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Modeling, Analysis and Design for CarrierAggregation in Heterogeneous Cellular Networks

Xingqin Lin, Jeffrey G. Andrews and Amitava Ghosh

Abstract—Carrier aggregation (CA) and small cells are twodistinct features of next-generation cellular networks. Cellularnetworks with different types of small cells are often referred toas HetNets. In this paper, we introduce a load-aware model forCA-enabled multi-band HetNets. Under this model, the impactof biasing can be more appropriately characterized; for example,it is observed that with large enough biasing, the spectralefficiency of small cells may increase while its counterpartina fully-loaded model always decreases. Further, our analysisreveals that the peak data rate does not depend on the basestation density and transmit powers; this strongly motivates otherapproaches e.g. CA to increase the peak data rate. Last but notleast, different band deployment configurations are studied andcompared. We find that with large enough small cell density,spatial reuse with small cells outperforms adding more spectrumfor increasing user rate. More generally, universal cochanneldeployment typically yields the largest rate; and thus a capacityloss exists in orthogonal deployment. This performance gapcan be reduced by appropriately tuning the HetNet coveragedistribution (e.g. by optimizing biasing factors).

I. I NTRODUCTION

Carrier aggregation (CA) is considered as a key enablerfor LTE-Advanced [1], which can meet or even exceed theIMT-Advanced requirement for large transmission bandwidth(40 MHz-100 MHz) and high peak data rate (500 Mbps inthe uplink and 1 Gbps in the downlink). Simply speaking,CA enables the concurrent utilization of multiple componentcarriers on the physical layer to expand the effective bandwidth[2]–[4]. The aggregated bandwidth can be as large as 100MHz, for example, by aggregating 5 component carriers ofbandwidth 20 MHz each. The bandwidth of these componentcarriers can vary widely (ranging from 1.4 MHz to 20 MHzfor LTE carriers [5]–[7]). The propagation characteristics ofdifferent component carriers may vary significantly, e.g.,acomponent carrier in the 800 MHz band has very differentpropagation characteristic from a component carrier in the2.5GHz band. Due to the significance and unique features ofCA, appropriate CA management is essential for enhancingthe performance of CA-enabled cellular networks.

A. Related Work and Motivation

Cellular networks are undergoing a major evolution ascurrent cellular networks cannot keep pace with user demandthrough simply deploying more macro base stations (BS) [8].

Xingqin Lin and Jeffrey G. Andrews are with Department of Electrical& Computer Engineering, The University of Texas at Austin, USA. (E-mail:xlin@utexas.edu, jandrews@ece.utexas.edu). Amitava Ghosh is with NokiaSiemens Networks. (E-mail: amitava.ghosh@nsn.com). Thisresearch wassupported by Nokia Siemens Networks.

As a result, attention is being shifted to deploying small,inexpensive, low-power nodes in the current macro cells; theselow power nodes may include pico [9] and femto [8] BSs,as well as distributed antennas [10]. Cellular networks withthem take on a very heterogeneous characteristic, and are oftenreferred to as HetNets [11]–[14]. Due to the heterogeneous andad hoc deployments common in low power nodes, the validityof adoption of the classical models such as Wyner model [15]or hexagonal grid [5] for HetNet study becomes questionable[16].

Not surprisingly, random spatial point processes [17], par-ticularly homogeneous Poisson point process (PPP) for itstractability, have been used to model the locations of thevarious types of heterogeneous BSs. Such a probabilisticapproach for cellular networks can be traced back to late1990’s [18], [19]. The PPP does not exactly capture everycharacteristic of cellular networks; for example, two points ina PPP can infrequently be arbitrarily close to each other, whichis not usually true in practice, especially for the deploymentof macro BSs. Nevertheless, as a null hypothesis, the PPPmodel opens up tractable ways of assessing the statisticalproperties of cellular networks, and it provides reasonablygood performance prediction. Indeed, the recent work [16],[20] have demonstrated that the PPP model is about as accuratein terms of downlink SINR distribution and handover rateas the hexagonal grid for a representative urban/suburbancellular network; the gap is even smaller for the uplink SINRdistribution with channel inversion [21]. Further, the PPPmodel may become even more accurate in HetNets due tothe heterogeneous and ad hoc deployments common in lowpower nodes [16].

Using the PPP model, the locations of different type ofBSs in a HetNet are often modeled by an independent PPP[22]. Due to the tractability of PPP model, many analyticalresults such as coverage probability and rate can be obtained[22]–[24]; more interestingly, these analytical results fairlyagree with industry findings obtained by extensive simulationsand experiments [25]. As a result, similar models have beenfurther used to optimize the HetNet design including spectrumallocation [26], load balancing [27], [28], spectrum sensing[29], etc. These encouraging progresses motivate us to adoptthe PPP model for CA study in this paper.

One major concern about deploying small cells is that theyhave limited coverage due to their low transmit powers. Asa result, small cells are often lightly loaded and would notaccomplish much without load balancing, while macro cellsare still heavily loaded. To alleviate this issue, a simple loadbalancing approach called biasing has been proposed [11];

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biasing allows the low power nodes toartificially increasetheir transmit powers. As a result, it helps expand the coverageareas of small cells and enables more user equipment (UE) tobe served by small cells [11], [25], [30]. It is expected thatbiasing can help balance network load and correspondinglyleads to higher throughput. However, a theoretical study ontheimpact of biasing is challenging. While [24] modeled biasingin a single-band HetNet, it assumes a fully-loaded HetNet,i.e., all the BSs are simultaneously active all the time. Similarfully-loaded model is also used in [27] for offloading study.

Sum rate is an important metric in wireless networks, partic-ularly CA-enabled HetNets. Unfortunately, from the sum-rateperspective, biasing does not help in a fully-loaded network.Thus, in order to appropriately examine the effect of biasingon the sum rate of CA-enabled HetNets, an appropriate notionof BS load is needed. While [31] proposed a load-aware modelin which low power nodes can be less active than macro BSsover thetime domain, it focuses onsingle-band HetNet andis still not sufficient for CA study which essentially involvesmulti-band modeling and analysis. Therefore, the main goalof this paper is to propose a multi-band HetNet model withan appropriate notion of load. With the proposed model, wewould like to theoretically examine the impact of biasing andstudy how to deploy the available bands in a HetNet to bestexploit CA.

B. Contributions and Main Results

The main contributions and outcomes of this paper are asfollows.

1) A new load-aware multi-band HetNet model:In SectionII, we introduce a load-aware model for CA-enabledmulti-band HetNets. Compared to [31] which uses a time-domainnotion of load, the new model uses a notion of fractionalload in the frequency domain. While the former is suitablefor bursty traffic, the latter is more suitable for static trafficlike VoIP. Moreover, the latter provides a different viewon load modeling in HetNets and thus can be viewed ascomplementary to the former. The proposed model is flexibleenough to capture the main unique features of both multi-and single flow CA (defined in Section II) but yet tractableenough for analysis. Under this load-aware model, the impactof biasing is appropriately characterized; for example, itisobserved that with large enough biasing, the spectral efficiencyof small cells can actually increase while its counterpart in afully-loaded model always decreases because of the signal-to-interference-plus-noise ratio (SINR) reduction from biasing.

2) Rate analysis in multi-band HetNets:Unlike rate anal-ysis in a single-band HetNet, there exists correlation amongthe signals and interference across the bands in a multi-bandHetNet. This correlation feature is highlighted in SectionIII.In this paper, we present a way to deal with this correlationfirst for single tier networks (c.f. Section III) and then extendit for general multi-tier HetNets (c.f. Section IV). This wayof dealing with the multi-band correlation contributes to thetractable approach to the performance analysis of cellularnetworks [16].

3) Design insights: From the analytical results, severalobservations may be informative for system design. In singletier interference-limited networks (where noise is ignored),it is found that the peak data rate does not depend on theBS density and transmit powers; this strongly motivates otherapproaches e.g. CA to increase the peak data rate. Further,if the aggregated carriers are sorted in ascending order basedon their path-loss exponents, the peak data rate scalessuper-linearly with the number of aggregated carriers; this providesa finer characterization for the common conjecture oflinearlyscaled peak data rate with increasing number of carriers [3].

Different band deployment configurations are studied andcompared for CA-enabled HetNets. We derive the rate ex-pressions of1-band-K-tier deployment andK-band-1-tierdeployment; these two deployments represent two popularapproaches for increasing rate in cellular networks: spatialreuse with small cells and adding more bandwidth. It is foundthat, if the densities of low power nodes are large enough,a 1-band-K-tier deployment can provide larger rate than theK-band-1-tier deployment. This gives additional theoreticaljustification for using small cells to solve the current “spectrumcrunch”. More generally, we find that universal cochanneldeployment – all the tiers use all the bands – typically yieldsthe largest rate. Correspondingly, there is a capacity lossinorthogonal deployment – different tiers use different bands.This performance gap can be reduced by appropriately tuningthe HetNet coverage distribution (e.g. by optimizing biasingfactors).

II. SYSTEM MODEL

We consider a general HetNet withK = {1, ...,K} denotingthe set ofK tiers which may include macro cells, pico cells,femtocells, and possibly other elements. BSs of different typesmay differ in terms of deployment density, spectrum resource,transmit power and supported modulation and coding scheme.In this paper, we focus on the downlink and assume openaccess for all the small cells. The other key aspects of thestudied model are described below.

A. Distributions of BSs and UEs

The BS locations are modeled asK independent homoge-neous PPPs [17]. Denote byΦk the set of PPP distributedBSs in tierk andλk its density. The UEs are also assumed tobe randomly distributed according to an independent PPP ofdensityλ(u). Or equivalently, the number of UEs in a certainregion follows Poisson distribution whose mean equalsλ(u)

multiplied by the area of the region, and given the number,UEs are independently and uniformly located in the region.Uniform random UE spatial distribution over a certain regionis often utilized by industry in system level simulations (seee.g. [32]).

B. Channel Model

We assume that there are a set ofM availablebandsdenotedasM = {1, 2, ...,M}. The bandwidth and path loss exponentof each bandi are denoted byBi andαi, respectively. Here we

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Multi flow: Single flow:

Macro BS Macro BS

Pico BS Pico BS

Desired signal

I t f

Bands used by macro cells

Interference Band used by pico cells

Fig. 1. System model: multi-flow versus single flow

use different path loss exponents for different bands to capturethe possibly large differences in propagation characteristicsassociated with each band’s carrier frequency. We furtherassume that each bandi is small enough to have relativelyconstant path loss exponentαi across it.

Suppose each tier-k BS transmits at constant powerPi,k inthe i-th band, provided that bandi is used by tierk. Then thereceived powerPi,k at the typical UE (assumed to be at theorigin) from the BS located atY ∈ R

¯2 is modeled as

Pi,k,Y = Pi,kHi,Y Ci‖Y ‖−αi , (1)

where‖Y ‖ denotes the Euclidean norm ofY , Ci is a constantthat gives the path loss in bandi when the link length is1,andHi,Y is a random variable capturing the fading value ofthe radio link from the BS atY to the typical UE in bandi.Note thatCi strongly depends on carrier frequency, e.g.Ci

∼=(µi/4π)

2 where µi denotes the wavelength. For simplicity,we ignore shadowing and consider Rayleigh fading only, i.e.,Hi,Y ∼ Exp(1). In fact, the randomness of the BS locationsactually helps to emulate shadowing: As shadowing varianceincreases, the resulting propagation losses between the BSsand the typical user in a grid network converge to those in aPoisson distributed network [33].

C. User Association Schemes

We introduce two types of CA, as shown in Fig. 1. The firsttype is calledmulti-flow CA, where UEs can be potentiallyassociated with all the available tiers simultaneously (but indifferent bands) and can aggregate data using all the availablebands. Then the typical UE is associated with tierk in bandi that provides the maximum biased received power, i.e.,k = argmax(ZℓPi,ℓ‖Yℓ0‖−αi : ℓ ∈ K), whereZℓ denotesthe biasing factor of tierℓ andYℓ0 denotes the location of thenearest BS in tierℓ. Biasing factorsZ = {Zk : k ∈ K} areused in HetNets for the purpose of load balancing: Adoptinglarger Zk expands the cell range of BSs in tierk and thusmore UEs can connect to tierk.

The second type is calledsingle flowCA, where UEs can beassociated with only one of the available tiers at a time, i.e.,one BS at some tier, though they still can aggregate data using

all the available bands used by that tier. For this type of CA,the typical UE scans over all the tiers and bands and connectsto the tier that provides the strongest biased received power insome band, sayi∗. Then the UE performs CA with respect tothis tier. Formally, the UE connects to tierk such that(k, i∗) =argmax(ZℓPi,ℓ‖Yℓ0‖−αi : (ℓ, i) ∈ K×M). Single flow maybe closer to how CA is normally supported in reality: UE isconfigured with a primary component carrier that provides thebest signal quality and other secondary component carriersareonly added when applicable.

In summary, multi-flow allows the UE to perform CA acrossall the available bands in (possibly) different tiers, while singleflow allows the UE to perform CA only across the availablebands used by the selected tier. In this paper, we focus onsingle flow and refer to [34] for the multi-flow study.

D. Load Modeling

We assume that each UE connecting to tierk in band irequires a basic sharebi,k of the bandwidth resourceBi.For example,bi,k may represent the basic frequency resourceallocation unit in LTE networks. This requirement can besatisfied if tierk in bandi is under-loaded, e.g.,Li,k ≤ Bi/bi,kwhereLi,k denotes the mean number of UEs attempting toconnect to a tier-k BS in band i. If Li,k > Bi/bi,k, thisimplies that too many UEs attempt to connect to tierk andcorrespondingly tierk becomes fully-loaded. In this case,some UEs within the coverage of tierk have to be blocked.Or equivalently, assuming all the UEs are admitted by tierk,they can only obtain a fraction of the time domain resource.1

In the following, we stick to the former interpretation andwill derive the admission probability while keeping in mindthat the admission probability can also be interpreted as thefraction of the time domain resource shared by the UEs withno blocking. For each BS in the under-loaded tierk, weassume that the location of the used bandwidthbi,kLi,k inbandi is uniformly and independently selected from bandi.Equivalently, frequency hopping can be assumed. We furtherdefine

θi,k = min

(

bi,kLi,k

Bi, 1

)

, ∀k ∈ K, ∀i ∈ M, (2)

and refer toθi,k as theload of tier k in bandi. Accordingly,the spectrum resources of bandi currently allocated by tierkequalsθi,kBi.

This bandwidth sharing model helps capture the fact that theload in HetNets is often unbalanced. In particular, small BSshave limited coverage and thus the mean numberLi,k of UEsserved by per small BS is small, yielding light loadθi,k ≪ 1.In contrast, macro BSs have much larger coverage and thusthe mean numberLi,k of UEs served by per macro BS islarge, yielding heavy loadθi,k ≈ 1. Further, the bandwidthsharing model is flexible enough to model different network

1This is essentially a round-robin scheduling mechanism. Other moresophisticated scheduling mechanisms (e.g. proportional fair scheduling) maybe considered. However, such extensions may significantly complicate theanalysis and we treat them as future work. Nevertheless, ourcurrent analysiscan provide a lower bound on the performance of the more sophisticatedscheduling mechanisms.

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scenarios: the largerbi,k is, the more data-hungry UEs existin the HetNet; in an extreme case, we can setbi,k = Bi,which makes all the available spectrum resources at a BS beconsumed as long as the BS serves at least one UE.

E. Performance Metric

The performance of the HetNet depends on how the avail-able bands are deployed. To model band deployment, weintroduce binary decision variables defined as

xi,k =

{

1 if band i is used by tierk;0 otherwise,

for all i ∈ M, k ∈ K. We groupxi,k ’s into a vectorx denotingthe band deployment configuration and make the followingassumption on band deployment.

Assumption 1.Each band is used by at least one tier andeach tier uses at least one band, i.e.,

∑

k∈K xi,k ≥ 1, ∀i, and∑

i∈M xi,k ≥ 1, ∀k.

There is no loss of generality in the above assumption sinceone can simply exclude the unused band and/or the useless tierfrom consideration. Given a band deployment configuration,the signal-to-interference-plus-noise ratio (SINR) in band iprovided by tierk can be computed as

SINRi,k =xi,kPi,kHi,k0 ‖ Yk0 ‖−αi

∑

ℓ∈K

∑

Y ∈Φi,ℓ\Yk0Pi,lHi,Y ‖ Y ‖−αi +wi,k

,

wherewi,k = bi,kN0/Ci with N0 being the power spectraldensity of the background noise, andΦi,k is the PPP of densityθi,kλk thinned fromΦk.

Rate, a function of the received SINR, is the paramountmetric in CA which aims to provide UEs with very high datarate. This motivates us to adopt theUE ergodic rateas themetric for CA study in this paper. UE ergodic rate measuresthe long term data rate attained by a typical UE and will bederived in the following sections.

Before ending this section, we would like to stress thatthough not modeled in this paper, interference managementis also a critical part of HetNets, especially for cochanneldeployment with biasing; otherwise, aggressive biasing mayresult in unacceptably poor SINR performance of cell-edgeusers. We treat the incorporation of interference managementas future work.

III. M ULTI -BAND ANALYSIS: SINGLE TIER CASE

CA study essentially involves multi-band analysis, whichis involved due to the spatial correlations in the multi-bandsignals and interference. To make this point explicit, let usconsider a single tier cellular network that consists of macroBSs only and uses two bands,1 and 2. In this case multi-flow and single flow CA coincide. Further, the general userassociation policies (c.f. Section II-C) reduce to the simplenearest BS association: the typical UE connects to its nearestBS. Clearly, the received signals in band1 and 2 are bothemitted from the nearest BS and thus are strongly correlated.By similar reasoning, the interference powers in band1 and2are also correlated due to the presence of common randomnessin the locations of the interferers.

In this section we analyze single tier cellular network todemonstrate how to cope with the spatial correlations in themulti-band analysis; we will extend it to generalK tiernetwork later. AsK = 1, we shall drop the subscriptk inthis section for ease of notation.

To begin with, we know that there areλ(u)/λ UEs2 onaverage associated with a typical BS and correspondinglyLi = λ(u)/λ, ∀i. Then by definition the load of each bandi is given by

θi = min

(

biLi

Bi, 1

)

= min

(

λ(u)biλBi

, 1

)

. (3)

It follows that the effective transmitter processΦi in bandi isa PPP with densityθiλ, thinned from the common groundtransmitter processΦ of density λ. Clearly, {Φi} are notindependent and thus spatial correlations are induced.

Now let us condition on the event that the typical UEconnects to the BSY0 located at a distancer from the UE.Here comes a tricky thing: conditioned on connecting to theBS Y0, this tagged BS has a coverage area containing thetypical UE and thus statistically it has a larger coverage cellthan that of a typical BS. This fact is known as Feller’s paradox(see e.g. [36]). Forx ∈ Φ, denote byC(x,Φ) the Voronoi cell(induced byΦ) centered atx. From [37], we know the pdf ofthe size‖C(0,Φ)‖ of the coverage area of the tagged BS isgiven by

f‖C(0,Φ)‖(x) =3.54.5

Γ(4.5)λ(λx)3.5e−3.5λx, x ≥ 0.

Conditioning on‖C(0,Φ)‖ = x, the number of UEs locatedin C(0,Φ) is Poisson with meanλ(u)x. So the mean numberof UEs located inC(0,Φ) is given by

N =

∫ ∞

0

λ(u)xf‖C(0,Φ)‖(x) dx =9λ(u)

7λ.

As expected, the above term is greater thanλ(u)/λ. Then theadmission probability of the typical UE in each bandi is givenby

pi = max

(

7λBi

9λ(u)bi, 1

)

. (4)

Now conditioning on the length‖Y0‖ = r of the typicallink and the point process{Φi}, the rate that the typicalUE experiences in bandi equalspi multiplied by spectralefficiencylog(1 +SINRi). Further, conditioning on‖Y0‖ = rand{Φi}, these rates are independent over theM bands sincerate in each bandi only depends on the fading field in bandiand the fading fields are assumed to be independent over theM bands.

Summing the rates over the bands, we obtain the ergodic

2This can be shown rigorously by using Neveu’s exchange formula (seee.g. [35]).

5

rate of the CA-enabled UE as

E¯H

[R|{Φi}, ‖Y0‖ = r]

= E¯H

[

M∑

i=1

pibi log(1 + SINRi)∣

∣

∣{Φi}, ‖Y0‖ = r

]

, (5)

where the expectation is over the fading fields. Denote byIΦi

=∑

Y ∈Φi\Y0PiHi,Y ‖ Y ‖−αi , the right hand side of (5)

equals

E¯H

[

M∑

i=1

pibi log

(

1 +PiHi,0r

−αi

IΦi+ wi

)]

=

M∑

i=1

pibiE¯

[

log

(

1 +PiHi,0r

−αi

IΦi+ wi

)]

=

M∑

i=1

pibi

∫ ∞

0

1

1 + tP¯

(

PiHi,0r−αi

IΦi+ wi

≥ t

)

dt, (6)

where in the last equality we use the fact: E¯[f(X)] =

∫∞

0 f ′(x)P¯(X ≥ x) dx, where f(·) is non-negative and

monotonically increasing function. Further,

P¯

(

PiHi,0r−αi

IΦi+ wi

≥ t

)

= P¯

(

Hi,0 ≥ trαiP−1i (IΦi

+ wi))

= e−twirαiP−1

i E¯H[

e−trαiP−1i

IΦi

]

= e−twirαiP−1

i E¯H[

e−trαi∑

Y ∈Φi\Y0Hi,Y ‖Y ‖−αi

]

= e−twirαiP−1

i E¯H

∏

Y ∈Φi\Y0

e−trαiHi,Y ‖Y ‖−αi

= e−twirαiP−1

i

∏

Y ∈Φi\Y0

E¯H[

e−trαiHi,Y ‖Y ‖−αi]

= e−twirαiP−1

i

∏

Y ∈Φi\Y0

1

1 + trαi ‖ Y ‖−αi, (7)

where in the second equality we useHi,0 ∼ Exp(1) and thusP¯(Hi,0 ≥ x) = e−x; the penultimate equality follows as the

fading fields are independent; and in the last equality we useHi,Y ∼ Exp(1) and thus E

¯[e−sHi,Y ] = 1

1+s . Plugging (7) into(6) yields

E¯[R|{Φi}, {‖Y0‖ = r}] =

M∑

i=1

pibi·∫ ∞

0

1

1 + te−twir

αiP−1i

∏

Y ∈Φi\Y0

1

1 + trαi ‖ Y ‖−αidt.

Now de-conditioning with respect to{Φi} yields

E¯[R|‖Y0‖ = r] = E

¯{Φi}

[

E¯[R|{Φi}, ‖Y0‖ = r]

]

=M∑

i=1

pibi

∫ ∞

0

1

1 + te−twir

αiP−1i

· E¯{Φi}

∏

Y ∈Φi\Y0

1

1 + trαi ‖ Y ‖−αi

dt

=

M∑

i=1

pibi

∫ ∞

0

e−twirαiP−1

i

1 + te−θiλ

∫

B(0,r)1− 1

1+trαi y−αidy

dt

=

M∑

i=1

pibi

∫ ∞

0

1

1 + te−wiP

−1i

trαie−πθiλρ(t,αi,1)r

2

dt, (8)

where in the penultimate equality the domain of integrationisfrom r to ∞ (because conditioning on the association with theBS atY0, the closest interferer has at least a distance‖Y0‖ = raway from the typical UE), and

ρ(t, α, β) =

∫ ∞

1

t

t+ βxα2

dx. (9)

The final step is to de-condition with respect to‖Y0‖ = r. Tothis end, we need the distribution function of the length‖Y0‖of the typical link; this is easy in single tier network [35]:

P¯(‖Y0‖ ≥ r) = P

¯(Φ(B(0, r)) = 0) = e−λπr2 , r ≥ 0, (10)

from which we havef‖Y0‖(r) = 2πλre−πλr2 , r ≥ 0. Usingf‖Y0‖(r) and de-conditioning with respect to‖Y0‖ = r in (8)yields

E¯[R] = E

¯‖Y0‖ [E¯[R|‖Y0‖]] =

∫ ∞

0

M∑

i=1

pibi

∫ ∞

0

· 1

1 + te−wiP

−1i

trαie−πθiλρ(t,αi)r

2

dt · 2πλre−πλr2 dr.

Applying Fubini’s theorem, we finally obtain UE ergodic ratein Prop. 1.

Proposition 1 (Single Tier UE Ergodic Rate). The single tierUE ergodic rateR = E

¯[R] is given by

R =M∑

i=1

2πλpibi

∫ ∞

0

∫ ∞

0

1

1 + te−wiP

−1i

trαi

· e−πθiλρ(t,αi,1)r2

e−πλr2r dr dt. (11)

Further simplification is possible when the noise is ignored,i.e.,wi → 0; we term this caseinterference-limitednetworks.This special case is of interest because, compared to interfer-ence, thermal noise is often not an important issue in moderncellular networks. Withwi → 0, (11) reads as follows.

E¯[R] =

M∑

i=1

pibi

∫ ∞

0

1

(1 + t)(1 + θiρ(t, αi, 1))dt.

If λ(u) ≥ λ, bi = Bi and the tagged BS only serves the typical

6

UE, the UE will have a peak data rate:

E¯[R] =

M∑

i=1

q(αi)Bi, (12)

whereq(α) =∫∞

01

(1+t)(1+ρ(t,α,1)) dt. Some remarks are inorder.

• q(α) is increasing withα asρ(t, α, β) is decreasing withα. This implies that bands of larger path-loss exponentscan offer higher rate per Hz, which seems a bit counter-intuitive. However, a careful thought reveals that bands oflarger path-loss exponents provide better spatial separa-tion for the wireless links, which is particularly importantin interference-limited networks.

• Interestingly, Eq. (12) does not depend on the BS densityλ and transmit powers{Pi}. This is intuitive because,increasing BS densityλ leads to increased signal powerbut also increased interference power; these two effectsexactly counter-balance each other when noise is ignored.Similar reasoning holds if one increases the transmitpowers.

• Following the previous remark, we see that increasingBS density in interference-limited networks does not leadto increased peak data rate. (But deploying more BSsdoes allow the network to serve more UEs at the sametime.) This strongly motivates other approaches e.g. CAfor increasing UE peak data rates.

• Intuitively, it is believed that peak data rate will scalelinearly with the number of aggregated carriers [3]. Eq.(12) provides a finer characterization of this statement:If the aggregated carriers are sorted in ascending orderbased on their path-loss exponents, the peak data ratein interference-limited networks will scalesuper-linearlywith the number of aggregated carriers.

IV. SINGLE FLOW CARRIER AGGREGATION

In this section we extend the results for single tier networksto the generalK tier HetNets with single flow CA. We makethe following additional assumption for tractability.

Assumption 2.The transmit power of each tierk is not band-dependent, i.e.,Pi,k = Pk, ∀i ∈ M.

A. Coverage, Admission Probability and Load

The key difference between HetNets and single tier cellularnetworks is that a typical UE in a HetNet can connect to anyone of theK tiers. In other words, UEs in different areas are(possibly) served by different kinds of BSs. For example, someUEs may connect to macro BSs while other UEs are servedby newly deployed low power nodes including pico and femtoBSs. As a first step, we characterize in Lemma 1 the coverageof each tierk, defined as the fraction of UEs served by BSsin tier k.

Lemma 1 (Single Flow Coverage). Letπ = (πk)k∈K,3 whereπk is the fraction of single flow UEs served by BSs in tierk.

3We useπ to denote both the single flow coverage and the constant Pi.The meaning should be clear from the context.

Then

πk = 2πλk

∫ ∞

0

rhk(r)dr, (13)

wherehk(r) = exp(−π∑

ℓ∈K λℓ(ZℓPℓ

ZkPk)

2αℓ⋆ r

2αk⋆

αℓ⋆ ) andk⋆ =argmini∈M:xi,k 6=0 αi .

Proof: See Appendix A.The coverage{πk} can be equivalently understood as the

tier connection probabilities of the typical UE. That is,πk

denotes the probability that the typical UE is “covered” by tierk. To gain some intuition about single flow coverage, let us sortthe path-loss exponents in ascending order:α1 ≤ ... ≤ αM .Suppose all the tiers use band1. Thenk⋆ = 1, ∀k ∈ M, and(13) reduces to the following:

πk = 1−∑

ℓ 6=k λℓ(ZℓPℓ)2

α1

λk(ZkPk)2

α1 +∑

ℓ 6=k λℓ(ZℓPℓ)2

α1

.

The above equality clearly shows that increasing biasingZk

(resp. BS densityλk) in tier k increases its coverageπk, andπk → 1 asZk → ∞ (resp.λk → ∞). Further, sinceα1 > 2,the coverageπk is more sensitive to the variation in the BSdensityλk than to the variation in the biased transmit powerZkPk, which is also true for the throughput in wireless packetnetworks [38].

With Lemma 1, we next derive the load of each tier, whichuniformly and independently blocks UEs in its coverage if itis fully-loaded.

Lemma 2 (Single Flow Load). With single flow CA, the loadθi,k of tier k in bandi is given by

θi,k =

2πbi,kλ(u)Gk

Biif xi,k 6= 0 and0 < 2πλ(u)Gk ≤ Bi

bi,k;

1 if xi,k 6= 0 and2πλ(u)Gk > Bi

bi,k;

0 if xi,k = 0.

whereGk =∫∞

0 rhk(r)dr.

Proof: See Appendix B.Finally, let us consider the admission probability. InK-tier

HetNets, the mean number of UEs served by the tagged BSdepends on which tier it belongs to and is hard to computeexactly. In this paper, we adopt the following approximationproposed in [27] for the mean number of UEs served by thetagged BS in tierk:

Nk∼= 1 +

1.28πkλ(u)

λk. (14)

With this approximation the admission probabilitypi,k of tierk in bandi is given by

pi,k =

1 if xi,k 6= 0 and0 < Nk ≤ Bi

bi,k;

Bi

bi,kNkif xi,k 6= 0 andNk > Bi

bi,k;

0 if xi,k = 0.

B. UE Ergodic Rate

In this subsection, we extend the proof technique used inSection III to derive single flow UE ergodic rate in generalK-tier networks. To begin with, letJ be the random tier that the

7

typical UE connects to. Conditioning onJ = k, ‖Yk0‖ = r,and{Φi,ℓ}, we obtain the ergodic rate of the CA-enabled UEas

E¯H

[R|{Φi,ℓ}, ‖Yk0‖ = r, J = k]

= E¯H

[

M∑

i=1

pi,kbi,k log(1 + SINRi,k)∣

∣

∣{Φi,ℓ}, ‖Yk0‖ = r, J = k

]

= E¯H

[

M∑

i=1

pi,kbi,k log

(

1 +PkHi,k0r

−αi

∑

ℓ∈K IΦi,ℓ+ wi,k

)]

=

M∑

i=1

pi,kbi,k

∫ ∞

0

1

1 + tP¯

(

PkHi,k0r−αi

∑

ℓ∈K IΦi,ℓ+ wi,k

≥ t

)

dt,

whereIΦi,ℓ=∑

Y ∈Φi,ℓ\Yk0PℓHi,Y ‖ Y ‖−αi and

P¯

(

PkHi,k0r−αi

∑

ℓ∈K IΦi,ℓ+ wi,k

≥ t

)

= e−twi,krαiP−1

k

∏

ℓ

∏

Y ∈Φi,ℓ\Y0

1

1 + trαiP−1k Pℓ ‖ Y ‖−αi

.

Now de-conditioning with respect to{Φi,ℓ} yields

E¯[R|‖Yk0‖ = r, J = k] =

M∑

i=1

pi,kbi,k

∫ ∞

0

1

1 + t

· e−twi,krαiP−1

k e−

∑

ℓθi,ℓλℓ

∫

B(0,ξk,ℓ)1− 1

1+trαiP−1k

Pℓ‖y‖−αi

dydt,

where ξk,ℓ is determined byZkPk · r−αk⋆ = ZℓPℓ · ξ−αℓ⋆

k,ℓ ,

from which we haveξk,ℓ = ( ZℓPℓ

ZkPk)

1αℓ⋆ r

αk⋆

αℓ⋆ . It follows that

∑

ℓ

θi,ℓλℓ

∫

B(0,ξk,ℓ)

1− 1

1 + trαiP−1k Pℓ‖y‖−αi

dy

=∑

ℓ

πθi,ℓλℓξ2k,ℓρ

(

t, αi,Pk

Pl

(

ξk,ℓr

)αi)

.

The next step is to de-condition with respect to‖Yk0‖ = r.To this end, we need to derive the distribution of the distance‖Yk0‖ conditioning onJ = k, which is given in Lemma 3.

Lemma 3.Conditioning onJ = k, the pdf of the distance‖Yk0‖ is given by

f‖Yk0‖|J(r) =

1

πk2πλkr · e−π

∑

ℓ∈K λℓ(ZℓPℓZkPk

)2

αℓ⋆ r

2αk⋆

αℓ⋆

, r ≥ 0.

(15)

Proof: See Appendix C.To gain some intuition about the conditional distribution of

‖Yk0‖, supposeα1 ≤ ... ≤ αM and that all the tiers use band1. Then (15) reduces to the following Rayleigh distribution:

f‖Yk0‖|J(r) =

2πr∑

ℓ∈K

λℓ(ZℓPℓ

ZkPk)

2α1 exp(−πr2

∑

ℓ∈K

λℓ(ZℓPℓ

ZkPk)

2α1 ), r ≥ 0.

In particular, its first moment is given by E¯[‖Yk0‖

∣

∣J ] =√πk · 1

2

√

1λk

. Note that 12

√

1λk

is the mean distance to theclosest BS in tierk from the typical UEwithout conditioning.

Interestingly, the mean length of the typical radio link in tier

k conditional onJ = k equals the unconditional mean12

√

1λk

multiplied by a factor√πk ≤ 1. Thus, conditioning reduces

the mean distance to the closest BS, agreeing with intuition.Using Lemma 3, we now can de-condition on‖Yk0‖ = r

and obtain that E¯[R|J = k] equals

M∑

i=1

pi,kbi,k2πλk

πk

∫ ∞

0

∫ ∞

0

1

1 + te−twi,kr

αiP−1k

· e−π∑

ℓλℓξ

2k,ℓ

(

θi,ℓρ(

t,αi,PkPl

(

ξk,ℓr

)αi)

+1)

r dr dt.

As a final step, we de-condition onJ = k and obtain thefollowing Prop. 2.

Proposition 2 (Single Flow UE Ergodic Rate). The singleflow UE ergodic rate is given by

R =∑

i∈M

∑

k∈K

2πλkbi,kpi,k

∫ ∞

0

∫ ∞

0

g(s)i,k (t, r)hk(r)r

1 + tdr dt,

(16)

whereg(s)i,k (t, r) equals

e−twi,krαiP−1

k e−π

∑

ℓ λℓθi,ℓ(ZℓPℓZkPk

)2

αℓ⋆ ρ(

t,αi,PkPl

(

ξk,ℓr

)αi)

r2αk⋆αℓ⋆

.(17)

with ξk,ℓ = ( ZℓPℓ

ZkPk)

1αℓ⋆ r

αk⋆

αℓ⋆ .

V. D ISCUSSIONS ONNETWORK DEPLOYMENT

In this section, we use the derived analytical results to studythe HetNet performance through examining two typical banddeployment scenarios: orthogonal and cochannel deployment.

A. Orthogonal Deployment

In this part we assume orthogonal deployment: differenttiers use different bands. Without loss of generality, supposeM = K and tier k is matched with bandk. Then Prop. 2reduces to the following Corollary 1.

Corollary 1. SupposeM = K and orthogonal deployment,i.e.,xi,k = 1 if i = k and0 otherwise. Then the single flow UEergodic rate is given byR =

∑

k∈K Rk, whereRk is the rateserved by tierk and equals

Rk = πkbk,kpk,k

∫ ∞

0

∫ ∞

0

1

1 + te−twk,kr

αkP−1k

· e−πλkθk,kρ(t,αk,1)r2

f‖Yk0‖|J(r) dr dt, (18)

wheref‖Yk0‖|J(r) is given in Lemma 3 withℓ⋆ replaced byℓ.

Further, if noise is ignored,

R =∑

k∈K

πkbk,kpk,k

∫ ∞

0

1

1 + t

1

1 + πkθk,kρ (t, αk, 1)dt.

With Corollary 1, we first discuss the impact of biasingby examining its impact on each term in (18). For ease ofexposition, we further assumeαk = α, ∀k.

8

1) Impact on coverage:As pointed out in Section IV-A,

πk = 1−∑

ℓ 6=k λℓ(ZℓPℓ)2α

λk(ZkPk)2α +

∑

ℓ 6=kλℓ(ZℓPℓ)

2α

. Thus, increasing biasing

factorZk increases the coverageπk of tier k. On the contrary,πℓ, ℓ 6= k, decreases withZk and thus the coverage areas ofthe other tiers shrink.

2) Impact on admission probabilities:Recall thatpk,k = 1if tier k has limited coverage such thatNk ≤ Bk

bk,k; otherwise,

pk,k = Bk

bk,kNk. So if tier k is under-loaded,pk,k remains

1 with increasingZk until Nk = Bk

bk,k, after which pk,k

decreases asZk further increases. On the contrary,pℓ,ℓ, ℓ 6= k,is monotonically non-decreasing with increasingZk and thusUEs connecting to tierℓ can be scheduled more often, at leastunchanged.

3) Impact on interference powers:The interfering transmit-

ter density equalsλkθk,k. From Lemma 2,θk,k =bk,kλ

(u)πk

Bkλkif

tier k has limited coverage such thatπk ≤ Bkλk

bk,kλ(u) ; otherwise,θk,k = 1. So if tierk is under-loaded,θk,k (resp. interference intier k) increases with increasingZk until πk = Bkλk

bk,kλ(u) , afterwhich θk,k ≡ 1 (resp. interference in tierk remains constant)as Zk further increases. Note the impact of interference onRk is shown by the terme−πλkθk,kρ(t,αk,1)r

2

in (18). On thecontrary, θℓ,ℓ, ℓ 6= k, is monotonically non-increasing withincreasingZk and thus interference in tierℓ gets decreased,at least unchanged.

4) Impact on the lengths of radio links:From f‖Yk0‖|J

given in Lemma 3, we know that

P¯(‖Yk0‖ ≥ r|J = k) = exp(−πr2

∑

ℓ∈K

λℓ(ZℓPℓ

ZkPk)

2α1 ), (19)

which is increasing asZk increases. So increasing biasingfactorZk makes tierk serve more UEs of longer radio links.On the contrary, P

¯(‖Yℓ0‖ ≥ r|J = k) is decreasing asZk

increases; and thus tierℓ, ℓ 6= k, serves more UEs of shorterradio links.

To sum up, the overall impact of biasing on UE ergodicrate depends on each tier’s coverage, admission probability,interference, and lengths of radio links. For a given arbitrarynetwork, it is nota priori clear whether biasing improves orhurts its performance. Nevertheless, it is generally believedthat increasing the biasing of small cells makes them accom-plish more as loads become more balanced over the tiers; thus,biasing UEs towards small cells can benefit the network as awhole.

B. Cochannel Deployment

In this part we focus on single band case and drop thesubscripti for ease of notation. Then Prop. 2 reduces to thefollowing Corollary 2.

Corollary 2. Suppose that a single band with path-loss expo-nentα is used by all theK tiers. Then the UE ergodic rate isgiven byR =

∑

k∈K Rk where

Rk = πkbkpk

∫ ∞

0

∫ ∞

0

1

1 + te−twkr

αP−1k

· e−r2·π∑

ℓ∈K θℓλℓ(ZℓPℓZkPk

)2α ρ(t,α,

ZℓZk

)f‖Yk0

‖|J(r) dr dt. (20)

wheref‖Yk0‖|J(r) is given in Lemma 3 withαℓ⋆ replaced byα.

Further, if noise is ignored,

R =∑

k∈K

πkbkpk

∫ ∞

0

1

1 + t

1

1 +∑

ℓ πℓθℓρ(t, α,Zℓ

Zk)dt.

With Corollary 2, we now discuss the impact of biasing byexamining its impact on each term in (20). It is not difficult tosee the impact of biasing on coverage, admission probabilitiesand the lengths of radio links is the same as the orthogonaldeployment case; the difference lies in the interference, onwhich we shall focus below.

In cochannel case, the interference power experienced bythe UEs served by tierk is proportional to the exponent of thethird term of the integrand in (20), which can be re-written as

r2π

θkλkρ(t, α, 1) +∑

ℓ 6=k

θℓλℓ(Pℓ

Pk)

2α

∫ ∞

ZℓZk

t

t+ xα2

dx

.

Here the first term in the above parentheses indicates the intra-tier interference level and is non-decreasing with increasingZk. If tier k is already fully-loaded, i.e.,θk = 1, increasingbiasing does not change the intra-tier interference; otherwise,increasing biasing makes BSs of tierk more active and thusincreases intra-tier interference.

As for the inter-tier interference e.g. from tierℓ, the impactof increasingZk is more subtle: asZk increases,θℓ decreasesbut

∫∞

Zℓ/Zk

t

t+xα2

dx increases. To explain this subtle phe-nomenon, we partition the UEs connecting to tierk into twogroups: Group I consists of the original UEs connecting to tierk before increasingZk and Group II consists of the new UEsconnecting to tierk which are biased from other tiers due tothe increasedZk. IncreasingZk makes other tiers less activeand thus UEs of Group I will experience less interference; incontrast, increasingZk brings UEs of Group II closer to theinter-tier interferers, though interferers are less active. To sumup, increasingZk can either increase or decrease the inter-tierinterference from tierℓ; the answer depends on the trade-offbetween the two factors mentioned above. As a special case,if tier ℓ remains fully loaded whenZk increases moderately,then the inter-tier interference from tierℓ increases.

Next we compare Corollary 2 (i.e.,1-band-K-tier de-ployment) to Prop. 1 (i.e.,K-band-1-tier deployment). Thiscomparison is interesting as it will demonstrate the advan-tages/disadvantages of two popular approaches for increasingrate in cellular networks: spatial reuse with small cells versusadding more bandwidth. To this end, the following result thatfollows from Prop. 1 and Corollary 2 is instrumental.

Proposition 3.Suppose the following assumptions are satis-fied:

1) The network is interference-limited, i.e., noise is ignored;2) The UE densityλ(u) is large enough;3) No biasing factors are applied, i.e.,Zk = 1, ∀k; 4

4) All the bands have the same path-loss exponentα andbandwidthB;

4This is a natural assumption in a fully-loaded network for maximizingsum rate.

9

Then the UE ergodic rates of1-band-K-tier andK-band-1-tierdeployments are respectively given by

R1−K =

K∑

k=1

λk · 0.78Bλ(u)

∫ ∞

0

1

1 + t

1

1 + ρ(t, α, 1)dt

RK−1 = Kλ1 ·0.78B

λ(u)

∫ ∞

0

1

1 + t

1

1 + ρ(t, α, 1)dt. (21)

Interestingly,K-tier-1-band deployment can have higherrate than that of1-tier-K-band deployment as long as∑

Kk=1 λk

Kλ1≥ 1 which can be satisfied by makingλk ≥ λ1, ∀k.

Further, the gain∑

Kk=1 λk

Kλ1increases proportionally with the

BS densitiesλk, k 6= 1; in contrast, the gain decreases ifλ1 increases, agreeing with intuition: small cells become lessuseful if the macro BSs are deployed more densely. To sum up,Prop. 3 gives a theoretical support to the current vast interestin adding small cells to cellular networks. In particular, itdemonstrates the potential of small cells in solving the current“spectrum crunch”.

With similar assumptions as in Prop. 3, we can also derivethe UE ergodic rate of orthogonalK-band-K-tier deploymentas

R(⊥)K−K =

K∑

k=1

λk · 0.78Bλ(u)

∫ ∞

0

1

1 + t

1

1 + πkρ(t, α, 1)dt,

where the superscript⊥ denotes orthogonal deployment. Simi-larly, the UE ergodic rateR(co)

K−K of cochannelK-band-K-tierdeployment is given by

R(co)K−K =

K∑

k=1

λk ·0.78B

λ(u)

∫ ∞

0

1

1 + t

K

1 + ρ(t, α, 1)dt.

Typically, R(co)K−K ≥ R

(⊥)K−K . However, the performance gap

can be reduced by exploiting the additional design dimensionof orthogonalK-band-K-tier deployment: One can appropri-ately tune{πk} (e.g. by optimizing biasing factors) to getlarger rate.

VI. SIMULATION AND NUMERICAL RESULTS

In this section, we provide numerical results to demonstratethe analytical results. The main set-up is a typical HetNetconsisting of 2 tiers (macro and small BSs) and 2 bands(800MHz and2.5GHz band). The specific parameters used aresummarized in Table I unless otherwise specified. We furtherintroduce a shorthand notation[x1,1, x2,1;x1,2, x2,2] to denotethe band deployment configuration for ease of description. Forexample, configuration[1, 1; 1, 0] denotes that tier 1 uses bothband 1 and band 2 while tier 2 only uses band 1.

A. Monte Carlo Simulations

We first validate the derived rate expressions via MonteCarlo simulations. The set-up is a square areaA of size|A| = 20× 20 km2, whereλ1 = 0.5× (π5002)−1, λ2 = 2λ1,and λ(u) ranges from5λ1 to 49λ1. With this set-up, themean numbers of macro BSs and small BSs are255 and510,respectively, and the mean number of UEs ranges from1275

TABLE ISYSTEM PARAMETERS

Density of macro BSsλ1 (π5002)−1 m−2

Density of small BSsλ2 2× (π5002)−1 m−2

Density of UEsλ(u) λ(u) = 12λ2

Max Tx power of macro BSs 40 WMax Tx power of small BSs 1 WNoise PSD −174 dBmNoise figure 6 dB

800MHz band’s wavelengthµ1 0.375 m800MHz band’s path loss exp.α1 3800MHz band’s bandwidthB1 9 MHz800MHz band’s biasingZ1 0 dB2.5GHz band’s wavelengthµ2 0.12 m2.5GHz band’s path loss exp.α2 42.5GHz band’s bandwidthB2 9 MHz2.5GHz band’s biasingZ2 0 dBUE min BW req.bi,k ≡ b 1.8 MHz

Band deployment [1, 0; 0, 1]

to 12495. Such a large network is simulated to make boundaryeffect negligible. Then for eachλ(u), the simulation steps areas follows.

1) Generate a random numberNk for each tierk such thatNk ∼ Poisson(λk|A|).

2) GenerateNk points that are uniformly distributed inA;theseNk points represent the BSs in tierk.

3) With a similar approach as in the previous two steps,generateN (u) points representing the UEs.

4) Generate independently the fading gain from each BSto each UE.

5) Associate each UE to some BS using the single flowassociation policy introduced in Section II-C.

6) For each BS, UE scheduling is done as follows: Ifthere are more thanBi/b = 5 associated UEs, the BSrandomly picks up5 out of them to serve; otherwise,there are more subchannels than the number of UEs(sayn) associated and correspondingly the BS randomlypicks upn out of the5 subchannels to serve thesen UEs.

7) Compute the rate of each UE: The rate equalsb log(1+SINR) if the UE is scheduled; otherwise, the rate is0.

8) Compute the average rateR(t) (wheret is the iterationindex) by averaging over all the UE rates.

9) Repeat the above steps for100 times and then computethe ergodic rate asR = 1

100

∑100t=1 R(t).

The comparison results are shown in Fig. 2, where wesimulate each band separately. This is justified by the analysisin Section III which shows that spatial correlation naturally de-couples over the bands when single flow CA is supported. Fig.2 shows that the analytical results match the empirical resultsfairly well in the single tier network. However, small gaps existbetween analysis and simulation in the two-tier network. Thesegaps arise mainly because of the approximation (14) used inthe rate expression of the multi-tier networks. Nevertheless,the analytical results provide a pretty good estimate for the

10

5 10 15 20 25 30 35 40 450

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Normalized UE Density: λ(u)/λ1

UE

Erg

odic

Rat

e (M

bps)

SimulationAnalysis

α = 3, one tier

α = 4, one tier

α = 3, two tiers

α = 4, two tiers

Fig. 2. Analytical results versus Monte Carlo simulation results

simulation results, and is especially attractive for studyinglarge networks where simulation can be quite time-consuming.

B. Biasing Effect

In this subsection, we explore the biasing effect on networkperformance. First, we show the network sum rate, whichequals UE ergodic rate multiplied by UE density, as a functionof normalized density of small cells in Fig. 3. It is shown thatnetwork capacity increases almost linearly as the density ofsmall cells increases, which shows the promise of small cells.Further, appropriate biasing values (e.g. 10dB in Fig. 3) canhelp small cells accomplish more by increasing the sum ate.Interestingly, biasing does not help when the densities of smallcells (proportional to UE densities) are either too small ortoolarge: In the former case the load of macro BSs is not heavyand thus offloading is not needed; in the latter case the loadof small BSs is already heavy enough and thus offloading canhardly increase the sum rate.

We next provide a finer view about the effect of biasingin Fig. 4 and 5. For brevity, we fix the UE density asλ(u)/λ1 = 24 and assume all the channels are in the 800MHzfrequency band. As shown in Fig. 4, the impact of biasing oncoverage and load is consistent with our previous high leveldiscussions in Section V-A. The impact of biasing on spectralefficiency, a function of interference powers and lengths ofradio links, is shown in Fig. 5. As implied by the discussionin Section V-A, with increasing biasing of small cells, thespectral efficiency of small cells decreases while the converseis true for the spectral efficiency of macro cells. However,the situation becomes more complicated when it comes tothe cochannel deployment case. For example, with increasingbiasing of small cells, the spectral efficiency of small cells firstdecreases and then increases. This subtle effect of biasingcanbe understood along the reasoning presented in Section V-B.

5 10 15 20 250

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Normalized Density of Small Cells: λ2/λ

1

Sum

Rat

e (M

bps

* m

−2 *

104 )

Z

2 = 0dB

Z2 = 10dB

Z2 = 20dB

Z2 = 30dB

Fig. 3. Impact of the density of small cells and biasing on network sum rateunder orthogonal deployment.

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Biasing of Small Cells Z2 (dB)

Cov

erag

e

Macro CellsSmall Cells

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Biasing of Small Cells Z2 (dB)

Load

Macro CellsSmall Cells

λ2/λ

1=8 λ

2/λ

1=2 λ

2/λ

1=32

λ2/λ

1=32

λ2/λ

1=8

λ2/λ

1=2

Fig. 4. Impact of biasing on coverage and load under orthogonal deployment.

C. Band Deployment

In this subsection, we study how different band deploymentconfigurations affect UE ergodic rate. Since in reality the800MHz band has to be deployed in macro cells to ensurecoverage, we only focus on those band deployments withx1,1 = 1.

Fig. 6 and 7 show the UE ergodic rates under band deploy-ments[1, 0; 0, 1] and [1, 1; 0, 1], respectively. Under these twodeployments, small cells do not use the 800MHz band and

11

0 5 10 15 20 25 30 350

1

2

3

4

5

6

7

Biasing of Small Cells Z2 (dB)

Spe

ctra

l Effi

cien

cy

Macro CellsSmall Cells

0 5 10 15 20 25 30 350

1

2

3

4

5

6

7

Biasing of Small Cells Z2 (dB)

Spe

ctra

l Effi

cien

cy

Macro CellsSmall Cells

λ2/λ

1=8 λ

2/λ

1=2

λ2/λ

1=32

λ2/λ

1=32

λ2/λ

1=2

λ2/λ

1=8

Fig. 5. Impact of biasing on spectral efficiency: The top subfigure isfor orthogonal deployment while the bottom subfigure is for cochanneldeployment.

thus have much smaller coverage than macro cells. Hence,biasing can increase the UE ergodic rate by expanding thecoverage areas of small cells; this is true for moderate densitiesof small cells, as shown in Fig. 6 and 7. For example, in thecase ofλ2/λ1 = 8 shown in Fig. 6, an almost2× rate gainis achieved with17dB biasing. Further, the optimal biasingvalue is sensitive to the densities of small cells; in general,it decreases asλ2/λ1 increases. Note that Fig. 6 and 7 agreewith Fig. 3 that biasing does not help when the densities ofsmall cells and UEs are either too small or too large.

In contrast, Fig. 8 shows the UE ergodic rates under banddeployment[1, 1; 1, 0], where small cells use the 800MHz bandinstead of the 2.5GHz band. In this case, small cells havemuch better coverage than their counterparts under[1, 0; 0, 1]and [1, 1; 0, 1]. Thus, biasing is not that helpful; indeed, zerobiasing value is optimal as shown in Fig. 8. However, theUE ergodic rates under[1, 1; 1, 0] are lower than those under[1, 0; 0, 1] or [1, 1; 0, 1]. That is, though better coverage canbe obtained under[1, 1; 1, 0], the small cells still do notaccomplish much as they experience severe intra and crosstier interference in the 800MHz band.

Fig. 9 and 10 show the UE ergodic rates under banddeployments[1, 0; 1, 1] and[1, 1; 1, 1], respectively. For similarreason as in the deployment[1, 1; 1, 0], biasing is not helpfulhere; indeed, zero biasing value is optimal as shown in Fig. 9and 10. However, here we mainly take a sum-rate perspective;biasing may still be useful under[1, 0; 0, 1] and[1, 1; 0, 1] fromother perspectives e.g. rate of cell-edge users. In addition,

0 5 10 15 20 25 30 350.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

Biasing of Small Cells Z2 (dB)

UE

Erg

odic

Rat

e (M

bps)

λ

2/λ

1 = 2

λ2/λ

1 = 4

λ2/λ

1 = 8

λ2/λ

1 = 16

λ2/λ

1 = 32

λ2/λ

1 = 64

Fig. 6. UE ergodic rate vs. biasing under band deployment[1, 0; 0, 1]

0 5 10 15 20 25 30 351

1.5

2

2.5

3

3.5

4

4.5

Biasing of Small Cells Z2 (dB)

UE

Erg

odic

Rat

e (M

bps)

λ

2/λ

1 = 2

λ2/λ

1 = 4

λ2/λ

1 = 8

λ2/λ

1 = 16

λ2/λ

1 = 32

λ2/λ

1 = 64

Fig. 7. UE ergodic rate vs. biasing under band deployment[1, 1; 0, 1]

0 5 10 15 20 25 30 350.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

Biasing of Small Cells Z2 (dB)

UE

Erg

odic

Rat

e (M

bps)

λ

2/λ

1 = 2

λ2/λ

1 = 4

λ2/λ

1 = 8

λ2/λ

1 = 16

λ2/λ

1 = 32

λ2/λ

1 = 64

Fig. 8. UE ergodic rate vs. biasing under band deployment[1, 1; 1, 0]

12

0 5 10 15 20 25 30 350.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

Biasing of Small Cells Z2 (dB)

UE

Erg

odic

Rat

e (M

bps)

λ2/λ

1 = 2

λ2/λ

1 = 4

λ2/λ

1 = 8

λ2/λ

1 = 16

λ2/λ

1 = 32

λ2/λ

1 = 64

Fig. 9. UE ergodic rate vs. biasing under band deployment[1, 0; 1, 1]

0 5 10 15 20 25 30 351

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

Biasing of Small Cells Z2 (dB)

UE

Erg

odic

Rat

e (M

bps)

λ2/λ

1 = 2

λ2/λ

1 = 4

λ2/λ

1 = 8

λ2/λ

1 = 16

λ2/λ

1 = 32

λ2/λ

1 = 64

Fig. 10. UE ergodic rate vs. biasing under band deployment[1, 1; 1, 1]

compared to the other 4 deployments, deployment[1, 1; 1, 1]yields the largest UE ergodic rates in nearly all the scenariosof small cell densities, while[1, 0; 1, 1] is almost as goodas [1, 1; 1, 1] except the low small cell density regime, i.e.,λ2/λ1 = 2.

Before ending this subsection, we remark that the abovecomparison of different band deployment scenarios is con-ducted purely from a sum-rate perspective, which is just oneaspect of the network design. In reality, deployment choicesare made by jointly considering many other factors as well.In addition, interference management, which may affect theeffect of biasing, is not incorporated.

VII. C ONCLUSIONS

This paper focuses on two central issues in CA-enabledHetNets: biasing and band deployment. To this end, we haveproposed a general yet tractableM -bandK-tier load-awaremodel. This new model provides a better characterizationof the biasing effect and yields new design insights. Fur-ther, different band deployment configurations have also been

studied and compared under the proposed model; the resultsreveal that universal cochannel deployment is the right goal topursue. This work can be extended in a number of ways. Inparticular, many other distinct features such as multiple-inputmultiple-output (MIMO) technique, coordinated multi-pointtransmission/reception (CoMP) and device-to-device (D2D)communications can be jointly studied with CA.

ACKNOWLEDGMENT

The authors thank Rapeepat Ratasuk and Bishwarup Mondalof Nokia Siemens Networks for their help in identifying theproblem and refining system model and numerical results.The authors also thank the anonymous reviewers for theirvaluable comments and suggestions, which helped the authorssignificantly improve the quality of the paper.

APPENDIX

A. Proof of Lemma 1

This proof follows the approach used in [24]. In single flowCA, the typical UE scans over all the tiers and bands andthen connects to the tierk that provides the strongest biasedreceived power in some band sayk⋆ ∈ M. Then the UEperforms CA with respect to this selected tier. LetJ be therandom tier that the UE connects to. Then by Assumption 2,we have

πk = P(J = k) = P(ZkPk‖Yk0‖−αk⋆

= max(ZℓPℓ‖Yℓ0‖−αi : ℓ ∈ K, i ∈ M))

= P(ZkPk‖Yk0‖−αk⋆ = max(ZℓPℓ‖Yℓ0‖−αℓ⋆ : ℓ ∈ K))

= P(ZkPk‖Yk0‖−αk⋆ ≥ ZℓPℓ‖Yℓ0‖−αℓ⋆ , ∀ℓ 6= k)

=∏

ℓ 6=k

P¯(π(ZℓPℓ)

− 2αℓ⋆ ‖Yℓ0‖2 ≥ π(ZkPk)

− 2αℓ⋆ ‖Yk0‖

2αk⋆

αℓ⋆ ),

(22)

where the last equality follows from the independence of{‖Yℓ0‖} (because theK ground PPPs are independent). It isknown that‖Yℓ0‖ is Rayleigh distributed (c.f. (10)) with pdff‖Yℓ0

‖(r) = 2πλℓre−πλℓr

2

, r ≥ 0. It follows that

P¯(π‖Yℓ0‖2(ZℓPℓ)

− 2αℓ⋆ ≥ x) = exp(−(ZℓPℓ)

2αℓ⋆ λℓx), ∀ℓ ∈ K.

Using the above equality and conditioning on‖Yk0‖ = r yields

P¯(π‖Yℓ0‖2(ZℓPℓ)

− 2αℓ⋆ ≥ π(ZkPk)

− 2αℓ⋆ r

2αk⋆

αℓ⋆ )

= exp(−πλℓ(ZℓPℓ

ZkPk)

2αℓ⋆ r

2αk⋆

αℓ⋆ ). (23)

De-conditioning on‖Yk0‖ = r yields

P (π(ZℓPℓ)− 2

αℓ⋆ ‖Yℓ0‖2 ≥ π(ZkPk)− 2

αℓ⋆ ‖Yk0‖2αk⋆

αℓ⋆ )

=

∫ ∞

0

exp(−πλℓ(ZℓPℓ

ZkPk)

2αℓ⋆ r

2αk⋆

αℓ⋆ ) · 2πλkr exp(−πλkr2) dr.

(24)

13

Plugging (24) into (22) yields thatπk equals∏

l 6=k

∫ ∞

0

exp(−πλℓ(ZℓPℓ

ZkPk)

2αℓ⋆ r

2αk⋆

αℓ⋆ ) · 2πλkr exp(−πλkr2) dr

= 2πλk

∫ ∞

0

r exp(−π∑

ℓ∈K

λℓ(ZℓPℓ

ZkPk)

2αℓ⋆ r

2αk⋆

αℓ⋆ ) dr.

This completes the proof.

B. Proof of Lemma 2

Let Π(i)k (x) denote the probability that the UE at position

x connects to tierk in bandi. Then the mean number of UEsattempting to connect to a BS of tierk is given by

Li,k =

∫

R¯

2 Π(i)k (x)Φ(u)(dx)∫

R¯

2 Φk(dx)

=

∫ 2π

0

∫∞

0Π

(i)k (r, θ)λ(u)rdrdθ

∫ 2π

0

∫∞

0 λkrdrdθ

=λ(u)πk

∫ 2π

0

∫∞

0rdrdθ

λk

∫ 2π

0

∫∞

0 rdrdθ= 2πλ(u)Gk,

where the third equality follows from the ergodicity of PPPΦ(u), i.e.,Π(i)

k (x) = πk, ∀x, and pluggingπ(i)k yields the last

equality. So ifxi,k 6= 0 and0 < 2πλ(u)Gk ≤ Bi

bi,k, tier k

is under-loaded in bandi and the load equalsbi,kLi,k/Bi. Ifxi,k 6= 0 and2πλ(u)Gk > Bi

bi,k, tier k is fully-loaded in band

i and the load equals1. If xi,k = 0, it is trivial that pi,k = 0.

C. Proof of Lemma 3

Note thatP (‖Yk0‖ ≥ r | J = k) =P (‖Yk0

‖≥r,J=k)

πkwhere

P(‖Yk0‖ ≥ r, J = k)

= P(‖Yk0‖ ≥ r

andZkPk · ‖Yk0‖−αk⋆ ≥ ZℓPℓ · ‖Yℓ0‖−αℓ⋆ , ∀ℓ)

=

∫ ∞

r

f‖Yk0‖(x)·

P¯(π‖Yℓ0‖2(ZℓPℓ)

− 2αℓ⋆ ≥ π(ZkPk)

− 2αℓ⋆ x

2αk⋆

αℓ⋆ , ∀ℓ) dx

=

∫ ∞

r

f‖Yk0‖(x)·

∏

ℓ 6=k

P¯(π‖Yℓ0‖2(ZℓPℓ)

− 2αℓ⋆ ≥ π(ZkPk)

− 2αℓ⋆ x

2αk⋆

αℓ⋆ ) dx

=

∫ ∞

r

f‖Yk0‖(x)

∏

ℓ 6=k

exp(−πλℓ(ZℓPℓ

ZkPk)

2αℓ⋆ r

2αk⋆

αℓ⋆ ) dx,

where the last equality follows from (23). Pluggingf‖Yk0

‖(x) = 2πλkr exp(−πλkr2), x ≥ 0, we get the ccdf

of ‖Yk0‖ conditional onJ = k as

P(‖Yk0‖ ≥ r | J = k)

=1

πk2πλk

∫ ∞

r

x · e−π∑

ℓ∈K λℓ(ZℓPℓZkPk

)2

αℓ⋆ x

2αk⋆

αℓ⋆

dx.

Differentiating 1 − P(‖Yk0‖ ≥ r | J = k) with respect toryields the desired conditional pdf.

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