A Circular Interference Model for Wireless Cellular Networks

Martin Taranetz, Markus Rupp

Vienna University of Technology, Institute of TelecommunicationsGusshausstrasse 25/389, A-1040 Vienna, Austria

Email: {mtaranet,mrupp}@nt.tuwien.ac.at

Abstract—In this work, we investigate downlink co-channelinterference in wireless cellular networks. Our main target isto facilitate statistical analysis in networks with regular gridlayout. In particular, we focus on the interference statisticsoutside the center of a grid scenario. First, a novel circularinterference model is introduced. The key idea is to spread thepower of the interferers uniformly along the circumcircle of thegrid-shaping polygon. We then propose to model the aggregateinterference statistics by a single Gamma random variate. Thecorresponding shape- and scale parameters are determined inclosed form by employing this circular model. The analysis yieldskey insights on the distribution’s formative components. Weverify the accuracy of the Gamma approximation by qualitative-and quantitative measures. A basic guideline for applying andextending our approach indicates that it considerably improvesanalytic accessibility of regular grid models.

Index Terms—Interference Modeling, Hexagonal Cellular Net-work, Hexagon System Model, aggregate Co-Channel Interfer-ence (CCI), Interference Statistics, Interference Distribution

I. Introduction and Contributions

The proposal of a cellular structure for mobile networksdates back to 1947. Two Bell Labs engineers, Douglas H.Ring and W. Rae Young were the first to mention the idea inan internal memorandum. Almost two decades later, in 1966,Richard H. Frenkiel and Philip T. Porter, shaped a ”hexagonalcellular array of areas” to propose the first mobile phonesystem [1]. Although never published officially, the hexagonmodel gained high popularity within the research communityand is still extensively utilized nowadays [2–7]. It serves eitheras the system model itself, or as a reference system for moreinvolved simulation scenarios.

On the other hand, its geometric structure renders closed-form analysis of aggregate interference statistics difficult [8].Hence, simulation results often lack a mathematical back up.

Recently, closed-form results have been reported with sys-tem models based on stochastic geometry [9–11]. However,these results are obtained only for particular parameter choices,typically assuming spatial stationarity and isotropy of theinterference scenario. Thus, its potential to consider non-symmetric interferer impact, e.g., when the receiver is locatedoutside the scenario center, is limited. Moreover, the stochasticapproach is based on an ensemble of network realizationsand is therefore not applicable when a fixed structure of thenetwork is given.

Recently published work on hexagonal grid models hasmainly focused on link-distance statistics [12, 13]. The authors

also account for fading and provide closed-form approxima-tions for the co-channel interference of a single link. However,closed-form expressions for the moments and the distributionof aggregate co-channel interference are not available yet.

The main contributions of this work are:1) We introduce a circular interference model to facilitate

interference analysis in cellular networks with regulargrid layout. In particular, we focus on the hexagonalgrid due to its ubiquity in wireless communicationengineering [2–7]. The key idea is to consider the powerof the interfering transmitters as being uniformly spreadalong the circumcircle of the hexagon.

2) We propose to model interference statistics in a hexago-nal scenario by a single Gamma Random Variate (RV).Its shape- and scale parameters are determined in closedform by employing the circular model. The analysisyields key insights on the formative components of theinterference distribution.

3) We provide a basic guideline for applying and extendingour approach. Next to its Multiple Input Multiple Output(MIMO) capabilities, we also demonstrate the general-ization to multiple tiers and arbitrary regular polygonmodels.

The remainder of this work is structured as follows: Sec-tion II specifies the hexagonal reference-system model. InSection III the circular interference model and its dual pendantare introduced. Section IV investigates Gamma-distributedinterference and its parametrization by the proposed circularinterference model. In Section V, the accuracy of the Gammaapproximation is verified. This section also provides applica-tions and extensions of the model. Section VI concludes thework.

II. SystemModel

The reference hexagonal setup is composed of a centralcell and six interfering transmitters, as shown in Figure 1.The interferers are equipped with omnidirectional antennasand are located at the edges of a hexagon with radius R(marked as ’+’ in Figure 1). The signal from the i-th interferingtransmitter with polar coordinates (R,Φi) to a receiver withpolar coordinates (r, φ) experiences:• Path loss

`(d(M)ρ,∆i

)=

c · dα0 , d(M)ρ,∆i

< d0

c ·(d(M)ρ,∆i

)α, d(M)

ρ,∆i≥ d0

(1)978-1-4799-0959-9/14/$31.00 c© 2014 IEEE

+ +

+ +

+ +

1

ρ

∆

*

y/R

x/R

φ

Φ

Fig. 1: System model: Center cell with receiver and interferingtransmitters at normalized distances ρ and 1, respectively. Thereceiver is marked by ’∗’, the interferers of the hexagonalreference system are denoted by ’+’, respectively.

where c is a constant, α is the path loss exponent and

d(M)ρ,∆i

=

√R2 + r2 − 2Rr cos (φ − Φi) (2)

= R√(

1 + ρ2 − 2ρ cos(∆i)),

with Φi =2πM

i, i = 1, . . . ,M,

and ρ =rR, ∆i = φ − Φi. (3)

In the hexagonal scenario, M = 6. The terms ρ and ∆i

denote the receiver’s normalized distance to the centerand its angle-difference to the i-th interfering transmitter,respectively.

• Fading, which is modeled by independent and identicallydistributed (i.i.d.) Gamma RVs Gi ∼ Γ[k0, θ0], withshape- and scale parameters k0 and θ0, respectively. ItsProbability Density Function (PDF) is defined as

Γ[k0, θ0] =1

Γ(k0) θk00

xk0−1e−x/θ0 . (4)

Gamma-fading includes Rayleigh and Nakagami-m asspecial cases. It allows to model multiuser-MIMO sys-tems and can accurately approximate composite fadingdistributions (e.g., Rayleigh-Lognormal). Therefore, itcovers a wide range of scenarios.

III. Circular InterferenceModelIn a one-tier hexagonal grid scenario, as presented in Sec-

tion II, the received aggregate interference power at position(r, φ) can be expressed as

I6(ρ, φ) =

6∑i=1

P Gi

`(d(6)ρ,∆i

) , (5)

where P denotes the transmit power, Gi is the fading and`(d(6)

ρ,∆i) refers to the path loss at distance d(6)

ρ,∆i, with d(6)

ρ,∆ifrom

(2).

Outside the cell-center, i.e., ρ > 0, distribution and momentsof I6(ρ, φ) can in general not be evaluated in closed-form.In this section, we propose a circular interference model tofacilitate and enable statistical analysis.

A. Proposed Model

In the circular interference model, the power of the sixreference transmitters is spread uniformly along a circle ofradius R.

This is achieved by equally distributing the total transmitpower 6 P among M equally spaced transmitters and consid-ering the limiting case M → ∞. With (5), we obtain

limM→∞

6 PM

M∑i=1

Gi

`(d(M)ρ,∆i

) =6 PE [Gi]

2π

π∫−π

1

`(dρ,∆

)d∆, (6)

with `(·) from (1) and dρ,∆i from (2). The terms dρ,∆ and ∆

denote distance and angle-difference between the receiver andan infinitesimal interfering circular segment, as indicated inFigure 1.

Assuming a path loss exponent α = 2, i.e., free spacepropagation, and R > d0, (6) can be evaluated as

IC(ρ) = 6 PE [Gi]1

cR2

11 − ρ2 . (7)

An intuitive interpretation of this result by the model’spendant is provided in the next subsection.

Note: In the remainder of this work, we stick to a path lossexponent α = 2. It represents the worst case of low interferenceattenuation. However, previously- as well as all subsequentlypresented analysis can be carried out in closed-form for α = 2nwith n ∈ N. Values α other than these require the evaluationof elliptic integrals [14].

B. The Dual Model

Consider a receiver in a hexagonal scenario, which is movedalong a circle of radius ρ from −π to π, as depicted in Figure 1.The average expected interference along the circle can becalulated as

I′C(ρ) =1

2π

π∫−π

E[I6(ρ, φ′)

]dφ′ (8)

=

6∑i=1

PE [Gi]1

2π

π∫−π

1`(dρ,∆)

dφ′. (9)

The result is obtained by plugging (5) into (8), exchanging sumand integral and exploiting the linearity of the expectation.

The term I′C(ρ) in (6) is equivalent to IC(ρ) in (9) and, con-sequently, also yields (7). Thus, the result is independent of thereceiver’s angle-position. It can be interpreted as the averageexpected interference, i.e., the interference experienced by atypical receiver in a hexagonal scenario at distance ρ.

Note that the circular interference model is not restricted tohexagons. By replacing ’6’ by ’N’ in (5)–(9), it can generallybe applied for substituting any convex regular N-polygonalmodel, as validated in Section V-A.

IV. Statistics of Aggregate Interference

In this section, we investigate aggregate interference in ahexagonal scenario with i.i.d. Gamma fading. We propose toapproximate its statistics by a single Gamma RV. Its distance-dependent shape- and scale parameters are determined byapplying the previously presented circular model.

The rationale for this approximation are (i) the accurate fit,as verified in Section V-B and (ii) its applicability to evaluateaverage rate and Bit Error Ratio (BER) in closed form [11, 15].

A. Interference Statistics at the Center

Assume i.i.d. Gamma fading with Gi ∼ Γ [k0, θ0]. Referringto (5), interference can be considered as a sum of RVs, whichare weighted by the received power without fading; P/`(d(6)

ρ,∆i)).

At the center of a hexagonal scenario (ρ = 0), all weightingfactors are equal, i.e., P/`(d(6)

ρ,∆i) = P/cR2. By virtue of

the scaling- and summation property of a Gamma RV, theresulting interference is distributed as

I6(0, φ) ∼ Γ

[6 k0, θ0

PcR2

]. (10)

B. Interference Statistics outside the Center

Outside the center (ρ > 0), a non-uniform impact of theinterferers is observed: The distances d(6)

ρ,∆iand, thus, also the

weighting factors P/`(d(6)ρ,∆i

) generally differ from each other.The resulting interference distribution can be evaluated byrecursive methods [16] or hypergeometric functions [8], whichhamper closed-form analysis of further performance metricssuch as average rate and BER.

Therefore, we propose to approximate the typically experi-enced interference distribution at distance ρ by

I(ρ) ∼ Γ[k(ρ), θ(ρ)]. (11)

The rationale for this model are findings in prior work, whereout-of-cell interference in stochastic networks is accuratelyapproximated by a Gamma distribution [11].

The distribution is fully determined by the distance-dependent shape- and scale parameters k(ρ) and θ(ρ), re-spectively. In order to evaluate the two parameters, we firstemploy the proposed circular interference model to determineexpectation and variance of I(ρ). Then, we exploit the fact thatE[I(ρ)] = k(ρ) θ(ρ) and Var[I(ρ)] = k(ρ) θ2(ρ):

a) Expectation: As discussed in Section III-A, the un-equal received powers from the interfering transmitters can beaveraged. The average impact of each interferer is calculatedas

P1

2π

π∫−π

1

`(dρ,∆

)d∆ =P

cR2

11 − ρ2 , (12)

and yields the typically expected aggregate interference atdistance ρ as

E[I(ρ)

]= 6 k0θ0

PcR2

11 − ρ2 . (13)

Transmit power P 1

Circle radius R 1

Path loss exponent α 2

TABLE I: System parameters

b) Variance: The variance of the aggregate interferencecomprises of two components:

1) Variance of the fading:

Var f

[I(ρ)

]= 6 k0

(θ0

PcR2

11 − ρ2

)2

. (14)

2) Variance of the received power without fading, which iscaused by the unequal distances dρ,∆i . With

12π

π∫−π

P

`(dρ,∆

) − PcR2

2

d∆ =

(P2

cR2

)2 2ρ2 + ρ4 − ρ6(1 − ρ2)3 ,

(15)

the second variance component is obtained as

Vard

[I(ρ)

]= 6k0

(θ0

PcR2

)2 2ρ2 + ρ4 − ρ6(1 − ρ2)3 (16)

Since the two components are statistically independent, theoverall variance is calculated as

Var[I(ρ)

]= Var f

[I(ρ)

]+ Vard

[I(ρ)

]= 6 k0

(θ0

PcR2

11 − ρ2

)2 (1 +

2ρ2 + ρ4 − ρ6

1 − ρ2

)(17)

where Var f [I(ρ)] and Vard[I(ρ)] refer to (14) and (16), respec-tively.

Finally, the distance-dependent shape- and scale parameterare derived from (13) and (17) as

k(ρ) = 6 k01 − ρ2

1 + ρ2 + ρ4 − ρ6 (18)

θ(ρ) = θ0P

cR2

11 − ρ2

(1 +

2ρ2 + ρ4 − ρ6

1 − ρ2

)(19)

V. Numerical Results and Discussion

In this section, the accuracy of the circular model and theproposed Gamma approximation are verified. Ideal systemparameters are chosen in order to facilitate traceability.

A. Validation of Expected Aggregate Interference

First, we compare the expected interference powers inthe hexagonal reference scenario and the proposed circularinterference setup. The employed system parameters are sum-marized in Table I and fading is assumed to be distributed asGi ∼ Γ[1, 1].

0 π/4 π/2 3π/4 π5.5

6

6.5

7

7.5

8

8.5

Receiver angle φ

Expec

ted a

ggr

egat

e in

terf

eren

ce

ρ = 0

ρ = 0.25

ρ = 0.5

Circular Hexagonal

(III)

(II)

(I)

Fig. 2: Expected aggregate interference experienced at position(ρ, φ) in circular- (IC(ρ)) and hexagonal model (E[I6(ρ, φ)]),respectively. Receiver distances ρ = {0, 0.25, 0.5} refer to cell-center, middle of cell and cell edge, respectively.

Consider a receiver which is moved along a semi circle{ (ρ, φ)| φ ∈ [0, π]}, as indicated in Figure 1. The expectedinterference in the hexagonal scenario is calculated as

E[I6(ρ, φ)

]= 6

6∑i=1

1

`(d(6)ρ,∆i

) , (20)

with I6(ρ, φ) from (5) and E[Gi] = 1. For the circular model,we obtain

E[IC(ρ)

]= IC(ρ) =

61 − ρ2 , (21)

with IC(ρ) from (7). Figure 2 depicts the evaluated results of(20) and (21) for various distances ρ:• At cell-center, i.e., ρ = 0, the expected interference

powers in the hexagonal- and circular scenario (E[I6(0, φ)]and IC(0)) are equal.

• Outside the center, i.e., ρ > 0, E[I6(ρ, φ)] fluctuatesaround IC(ρ). The deviation is weak in the middle of thecell (ρ = 0.25), and strong at cell edge (ρ = 0.5). Notethat E[I6(ρ, φ)] is not symmetric about IC(ρ) due to theconcavity of the path loss model.

The relative error of the circular interference model iscalculated as

ε (ρ, φ) =

∣∣∣∣∣∣E[I6(ρ, φ)

]− IC(ρ)

E[I6(ρ, φ)

] ∣∣∣∣∣∣ , (22)

with E[I6(0, φ)] and IC(0) from (20) and (21), respectively. Thelargest error occurs at cell edge, i.e.,

maxρ,φ

ε (ρ, φ) = maxφε (0.5, φ). (23)

However, the expected interference powers of the circular- andthe hexagonal model deviate by no more than 3.07 %, as shownin Figure 3 (corresponding to marker (I) in Figure 2).

2 3 4 5 6 7 8 9 10 11 120

5

10

15

20

25

30

35

40

45

Number of interferers N

Max

imum

err

or o

f ci

rcula

r in

terf

eren

ce m

odel

[%

]

Tri

angl

e

Squar

e

Hex

agon

Fig. 3: Maximum error of circular interference model fromexpected interference in convex regular N-polygonal models.The labeled cell-shapes can be arranged in a grid withoutoverlapping areas.

+

+

y/R

x/R0.5

*

***

0 1

φ(II)

φ(III)

0.25

*** φ(I)

Fig. 4: Setup for evaluation. Cutout of Figure 1 (upper rightquadrant).

B. Validation of Gamma Approximation

In this subsection, we verify the accuracy of the Gammaapproximation in (11). The exact position-dependent distribu-tions of I6(ρ, φ) (cf. (5)) are obtained by numerically evaluatingthe approach in [16].

In order to obtain a representative profile of distributions,a receiver is moved along a straight line from ρ = 0 toρ = 0.5. The procedure is carried out for three receiver angle-positions Φ(I),Φ(II) and Φ(III), as shown in Figure 4. The anglescorrespond to the markers (I), (II) and (III) in Figure 2:• φ(I) = 0 represents a receiver, which is moved directly

towards its strongest interferer.• φ(II) is implicitely given by E

[I6(0.5, φ)

]= IC(0.5), i.e.,

the angle where the expected interference in the hexago-nal grid equals interference in the circular scenario.

• φ(III) = π6 represents the case, where a receiver is moved

centrally between its two strongest interferers.Fading is chosen as Gi ∼ Γ(2, 1). This corresponds to a 1 ×2 MIMO system with Rayleigh-fading and Maximum Ratio

0 0.1 0.2 0.3 0.4 0.50

0.01

0.02

0.03

0.04

Receiver distance ρ

Kol

mog

orov

-Sm

irnov

sta

tist

ic δ(ρ,φ)

φ(II)

φ(I)

φ(III)

**

*

*

*

**

Fig. 5: Kolmogorov-Smirnov statistic at position (ρ, φm): Com-parison of Gamma CDF and exact distributions. Receiverdistances ρ = 0 and ρ = 0.5 refer to cell-center and cell-edge,respectively.

Combining (MRC) at the receiver, or, equivalently, a 2 × 1MIMO system with MRC at the transmitter.

Then, for each distance ρ and angle φm ∈ {φ(I), φ(II), φ(III)}

we evaluate

• the Cumulative Distribution Functions (CDFs) of theGamma approximation, F I(x; k(ρ), θ(ρ)), where k(ρ) andθ(ρ) refer to (18) and (19), respectively.

• the CDFs F6(x; ρ, φm) of I6(ρ, θm), using the approach in[16].

In order to quantify the accuracy of the Gamma approximation,the Kolmogorov-Smirnov statistic is employed. It formulatesas

δ(ρ, φm) = supx

∣∣∣∣F I

(x; k(ρ), θ(ρ)

)− F6 (x; ρ, φm)

∣∣∣∣ . (24)

The results are depicted in Figure 5: The Gamma approx-imation most closely resembles the experienced interferencedistributions at φ(II), i.e., the typical receiver (lower curve).In this case, the difference between exact CDFs and Gammaapproximation is less than 1 % for ρ < 0.39 and 1.8 % at celledge (ρ = 0.5). The largest deviation occurs at φ(III), whenthe receiver is moved directly towards its strongest interferer(upper curve). Then, the distributions differ by less than 1 %for ρ < 0.34 and by 4.4 % at cell edge.

For qualitative evaluation, the exact CDFs and the corre-sponding Gamma approximations at representative receiverpositions are depicted in Figure 6. These positions are shownin Figures 4 and 5 and denoted as ’*’. The Gamma CDFsperfectly fit at cell center (ρ = 0) and in the middle of the cell(ρ = 0.25). At cell edge (ρ = 0.5), the Gamma approximationclosely resembles the experienced interference of a receiverat φ(II). The probability of high interference values at φ(III) isslightly underestimated by at most 4.4 % (cf. Figure 5).

0 5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Aggregate interference

CD

F

ρ=0

ρ=0.25

ρ=0.5

φ(I)

φ(II)

φ(III)

exact CDF

Gamma approximation

Fig. 6: Aggregate interference at particular receiver positions(see ’∗’ in Figure 4): Exact CDFs, as obtained by numericallyevaluating [16] for a hexagon scenario (dashed lines) andcorresponding Gamma approximations (solid lines).

C. Applications and Extensions

This subsection provides a basic guideline for applying andextending the circular interference model.

1) Average Rate and BER: The proposed Gamma approx-imation yields an input for quotient distributions, as typicallyappearing in Signal-to-Interference Ratio (SIR) metrics. Inparticular cases, these distributions are well known and allowto evaluate BER- and achievable rate statistics [11, 15].

2) Heterogeneous Networks: By fully characterizing thecellular contribution, the Gamma approximation considerablyfacilitates interference analysis in heterogeneous networks.

3) Multiple Tiers: The second tier of interferers in ahexagonal model consists of 12 transmitters, which are locatedon two hexagons with radius

√3R and 2R, respectively. For

these interferers, the cell edge of the center cell is located atρ = 1/(2

√3) and ρ = 1/4, respectively. According to (22), the

corresponding interference powers deviate at most 0.12 % and0.049 % from the exact solution.

Interference statistics in a multi-tier network can be cal-culated by applying the Gamma approximation for each tierseparately. This yields a sum of Gamma RVs [16], where theamount of sum terms is determined by the number of tiers andnot by the total number of interfering transmitters, as, e.g., in[17].

4) Regular N-polygonal Models: As stated in Section III,the circular model can substitute any convex regular N-polygonal system model, also denoted as N-gon1. This isverified by determining the error between circular model andvarious N-gon models, i.e., maxφ ε (0.5, φ), as depicted in

1The term N denotes the number of edges of the polygon and correspondsto the number of interfering transmitters.

Figure 3. The results are obtained by replacing ’6’ by ’N’in Equations (5) to (9) and (22). It is observed that the errordecreases for a higher number of interferers N, since the N-gons converge towards a circle.

5) Non-uniform Power Spreading: Non-symmetric impactof the interferers can be modeled by employing non-uniformpower distributions along the circle. This also allows to takeheterogeneous scenarios into account.

6) Uplink: Similar to [18], our approach is applicable as aframework for modeling interference in the uplink.

VI. ConclusionIn this work, we introduce a novel circular interference

model. We determine its distance-dependent aggregate inter-ference and show that the results deviate by no more than 3 %from the position-dependent interference powers in a hexag-onal grid. Then, we approximate the interference statisticsoutside the center of a hexagonal scenario by a single Gammarandom variate. We determine its distance-dependent shape-and scale parameters in closed form and identify the two keyformative components of the distribution: (i) the variance ofthe fading and (ii) the variance due to the eccentric receiverposition. A qualitative- and quantitative comparison with theexact distributions confirms the accuracy of the approximation.Gamma distributed interference enables statistical analysis offurther metrics such as average rate and bit error ratio in closedform. Therefore, our approach considerably widens the scopeof performance evaluation in regular grid models.

AcknowledgmentsThe authors would like to thank the LTE research group

for continuous support and lively discussions. This workhas been funded by the Christian Doppler Laboratory forWireless Technologies for Sustainable Mobility, KATHREIN-Werke KG, and A1 Telekom Austria AG. The financial supportby the Federal Ministry of Economy, Family and Youthand the National Foundation for Research, Technology andDevelopment is gratefully acknowledged.

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