,thereare two kinds of backhauls in small cell networks: wiredbackhaul (e.g., optical [13] or DSL [14]) and wireless backhaul(e.g., microwave [15] or millimeter waves [16]). Since thesetraditional backhauls are very expensive for infrastructuredeployments, self-backhauled small cells have attracted greatattentions from both academia and industry [17], [18]. Wire-less self-backhauling can improve reachability and coverageby easing connectivity between nodes [18]. In [17], a self-backhauling scheme was proposed, in which the self-backhaullink uses the same spectrum with the small cell downlink, buton different time slots.

Another emerging technology for next generation wirelessnetworks is in-bandfull duplex (FD) communications. Withthe recent advances of self-interference cancellation techniques[19], [20], it is possible for radios to transmit and receivesi-

2

multaneously in the same frequency band. Due to its promisingperformance, FD communication has been considered as aneffective way to achieve self-backhauls for small cells. Theauthors of [21] and [20] made fine attempts in this direction.Nevertheless, no detail has been reported.

Despite the potential vision of small cell networks with FDself-backhauls and virtualization, many research challengesremain to be addressed. One of the main research challengesis resource allocation, which plays an important role in tradi-tional wireless networks [22]–[24]. When wireless virtualiza-tion and FD self-backhauls are jointly considered, the problemof resource allocation becomes even more challenging, sincethe backhaul and access links are coupled, which depend onvirtual resource allocation and self-interference cancellationperformance. To the best of our knowledge, the problem ofvirtual resource allocation in small cell networks with FD self-backhauls and virtualization has not been studied in previousworks. The distinct features of this paper are summarized asfollows.

• We introduce wireless virtualization into small cell net-works, and propose a virtualized small cell networkarchitecture, where multiple InPs and multiple MVNOscoexist. In the proposed virtualized small cell networks,in addition to radio spectrum, both macro base stations(MBSs) and small cell base Stations (SBSs) from differ-ent InPs are virtualized as virtual resources, which canbe dynamically shared by users from different MVNOs.

• We formulate the virtual resource allocation problem invirtualized small cell networks with FD self-backhaulsas an optimization problem, which maximizes the totalutility of all MVNOs, considering not only the revenueearned by serving users but also the cost paid to InPsfor consuming power, spectrum, and backhaul resources.In addition, we take the residual self-interference of FDcommunications into account in the formulated problem.

• Since the formulated problem is a mixed combinatorialand non-convex optimization problem, its computationalcomplexity is high. Moreover, the centralized schememay suffer from signaling overhead, outdated dynamicsinformation, and scalability issues. To solve it efficiently,we divide the original problem into two subproblems.For the first subproblem, we transfer it to a convexoptimization problem, and then solve it by an efficiental-ternating direction method of multipliers[25] (ADMM)-based distributed algorithm, in which the InPs and virtualresource manager (VRM) only need to solve their ownproblems without exchange of channel state informationwith fast convergence rate. The second subproblem is aconvex problem, which can be solved by each InP.

• Extensive simulations are conducted with different sys-tem configurations to verify the effectiveness of the pro-posed scheme. It’s shown that we can take the advantagesof both wireless network virtualization and FD self-backhauls with the proposed distributed virtual resourceallocation algorithm. InPs, MVNOs, and users can benefitfrom the proposed resource allocation scheme.

The rest of paper is organized as follows. The proposed

virtualized small cell networks architecture and the FD self-backhaul mechanism are described in II. The resource al-location problem is formulated in III. Then we divide theformulated problem into two subproblems and the solutiondetails are described in IV. Simulation results are discussed inV. Finally, we conclude this study in Section VI.

II. SYSTEM MODEL

In this section, we first describe the virtualized small cellnetwork architecture. Then we present the FD self-backhaulingmechanism where the SBSs can transmit and receive data onthe same spectrum simultaneously.

A. Virtualized Small Cell Network Architecture

We present a virtualized small cell network architecturewith multiple InPs and multiple MVNOs, as shown in Fig.1. There areM InPs offering wireless access services in acertain geographical area. Each InP deploys and manages acellular network with one MBS and several self-backhauledSBSs. There areN MVNOs, which provide various servicesto their subscribers through the same substrate networks.Following the general frameworks of wireless network virtual-ization [5], [26], the virtualized small cell network architectureconsists of three layers: thephysical resource layer(PRL),the control and management layer(CML), and theMVNOlayer. The PRL, including base stations (BSs), spectrum,power and backhauls from different InPs, is responsible forproviding available physical resources. Moreover, the PRLalso provides CML with the interfaces needed to controlresources. The CML virtualizes the physical resources fromdifferent InPs and enables the sharing for MVNOs. Then, theCML manages and allocates the virtual resources to differentMVNO users. The resource management functions in CMLare realized by avirtual network controller and a virtualresource manager(VRM). The virtual network controller ofMVNOs is responsible to collect the resource consumptionprices negotiated with InPs, and the users’ information (e.g.,payment information and QoS requirements) from MVNOs,then feedback the resource allocation results to MVNOs forthe purpose of finishing the settlement between MVNOs andInPs. To maximize the total utility of all MVNOs, the VRMis responsible to dynamically allocate the virtual resourcesfrom multiple InPs to different MVNO users. Through thevirtualization architecture above, each MVNO can have avirtual network composed of the substrate networks frommultiple InPs. Hence each user can get services via differentaccess points (either MBSs or SBSs) from different InPs.

We assume that the spectrum bandwidth of them-th InP isBm. The transmit power of the MBS and the transmit powerof the SBSs, arePm and P s

m, respectively.Sm is used torepresent the set of SBSs that belong to them-th InP. LetSjm be thej-th SBS inSm. For ease of presentation, we use

j ∈ Sm to representSjm ∈ Sm in this paper. The set of

users of MVNOi is denoted asUi. Then, letU = ∪Ni=1Ui

be the set of all the users and‖U‖ be the total number ofusers. For the users, there are two access choices: MBS or self-backhauled SBS. Those users who are associated to them-th

3

N

M

Fig. 1: A virtualized small cell network architecture.

MBS is denoted byUm. Similarly, those users who access tothe j-th SBS in them-th InP is denoted byUsj

m . To facilitatethe formulation of the virtual resource allocation problem, theself-backhauling mechanism via in-band FD communicationswill be introduced in the next subsection.

B. Small Cell Self-backhauling Mechanism Based on FullDuplex Communications

As shown in Fig. 2(a), SBSs are equipped with FD hard-ware, which enables them to backhaul data for themselves. Inthe downlink (DL), a SBS can receive data from the MBS,while simultaneously transmitting to its users on the samefrequency band. In the uplink (UL), a SBS can receive datafrom the users, while simultaneously transmitting data to theMBS on the same frequency band. In this mechanism, theSBS can effectively backhaul itself, eliminating the need fora separate backhaul solution and a separate frequency band.Therefore, self-backhauling can significantly reduce the costand complexity of rolling out small cell networks. In order todistinguish DL from UL in access and backhaul transmissions,we call the relevant links as access UL, access DL, backhaulUL, and backhaul DL, respectively. Due to the limitation ofself-interference cancellation technologies, the backhaul DLand access UL will suffer some self-interference from accessDL and backhaul UL, respectively. Different from the FD relaymechanism in [20], the spectrum can be reused by differentSBSs and the SBSs can allocate resource to their users flexiblyin our self-backhaul scheme. Compared to DL, UL usuallyhas less traffic. If the transmission of DL is satisfied, thetransmission of UL will also be satisfied. As a result, we focuson the transmission of DL in this paper.

In this paper, the orthogonal spectrum reuse pattern isadopted, where the spectrum is divided for the MBS and SBSsto avoid the inter-layer interference [27]. As shown in Fig.2(a), we divide the spectrum of InPm into two parts:αmBm

as f1 for the MBS and(1 − αm)Bm as f2 for the SBSs. InDL, the MBS transmits data to not only macro users onf1 butalso SBSs onf2 . Similarly, the MBS receives the data from

Fig. 2: A small cell self-backhauling mechanism based on fullduplex communications.

macro users onf1 and SBSs onf2 in UL. At the same time,SBSs transmit and receive data to their users onf2 . Obviously,the spectrum indicator vectorα = (α1, α2, ..., αM ), whichdecides the throughput of backhaul and access link, plays animportant role in achieving small cell self-backhauls.

For useru ∈ Ui, the VRM will decide its associationBS and allocate some resources to it from the BS. Wedenote byxm,j

u andym,ju the user’s association indicator and

allocated resources ratio (Note that this resources mean timeslot resources), respectively. When a user is associated withone BS,xm,j

u = 1, otherwisexm,ju = 0. Further, j = 0

means the user is associated to them-th MBS, j 6= 0 meansthe user is associated toSj

m. Similarly, ym,0u denotes the

resource ratio that the user gets from them-th MBS, andym,ju (j 6= 0) denotes the resource ratio that the user gets fromSjm. Obviously,ym,j

u is related toxm,ju . Only whenxm,j

u = 1,ym,ju will be meaningful.For the access DL, we denote byRm,j

u the achievable linkrate of one user. Generally,Rm,j

u and Rsjm are logarithmic

functions of signal to interference and noise ratio (SINR).Formacro users, they do not suffer interference from other BSsbecause of the different spectrum bands among InPs and theOSR pattern between macro and SBSs, so the achievable linkrate can be expressed as

Rm,0u = αmBm log

{

1 +Pmhm,0

u

σ2

}

. (1)

For small cell users, they suffer co-channel interference fromother SBSs in the same InP, so the achievable link rate can beexpressed as

Rm,ju =

(1− αm)Bm log

1 +P smhm,j

u∑

k 6=j,k∈Sm

P smhm,k

u + σ2

, j 6= 0.

(2)

In our virtualized small cell networks, each user from anyMVNO can access to either a MBS or a SBS in any InP.

4

Hence, for a given useru, we defineCu as its overall long-term rate, which can be expressed as follows.

Cmu = xm,0

u ym,0u Rm,0

u (αm)︸ ︷︷ ︸

Cm,0u

+∑

j∈Sm

xm,ju ym,j

u Rm,ju (αm)

︸ ︷︷ ︸

Cm,ju

,

(3)whereCm,0

u andCm,ju are the overall long-term rates of user

u getting from them-th MBS andSjm, respectively. For the

backhaul DL, the MBS transmits data to SBSs onf2 , andthe SBSs buffer these data to transmit to small cell users.We define the backhaul link rate ofSj

m by Rsjm . When the

SBS receives data from the MBS, it transmits data to its usersat the same time, which results in the self-interference (SI).The value of SI is determined by self-interference cancellationtechnologies, and is proportional to the transmission powerof SBS DL, which could be expressed as SI= ϑmP s

m. ϑm

represents the residual self-interference gain. Different self-interference cancellation technologies may result in differentϑm. Any self-interference cancellation technology can beapplied at the SBSs, and the analysis in this paper is a generalcase. In addition, a SBS also suffers co-channel interferencefrom other SBSs because they use the same spectrumf2 , sothe achievable link rate can be expressed as

Rsjm =

(1− αm)Bm log

1 +Pmh

sjm

ϑmP sm +

∑

k 6=j,k∈Sm

P smhsk

sj + σ2

.

(4)

In each InP, SBSs share the spectrumf2, but the spectrumresource must be divided in time domain or frequency domain(TDD or FDD) to avoid interference. If FDD is applied,the SBS will have less spectrum receiving data than thattransmitting data, which increases the difficulty to investigateSI. As a result, TDD model is adopted in this paper, as shownin Fig. 2(b). We usezsjm to denote the time slot ratio occupiedby SBSSj

m. If we denote byCsjm the overall long-term rate of

backhaul DL from them-th MBS toSjm, it can be expressed

asCsjm = z

sjmR

sjm .

In (1), (2), and (4),σ2 denotes the noise power level,hm,ju denotes the channel gain between one user and macro

(small) cell BS andhsjm denotes the channel gain between

MBS and SBS. In general, the channel gain includes path loss,shadowing and antenna gain. The association is assumed tobe carried out in a large time scale compared to the dynamicsof channels. The SINR for association is averaged over theassociation time, and thus it is a constant regardless of thedynamics of channels (i.e., fast fading is averaged out). Asfor resource allocation, we assume that resource allocation iscarried out well during the channel coherence time, and thusthe channel can be regarded as static during each resourceallocation period. This model is applicable for low mobilityenvironment.

The notations used in this paper are presented in Table I.

III. PROBLEM FORMULATION

In this section, we formulate the virtual resource allocationproblem in virtualized small cell networks with FD self-

TABLE I: Notation

Notation Definition

U set of usersS set of SBSsα spectrum allocation indicator vectorM the number of InPsN the number of MVNOsm index of MBSsi index of MVNOsj index of SBSsPm transmit power spectrum density of macro BS of InPmP sm transmit power spectrum density of SBSs of InPm

Bm the bandwidth of them-th InP’ spectrumhsjm channel gain betweenSj

m to them-th macro BShm,ju channel gain between useru to the j-th SBS in InPm

Rm,ju achievable link rate between useru to BSs of InPm

Rsjm achievable backhaul link rate of thej-th SBS in InPm

xm,ju user cell association indicator,

ym,ju user resource allocation indicator

zsjm SBS resource allocation indicatorC

m,ju rate of useru getting from BSs ofm-th InP

Cmu sum rate of useru getting from all BSs ofm-th InP

Csjm rate of the backhaul link of thej-th SBS in InPm

backhauls. Firstly, we present the utility functions for users,MVNOs, and InPs. Then, the problem of virtual resourceallocation is formulated to maximize the total utility of allMVNOs in a centralized manner at the VRM.

A. Utility Function Definition

In our virtualized small cell networks, users pay to theirMVNO for getting services. Hence, for a given useru, theutility function can be defined as the difference betweenher/his service rate and the money she/he paid, which isexpressed as

Gu =

M∑

m=1

ςCmu − δu, (5)

whereς means the profit of users for per bit rate. For simplic-ity, we assumeς = 1. MVNOs purchase resources (includingspectrum,time slot, power and backhaul) from InPs to provideservices to their users. The responsibility of the VRM is toefficiently allocate the virtual resources to maximize the totalutility of all MVNOs. In this paper, we consider a long-termrate-aware utility function for MVNOs, which is defined asthe difference between the total income of all MVNOs earnedby serving the users and the total resource consumption cost.For thei-th MVNO, its utility can be expressed as:

GMV NO =M∑

m=1

∑

u∈Ui

δuCmu −

M∑

m=1

γmTmi −

M∑

m=1

Qmi . (6)

The first term represents the income of the MVNO fromproviding services to users, andδu represents the money useru has paid to its subscribed MVNO. The second term definesthe cost of MVNO for using the spectrum and power resource,andγm represents the deliberated price between them-th InP

5

and MVNOs.Tmi can be expressed as

Tmi =

∑

u∈Um&u∈Ui

{xm,0u ym,0

u (αmBm · Pm)

+wm

∑

j∈Sm

xm,ju ym,j

u {(1− αm)Bm · P sm}

, (7)

wherebandwidth-power productis used to quantify the con-sumed wireless resources of the access cell [28], coefficientwm specifies the weight of small cell resources with respectto the MBS resources. As is well known, the load unbalanceproblem is a serious problem in small cell networks, whichleads to low utilization of SBSs [29], [22]. As a result,we choosewm < 1 to attract more users to SBSs. Thismethod was also used in [30]. The last term represents thecost of MVNO for using the backhaul resource of InPs,which depends on the amount of backhaul data of MVNOand the type of backhaul technique. It’s obvious that theamount of backhaul data for thei-th MVNO in Sj

m canbe expressed as

∑

u∈Ui&u∈Usjm

xm,ju ym,j

u Rm,ju . We define the

price of different backhaul techniques for backhaulone bitbythe price functiong(ηm), whereηm represents the backhaultechnique type of them-th InP (e.g., optical, micowave, DSL,cell communication backhaul). Thus, the cost of MVNO forusing backhaul resources can be expressed as

Qmi =

∑

j∈Sm

g(ηm)∑

u∈Ui&u∈Usjm

xm,ju ym,j

u Rm,ju . (8)

Actually, the FD self-backhaul just utilizes part of MBSpower on SBS access spectrum to achieve backhaul ratherthan additional spectrum or infrastructure, which is cheaperthan traditional backhaul methods. Mathematically,g(ηm) ofour proposed FD self-backhaul is expressed asg(ηm) =(1− αm)Pmz

sjm .

As mentioned earlier, the responsibility of the VRM isto efficiently allocate the virtual resources for the purposeof maximizing the total utility of all MVNOs. In order toguarantee the fairness during the resource allocation process,we change the utility of MVNO as follows by adopting themethod in [29],

G′MV NO =

M∑

m=1

∑

u∈Ui

δuU(Cmu )−

M∑

m=1

γmTmi −

M∑

m=1

Qmi ,

(9)

where functionU(Cmu ) is defined as

U(Cmu ) = xm,0

u log{ym,0u Rm,0

u }+∑

j∈Sm

xm,ju log{ym,0

u Rm,ju }.

(10)Then the utility function ofVRM will be changed as

G′V RM =

N∑

i=1

G′MV NO

=

N∑

i=1

{M∑

m=1

∑

u∈Ui

δuU(Cmu )−

M∑

m=1

γmTmi −

M∑

m=1

Qmi

}

.

(11)

For an InP, it not only leases its spectrum and powerresource to MVNOs and earn money from them, but alsorents or deploys infrastructure to backhaul data. So the utilityfunction of them-th InP can be expressed as

GInP = γm

N∑

i=1

Tmi +

N∑

i=1

Qmi −O, (12)

where O means the cost of renting or deploying backhaulinfrastructure. For simplicity, we assume that all the backhaulincome from MVNOs is used to pay for the rent or deploymentof backhaul infrastructure in non-self-backhauled small cell

networks, which meansN∑

i=1

Qmi = O.

B. Virtual Resource Allocation Problem Formulation

In each resource allocation cycle, the VRM needs to dynam-ically allocate the virtual resources, including MBSs, SBSs,and spectrum, to MVNOs. The objective of this paper is todevelop a slot-by-slot virtual resource allocation algorithmthat maximizes the total utility in (11) under the followingconstraints to determineα, X, Y andZ.

Firstly, in our proposed small cell self-backhaul mechanism,the DL data of small cell users are transmitted from MBS toSBSs firstly, and then the SBSs store and forward it to theirusers. In this process, the throughput of backhaul DL decidesthe throughput of access DL. Hence, we have the throughputconstraint of SBSs

C1 :∑

u∈Usjm

xm,ju ym,j

u Rm,ju ≤ zsjmRsj

m j 6= 0. (13)

Secondly, for simplicity, each user can only be served eitherby one MBS or by one SBS. Therefore, the cell associationindicator should also satisfy the following constraints.

C2 : xm,ju ∈ {0, 1}, (14)

C3 :M∑

m=1

∑

j=0orj∈Sm

xm,ju ≤ 1. (15)

Thirdly, one BS schedules its associated users on timedimension andym,j

u represents the schedule ratio of useru,the SBSs in the same InP share the frequencyf2 on timedimension to access and backhaul.z

sjm defines the schedule

ratio of SBSSjm to backhaul. So, the following condition must

be satisfied.

C4 : 0 ≤ ym,ju ≤ 1 ∀m, j, (16)

C5 :∑

u∈U0morU

sjm

xm,ju ym,j

u ≤ 1, (17)

C6 : 0 ≤ zsjm ≤ 1 ∀m, j, (18)

C7 :∑

j∈Sm

zsjm ≤ 1 ∀m. (19)

Furthermore, the spectrum allocation indicatorαm will takea value in the interval of[0, 1]. Mathematically, it can beexpressed as follows.

C8 : 0 ≤ αm ≤ 1. (20)

6

Then, based on the above constraints, the problem ofresource allocation can be formulated as follows

P : maxX,Y ,Z,α

G′V RM (21)

s.t. C1, C2, C3, C4, C5, C6, C7, C8.

IV. D ISTRIBUTED V IRTUAL RESOURCEALLOCATION

ALGORITHMS

It can be observed that the considered problem is combina-torial and non-convexity. The combinatorial nature comes fromthe integer constraintC2. The non-convexity is caused by theobjective function and constraintC1. As a result, a brute forceapproach can be used to obtain the optimal virtual resourceallocation policy. However, such a method is computationallyinfeasible for a large system and does not provide useful sys-tem design insight. To reduce the computational complexity,firstly, we assume the spectrum allocation indicator vectorα isfixed, and then we convert the original problem into a convexproblem by variable transformation and perspective functiontheory to solveX, Y andZ. Secondly, based on the obtainedresults, we can prove that the problem is convex problem aboutα, then it’s easy to get the optimizedα by some convexoptimization algorithms. Furthermore, we come back to firststep with the result ofα to get the new values ofX, Y , andZ. By iterations like this for a number of times, the valuesof X, Y , Z andα will converge, which can be seen as thesolution of original problemP. In Subsection IV-A, we solveX, Y andZ by assumingα is given. Subsection IV-B willintroduce how to solveα when X, Y andZ are fixed. InSubsection IV-C, we describe the whole process of solutionand the convergence of the proposed algorithms.

A. SolvingX, Y and Z under Fixed Spectrum AllocationIndicatorsα

For any given spectrum allocation indicator vectorα, theoriginal problem reduces to the following problem:

P1 : maxX,Y ,Z

N∑

i=1

{M∑

m=1

∑

u∈Ui

δuU(Cmu (X,Y ))

−M∑

m=1

γmTmi (X,Y )−

M∑

m=1

Qmi (X,Y ,Z)

}

(22)

s.t. C1 :∑

u∈Usjm

xm,ju ym,j

u Rm,ju ≤ zsjmRsj

m ∀j,

C2, C3, C4, C5, C6, C7

Theorem 1: If the problem P1 is feasible, the objec-tive function achieves the optimal solution only when theconstraintC1 is tight, which means

∑

u∈Usjm

xm,ju ym,j

u Rm,ju =

zsjmR

sjm , ∀j ∈ Sm must hold when the objective function gets

the maximum value.Proof: We will prove Theorem 1 by a simple contradic-

tion statement. Assume that the optimal resource allocationscheme (e.g.,xm,j

u

∗, ym,j

u

∗, zsjm

∗) for problemP1 is obtained

when∑

u∈Usjm

xm,ju

∗ym,ju

∗Rm,j

u < zsjm

∗R

sjm . Now all SBSs

can achieve backhaul data well because the backhaul rateis higher than the access rate. However, there must exist a∆p > 0 such that

∑

u∈Usjm

xm,ju

∗ym,ju

∗Rm,j

u = (zsjm

∗−∆p)R

sjm .

According to the definition ofQmi (X,Y ,Z), it’s obvious

thatQmi (xm,j

u

∗, ym,j

u

∗, z

sjm

∗) > Qm

i (xm,ju

∗, ym,j

u

∗, z

sjm

∗−∆p).

That’s to say, a larger utility for the VRM can be obtained atxm,ju

∗, ym,j

u

∗, z

sjm

∗−∆p, which also satisfies other constraints.

Therefore, there is a contradiction between this conclusionand our assumption. In other words, the optimal cell asso-ciation and resource allocation scheme is not obtained when∑

u∈Usjm

xm,ju

∗ym,ju

∗Rm,j

u < zsjm

∗R

sjm . As a result, Theorem 1 is

proved.Based on Theorem 1, we can getzsjm =

1

Rsjm

∑

u∈Usjm

xm,ju ym,j

u Rm,ju . Namely, we can replaceZ in

objective functionP1 by X and Y , which means that theoriginal problem will be transformed from a three variablesproblem into a two variables problem as following:

P1 : maxX,Y

N∑

i=1

M∑

m=1

∑

u∈Ui

δuU(Cmu (X,Y ))

−

N∑

i=1

M∑

m=1

γmTmi (X,Y )

−

M∑

m=1

∑

j∈Sm

1

Rsjm

(1− αm)Pm(∑

u∈Usjm

xm,ju ym,j

u Rm,ju )2

(23)

s.t. C2, C3, C4, C5

C6 :1

Rsjm

∑

u∈Usjm

xm,ju ym,j

u Rm,ju ≤ 1

C7 :∑

j∈Sm

1

Rsjm

∑

u∈Usjm

xm,ju ym,j

u Rm,ju ≤ 1

Although problemP1 is simplified based on Theorem 1, theabove problemP1 is still difficult to solve based on thefollowing observations:

• The feasible set ofP1 is non-convex as a result of thebinary variablesxm,j

u .• The objective function is not convex due to the product

relationship betweenxm,ju and convex function ofym,j

u .

As a result, we first transfer this problem into a convexproblem and solve it via a distributed ADMM-based algorithm.

1) Transferring P1 into convex problem :As is wellknown, a mixed discrete and non-convex optimization problemis expected to be very challenging to find its global optimum.Thus, we have to simplify problemP1. Following the ap-proach in [29], we relaxxm,j

u (j = 0 is included) inP1 C2and C3 to be real value variables such that0 ≤ xm,j

u ≤ 1.The relaxedxm,j

u can be interpreted as the time sharing factorthat represents the ratio of time when useru associates tothe m-th MBS or SBSSj

m. However, even after relaxing thevariables, the problem is still non-convex due to the non-convex objective function. Thus, to make the problemP1

7

tractable and solvable, a second step is necessary. Next, wegive a proposition of the equivalent problem ofP1.

Proposition: If we define ym,ju = ym,j

u xm,ju , ∀j and

xm,ju log

ym,jRm,ju

u

xm,ju

= 0 for xm,ju = 0, there exists an equivalent

formulation ofP1 as shown in (24).The relaxed problemP ′

1 can be recovered by substitution ofvariableym,j

u = xm,ju ym,j

u into problemP1 exceptxm,ju = 0.

Due to the loss of definition whenxm,ju = 0, it is not a one-to-

one mapping. However,xm,ju = 0 certainly holds because of

the optimality. Obviously, BS does not allocate any resourceto any user if the user does not associate with the BS. Thus, itbecomes a one-to-one mapping when the completed mappingbetween{xm,j

u , ym,ju } and{xm,j

u , ym,ju } is defined as

ym,ju =

{ym,ju

xm,ju

if xm,ju > 0

0 if otherwise. (24)

Holding thePropositionand well-known perspective func-tion in convex optimization theory [31], we have the followingtheory that gives the convexity ofP ′

1.Theorem 2: If problemP ′

1 is feasible, it is jointly convexwith respect to all optimization variablesxm,j

u and ym,ju .

Proof: Since the constraintsC2′, C3′, C4′, C5′, C6, C7 arelinear, they are obviously convex. Due to the fact that objectivefunctionP ′

1 is a linear sum ofU(Cmu

′), Tmi

′ andQmi

′, we justneed to prove the convexity ofU(Cm

u′), Tm

i′ andQm

i′, based

on the following principle:the linear sum of convex functionsis still convex[31].

The convexity proof ofU(Cmu

′) is similar to [32], whichwill be described briefly as follows. Firstly, we prove thecontinuity of functionf(t, x) = x log(t/x), t ≥ 0, x ≥ 0at the point of x = 0. Let s = t/x, then f(t, 0) =limx→0

x log tx

= lims→∞

tslog s = t lim

s→∞

log ss

= 0. Function

f(t, x) = x log(t/x), t ≥ 0, x ≥ 0 is the well-knownper-spective operationof logarithmic function, and the perspectivefunction of a convex function is also a convex function basedon [31]. Sincexm,j

u logym,ju

xm,ju

is th perspective function of

log ym,ju , and the functionlog ym,j

u is convex aboutym,ju ,

xm,ju log

ym,ju

xm,ju

is convex. What’s more, functionU(Cmu

′) is a

weighted sum of series of convex functionxm,ju log

ym,ju

xm,ju

, so

it is also a convex function. FunctionTmi

′ is linear functionof ym,j

u , and the convexity of it is obvious. Since the functionQm

i′ is a sum of quadratic function about(ym,j

u )2, we caneasily know it is a concave function by solving the secondderivative of(ym,j

u )2. With the theory that a negative concaveproblem is a convex problem, we can conclude that the formof the objective function inP ′

1 is a linear sum of convexproblems. With the addition that all the constraints are convex,the convexity of theP ′

1 is proved.Since problemP ′

1 is a convex problem, a lot of methods(e.g., interior point method) can be used to solve it. However,with the increase of the number of BSs and users, thesize of problem will be very large. Practically, even with apowerful computing center, the overhead of deliver enoughlocal information (e.g., channel status information (CSI)) tothe global center is extremely inefficient. Therefore, for the

purpose of implementation, a distributed algorithm runningon each InP should be adopted.

2) ADMM-based solution algorithm ofP1: Due to con-straintC3′ in P ′

1, the problem is not separable with respectto different InPs. To apply ADMM to the resource allocationproblem P ′

1, this coupling must be handled appropriately.Therefore, we introducelocal copyX‡

m of the relatedglobalcell association variableX for them-th InP. Roughly speak-ing, eachlocal variable can be interpreted as the InP’s opinionabout the correspondingglobal variable. Naturally, variablesYm is also local variables for InPm, since the InP operateswithout any limits from other InP. For the sake of brevity,let us define vectorAm = (X‡

m, Ym) to represent thelocalvariables of them-th InP. Inspired by [31], to deal with theconstraints, we introduce an indicator functiong(Am) suchthat g(Am) = 0 whenAm ∈ Φ; otherwise,g(Am) = +∞,whereΦ represents the feasible set of problemP ′

1. With thesenotations above, problemP ′

1 of maximizingGV RM on setΦis equivalent to

P′′1 : min

Am

{−GmVRM (Am) + g(Am)} (25)

s.t.X‡m −X = 0

where

GmV RM (Am) =

N∑

i=1

∑

u∈Ui

δuU(Cmu

′(X‡m, Y )

−

N∑

i=1

γmTmi

′(X‡m, Y )−Qm′(X‡

m, Y ) (26)

It can be seen that the objective function is separable acrossInPs in the virtualized small networks. However, theglobalcell association variables involved in the consensus constraintscouple the problem with respect to the InPs. Therefore, thebasic idea to solve problemP ′′

1 is that each InP only deter-mines itslocal variables based on the local information, andVRM is responsible for achieving consensus between thelocalvariables and theglobal variable according to the consensusconstraint.

We apply ADMM [33] for solving the problem inP ′′1 in a

distributed way. The machinery of the ADMM applied toP ′′1

is initiated by forming anaugmented Lagrangianwith respectto the consensus constraints. The augmented Lagrangian notonly includes a set of consensus constraints weighted byLagrange multipliers (conventional Lagrangian), but it alsoinvolves an additional regularized quadratic term: a squared‖L‖2 norm with respect to the consensus constraints.

Let λm be the Lagrange multipliers set associated withconsensus constraints ofP ′′

1 . Thus, the augmented Lagrangianfor P ′′

1 can be written as

Lρ(Am,X) = −GmVRM + g(Am) + λm(X‡

m −X)

+ρ

2

M∑

m=1

‖X‡m −X‖22 (27)

whereρ ∈ R++ is a positive constant parameter for adjustingthe convergence speed of the ADMM. Note that due tothe structured interconnections in the virtualized small cell

8

P′1 : max

X,Y

N∑

i=1

M∑

m=1

∑

u∈Ui

δu

xm,0u log

ym,0u Rm,0

u

xm,0u

+∑

j∈Sm

xm,ju log

ym,ju Rm,j

u

xm,ju

︸ ︷︷ ︸

U(Cmu

′)

−

N∑

i=1

M∑

m=1

γm∑

u∈Um&u∈Ui

ym,0u (αmBm · Pm) + wm

∑

j∈Sm

ym,ju (1− αm)Bm · P s

m

︸ ︷︷ ︸

Tmi

′

(24)

−

M∑

m=1

∑

j∈Sm

1

Rsjm

(1− αm)Pm(∑

u∈Usjm

ym,ju Rm,j

u )2

︸ ︷︷ ︸

Qm′

s.t. C2′ : 0 ≤ xm,ju ≤ 1, ∀j, C3′ :

M∑

m=1

∑

j=0&j∈Sm

xm,ju = 1, C4′ : 0 ≤ ym,j

u ≤ xm,ju , C5′ :

∑

u∈U

ym,ju = 1, ∀m, j

C6′ :1

Rsjm

∑

u∈Usjm

ym,0u Rm,j

u ≤ 1, C7′ :∑

j∈Sm

1

Rsjm

∑

u∈Usjm

ym,0u Rm,j

u ≤ 1.

networks, the augmented Lagrangian is separable with respectto the InPs.

Fundamentally, the augmentation can be seen as a penaltyterm added to the primal objective function [25]. The regular-ization term facilitates the algorithm to drive thelocal and therelatedglobalvariables into consensus. It’s worth emphasizingthat the solution of the original optimization problemP ′′

1

is not affected by adding the quadratic regularization termto the Lagrangian since it vanishes for any set of primalfeasible variables. The ADMM method consists of sequentialoptimization phases over the primal variables followed by themethod of multipliers update for the dual variables [25]. Byapplying the ADMM to the problem inP ′′

1 , we first minimizethe augmented Lagrangian in (27) over thelocal variables,then over theglobal variables, and finally, perform the dualvariable update. Thus, the ADMM method consists of thefollowing steps:

At+1m =arg min

X‡m,Ym

{

−GmVRM (Am) + g(Am)

+[λTm]t(X‡ − [X]t) +

ρ

2‖X‡ − [X]t‖22

}

(28)

Xt+1 =arg min

X

{M∑

m=1

[λTm]t([X‡

m]t+1 −X)

+ρ

2

M∑

m=1

‖[X‡m]t+1 −X‖22

}

(29)

[λm]t+1 = [λm]t + ρ([X‡m]t+1 − [X]t+1) (30)

where t stands for the iteration index of the ADMM algo-rithm. The basic idea behind the ADMM iterations is thatwe first minimize the augmented Lagrangian with respect tolocal variables(X‡

m, Ym) in A-update, and then the VRM

calculates theglobal variableX based on alllocal variables,finally update the Lagrange multipliers. In the following, wewill deal with how to updateAm, X andλ in each iteration.Moreover, the distributed implementation of each update willbe also discussed.

a) Am-Update: Interestingly, as mentioned earlier, it can befound thatAm-update is separable across each InP. Therefore,Am-update can be decomposed intoM subproblems, whichcan be solved locally at each InP. After dropping off theconstant terms in (28) which do not affect the solution, thesubproblem to be solved at InPm can be given as

minX

‡m,Ym

− GmV RM (X‡

m, Ym)

+∑

u∈Um

∑

j=0&j∈Sm

{(x‡m,j

u − [xm,ju ]t)2 + λm,j

u x‡m,j

u }

(31)

s.t.Am ∈ Φ

Obviously, problem (31) is a convex problem because of theconvexity of problemP ′

1. For the purpose of fast convergence,steepest descent method [31] will be adopted at each InP torealizeAm-update.

Recall that we have relaxed the cell association indicator tobe a real value between zero and one instead of a Boolean atthe beginning of this subsection, hence we have to recover itto a Boolean after we get the optimal solution of subproblem(31). Assumexm,j

u

∗, ym,j

u

∗(including j = 0 ) and z

sjm

∗

are the optimal solution of (31) obtained directly from thesteepest descent method. Inspired by [34], we first compute themarginal benefit for eachxm,j

u as follows.Dm,ju =

∂Lρ(Am)

∂xm,ju

,where∂Lρ(Xm) is the objective function of problem (31).

9

Then, the indicatorxm,ju

∗can be recovered to zero or one by

xm,ju

∗=

{

1 if xm,ju

∗= max

m,jDm,j

u andDm,ju ≥ 0

0 otherwise.

(32)After recovering the indicatorxm,j

u

∗, we resolve the problem

(31) to get the optimal solution ofym,ju according to the

known recoveredxm,ju

∗, which is easy due to the fact that

the problem (31) will be concave respect toym,ju . Note that

this is a common method to deal with the indicator variables inresource allocation, which is widely adopted in the literature(e.g., [34]).

b) X-Update: Due to the added quadratic regularizationterm in the augmented Lagrangian (27), the objective in (29)isstrictly convex inxm,j

u . Therefore, the unique optimal solutionis found by setting the derivative to zero that results in

X =1

M

M∑

m=1

[X‡m]t+1 +

1

Mρ

M∑

m=1

[λm]t (33)

By usingM∑

m=1[λm]t = 0, it will result that X =

1M

M∑

m=1[X‡

m]t+1 at each iteration [25]. Namely, theglobal

variables are obtained at each iterationt by averaging out thecorresponding updatedlocal copies. This introduces a philos-ophy of interpreting the currentlocal copies as InP’s opin-ions about the optimalglobal variables. Since the Lagrangemultipliers are not involved in (33), the local communicationoverhead is reduced in this step.

c) λ-Update : Compared toAm-update andX-update,the process ofλ-update is quite simple. After receiving theupdatedglobal variablesXt+1, λ-update can be performeddirectly via (30) at each InP in each iteration.

The iteration process of ADMM-based solution algorithmis concluded inAlgorithm 1 .

Algorithm 1 ADMM-based solution algorithm

1: Initializationa) At each InPm, collect channel state information of allusers within its coverage;b) Initialize X

0 ∈ Φ, λ0 > 0 and a stop criterion

thresholdξ2 at the VRM2: while ‖Xt+1 −X

t‖ > ξ2a) BroadcastXt andλt to each InP;b) At each InPm, calculateAm

t+1;c) At the VRM, updateXt+1 by combing the results ofX

‡m

t+1from each InP;

d) UpdateXt+1 via (33);e) Updateλt+1 via (30) at VRM;End

3: RecoverY andZ4: Output the optimal resource allocation schemeX

∗, Y ∗

andZ∗

B. Solvingα under Fixed Resource Allocation SchemeX, Y ,andZ

For any fixed feasible resource allocation schemeX, Y ,andZ, the original problemP will be reduced to a simplifiedoptimized problem with variablesα as follows:

P2 : maxα

N∑

i=1

{M∑

m=1

∑

u∈Ui

δuU(Cmu (αm))

−

M∑

m=1

γmTmi (αm)−

M∑

m=1

Qmi (αm)

}

(34)

s.t. C1 :∑

u∈Usjm

xm,ju ym,j

u Rm,ju (αm) ≤ zsjmRsj

m(αm) ∀j & j 6= 0

C8

Then we have the following property.Property : If problem P2 is feasible, it’s jointly convex

with respect to the optimization variables{αm, ∀m}.Proof: Since the constraintsC1 and C8 are linear, they

are obviously convex. Nextly, we prove the convexity of theobjective function. For convenience, we usef(α) to representthe objective function inP2, it is easy to know ∂2f

∂αm2 < 0. So,

the objective function ofP2 is convex and then the convexityof problemP2 is proved.

Since problemP2 is a convex problem, many solutions canbe used to solve this problem. In this paper, considering thefast convergence, we still adopt the steepest descent methodto solve this problem. What’s more, it can be observed thatthere are no coupled relationship among differentαm, so thesolution ofP2 can be divided intom subproblems and thesesubproblems can be solved in each InP.

C. Overall Algorithm: The Distributed Virtual Resource Allo-cation Algorithm

Based on the analysis of Subsections IV-A and IV-B,the distributed virtual resource allocation algorithm canbesummarized asAlgorithm 2 . In this algorithm,α is a longtime-scaleoptimization variable, whileX, Y andZ areshorttime-scaleoptimization variables. In a relatively long period,the InPs do not change the values ofα, and they solve theircorresponding subproblems in parallel in each iteration tooptimize their local variables by using local CSI informationand transmit their local results to the VRM. Actually, eachlocal variable can be interpreted as InP’s opinion about thecorresponding global variable. Then the VRM collects all thelocal results and coordinates all the InPs to achieve the globalconsensus based on the consensus constraint and the regular-ization function. Upon the convergence of the cell associationand resource scheme, the InPs attempt to change the value ofα to get the optimal gain, and then the network will start thenext round cell association and resource allocation adjustment.By this way, there is no need to exchange CSI informationbetween InPs and VRM, which will reduce the signalingoverhead significantly. Furthermore, since the combinatorialnature and non-convexity of considered problem have beenremoved through the problem transformation in Subsections

10

IV-A and IV-B, the computational complexity to solve theoriginal problem has been reduced to a reasonable level.

Algorithm 2 Distributed virtual resource allocation algorithmof small cell networks with FD self-backhauls and virtualiza-tion

1: Initializationa) At each InPm, collect channel state information of allusers within its coverage;b) Initializeα0 = 0.5, o = 0 and a stop criterion thresholdξ1 at the VRM;

2: while ‖G′V RM

o+1−G′

V RMo‖22 > ξ1

Push the value ofαom to them-th InP

Run Algorithm 2Updateαo → α

o+1 based on the result ofX∗, Y ∗

End3: Output the optimal resource allocation schemeX

∗, Y ∗,Z

∗ andα∗

V. SIMULATION RESULTS AND DISCUSSIONS

In this section, the effectiveness of our proposed virtualizedsmall cell networks with FD self-backhauls and distributedvirtual resource allocation algorithm will be demonstratedby computer simulations. In the simulations, we consider a1Km × 1Km square area covered by two InPs and twoMVNOs. In each InP, there are one macro BS and four SBSs.Each MVNO owns some subscribed users. The number ofsubscribed users in each MVNO will be varied in differentsimulation scenarios, and they are randomly located in thewhole area. The available bandwidth of both of the twoInPs are10 MHz. The transmit power of the macro BS andthe transmit power of the SBS are46dBm and 20dBm,respectively. The channel propagation model refers to [35].One macro BS is located at the center of the left half of thearea, while the other macro BS is located at the center of theright half. The SBSs are randomly deployed in the area. Thisdeployment of BSs is based on the consensus that the locationof macro BS is usually calculated by network planning but thelocation of SBS (e.g., femtocell) may depend on the users.

A. Performance Evaluation of the Proposed Virtualized SmallCell Networks with FD Self-backhauls

In this subsection, we evaluate the performance of theproposed virtualized small cell networks with self-backhaulsby comparing the following schemes: (a) A traditional smallcell network without FD self-backhaul and virtualization,which is similar to that in [36]; (b) a small cell network withvirtualization but without FD self-backhaul, which is similarto that in [37]; (c) a small cell network with FD self-backhaulbut without virtualization, which is similar to that in [19]. Eachscheme has the similar system configurations as describedabove. For the schemes with virtualization, we optimize thetotal utility function defined in (11). For the schemes withoutvirtualization, MVNOs demote to general network operationswith one InP, and the utility function is optimized separately.In this subsection, the augmented Lagrangian parameterρ is

10 20 30 40 50 60 70 80 90 1000

0.5

1

1.5

2

2.5

3

3.5

4x 10

7

The number of users

The

tota

l util

ity o

f MV

NO

s

Proposed scheme with FD self−backhaul and virtualizationExisting scheme without FD self−backhaul but with virtualizationExisting scheme with FD self−backhaul but without virtualizationExisting scheme without FD self−backhaul and virtualization

Fig. 3: The total utility of MVNOs withδu = 106 andγi = 5.

set to5 ∗ 107, and the small cell discounting pricew is set to10−3.

In Figs. 3-6, we compare the utility of MVNOs, users, andInPs, as well as the resource utilization ratio of networks withdifferent schemes. As shown in these four figures, the schemewith FD self-backhaul always outperforms the schemes with-out FD self-backhaul. This is because the proposed schemewith FD self-backhaul is able to reduce the cost of backhauland improve the SBS utility ratio, which means that MVNOscan get more available resources at a lower price. As a result,utility values of MVNOs and users are improved. For InPs, themore users access to SBSs, the more backhaul revenue theywill get in the small cell network with self-backhaul. However,in traditional small cell networks, the backhaul revenue isusedto pay for the backhaul infrastructure rent or construction,and then InPs can only get the resource consumption revenuein access links. This is the reason why the utility of InPwith FD self-backhaul is higher than that without FD self-backhaul. Furthermore, there is appreciable performance gainof our proposed scheme compared to traditional schemeswithout virtualization. The reason is that, with infrastructurevirtualization, a user is able to connect to a better access pointwith better channel conditions and lower resource consumptionprice. That is to say, access point selection gain and spectrumselection gain can be obtained from our proposed virtualizedsmall cell networks. Therefore, our proposed virtualized smallcell networks with FD self-backhauls can take the advantageofboth wireless networks virtualization and FD self-backhauls.

In addition, as shown in Figs. 3-6, with the growth ofthe number of users, the total utility of MVNOs increaseslinearly, the average utility of users decreases slowly, andthe total utility of InPs and the resource utilization ratioofnetworks will increase, but the increase rate becomes moreand more slow. Because MVNOs have to pay money to InPsfor using resources, the VRM only allocates optimal resourceamount to users rather than all resources. As a result, thetotal utility of MVNOs will grow since more users will bringmore income, and the total utility of InPs and the resourceutilization ratio will also go up due to the fact that moreresources are consumed. Meanwhile, the average utility ofusers will descend because some users with bad channel

11

10 20 30 40 50 60 70 80 90 1000.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4x 10

5

The number of users

The

util

ity o

f use

rs

Proposed scheme with FD self−backhaul and virtualizationExisting scheme without FD self−backhaul but with virtualizationExisting scheme with FD self−backhaul but without virtualizationExisting scheme without FD self−backhaul and virtualization

Fig. 4: The average utility of users withδu = 106 andγi = 5.

1 2 3 4 5 6 7 8 9 100

1

2

3

4

5

6

7

8

9x 10

5

The number of users

The

tota

l util

ity o

f InP

s

Proposed scheme with FD self−backhaul and virtualizationExisting scheme without FD self−backhaul but with virtualizationExisting scheme with FD self−backhaul but without virtualizationExisting scheme without FD self−backhaul and virtualization

Fig. 5: The total utility of InPs withδu = 106 andγi = 5.

10 20 30 40 50 60 70 80 90 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

The number of users

The

res

ourc

e ut

iliza

tion

ratio

Proposed scheme with FD self−backhaul and virtualizationExisting scheme without FD self−backhaul but with virtualizationExisting scheme with FD self−backhaul but without virtualizationExisting scheme without FD self−backhaul and virtualization

Fig. 6: The resource utilization ratio withδu = 106 andγi = 5.

condition access to the network. However, when the numberof users is large enough, the ratio of users with bad channelcondition will increase and the average link rate of users willdecrease accordingly. Considering the resource consumptionprice, the VRM will not allocate more resources to usersbecause the service rate gain will be lower than the cost ofconsuming resources. So, the resource utilization ratio and thetotal utility of InPs will grow no more. Nevertheless, the total

−50 −40 −30 −20 −100

0.5

1

1.5

2

2.5x 10

7

The residual self−interference gain(dB)

The

tota

l util

ity o

f MV

NO

s

The number of users is 60The number of users is 40The number of users is 20

Fig. 7: The total MVNO utility of different residual self-interference gain withδu = 106 andγi = 5.

−50 −40 −30 −20 −100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

The residual self−interference gain(dB)

The

rat

io o

f use

rs a

cces

sing

to s

mal

l cel

l BS

s

The number of users is 60The number of users is 40The number of users is 20

Fig. 8: The ratio of users accessing to SBSs of differentresidual self-interference gain withδu = 106 andγi = 5.

utility of MVNOs will keep increasing because of the multi-user diversity gain. This is also the reason why the averageutility of users does not decline sharply.

B. The Effect of Self-interference on Our System Performance

In Fig.7 and Fig. 8, we evaluate the effect of self-interference on the total utility of MVNOs and the ratio ofusers accessing to SBSs, respectively. As shown in Fig.7, thetotal MVNO utility will decrease and the rate of decreasespeeds up with the increase of the residual self-interferencegain. This is because that it will consume more resourcesfor users to access to SBSs because of the terrible backhaullink leading by high residual self-interference gain, and thenthe users have to access to the relatively expensive MBS,which can be find in Fig. 8. What’s more, due to the limitof MBSs’ resource, the users’ service rate getting from MBSswill become lower and lower with the users who access toMBSs increasing. This also is why the total MVNO utility ismore sensitive to the residual self-interference gain whenthereare more users. As shown in Fig. 8, there no users accessingto SBSs when the residual self-backhaul gain is−10 dB,

12

10 20 30 40 50 60 70 80 90 1004

4.2

4.4

4.6

4.8

5

5.2x 10

8

Iteration Steps

The

tota

l Util

ity o

f MV

NO

Centrilized Algorithm

ADMM−based Algorithm with ρ=8*107

ADMM−based Algorithm with ρ=5*107

Fig. 9: The convergence process of ADMM and the effectof parameterρ (2 ∗ 4 SBSs,40 users,α = [0.5, 0.5], andw = [1, 1]).

2 4 6 8 10 12 14 16 180.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Iteration Steps

The

Val

ue o

f α1

Initial α

1=0.9, w=1

Initial α1=0.1, w=1

Initial α1=0.9, w=10−3

Initial α1=0.1, w=10−3

Fig. 10: The convergence process ofα and the effect ofparameterw (2 ∗ 4 SBSs,40 users, andρ = 5 ∗ 107).

which waste the resource of SBSs and decrease the spectrumreuse ratio. Therefore, a good self-interference cancellationtechnology is critical to our FD self-backhaul scheme forSBSs.

C. The Convergence of the Algorithms

To demonstrate the performance of our proposed schemefurther, we show the good convergence performance of ourproposed distributed virtual resource allocation algorithm invirtualized small cell networks. The convergence performanceincludes not only the ADMM-based algorithm inAlgorithm1 but also the overall algorithm inAlgorithm 2 .

Fig. 9 shows the convergence of the proposed ADMM-basedAlgorithm 1 and the effect of parameterρ in ADMM. Asshown in this figure, the gap between the ADMM-based al-gorithm and the centralized algorithm is narrow, which meansthe effectiveness of ADMM-based algorithm is equivalent tothe centralized algorithm in terms of the overall utility. It canbe found that the results with differentρ finally converge toalmost the same utility value with only a small gap. However,

2 4 6 8 10 12 14 16 18 205.09

5.095

5.1

5.105

5.11

5.115

5.12

5.125

5.13

5.135

5.14x 10

8

Iteration Steps

The

Tot

al U

tility

of M

VN

O

Initial α1=0.9, w=1

Initial α1=0.1, w=1

Initial α1=0.9, w=10−3

Initial α1=0.1, w=10−3

Fig. 11: The convergence process of overall algorithm and theeffect of parameterw (2 ∗ 4 SBSs,40 users, andρ = 109).

the value ofρ affects the rate of convergence.ρ = 5 ∗ 107

gives higher convergence rate thanρ = 8 ∗ 107 especiallybefore the10-th iteration. Furthermore, a significant decreaseof utility gap between centralized algorithm and proposedADMM-based algorithm can be found from the1-st iterationto the10-th iteration. After the10-th iteration, the gain of moreiterations is still increasing but with less rate. Thus, a tradeoffexits between acceptable utility value and iteration steps.What’s more, after multiple experiments, we find the orderof magnitudes ofρ must approach that ofGm

VRM ; otherwise,the proposed ADMM-based algorithm will not converge.

Fig. 10 and Fig. 11 show the convergence of the overallalgorithm in Algorithm 2 and the influence factors of con-vergence. As shown in Fig. 10, the final value ofα1 almostconverges to one constant, although they begin from differentinitial values. Similarly, the total utility of MVNOs withdifferent initial (α1, α2) are very close in Fig. 11. This showsthe robustness of our proposed overall algorithm with differentinitial values of α. InPs can derive new cell associationand resource allocation policies when the network situationchanges (e.g., new users arrive), rather than waiting for therelated message from VRM, which presents the improvementof self-adaption ability of the network. What’s more, it canbe found in Fig. 10 that the value ofw has an influence onthe final value ofα. Whenw = 1, it means no discount forMVNO to consume small cell resources, and then the MBShas some superiority to associate more users compared toSBSs because the difference of transmission parameters (e.g.,transmission power, antenna gain, the height of BS tower, etc.).However, whenw = 10−3, this superiority of the MBS willbe counteracted by the price discount of SBSs, and then somemacro users will tend to access to SBSs. This is the reasonwhy the final convergence value ofα1 whenw1 = 10−3 ishigher than that whenw1 = 1. Corresponding to Fig. 10,the total utility of MVNOs whenw = 10−3 is higher thanthat whenw = 1. One reason is that the cost of MVNOsfor consuming resources decreases as we described in Fig.10. Another reason is that the load of the network becomesmore balanced by adjustingw, and then the utilization ratio

13

of SBSs improves, which results in the average throughputimprovement of users.

VI. CONCLUSION AND FUTURE WORK

In this paper, we investigated the virtual resource allocationissues in small cell networks with FD self-backhauls andvirtualization. We first introduced the idea of wireless networkvirtualization into small cell networks, and proposed a virtualresource management architecture, where radio spectrum, timeslots, MBSs, and SBSs are virtualized as virtual resources.After virtualization, users can access to different InPs toget performance gain. In addition, we proposed to use FDcommunications for small cell backhauls. Furthermore, weformulated the virtual resource allocation problem as an opti-mization problem by maximizing the total utility of MVNOs.In order to solve it efficiently, the virtual resource allocationproblem is decomposed into two subproblems. In this process,we transferred the first subproblem into a convex problemand solved it by our proposed ADMM-based distributed al-gorithm, which can reduce the computation complexity andoverhead. The second subproblem can be solved by each InPeasily because of its convexity and incoherence among InPs.Simulation results showed that the proposed virtualized smallcell networks with FD self-backhauls are able to take theadvantages of both wireless network virtualization and FDself-backhauls. MVNOs, InPs, and users could benefit fromit, and the average throughput of the small cell networks canbe improved significantly. In addition, simulation resultsalsodemonstrated the effectiveness and good convergence perfor-mance of our proposed distributed virtual resource allocationalgorithm. Future work is in progress to consider information-centric networking [38] in our proposed framework.

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Lei Chen received the B.S. degree in Communica-tion Engineering from Beijing University of Postsand Telecommunications (BUPT), Beijing, China,in 2011. He is currently working toward the Ph.D.degree with the School of Information and Com-munication Engineering, BUPT, Beijing, China. Heis also with the University of British Columbiaas a visiting scholar since Nov. 2014. His currentresearch interests include 5G cellular network, full-duplex wireless, resource management, cross-layerdesign, and small cell networks.

F. Richard Yu (S’00-M’04-SM’08) received thePhD degree in electrical engineering from the Uni-versity of British Columbia (UBC) in 2003. From2002 to 2004, he was with Ericsson (in Lund,Sweden), where he worked on the research anddevelopment of wireless mobile systems. From 2005to 2006, he was with a start-up in California, USA,where he worked on the research and developmentin the areas of advanced wireless communicationtechnologies and new standards. He joined Car-leton School of Information Technology and the

Department of Systems and Computer Engineering at CarletonUniversityin 2007, where he is currently an Associate Professor. He received theIEEE Outstanding Leadership Award in 2013, Carleton Research AchievementAward in 2012, the Ontario Early Researcher Award (formerlyPremiersResearch Excellence Award) in 2011, the Excellent Contribution Award atIEEE/IFIP TrustCom 2010, the Leadership Opportunity Fund Award fromCanada Foundation of Innovation in 2009 and the Best Paper Awards at IEEEICC 2014, Globecom 2012, IEEE/IFIP TrustCom 2009 and Int’l Conferenceon Networking 2005. His research interests include cross-layer/cross-systemdesign, security, green IT and QoS provisioning in wireless-based systems.

He serves on the editorial boards of several journals, including Co-Editor-in-Chief for Ad Hoc & Sensor Wireless Networks, LeadSeriesEditor for IEEE Transactions on Vehicular Technology, IEEECommuni-cations Surveys & Tutorials, EURASIP Journal on Wireless Communica-tions Networking, Wiley Journal on Security and Communication Networks,and International Journal of Wireless Communications and Networking, aGuest Editor for IEEE Transactions on Emerging Topics in Computingspecial issue on Advances in Mobile Cloud Computing, and a Guest Editorfor IEEE Systems Journal for the special issue on Smart Grid Commu-nications Systems. He has served on the Technical Program Committee(TPC) of numerous conferences, as the TPC Co-Chair of IEEE Green-Com15, INFOCOM-MCV15, Globecom14, WiVEC14, INFOCOM-MCC14,Globecom13, GreenCom13, CCNC13, INFOCOM-CCSES12, ICC-GCN12,VTC12S, Globecom11, INFOCOM-GCN11, INFOCOM-CWCN10, IEEEIWCMC09, VTC08F and WiN-ITS07, as the Publication Chair of ICSTQShine10, and the Co-Chair of ICUMT-CWCN09. Dr. Yu is a registeredProfessional Engineer in the province of Ontario, Canada.

Hong Ji received the B.S. degree in communicationsengineering and the M.S. and Ph. D degrees ininformation and communications engineering fromthe Beijing university of Posts and Telecommunica-tions (BUPT), Beijing, China, in 1989, 1992, and2002, respectively. From June to December 2006,she was a Visiting Scholar with the University ofBritish Columbia, Vancouver, BC, Canada. She iscurrently a Professor with BUPT. She also works onnational science research projects, including the Hi-Tech Research and Development Program of China

(863 program), The National Natural Science Foundation of China, etc. Herresearch interests include heterogeneous networks, peer-to-peer protocols,cognitive radio networks, relay networks, Long-Term Evolution/fifth gener-ation, and cooperative communications.

Gang Liu received the B.S. degree in Electronicsand Information Engineering from Sichuan Uni-versity, Sichuan, China, in 2010. He is currentlyworking toward the Ph.D. degree with the School ofInformation and Communication Engineering, Bei-jing University of Posts and Telecommunications,Beijing, China. He is also with the University ofBritish Columbia as a visiting scholar since Nov.2013. His current research interests include 5G cel-lular network, full-duplex wireless, resource man-agement, cross-layer design, and TCP/IP protocol

in satellite communication systems. He won the second prizein the 2009National Undergraduate Electronic Design Contest and BestPaper Awardin IEEE ICC’2014. He has served as reviewer for International Journal ofCommunication system, China Communications, ICC’2012, Globecom’2013and so on.

Victor C.M. Leung (S75-M89-SM97-F03) receivedthe B.A.Sc. (Hons.) degree in electrical engineeringfrom the University of British Columbia (U.B.C.)in 1977, and was awarded the APEBC Gold Medalas the head of the graduating class in the Facultyof Applied Science. He attended graduate schoolat U.B.C. on a Natural Sciences and Engineer-ing Research Council Postgraduate Scholarship andcompleted the Ph.D. degree in electrical engineeringin 1981.

From 1981 to 1987, Dr. Leung was a SeniorMember of Technical Staffin the satellite systems group at MPR Teltech Ltd.He started his academic career in the Department of Electronics at the ChineseUniversity of Hong Kong in 1988. He returned to U.B.C. as a faculty memberin 1989, where he currently holds the positions of Professorand TELUSMobility Research Chair in Advanced Telecommunications Engineering in theDepartment of Electrical and Computer Engineering. He is a member of theInstitute for Computing, Information and Cognitive Systems at U.B.C. He alsoholds adjunct/guest faculty appointments at Jilin University, Beijing JiaotongUniversity, South China University of Technology, the HongKong PolytechnicUniversity and Beijing University of Posts and Telecommunications in China.Dr. Leung has co-authored more than 500 technical papers in internationaljournals and conference proceedings, and several of these papers had beenselected for best paper awards. His research interests cover broad areas ofwireless networks and mobile systems.