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Backhaul-Constrained Small Cell Networks:Refunding and QoS Provisioning

Yufei Yang, Student Member, IEEE, Tony Q. S. Quek, Senior Member, IEEE, and Lingjie Duan, Member, IEEE

Abstract—Small cell access points (SAPs) can offload macro-cell traffic, improve indoor coverage and cell-edge user perfor-mance, and boost network capacity. In this paper, we investigatethe problem faced by the mobile network operator (MNO)on how to properly incentivize the existing private SAPs toserve extra roaming macrocell users. We propose a refundingframework for small cell networks with limited-capacity back-haul, where small cell holders (SHs) receive refunding fromthe MNO and then admit macrocell users. Specifically, weformulate a two-stage refunding-admission game with MNObeing the leader and SHs being followers. Our results canbe summarized as follows: 1) we formulate a revenue maxi-mization problem by allowing the MNO to set individualizedrefunding and interference temperature constraints to SAPs. Wepropose a lookup table approach to solve it; 2) for small cellswith guaranteed QoS provisioning, we consider access-basedrefunding and propose a near-optimal joint user admission andpower allocation algorithm to solve the utility maximizationproblem at each SAP; and 3) for small cells with best-effortQoS provisioning, we consider usage-based refunding and pro-pose a majorization method based power allocation algorithm.Extensive numerical results show that our proposed frameworkand algorithms yield significant improvements on the MNO’s netrevenue and SHs’ utilities compared with non-refunding case.Our research highlights the possibility of enhancing the MNO’snet revenue without changing the current network structureand the importance of incentivizing SHs by taking the limited-capacity backhaul into account.

Index Terms—Small Cell Network, Limited-Capacity Back-haul, Refunding, Two-Stage Game, Admission and Power Con-trol

Manuscript received September 1, 2013; revised November 26, 2013;accepted May 5, 2014. Date of publication (to be added); date of cur-rent version May 10, 2014. This work was supported in part by theSRG ISTD 2012037, SUTD-MIT International Design Centre under GrantIDSF1200106OH, and the A*STAR SERC Grant 1224104048. The editorcoordinating the review of this manuscript and approving it for publicationwas Prof. Zhengdao Wang.

This paper was presented, in part, at the IEEE International Conferenceon Acoustics, Speech and Signal Processing, Vancouver, BC, Canada, May2013, and the IEEE/CIC International Conference on Communications inChina , Xi’an, China, Aug. 2013.

Y. Yang and L. Duan are with the Engineering Systems & DesignPillar, Singapore University of Technology & Design (e-mail: {yufei,lingjie_duan}@sutd.edu.sg).

T. Q. S. Quek is with the Information Systems Technology & DesignPillar, Singapore University of Technology & Design and the Institute forInfocomm Research, Singapore (e-mail: [email protected]).

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

Digital Object Identifier (to be added)

I. INTRODUCTION

SMALL cell access points (SAPs) are low-power accesspoints operated in licensed spectrum, which encompass

femtocells, picocells, microcells, and carrier Wi-Fi accesspoints. Nowadays, SAPs are increasingly installed in publicareas as well as residential sites to cope with the rapid growthof mobile data traffic [1]-[3]. The commercialized SAPsare either deployed by mobile network operators (MNOs)or purchased by customers. The MNOs deploy small cellnetworks on the existing macrocell networks to offload dataand extend service coverage. It is easier for a centralizedmanagement and network optimization. While the majorityof small cells are purchased by customers or enterprisesto meet their data requirements. Therefore, the small cellstandard supports three usage models according to differentdeployment scenarios, i.e., open access, closed access, andhybrid access [4].

Till now, the small cell technique has emerged to be amarket success. It is predicted that the worldwide market forsmall cells will be worth $14.3 billion by 2017 accordingto an Allied Business Intelligence (ABI) Inc. report. TheLTE small cells sold will surpass the number of LTE macro-cells, forecasting at 127, 000, as early as 2014. Multitudesof scholarly works on the economic issues of femtocellsare considered under various pricing and spectrum sharingschemes in [5]-[9]. In [6], the authors consider pricing issuesfor femtocells under different usage models. In [7], theauthors investigate the economic incentive for a MNO toadd femtocell service on the top of its existing macrocellservice. In [8], the authors propose a utility-aware refundingframework for hybrid access femtocell networks. On theother hand, there are still many technical constraints, whichmay potentially barricade the further prosperity of small cellmarket. Among which, backhaul constraint is arguably thekey challenge for small cells [10]. Currently, the majority ofbackhaul are finite wired links, e.g., 70%-80% in US and40% worldwide in 2010. It is predicted that mobile networkswill require 10x fatter backhaul in 2016. In literature, thelimited-capacity backhaul usually appears as constraints tothe sum-rate or transmission delay [11]-[14].

With respect to different usage models, the joint admissionand power control (J-APC) problem and sum-rate maximiza-tion (SRMax) problem are closely related to the quality ofservice (QoS) of subscribers. In literature, the main objectivefor the J-APC problem is to find the maximum admission set




Two-stage refunding-admission game:

Stage I: or ;

Stage II: or . PGQoSMNO PBQoS

MNO

PGQoSm

PBQoSm

Algorithm 1: Joint GU admission & power control

Algorithm 4: Utility maximization via

majorization theory

Algorithm 3: Look-up table

approach at the MNO

1.Equal fading gain &

Algorithm 2

2.Equal target SINR:

3.LP relax:

PEFGm

PESINRm

PLPm

I

m

g

m

(p

m

0 + p

MAX)

Case when :

PIBQoSm

Access-based refunding with guaranteed QoS provisioning

Usage-based refunding with best-effort QoS provisioning

PGQoSm

(pm

⌦m)

Fig. 1. The organization of this paper

while minimizing the total transmission power. The state ofthe art on J-APC has focused on gradual admission [15],gradual removal [16], linear programming (LP) deflation[17], semidefinite relaxation [18], and distributed prime-dual implementation [19]. For SRMax problem, a completesurvey can be found in [20]. It has been shown that theSRMax problem is generally a non-convex problem in termsof transmission power. In high SINR regime, the sum-ratecan be approximated as a convex problem in the form ofgeometric programming [21]. In [22], the authors proposea more general multiplicative linear fractional programmingbased power allocation algorithm to solve the long-standingSRMax problem globally, which considers minimum datarate and peak power constraints for users.

Since the macrocell networks have become more and morecongested, it often happens that the MNO cannot satisfy someof its macrocell subscribers’ data rates. These subscribers willstop paying to the MNO if they cannot receive a desired QoS.As a result, it reduces the MNO’s net revenue and reputationon service provisioning. Therefore, we are interested in howto incentivize the existing SAPs to serve these subscribers.Since SAPs are usually configured as closed access by SHs,the MNO needs to provide refunding to SHs as incentivesto encourage them to admit extra macrocell users. In thesequel, we refer pre-registered small cell subscribers ashome users (HUs) and macrocell subscribers as guest users(GUs). To increase net revenue, the MNO is willing toprovide certain amount of refunding to SHs. While for SHs,with expectation of receiving refunding, they are willing toswitch closed access to hybrid access and admit a limitednumber of GUs. To investigate the interactions betweenthe MNO and SHs, we will formulate a Stackelberg gamewith MNO being the leader and SHs being followers. AtSAPs, we consider guaranteed QoS provisioning and best-effort QoS provisioning, respectively. For guaranteed QoSprovisioning, the optimization problem at SAPs is similarto J-APC problem. But the key difference is that in ourproblem we aim at balancing the number of admitted GUswith HU’s QoS triggerd by refunding. For best-effort QoSprovisioning, the optimization problem at SAPs is similar

TABLE IPARAMETERS USED THROUGHOUT THE PAPER.

Notation Definition

M Total number of SHs (SAPs)Nm Total number of GUs in the m-th SAPpmi Transmission power of the i-th user in the m-th SAPhmni Fading gain from the i-th user in the n-th SAP to the m-th SAP�mi Target SINR threshold of the i-th user in the m-th SAP

�GQoSm Access-based refunding to the m-th SH

�BQoSm Usage-based refunding to the m-th SH⇡ Payment by each admitted GU⌫m QoS valuation parameter of the m-th SHIm Interference temperature constraint on the m-th SAPQ Upper Bound of the aggregate interference in the network⌦m Admission set of the m-th SAP

to SRMax problem. But the key difference is that in ourproblem we aim at balancing the GUs’ achievable sum-ratewith HU’s QoS triggerd by refunding. The organization ofthis paper is illustrated in Fig. 1 and the main contributionsare summarized as follows:

• Two-Stage Refunding-Admission Game Formulation: Weformulate a novel refunding-admission game combiningeconomic refunding and technical specifications. Atthe refunding game in Stage I, the MNO maximizesnet revenue subject to the aggregate interference con-straint. At the admission game in Stage II, each SAPtradeoffs between refunding and HU’s QoS subject totransmission power, interference temperature, signal tointerference and noise ratio (SINR), and backhaul ca-pacity constraints. Backward induction is used to obtainsubgame perfect equilibrium.

• Access-Based Refunding with Guaranteed QoS Provi-sioning [23]: Each SAP is to tradeoff between refundingin terms of the number of admitted GUs and HU’sQoS metric, i.e., proportional fairness throughput. Wepropose a near-optimal joint GU admission and powercontrol algorithm. Three simplified cases are consid-ered, i.e., equal fading gain, equal target SINR, andLP approximation, to further understand the admissionprocess.

• Usage-Based Refunding with Best-Effort QoS Provision-ing [24]: Each SAP is to tradeoff between refunding interms of GUs’ achievable sum-rate and HU’s propor-tional fairness throughout. We obtain the optimal powerallocations for GUs through majorization theory.

• Individualized Refunding and Interference Temperaure:We propose a look-up table approach to solve therevenue maximization problem in Stage I. The MNOdecides individualized refunding and interference tem-perature constraints to different SAPs.

The layout of this paper is as follows. In Section II,we elaborate the system model and performance metrics.In Section III, we present the MNO refunding frameworkwith guaranteed QoS provisioning. In Section IV, we presentthe MNO refunding framework with best-effort QoS provi-




Rejected GUAdmitted GUHU

Backhaul

Backhaul

2

1

Fig. 2. System Model: one macrocell with two SAPs network. The firstSH admits 1 GU and rejects 1 GU. The second SH admits 1 GU and rejects2 GUs.

sioning. Numerical results are presented in Section V andconclusions are given in Section VI. Throughout the paper,we use the following notations and parameters listed in TableI unless otherwise stated.

• Boldface uppercase letters denote matrices, boldfacelowercase letters denote column vectors, italics denotescalars. For a matrix X (or a vector x), XA (or xA)denote the submatrix (or subvector) with index set A.

• For a vector x in RN , we denote its ordered coordinatesby x(1) � x(2) � · · · � x(N) and x[1] x[2] · · · x[N ]. For x and y in RN , we say x majorizesy, i.e., x �

M

y asP

k

i=1 x(i) � P

k

i=1 y(i), 8k =

1, 2, · · · , N � 1 andP

N

i=1 x(i) =

P

N

i=1 y(i). For afunction f : RN 7! R is said to be strictly Schur-convex if x �

M

y implies f (x) � f (y) for all xand y which are not a permutation of each other. Fora function g : IN 7! R, where interval I ⇢ R, is saidto be a separable convex function if g is in the formof g (x) =

P

N

i=1 fi (xi

), where f

i

is a convex functionon I. Any separable convex function is Schur convexfunction.

II. SYSTEM MODEL AND PERFORMANCE METRICS

We consider an uplink two-tier network, which consists ofone macrocell operated by the MNO overlaid with M SAPsinstalled by different SHs indexed as M = {1, 2, · · · ,M},shown in Fig. 2. All the SAPs are connected to the macrocellthrough finite capacity-limited backhaul links. The MNO re-funds SHs whenever they allow GUs to share their SAPs. Themacrocell and SAPs operate in two separate frequency bandsso there is no cross-tier interference [25-26]. However, SAPsare densely deployed and all HUs and GUs are sharing acommon bandwidth, thus resulting in intra-cell interference1.In the following, we assume in the m-th SAP there is one

HU2 and N

m

GUs who are requesting to connect to it at

1While it is possible to allocate orthogonal frequency bands to GUs toavoid interference, this will decrease the spectrum efficiency of SAP. Thus,this work aims to maximize the number of GUs to be admitted into eachSAP while maintain the HU’s QoS.

2It can be exteneded to multiple HUs case via TDMA or OFDMA scheme.

each time slot. The HU is indexed as 0 and GUs are indexedas N

m

= {1, 2, · · · , Nm

}. Note that N

m

can vary duringdifferent time slots. The instantaneous SINR of the HU inthe m-th SAP is given by

SINRm

0 (pm

) (1)

=

h

mm

0 p

m

0P

Nm

i=1 hmm

i

p

m

i

+

P

M

n=1,n 6=m

P

Nn

k=0 hmn

k

p

n

k

+ �

2m

where h

mn

i

denotes the fading gain, including slow and fastfading gains from the i-th user belonging to the n-th SAPto the m-th SAP, p

m

0 and pm

=

⇥

p

m

1 , p

m

2 , · · · , pmNm

⇤

T arethe HU’s and GUs’ transmission powers in the m-th SAP,and �

2m

is the variance of the additive white Gaussian noise(AWGN) at the m-th SAP. In the denominator of (1), the firstterm is the aggregate interference generated by GUs in them-th SAP and the second term is the aggregate interferencegenerated by all the other users in M � 1 SAPs. Similarly,the instantaneous SINR of the i-th GU (8i 2 N

m

) in them-th SAP can be expressed as

SINRm

i

(pm

) (2)

=

h

mm

i

p

m

i

P

Nm

j=0, j 6=i

h

mm

j

p

m

j

+

P

M

n=1,n 6=m

P

Nn

i=0 hmn

k

p

n

k

+ �

2m

In this work, we assume that the interference generated bya SAP to other SAPs is upper bounded by a threshold and wemodel it as max

n2M\{m}

�

P

Nn

i=0 hnm

i

p

m

i

� g

m

�

P

Nm

i=0 pm

i

� I

m

, 8m 2 M, where g

m

= max

n2M\m,i2Nm[{0}{hnm

i

} is the

maximum fading gain from users in the m-th SAP to theother (M � 1) SAPs. The MNO has the authority to set{I

m

}, whereP

M

m=1 Im Q. With the above assumption, thesecond term

P

M

n=1,n 6=m

P

Nn

k=0 hmn

k

p

m

k

in (1) and (2) can beapproximated as

P

M

n=1,n 6=m

I

n

. The main motivation of thisapproximation is to enable the SAPs to perform distributedoptimization as shown later in Section III. With Rayleighfading assumption on {hmm

i

}, the outage probability of theHU in the m-th SAP is given by [27]

Pm

out (pm

) = P (SINRm

0 (pm

) �

m

0 ) (3)

= 1� P

h

mm

0 p

m

0P

Nm

i=1 hmm

i

p

m

i

+ n

m

� �

m

0

!

= 1� e

⇣� �m

0 nmpm0

⌘NmY

i=1

✓

1 +

�

m

0 p

m

i

p

m

0

◆�1

where �

m

0 � 1 is the fixed target SINR of HU and n

m

=

P

M

n=1,n 6=m

I

n

+�

2m

. In the calculation of (3), the fading gains{hmm

i

} in each SAP are assumed to be independent andidentically exponentially distributed random variables withunit mean. Furthermore, since SAPs have a relative smallercoverage radius, we assume that the slow fading gains forGUs within a common SAP are the same and normalized to1.

For the HU, the chosen QoS performance metric isthroughput, which is a function of outage probability, and, in




turn, depends on the transmission powers pm [28]. Specifi-cally, the HU’s throughput is given by

c

m

= [1� Pm

out (pm

)] log (1 + �

m

0 ) (4)

and the HU’s QoS utility is then defined as [29]

T

m

= log (c

m

) = �NmX

i=1

log

✓

1 +

�

m

0 p

m

i

p

m

0

◆

+A

m

(5)

where A

m

= ��

m0 nm

p

m0

+ log [log (1 + �

m

0 )] is a constantfor a given {I

m

} and T

m

can be viewed as proportionalfairness throughput of the HU. It is apparent that each GUcontributes to the HU’s QoS utility degradation separately,e.g., if the i-th GU is rejected by the HU, the correspondingterm � log

⇣

1 +

�

m0 p

mi

p

m0

⌘

vanishes without affecting the HU’sutility.

For SAPs with guaranteed QoS provisioning, we considera linear refunding function �

GQoSm

in terms of the number ofadmitted GUs as follows:

�

GQoSm

= �

GQoSm

|⌦m

| ,m 2 M (6)

where �

GQoSm

is the access-based refunding and |⌦m

| is thecardinality of the admission set ⌦

m

such that ⌦

m

⇢ Nm

.The total refunding is proportional to the number of admit-ted GUs, which implies that the MNO treats all the GUswithin the same SAP equally. From the HU’s perspective, itprefers to admit the GUs causing the least amount of HU’sQoS utility degradation, i.e., requiring the least amount oftransmission powers. Therefore, we will capture this tradeoffin the design of the utility function for SH later.

On the other hand, for SAPs with best-effort QoS provi-sioning, the refunding function �

BQoSm

is defined in terms ofGUs’ sum-rate as follows:

�

BQoSm

= �

BQoSm

R

m

(pm

) ,m 2 M (7)

where �

BQoSm

is the usage-based refunding and R

m

(pm

) isthe GUs’ approximated sum-rate as

R

m

(pm

) ⇡NmX

i=1

log

1 +

h

mm

i

p

m

i

P

Nm

j=1,j 6=i

h

mm

j

p

m

j

+ n

0m

!

(8)

where n

0m

= h

mm

0 p

m

0 +

P

M

n=1,n 6=m

I

n

+ �

2m

.At the m-th SAP, we define the total utility function as

the summation of refunding from the MNO and HU’s QoSutility. For guaranteed QoS provisioning, it is written as

U

GQoSm

= �

GQoSm

+ ⌫

m

T

m

,m 2 M (9)

and, for best-effort QoS provisioning, it is written as

U

BQoSm

= �

BQoSm

+ ⌫

m

T

m

,m 2 M (10)

where ⌫

m

is the unit converter between monetary utility�

GQoSm

�

�

BQoSm

�

and QoS utility T

m

. Given a ⌫m

, serving GUscan receive refunding from the MNO but, as a consequence,cause HU’s QoS utility degradation. However, rejecting GUscan secure a better HU’s QoS at the expense of refunding. If

⌫

m

is extremely large, the SH is unlikely to serve any GUssince it weighs the HU’s QoS more. On the contrary, if ⌫

m

is a small, the SH tends to serve as many GUs as possible,i.e., receiving as much refunding as possible regardless ofthe HU’s QoS.

III. ACCESS-BASED REFUNDING WITH GUARANTEEDQOS PROVISIONING

A. Two-Stage Refunding-Admission Game FormulationIn Stage I, the refunding game at the MNO is formulated

as

PGQoSMNO :=

8

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

:

max{�GQoS

m },{Im}U

GQoSMNO =

M

X

m=1

�

⇡ � �

GQoSm

� |⌦m

|s.t. I

m

� g

m

p

m

0 ,m 2 MM

X

m=1

I

m

Q

0 �

GQoSm

⇡,m 2 M(11)

where ⇡ is the payment by the admitted GUs and the GUsonly need to pay to the MNO when they are connected to theSAPs; Q is the upper bound of the aggregated interferencegenerated by all SAPs. The first constraint ensures that I

m

is at least equal to g

m

p

m

0 , which allows for the transmissionof HUs since they have subscribed to the small cell service.If the MNO sets a larger I

m

, it means the corresponding m-th SAP can potentially admit more GUs. If the MNO sets asmaller I

m

, it means the MNO prohibits the m-th SAP toadmit GUs. Since the MNO has been authorized to manageits own licensed spectrum, it can assign {I

m

} to differentSAPs which are operated on the MNO’s licensed spectrum.The second constraint ensures that the total interferencegenerated by all SAPs should be less than or equal to Q.Together with {�GQoS

m

}, the MNO controls the admissionpreocess at each SAP and U

GQoSMNO is the MNO’s net revenue,

which consists of two terms: the first term,P

M

m=1 ⇡|⌦m

|,is the total gain due to GU admission and the second term,P

M

m=1 �GQoSm

|⌦m

|, is the total refunding to SHs. In (11),the MNO determines

��

�

GQoSm

, I

m

�

to maximize its netrevenue. Since |⌦

m

| is an implicit integer function of �GQoSm

and I

m

, PGQoSMNO is a mixed integer optimization problem and

it is NP hard.In Stage II, the admission game at the m-th SH (8m 2 M)

is formulated as

PGQoSm

:=

8

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

:

max⌦m,pm

U

GQoSm

= �

GQoSm

|⌦m

|+ ⌫

m

A

m

�⌫

m

NmX

i=1

log (1 + a

m

p

m

i

)

s.t. 0 p

m

i

p

MAX, 8i 2 ⌦

m

SINRm

i

(pm

) � �

m

i

, 8i 2 ⌦

m

g

m

X

i2⌦m

p

m

i

+ p

m

0

!

I

m

X

i2⌦m

log (1 + �

m

i

) ¯

C

m

(12)




where a

m

=

�

m0

p

m0

, pm0 is the HU’s fixed transmission power,p

MAX is the maximum transmission power, �

m

i

is the i-thGU’s target SINR, and ¯

C

m

= C

m

� log (1 + �0) is theremaining backhaul capacity after serving the HU. The firstconstraint is the transmission power constraint. The secondconstraint is the SINR constraint, where the SH will guaran-tee each admitted GU’s target SINR. The third constraint isthe interference temperature constraint, where the aggregateinterference generated by the m-th SAP should be less thanor equal to I

m

. The last constraint is the backhaul constraint,where the GUs’ total data rate should be less than or equalto the remaining backhaul capacity. The total utility functionU

GQoSm

consists of two terms: the first term, �GQoSm

|⌦m

|, isthe total refunding from the MNO, and the second term,�⌫

m

P

Nm

i=1 log (1 + a

m

p

m

i

)+⌫

m

A

m

, is the HU’s QoS utility.The goal is to tradeoff between refunding and performanceof HU by choosing ⌦

m

and pm. Since {Im

} decouple theintra-cell interference, it allows each SAP to make decisionnon-cooperatively and distributedly. If we directly work onthe instantaneous SINRs in (1) and (2), it may incur ping-pong effect among SAPs. Consider a simple example of twoSAPs A and B: both of them have two feasible admission setchoices {A1, A2} and {B1, B2} such that |A1| = |A2| and|B1| = |B2|. If we use the instantaneous SINR in (1) and(2), the ping-pong effect may happen between A and B thatA1

B�! B1A�! A2

B�! B2A�! A1, where A1

B�! B1

means if A chooses A1, B will choose B1. Hence, oneSAP’s decision can trigger oscillations between A and B.However, the MNO may not be aware of the ping-pongeffect since it only cares about the cardinality of admissionsets. By introducing {I

m

}, we can alleviate ping-pong effectat the expense of net revenue. Finally, from our refundingframework, we can observe that the network performanceimprovement is equal to the GUs’ aggregate rates from allthe SAPs, i.e.,

P

M

m=1

P

i2⌦mlog(1 + �

m

i

).The problem PGQoS

MNO and PGQoSm

form a two-stagerefunding-admission game and the subgame perfect equilib-rium (SPE) is defined as follows.

Definition 1. Denote⇣

�GQoS?, I?

⌘

be a feasible solution inStage I and P? be a feasible solution in Stage II. Then, thepoint

⇣

�GQoS?, I?,P?

⌘

is a SPE for the formulated Stackel-

berg game if for any other feasible solution⇣

�GQoS0, I0,P0

⌘

,the following conditions are satisfied:

U

GQoSMNO

⇣

�GQoS?, I?,P?

⌘

� U

GQoSMNO

⇣

�GQoS0, I0,P?

⌘

U

GQoSm

⇣

�GQoS?, I?,P?

⌘

� U

GQoSm

⇣

�GQoS?, I?,P0

⌘

, 8m 2 M(13)

where I = [I1, I2, · · · , IM ]

T , P = [p1,p2

, · · · ,pM

], and�GQoS

= [�

GQoS1 ,�

GQoS2 , · · · ,�GQoS

M

]

T .

To obtain SPE, the two-stage refunding-admission game isanalyzed by backward induction and the goal is to obtain thesubgame perfect equilibrium. The Stage II problem is solvedfor a given

⇣

�GQoS, I⌘

first. Then we can achieve the SPE

by solving the Stage I problem based on the best responsefunctions from Stage II.

B. Joint GU Admission and Power Control AlgorithmWe propose a joint GU admission and power allocation

algorithm. Firstly, we have the following theorem given afeasible ⌦

m

.

Theorem 1. For any feasible admission set ⌦m

, the minimumtransmission powers are

pm

⌦m= n

0m

�

Hm

⌦m

��11|⌦m|, (14)

where [Hm

]

ij

=

⇢

h

m

i

/�

m

i

if i = j

�h

m

j

if i 6= j

.

Proof: For any feasible admission set ⌦m

, it receives afixed refunding �

GQoSm

|⌦m

| from the MNO. Then, we simplyneed to obtain the minimum sum-log power of the followingoptimization problem

PGQoSm

(pm

⌦m) :=

8

<

:

minpm

⌦m

X

i2⌦m

log (1 + a

m

p

m

i

)

s.t. Hm

⌦mpm

⌦m⌫ n

0m

1|⌦m|

(15)

Minimizing a concave function is NP hard and the optimalsolution lies in the extreme points of the feasible domain. It iseasy to verify that pm

⌦m= n

0m

�

Hm

⌦m

��11|⌦m| is the extreme

point which minimizes the objective function of PGQoSm

(pm

⌦m)

[30].Theorem 1 states the minimum transmission powers for

any feasible admission set in (11). Then in the followingtheorem, we derive a power update rule by adding a newGU to the current admission set.

Theorem 2. For simplicity, denote K , instead of ⌦m

, as thecurrent admission set (|K| = K), if the SH adds a new GUK + 1 to the set K (we assume all the constraints in (11)are still valid), then the transmission power for each GU isupdated as follows:

pm

K+1 (K + 1) = D

m

⇣

(hm

K )

T

¯pm

K + n

0m

⌘

, (16)

pm

K+1 (k) = ¯pm

K (k)

✓

1 +

h

m

K+1

n

0m

pm

K+1 (K + 1)

◆

, k 2 K(17)

where hK = [h

m

1 , h

m

2 , · · · , hm

K

]

T , ¯pm

K are the transmissionpowers before adding the new (K + 1)th GU, pm

K+1 are theupdated transmission powers after adding the new (K+1)th

GU, and D

m

=

⇣

h

mK+1

�

mK+1

� h

mK+1

n

0m

(hm

K )

T

(Hm

K )

�1¯pm

K

⌘�1

.

Proof: See Appendix A.Theorem 2 states how to update transmission powers if the

SH adds GUs gradually. From Theorem 2, by adding one newGU, the transmission powers of previous GUs are amplifiedby the same scaling factor, i.e., 1 +

h

mK+1

n

0m

pm

K+1 (k + 1),and the transmission power of the newly admitted GU isjointly determined by h

mK+1

�

mK+1

, ¯pm

K and Hm

K . In spite of otherconstraints, the transmission power becomes infeasible when




h

mK+1

�

mK+1

h

mK+1

n

0m

(hm

K )

T

(Hm

K )

�1¯pm

K . It is difficult to obtaina simple admission criterion. To cope with this difficulty,we propose a gradual GU admission and power allocationalgorithm. The idea is to add GUs in terms of the leastincrement of sum-log-power until we reach the suboptimalsolution.

Algorithm 1 consists of two-phases: it first sorts feasibleGUs in terms of least increment of sum-log-powers andthen determines the optimal admission set which maximizesU

GQoSm

. Algorithm 1 significantly reduces computational com-plexity from O �

2

Nm�

(enumeration search) to O �

N

2m

�

. Italso avoids recalculation of the matrix inverse when addinga new GU.

C. Special Cases1) Equal Fading Gain: Within the coverage of the m-th

SAP, we assume all the users experience the equal fading gain˜

h

m. Then the outage probability of HU in (3) is rewritten as

Pm

out (p̂m

K ) = P

˜

h

m

p

m

0

˜

h

m

p̂

m

K + n

m

�

m

0

!

= 1� exp

✓

� �

m

0 n

m

p

m

0 � �

m

0 p̂

m

K

◆

(18)

where p̂

m

K is the sum-power of GUs in K and p̂

m

K is givenby

p̂

m

K =

P

i2K

⇣

11+1/�m

i

⌘

n

0m

h̃

m

1�P

i2K

⇣

11+1/�m

i

⌘

. (19)

Then the problem PGQoSm

becomes

PEFGm

:=

8

>

>

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

>

>

:

max⌦m,pm

U

GQoSm

= �

GQoSm

|K|� ⌫m�

m0 nm

p

m0 ��

m0 p̂

mK

+⌫

m

log log (1 + �

m

0 )

s.t. K ⇢ Nm

0 p

m

i

p

MAX, 8i 2 K

SINRm

i

� �

m

i

, 8i 2 KX

i2Klog (1 + �

m

i

) ¯

C

m

p̂

m

K min⇣

p

m0

�

m0,

Imgm

� p

m

0

⌘

(20)

In this case, we propose a simple algorithm to obtain theoptimal solution of PEFG

m

, which is denoted as Algorithm 2. Itterminates if any constraints in (20) become invalid or addinga new GU begins to decrease U

GQoSm

. It chooses a subset ofordered GUs with �

m

[1] �

m

[2] · · · �

m

[Nm] to maximizeU

GQoSm

. Intuitively, by adding a GU, the refunding incrementis a constant �

GQoSm

while the difference p̂

m

K+1 � p̂

m

K is anincreasing function, hence, there exists a unique admissionset which maximizes U

GQoSm

.

Theorem 3. Algorithm 2 attains the optimal set K?, wherethe admitted GUs have smaller target SINRs compared withrejected GUs. The cardinality of the optimal set is equal to K

if the access-based refunding satisfies �

GQoSm

[K] �

GQoSm

<

�

GQoSm

[K +1], where �

GQoSm

[K] =

⌫m(�m0 )2nm(p̂

mK+1�p̂

mK )

(

p

m0 ��

m0 p̂

mK+1)(p

m0 ��

m0 p̂

mK )

.

Algorithm 1 Gradual GU Admission and Power Control

Initialize ⌦

0m

= ;, �

m

= Nm

, U

GQoS0

m

[0] = A

m

, andp

m

i

=

�

mi n

0m

h

mi

, 8i 2 Nm

for i = 1 to N

m

doif p

m

k

> min

n

p

MAX,

Imgm

� p

m

0

o

or log (1 + �

m

k

) >

¯

C

thenp

m

k

= 1end if

end forFind k = argmin

i2�m p

m

i

if pmk

6= 1 then⌦

0m

= ⌦

0m

[ {k}, �m

= �

m

\ {k}else

return No GU can be served.end iffor i = 1 to N

m

� 1 doFind the GU l in �

m

with the minimum increment oflog-sum-power using (16) and (17).if all the constraints in (12) are valid then

⌦

0m

= ⌦

0m

[ {l}, �m

= �

m

\ {l}end if

end forfor i = 1 to |⌦0

m

| doCalculate U

GQoS0

m

[i] in (12)end forFind t = arg max

i=0:1:|⌦m|U

GQoS0

m

[i] .

return U

GQoS?

m

= U

GQoS0

m

[t] and ⌦

?

m

=

{the first t ordered GUs in ⌦

0m

}

Proof: See Appendix B.2) Equal Target SINR: We assume that all GUs have the

same target SINR �

m. The transmission power of the (i)thGU in K is given by

p

m

(i) =n

0m

h

m

(i)

⇣

1 +

1�

m �K

⌘

, i 2 K (21)

which implies that the admitted GUs have better channelgains. Then, PGQoS

m

can be rewritten as

PESINRm

:=

8

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

:

maxK

U

GQoSm

= �

GQoSm

K + ⌫

m

A

m

�⌫

m

K

X

i=1

log

0

@

1 +

a

m

n

0m

h

m

(i)

⇣

1 +

1�

m �K

⌘

1

A

s.t. K min

n

N

m

,

C̄mlog(1+�

m)

o

0 n

0m

h

m(i)(

1+ 1�m

�K

)

p

MAX, 8i 2 K

K

X

i=1

n

0m

h

m

(i)

⇣

1 +

1�m

�K

⌘ I

m

g

m

� p

m

0

(22)We can find the optimal set K? using a similar method asthe equal fading gain case. The SH prefers to serve the GUswith better channel gains since all GUs’ target SINR are the




Algorithm 2 Equal Power GainInitialize K = ;, ⇧

m

= Nm

and U

GQoS?

m

= ⌫

m

A

m

for i = 1 to N

m

doFind k = arg min

i2⇧m

�

i

and calculate p

K+{k}sum in (19)

if all constraints in (20) are valid thencalculate U

GQoS0

m

= �

GQoSm

(|K|+ 1)� ⌫m�

m0 nm

p0��

m0 p

K+{k}sum

+

⌫

m

log log (1 + �

m

0 )

elsereturn K

end ifif U

GQoS0

m

� U

GQoS?

m

thenU

GQoS?

m

= U

GQoS0

m

, K = K + {k}, ⇧m

= ⇧

m

� k

elsereturn K

end ifend for

same. It is equivalent to choosing a subset of ordered GUswith h

m

(1) � h

m

(2) � · · · � h

m

(Nm) which maximizes U

GQoSm

.3) LP approximation: We assume that a

m

p

MAX ⌧ 1, thenlog (1 + a

m

p

m

i

) can be approximated as am

p

m

i

. Then, PGQoSm

can be approximated as

PLPm

:=

8

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

:

max{um

i },{pmi }

U

GQoSm

= �

GQoSm

NmX

i=1

u

m

i

�⌫

m

NmX

i=1

a

m

p

m

i

+ ⌫

m

A

m

s.t. 0 � pm � p

MAX10 � um � 1p

mi h

mi +(!m

i )�1(1�u

mi )PNm

j 6=i p

mj h

mj +n

0m

� �

m

i

NmX

i=1

p

m

i

I

m

g

m

� p

m

0

N

X

i=1

u

m

i

log (1 + �

m

i

) ¯

C

m

(23)where u = [u

m

1 , u

m

2 , · · · , um

Nm]

T are auxiliary variables toindicate the GUs admission status and !

m

i

is an auxiliary con-stant such that 0 !

m

i

1

�

mi (

Pj 6=i p

MAXh

mj +n

0m)

, 8i 2 Nm

.

Since PLPm

is a LP, we can solve it globally and efficiently.The GUs are gradually removed if u

m

i

6= 1. For iterativeGU removal, the performance is heavily dependent on thecriterion to remove GUs. In [18-19], they remove the userwith the largest gap to its target SINR. In this case, it isequivalent to removing the GU with the smallest um

i

.

D. Look-up Table Approach at the MNOThe problem PGQoS

MNO is generally difficult to solve since|⌦

m

| is an implicit integer function of �

GQoSm

and I

m

.Therefore, we propose a look-up table approach to deter-mine individualized refunding and interference temperatureconstraints to different SHs. The MNO divides the feasible

Algorithm 3 Look-up Table Approach• Each SH feedbacks {hmm

i

} and the MNO estimates{g

m

};• On behalf of each SH, the MNO calculates the

best response function table ⌦

m

for different pair of(t�⇡, I0 + l�I) using Algorithm 1, t = 1, 2, · · · , Tand l = 1, 2, · · · , L;

• Then, it searches for the near-optimal strategies by anyfast search method;

• Finally, the MNO feedbacks the refunding and GUadmission results to the corresponding SHs.

refunding interval [0, ⇡] into T equal intervals with stepsize �⇡ =

⇡

T

and the interference temperature interval[I0, Q] into L equal intervals with step size �I =

Q�I0

L

,where I0 = max (g

m

p

m

0 ). Then, on behalf of each SH, theMNO calculates a table of ⌦

m

in terms of different pairsof (t�⇡, I0 + l�I), t = 1, 2, · · · , T and l = 1, 2, · · · , L.Finally, the MNO decides its strategy through a look-up tableapproach. The performance of Algorithm 3 is dependent onstep size and search method, which is a tradeoff betweenoptimality and computational complexity. As �⇡ ! 0 and�I ! 0, the solution converges to the SPE. Implicitly, weassume that the MNO has enough computational resourcesto employ this look-up table approach. However, to ensurethe scalability of the approach, the MNO can constrain onthe number of SAPs that are available to admit GUs.

IV. REFUNDING WITH BEST-EFFORT QOS PROVISIONING

In this section, we discuss the refunding with best-effortQoS provisioning, which is different from Section III. InStage I, the refunding game at the MNO is formulated as

PBQoSMNO :=

8

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

>

>

>

>

>

>

:

max{Im},{�BQoS

m }U

BQoSMNO

=

M

X

m=1

⇡min

�

¯

C

m

, R

m

(pm

)

�M

X

m=1

�

BQoSm

R

m

(pm

)

s.t.M

X

m=1

I

m

Q

I

m

� g

m

p

m

0 , 0 �

BQoSm

⇡

(24)The objective function of PBQoS

MNO is the MNO’s net revenue.It consists of two terms: the first term is the total servicegain, where the MNO can, at most, charge a total of⇡min

�

¯

C

m

, R

m

(pm

)

from the GUs in the m-th small celland the second term is the total refunding to the SHs. Fromour refunding framework, we can observe that the networkperformance improvement is equal to the GUs’ aggregaterates from all the SAPs, i.e.,

P

M

m=1 Rm

(pm

).In Stage II, the admission game at the m-th SH is




formulated as

PBQoSm

:=

8

>

>

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

>

>

:

maxpm

U

BQoSm

= �

BQoSm

R

m

(pm

)

�⌫

m

NmX

i=1

log (1 + a

m

p

m

i

) + ⌫

m

A

m

s.t. 0 p

m

i

p

MAX

NmX

i=1

p

m

i

I

m

g

m

� p

m

0

(25)

Lemma 1. In SPE, the sum-rate R

?

m

is always less than orequal to ¯

C

m

for all small cells.

Proof: By contradiction, in SPE, if R?

m

>

¯

C

m

, then wecan always find a achievable R

0m

=

¯

C

m

such that UGQoS0

MNO >

U

GQoS?

MNO . Hence, the term min

�

¯

C

m

, R

m

(pm

)

in PBQoSMNO is

equivalent to inserting a backhaul constraint Rm

(pm

) ¯

C

m

to PBQoSm

.

A. Utility Maximization via MajorizationWe focus on solving PBQoS

m

.

Lemma 2. Denote x

m

i

= h

m

i

p

m

i

, the GUs’sum-rate R

m

(xm

) is a strictly schur-convexfunction on the feasible domain D

m

=

n

xm|0 x

m

i

h

m

i

p

MAX,

P

Nm

i=1x

mi

h

mi

Imgm

� p

m

0

o

.

Proof: FixP

Nm

i=1 xm

i

= X

m, we can write R

m

(xm

) asP

Nm

i=1 log

⇣

X

m+n

0m

X

m�xi+n

0m

⌘

. It is easy to prove that Rm

(xm

) =

P

Nm

i=1 log

⇣

X

m+n

0m

X

m�xi+n

0m

⌘

is a separable convex function on

[0, X

m

]

Nm , which implies that Rm

(xm

) is a strictly schur-convex function on D

m

.Based on the Schur-convexity of sum-rate function

R

m

(xm

) and concavity of logarithmic function, i.e.,P

Nm

i=1 log (1 + a

m

p

m

i

), we have the following theorem toobtain the optimal transmission powers of GUs for a givenB

m

and I

m

.

Theorem 4. At the optimal solution of PBQoSm

,1) if more than two GUs can transmit, then N

m

= Am

1 [{k} [Am

2 and the GUs’ transmission powers are

p

m

i

=

8

>

<

>

:

p

MAXi 2 Am

1

0 or min

n

p

MAX,

Imgm

� p

m

0 � lp

MAXo

i = k

0 i 2 Am

2(26)

where h

mm

i

� h

mm

k

� h

mm

j

, 8i 2 Am

1 and 8j 2 Am

2 .2) If only one GU transmits, then h

mm

k

� h

mm

i

, 8i 2N

m

and its transmission power p

m

k

lies in one of thefollowing three discrete power points

(

0,

�

BQoSm h

m(1)�⌫mamn

0m

(

⌫m��

BQoSm )

a

mh

m(1)

�min{pMAX,

Imgm

�p

m0 }

0

,

min

n

p

MAX,

Imgm

� p

m

0

oo

(27)

which maximizes UBQoSm

. [a]cb

is the projection of a ontothe interval [b, c].

Algorithm 4 Utility Maximization via MajorizationSort the GUs with h

m

i

in the descending orderInitialize U

BQoS?

m

and R

?

m

using (24) and (25)for i = 2 to N

m

doif (i� 1) p

MAX Imgm

� p

m

0 thenfor j = 1 to N

m

doif j i� 1 thenp

m

(j) = p

MAX

else if j = i thenp

m

(j) = min

n

p

MAX,

Imgm

� p

m

0 � (i� 1) p

MAXo

elsep

m

(j) = 0

end ifend for

end ifCalculate U

BQoS0

m

= U

BQoSm

and R

0m

= R

m

(pm

)

if UBQoS0

m

� U

BQoSm

thenU

BQoS?

m

= U

BQoS0

m

and R

?

m

= R

0m

end ifend forreturn U

BQoSm

and R

m

Proof: See Appendix C.Based on Theorem 4, we propose Algorithm 4 to maximize

U

BQoSm

. For the case Imgm

� p

m

0 < p

MAX, no GU can transmitat full power. We have the following proposition.

Proposition 1. Consider the problem PBQoSm

only with inter-ference temperature constraint as

P IBQoSm

:=

8

>

>

>

>

>

>

>

<

>

>

>

>

>

>

>

:

maxpm

U

BQoSm

= �

BQoSm

R

m

(pm

) + ⌫

m

A

m

�⌫

m

NmX

i=1

log (1 + a

m

p

m

i

)

s.t.NmX

i=1

p

m

i

I

m

g

m

� p

m

0

(28)

At the optimal solution of P IBQoSm

, only the GU with the bestchannel condition can transmit and the transmission poweris one of the following two discrete power points

8

>

<

>

:

0,

2

4

�

BQoSm

h

m

(1) � ⌫

m

a

m

n

0m

⇣

⌫

m

� �

BQoSm

⌘

a

m

h

m

(1)

3

5

Imgm

�p

m0

0

9

>

=

>

;

(29)

which maximizes U

BQoSm

.

Proof: From Theorem 4, we know that the GU withbetter channel gain has a higher priority to transmit. At theoptimal solution of problem P IBQoS

m

, suppose that there aretwo GUs transmitting. Without loss of generality, we assumex

m

(1) � x

m

(2) and h

m

(1) � h

m

(2) (we can always have suchassumptions through a feasible update used in the proof ofTheorem 4). Therefore, it is easy to prove that there exists a✏ such that x̃m

(1) = x

m

(1) + ✏ and x̃

m

(2) = x̃

m

(2) � ✏, it achievesa higher value of objective function. It implies that only the




0 1 2 3 4 5 6 70

2

4

6

8

10

12

14

16

Exhaustive SearchAlgorithm 1

E(U

GQoS

m

)

�

GQoSm

Exhaustive SearchAlgorithm I

Fig. 3. The comparison of the expected utility using Algorithm 1 andexhaustive search method versus refunding, where vm = 5, C̄m = 1, andIm = 3⇥ 10�3 W.

GU with the best channel gain is allowed to transmit. Thetransmission power is derived in the same way as that of caseII in Theorem 4.

Finally, each SH feedbacks {hmm

i

} and the MNO calcu-lates the best response function table of R

m

(pm

) in termsof different pairs of (t�⇡, I0 + l�I) on behalf of each SH.Then, the MNO decides the individualized refunding andinterference temperature constraints to different SHs throughthe same look-up table approach using Algorithm 4.

V. NUMERICAL RESULTS

In the following, we perform computer simulations toillustrate the utility gains for the MNO and SHs under ourproposed refunding framework. We consider two settings:firstly, a typical SH with different parameter settings andcompare the performance of Algorithm 1 with exhaustivesearch method; and second, a simple network consisting ofone MNO and two SAPs. Each figure shows the Monte-Carloaverage results for 5000 realizations. The simulation settingsare as follows: each SAP has a circular coverage area and theSAPs are separated from each other by 10 m. The varianceof AWGN �

2, the upper bound of the aggregate interferenceQ, and the maximum transmission power p

MAX are set to�40 dBm, 8 ⇥ 10

�3 W, and 3 W, respectively. For eachMonte-Carlo realization, the HUs’ and GUs’ fading gainsare generated as independent and exponentially distributedrandom variables with unit mean. The GUs’ target SINRs aregenerated as independent and uniformly distributed randomvariables in [0, 1] dB. Note that the HU’s transmission power

0 1 2 3 4 5 6 70

2

4

6

8

10

12

14

16

18

vm=1, cm=2, Im=2data2data3data4data5data6data7data8

Type 1

Type 2

Type 3

Type

⌫

m

= 5,

¯

C

m

= 0.5, I

m

= 2⇥ 10

�3

⌫

m

= 5,

¯

C

m

= 0.5, I

m

= 5⇥ 10

�3

⌫

m

= 5,

¯

C

m

= 3, I

m

= 2⇥ 10

�3

⌫

m

= 5,

¯

C

m

= 3, I

m

= 5⇥ 10

�3

⌫

m

= 1,

¯

C

m

= 0.5, I

m

= 2⇥ 10

�3

⌫

m

= 1,

¯

C

m

= 0.5, I

m

= 5⇥ 10

�3

⌫

m

= 1,

¯

C

m

= 3, I

m

= 2⇥ 10

�3

⌫

m

= 1,

¯

C

m

= 3, I

m

= 5⇥ 10

�3

E(U

GQoS

m

)

�

GQoSm

(a)

0 1 2 3 4 5 6 70

0.5

1

1.5

2

2.5

3

vm=1, cm=2, Im=2data2data3data4data5data6data7data8

⌫

m

= 5,

¯

C

m

= 0.5, I

m

= 2⇥ 10

�3

⌫

m

= 5,

¯

C

m

= 0.5, I

m

= 5⇥ 10

�3

⌫

m

= 5,

¯

C

m

= 3, I

m

= 5⇥ 10

�3

⌫

m

= 5,

¯

C

m

= 3, I

m

= 2⇥ 10

�3

⌫

m

= 1,

¯

C

m

= 0.5, I

m

= 2⇥ 10

�3

⌫

m

= 1,

¯

C

m

= 0.5, I

m

= 5⇥ 10

�3

⌫

m

= 1,

¯

C

m

= 3, I

m

= 2⇥ 10

�3

⌫

m

= 1,

¯

C

m

= 3, I

m

= 5⇥ 10

�3

E(|⌦

m

|)

�

GQoSm

(b)

Fig. 4. Expected utility (a) and expected number of admitted GUs (b) ina typical SH versus refunding with GQoS provisioning.

p

m

0 and target SINR �

m

0 are fixed to p

MAX and 2 dB forall the realizations. Furthermore, we approximate g

m

asmax

n2M\m[d

nm

]

�↵, where ↵ is the path loss exponent and d

nm

is the distance between the m-th SAP and the n-th SAP.Hence, g

m

is the largest slow fading gain from the m-thSAP to the other M � 1 SAPs.




A. Typical SH

Here, we investigate the utility of a typical SH in terms ofrefunding with GQoS and BQoS provisioning, respectively.By looking at PGQoS

m

and PBQoSm

carefully, we can categorythe SH type by (⌫

m

,

¯

C

m

): type 1 as (low ⌫

m

, high ¯

C

m

), type2 as (low ⌫

m

, low ¯

C

m

), type 3 as (high ⌫

m

, high ¯

C

m

), andtype 4 as (high ⌫

m

, low ¯

C

m

). In the simulation settings, wedefine ⌫

m

= 5 as high, ⌫m

= 1 as low, ¯

C

m

= 3 as high, and¯

C

m

= 0.5 as low. There are N

m

= 6 GUs who are seekingto access to the SAP simultaneously. Each curve has beentranslated down by ⌫

v

A

m

to make it clearer and easier forcomparison, wherev

m

A

m

is the SH’s utility baseline and wemove it to the origin.

In Fig. 3, we compare a typical SH’s utility using Al-gorithm 1 and exhaustive search method versus refunding,where v

m

, ¯

C

m

, and I

m

are 5, 1, and 3⇥10

�3 W, respectively.We can observe that our proposed Algorithm 1 is close tooptimal but with less computational complexity. In terms ofworse-case complexity, it reduces from O(2

Nm) to O(N

2m

),where we only count the number of possible GU admissioncombinations. One of the advantages of our proposed Algo-rithm 1 is that it can be adapted to the dynamics of admissionprocess at SH. At each transmission slot, the rejected GUleaving the SAP will not affect the system. If an admittedGU leaves, the SH can keep admitting GUs using Algorithm1.

In Fig. 4(a) and Fig. 4(b), we illustrate a typical SH’sexpected utility E(UGQoS

m

) and expected number of admittedGUs E(|⌦

m

|) versus �

GQoSm

with GQoS provisioning, wherethe results are generated using Algorithm 1. From the nu-merical results, we obtain the following pieces of managerialconclusions: 1) For fixed (�

GQoSm

, I

m

), the expected utility oftype 1 SH is higher than that of type 2, than type 3, than type4. 2) The backhaul constraint is more crucial than ⌫

m

. WhenI

m

= 5 ⇥ 10

�3 W, for instance, by comparing type 2 andtype 3, the expected utility of type 3 nearly approaches tothat of type 2 in the high �

GQoSm

regime. In Fig. 4(b), we canclearly observe that the expected number of admitted GUs oftype 1 and type 3 as well as type 2 and type 4 converge to thesame stable level regardless of v

m

. That’s to say, providedthat a typical SH has enough backhaul capacity the MNO canset high refunding to achieve its net revenue if needed whileignore v

m

. 3). For fixed I

m

, the curve of expected utilityexperiences two phases: firstly, when �

GQoSm

is smaller thana threshold, it is nonlinear increasing; and then it is linearincreasing when larger than it. The reason is that when �

GQoSm

is smaller than the threshold E(|⌦m

|) keeps increasing whileE(|⌦

m

|) becomes stable after that. Obviously, in Fig. 4(b),the threshold is �

GQoSm

= 1 for type 1 while �

GQoSm

= 5 fortype 3 when I

m

= 5⇥ 10

�3 W.Similarly, in Fig. 5(a) and Fig. 5(b), we illustrate a

typical SH’s expected utility E(UBQoSm

) and expected numberof transmitted GUs versus �

GQoSm

with BQoS provisioning,where the results are generated using Algorithm 4. Due toformulation difference PBQoS

m

, we only consider high ⌫

m

! " # $ % & ' (!

"

#

$

%

&

'

(

)

*+",-./+#0"! $1232#1232$1232%

⌫

m

= 1, I

m

= 2⇥ 10

�3

⌫

m

= 1, I

m

= 5⇥ 10

�3

⌫

m

= 5, I

m

= 2⇥ 10

�3

⌫

m

= 5, I

m

= 5⇥ 10

�3

�

BQoSm

E(U

BQoS

m

)

(a)

! " # $ % & ' (!

!)#

!)%

!)'

!)*

"

")#

+,"-./0,#1"! $2343#2343$2343%

Expe

cted

Num

ber o

f Tra

nsm

itted

GU

s

⌫

m

= 1, I

m

= 2⇥ 10

�3

⌫

m

= 1, I

m

= 5⇥ 10

�3

�

BQoSm

⌫

m

= 5, I

m

= 2⇥ 10

�3

⌫

m

= 5, I

m

= 5⇥ 10

�3

(b)

Fig. 5. Expected utility (a) and expected number of admitted GUs (b) ina typical SH versus refunding with BQoS provisioning.

and low ⌫

m

in this case. Nevertheless, we can still obtainsimilar conclusions as the GQoS provisioning case. Thoughit is difficult to obtain a fair comparison, the most obviousdifference is that the expected utility with BQoS provisioningis less than that of GQoS provisioning since there are lesstransmitted GUs or equivalently less admitted GUs. In Fig.




0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

2

4

6

8

10

12

Im=3*10data2data3data4data5data6data7data8

�

GQoSm

I1 = 0

I1 = 3⇥ 10

�3

I1 = 5⇥ 10

�3

I1 = 2⇥ 10

�3

I2 = 0

I2 = 2⇥ 10

�3

I2 = 3⇥ 10

�3

I2 = 5⇥ 10

�3

SAP 1

E(U

GQoS

MNO)

SAP 2

Fig. 6. The MNO’s expected net revenue versus refunding with GQoSprovisioning, where (⌫1, C̄1) = (1, 3) and (⌫2, C̄2) = (5, 0.5).

5(b), it is interesting to note that there is no more thanone transmitted GU regardless of ⌫

m

and I

m

. The backhaulconstraint will take effect in the refunding game at the MNOas over-refunding.

B. Single MNO with two SAPsNext, we investigate a simple network consisting of one

MNO and two SAPs. The parameters (⇡, �⇡, �I) are 5, 0.5,10

�3 W, respectively. For illustrative purpose, we considerthe following two SHs, where the first SH is type 1 with(⌫1 = 1, ¯

C1 = 3), the second SH is type 4 with (⌫1 = 5,¯

C1 = 0.5), and g1 = g2 = 10

�3.In Fig. 6, we illustrate the MNO’s expected net revenue

E(UGQoSMNO ) versus �

GQoSm

(m = 1, 2) with SAPs in the GQoSprovisioning. From our results, we can see that the SHcontributes more to the MNO’s expected net revenue if ithas a low ⌫

m

or high ¯

C

m

or both. For fixed I

m

, the curveof E(UGQoS

MNO ) is a concave function thus existing an uniqueoptimal point. For fixed �

GQoSm

, increasing one SAP’s I

m

canincrease its contribution to the MNO’s net revenue whiledecreasing the other SAP’s contribution. In Fig. 6, we canobserve that the best strategies for the MNO are to setI1 = 3 ⇥ 10

�3 W and �

GQoS1 = 0.5 to the first SH and

I2 = 2 ⇥ 10

�3 W and �

GQoS1 = 1.5 to the second SH,

thus achieving 15.33 in the expected net revenue. Since theyare averaging results, the MNO do not need to adjust itsstrategies provided that the types of SHs remain unchangedand then it can achieve the net revenue in the long run.Similarly, in Fig. 7, we illustrate the MNO’s expected net

! !"# $ $"# % %"# & &"# ' '"# #'

&

%

$

!

$

%

&

'

#

()*&+$!,-.-%,-.-&,-.-',-.-#,-.-/,-.-0,-.-1

I1 = 0

I1 = 3⇥ 10

�3

I1 = 5⇥ 10

�3

I1 = 2⇥ 10

�3

I2 = 0

I2 = 2⇥ 10

�3

I2 = 3⇥ 10

�3

I2 = 5⇥ 10

�3

SAP 1

SAP 2

�

BQoSm

E(U

BQoS

MNO)

Fig. 7. The MNO’s expected net revenue versus refunding with BQoSprovisioning, where (⌫1, C̄1) = (1, 3) and (⌫2, C̄2) = (5, 0.5).

revenue E(UBQoSMNO ) versus �BQoS

m

(m = 1, 2) with SAPs in theBQoS provisioning. In the second SAP, the negative part isdue to the backhaul constraint. To be more specific, givena �

BQoSm

, the GUs’ achievable sum-rate is larger than thebackhaul capacity in PBQoS

MNO , which results in over-refunding.Furthermore, it may be wise of the MNO to safely removethe second SAP since it contributes negligible net revenuegain. Obviously, the best strategies for the MNO are to setI1 = 3 ⇥ 10

�3 W and �

BQoS1 = 1 to the first SH and

I2 = 2 ⇥ 10

�3 W and �

BQoS1 = 1.5 to the second SH, thus

achieving 4.28 in the expected net revenue, which is 3.6 timesless than that of the GQoS case.

Finally, we generalize two possible ways to simplifythe refunding framework: firstly, by assuming the MNOhas abundant computational resources and it is a trustedorganization, the optimal {�GQoS

m

} or {�BQoSm

} can be pre-calculated offline by averaging enough channel and targetSINR realizations. The only information required from SHsare their type information, which can be collected when SHsenter the refunding framework. Hence, the MNO and SHscan achieve the optimal expected utility gains in the long-run. It can also avoid unnecessary information exchangesand synchronization problems between the MNO and SHs.Secondly, by grouping the SAPs by their geographical loca-tions, we can scale the lookup table approach at a reasonablecomplexity. That is to say, the MNO runs the lookup tableapproach for different SAP groups separately. Furthermore,by constraining

P

M

m=1 Im = Q, we can further reduce thesearching complexity as well.




VI. CONLUSION

In this paper, we consider the refunding and QoS provi-sioning for SAPs with limited-capacity backhaul. We proposea novel refunding framework which significantly improvesthe MNO’s expected net revenue and the SHs’ expectedutilities. It is a win-win strategy to provide refunding by theMNO and SHs’ commitments to share accesses with GUs.In this paper, we only focus on linear refunding, it is alsoworthwhile to consider nonlinear refunding functions. Oneof them is A

m

1

�

1� e

�A

m2 |⌦m|�, where A

m

1 is the maximumamount of refunding to a SAP from the MNO and A

m

2 is thedegree of aversion to admit GUs. It is referred as risk aversionfunction in economic literature. Besides, it is also worthwhileto consider SHs adaptively choose QoS provisioning basedon the instantaneous channel conditions and target SINRs.Besides, a distributed implementation is desired. In the long-run, the MNO can immediately determine the individualizedrefunding and interference temperature constraint to eachSH merely based on its type information. It also implies apotential advertising strategy for the MNO to attract SHs:providing SHs a mapping table from type information torefunding, then they can strategically choose to join or notand what type they prefer to be. In the final analysis, ourwork provides an initial look into the importance and benefitof motivating SAPs to serve GUs while considering the effectof limited-capacity backhaul.

APPENDIX

A. Proof of Theorem 2The minimum transmission powers for the admitted GUs

in K are written as

¯pm

K = n

0m

(Hm

K )

�11K

.

Adding one new GU K + 1, the coefficient matrix can bewritten as

Hm

K+1 =

"

Hm

K �h

m

K+11K

� (hm

K )

T

h

mK+1

�

mK+1

#

.

By using block matrix inverse, we obtain

�

Hm

K+1

��1=

Km

K+1 D

m

(Hm

K )

�1h

m

K+11K

D

m

(hm

K )

T

(Hm

K )

�1D

m

�

where D

m

=

⇣

h

mK+1

�

mK+1

� h

mK+1

n

0m

(hm

K )

T

(Hm

K )

�1¯pm

K

⌘�1

and

Km

K+1 = (Hm

K )

�1+D

m

(Hm

K )

�1h

m

K+11K

(hm

K )

T

(Hm

K )

�1.Hence, the minimum transmission powers after admitting(K + 1)th GU are calculated as

pm

K+1 = n

0m

�

Hm

K+1

��11K+1.

Therefore, we have

pm

K+1 (1 : K) = n

0m

�

(Hm

K )

�11K

+D

m

(Hm

K )

�1h

m

K+11K

(hm

K )

T

(Hm

K )

�11K

+D

m

(Hm

K )

�1h

m

K+11K

�

,

pm

K+1 (K + 1) = D

m

(hm

K )

T

(Hm

K )

�1n

0m

1K

+D

m

n

0m

.

Substituting D

m

(hm

K )

T

(Hm

K )

�1n

0m

1K

= pm

K+1[K + 1] �D

m

n

0m

and pm

K = n

0m

(Hm

K )

�11K

into above, we obtainthe power update rule in (16) and (17).

B. Proof of Theorem 31. Sort the GUs with SINR in ascending order �

m

[1] �

m

[2] · · · �

m

[Nm], we have

p̂

m

K+1

p̂

m

K=

�

P

K

i=1 bm

[i] + b

m

[K+1]

��

1�P

K

i=1 bm

[i]

�

�

P

K

i=1 bm

[i]

��

1�P

K

i=1 bm

[i] � b

m

[K+1]

�

� 1

where b

m

[i] =1

1+1/�m[i]

and p̂

m

K is the sum-power of the firstK sorted GUs. It implies p̂

m

K+1 � p̂

m

K and then

p̂

m

K+1 � p̂

m

K =

b

m[K+1]n

0m/h̃

m

⇣1�

PK+1i=1 b

m[i]

⌘⇣1�

PKi=1 b

m[i]

⌘ )p̂

mK+2�p̂

mK+1

p̂

mK+1�p̂

mK

=

b

m[K+2](1�

PKi=1 b

m[i])

b

m[K+1]

⇣1�

PK+2i=1 b

m[i]

⌘ � 1

which implies p̂

m

K+2 � p̂

m

K+1 � p̂

m

K+1 � p̂

m

K .2. In step 4, Algorithm 2 stops if UGQoS

m

begins to decrease.Suppose at kth iteration, the SH has already admitted K GUsand admitting (K + 1)th GU begins decreasing the utility, wehave

�

GQoSm

⌫

m

(�

m

0 )

2n

m

�

p̂

m

K+1 � p̂

m

K�

�

p

m

0 � �

m

0 p̂

m

K+1

�

(p

m

0 � �

m

0 p̂

m

K )

.

Suppose all constraints in (20) are still valid by admitting the(K + 2)th GU, we have

�

GQoSm

⌫m(�m0 )2nm(p̂

mK+2�p̂

mK+1)

(

p

m0 ��

m0 p̂

mK+2)(p

m0 ��

m0 p̂

mK+1)

)�

GQoSm

[K + 2]� ⌫m�

m0 nm

p

m0 ��

m0 p̂

mK+2

�

GQoSm

[K + 1]� ⌫m�

m0 nm

p

m0 ��

m0 p̂

mK+1

which implies admitting the (K + 2)th GU will furtherdecrease U

GQoSm

. Hence, the algorithm obtains the optimalK?. The relationship between �

GQoSm

and K? is specified asfollows

|K?| := K if �GQoSm

[K] �

GQoSm

< �

GQoSm

[K + 1]

where �

GQoSm

[K] =

⌫m(�m0 )2nm(p̂

mK+1�p̂

mK )

(

p

m0 ��

m0 p̂

mK+1)(p

m0 ��

m0 p̂

mK )

.

C. Proof of Theorem 41. (By contradiction) Suppose there are more than two GUs

scheduled to transmit at optimal solution. Assume there existstwo GUs transmit at 0 < p

m

i

< p

MAX and 0 < p

m

j

< p

MAX.Without loss of generality, we assume x

m

i

� x

m

j

. If h

m

i

�h

m

j

, we update transmission powers with a small constant ✏as

x̃

m

i

= x

m

i

+ ✏ $ p̃

m

i

= p

m

i

+

✏

h

mi

x̃

m

j

= x

m

j

� ✏ $ p̃

m

j

= p

m

j

� ✏

h

mj

x̃

m

k

= x

m

k

$ p̃

m

k

= p

m

k

, k 6= i, j

It is a feasible update becauseP

Nm

i=1 pm

i

+ ✏

⇣

1h

mi

� 1h

mj

⌘

Imgm

� p

m

0 and x̃

m

i

h

m

i

p

MAX. Due to the schur-convexity of




sum-rate, we have R

m

(

˜xm

) � R

m

(xm

). Denote g(pm

) =

�P

Nm

i=1 log (1 + a

m

p

m

i

), we show

(1 + a

m

p

m

i

)

�

1 + a

m

p

m

j

�� (1 + a

m

p̃

m

i

)

�

1 + a

m

p̃

m

j

�

= a

m

✏

⇣

1h

mj

� 1h

mi

⌘

+

a

2m✏

(

x

mi �x

mj )+a

2m✏

2

h

mi h

mj

� 0

which implies g (

˜pm

) � g (pm

), hence, the update leads toU

m

(

˜pm

) � U

m

(pm

). If h

m

j

> h

m

i

, we update p

m

i

and p

m

j

as followsp̃

m

i

=

h

mj p

mj

h

mi

< p

MAX

p̃

m

j

=

h

mi p

mi

h

mj

< p

MAX

It is a feasible update since p

m

i

+ p

m

j

� p̃

m

i

� p̃

m

j

=

⇣

1h

mi

� 1h

mj

⌘

�

x

m

i

� x

m

j

� � 0. Note that the sum-rate isinvariant in this update, i. e. R

m

(

˜pm

) = R

m

(pm

). We showthat

(1 + a

m

p

m

i

)

�

1 + a

m

p

m

j

�� (1 + a

m

p̃

m

i

)

�

1 + a

m

p̃

m

j

�

= a

m

⇣

1h

mi

� 1h

mj

⌘

�

x

m

i

� x

m

j

� � 0

which implies g (

˜pm

) � g (pm

). It also implies the trans-mitted GUs have better channel gains. Now we need toprove that there is one GU transmits at 0 or p̃

MAX=

min

n

p

MAX,

Imgm

� p

m

0 � lp

MAXo

. We denote

z (x) = �

BQoSm

l

X

i=1

log

1 +

h

m

(i)pMAX

P

j 6=i

h

m

(j)pMAX

+ h

m

k

x+ n

0m

!

+ �

BQoSm

log

1 +

h

m

k

x

P

l

i=1 hm

(i)pMAX

+ n

0m

!

� ⌫

m

log(1 + a

m

x).

Taking the first order derivative of z (x), we have

z

0(x) =

h

m

k

P

l

i=1 hm

(i)pMAX

+ h

m

k

x+ n

m

f (x)

where

f (x) = �

BQoSm

1�l

X

i=1

h

m

(i)pMAX

P

j 6=i

h

m

(j)pMAX

+ h

m

k

x+ n

0m

!

�a

m

⌫

m

⇣

P

l

i=1 hm

(i)pMAX

+ h

m

k

x+ n

0m

⌘

h

m

k

(1 + a

m

x)

.

Since h

m

(i) � h

m

k

and �

m

0 � 1, it is easy to showf (x) is a increasing function in terms of x. Therefore, 1).z

0(0) � 0, then z

�

p̃

MAX� � z (x); 2). z

0(p̃

MAX) 0,

then z (0) � z (x); 3). z0(0) 0 and z

0(p̃

MAX) � 0, then

max{z �p̃MAX�

, z (0)} � z (x).2. Suppose only one GU is scheduled to transmit. Then the

one has the best channel gain as proved above. Consider thefunction t (x) = �

BQoSm

log

⇣

1 +

h

m(1)x

n

0m

⌘

� ⌫

m

log (1 + a

m

x),taking the first-order derivative, we have

t

0(x) =

⇣

�

BQoSm

� ⌫

m

⌘

amh

m(1)x

n

0m

+

✓

�

BQoSm h

m(1)

n

0m

� ⌫

m

a

m

◆

⇣

1 +

h

m(1)

x

n

0m

⌘

(1 + a

m

x)

.

It is easy to find that the optimal transmission power is oneof the following three power points

(

0,

�

BQoSm h

m(1)�⌫mamn

0m

(

⌫m��

BQoSm )

a

mh

m(1)

�min{pMAX,

Imgm

�p

m0 }

0

,

min

n

p

MAX,

Imgm

� p

m

0

oo

which maximizes U

BQoSm

.

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[4] W. C. Cheung, T. Q. S. Quek, and M. Kountouris, “ThroughputOptimization, Spectrum Allocation, and Access Control in Two-TierFemtocell Networks,” IEEE J. Select. Areas Commun., vol. 30, no. 3,pp. 561-574, Apr. 2012.

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[6] S.-Y. Yun, Y. Yi, D.-H. Cho, and J. Mo, “The Economic Effects ofSharing Femtocells,” IEEE J. Select. Areas Commun., vol. 30, no. 3,pp. 595-606, Apr. 2012.

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Yufei Yang (S’13) received the B.Eng degreein Information Engineering from Southeast Uni-versity, Nanjing, China. He is currently workingtowards the PhD degree at Singapore Universityof Technology and Design. His research interestsspan mathematical finance, nonlinear and convexoptimization, and wireless systems.

Tony Q. S. Quek (S’98-M’08-SM’12) received theB.E. and M.E. degrees in Electrical and ElectronicsEngineering from Tokyo Institute of Technology,Tokyo, Japan, respectively. At Massachusetts In-stitute of Technology, he earned the Ph.D. inElectrical Engineering and Computer Science. Cur-rently, he is an Assistant Professor with the In-formation Systems Technology and Design Pillarat Singapore University of Technology and Design(SUTD). He is also a Scientist with the Institute forInfocomm Research. His main research interests

are the application of mathematical, optimization, and statistical theoriesto communication, networking, signal processing, and resource allocationproblems. Specific current research topics include sensor networks, het-erogeneous networks, green communications, smart grid, wireless security,compressed sensing, big data processing, and cognitive radio.

Dr. Quek has been actively involved in organizing and chairing sessions,and has served as a member of the Technical Program Committee aswell as symposium chairs in a number of international conferences. Heis serving as the TPC co-chair for IEEE ICCS in 2014, the WirelessNetworks and Security Track for IEEE VTC Fall in 2014, the PHY &Fundamentals Track for IEEE WCNC in 2015, and the CommunicationTheory Symposium for IEEE ICC in 2015. He is currently an Editor forthe IEEE TRANSACTIONS ON COMMUNICATIONS, the IEEE WIRELESSCOMMUNICATIONS LETTERS, and an Executive Editorial Committee Mem-ber for the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS. Hewas Guest Editor for the IEEE COMMUNICATIONS MAGAZINE (SpecialIssue on Heterogeneous and Small Cell Networks) in 2013 and the IEEESIGNAL PROCESSING MAGAZINE (Special Issue on Signal Processing forthe 5G Revolution) in 2014.

Dr. Quek was honored with the 2008 Philip Yeo Prize for OutstandingAchievement in Research, the IEEE Globecom 2010 Best Paper Award, the2011 JSPS Invited Fellow for Research in Japan, the CAS Fellowship forYoung International Scientists in 2011, the 2012 IEEE William R. BennettPrize, and the IEEE SPAWC 2013 Best Student Paper Award.

Lingjie Duan (S’09-M’12) is an Assistant Pro-fessor in the Pillar of Engineering Systems andDesign, Singapore University of Technology andDesign (SUTD). He received Ph.D. degree in Infor-mation Engineering from The Chinese Universityof Hong Kong in 2012. During 2011, he was avisiting scholar in the Department of ElectricalEngineering and Computer Sciences at Universityof California at Berkeley. His research interests arein the area of network optimization and economics,game theory, and resource allocation. He has

served as a technical program committee (TPC) co-chair for INFOCOM2014Workshops on GCCCN, and a TPC member for multiple top-tier conferences(e.g., ICC, GLOBECOM, VTC, PIMRC, WCNC, SmartGridComm, andMobiArch).

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