Network Configuration for Two-tier Macro-FemtoSystems with Hybrid AccessBinglai Niu and Vincent W.S. Wong, Senior Member, IEEE

Abstract—In this paper, we study uplink network configurationin a two-tier macro-femto heterogeneous system with hybridaccess control. We consider a system where one macro basestation and a cluster of adjacent femto base stations togetherserve a number of mobile users. In this system, base stations andusers make decisions in various network configuration processeswith different optimization objectives. Such decision makingprocesses are usually correlated and an efficient mechanism isneeded to coordinate the decision makers. In this work, wepropose a five-stage network configuration mechanism whereaccess control, resource allocation and power management areperformed sequentially at the base stations and users, respec-tively. We show that this mechanism provides incentive forthe FBSs to operate at the hybrid access mode. We modelthe configuration mechanism as a multi-stage decision makingprocess and formulate a multi-level optimization problem. Weanalyze the problem in a bottom-up manner, and propose efficientalgorithms to solve the optimization problem in each levelsequentially. Simulation results show that the proposed networkconfiguration mechanism achieves a higher system utility thanconfiguration mechanisms with topology-based hybrid access orclosed access.

Index Terms—Network configuration, macro-femto heteroge-neous systems, hybrid access, multi-level optimization.

I. INTRODUCTION

THE increasing demand of mobile data service has trig-gered the evolution of cellular architecture from single-tier networks towards multi-tier heterogeneous networks [2].Recently, there is an emerging trend of deploying low costand low power base stations overlay the existing cellularinfrastructure to extend the coverage area and increase thesystem capacity. In particular, in urban residential area, in-stalling femto base stations (FBSs) in houses and apartmentshas been considered as an efficient approach to improve theindoor signal quality [3]. FBSs are inexpensive low power basestations referred as home nodeB or home eNodeB in the ThirdGeneration Partnership Project (3GPP). FBSs are installed byusers and can serve a small number of subscribers at theoperator’s licensed spectrum [4]. By deploying FBSs, homeusers’ devices can experience better indoor signal quality and

Manuscript received on Mar 3, 2014; revised on August 21, 2014, February14, 2015; and accepted on March 29, 2015. This work was supported by theNatural Sciences and Engineering Research Council (NSERC) of Canada. Partof this paper was presented at the IEEE Global Communications Conference(Globecom’13), Atlanta, GA, Dec. 2013 [1]. The review of this paper wascoordinated by Dr. Berk Canberk.

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.

B. Niu and V. W. S. Wong are with the Department of Electrical andComputer Engineering, The University of British Columbia, Vancouver, BC,Canada, V6T 1Z4, e-mail: {bniu,vincentw}@ece.ubc.ca.

longer battery life due to the close proximity. The operatorcan also offload the data traffic at the macro base stations(MBSs) to improve the system capacity [5]. Despite thepromise, the deployment of FBSs also introduces technicalchallenges, such as interference management between themacrocell and femtocells and adaptive configuration of thesystem parameters [6]. Several approaches have been proposedto address the interference issue in heterogeneous networks,such as frequency planning, orthogonal resource allocation,and power control [7].

Recent attention on two-tier heterogeneous systems hasbeen drawn to designing efficient network configuration mech-anisms (i.e., resource allocation, power management) withaccess control at the FBSs [8]–[20]. Since FBSs are userdeployed, typically they are configured to operate in closedaccess mode and only serve authorized subscribers, referredas femtocell users (FUs). However, it has been shown thatallowing non-subscribers, i.e., macrocell users (MUs), to ac-cess FBSs can improve the overall system performance whenthe MUs experience poor signal quality from the MBS [8].As a result, open access and hybrid access modes have beenproposed to deal with this issue. In open access mode, MUscan access the FBSs freely and share the same resources withFUs. Thus, the available radio resources can be allocated ina flexible manner. The overall system performance can beimproved at the expense of performance degradation at someFUs due to the altruistic resource sharing. In hybrid accessmode, FBSs may grant access to MUs with limited resourceswhile serving their subscribed FUs with higher priority [9].This access mode aims to seek a balance between improvingthe overall system performance and preserving the quality ofservice (QoS) at the FUs, and has become a popular choicein designing novel network configuration mechanisms [10].

The emergence of open and hybrid access modes bring newchallenges to designing network configuration mechanismsfor heterogeneous networks. First, since an FBS is ownedby its FUs, incentives should be provided to motivate theoperation of open or hybrid access mode. In [11], a spectrumleasing mechanism has been proposed, where the MBS leasethe spectrum to the FBSs, and the FBSs further lease theirspectrum to nearby MUs. The mechanism is formulated asa Stackelberg game and the optimal spectrum leasing pricesare determined via equilibrium analysis. Similar approacheshave been employed in [12] and [13] with different opti-mization objectives. An access permission based incentivemechanism has been proposed in [14], where the macrocellservice provider purchases access permissions from multiplecompeting femtocell service providers. In [15], the authors

propose a spectrum-sharing rewarding framework to encouragehybrid access at FBSs, where femtocell owners determine theamount of resources shared with public users and the operatordetermines the ratio of revenue distribution to femtocell own-ers. The aforementioned pricing schemes are developed basedon specific business models, which may not be available inthe existing systems. In practice, an MU may pay access feeto the service provider instead of making separate paymentsto FBSs. Therefore, it is interesting to study how to encouragethe FBSs to adopt hybrid access without changing the businessmodel, which motivates the research in this paper.

In addition to providing incentives for using open or hybridaccess mode at the FBSs, an efficient access control schemeis needed to determine the set of MUs to be served at eachFBS. Several access control schemes have been proposed inrecent works. In [16], FBSs grant access to the MUs whogenerate significant interference to the FUs in the uplink, sothat the total number of interferers is reduced below a certainthreshold. In [17], an MU is selected to be associated withits geographically closest FBS if the distance to this FBS issmaller than a threshold. However, these schemes are eitherheuristic or designed independently from other network con-figuration processes. In practice, the access control decisions atFBSs may affect other network configuration processes such asresource allocation at the MBS and power management at theusers. For instance, when allocating channels, the MBS mayneed the user association decisions from the FBSs to optimizethe overall network performance. Therefore, optimization ofaccess control, resource allocation and power managementshould be jointly considered when designing the networkconfiguration mechanism for heterogeneous systems. In [18],joint admission control and resource allocation has beenstudied using the theory of semi-Markov decision process,and a distributed power adaptation algorithm is proposed toreduce energy consumption at femtocells. In [19], the authorspropose optimal access control, subcarriers allocation andpower control algorithms in an orthogonal frequency divisionmultiplexing (OFDM) macrocell/femtocell network assuminghybrid access mode at the FBSs. However, the incentivefor private owners to use hybrid access is not discussed inthe aforementioned work. The work in [20] proposed anon-demand waterfilling type channel allocation scheme withthe consideration of user grouping for MBS and FBSs. Theaforementioned joint optimization problems only consider thescenario where open access mode is used at the FBSs, whileleaving the hybrid access control scenario unexplored. In ourprevious work [1], we proposed a resource management mech-anism with hybrid access control for a macro-femto system.However, the mechanism is designed for a simple system withone MBS and one FBS, which limits its application in practice.

In this paper, we study uplink network configuration of atwo-tier macro-femto system, where an MBS and a clusterof adjacent FBSs together serve mobile users with differentQoS requirements. Different from existing works, we considerbase stations and users are individual decision makers withdifferent optimization objectives. We aim to design an efficientmechanism which provides incentive for using hybrid accessat the FBSs, and jointly optimizes the access control, resource

allocation and power management processes. The major con-tributions are summarized as follows:• We propose a five-stage network configuration mecha-

nism for a two-tier macro-femto system. In this mech-anism, FBSs make access control decisions at the be-ginning. Then, the scheduler at the operator allocateschannels to the base stations adaptively according tothe access control decisions. Next, the MBS and FBSsallocate their available resources to their associated users.Finally, the users determine their transmission power. Weshow that this mechanism provides incentives for theFBSs to operate in hybrid access mode without changingthe business model.

• We model the decision making process in the proposednetwork configuration mechanism as a multi-level op-timization problem and design efficient algorithms tofind the optimal solution. In this problem, each levelcorresponds to a stage in the configuration mechanism,and the optimization in one level requires the knowledgeof the optimality of the following levels. We analyze themulti-level optimization problem in a bottom-up manner,and develop efficient algorithms to find the solution foreach level.

• We evaluate the performance of the proposed config-uration mechanism under various scenarios. Simulationresults show that our proposed mechanism achieves ahigher system utility and throughput compared to themechanisms with closed access scheme or a topology-based access control with orthogonal channel allocationscheme at the FBSs. The results also show that ourproposed mechanism can provide incentives for usinghybrid access mode at the FBSs.

The rest of this paper is organized as follows. In SectionII, we describe the system model and network configurationmechanism, and formulate the multi-level optimization prob-lem. In Section III, we analyze the multi-level optimizationproblem and design the network configuration mechanismbased on the algorithm developed in each optimization level.Simulation results are presented in Section IV, and conclusionsare drawn in Section V.

II. SYSTEM MODEL AND PROBLEM FORMULATION

We consider the uplink of a two-tier macro-femto wirelesssystem, where an MBS and a cluster of J > 1 FBSs (whichare close to each other) together serve a number of mobileusers as shown in Fig. 1. This model represents the cellularnetwork in a residential area (e.g., in a building) where FBSsare installed by home users to improve the service quality. Insuch networks, each FBS has a dedicated group of subscribedFUs, and the MBS has a number of subscribed MUs thatare also close to the FBSs. The FBSs connect to the MBSand the operator’s core network via wired backhaul links. Weassume the capacity of the backhaul link for each FBS is largeenough to handle the data transmission for its associated users.A central scheduler located at the operator’s core networkis responsible for allocating the available spectrum resourcesto all the base stations. We denote the set of base stations

MBS

FBS

FBS

Femtocell Users

Macrocell Users

Wired

Backhaul links

Scheduler

Fig. 1. Two-tier macro-femto system model: the MBS has a set of subscribedMUs and each FBS has a set of subscribed FUs.

as I = {0, 1, . . . , J}, where 0 indicates the MBS. For basestation j ∈ I, we define the set of associated MUs as Smj ,and the set of associated FUs as Sfj . Thus, the set of all usersassociated with base station j is denoted as Sj = Smj

⋃Sfj .

The set of MUs in the system is denoted as SM =⋃j∈I Smj .

A list of key notations is provided in Table I.We consider a time slotted system, where the time frame

is divided into slots of equal length. We denote the set ofavailable channels as N = {1, 2, . . . , N}, where each channelhas a bandwidth of B. We assume a frequency flat block fadingwireless channel model [21], where for any of the N channels,the channel fading between a user and its base station remainsconstant during a time slot and is independent and identicallydistributed (i.i.d.) in different time slots with zero mean andunit variance complex Gaussian distribution. Since the MUsare close to the FBSs, to avoid the cross-tier interference(from the MUs to the FBSs and from the FUs to the MBS),the scheduler allocates orthogonal channels to the macrocelltier and the femtocell tier. However, in the femtocell tier, thesame channel may be reused by multiple femtocells under thecondition that the maximum aggregate interference over thatchannel at each FBS is less than a predefined threshold θ.Users associated with the same base station may share theavailable resources using time division multiple access, and atmost one user in a femtocell can access a particular channelat a time slot.

Each FBS can serve at most L users and is configuredto operate in the hybrid access mode. In this access mode,the FBS may allow some MUs to access it but reserves aproportion of the available resources for its own FUs. Tofacilitate the access control process, we introduce an accessratio 0 ≤ �j < 1, which denotes the maximum proportionof available resources that FBS j ∈ I\{0} can allocate tothe MUs associated with it. Note that �j = 0 correspondsto closed access mode where the FBSs only serve their ownFUs. We assume users’ devices have the capability to sensethe signal quality from nearby base stations and may sendrequests to access the FBS that provides the best signal quality.We consider each user requests a service (e.g., video chatting)that requires a minimum average data rate (over a transmissionperiod) to guarantee its QoS. We assume the total amountof resources in the system is sufficient to satisfy the QoSrequirements for all users. We consider there is a networkconfiguration phase to setup system parameters prior to users’

TABLE ILIST OF KEY NOTATIONS

aij Indicator for association between MU i and FBS j.aj User association profile at FBS j.Aj Feasible set of association profiles at base station j.A′j Set of association profiles with which FBS j obtains

more resources than the maximum amount it needs.B Bandwidth of each channel.ci(·) Function that characterizes the energy consumption of

user i’s device.d Distance between MBS and the region center where MUs

and FBSs are located in the simulation.fi(·) Satisfaction function of user i defined over [0,+∞).gij Average channel gain from user i to base station j due to

path loss.hij Instantaneous small scale channel fading from user i to

base station j.I Set of base stations including MBS and FBSs.Ik Set of base stations to which channel k has been allocated.J Total number of FBSs in the system.L Maximum number of users an FBS can serve.N Set of available channels for the system.N Total number of channels for the system.nj Number of channels allocated to base station j.ncj Optimal number of channels allocated to base station j

under closed access.Pi User i’s transmission power.rij Indicator for access request from MU i to FBS j.Rij , R̃ij Average data rate and approximate data rate from user i

to base station j.SM Set of MUs in the system.Smj ,S

fj Set of MUs and set of FUs associated with

base station j, respectively.T Number of time slots in one transmission period.uij Utility of user i associated with base station j.∆uij Utility increment of user i associated with FBS j given one

additional time slot.Uj Total utility at base station j∆Uj Base station j’s utility increment given one additional

channel.wij The proportion of time that user i is allocated one channel

from base station j.wci0 Optimal resource sharing factor for MU i under closed

access.xkj Channel allocation indicator of channel k for base station j.δ(k, xk) Interference condition indicator for channel k given channel

allocation profile xk .∆ϕj Additional channels that base station j can obtain from

MBS.Φi(·) Satisfaction function of user i defined over [Rmini ,+∞).�j Access ratio for FBS j.θ Interference threshold at FBSs.β Weighting factor in utility function.σ2 Additive white Gaussian noise power.ξj Maximum interference allowed at base station j.

data transmission, which includes access control, resourceallocation, and power management at the base stations andusers. Such network configuration is performed periodically,i.e., every T time slots.

A. Utility Function

In this work, we define utility functions to characterize theobjectives at the users and base stations. In practice, a wirelessuser prefers receiving high QoS with low energy consumption.Therefore, we define the utility function for user i ∈ Sj who

)( iji Rf

C

min

ijR0

)( iji RF

ijR

Fig. 2. User’s satisfactory function fi(Rij) with respect to the average datarate Rij . Rmini is the minimum rate requirement, C is a constant, and Φi(·)is a concave function defined in [Rmini ,+∞).

is associated with base station j ∈ I as

uij = fi(Rij)− ci(P i), (1)

where fi(·) characterizes the satisfaction of user i with respectto the average data rate over one transmission period (Rij),and ci(·) is a function of the average transmission power (P i)that characterizes the energy consumption of user i’s device.

According to [22], the satisfaction of a user is an increasingfunction of the data rate, which has a decreasing marginalimprovement as the data rate increases. Such property can bemodeled using concave functions [23] [24]. In this work, tocharacterize users’ satisfaction under different QoS require-ments, we consider a satisfaction function fi(·) as shown inFig. 2. For user i associated with base station j, we have

fi(Rij) =

{Φi(Rij), if Rij ≥ Rmini ,0, otherwise,

(2)

where Φi(·) is a concave function defined in [Rmini ,∞), andRmini is the minimum average data rate required to guaranteethe basic QoS for user i. To simplify the analysis, we assumeusers have the same satisfaction at their QoS threshold, whereΦi(R

mini ) = C,∀ i ∈ Sj ,∀ j ∈ I and C is a constant

that guarantees positive utility for users. Equation (2) impliesthat user i gains nothing when its data rate is less than thebasic QoS requirement. Once the basic QoS is guaranteed,the user’s satisfaction becomes a concave function of thedata rate. Equation (2) can be used to characterize the user’ssatisfaction when running delay sensitive applications suchas video chatting. Note that user’s satisfaction function mayvary with respect to different applications. In this paper, ouranalysis and results can be applied to satisfaction functiondefined in (2) with any concave function Φi(·).

We assume each user chooses a fixed transmission powerfrom [0, Pmax] to optimize their utilities during the transmis-sion phase (T time slots), that is P i = Pi for i ∈ Sj , j ∈ I.We propose to use

ci(Pi) = βwijPi, (3)

where β is a constant weighting factor and wij ∈{ 1T ,

2T , . . . , 1} is the resource sharing variable, which is de-

fined as the proportion of time that user i ∈ Sj is grantedaccess to base station j ∈ I during the transmission phase. β is

used to characterize the importance of the energy consumptionin the utility function. It can be seen that when β is smallsuch that ci(Pi) < fi(Rij), users care more about data raterather than their energy consumption. However, when β issufficiently large such that ci(Pi) > fi(Rij), energy con-sumption becomes more critical when making the transmissiondecisions. Therefore, we can achieve different optimizationobjectives by adjusting the weighting factor β. Note that inthe configuration phase, the actual data rate of a user forthe transmission phase cannot be achieved since the channelgain in the future T time slots is unknown. Therefore, weuse an approximate data rate to calculate the utility whenoptimizing the decisions. Based on the channel capacity in[25], we approximate the average data rate for user i whocommunicates with base station j using the average channelgain between them, where the approximate data rate is

R̃ij = wijB log2

(1 +

E[gij |hij |2

]Pi

σ2 + ξj

)(4)

= wijB log2

(1 +

gijPiσ2 + ξj

),

where gij is the average channel gain from user i to basestation j, hij is the corresponding small scale fading thatfollows a complex Gaussian distribution with zero mean andunit variance, σ2 is the additive white Gaussian noise power,and ξj is the maximum interference allowed at base stationj. Here, we use the fact that E[|hij |2] = 1 for complexGaussian variable hij with unit variance. Note that in (4) weuse the maximum allowed interference instead of the actualinterference, since the actual interference experienced at abase station is unknown prior to the transmission phase. Sinceorthogonal channels are allocated to the macrocell tier andfemtocell tier, there is no interference at the MBS. Therefore,we have

ξj =

{θ, if j ∈ I \ {0},0, otherwise. (5)

Thus, the approximate data rate of user i to base station jis a function of the resource sharing variable wij and thetransmission power Pi. In the remaining of the work, we usethe approximate data rate to calculate the utility for each userwhen designing the network configuration mechanism.

We assume the MBS aims to optimize the total utility of theMUs. Each FBS allocates the resource to maximize the totalutility of its own FUs, while guaranteeing the QoS of all itsassociated users (FUs and MUs). Since users receive wirelessservice from the same provider, and the wireless spectrum islimited, the operator may make channel allocation decisionsto optimize the performance of the whole network, i.e., tomaximize the system throughput [17] or total utility [26]. Weconsider the operator’s objective at the scheduler is to optimizethe total utility of all users in the system. Note that theproposed configuration mechanism and corresponding analysisin this work can also be applied to other utility functions withproper adjustment.

FBSs select desired set of MUsFBSs select desired set of MUs

FBSs determine access ratiosFBSs determine access ratios

Scheduler allocates channels to base stationsScheduler allocates channels to base stations

MBS and FBSs allocate resource to usersMBS and FBSs allocate resource to users

Users determine transmission powerUsers determine transmission power

Stage I

Stage II

Stage III

Stage IV

Stage V

( !, )

( !, !, )

( , !, !, )

!

Fig. 3. Five-stage network configuration mechanism: Stages I and II corre-spond access control process, Stages II and IV are the resource allocationprocess. Stage V is the power management process.

B. Network Configuration Mechanism

In this work, we aim to design a network configurationmechanism that includes access control, resource allocation,and power management between base stations and users. Weconsider the FBSs perform access control at the beginning andthe scheduler allocates the channels to the FBSs adaptivelybased on their access control decisions. This is to encouragethe FBSs to use hybrid access mode to serve MUs who receivelow QoS from the MBS. Specifically, if an FBS grants accessto some MUs, it may obtain more channels and thus mayimprove its own utility. Note that to determine the transmissionpower that optimizes the utility, a user may require theknowledge of which base station it is associated with andthe amount of resource it is granted from that base station.Therefore, it is reasonable to consider power management afterthe resource allocation process.

Based on the above discussion, we propose a networkconfiguration mechanism that follows the procedure as shownin Fig. 3, which consists of five stages. The first two stagescorrespond to the access control process at the FBSs. InStage I, after receiving the access requests from the MUs,each FBS j (j ∈ I\{0}) determines its association profileaj = (aij , i ∈ SM ), where aij = 1 indicates that MUi is granted access to FBS j. Note that each MU canonly be associated with one base station, which leads to∑j∈I\{0} aij ≤ 1. Then, FBS j determines its access ratio

�j in Stage II. After the access control process, the FBSssend the access control decisions (including the associationdecision profile A = {aj , j ∈ I\{0}} and the access ratios� = (�j ,∀ j ∈ I\{0}) to the scheduler. In Stage III, thescheduler performs channel allocation. We define xkj ∈ {0, 1}as the indicator whether channel k is allocated to base stationj (xkj = 1) or not (x

kj = 0). The scheduler determines the

set of channel allocation profiles x = (xj ,∀ j ∈ I), wherexj = (xkj ,∀ k ∈ N ) denotes the channel allocation profileat base station j. Then, the MBS and the FBSs allocate theiravailable resources (channels and time slots) to their associatedusers in Stage IV, i.e., base station j ∈ I determines itsresource allocation profile wj = (wij ,∀ i ∈ Sj). Finally,each user i determines its transmission power Pi based onthe available resource from its associated base station.

The proposed network configuration mechanism can beviewed as a multi-stage decision making process, where base

stations and users make decisions sequentially. Our objectiveis to design efficient decision making strategies for the basestations and users in order to optimize their own utilities.

C. Multi-level Optimization Problem

In the proposed network configuration mechanism, basestations and users have different objectives. Therefore, insteadof formulating a single utility maximization problem whichonly targets at one objective, in this paper, we adopt a multi-level optimization approach [27] and formulate the networkconfiguration process as a five-level optimization problem.The multi-level optimization approach models each decisionmaking stage as an optimization problem with a specificobjective function. It finds the optimal decision at one stageby considering the decisions made in the previous stages andthe decisions to be made in the following stages. We definerij as the indicator whether FBS j receives a request fromMU i, where rij = 1 indicates the request is received andrij = 0 otherwise. We define the set of feasible associationdecisions for FBS j ∈ I\{0} as Aj , {aj | aij ≤ rij ,∀ i ∈SM ,

∑i∈SM aij ≤ L− |S

fj |}, where | · | is the cardinality of

the set. We further denote Pj = (Pi, i ∈ Sj) for j ∈ I\{0},and define �0 = 0 and a0 = (ai0, i ∈ S0). We define Ujas the utility of base station j ∈ I. As mentioned in SectionII-A, the utility of MBS is the total utility of MUs, which isU0 =

∑i∈SM ui0. The utility of FBS j ∈ I \ {0} is the total

utility of its FUs, which is Uj =∑i∈Sfj

uij . Note that a user’sutility depends on its transmission power Pi and the amountof allocated resources wij , while the determination of Pi andwij is also related to the access control decision aj , �j and thechannel allocation decision xj . Therefore, the utility at eachbase station j ∈ I is a function of the variables aj , �j , xj ,wjand Pj . In the following, we use Uj(aj , �j , xj ,wj ,Pj) as theutility function at base station j during the problem formula-tion. Then, the multi-level optimization problem, denoted asproblem P, can be represented as

Level I : For j ∈ I\{0},maximize

ajUj(aj , �j , xj ,wj ,Pj)

subject to aj ∈ Aj ,where �j (j ∈ I\{0}) is the solution ofLevel II : For j ∈ I\{0},

maximize�j

Uj(a∗j , �j , xj ,wj ,Pj)

subject to �minj ≤ �j < 1,where xj (j ∈ I\{0}) is the solution of

Level III :maximize

xj ,j∈I

∑j∈I Uj(a

∗j , �∗j , xj ,wj ,Pj)

subject to nminj ≤∑k∈N x

kj ≤ nmaxj , ∀ j ∈ I,

xkj∑l∈J\{j,0} x

kl maxi∈Sl{gijPmax} ≤ θ,∀ k ∈ N , j ∈ I \ {0},

where wj (j ∈ I) is the solution ofLevel IV : For j ∈ I,

maximizewj

Uj(a∗j , �∗j , x∗j ,wj ,Pj)

subject to∑i∈Sj wij ≤

∑k∈N x

kj ,

wij ≥ wminij , ∀ i ∈ Sj ,where Pj (j ∈ I) is the solution of

Level V : For i ∈ Sj , j ∈ I,maximize

Piuij(w

∗ij , Pi)

subject to Pi ∈ [0, Pmax],

where a∗j , �∗j , x∗j and w∗ij represent the optimal decisions at the

corresponding optimization level. In the above formulation, ateach level, we need to solve an optimization problem takinginto account the parameter(s) obtained from the upper level(s)and the optimality of problem(s) in the following level(s). Forexample, in Level II, FBS j determines its access ratio �j basedon the user association decision aj obtained in Level I and theother optimal decisions (i.e., xj) obtained from lower levels.The determination of access ratio in this stage is importantsince �j works as an indicator for how many channels shouldbe allocated to FBS j in order to guarantee the minimumQoS requirement(s) for the MU(s) accepted by FBS j. Werequire �j be no smaller than a certain value, denoted as �minj .It also indicates how much resources the FBS would like toshare with its associated MUs. In Level III, nminj and n

maxj

denote the minimum and maximum number of channels thatcan be allocated to FBS j, respectively. Given the associationdecisions from the FBSs, nminj and n

maxj should be properly

selected to guarantee the feasibility of optimization in thefollowing levels. The channel allocation decision at this levelalso satisfies the maximum allowed interference constraint ateach FBS. In Level IV, wminij represents the minimum resourcethat user i requires from base station j to guarantee its QoSrequirement. The determination of the parameters (�minj , n

minj ,

nmaxj , and wminij ) is discussed in Section III. The solution

for each optimization problem corresponds to the optimaldecision(s) for the decision maker(s) at that stage. Thus, ourobjective becomes designing efficient algorithms to find theoptimal solution for each optimization level.

III. ANALYSIS OF THE MULTI-LEVEL OPTIMIZATIONPROBLEM

In this section, we analyze the multi-level optimizationproblem and design efficient algorithms to achieve the optimalsolution at each level. The multi-level optimization problemcan be analyzed by a bottom-up approach [27], which followsthe order from the lowest to the highest level sequentially.

A. Power Management at Users

We first consider the last level of problem P. Our objectiveis to find the optimal transmission power for each user i ∈ Sjgiven the the resource sharing variable wij from its associatedbase station j ∈ I. In this level, the utility of a user becomesa function of the transmission power. The correspondingoptimization problem is

maximizePi

uij(wij , Pi)

subject to Pi ∈ [0, Pmax].(6)

We analyze problem (6) with respect to different values ofwij . Note that the minimum amount of resources that useri requires from base station j to guarantee its QoS can becalculated as wminij = R

mini

/(B log2(1+

gijPmaxσ2+ξj

)). First, whenwij < w

minij , according to equations (1), (2) and (3), we have

uij(Pi) = −βwijPi, and the optimal power is P ∗i = 0 in thisscenario. Next, when wij ≥ wminij , we define Pmini ≤ Pmaxas user i’s minimum power that satisfies its QoS requirement,which is given by Pmini =

σ2+ξjgij

(2Rmini /(wijB) − 1). In this

scenario, to solve problem (6), we only need to consider thefeasible region [Pmini , Pmax]. For Pi ∈ [Pmini , Pmax], we haveuij(Pi) = Φi(R̃ij)− βwijPi and

duijdPi

=dΦi

dR̃ij

dR̃ijdPi

− βwij (7)

= wij

(dΦi

dR̃ij

Bgijln 2(σ2 + ξj + gijPi)

− β

).

Note that Φi(·) is a concave increasing function of R̃ij ,which implies that dΦi

dR̃ijis a decreasing function of R̃ij

(or d2ΦidR̃2ij

< 0). Since R̃ij is increasing with Pi, dΦidR̃ij is a

decreasing function of Pi. From (7), we conclude thatduijdPi

is also decreasing with Pi in (Pmini , Pmax). Based on thedecreasing property of duijdPi , if

duijdPi

∣∣∣Pi=Pmini

≤ 0, we haveduijdPi

< 0 for Pi ∈ (Pmini , Pmax). Therefore, the optimalpower in this case is P ∗i = P

mini . On the other hand, if

duijdPi

∣∣∣Pi=Pmini

> 0, since duijdPi

∣∣∣Pi→∞

= −βwij < 0 andduijdPi

is decreasing with Pi, equationduijdPi

= 0 has a uniqueroot Pi = P̃i in (Pmini ,∞), where we have

duijdPi

> 0 in(Pmini , P̃i) and

duijdPi

< 0 in (P̃i,∞). Therefore, in this case,the optimal power is P ∗i = min(P̃i, Pmax). In summary, theoptimal solution to problem (6) can be represented as

P ∗i

=

I{ duij

dPi

∣∣∣Pi=P

mini

≤0}Pmini

+I{ duijdPi

∣∣∣Pi=P

mini

>0}min(P̃i, Pmax), if wij ≥ wminij ,

0, otherwise,(8)

where indicator function IX = 1 if X is true, and is equalto zero otherwise. It can be seen that given the resourcesharing factor wij from base station j ∈ I, the optimal powermanagement decision for user i ∈ Sj is provided in (8).

B. Resource Allocation at Base Stations

Level IV of problem P corresponds to resource allocation atthe base stations. As mentioned in Section II, the channel gainbetween a user and a base station is i.i.d.. Therefore, whenoptimizing the resource allocation decisions, we only deter-mine the resource sharing variables wij , but do not considerwhich channel or time slots are allocated to each user. Afterdetermining the resource sharing variables, the number of timeslots for each user is fixed. The set of time slots for each usercan be determined by applying a scheduling algorithm, suchas round robin scheduling algorithm or random schedulingalgorithm. We first consider resource allocation at FBS jgiven its access control decision (aj and �j) and the channelallocation decision xj from the scheduler. Note that we require

the FBS’s decision guarantee the QoS requirements of all itsassociated users, and we assume such decision is feasible. Theoptimal resource allocation decision wj = (wij ,∀ i ∈ Sj)that maximizes FBS j’s utility can be found by solving thefollowing problem:

maximizewij , i∈Sj

∑i∈Sfj

uij(wij , P∗i (wij))

subject to∑i∈Smj

wij ≤ �j∑k∈N

xkj ,∑i∈Sfj

wij +∑i∈Smj

wij ≤∑k∈N

xkj ,

wij ≥ wminij , ∀ i ∈ Sj ,wij ∈

{1T ,

2T , . . . , 1

}, ∀ i ∈ Sj ,

(9)

where P ∗i (wij) is the optimal transmission power for user igiven wij . The first constraint indicates that the FBS allocatesat most �j of the total resource to the MUs. The secondconstraint implies that the total amount of resources allocatedto users should not exceed the total number of channelsavailable at the FBS. The third constraint guarantees the QoSof all users associated with this FBS. The last constraintimplies that a user can only obtain an integer number of timeslots and cannot access more than one channel simultaneously.Since P ∗i (wij) also depends on wij , from (7) and (8), it canbe seen that problem (9) is a nonlinear discrete optimizationproblem, which is hard to solve in general. In the rest of thissection, we design an efficient algorithm to achieve the optimalsolution.

Since the objective function in problem (9) is only related toFUs associated with FBS j, problem (9) can be solved in twosteps, where we first determine the resource sharing variablesfor MUs associated with FBS j, and then we determine theresource sharing variables for FUs. Intuitively, to maximizeFUs’ total utility, the FBS should allocate as much resourcesas possible to its FUs, or equivalently, to minimize the amountof resources allocated to the MUs while satisfying their QoSconstraints. Therefore, in the first step, the optimal resourcesharing variables for the MUs can be found by solving thefollowing subproblem:

minimizewij ,i∈Smj

∑i∈Smj

wij

subject to∑i∈Smj

wij ≤ �j∑k∈N

xkj ,

wij ≥ wminij , ∀ i ∈ Smj ,wij ∈ { 1T ,

2T , . . . , 1}, ∀ i ∈ S

mj .

(10)

In the second step, with the solution to subproblem (10),w∗ij ,∀ i ∈ Smj , problem (9) is transformed into the followingsubproblem:

maximizewij ,i∈Sfj

∑i∈Sfj

uij(wij , P∗i (wij))

subject to∑i∈Sfj

wij ≤∑k∈N

xkj −∑i∈Smj

w∗ij ,

wij ≥ wminij , ∀ i ∈ Sfj ,

wij ∈ { 1T ,2T , . . . , 1}, ∀ i ∈ S

fj .

(11)

By solving subproblem (11), we obtain the optimal resourcesharing variables for the FUs. It can be seen that the solu-

Algorithm 1 Optimal resource allocation algorithm at FBS j withfixed user association

1: Basic resource allocation:2: Calculate the minimum amount of resource required by each

user, wminij , ∀ i ∈ Sj .3: Set wij := wminij , ∀ i ∈ Sj .4: Remaining resource allocation:5: Determine the total remaining time slots TR := T (

∑k∈N x

kj −∑

i∈Sj wminij ).

6: Calculate the initial value of ∆uij(wij + 1/T ), ∀ i ∈ Sfj .7: Repeat8: Find the set of FUs, St, that satisfies wij < 1,∆uij(wij +

1/T ) > 0, ∀ i ∈ St.9: if St 6= ∅

10: Find user i ∈ St that has the largest ∆uij(wij + 1/T ) andset wij := wij + 1/T , TR := TR − 1.

11: Update ∆uij(wij + 1/T ) for user i.12: endif13: Until TR = 0 or St = ∅.

tions to subproblems (10) and (11) constitutes the solutionto problem (9). We define ∆uij(wij)

4=uij(wij , P

∗i (wij)) −

uij(wij − 1T , P∗i (wij − 1T )) as the changes of utility when

one time slot is assigned to user i. Based on the previousdiscussion, we propose a two-step greedy allocation algorithmto find the optimal solution for (9), as shown in Algorithm 1.In the first step, we allocate the minimum number of timeslots (wminij T ) for each user to satisfy their QoS constraints.Intuitively, this solves subproblem (10). In the second step, weallocate the remaining time slots one by one, where each timeslot is allocated to the FU who has the largest utility increment(∆uij(wij)). With Algorithm 1, the following result can beobtained.

Theorem 1: The resource allocation profile wj obtainedusing Algorithm 1 constitutes the optimal solution to problem(9).

The proof of Theorem 1 is provided in Appendix A. Forthe MBS, the resource allocation algorithm is similar to thatat the FBS, except that the MBS only serves MUs. Thisalgorithm can be obtained by replacing the terms FU by MU,and Sfj by S0, respectively, in Algorithm 1. Thus, we havefound the desired strategy for FBSs and MBS in Level IV.Note that we can apply other objective functions to realizealternative optimization objectives. For example, we can use∑i∈Sj log(Rij) to achieve proportional fairness of data rate

among the users. The optimality of the proposed algorithm isstill valid as long as the objective function is concave withrespect to the data rate.

C. Channel Allocation at MBS

Next, we consider the channel allocation problem in LevelIII of problem P. In this level, the scheduler allocates the avail-able channels (N ) to each base station in order to maximizethe total system utility. The channel allocation decision shouldguarantee the feasibility of the decisions in the followinglevels, which is to guarantee the QoS requirement for each userin the system. To achieve this, we require that the number ofchannels allocated to FBS j ∈ I, defined as nj ,

∑k∈N x

kj ,

satisfies nminj ≤ nj ≤ nmaxj . We define nminj and nmaxj based

on the access control decisions at base station j as follows.First, since we assume each user can only access one channelat a time, the total number of channels allocated to a basestation should not exceed the number of its associated users.Therefore, it is reasonable to define nmaxj = |Sj |. For basestation j which only serves its own subscribed users (eitherMUs or FUs), nminj is defined as n

minj = d

∑i∈Sj w

minij e,

where d·e is the ceiling function. For FBS j, which servesboth FUs and MUs, to guarantee the QoS of all its associatedusers, nminj should satisfy

�jnminj ≥

∑i∈Smj

wminij , and (1−�j)nminj ≥∑i∈Sfj

wminij . (12)

That is nminj ≥ max(d 1�j∑i∈Smj

wminij e, d 11−�j∑i∈Sfj

wminij e).Note that if FBS j intentionally selects a small �j that is closeto zero, nminj may become a large number, which makes theallocation problem infeasible. To prevent such behavior, weset an upper bound for nminj as follows. We denote n

cj as

the optimal number of channels allocated to base station jwhen all base stations use closed access mode, and representthe corresponding optimal resource sharing variable for MUi ∈ SM (associated with MBS) as wci0. Then, for FBS j whoaccepts to serve the set of MUs Smj , the minimum number ofchannels to be allocated satisfies

nminj ≤ ncj +⌊ ∑i∈Smj

wci0⌋. (13)

Inequality (13) implies that if FBS j accepts to serve MUs, itis guaranteed to obtain up to b

∑i∈Smj

wci0c additional channelsfrom the scheduler (depending on the selected access ratio).Note that allocating these additional channels to the FBS doesnot reduce the number of channels allocated to other basestations when the set of MUs switch from accessing the MBSto accessing the FBS.

Based on the previous discussion, we determine nminj as

nminj

=

min{max

(d 1�j∑i∈Smj

wminij e, d 11−�j∑i∈Sfj

wminij e),

ncj + b∑i∈Smj

wci0c}, if �j > 0,d∑i∈Sj w

minij e, otherwise.

(14)

With the above definition, the optimal channel allocationstrategy that maximizes the system utility can be found bysolving the following problem

maximizexj ,j∈I

∑j∈I

∑i∈Sj uij((w

∗ij(xj), P ∗i (w∗ij(xj)))

subject to nminj ≤∑k∈N

xkj ≤ nmaxj , ∀ j ∈ I,

xkj∑

l∈I\{j,0}

xkl maxi∈Sl{gijPmax} ≤ θ,

∀ k ∈ N , j ∈ I \ {0},xkj ∈ {0, 1}, ∀ k ∈ N , j ∈ I,

(15)where P ∗i (w

∗ij(xj)) and w∗ij(xj) can be found for a given xj

using algorithms in Sections III-A and III-B, respectively. Notethat xkj takes integer values and the objective function in (15)

does not have an explicit form with respect to xkj . Therefore, itis difficult to find the optimal solution. Similar to Algorithm 1,we propose a greedy allocation algorithm to find a suboptimalsolution.

We define ∆Uj(nj) , Uj(nj) − Uj(nj − 1) as the utilityincrement by assigning one additional channel to base stationj. We define δ(k, xk) ∈ {0, 1} as the indicator whetherthe interference constraint for all FBSs over channel k issatisfied or not, given the channel allocation profile xk =(xkj , j ∈ I \ {0}). We further denote xk|j as the channelallocation profile obtained by setting xkj = 1 from xk, andδ(k, xk|j) indicates whether the interference constraints aresatisfied when allocating channel k to FBS j. We determine thevalue of δ(k, xk|j) as follows. At the beginning of the channelallocation process, we determine a J×J matrix Γf , where Γfi,jrepresents the maximum interference from users associatedwith FBS i to FBS j. That is, Γfi,j = maxu∈Si{Puguj}.Note that Γf characterizes the interference between any pairof femtocells and Γfi,j is different from Γ

fj,i. We also maintain

a J × N interference matrix Γc, where Γcj,k represents theaggregate interference experienced at FBS j on channel k.To determine δ(k, xk|j) when allocating a channel k to anFBS j, we first find the set of FBSs that have alreadybeen allocated channel k, denoted as Ik. Then, we calculatethe temporary aggregate interference for FBS i ∈ Ik overchannel k assuming channel k is allocated to FBS j, whichis Γc′i,k = Γ

ci,k + Γ

fj,i, where we add the new interference

generated from users associated with FBS j (Γfj,i) to existinginterference at FBS i ∈ Ik (Γci,k). We also calculate themaximum aggregate interference experienced at FBS j fromother femtocells on channel k, which is Γc′j,k =

∑i∈Ik Γ

fi,j .

Then, the interference condition indicator can be determinedas

δ(k, xk|j) ={

1, if Γc′i,k < θ,∀ i ∈ Ik⋃{j},

0, otherwise.(16)

Once channel k is allocated to FBS j, we update Γci,k and Γfi,j

∀ i, j ∈ I accordingly. By introducing the indicator functionδ(k, xk|j), the proposed greedy channel allocation algorithmis shown in Algorithm 2. The main idea of Algorithm 2is to allocate the channels one by one, where each channelis allocated to as many base stations as possible under theinterference constraints. The available channels are indexedfrom 1 to N . Since MBS and FBSs use different set ofchannels, we allocate the channels to MBS starting from indexN in a decreasing order, while allocating channels to FBSsstarting from index 1 in an increasing order.

Algorithm 2 contains two main steps. In the first step, weallocate the minimum number of channels to each base stationso that nj = nminj ,∀ j ∈ I. Specifically, we allocate nmin0channels, which indexed from N to N−nmin0 +1, to the MBSin Line 3. Then, we allocate the minimum number of channelsto FBSs in Lines 4 to 14 starting from channel indexed at 1,where for each channel, we try to allocate it to as many FBSsas possible by checking the interference constraint for eachFBS (from Lines 8 to 12). After allocating the minimum num-ber of channels, in the second step, we allocate the remaining

Algorithm 2 Greedy channel allocation algorithm1: Basic channel allocation:2: Calculate nminj and n

maxj , ∀ j ∈ I.

3: Set xk0 := 1,∀ k = N − nmin0 + 1, . . . , N.4: Set k := 1.5: Find the set of FBSs Ib that satisfies nj < nminj , ∀ j ∈ Ib.6: while( Ib 6= ∅ )7: Set k := k + 1.8: for each FBS j ∈ Ib9: if δ(k, xk|j) = 1

10: Set xkj := 1, nj := nj + 1.11: endif12: endfor13: Find the set of FBSs Ib that satisfies nj < nminj , ∀ j ∈ Ib.14: endwhile15: Remaining channel allocation:16: Set the starting point of remaining channels for FBS and MBS

nFBS := 2, nMBS := N − nmin0 .17: Calculate the initial value of ∆Uj(nj + 1), ∀ j ∈ I based on

Algorithm 1.18: Find the base station set I′b that satisfies nj < nmaxj and

∆Uj(nj + 1) > 0.19: while( I′b 6= ∅ and nFBS < nMBS )20: Find base station j ∈ I′b that has the largest ∆Uj(nj + 1).21: if j = 0 and nMBS > k22: Set xnMBS0 := 1, n0 := n0 + 1, nMBS := nMBS − 1.23: Update ∆U0(n0 + 1) by computing U0(n0 + 1) and U0(n0)

according to Algorithm 1.24: else25: for FBS j in decreasing order with respect to ∆Uj(nj + 1)26: if δ(nFBS, xnFBS |j) = 1 and xnFBSj = 027: Set xnFBSj := 1, nj := nj + 1.28: Update ∆Uj(nj + 1) by computing Uj(nj + 1) and

Uj(nj) according to Algorithm 1.29: endif30: endfor31: endif32: Set nFBS := nFBS + 1.33: Find the base station set I′b that satisfies nj < nmaxj and

∆Uj(nj + 1) > 0,∀ j ∈ I′b.34: endwhile

channels one by one to the MBS and FBSs, respectively, fromLines 16 to 33. We introduce two variables nFBS and nMBSto denote the index of channels to be allocated to FBSs andMBS, respectively. We initialize nFBS = 2 in Line 16. Thisis because in the basic allocation process we only considerthose FBSs whose minimum requirements are not satisfied,and the channels indexed from 2 to k may still be allocatedto some FBSs without violating the interference constraints.We initialize nMBS = N − nmin0 and allocate channels toMBS following a decreasing order of channel index. Then, ineach iteration, we find the base station with the largest utilityincrement assuming one additional channel is allocated to it inLine 20. If it is the MBS, we allocate channel nMBS to MBSin Lines 22 to 23. Otherwise, we allocate channel nFBS to asmany FBSs as possible under the interference constraints fromLines 25 to 30 according to the order of utility increment atthe FBSs. The allocation terminates when no more channelsare available (nFBS ≥ nMBS) or all the base stations have beenfully allocated (nj = nmaxj ,∀ j ∈ I).

Note that the aforementioned closed access scenario (where�j = 0,∀ j ∈ I) is a special case of the hybrid access scenario

and the values of ncj (∀ j ∈ I) and wcij (∀ i ∈ Sj , j ∈ I) canbe determined by applying Algorithm 2 at the beginning of thenetwork configuration (before the access control stages). Wedenote N cFBS = {1, . . . , ncFBS} and N cMBS = {ncMBS, . . . , N}as the set of channels allocated to FBSs and the MBS in theclosed access scenario, respectively. Then, in the hybrid accessscenario, we only need to determine the additional channelsthat should be reallocated from the MBS to FBSs who acceptto serve MUs, denoted as Na, where Na =

∑j∈I\{0}(n

minj −

ncj) according to (14). Then, we allocate these Na channelsto FBSs using Algorithm 2 by setting k = ncFBS + 1, nFBS =ncFBS + 1 and nMBS = n

cMBS + Na in Line 4 and Line 16,

respectively. Thus, the desired channel allocation strategy isprovided in Algorithm 2.

D. Access Control at FBSs

In this subsection, we study the first two levels of problemP, and find the desired access control decisions for FBSsassuming the resource allocation and power managementalgorithms proposed in previous sections are adopted. We firstdetermine the access ratio �j at FBS j ∈ I\{0}. Obviously, ifFBS j does not serve any MU, its access ratio �j should be setto zero. If FBS j serves both FUs and MUs, according to (12),given the set of MUs associated with FBS j (Smj ), a smalleraccess ratio �j may result in a larger number of channelsallocated to this FBS (nj), since the channel allocation shouldsatisfy �jnj ≥

∑i∈Smj

wminij . Moreover, a smaller access ratiomeans sharing fewer resources with the MUs. Therefore, inorder to maximize the utility, FBS j prefers selecting the min-imum achievable access ratio. However, an FBS should alsoguarantee the minimum QoS requirements for its associatedMUs. According to (13), the maximum number of channelsthat FBS j can be guaranteed is ncj +b

∑i∈Smj

wci0c. Thus, theminimum access ratio for FBS j to guarantee its associated

MUs’ QoS requirements is �minj =∑

i∈Smjwminij

ncj+b∑

i∈Smjwci0c

. Note that

ncj and wci0 can be calculated by the scheduler and broadcast

to the FBSs and MUs at the beginning of the configurationperiod. In summary, the desired strategy for base station j ∈ Iin Level II is

�∗j =

{0, if Smj = ∅,�minj , otherwise.

(17)

Finally, we study the user association decision at FBS jwhen multiple access requests are received from the MUs. Asmentioned before, the FBSs are selfish and they select thedesired MUs to maximize their own utilities. For FBS j, theoptimal user association decision can be obtained by solvingthe following problem

maximizeaj

Uj

subject to aj ∈ Aj ,(18)

where Uj =∑k∈Sfj

ukj(w∗kj(x∗j (�∗j (aj),aj)), P ∗k (w∗kj(x∗j (�∗j (

aj),aj)))) and Aj is the set of feasible associationprofiles defined in Section II-C. P ∗k (w

∗kj(x∗j (�∗j (aj),aj))),

w∗kj(x∗j (�∗j (aj),aj)), x∗j (�∗j (aj),aj) and �∗j (aj) are obtainedusing strategies proposed in Sections III-A, III-B, and III-C,

and equation (17), respectively, for a given association profileaj . Note that the FBS prefers obtaining more resources fromthe MBS to serve its FUs. The optimal values of aj in (18) canbe found by searching among the association profiles in Ajand selecting the one that brings the largest amount of addi-tional resources to the FBS. Specifically, for each associationprofile, i.e., aj , the guaranteed number of channels allocatedto FBS j is nj = ncj + b

∑i∈SM aijw

ci0c, and the additional

resources FBS j obtains by accepting MUs can be representedas ∆ϕj(aj) , b

∑i∈SM aijw

ci0c−

∑i∈SM aijw

minij , where the

first term is the additional channels obtained from the MBSand the second term is the amount of resources to be allocatedto the MUs. Therefore, the association profile that maximizes∆ϕj(aj) at FBS j is arg maxaj∈Aj ∆ϕj(aj). Note that itis not necessary for the FBS j to obtain more than |Sfj |channels to serve its FUs, since we assume each FU can onlyaccess one channel at a time. We denote A′j = {aj | aj ∈Aj , ncj + ∆ϕj(aj) > |S

fj |}. It can be seen that if A′j 6= ∅,

accepting any association profile in A′j maximizes the totalutility of FUs at FBS j, since the resources available forthe FUs are sufficient to guarantee that each FU obtains onechannel. In this case, we choose the profile that contains theminimum number of MUs as the association decision. On theother hand, if A′j = ∅, the optimal association decision canbe selected as the association profile that gives the largest∆ϕj(aj). Therefore, the optimal user association decision forFBS j can be represented as

a∗j =

{arg minaj∈A′j

∑i∈SM aij , if A′j 6= ∅,

arg maxaj∈Aj ∆ϕj(aj), otherwise.(19)

E. Procedures of the Proposed Mechanism

We have derived the desired strategies for base stations andusers in each network configuration stage. The proposed net-work configuration mechanism can be summarized as follows.(1) Initialization: The scheduler collects global information ofthe network, and calculates the resource allocated to the FBSsand MUs using Algorithms 2 and 1, respectively, assumingclosed access is used at all FBSs. Then, it sends the infor-mation ncj and w

ci0,∀ i ∈ SM to each FBS j ∈ I\{0}. The

scheduler also obtains the channel allocation decisions in theclosed access mode, where the sets of channels allocated tothe FBSs and the MBS are {1, . . . , ncFBS} and {ncMBS, . . . , N},respectively.(2) Access Control Process: Each MU sends an access requestto the FBS from which the MU can obtain the largest utility.Then, each FBS j determines its association decision accord-ing to equations (19) and (17), and sends the information ofaj and �j to the scheduler.(3) Resource Allocation Process: The scheduler first deter-mines the number of additional channels (Na) to be reallocatedfrom MBS to FBSs based on the resource allocation decisionsin closed access mode (obtained in the initialization step) andthe access control decisions. Next, the scheduler calculates theminimum number of channels required at each base stationsaccording to (14) and allocates the additional channels Naone by one using Algorithm 2 by setting k = ncFBS + 1,

nFBS = ncFBS+1 and nMBS = n

cMBS+Na in Line 4 and Line 16,

respectively. Then, the MBS and FBSs allocate their availableresources to each associated user according to the resultsobtained using Algorithm 1 during the channel allocationprocess.(4) Power Management Process: Each user determines its op-timal power according to equation (8) based on the resourcesallocated by its associated base station.

Note that the proposed mechanism explores channel reuseamong femtocells, which significantly improves the systemperformance. Moreover, the proposed mechanism has a niceproperty, it encourages an FBS to use hybrid access controlin order to improve the total utility of its FUs. On one hand,an FBS may obtain additional resources to serve its own FUsby accepting to serve nearby MUs, as discussed in SectionIII-D. On the other hand, an MU who communicates withthe MBS with poor signal quality can improve its utility byaccessing the nearby FBS without additional cost (i.e., extrapayment). This type of incentive is important since our modelconsiders FBSs are installed by selfish private users who maynot share their services for free. Without such incentive, FBSsmay refuse to serve MUs and always use closed access mode.

F. Complexity of the Proposed Algorithms

In the proposed configuration mechanism, the optimizationin Levels 2 and 5 takes constant time, as can be seen from (17)and (8), respectively. From (19), the computational complexityof Level 1 is proportional to the cardinality of the feasible set|Aj |. Therefore, the major computation complexity originatesfrom Algorithms 1 and 2. Both algorithms use iterative ap-proaches. In the following, we provide an analysis of thesetwo algorithms with respect to the number of iterations. InAlgorithm 1, the basic allocation step from Line 2 to Line 3can be computed in constant time. Therefore, the complexitymainly depends on the loop from Line 7 to Line 13. Notethat in each iteration, we allocate one time slot to a user.There are TR = T (nj −

∑i∈Sj w

minij ) iterations in this loop.

Within each iteration, Line 10 searches for the user with thelargest utility increment, which can be implemented usingbinary search. The average number of searching iterations forthis step is log(|Sj |). Therefore, the complexity of Algorithm1 with respect to the number of iterations is TR log(|Sj |), andeach iteration takes constant time for computation. Algorithm2 also contains two steps, and the major complexity dependson the second step from Line 17 to Line 33. We denote thenumber of remaining channels after the basic allocation asNR. In the second step, we allocate the channels to eachbase station one by one iteratively. For simplicity, we assumeall FBSs have the same number of subscribed FUs. In thiscase, the maximum number of iterations for the outer loop isNR. Within each iteration for channel allocation, we checkall base stations one by one. Each time we need to invokeAlgorithm 1 to update the utility increment with one additionalchannel, which further takes T log(|Sj |) iterations. Thus, thecomplexity with respect to the number of iterations in thisworst case is NR(|I|)T log(|Sj |). It can be seen that thecomplexity increases linearly with the number of remaining

26 28 30 32 34 36 38 401.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1x 10

4

Number of channels

Uti

lity

of

the

syst

em

Proposed mechanism with hybrid access

Proposed mechanism with closed access

Orthogonal channel allocation mechanism

Fig. 4. Total utility of the system versus total number of channels N , withr = 40 m and d = 150 m.

channels, the number of FBSs, and the number of time slotsin the transmission phase.

IV. PERFORMANCE EVALUATION

In this section, we evaluate the performance of the proposednetwork configuration mechanism using simulation. We con-sider a system that consists of an MBS and 10 FBSs whichare close to each other. The FBSs are randomly located withina circular region whose radius is r. The distance between theMBS and the region center is d. The radius of each femto-cell is 10 m, and four FUs are randomly distributed withineach femtocell. The MBS serves 15 MUs who are randomlydistributed in the region with the constraint that at least 8of them are indoor users who are located within at least onefemtocell. The maximum number of users an FBS can serveis L = 8. There are N channels available for the system, eachwith a bandwidth of 180 kHz. The wireless channel modelfollows [28], where the path loss exponent between a userand the MBS (or an FBS) is 4 (or 3), and we choose the wallpenetration loss as 8 dB. The users’ maximum transmissionpower is Pmax = 250 mW and the noise power is −120 dBm.The interference threshold θ is −90 dBm. Each time slotis 1 ms and one transmission period consists of T = 1000time slots. We consider two types of applications, the MUsare requesting regular video chatting with the minimum raterequirement (Rmini ) of 256 kbps. The FUs are requesting highdefinition video chatting with minimum rate requirement of400 kbps. Similar to [24], we implement the user satisfactionfunction with Φi(R̃ij) = α ln(1 + (R̃ij −Rmini )) + C, whereα = 200 and C = 50. We choose β = 0.1. In this section,the utility of a user is calculated using the actual average datarate, which is obtained by averaging the instantaneous datarate over the transmission phase (1000 time slots). We adopta random scheduling algorithm to determine the set of timeslots for each user based on the resource sharing variables.Specifically, at each time slot, FBS j randomly selects njusers from the scheduling set, which initially contains all theusers to be served. Once the number of time slots allocated toa user reaches the desired number, the user is removed fromthe scheduling set. We simulate the proposed configurationmechanism and compare its performance with configuration

26 28 30 32 34 36 38 404

4.5

5

5.5

6

6.5

7

7.5

8x 10

4

Number of channels

Th

rou

gh

pu

t o

f th

e sy

stem

(k

bp

s)

Proposed mechanism with hybrid access

Proposed mechanism with closed access

Orthogonal channel allocation mechanism

Fig. 5. System throughput versus total number of channels N , with r = 40m and d = 150 m.

mechanism with closed access and configuration mechanismwith topology-based access introduced in [17], respectively.We first evaluate the performance of the aforementionedconfiguration mechanisms with respect to different number ofchannels N , and the results are averaged over 50 simulationruns with r = 40 m and d = 150 m. Fig. 4 shows thetotal system utility achieved by different network configurationmechanisms, and Fig. 5 shows their corresponding systemthroughput. It can be seen that the proposed mechanismachieves the largest total utility and the highest throughputamong the three mechanisms, and the performance gap be-tween the proposed mechanism and the other two mechanismsbecomes smaller as the number of channels increases. Thereason is as follows: Using the proposed mechanism, whenthe number of channels is small, these MUs who receive lowservice quality may switch to a nearby femtocell to improvetheir utility, which may also improve the total utility (orthroughput) of the FUs in that cell. Similarly, the mechanismwith topology-based access can also improve the system utility(or throughput) by accepting MUs at the FBSs. However, thisapproach allocate orthogonal channels to each FBS, whichneglect the benefit of channel reuse among the FBSs. Inaddition, since the access control is purely based on topologyinformation, it is possible that an FBS may reject the requestsof some MUs and miss the opportunity to improve its ownperformance. It is also possible that an FBS accepts to servesome MUs but does not obtain additional channels from theMBS, which may degrade the utility of FUs due to resourcesharing. Therefore, the performance of the mechanism withtopology-based access control is not as good as the proposedmechanism. As the number of channels increases, some FBSsor MUs may obtain sufficient resources, and access controlbetween these MUs and FBSs are not necessary. Thus, theperformance gap between the proposed mechanism and theother mechanisms becomes smaller.

In Table II, we show the average running time of theproposed mechanism and the topology-based mechanism withrespect to different number of channels using MATLAB. Asthe number of channels increases, the average running timeof these two mechanisms increases, since the computationalcomplexity increases due to the increased number of variables.

TABLE IIAVERAGE RUNNING TIME VERSUS NUMBER OF CHANNELS

N 26 28 30 32 34 36 38 40Proposed mechanism 6.33 sec 6.36 sec 6.95 sec 7.09 sec 7.13 sec 7.23 sec 7.48 sec 7.63 sec

Topology-based mechanism 9.00 sec 10.22 sec 10.93 sec 11.33 sec 11.62 sec 11.87 sec 12.13 sec 12.70 sec

30 35 40 45 50 55 601.4

1.5

1.6

1.7

1.8

1.9

2

2.1x 10

4

Radius of the region (m)

Uti

lity

of

the

syst

em

Proposed mechanism with hybrid access

Proposed mechanism with closed access

Orthogonal channel allocation mechanism

Fig. 6. Total utility of the system versus radius of the region, with N = 30,and d = 150 m.

It is implied in Table II that these two mechanisms have similarcomputational complexity. However, the proposed mechanismachieves better total utility and system throughput, as shownin Figs. 4 and 5.

Next, we evaluate the performance of the three mechanismswith respect to different radius of the circular area r. Itis shown in Fig. 6 that the system utility of the proposedmechanism and the closed access mechanism increase asthe radius of the region increases. This is because as theradius increases, the average distance between two randomlydistributed FBSs becomes larger, which increases the pos-sibility of channel reuse among the FBSs. The topology-based mechanism remains almost constant as r changes.This is because this mechanism uses an orthogonal channelallocation scheme without the consideration of interferenceamong the femtocells, and the decision is not affected by theradius of the circular region. Therefore, when the radius issmall, the proposed mechanism with hybrid access achievesa similar performance to the topology-based mechanism sincechannel reuse rarely happens due to the close proximity ofthe FBSs, as the radius increases, the proposed mechanismachieves significant utility improvement and the performancegap between the two mechanism becomes larger.

Then, we adjust the distance between the MBS and theconsidered region center, and evaluate the performance ofthe configuration mechanisms with respect to different valuesof d. Fig. 7 shows that the system utility decreases as dincreases for the proposed mechanism and the closed accessmechanism. However, the utility decrement step size of theproposed mechanism becomes smaller when d increases from100 m to 140 m and then increases thereafter. This is becausewhen d is small, the signal quality from the MBS to theMUs are good enough to satisfy their QoS requirements andaccessing the MBS may achieve a larger utility for each MU.As d increases, the signal quality from the MBS to the MUs

80 100 120 140 160 180 2001.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2x 10

4

d (m)

Uti

lity

of

the

syst

em

Proposed mechanism with hybrid access

Proposed mechanism with closed access

Orthogonal channel allocation mechanism

Fig. 7. Total utility of the system versus d, with N = 30 and r = 40 m.

2 4 6 8 10 121.7

1.75

1.8

1.85

1.9

1.95

2x 10

4

Number of MUs in femtocells

Uti

lity

of

the

syst

em

Proposed mechanism with hybrid access

Proposed mechanism with closed access

Fig. 8. Total utility versus number of MUs moved into the femtocell regionwith N = 30, d = 150 m and r = 40 m.

degrades accordingly, which decreases the total utility. Whend is greater than a certain value, i.e., 100 m, the signal qualityfrom the MBS to the MUs is poor and the MUs may requestto access their nearby FBSs to improve their utilities. With theproposed mechanism, the FBSs can accept to serve the MUswho experience severe channel fading from the MBS, whichimproves both the utilities of MUs and FUs when d variesfrom 100 m to 140 m. After that, FBSs may not accept moreMUs, and the total utility decrease linearly as d increases. Asimilar trend can be seen for the topology-based mechanism.

In Fig. 8, we adjust the locations of MUs and evaluate theirimpact on the performance of the proposed mechanism. Werandomly position MUs in a ring area and the average distancebetween MU to the region center is 80 m. We then movea number of MUs into the circular region where the FBSsare located. It is shown in Fig. 8 that the system utility ofthe proposed mechanism first increases and then decreasesas more MUs move into femtocells, and the performancegap between the proposed mechanism and the closed access

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

1.1

1.6

2.1

2.6

3.1

x 104

β

User

Satisfa

ction

4

7

10

13

16

19

22

25

28

31

Energ

y c

onsum

ption (

J)

Proposed mechanism with Φi(·)

Proposed mechanism with Φ†i(·)

Proposed mechanism with Φi(·)

Proposed mechanism with Φ†i(·)

Fig. 9. Energy consumption and user satisfaction versus β, with N = 28,d = 120 m and r = 40 m.

mechanism becomes larger. The reason is that as the MUsmoves into the femtocells, their minimum resource demandfrom the MBS becomes larger due to the wall penetrationloss of signal, which decreases their utility in the closedaccess mode. However, as MUs move closer to the FBSs, theyrequire less resources from the FBSs to satisfy their minimumrate requirements. In this case, the proposed mechanism im-proves the performance of MUs and FUs by redistributingthe channels among the MBS and FBSs, which results in anincrease of the performance gap. As more MUs moves into thefemtocells, once the utility improvement with hybrid accessdoes not compensate the utility decrease (for these MUs), thesystem utility starts to decrease. Nevertheless, the proposedmechanism achieves superior performance than the mechanismwith closed access mode.

Finally, we evaluate the performance of the proposed mech-anism with respect to different values of the weighting factorβ. We select β ≤ C/Pmax to guarantee positive utility ofeach user in the system. We also implement another usersatisfaction function [29] as Φ†i (R̃ij) = α(1 − exp(−(R̃ij −Rmini )/R

mini )) + C. Fig. 9 shows the users’ satisfaction and

the energy consumption decrease as β increases for bothscenarios. The reason is that when β is small, improving users’satisfaction with respect to data rate is more important thansaving energy, and users may use large transmission power toachieve high data rates. However, as β increases, the energyconsumption component in the utility function becomes morecritical. To maximize the utility, users may consider reducingtransmission power to save energy. Therefore, we can adjustthe parameter β accordingly to balance the trade-off betweenusers’ satisfaction and energy consumption, and to realizedifferent optimization objectives. Note that this result is validfor other types of concave satisfaction functions as well, sincethe proposed mechanism is shown to achieve optimal resourceallocation solution with any concave satisfaction function.

V. CONCLUSION

In this paper, we studied uplink network configuration ina two-tier macro-femto system with hybrid access control.To motivate hybrid access at the FBSs and allocate resourceefficiently, we proposed a network configuration mechanism

which consists of access control, channel allocation andpower management processes. We modeled the configura-tion mechanism as a multi-stage decision making process,where base stations and users make decisions sequentiallyto optimize their own utilities. To find the optimal strategyat each stage, we formulated the sequential decision makingprocess as a multi-level optimization problem, and analyzedeach optimization level in a bottom-up manner. We proposedefficient algorithms to solve the optimization problems sequen-tially. Simulation results showed that the proposed networkconfiguration mechanism achieves higher system utility thanconfiguration mechanism with topology-based hybrid accessor closed access, especially when the number of availablechannels in the system is small.

APPENDIX

A. Proof of Theorem 1

According to the discussion in Section III-B, we only needto prove that Algorithm 1 solves subproblems (10) and (11).First, it is clear that the solution to subproblem (10) is toallocate the minimum amount of resource to the MUs thatguarantees their QoS constraints, which corresponds to Lines1 and 2 in Algorithm 1.

Next, we show that Algorithm 1 also solves subproblem(11) optimally. We denote ∆ũij(k) = ∆uij( kT ) as the utilityincrement when FU i obtains its kth time slot from the FBS.It can be shown that ∆ũij(k) is a non-decreasing function ofk, which gives

∆ũij(k) ≥ ∆ũij(k + 1) if wminij T < k < T. (20)

The proof of (20) is shown in Appendix B. We denote set K ={1, 2, . . . , T}, and define W kij as an indicator such that W kij =1 indicates user i obtains its kth time slot (W kij = 0 otherwise).Then, subproblem (11) is equivalent to the following problem

maximizeWkij ,i∈S

fj ,k∈K

∑i∈Sfj

∑k∈KW

kij∆ũij(k)

subject to∑i∈Sfj

∑k∈K

W kij ≤ T( ∑k∈N

xkj −∑i∈Smj

wij

),

W kij = 1, ∀ k ≤ wminij T, ∀ i ∈ Sfj ,

k∏t=1

W tij = Wkij , ∀ k ∈ K, ∀ i ∈ S

fj .

(21)The first constraint indicates the total number of time slotsallocated to the FUs is no larger than T (

∑k∈N x

kj −∑

i∈Smjwij). The second constraint indicates that user i

should obtain at least wminij T time slots from the FBS. Thethird constraint implies that if user i obtains its kth timeslot from the FBS (W kij = 1), it should also obtain all theprevious time slots (W tij = 1,∀ t ≤ k). We define the set∆Uj = {∆ũij(k),∀ k ∈ {wminij T + 1, . . . , T},∀ i ∈ S

fj } and

denote ∆U ′j as the set of the largest TR (defined in Algorithm1) elements from ∆Uj . With property (20), it can be shownthat if ∆ũij(k) ∈ ∆U ′j , then ∆ũij(t) ∈ ∆U ′j ,∀ t < k.Therefore, the corresponding values W kij = 1 associated withthe positive elements ∆ũij(k) in ∆U ′j constitute the solution

to (21). Note that in Algorithm 1, for each remaining timeslot after basic allocation, we allocate it to the user with thelargest positive ∆uij(wij+1/T ). Based on the non-increasingproperty of ∆uij(wij) = ∆ũij(wijT ), Algorithm 1 exactlyfinds the positive elements among the first largest TR elementsin set ∆Uj . Therefore, Algorithm 1 also solves problem (21),which implies that it solves subproblem (11). This completesthe proof.

B. Proof of (20)

According to the definition of ∆ũij(k), to prove property(20), we only need to prove that uij(wij , P ∗i (wij)) is aconcave function with respect to wij in (wminij , 1) (where werelax the integer constraint on wij). We define φ(wij) =

duijdwij

and show that dφdwij < 0,∀ wij ∈ (wminij , 1). Note that

according to (8) in Section III-A, the optimal power P ∗i isa function of wij , which may be either Pmini , Pmax or P̃i.We define W1,W2 and W3 as the intervals of wij whereP ∗i is P

mini , Pmax and P̃i, respectively. We show that for

wij ∈ Wi, i ∈ {1, 2, 3}, we always have dφdwij < 0.First, if W1 6= ∅, we have P ∗i = Pmini and R̃ij =

Rmini for wij ∈ W1. In this case, uij = C −βwij

σ2+ξjgij

(2R

mini /(wijB) − 1

). Then, we have φ(wij) =

−β(σ2+ξj)gij

2Rmini /(wijB)(1 − R

mini ln 2Bwij

) + 1, and dφdwij =

−β(σ2+ξj)gij

(Rmini )2(ln 2)2

B2w3ij2

(RminiBwij

)< 0. Next, if W2 6= ∅, we

have P ∗i = Pmax, and R̃ij = wijB log2(1 +gPmaxσ2+ξj

) for

wij ∈ W2. Then, we have uij = Φi(R̃ij) − βwijPmax andφ(wij) =

dΦidR̃ij·B log2(1 +

gPmaxσ2+ξj

)− βPmax. Since Φi(·) is aconcave function of R̃ij , dΦidR̃ij decreases with R̃ij . Note that

in this case, R̃ij is an increasing function of wij . Therefore,φ(wij) decreases with wij , which gives dφdwij < 0. Finally, if

W3 6= ∅, we have P ∗i = P̃i for wij ∈ W3, where P̃i satisfiesdΦidR̃ij

dR̃ijdPi− βwij = 0. Then, we have

dΦi

dR̃ij

∣∣∣∣∣Pi=P̃i

B

ln 2(P̃i + (σ2 + ξj)/gij)− β = 0. (22)

In this case, we first show that P̃i decreases with wij . Assumew′ij and P̃

′i also satisfy (22) and w

′ij > wij . If P̃

′i ≥ P̃i, we

have R̃′ij > R̃ij . SincedΦidRij

decreases with R̃ij , we have

dΦi

dR̃ij

∣∣∣∣∣Pi=P̃i

B

ln 2(P̃i + (σ2 + ξj)/gij)− β

>dΦi

dR̃ij

∣∣∣∣∣Pi=P̃ ′i

B

ln 2(P̃ ′i + (σ2 + ξj)/gij)

− β = 0, (23)

which contradicts (22). Therefore, for w′ij > wij , we musthave P̃ ′i < P̃i, which implies that

dP̃idwij

< 0.

We further have

φ(wij)

=dΦi

dR̃ij

∣∣∣∣∣Pi=P̃i

(B log2

(1 +

gijP̃iσ2 + ξj

)

+wijB

ln 2(P̃i + (σ2 + ξj)/gij

) · dP̃idwij

)

−β

(P̃i + wij

dP̃idwij

)

= β ln 2

(P̃i +

σ2 + ξjgij

)log2

(1 +

gijP̃iσ2 + ξj

)− βP̃i.

(24)

From (24), we have dφdwij =dφ

dP̃i

dP̃idwij

= β ln 2 log2(1 +

gij P̃iσ2+ξj

) dP̃idwij < 0 according todP̃idwij

< 0.In summary, we have shown that dφdwij < 0 for wij ∈W1

⋃W2

⋃W3. Note that (wminij , 1) ⊆ W1

⋃W2

⋃W3, and

φ(wij) is a continuous function over (wminij , 1). Therefore, weconclude that dφdwij < 0, ∀ wij ∈ (w

minij , 1), which implies

uij(wij , P∗i (wij)) is a concave function of wij in (w

minij , 1).

This completes the proof.

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Binglai Niu (S’11) received the B.S. degree in elec-tronics engineering from Fudan University, Shang-hai, China, in 2008 and the M.Sc. degree in elec-trical engineering from the University of Alberta,Edmonton, Alberta, Canada, in 2010. He is currentlyworking towards his Ph.D. degree at the Universityof British Columbia, Vancouver, British Columbia,Canada. His research interests include optimization,game theory, resource allocation and interferencemanagement in wireless networks, and incentivemechanism design for cooperative communications.

Vincent W.S. Wong (SM’07) received the B.Sc.degree from the University of Manitoba, Winnipeg,MB, Canada, in 1994, the M.A.Sc. degree from theUniversity of Waterloo, Waterloo, ON, Canada, in1996, and the Ph.D. degree from the University ofBritish Columbia (UBC), Vancouver, BC, Canada,in 2000. From 2000 to 2001, he worked as asystems engineer at PMC-Sierra Inc. He joined theDepartment of Electrical and Computer Engineeringat UBC in 2002 and is currently a Professor. Hisresearch areas include protocol design, optimization,

and resource management of communication networks, with applications towireless networks, smart grid, and the Internet. Dr. Wong is an Editor ofthe IEEE Transactions on Communications. He has served on the edito-rial boards of IEEE Transactions on Vehicular Technology and Journal ofCommunications and Networks. He has served as a Technical Program Co-chair of IEEE SmartGridComm’14, as well as a Symposium Co-chair ofIEEE SmartGridComm’13 and IEEE Globecom’13. He received the 2014UBCs Killam Faculty Research Fellowship. He is the Chair of the IEEECommunications Society Emerging Technical Sub-Committee on Smart GridCommunications and IEEE Vancouver Joint Communications Chapter.