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3798 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 9, NO. 12, DECEMBER 2010

Open vs. Closed Access Femtocells in the UplinkPing Xia, Vikram Chandrasekhar, and Jeffrey G. Andrews

AbstractFemtocells are assuming an increasingly important

role in the coverage and capacity of cellular networks. In contrastto existing cellular systems, femtocells are end-user deployed andcontrolled, randomly located, and rely on third party backhaul(e.g. DSL or cable modem). Femtocells can be configured to be ei-ther open access or closed access. Open access allows an arbitrarynearby cellular user to use the femtocell, whereas closed accessrestricts the use of the femtocell to users explicitly approvedby the owner. Seemingly, the network operator would preferan open access deployment since this provides an inexpensiveway to expand their network capabilities, whereas the femtocellowner would prefer closed access, in order to keep the femtocellscapacity and backhaul to himself. We show mathematicallyand through simulations that the reality is more complicatedfor both parties, and that the best approach depends heavilyon whether the multiple access scheme is orthogonal (TDMAor OFDMA, per subband) or non-orthogonal (CDMA). In aTDMA/OFDMA network, closed-access is typically preferable athigh user densities, whereas in CDMA, open access can providegains of more than 300% for the home user by reducing thenear-far problem experienced by the femtocell. The results ofthis paper suggest that the interests of the femtocell ownerand the network operator are more compatible than typicallybelieved, and that CDMA femtocells should be configured foropen access whereas OFDMA or TDMA femtocells should adaptto the cellular user density.

Index TermsTwo-tier femtocell network, closed access, openaccess, normalized ergodic rate, TDMA/OFDMA, CDMA.

I. INTRODUCTION

FEMTOCELL access points (FAPs), also known as HomeNodeBs (HNBs), are short-range low-power and ex-tremely low-cost base stations with third party backhaul

(e.g. DSL or cable modem). They are usually deployed

and controlled by end-users who desire better indoor signal

transmission and reception. With the help of such FAPs, the

network operator is able to extend high quality coverage inside

peoples houses without the need of additional expensivecellular towers. At the same time, FAPs offload traffic from the

cellular network and subsequently improve network capacity[1], [2]. Not surprisingly, two-tier femtocell networks that is,

a macrocell network overlaid with femtocell access points are under intense investigation and rapid deployment [3][8].

Manuscript received February 15, 2010; revised July 2, 2010 andSeptember 3, 2010, accepted September 19, 2010. The associate editorcoordinating the review of this paper and approving it for publication wasO. Dabeer.

P. Xia and J. G. Andrews are with the Wireless Networking andCommunications Group, Department of Electrical and Computer Engineering,The University of Texas at Austin, 1 University Station C0803, Austin, TX78712 (e-mail: [email protected], [email protected]).

V. Chandrasekhar is with Texas Instruments, Dallas, TX.

Digital Object Identifier 10.1109/TWC.2010.101310.100231

A. Interference Management Issues in Femtocells

Despite FAPs promise, many concerns still remain, espe-

cially cross-tier interference [9][12]. Two particular aspects

of FAPs give rise to serious interference issues: 1) the co-

channel spectrum sharing between femtocells and macrocells;

2) the random placement of FAPs. First, unlike Wi-Fi

access points, FAPs serve users in licensed spectrum, toguarantee Quality-of-Service (QoS) and because the devices

they communicate with are developed for these frequencies.

Compared to allocating separate channels inside the licensedspectrum exclusively to FAPs, sharing spectrum would be

preferred from an operator perspective [1], [13]. Secondly,

FAPs are installed by end-users in a plug-and-play manner,which translates into randomness in their locations: they can

be deployed anywhere inside the macrocell area with no prior

warning [1], [14]. For these two reasons, interference in two-

tier networks is quite different than in conventional cellular

networks, and endangers their successful co-existence [11],[15], [16]. A typical scenario is the Dead Zone or Loud

Neighbor problem, where mobile users transmit and receive

signals at positions near FAPs but far from the macrocell BSs,

causing significant macro-to-femto interference in the uplink.

In the downlink, these users likewise suffer from low signalto interference ratios (SIRs) because of the strong interference

from the FAPs. These affects are akin to the well known near-

far problem, but exacerbated by the de-centralization and lack

of coordinated power control inherent in a two-tier network.

Because of the presently non-existent coordination betweenFAPs and macrocell BSs, centralized cooperation to mitigate

cross-tier interference is infeasible in the near future, andso in this paper we assume that a two-tier network needs

to adopt decentralized strategies for interference management

[17], [18] such as femtocell access control [11], [13], [16],

[19]. Femtocell access control schemes can be divided into

two categories: closed and open access. FAPs only provide

service to specified subscribers in closed access, to ensurethey can monopolize their own femtocell and its backhaul

with privacy and security. However this potentially leadsto severe cross-tier interference as described above. On the

contrary, open access allows arbitrary nearby cellular users

to use the femtocell. Seemingly, open access is beneficial tonetwork operators, by providing an inexpensive way to expand

their network capacities by leveraging third-party backhaul for

free. Open access also reduces macro-to-femto interference

by letting strong interferers simply use the femtocell and

coordinate with the existing users through it. However, in

order to attain a certain target receive power at the FAP, the

handed over cellular user in open access generally transmits

with higher power to the FAP (thereby creating increased

femto-to-macro interference) as compared to the in home

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XIA et al.: OPEN VS. CLOSED ACCESS FEMTOCELLS IN THE UPLINK 3799

user. Thus open access potentially deteriorates QoS provided

to cellular users remaining in the macrocell (arising due to

increased interference). Crucial unanswered questions remainin femtocell access control, such as:

1) Which mode meets the interests of femtocell owners?

Which mode is preferable to the network operator? Are

these two choices the same or different?

2) How does the answer depend on factors such as multiple

access protocol (e.g. OFDMA, CDMA), user densities,

user scheduling, and femtocell backhaul constraints?

B. Related Work

The uplink interference in two-tier femtocell networks was

evaluated in [11], showing that tier-based open access can

reduce the interference and offer an improvement in the

network-wide area spectral efficiency the feasible number

of femtocells and macrocell users per cell-site. Similar con-

clusions were presented in many simulation-centric studies

accomplished by the 3GPP RAN 4 group [12], [20], [21].Downlink network capacities under open and closed access

were explored in [20]; Feasible combinations of femtocells

and macrocells under the constraint of network interference

were examined in [12]; Various scenarios were presented in[21] to compare femtocell open and closed access. All these

simulations show that with adaptive open access, the interfer-

ence in two-tier networks is mitigated and the deployment

of co-channel femtocells becomes feasible. However, since

femtocells are installed and paid for by their owners, it is

necessary to evaluate their loss of femtocell resources in

open access. It is important that the benefits of mitigated

interference are not undermined by the loss of femtocellresources, such as over-the-air (OTA) and backhaul capacity.

The issues of femtocell backhaul sharing in open access

were examined in [13], which simulated open and closedaccess in HSDPA, with the thesis that completely open access

is problematic because of sharing limited femtocell backhaul

among a potentially large number of mobile users. Based on

simulations incorporating femtocell backhaul issues and cross-

tier interference, this work concludes that open access with arestriction on the number of supported users at the FAP is the

preferred approach. We derive the same conclusion in uplink

based on both analytical and simulations results. Moreover, we

show that such a conclusion strongly depends on whether the

multiple access scheme is orthogonal (TDMA or OFDMA) or

non-orthogonal (CDMA).

The increased handover frequency and hence overhead

signaling in open access is a possible challenge to its im-

plementation. A technique combining intracell handovers with

power control was proposed in [16], and a hybrid access model

open access with a cap on the amount of resources allocated

to the cellular users was simulated in [19]. Both of these

approaches substantially reduce the number of handovers in

open access while mitigating the cross-tier interference. In this

paper, we simply call the hybrid access model open access,

since our open access approach has an upper limit of users,where could become arbitrarily large to conform to fullyopen access.

C. Contributions

This paper evaluates the performance of femtocell open

and closed access in the uplink, from the viewpoints of both

the femtocell owner (owners achieved rate) and the network

operator (cellular users sum throughput).

First, we derive the cumulative distribution function (CDF)

for uplink cross-tier interference in orthogonal multiple access

schemes (TDMA or OFDMA). The capacity tradeoffs for boththe femtocell owner and cellular users are then presented. In

TDMA or OFDMA, the preferences of the femtocell ownerand the network operator are highly dependent on cellular user

density: Their choices are incompatible/ open access/ closed

access in low/ medium/ high cellular user density respectively.

Thus, for 4G networks (LTE & WiMAX) that use OFDMA,

the femtocell access control should likely be adaptive to thecellular user density.

Second, by deriving lower bounds on the performance ofopen access, we show that in non-orthogonal multiple access

(i.e. CDMA) open access is a strictly better choice for the

home user. In typical propagation scenarios, it provides morethan a factor of3 rate gain to the femtocell owner by loweringinterference. From the viewpoint of the network operator,open access is also preferred. In the less important regime

of low cellular user density, open access achieves almost the

same performance as closed access, while in the important

regime of high cellular user density, it improves performance

significantly. We suggest that femtocell open access is the

preferred approach for CDMA femtocells (i.e. 3G), from theviewpoints of both femtocell owners and network operators.

The rest of this paper is organized as follows. In Section

II, we describe the system model and assumptions in detail.

The capacity contours in orthogonal multiple access schemesare presented in Section III, which in non-orthogonal multiple

access scheme are derived in Section IV. Numerical results are

used to illustrate the main takeaways in Section V.

I I . SYSTEM MODEL

In the interior area of a macrocell of radius , the macrocellBS is located at the center, with a single femtocell access

point (FAP) at a distance away from it. Suppose there are cellular users, denoted as 1, 2, . . . , , roaming insidethe macrocell. Their positions are i.i.d. random variables,

uniformly distributed in the macrocell area. The femtocell

owner, or alternatively the home user, is denoted as 0. Sincethe home user is transmitting and receiving inside the small

area of a house, we could assume it is located at a deterministic

position, with a distance of from the FAP. As the subscriber,the home user would always talk to the FAP. On the other

hand, cellular users can be served by the macrocell BS, or theFAP if it employs open access.

Femtocell-to-femtocell interference has been neglected for

reasons of analytical tractability. However, because uplink

femtocell transmissions typically originate and terminate in-

doors and are of low power, their contribution to the overallinterference is expected to be negligible compared to the more

numerous and high power outdoor (macro-cellular) users.

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A. Channel Model and Interference

We consider path loss attenuation effects only and ignore

short-term fading in our channel model. This assumption is

reasonable because fading does not have a large effect in a

wideband system with sufficient diversity, e.g. RAKE receiver

(CDMA), or multi-antenna diversity or distributed subcarrier

allocation (OFDMA). The path-loss exponent of outdoor (in-

door) transmission is denoted by (). In particular, thechannel model is given by

() =

{ outdoor & cross-wall transmission

indoor transmisstion(1)

Here, is the distance from the transmitter to the respectivebase station. Setting > incorporates wall penetration lossin our channel model.

Assumption 1: We assume that there is no coordination

between the FAP and the macrocell BS, nor between different

FAPs, in terms of power control or resource scheduling.

In the uplink, denote and as the received power at the

macrocell BS and the FAP respectively. Through uplink powercontrol, a macrocell user causes interference of / tothe FAP, where ( ) is its channel to the FAP (the macrocellBS). Conversely, a femtocell user causes interference of/ to the macrocell BS.

Definition 1: For {0, 1, . . . , }, its interferencefactor is defined as / . {1, 2, . . . , } are i.i.d.random variables, and we define their ordered statistics as

(1) = min(1, . . . , ), () = max(1, 2, . . . , )

and for 1 < < ,

() = min({1, . . . , }{(1), . . . , (1)})

Correspondingly cellular users are reordered as

{(1), (2), . . . , ()}.Assumption 2: We assume 0 () holds, because the

home user is closer to the FAP, and the indoor channel has a

smaller path loss exponent.

Lemma 1: The cumulative distribution function for the in-

terference factor of each cellular user is

() =

(/)2 0 < ( + )

() ( + ) < 1

+0.5 sin(2) = 1

1 () 1 < (

)

1 (/)2 ( ) <

(2)

where r, L(i), are given by:

=1/

1 2/, = arccos

(

2

),

() =( 0.5sin2)2 + ( 0.5sin2)2

2,

where =2/

12/, = arccos

2+2

2

2

and =

arccos

2+

22

2 .

Proof: See Appendix A

In non-orthogonal multiple access (CDMA), the inter-

ference is additive. For a set of interference factors

{1 , 2 , . . . , }, denote function (, ) as the CDF of=1

, which is given by

(, ) = (

=1

).

(, ) is the same for whatever interference factors, sincethe interference factors are i.i.d..

Lemma 2: An upper bound on the CDF function (, )is given by

(, ) (, ) = (())

() (3)

Proof: Without loss of generality, we assume 1 is themaximum interference factor of {1 , . . . , }. Therefore(1 ) =

(), which provides an upper bound on

(, ).

(, ) =

=1

= 1

=1

>

1 (1 > ) =

() (4)

To summarize, in this subsection we derived the CDF ()for each interference factor and an upper bound on the CDF

(, ) for the sum of these interference factors. These twoCDFs will be will be used to calculate outage probability for

orthogonal and non-orthogonal multiple access respectively.

B. Hand over Metric and Procedure

When the FAP deploys open access control, it can choose

to serve cellular users based on certain metrics. A typical

metric is that it provides service to cellular users if bothof the following two conditions hold: 1) these cellular users

cause outage to the home user, and 2) the FAP has available

resources. Such a metric allows cellular users to share the

femtocell resources when they can potentially boost the capac-

ity of femtocell owners by reducing co-channel interference.Suppose the maximum number of additional cellular users that

the FAP can serve is .Assumption 3: When cellular users cause outage to the

home user, the FAP picks the most noisy interferer from the

macrocell to serve. This hand over procedure continues as

long as the home user still experiences outage and the number

of handed over cellular users does not exceed .Based on this assumption, when the FAP provides service

to cellular users, these users must be the strongest in-terferers {(), . . . (+1)}, which reduce the macro-to-femto interference to the largest extent possible. When

served by the FAP, these cellular users cause interferenceof{/(), . . . /(+1)} to the macrocell respectively,which are also the smallest possible. Therefore, the proposed

procedure is preferable for the two parties, since it maximally

reduces the interference they experience after the handover.

C. Resource Allocation and Ergodic Rate

From Assumption 1, we consider distributed resource allo-

cation in two-tier femtocell networks.

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Backhaul Allocation. The macrocell BS usually has a

large backhaul capacity. So when a cellular user is served

by the macrocell BS, its rate will not be constrained by thebackhaul. However, the FAP backhaul capacity, denoted as ,is typically modest and often shared, common examples being

DSL and cable modem. Thus it is necessary to incorporate the

FAPs backhaul allocation into the analysis. As the FAP serves

additional (0 ) cellular users, the home user is

allocated with a portion of this backhaul capacity, whileeach of the cellular users is assigned a portion of theFAP backhaul capacity. In both closed and open access, when

the FAP does not serve any cellular users, there is no backhaul

issue, i.e. 0 = 1 and 0 = 0.Assumption 4: For {0, 1, . . . , } defined above, the

following inequality holds:

0 1 . . . (5)

Because as users are added, the fraction of resources allocated

to the home user should not increase.

Time Allocation (in TDMA/OFDMA per subband). In

the macrocell network, since all the users have the same raterequirement and they are i.i.d. located inside the macrocell, the

time resources will be fairly allocated among them. Therefore,

when the macrocell BS is serving cellular users, each userenjoys an average time fraction1 of 1/. In the femtocellnetwork, when the FAP serves additional cellular users, thetime fraction allocated to the home user and each of the cellular users should be and respectively, according tothe allocation in backhaul capacity.

Assumption 5: Each macrocell user has a target rate

, while each femtocell user has a rate requirement ofmin{, }, where is its allocated backhaul capacity.

Assumption 6: According to its rate requirement, each userhas a SINR target. The user achieves its required rate when

the received SINR (we assume additive white Gaussian noise

with variance of 2) at or above its SINR target. Otherwiseit is in outage and the rate is zero.

Definition 2: The event ( {0, 1, 2, . . . , }) is de-fined as the FAP provides service to additional cellular users.In event , denote the SINR targets of the home user, handedover cellular users and the remaining cellular users as, , , and , respectively. Their success probabilitiesare denoted as ,, , and , accordingly.

Definition 3: A users ergodic rate is its rate requirement

multiplied by its success probability.In this paper, we evaluate open vs. closed access from

the viewpoints of the femtocell owner the home users

ergodic rate 0, and the network operator cellular users sumthroughput , which is defined as the sum of all cellularusers ergodic rates. Although we use the hybrid model in

[19] as a more general form of open access, the overheadsignaling from handovers still would affect the rates of mobile

and femtocell users. However, since it is difficult to quantify

precisely, often involves separate overhead channels, and the

exact implementation varies significantly from protocol to

1Although in practice the time slot assigned to a mobile user can only

be a group of discrete values, the average time fraction of the mobile usercan be any value between 0 and 1. The same argument is applicable to thevalue of and .

protocol, we do not include the impact of handover signalling

in the analysis.

III. CAPACITY CONTOURS IN ORTHOGONAL MULTIPLE

ACCESS SCHEMES

In LTE and WiMAX, which both use a similar form of

OFDMA, the end-user is assigned a portion of the spectrum

for a (sub)frame, which is identical to being allocated the

entire spectrum for certain time slots (i.e. TDMA) from ananalysis perspective. Besides, each subband in OFDMA is

orthogonal and allocated in a TDMA fashion along the time

axis. Therefore in this section, we analyze a TDMA scenario,which can also be viewed as OFDMA on a per subband basis.

We first consider the scenario when the FAP can serve cellular users. We then focus on the important special case of

= 1.In an arbitrary time slot, suppose users and are

active at the femtocell and the macrocell respectively, causinginterference of / and at macro and femto BSsaccordingly.

Theorem 1: In TDMA or OFDMA, the home users ergodicrate and the cellular users sum throughput in femtocell closed

access are given by

0 = min(, 0)

(

,0

2

)(6)

=

(

/0 + 2 ,0

)(7)

Proof: Since at the macrocell BS, each cellular user

causes interference to the home user during its time slot,

namely 1/, the ergodic rate of the home user is

0 = min(, 0)

=1

1(

+ 2

,0)

= min(, 0)

(

,0

2

)(8)

On the other hand, each cellular user experiences an interfer-

ence of /0 from the home user. Their sum rate is

= =

(

/0 + 2 ,0

)(9)

Remark 1: In TDMA the macro-to-femto interference istime shared, so the home users probability of success is aver-

aged over time and consequently not scaled by . Therefore,the home users ergodic rate in closed access is independent of

the number of cellular users. Things are different in CDMA,

as shown in the next section.

It is an important fact that the SINR target of cellular

users in the macrocell (in both open and closed access) is

an increasing function of their density. Intuitively, when the

macrocell BS serves more mobile users, each of them hasa smaller time fraction and must increase its SINR target to

achieve a given rate requirement.In closed access, since the received SINR of a cellular

user in the macrocell is a constant value of

/0+2 , there

is a cutoff user loading , such that: 1) when ,each cellular users SINR target constraint is satisfied and

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their sum throughput is the maximum possible, = ;2) when > , each cellular users SINR target isinfeasible and = 0. The value of is governed by theinequality /0+2 (SINR target), which for example in

a Gaussian channel is = 1 log2(1 +

/0+2

).It is less clear if there is such a cutoff value in open

access, because the received SINR of each cellular user in

the macrocell is not constant, due to the random interference

from the cellular users supported at the FAP. The simulation

results in Section V show that such a cutoff value occursunder practical network configurations, which is essentially

due to two particular aspects of open access. First, the FAP

predictably allocates a large portion of OTA resources to the

home user. Therefore the femto-to-macro interference is still a

constant value for a large portion of time. Second, the cellular

users served by the femtocell must be very close to the FAPaccording to the handover criteria, which greatly reduces the

randomness of their locations. As a result, their interferenceto the macrocell is also nearly deterministic. The numerical

relation between and is discussed later in Remark 3.

In femtocell open access, events {, = 0, 1, 2, . . . , }can occur. From Assumption 3, FAP will pick the strongest

interferers sequentially, so

() =

(

1=0

, ) <

(1=0

) =

(10)

where for 0 , event = {

()+2< , } and

its complementary event is denoted as .Lemma 3: In TDMA or OFDMA, the ergodic rate of the

home user and sum throughput of the cellular users in open

access are given by

0 =

=0

min(, ), (11)

= ,0 +

=1

{( ), + min(, ),}

(12)

where ,,, and, are success probabilities of the homeuser, the supported cellular users at the femtocell and the

remaining cellular users at the macrocell respectively, whichare given by

, =

{() <

1

=1

()+2 ,,

=

(13)

, =1

=1

(

() + 2 ,,

)(14)

, =

(

/0 + 2 ,

)()

+

=+1

(

/() + 2 ,,

)(15)

Proof: Denote as the event that the home usersucceeds in its communication process. When < , wehave

, = (, ) = ()() = () (16)

When the FAP serves only ( < ) cellular users, it impliesthe home user experiences no outage at this point based on

Assumption 3. Therefore () = 1 and the last equalityholds. When = , the remaining cellular users inthe macrocell, which correspond to {(1), (2), . . . , ()},can possibly cause outage to the home user. Since they are

fairly scheduled, they are equally likely interfering the home

user with probability 1

. So in the event of, the successprobability of the home user is

, =

=1

1

(

() + 2 ,,

)(17)

Similar arguments hold for , and ,.In open access, due to the random macro-to-femto andfemto-to-macro interference, the cellular users sum through-

put is strictly between 0 and , which are two possible sumthroughput in closed access. Therefore, the network operators

choice between open vs. closed access is fairly clear.

Remark 2: According to the value of , the network op-erator prefers closed access when , while embracingopen access when > .

The reason why open access reduces the sum throughput

when is explained as follows. The femto-to-macrointerference is /0 in closed access for all time slots,

which in open access after handover will increase to (due toAssumption 2) /() in the time slot of(), the cellular userserved by the FAP. The increased femto-to-macro interference

from the handed over cellular users indeed bottlenecks the

performance of open access by reducing sum throughput.

Remark 3: When the amount of cellular users in the macro-cell is over , the femto-to-macro interference in closedaccess causes their sum throughput to be zero, which should

also be true in open access due to the increased femto-to-

macro interference. Considering the at most cellular usersserved by the FAP, the cutoff value should be given by

+ .

In the following, we focus on a special case of = 1.Such a case is important because femtocell owners can bereasonably expected as selfish users with their infrastructure.

Theorem 2: In TDMA or OFDMA, when the FAP is set to

serve at most one cellular user, namely = 1, the ergodicrate of home user and the sum throughput of cellular users in

open access are given by

0 = min(, 0)

(

,0

2

) + min(, 1),1

(18)

= ( /0 + 2

,0) ( ,0

2

)

+ ( 1),1 + min(, 1),1 (19)

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Where ,1, ,1 and ,1 are given by

,1 =

1(

,1

2

)

(1 1 (

,0

2

)

)

,1 = 1

(

/0 + 2 ,1

)

+1 [1 max( ( ,0

2

), (,1

,12

))],1 =

1(

,1

2

)

(1 1 (

,0

2

)

)

Proof: See Appendix B

Note that the SINR target of a femtocell user is a non-

increasing function of its allocated time fraction. For example,in a Gaussian Channel, a femtocell user has a SINR target = 2min(,)/ 1 = 2min(/,) 1 as is its timefraction. Thus, the observation below follows.

Remark 4: From Theorem 2, it is seen that the ergodic rate

of the home user in open access is an increasing function

of 1: As stated above ,1 dose not increase when 1 getslarger, then both (

,1

2

) and min(, 1) are non-

decreasing functions w.r.t. 1.

The remark above implies that with a large enough valueof 1, the home users ergodic rate can possibly be higherthan that in closed access. However, the following corollary

shows that such a rate gain in open access is not possible in

the regime of very large cellular user density.

Corollary 1: If the values of 1 is independent of , thenas the number of cellular user goes to arbitrarily large, that

is , the ergodic rate of home user in TDMA becomes

0 ={min(,

0

)

(

,0

2

) Closed Access

min(, 1)(

,1

2

) Open Access

(20)

Remark 5: Since 1 0 = 1, ,1 is greater than ,0.Therefore, as shown in Corollary 1, with infinitely large user

density, open access is inferior to closed access in terms of

home users rate.

Simulations in Section V show that open access provides

a very marginal rate gain to the home user in high user

density (e.g. on the order of hundred users per macrocell). This

observation, along with Corollary 1, indicates that open access

is not a suitable choice in densely populated scenarios. The

reason is explained as follows. When the number of cellularusers increases, the amount of time occupied by each interfererdecreases. Thus, in high cellular user density, handing over

a small group of interferers (Corollary 1 is derived in the

case of = 1, but the argument can be extended) lowersthe macro-to-femto interference merely for a small portion

of time, i.e. very short time length originally occupied by

them. The home users signal quality is still inhibited by the

residual interference from the remaining cellular users. Thus in

high user density, the femtocell will be reluctant to admit the

interferers even if they cause outage. Note that this conclusion

is possibly contingent on the assumption of no coordination

between femtocell and macrocell BS. Our conjecture is thatopen access with inter-BS coordination will be the appropriate

solution in high user density.

IV. CAPACITY CONTOURS IN NON -ORTHOGONAL

MULTIPLE ACCESS SCHEME

3G CDMA networks have been launched worldwide in

recent years and will be in wide service for at least a

decade. This necessitates research and standardization for

incorporating femtocells in CDMA cellular networks [3], [4],

[6], [7]. Even if both TDMA and CDMA are part of the

medium access (e.g. HSPA in 3GPP and EVDO in 3GPP2),we restrict our attention to the CDMA aspect here, and this

analysis would thus be valid per time or frequency slot. We

show that in CDMA the interests of the femtocell owner

and the network operator are compatible: Open access is theappropriate approach for both two parties.

In CDMA, suppose cellular users are served by the FAP,and the received SIRs at the two BSs are

+=1 ()+

2SINR at the FAP

/0+

=+1

/()+(1)+2 SINR at macro BS

(21)

To be consistent with previous analysis in TDMA, we usethe same notations of users target SIRs, but note that their

values change as the rate-SINR mapping function in CDMA

is different due to spreading.

Theorem 3: In CDMA, the ergodic rate of the home user

and the sum throughput of cellular users in femtocell closed

access are given by

0 = min(, 0)(,

,0

2

) (22)

=

(

/0 + ( 1) + 2 ,0

)(23)

Proof: In closed access, no cellular user is served by

the FAP, meaning that the value of in equation (21) iszero. Thus, the success probabilities of the home user and

the cellular users are given by

=

=1

,0

2

=

(,

,0

2

)

=

(

/0 + ( 1) + 2 ,0

)(24)

Then the results of 0 and follow.

Similar to TDMA or OFDMA, there exist cutoff userloadings and

for sum throughput in CDMA as well.

For example, in a Gaussian channel, the value is governedby

2 1

=

/0 + ( 1) +

2(25)

In femtocell open access, the mathematical expression of

{, = 0, 1, 2, . . . , } is given by

() =

(

1

=0 , ) <

(1

=0 ) =

(26)

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where for 0 , event is defined as

=

{

+

=1 () + 2

< ,

}

=

{=1

() >

,

2 +

}(27)

and its complementary event is denoted as .The general form of capacity contours in open access inCDMA are the same as those in Lemma 3, however in which

the success probabilities are different.

Lemma 4: In CDMA, the success probabilities of the home

user, the supported cellular users at the femtocell and the

remaining cellular users at the macrocell are given by

, =

{() <

=1 ()++2

,,

=

(28)

, =

=1 () + + 2

,,

(29)

, =

(

+ 2 ,,

)(30)

where = ( 1) + /0 +

=+1

/().

Proof: The proof is very similar to Lemma 3, so isomitted.

Based on Lemma 4, we derive two helpful lower bounds in

the following theorems.

Theorem 4: In CDMA, the home users ergodic rate in open

access is

0 = min(, 0)(,

,0

2

)

+

=1

min(, ), (31)

a lower bound on , is given by

,

[1

(

) ((); + 1, )

] ( , ) (32)

Where =

,1 2

, =

(1,)2,

, and thefunction (; , ) is the incomplete beta function

(; , ) =

0

1(1 )1 (33)

Proof: See Appendix C.

Remark 6: Theorem 4 shows that in CDMA, open access

has strictly better performance than closed access in terms of

the home users rate, irrespective of the femtocell resource

allocation after handover.

In CDMA, interference is additive. So handing over a

small group of strongest interferers always means a significant

reduction in macro-to-femto interference and consequentlyan improvement in the home users rate. On the contrary,

interference is time shared in TDMA. Thus handing over a

small group of interferers for just part of the time does not

guarantee an appreciable reduction of cross-tier interference.

Theorem 5: In CDMA, when the FAP is set to serve one

cellular user at most, namely = 1, the sum throughput ofcellular users in open access is given by

= ( 1),1 + min(, 1),1

+ (

/0 + ( 1) + 2 ,0) (, ) (34)

where lower bounds of ,1 and ,1 are given by

,1

1 ()

( 1, ) (35)

,1 1

(

2

)(36)

where and are given by

=

,0

2

=(1 ,1) 2,1

,1

= max

(,1

( 2),1 ,1/0,

,0

)Proof: we first deploy the same technique as in the proof

of theorem 4 in deriving the lower bound of ,1. Accordingto equation (29), we have

,1 =

=1

,1=1

()

() >

( 1, )

=

1 ()

( 1, ) (37)

A similar proof applies to the lower bound of ,1After the FAP serves cellular users, the femto-to-

macro interference in CDMA is ( ) + /0 +=+1 /(), smaller than ( 1) + /0 in

closed access with high probability (according to Definition

1 of ordered interference factors). However, due to the result-ing variance, the femto-to-macro interference in open access

can exceed a certain threshold with a positive possibility,

consequently causing outage to cellular users remaining in

the macrocell. Open access thus causes a minor loss of sum

throughput in CDMA, as shown in Section V.Remark 7: In CDMA, open access is also the preferred

choice for the network operator, since it is almost as goodas (strictly better than) closed access in the regime of small

(large) .

V. NUMERICAL RESULTS AND CONCLUSION

Notations and system parameters are given in Table I. Notethat in our plots, the home users ergodic rate and cellular

users sum throughput are normalized by .

A. TDMA or OFDMA Access

Cellular User Density. Fig. 1 and 2 depict the home users

ergodic rate and cellular users sum throughput w.r.t. cellularuser density. For the purpose of fair comparison, the values

of and in these two plots are set as1

and 1

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TABLE INOTATIONS AND PARAMETERS

Symbol Description Sim. Value

macrocell radius 300 meters Distance between macro & femto BSs 150 meters Distance between home user and femto BS 5 meters, Path loss exponents 4, 2/

2 Femtocell BS received SNR 20 dB/2 Macrocell BS received SNR 20 dB

Spreading factor (for CDMA) 64 User rate requirement 0.5 bps/Hz Femtocell backhaul capacity 2 bps/Hz The home users portion of FAP resources

with handed over cellular usersN/A

Portion of FAP resources allocated to eachof the handed over cellular users

1

,home users SINR target in TDMA 2

min(

,) 1

home users SINR target in CDMA 1

(2min(,)1)

,SINR target of cellular user supported inthe femtocell in TDMA

2min(

,) 1

SINR target of cellular user supported inthe femtocell in CDMA

1

(2min(,)1)

, SINR target of cellular user remaining inthe macrocell in TDMA 2

()

1

SINR target of cellular user remaining inthe macrocell in CMDA

1

(2 1)

0 Ergodic rate of the home user N/A Sum throughput of cellular users N/A Cutoff user loadings for sum throughput in

closed accessN/A

Cutoff user loadings for sum throughput inopen access

N/A

0 20 40 60 80 100 120 1400.7

0.75

0.8

0.85

0.9

0.95

1

Number of cellular users

Ergodicrateofthe

homeuser(bps/Hz

)

Closed Access

Open Access (K=1)

Open Access (K=3)

Open Access (K=5)

Fig. 1. The home users ergodic rate versus cellular user density in TDMA.We have = 1

and =1

for fair comparison, 0 .

respectively.In low user density ( = 49), open accessprovides an appreciable rate gain to the home user, however

which also causes a noticeable decrease in cellular users sumthroughput. It is seen that the rate gain and loss are about the

same in terms of percentage: For = 3 case at = 20, as anexample, the rate gain of the home user is about 15%, and the

rate loss of cellular users is almost 20%. Indeed, the choices of

the two parties in low cellular user density are incompatible.

As predicted previously (below Remark 1), there is a cutoff

0 10 20 30 40 50 600

5

10

15

20

25

30

35

40

45

50


Cellularusers'su

mt

hroughput(bps/Hz)

Closed Access

Open Access (K=1)

Open Access (K=3)

Open Access (K=5)

(a) Regime of low cellular user density

60 70 80 90 100 110 1200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18


Cellularusers'sumt

hroughput(bps/Hz)

Closed Access

Open Access (K=1)

Open Access (K=3)

Open Access (K=5)

(b) Regime of high cellular user density

Fig. 2. Cellular users sum throughput versus cellular user density in TDMA.We have = 1

and =1

for fair comparison, 0 .

user loading in open access. Indeed, further simulationsshow as long as 30%, equals

+ . In other

words, single open access femtocell expands the macrocell

network capacity by . Therefore when , open

access is appropriate for both parties, especially to the network

operator by offloading traffic from overloaded macrocell.

In high user density ( = 55), open access providesonly a very small rate gain to the home user. Its rate gain to the

cellular user is also marginal because the macro BS remains

overloaded even after handover users to the femtocell. Notethat we assume that the overloaded macro BS does not turn

off any cellular users even when the outage becomes high.

Obviously admission control could be used to improve the

sum rate, so this can be considered a worst case scenario for

closed access since the sum throughput tends to zero. Even

with this assumption favoring open access which maintains

a low but strictly positive sum throughput due to the handoffs not much gain is observed for open access. If the overloaded

macrocell BS randomly blocks users to avoid congestion,

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TABLE IICHOICES OF T WO PARTIES W.R.T. CELLULAR USER DENSITY

TDMA or OFDMA CDMAFemtocell Owner Network Operator Femtocell Owner Network Operator

Low Cellular User Density ( < ) Open Access Closed Access Open Access IndifferentMedium Cellular User Density (

) Open Access Open Access Open Access Open Access

High Cellular User Density ( > ) Closed Access Indifferent Open Access IndifferentChoices Highly Dependent on Cellular User Density Open Access

0 10 20 30 40 50 60 70 80 90 100 1100.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


themin

imalvalueofr

equiredbythehomeuser

Open Access (K=1)

Open Access (K=3)

Open Access (K=5)

Fig. 3. The value of, i.e. the minimal proportion of femtocell resourcesrequired by the home user in TDMA open access, versus cellular user density.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

The value of , i.e. the proportion of femtocell resources allocated to the home user

Ergodicrateofthehomeuser

Open Access (K=1)

Open Access (K=3)

Open Access (K=5)

Fig. 4. The home users ergodic rate versus the portion of femtocell resourcesallocated to him in TDMA, i.e. the value of in open access. The cellularuser density = 30.

the curves in Fig. 2 would flatten beyond the cut-off, which

implies lower sum throughput to the cellular users in open

access. We summarize our observations in Table II.

Femtocell Resource Allocation. In this subsection, we

identify the appropriate fraction of femtocell resources al-

located to the home user in open access. Note that in thissubsection, is assumed to be the same for all and wesimply denote it as here. As stated in Remark 4, the home

users ergodic rate is an increasing function of . Therefore,to ensure the home users ergodic rate is not reduced after the

handover, the value of must be above some threshold . Itis shown in Fig. 3 that is an increasing function of cellularuser density . In higher cellular user density , the reductionof macro-to-femto interference is diminishing (as stated below

Remark 5), the home user will require more time resources,

and therefore larger value of , to lower its SINR targetand consequently be more tolerant to the interference. Hence

the FAP resource allocation in TDMA or OFDMA should be

adaptive to cellular user density, which is potentially difficult

due to no coordination between the FAP and the macrocellBS. Otherwise, the performance of open access is very likely

to degrade sharply, because the home users ergodic rate issensitive to the value of , as shown in Fig. 4.

Summary for TDMA/OFDMA Access. In orthogonalmultiple access, the choices of the two parties are highly

dependent on the cellular user density, with both preferringopen access in medium density, closed access in high density,

and they are in disagreement at low density. Therefore, our

results suggest that when deploying OFDMA, femtocell access

control should be adaptive based on the estimated cellular user

density. Note that this conclusion is possibly contingent on the

assumption of no coordination between the femtocells andthe macrocell BS as well as possibly the lack of admission

control at the macrocell BS. Future work should consider

inter-BS coordination both among and across the two tiers

in the context of both closed and open access. Investigations

on admission control and handoff policies are also importantrelated topics for future research.

B. CDMA Access

Cellular User Density. Fig. 5 and 6 present the home users

ergodic rate and cellular users sum throughput in CDMA. To

be consistent with TDMA/OFDMA case, the values of and in these two plots are also set as1

and 1 respectively.

Theorem 4 states that the home user will always experience

a rate gain in open access in CDMA, which is over 300%

(5 dB) for a vast range of cellular user density, as shown in

Fig. 5. In the regime of small ( = 155), openaccess in CDMA only leads to a negligible loss of cellular

users sum throughput, as shown in Fig. 6. Open access in

CDMA will strictly improve cellular users sum throughputwhen = 155, for the same reason in TDMA orOFDMA. Therefore, in CDMA open access is an appropriateapproach for both parties in the whole range of cellular user

density.

Femtocell Resource Allocation. As stated in Theorem 4

and Remark 6, open access improves the home users rate,

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0 10 20 30 40 50 60 70 800

5

10

15

20

25

Rategainofope

naccessindB(dashlines)


0 10 20 30 40 50 60 70 80

0

0.2

0.4

0.6

0.8

1

Ergodicrateofthehom

euserinclosedaccess(solidline)

Closed Access

Rate Gain in Open Access (K=1)



Fig. 5. The home users rate gains in dB (dash lines) in CDMA open accesscompared with closed access (solid line). For the purpose of comparison withTDMA/OFDMA case, we have = 1

and =1

, 0 .

60 70 80 90 100 110 120 130 140 150 1600

20

40

60

80

100

120

140

160


Cellularusers'sumt

hroughput(bps/Hz)

Closed Access

Open Access (K=1)

Open Access (K=3)

Open Access (K=5)

Fig. 6. Cellular users sum throughput versus cellular user density in TDMA.For the purpose of comparison with TDMA/OFDMA case, we have =1 and =

1

, 0 .

no matter how the femtocell backhaul is shared among users

and what the cellular user density is. Indeed, when ,

the home users rate is not affected by the femtocell resourceallocation. This observation provides insight into the optimal

value of , which is the maximum number of additionalcellular users the FAP can support. Compared with smaller

values of , the choice of = max{ :

} ispreferred by the home user, which can significantly reduce theinterference while still not affecting its available resources.

In short, the optimal value of should be not less thanmax{ :

}.

Summary for CDMA Access. Open access in CDMA

benefits both parties in almost the whole range of cellular user

density. Moreover, these appreciable benefits do not require theFAP to deploy adaptive resource allocation. Therefore, open

access is conclusively preferred in 3G CDMA networks.

0 20 40 60 80 100 120 140 1600.65

0.7

0.75

0.8

0.85

0.9

0.95

1


Ergodicrateofthehomeuser

Closed Access

Open Access (K=1)

Open Access (K=3)

Open Access (K=5)

Fig. 7. the home users ergodic rate in TDMA by incorporating theshadowing effect into the channel model. We assume a lognormal shadowingwith standard deviation = 10 dB. For the purpose of comparison withFig. 1 (which includes path loss attenuation only), we have = 1

and =1

, 0 .

C. Discussion on Shadowing

We investigate the impact of large-scale random channeleffects in this subsection by incorporating lognormal shad-

owing with standard deviation = 10 dB in the channelmodel. Simulation results show that shadowing affects the

performance of open and closed access as one would expect,

but it does not change the main conclusions about the optimal

access policy. Consider the home users ergodic rate in TDMA

for instance. By comparing Fig. 1 (path loss only) and Fig. 7

(path loss and shadowing), it is seen that shadowing lowersthe rates by approximately 5% 10%, but the main trendsof the curves are preserved: open access in TDMA still has a

diminishing rate gain for the home user as cellular user densitybecomes large. Therefore, in the presence of shadowing, open

access in TDMA is preferred by the femtocell owner in lowuser density, the same conclusion as in the path loss only

case. Further simulation results (not included due to space

limitations) confirm that our main conclusions listed in Table

II are unchanged in view of shadowing.

APPENDIX

A. Proof of Lemma 1Denote (, ) as the location of a cellular user. Thus the

CDF of its interference factor is

() = ( )

=

(1 2 )2 + (1

2 )2 + 2

2 2

2 0

= /(2). (38)

is the area inside the macrocell, and governed by

(1 2/)2 + (1 2/)2 + 22/ 22/ 0

When = 1, the above equation defines a circle area, with

center of = 2/12/ and radius of =1/12/

. Moreover,

when < 1, is the area inside the circle, while > 1, is

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the area outside the circle. Therefore, according to the range

of , () can be divide into 5 segments:

1) when 0 < ( + ) (note that interference factor is

a non-negative r.v., we must have 0), the circle iscontained in macrocell ( + ). Thus = 2.

2) when ( + ) < 1, the circle intersects with

macrocell ( + ). Using the method in [22],we get:

= ( 0.5sin2) 2 + ( 0.5sin2) 2

where = arccos(2+2

2

2) and =

arccos(2+

22

2).

3) when = 1, the area of is a half plane ( 2 )intersected with the macrocell. Thus = ( +0.5sin2)2, where = arccos( 2 ).

4) when 1 < ( ), the circle intersects with

macrocell ( + ). Note that now is the areaoutside the circle. Therefore

= 2

( 0.5sin2) 2

( 0.5sin2) 2

5) when ( ) < , the circle is contained in macrocell

( + ). Similarly, = (2 2).

B. Proof of Theorem 2

As = 1, there are only two events 0 and 1, withprobabilities of

(0) =

(

,0

2

)

(1

) = 1

(

,0

2

)

The key in the proof is the calculation of ,1, ,1 and ,1.Applying (13), we have

,1 =

1=1

()

,1

2

, () >

,0

2

1

Denote (

,1

2

) as , and (

,0

2

) as

, then we have

1

=1 (()

,1

2

, () >

,0

2

)

=

1=1

1=

(

) (1 )

(

) ()

= (1 1)

Substituting back for and , ,1 is given by

,1 =

1(

,1

2

)

(1 1 (

,0

2

)

)(39)

The success probability ,1 of handover user () followsby applying the same technique.

In the femtocell, the home user is allocated a time fraction

of 1, and the handed over user () is assigned a time

fraction of 1. Therefore, the success probability of a cellularuser in the macrocell network is

,1 = 1

(

/0 + 2 ,1

)

+ 1

(

/() + 2 ,1,

() + 2

< ,0

)

= 1(

/0 + 2 ,1)

+1

[1 max

( (

,0

2

), (

,1 ,12

)

)](40)

C. The Proof of Theorem 4

As = 0, we have

,0 =

=1 () +

2 ,0

= (

, ,0

2

)(41)

For 1 , it is easy to check that , has the sameform, which are lower bounded as below. The constants and are as defined in the statement of Theorem 4.

, = (

1=0

, )

()

1=0

{=1

() >

,

2

},

=1

()

()

+1=1

() > ,

=1

()

(+1) > ,

=1

()

()

(+1) >

( , )

=

[1

(

) ((); + 1, )

] ( , )

(42)

The inequality (a) comes from the lower bound on the proba-

bility of (see equation (27)). It is important to note that inCDMA , is a non-decreasing function of . In a Gaus-sian channel, for example, , =

2min(,) 1

/.

Since does not increase as goes larger, , is alsoa non-increasing function of . Thus the inequality (a)holds. Instead of making

=1 () smaller than a cer-

tain constant, we randomly pick elements from theset of {(1), (2), . . . , ()}, which are also independent of(+1). Therefore inequality (b) holds.

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Ping Xia (S08-M10) received the B.S. in Elec-tronic Engineering with high honor from TsinghuaUniversity, China in 2008 and the M.S. in ElectricalEngineering from University of Texas at Austin (UTAustin) in 2010. Currently, he is pursuing the Ph.D.degree in Wireless Networking and CommunicationsGroup (WNCG) at the Department of Electricaland Computer Engineering, UT Austin. His researchfocuses on the heterogeneous networks comprisingmacro and femto cells, withe an emphasis on limits

and algorithms for base station coordinations ininterference cancelation and throughput enhancement in these networks.

Vikram Chandrasekhar is a systems and standardsengineer in the Communications Infrastructure di-vision at Texas Instruments. He is TIs lead 3GPPRAN1 delegate for on-going LTE-standardizationin heterogeneous network topics related to femto-cells, picocells and wireless relays. He is a co-recipient of the best paper awards in the 2009 IEEEGlobecom conference (communication symposium)and the overall runner-up (and best communicationspaper) during the 2008 IEEE Asilomar Conferencein Signals, Systems and Computers. He is a recipient

of the National Talent Search scholarship awarded by the Government of Indiain 1994.

Mr. Chandrasekhar received his Ph.D. from the University of Texas atAustin in May 2009. His PhD dissertation focused on co-existence and inter-ference management in femtocell-based cellular architectures. He completedhis B.Tech (honors) at Indian Institute of Technology, Kharagpur in 2000and received his M.S. at Rice University in 2003. After his M.S., he held astaff engineer position at National Instruments for two and half years wherehe was a finalist for the best new employee of the year. He was an internat Texas Instruments in the summers of 2007 and 2008, working on powercontrol schemes in femtocell networks and maximizing sum rates in multi-cellsystems.

Jeffrey G. Andrews (S98, M02, SM06) receivedthe B.S. in Engineering with High Distinction fromHarvey Mudd College in 1995, and the M.S. andPh.D. in Electrical Engineering from Stanford Uni-versity in 1999 and 2002, respectively. He is anAssociate Professor in the Department of Electrical

and Computer Engineering at the University ofTexas at Austin, and the Director of the WirelessNetworking and Communications Group (WNCG),a research center comprising 17 faculty and 10industrial affiliates. He developed Code Division

Multiple Access systems at Qualcomm from 1995-97, and has consulted forentities including the WiMAX Forum, Microsoft, Apple, Clearwire, Palm,ADC, and NASA.

Dr. Andrews is co-author of two books, Fundamentals of WiMAX (Prentice-Hall, 2007) and Fundamentals of LTE (Prentice-Hall, 2010), and holds theEarl and Margaret Brasfield Endowed Fellowship in Engineering at UTAustin, where he received the ECE departments first annual High Gainaward for excellence in research. He is a Senior Member of the IEEE, andserved as an associate editor for the IEEE T RANSACTIONS ON WIRELESSCOMMUNICATIONS from 2004-08.

Dr. Andrews received the National Science Foundation CAREER award in

2007 and is the Principal Investigator of a 9 university team of 12 facultyin DARPAs Information Theory for Mobile Ad Hoc Networks program.He has been co-author of four best paper award recipients, two at IEEEGlobecom (2006 and 2009) one at Asilomar (2008), and the 2010 IEEECommunications Society Best Tutorial Paper Award. His research interestsare in communication theory, information theory, and stochastic geometryapplied to wireless ad hoc, femtocell and cellular networks.

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