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IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 30, NO. 7, SEPTEMBER 2012 1

Relay Deployment in Cellular Networks:Planning and Optimization

Weisi Guo, Tim O’FarrellDepartment of Electrical and Electronic Engineering

University of Sheffield, United KingdomEmail: {w.guo, t.ofarrell}@sheffield.ac.uk

Abstract— This paper presents closed-form capacity expres-sions for interfere-limited relay channels. Existing theoreticalanalysis has primarily focused on Gaussian relay channels, andthe analysis of interference-limited relay deployment has beenconfined to simulation based approaches. The novel contributionof this paper is to consolidate on these approaches by proposinga theoretical analysis that includes the effects of interferenceand capacity saturation of realistic transmission schemes. Theperformance and optimization results are reinforced by matchingsimulation results.

The benefit of this approach is that given a small set of networkparameters, the researcher can use the closed-form expressions todetermine the capacity of the network, as well as the deploymentparameters that maximize capacity without committing to pro-tracted system simulation studies. The deployment parametersconsidered in this paper include the optimal location and numberof relays, and resource sharing between relay and base-stations.The paper shows that the optimal deployment parameters arepre-dominantly a function of the saturation capacity, pathlossexponent and transmit powers.

Furthermore, to demonstrate the wider applicability of thetheoretical framework, the analysis is extended to a multi-room indoor building. The capacity improvements demonstratedin this paper show that deployment optimization can improvecapacity by up to 60% for outdoor and 38% for indoor users.The proposed closed-form expressions on interference-limitedrelay capacity are useful as a framework to examine how keypropagation and network parameters affect relay performanceand can yield insight into future research directions.

I. INTRODUCTION

Relays have been proposed as a solution to solving thechallenge of improving local capacity in Long-Term-EvolutionAdvanced (LTE-A) and 802.16 j/m standards. Its primarypurpose is to either increase the capacity of an existing area orto extend the coverage area of the parent cell-site. The Quality-of-Service (QoS) provided by an operator is not necessarilydetermined by the average performance, but by that achievedby a certain bottom percentage of customers. This is generallycustomers operating either on the interference limited cell-edge or indoors. Statistically, over 70% of the mobile trafficis carried to indoor users, therefore there is an urgency inaddressing how to enhance capacity for users in both of thesescenarios. This paper presents a novel closed-form capacityexpression for an interference-limited relay-assisted cellularnetwork, with consideration to the capacity saturation of real-istic transmission schemes, as well as both outdoor and indoorusers. The benefit of this approach is that given a small set ofnetwork parameters, the researcher can determine the capacity

of the network, as well as the deployment parameters thatmaximize capacity without committing to protracted systemlevel simulation studies.

A. Review

The topic of relays in cellular networks has been wellstudied in the past [1] [2]. In terms of analysis, existingtheory has largely focused on extending the original work onGaussian relay channels. Closed-form expressions on optimalrelay deployment in Gaussian channels was proven in [3]for location and in [4] for resource allocation. In a realisticcellular system, the effects of inter-cell interference [5] andcapacity saturation of realistic modulation schemes have asignificant impact on both the capacity of the system and theoptimal solutions, as shown in [6]. For relay deployment ina multi-cell interference-limited network, the characterizationof capacity and outage performance has been limited tosimulation based studies [5] [7]. Interference-limited stochasticgeometric theoretical methods have not yet been extendedto relay channels [8]. The optimization of relay deploymentlocation [9], resource allocation [1] [5], and cost efficiency[10] is conducted using iterative numerical approaches onsimulation results.

From a system designer perspective, there is a dichotomyin the theoretical and simulation approaches. The lack oftractable interference-limited relay capacity expressions meansthat one either has to rely on closed-form Gaussian channelexpressions or extensive multi-cell simulation results. Thishas restricted the insight into how and why key network andchannel parameters affect the relay performance and optimaldeployment solutions.

B. Contribution

The novelty of this paper is to consolidate the theoreticaland simulation based approaches by proposing an interfere-limited theoretical framework that considers capacity satu-ration. The paper presents closed-form capacity expressionsfor a relay-assisted base-station and maximizes the capacitywith respect to deployment parameters. The benefit of thisapproach is that, for any set of network parameters, the systemdesigner is able to characterize and optimize the multi-cellnetwork performance without resorting to extensive multi-cell simulations. The proposed analysis is also validated bya multi-cell simulator. Furthermore, to demonstrate the wider


Fig. 1. Example system setup for a particular relay-node (RN) configuration(3-sector BS with 5-relays per sector): a) 19-BS RAN model; b) BS withRNs; c) Mean received SINR plot.

applicability of the theoretical framework, the analysis isextended to a multi-room indoor building. The theoreticalanalysis in this paper are validated by multi-cell simulationresults from our own simulator and existing work.

II. SYSTEM MODEL

The system considers a Multi-Cell-Multi-User (MCMU)Radio-Access-Network (RAN), where the base-stations (BSs)are homogeneously deployed, and are assisted by relay-nodes(RNs). The homogeneous deployment offers an upper-boundto the RAN performance, in comparison to other irregularcell distribution methods such as spatial poisson-point-process(SPPP) distribution [8]. The RN-assisted BS establishes a highquality BS-RN channel by providing a LOS BS-RN channel,as recommended by 802.16j/m specifications [11]. The paperconsiders non-cooperative Decode-and-Forward (DF) relayingprotocol due to its low CSI estimation complexity and rela-tively strong system level performance compared to AF relay-ing [12]. The relays in this paper operate in the transparentmode, whereby the parent BS is responsible for scheduling thepacket transmission of each user with knowledge of the relaydeployment. The traffic model assumes that it is full-bufferand given that relays can yield a higher overall capacity, theauthors expect that the relays can in fact reduce congestiondelay.

The system layer simulation results are derived from ourown proprietary VCESIM LTE Dynamic System Simulator[13], which is bench-marked against 3GPP tests and hasbeen verified by our sponsors Fujitsu and Nokia SiemensNetworks. Each BS’s throughput considers 2-tiers of inter-BSinterference, which is sufficiently accurate [5]. The simulationsystem model is shown in Fig. 1a, where a 19 BS networkis created with wrap-around [5]. The relays are deployed nearthe cell-edge of each BS, where the cell-edge is defined asthe region where the interference power from other cells issimilar or stronger than the signal power from the servingBS. An illustration of one form of relay deployment is shownin Fig. 1b, and the corresponding average received SINR isshown in Fig. 1c.

In simulations, the instantaneous received signal to interfer-ence plus noise ratio (SINR) of a single sub-carrier of a singleuser is a function of: the transmit power of BS (P ), pathloss

(λ), AWGN power (n) and antenna gain (A):

γs,i =|hi|2λi10

Si+A(θi)

10 Ps,i

n+∑Ncellj=1,j 6=i |hj |2λj10

Sj+A(θj)

10 Ps,j, (1)

where the values of each parameter is given in Table. I inthe Appendix. The parameters h and S are the multi-pathand log-normal shadow fading components, defined in [14].The pathloss component can be expressed as a function of thedistance x: λ = Kx−α; where K is the frequency dependentpathloss constant and α is the pathloss exponent. The downlinkthroughput employs an adaptive-modulation-coding results aretaken from a physical link layer simulator [15].

III. ANALYTICAL MODEL

A. Interference-Limited Capacity

The analytical model considers a simplified and tractableSINR expression of (1), based on a serving cell (i) anddominant interference cell (j), as shown in Fig. 2. This issimilar to the analytical models in [16], whereby it has beenshown that the effects of fading on capacity, when averagedover time, are small compared to the effects of pathloss. Thecomplete SINR expression in (1) can be approximated to:γi ≈

x−αi Pi

x−αj Pj, by not assuming that the interference power is

greater than AWGN power.The downlink throughput is found using the Shannon ex-

pression, with consideration to mutual information saturation:

Ci =

{Cs for: 0 < xi < dsalog2(1 + bγi) for: ds < xi < r

(2)

where at the distance ds or less away from the serving cell,the the maximum achievable capacity in LTE is Cs = 4.3bit/s/Hz, for a modulation and coding scheme of 64QAMand 6/7 Turbo Code. The factors a = 0.8 and b = 0.6are the Shannon adjustment values to compensate for codinglosses in mutual information [17]. It has been shown thatif the capacity saturation of channels is not considered, theoptimization solution can often be skewed towards awardinghigh capacity links with more resources, when in reality thesechannels have already been saturated [6]. The value of thedistance at which capacity saturation occurs (ds) can be foundas a function of the inter-BS distance (2r), pathloss exponent(α) and the saturation capacity (Cs):

ds ≤2r

1 + ((2Csa −1)PjbPi

)1α

, (3)

which is proven in Lemma 1 in the Appendix.The paper’s simulation results and analytical model consid-

ers both the mean capacity and the edge capacity, as definedby:• Mean Capacity: the mean capacity achieved is domi-

nated by BS centre users that achieve a large receivedSINR. By stating that SINR (γ) of each position is largeenough for the approximation log(1 + γ) ≈ log(γ) tohold, the margin of error averaged across the BS is small(0.1%). This is proven in Lemma 2 of Appendix.


Fig. 2. Analytical (top) System Model and Simulated (bottom) Capacity for: a) Homogeneous BS; and b) Heterogeneous Outdoor-Outdoor Network withBS and Relays.

Fig. 3. Baseline downlink throughput results for simulation (symbols) andtheory (lines) with BS coverage sizes.

• Edge Capacity: the capacity achieved by cell-edge usersthat has the lowest received SINR, which determine theQuality-of-Service (QoS).

The paper will now consider the cellular capacity for a baselineoutdoor system in Section III.

B. No Relay

In the baseline outdoor capacity analysis, the paper consid-ers an omni-directional 1-sector BS, as shown in Fig. 2. Thecapacity of a single sector BS with co-frequency interferencefrom other BSs can be expressed as:

C =

Cs for: 0 < x < 2r

1+γ1αs

alog2(1 + bx−α

(2r−x)−α ) for: 2r

1+γ1αs

< x < r(4)

The mean capacity achieved is the average capacity achievedfrom edge of BS coverage (x = r) to the base of the BS

(x = 0):

C̄ =1

r

∫ 0

r

Cdx ≈1

r{Σ + a | dslog2[(

ds2r − ds

)−α]

−2αrlog2(2r − ds

r) |},

(5)

where ds = 2r

1+γ1αs

is given by expression (3) and Σ = Csds.

The full proof is given in Lemma 3.The edge capacity achieved is between 2 closest BSs

deployed, as illustrated in Fig. 2 at location 1. The minimumcapacity (edge) occurs when (x = r):

Cedge = alog2(1 + br−α

(2r − r)−α). (6)

The results of the theoretical analysis and the simulation modelis presented in Fig. 3, where similar to the conclusion in [18], areasonably good match was found (margin of error is 1.2% formean capacity and 16% for edge-capacity). The paper will nowintroduce the interference-limited relay capacity and maximizethe capacity with respect to: RN location, number of RNs andresource shared between BS and RN.

IV. OUTDOOR RELAYING (OR)

The paper assumes that the RNs are deployed in placeswhere the channel between the parent BS and the RN is Line-of-Sight (LOS) based [14]. All other channels are assumed tobe NLOS-based in the theoretical framework, and probabilitybetween NLOS and LOS based in the simulation framework(see Section I). When co-frequency RNs are inserted into aBS to improve outdoor capacity, the dominant interferencecoupling is between the parent BS and the RN, and not fromthe neighbouring BSs. This is due to the fact that signalpower is dominated by propagation distances, despite the RN’ssignificantly lower transmit power.

For an UE in the BS, there is a handover point where themean received SINR from the BS-UE and the RN-UE is equal,


Fig. 4. Relaying downlink capacity variation with RN distance: Mean andEdge capacity for Gaussian relay channels without interference and capacitysaturation (simulation in symbols and theory in lines).

as shown in Fig. 2. This distance from the BS, is denoted dHOand is shown to be the following in Lemma 4:

dHO =dr

1 + (PrPc )1α

, (7)

where dr is the distance of the RN from the serving BS. If theRN and BS transmit at the same power, the handover distanceis half-way between the RN and the BS.

The capacity of co-frequency outdoor-relaying (CF-OR) cantherefore be represented as:

CCF-OR =

Cs for: 0 < x < ds1

alog2(1 + bPcx−α

Pr(dr−x)−α ) for: ds1 < x < dHO

alog2(1 + bPr(dr−x)−α

Pcx−α) for: dHO < x < ds2

Cs for: ds2 < x < ds3

alog2(1 + bPr(x−dr)−α

Pcx−α) for: ds3 < x < r

(8)

which is for dHO > ds1 and holds true if the BS-RN channelis always stronger than the RN-UE channel, as proven inLemma 5. The values for the break-point distances ds1, ds2and ds3 can be found via Lemma 1 and Lemma 4 and ispresented in Lemma 6. Figure 4 shows the downlink capacity(CCF-OR) variation with BS-RN distance dr. In Fig. 4, theclassical Gaussian relay channel is considered, whereby nocapacity saturation and interference are modelled. The resultsshow that the optimal RN location is generally less than half-way (dr/r < 0.5), similar to the results obtained in for acooperative DF relaying [3] [4]. By introducing the co-channelinterference and capacity saturation, the optimization solutionshifts to deploying the RN away from the BS to dr/r ' a.This will be more closely explained in the next section. Themean capacity results show a good match between simulationand theoretical capacity. The detailed proof on optimizationfor mean and edge capacity is given below.

Fig. 5. Relaying downlink capacity variation with RN distance: Meanand Edge capacity for interference-limited network with capacity saturation(simulation in symbols and theory in lines).

A. Mean Capacity Optimization

The mean capacity achieved is the average capacityachieved from edge of BS (x = r) to the base of the BS(x = 0):

C̄CF-OR =1

r

∫ 0

r

CCF-ORdx ≈ 1

r{ΣCF-OR+ | aFΣ |}, (9)

where ΣCF-OR contains the BS and RN saturation capacityterms, and FΣ is a composite logarithm term that containsthe non-saturated terms, as explained in Lemma 6.

In order to maximize the mean capacity with respect tothe location of the RN, expression (9) is differentiated withrespect to dr. The optimal BS-RN distance that maximizesmean capacity is:

d∗r,CF-OR,mean-opt. ≈ 0.4FΣ[1 + (γsPrPc

)−1α ], (10)

where FΣ is a constant and the full optimization proof isshown in Lemma 6 of the Appendix.

The conclusion is that the optimal RN location is almostentirely dependent on the power ratio between the RN andthe BS (PrPc ), the pathloss exponent (α), and the saturationSINR threshold (γs):• A lower γs (worse transmission technique) means the

RN should be placed further to the parent-BS, becausemost of the coverage area near the BS is saturated.Therefore the RN is only beneficial at the cell-edge andthe effectiveness of RN deployment is also reduced.

• A lower RN to BS transmit power ratio means the RNshould be placed further from the parent-BS, because thestronger the BS transmit power, the stronger the effect ofthe BS’s coverage.

• A lower α (more LOS based propagation) means the RNshould be placed further from the BS, because the weakerthe pathloss effect, the stronger the effect of the servingBS’s coverage.

The results in Fig. 4 show that the optimal RN location forGaussian channels is generally less than half-way (dr/r <


Fig. 6. Optimal Number of RNs for simulation and theory for a variety ofRN and BS transmit powers.

0.5) [3] [4]. By introducing the co-channel interference andcapacity saturation (Fig. 5), the optimization solution in (10)shifts to deploying the RN to dr/r ' 0.8, yielding a capacityimprovement of 55%. This generally agrees with existingrecommendations (dr/r ' 0.66 to 0.8) on relay deploymentin a multi-cell environment [7] [9].

The results in Fig. 5 also show that the mean capacity issignificantly diminished when the RNs are placed too close tothe BS, but the edge-capacity generally remains undiminished.This is because regions close to the BS are already operatingwith a saturated capacity and by adding RNs, the capacitycan only be degraded via increased interference. However, theedge-capacity is at the inter-BS location and is thus largelyunaffected by RNs placed near the BS.

B. Edge Capacity Optimization

For a BS with coverage radius (r), the edge capacity occursat one of the 2 possible low SINR locations, as illustrated inFig. 2:

1) the traditional inter-BS edge (x = r),2) the handover point between BS and RN (x = dHO),

whereby the minimum of the capacity at these 2 locationsdetermines the edge-capacity:

CCF-OR,edge = min[CCF-OR(x = dHO), CCF-OR(x = r)]. (11)

In order to maximize the edge capacity, neither of the afore-mentioned capacity terms can be smaller than the other andthus, equating the terms in expression (11) leads to the optimalRN location and maximum edge-capacity to be:

d∗r,CF-OR,edge-opt. = r[1− (PrPc

)1α ]. (12)

The insight here is that the optimal RN location is almostentirely dependent on the power ratio between the RN and theBS (PrPc ) and the pathloss exponent (α):• A lower RN to BS transmit power ratio means the RN

should be placed further from the parent-BS, because the

Fig. 7. Downlink Capacity for Non-Co-Frequency Relaying (NCF).

stronger the BS transmit power, the stronger the effect ofthe serving BS’s coverage.

• A lower α means the RN should be placed further fromthe parent-BS, because the weaker the pathloss effect, thestronger the effect of the serving BS’s coverage.

The relation’s insight is very similar to that found for the meancapacity in expression (10), without the capacity saturationparameter. This is because capacity saturation can not beachieved on the cell-edge and poor coverage areas. Thereis a conflict of interest between maximizing mean-capacityand edge-capacity by deploying the RN at either (10) or (12)respectively. However, as the results in Fig. 5 indicate, bysacrificing a small percentage of edge-capacity, the maximummean-capacity can be achieved. This sacrifice in edge-capacityis negligible in the theoretical framework and approximately4% from the simulation results.

C. Number of Relays

So far, the paper’s analytical framework has consideredrelaying only on a 1-dimensional level (distance away fromparent BS). In order to consider the impact of increasing thenumber of RNs (Nr) evenly distributed around the parent-BS,the analytical model is expanded in the following logic:

• Increasing the number of RNs can improve the capacity,provided that the mutual interference between RNs doesnot exceed the interference from the BS.

• The inter-RN interference dominates performance whenthe inter-RN distance is small (RN density is large).

Furthermore, the paper defines that additional RNs are de-ployed equal distant to the BS and that the propagation channelbetween RNs is NLOS based. Given these assumptions andconditions, the theoretical maximum number of beneficial RNscan be shown to be the following:

N∗RN,opt. < bπ(2PrPc

)−1α c, (13)

with the full proof in Lemma 7. The insight here is thatthe optimal RN number is almost entirely dependent on the


Fig. 8. Mean and Edge capacity tradeoff for different relaying techniques.

power ratio between the RN and the BS (PrPc ) and the pathlossexponent (α):• A higher RN transmit power means fewer RNs should be

placed to reduce mutual interference.• A lower α means more RNs should be placed.• The optimal number of RNs is largely independent of the

BS coverage size (r).In Fig. 6, the results show that the theory matched verywell with the simulation results for the optimal number ofRNs that maximizes mean capacity for a variety of transmitpower levels. The theoretical results are validated by our ownsimulation results and existing literature [10].

D. BS-RN Resource Sharing

This section of the paper examines non-co-frequency (NCF)relaying and what the resource block sharing ratio should bebetween the parent BS and RNs. Generally speaking WCDMAand LTE cells are deployed with frequency reuse pattern 1[14] and there is debate on what the optimal frequency reusepattern is for RNs in different scenarios [1] [5]. In order tomaximize the edge-capacity, the edge-capacity of the BS andthe RN should be equal:

B∗r,opt. ≈

{0.5 for low density of RNs(1 + log(Φ)

log(Ψ) )−1 for high density of RNs ,

(14)

where Φ = dHO2r−dHO

and Ψ = 2−1α dr√

1+ 4π2

NRNd2r

. If the RN density

is low, such that the worst coverage area is at the inter-BSedge, then the optimal resource block sharing fraction is ap-proximately 0.5. As shown in Fig. 7, the theoretical expressionin (14) is validated with our own multi-cell simulations andthose in existing literature [5]. The numerical search methodsin simulations found that Br is optimally between 0.45-0.6 inorder to maximize edge capacity.

A further capacity enhancing technique is to grant users thatbenefit from relaying constant channel capacity parity betweenthe BS-RN and RN-UE channels. For example, it has been

Fig. 9. Directional and Omni-Directional RN Deployment SINR Plots.

shown in Lemma 5 that if the user position is smaller than athreshold from the RN, the 2-hop relay capacity becomes BS-RN limited. Adaptive NCF (A-NCF) dynamically balancesthe BS-RN and RN-UE capacity to be equal, by prioritizing2-hop relay transmission over direct BS-UE transmission. Withthe aid of Lemma 5, the proportional increase in bandwidth ofthe BS-RN channel Bδ required for capacity parity betweenthe 2-hop relay channels is:

Bδ 'log( xr

dr−xr )

log( dβαr

2r−xr )

, (15)

where the value of Bδ can be above or below unity dependingon the user location (xr). The results in Fig. 8 show that the CFrelaying offers the greatest mean capacity. The NCF relayingyields a mean capacity degradation, but an improvement inedge-capacity compared to no relaying. The A-NCF relayingyields the greatest edge-capacity improvement due to itschannel parity scheme, but it does suffer a mean capacitydegradation compared to the CF relaying, due to the resourcessacrificed in the BS-UE channel (15).

The paper also extends the research to directional BS andRN antennas. The rationale is whether directional RNs can notonly achieve a capacity improvement for the relaying region,but also reduce the radiated interference to the other regionsof the BS that do not require relaying. It was found thatdirectional RNs can further enhance the capacity by 26%,when the bore-sight of the RNs are directed along the cell-edge, as shown in Fig. 9.

V. INDOOR RELAYING (IR)A. Indoor System Setup

Previously, the paper discussed how the worst performanceusers are usually on the inter-BS edge or indoors. This


Fig. 10. Analytical System Model for Indoor Network with Outdoor BS andRN.

section shows that the interference-limited relay analysis canbe extended to analyze the indoor capacity. The analysisdemonstrates that the interference-limited framework is robustand can model a variety of novel scenarios.

The capacity of a user at a distance of x from the nearestwall to the parent BS is:

Cindoor ' alog2(1 + bPc(D + x)−αWoutW

k−1in 10−

x20

Pc(2r −D)−αWout), (16)

assuming that the serving BS’s signal power is predominantlyfrom one direction and the interference powers are fromall directions. The indoor capacity is generally below thesaturation capacity Cs, unless the building has no internalwalls (Nwall = 0) and is very close to the serving cell.

The paper now considers the addition of a closed-access co-frequency relay to improve coverage to the indoor users. Themain research theme of this section considers whether the RNshould be placed outside or inside the building, as a functionof building size (L) and distance from the serving BS (D). Byusing the same analysis as previously shown in Lemma 5, thepaper’s indoor analytical model assumes the following:• RN is outside building: the BS-RN channel is always

stronger than the RN-UE channel• RN is inside building: the BS-RN channel is always

weaker than the RN-UE channel• The rooms in the building are equally spaced in this

analysisNote that the simulation results make no assumptions regard-ing the strength of channels. The results below will show thatthese theoretical assumptions yield accurate approximations tosimulated results.

The paper considers a building with length L with Nwallevenly spaced internal walls and is located at a distance Dfrom the serving BS (as shown in Fig. 10. The mean indoorcapacity achieved is the average capacity achieved from oneend of the building x = 0 to the other x = L, and it can beshown to be approximately:

C̄IR '{alog2(bPrPc ( LD )−α) outside RNalog2(1 + b( D

2r−D )−α) inside RN, (17)

where the full mean capacity proof is given in Lemma 8.

B. Indoor Relay PlacementIn order to maximize mean capacity, the location of the RN

(outside or inside the building) depends on the distance of the

Fig. 11. Indoor Mean Capacity as a function of Building Location for aBS-size r = 500m and L = 20m, with Simulation (Symbols) and Theory(Lines).

building (D) from the serving BS. From expression (17), thetwo capacity expressions are equal when:

D∗ = 0.5[

√8rL(

PrPc

)−1α − L(

PrPc

)−1α ]. (18)

Therefore, the adaptive deployment guideline that maximizesindoor capacity for a multi-room building is:

• Deploy RN inside if the building is at D < D∗.• Deploy RN outside if the building is at D > D∗.

The results in Fig. 11 show that deploying a RN can signif-icantly improve the indoor capacity, especially for buildingsthat are far away from the parent-BS. In order to maximizethe benefit of RNs, the RNs should be placed adaptively eitherinside or outside the building depending on the location ofthe building (D), as given by expression (18). The resultsshow that a mean capacity improvement of up to 38% can beobtained in the adaptive RN deployment strategy. It should benoted that if the pathloss exponent is different for when the RNis inside compared to outside, the optimization expression (18)can be adjusted relatively easily. The challenge of addressingnon-uniform distribution of rooms and users is non-trivial andis not considered in the scope of this paper.

The results in Fig. 12 show that by deploying a fixed omni-directional closed-access RN outside the building to coverindoor users, the interference it causes to the outdoor networkleads to a mean capacity degradation of 44%. By adoptingan adaptive deployment based on the guideline devised in(18), the outdoor mean capacity degradation is 31%, animprovement of 20% over the fixed strategy. That is to say, notonly does the adaptive indoor RN deployment benefit indoorusers (30% improvement), it also benefits outdoor users (20%improvement). By employing a directional RN, whereby aRN radiates the directional bore-sight (4dBi) into the targetedbuilding and radiates the backside (-10dBi) towards the out-door network, a further 13% capacity gain can be obtainedcompared to the omni-directional RNs.


Fig. 12. Simulated outdoor capacity as a function of the distance of theinterfering RN to the parent-BS.

VI. CONCLUSIONS

The purpose of this paper is to propose a tractable the-oretical framework that can characterize and optimize theperformance of a relay-assisted network. The novelty is thatthe theory considers the effects of cellular interference andcapacity saturation of realistic transmission schemes. The pa-per distinguishes itself from existing work, which have largelyconsidered theoretical Gaussian relay channels or multi-cellresults based on simulations.

The benefit of this approach is that given a set of essentialnetwork parameters, the researcher can use the expressionsto determine the capacity of the network, as well as therelay deployment parameters that maximize capacity. Theframework can therefore be used to dimension and plan anetwork before committing to protracted system level sim-ulations. Compared to Gaussian relay channel optimization,the interfere-limited analysis yields a 55% improved capacityin a cellular environment. The theory in this paper have beenvalidated by multi-cell simulation results.

Furthermore, the theoretical framework is general enoughto be extended to optimizing resource sharing between base-station and relay, as well as optimizing the relay location foran indoor multi-room building. The capacity improvementsdemonstrated in this paper show that optimization of theaforementioned parameters can improve capacity by up to 60%for outdoor and 38% for indoor users. The novel interference-limited relay capacity expressions are useful as a framework toexamine how key propagation and network parameters affectrelay performance and can yield insight into future researchdirections.

APPENDIX

A. Lemma 1: Capacity Saturation Range

In order to find the maximum distance away from theserving BS where saturated spectral efficiency (Cs) can still

be achieved, the following must hold true:

alog2(1 + bγs) = alog2(1 + bPid−αs

Pj(d− ds)−α)

ds =d

1 + (γsPjPi

)1α

.(19)

B. Lemma 2: Integral Approximation for High SINR

By encompassing RNs, the mean capacity expression needsto be approximated with log(1 + x) ' log(x). This has amargin of error of ε, which is:

ε =1

γslog2(

(1 + γs)1+γs

γγss) (20)

where γs is the saturation SINR for a realistic modulation andcoding scheme and R is the coverage radius of the consideredBS. For a LTE BS operating with 2x2 SFBC MIMO, the valueof γs = 18dB, which achieves a saturated spectral efficiencyof Cs = 4.3 bit/s/Hz. The resulting expression for margin oferror is: ε = 0.1 %.

C. Lemma 3: Mean Capacity (No Relays)

The mean capacity achieved is the average capacityachieved from edge of BS (r) to the base of the BS:

C̄outdoor =1

r

∫ 0

r

Cdx =1

r{Csds +

∫ ds

r

Coutdoordx}

=1

r{Σ+ | a(dslog2[(

ds2r − ds

)−α]

− 2rαlog2(2r − ds

r)) |},

(21)

where ds = 2r

1+γ1αs

and Σ = Csds.

D. Lemma 4: BS-RN Handover Location

The handover distance from the serving BS, when a co-frequency RN is deployed dr away from the BS is (γBS-UE =γRN-UE):

dHO =dr

1 + (PrPc )1α

, (22)

where Pc and Pr are the transmit power for the BS and RNsrespectively.

E. Lemma 5: Relay Channels

It can be proved that the BS-RN channel is always superiorto the RN-UE channel due to the LOS propagation character-istic of the system setup. Consider a RN located at dr from theparent-BS and a UE located at xr from the RN. The followingmust hold true for the BS-RN to be always equal or greaterthan the RN-UE capacity (CBS-RN ≥ CRN-UE):

xr ≥dr

1 + d− βαr (2r − dr)(PrPc )−

1α

(23)

which assumes that from an interference perspective, the usersare close to the RN so that 2r−xr ∼ 2r−dr. The value of xr


as a percentage of BS coverage radius (r) is no more than 2%for system values given in Table I. That is to say the relayingcapacity is limited by the RN-UE channel.

F. Lemma 6: Outdoor Mean Capacity (Relays)

The values for the break-point distances are as follows,using Lemma 1 and Lemma 4: dHO = dr

1+(ρ−1)1α

, ds1 =

dr

1+(γsρ−1)1α

, ds2 = dr

1+(γsρ)1α

, and ds3 = dr

1−(γsρ)1α

. The mean

capacity of a co-frequency BS and RN setup is therefore:

C̄CF-OR =1

r

∫ 0

r

CCF-ORdx =1

r{Cs(ds1 + ds3 − ds2)

+ | alog2[F1(ds1, dHO)F2(ds2, dHO)F3(r, ds3)]

− aαdrlog2((ds1 − dr)(ds2 − dr)(ds3 − dr)

(dHO − dr)2(r − dr)) |},

(24)

where F1(a, b) = (( badr−a )a

( bbdr−b )b

)−α, F2(a, b) = (( badr−a )a

( bbdr−b )b

)α,

aF3(a, b) = (( baa−dr )a

( bbb−dr )b

)α; and ΣCF-OR = Cs(ds1 + ds3 − ds2).

The below work shows the proof for optimal BS-RN dis-tance by differentiating the mean capacity term with respectto the BS-RN distance parameter (dr):

dC̄CF-OR

ddr=

1

r{ 1

1 + (γsρ−1)1α

+2(γsρ)

1α

1− (γsρ)1α

+ aFΣ

dr}. (25)

The optimal BS-RN distance that maximizes mean capacityis:

d∗r,CF,mean−opt. ≈ 0.4FΣ[1 + (γsρ)−1α ]. (26)

G. Lemma 7: Optimal Number of Relays

Assume a BS with NRN RNs deployed on the circumferenceof a circle around the BS with radius dr. At each RN, in orderfor the dominant interference power of 2 nearby RNs to bestronger than the interference power from the serving BS, thefollowing must hold:

2PrKd−αrr > PcKd

−αr where: drr ∼

2πdrNRN

∴ NRN < 2π(2PrPc

)−1α ,

(27)

which is only accurate when the number of RNs is above 2and high.

H. Lemma 8: Indoor Mean Capacity (Relays)

For indoor users being served by a RN on the outside, themean capacity (IR,o) is:

C̄IR,o =1

L

∫ 0

L

alog2(1 +bPrx

−α

Pc(D + x)−α)dx

' alog2(bPrPc

(L

D)−α), for: D � L,

(28)

which holds true for when the building size (L) is significantlysmaller than the distance from the serving BS (D), so that

TABLE ISYSTEM PARAMETERS FOR LTE SIMULATOR

Parameter Symbol ValueLTE Operating Frequency f 2600MHzLTE System Bandwidth BW 20MHzPath-loss Model λ [14]NLOS Pathloss Exponent α 3.67LOS Pathloss Exponent β 2.2Shadow Fading variance σ2

sd 9dBCapacity Saturation Cs 4.3 bit/s/HzSINR Saturation γs 18dBAWGN power per subcarrier n 6× 10−17 WBS Transmit Power Pc 10-40WDirectional Antenna Pattern Acell [14]Relay Transmit Power Pr 0.5-5WBuilding Length L 20mBuilding Distance to BS D 50-500mExternal Wall Loss Wout 20dBInternal Wall Loss Win 10dBNumber of Internal Walls Nwall 4

D ' D + L. For indoor users being served by a RN on theinside, the mean capacity (IR,i) is:

C̄IR,i = alog2(1 + b(D

2r −D)−α). (29)

I. System Modeling Parameters

The parent-BSs and RNs are assumed to be on roof-tops and have Line-of-Sight (LOS) propagation, whereas theinterference from adjacent BSs and RNs are assumed to beNon-Line-of-Sight (NLOS) based [14]. The parameter |h| isthe magnitude of the complex fading coefficient h, whichRayleigh distributed and generated from an auto-regressiveAR(n) process, where by the value of n is dependent on thedelay spread [14] [19]. The BS-UE and the RN-UE channelsare assumed to be based on a probabilistic model, wherebythe probability of being in LOS:

℘LOS = min(1,18

x)(1− e− x

36 ) + e−x36 , (30)

where x is the distance from the serving BS. The servingBS-RN channel is assumed to be in LOS [11] [20].

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Weisi Guo received his B.A., M.Eng.,M.A. and Ph.D. degrees from theUniversity of Cambridge. He is currentlyan Assistant Professor at the Universityof Warwick and is the author of theVCESIM LTE System Simulator. Hisresearch interests are in the areas ofself-organization, energy-efficiency, and

multi-user cooperative wireless networks.

Tim O’Farrell holds a Chair in Wire-less Communication at the Universityof Sheffield, UK. He is the AcademicCoordinator of the MVCE Green Ra-dio Project. His research encompass re-source management and physical layertechniques for wireless communicationsystems. He has led over 18 research

projects and published over 200 technical papers including8 granted patents.

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