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AN APPROACH FOR DOWNLINK VOIP CAPACITY

ESTIMATION IN OFDM-BASED FDD NETWORKSIana Siomina

Ericsson Research, Ericsson AB, SE-164 80 Stockholm, SwedenE-mail: [email protected]

Abstract— The paper presents a methodology for estimatingdownlink VoIP capacity in OFDM-based FDD networks that door do not allow for adaptive transmit power. The underlying ideais to formulate an optimization problem for a user snapshot andutilize the developed system model to enable statistical capacityestimation by adopting the results of the central limit theorem.The approach is applied to numerically study VoIP capacity innetworks with LTE-like parameter settings and various inter-site distances, frequency reuse of one and three, and SISO andMIMO antennas. The study demonstrates that adjustable transmitpower in downlink may improve VoIP capacity by up to 30 %in interference-limited environments. In networks with frequencyreuse of three and MIMO antennas, however, the effect maybe lower unless a larger bandwidth is available. The paper alsoaddresses a trade-off between the cell coverage degree and capacityby showing a marginal coverage cost in terms of VoIP capacity.

Index terms – VoIP, capacity, downlink, OFDM, LTE.

I. INTRODUCTION

Voice is one of the most important means of communicationover cellular networks today. With the advent of all-IP networks,voice will be transported over IP and hence Voice over IP (VoIP)will remain equally important in such future networks. Ensuringa high quality VoIP service, good enough to replace today’scircuit-switched telephony, requires a lot of radio networkresources due to low latency and jitter requirements. VoIPcapacity is therefore in the focus of this paper.

We develop a system model and present an approach fortheoretical estimation of downlink (DL) VoIP capacity in Fre-quency Division Duplex (FDD) networks based on OrthogonalFrequency-Division Multiplexing (OFDM). In particular, weaim at finding the critical number of users when a networkuses available radio resources at their maximum. The systemand optimization models are presented in Sections II and III,respectively. The solution approach is described in Section IV.In Section V, we discuss a numerical study for networkswith parameter settings designed for Long-Term 3G Evolution(LTE) [4], a new radio access technology being currentlyspecified by 3GPP. The conclusions are drawn in Section VI.

II. SYSTEM MODEL

Consider an OFDM-based FDD network with a set of cellsdenoted by I. In this study, we focus on DL VoIP capacity ofa single cell i∗ surrounded by a set {i ∈ I \ i∗} of interferingcells. The serving area of a cell is assumed to be known, i.e., itis statically defined based, for example, on the received signalquality or geographical area partitioning.

Let the entire network service area be represented by agrid of bins, where a bin is a small area unit of eitherregular or irregular shape, and let J denote the set of bins.The same propagation conditions are assumed across a bin.Given equipment specifications, antenna configurations, andradio propagation environment characteristics, we can predictend-to-end attenuation for each pair (i, j) and obtain a set ofpower gain values {gij , i ∈ I, j ∈ J }. We define a system

model assuming SISO antenna systems and then adapt it forMIMO. No channel-dependent fading is assumed.

Frequency-time domain. Given frequency spectrum B anda sub-carrier spacing ∆f , we find the total number of sub-carriers as Nf = κ·B

∆f , where κ is the effective spectrumutilization ratio [1], [2]. In the frequency domain the DL sub-carriers are grouped into sub-bands, each of which consists ofnf consecutive sub-carriers. A resource block (RB) is a two-dimensional unit that spans over a sub-band and one 0.5 msslot. The latter consists of a number of OFDM symbols (7or 6 symbols for normal or extended cyclic prefix in LTE,respectively). The number of available RBs in a time slot isNSB =

⌊Nf

nf

⌋. The minimum transmission time interval (TTI)

is 1 ms. Therefore, the minimum transmission block consists oftwo RBs adjacent in time (the pair of RBs is further referred toas a chunk).

Interference model and adaptive power. Let ρ be theaverage transmit power per chunk in an interfering cell. Letρjm be the transmit power in the studied cell i∗ allocated for auser located at bin j and using modulation and coding scheme(MCS) m selected by a link adaption algorithm. To achieve acertain block-error rate (BLER) for a given MCS, the signal-to-interference-plus-noise ratio (SINR) of a transmission over asub-band to a user in bin j at time t must satisfy

ρjm · gi∗j∑i∈I(i∗,t) ρ · gij + νj

≥ γm , (1)

where γm is the target SINR for MCS m, I(i∗, t) ⊆ I is theset of cells interfering with transmissions in i∗ at time t, and νj

is the thermal noise power at the user’s receiver. The minimumtransmit power ρjm needed for a user in bin j with MCS m isthus as follows,

ρjm = γm ·∑

i∈I(i∗,t) ρ · gij + νj

gi∗j. (2)

Let Pmax denote the maximum transmit power available ina cell (to simplify the presentation, we assume it to be the samefor all cells in I). Under the assumption of full and balancednetwork load, the average transmit power per sub-band in aninterfering cell is ρ = P max

NSB . The amount of transmit powerin cell i∗ is limited by Pmax and depends on the number ofscheduled users in the cell and selected MCSs.

Spectrum utilization. The minimum number of chunksneeded to transmit one VoIP frame with MCS m depends on anumber of parameters and can be defined as follows,

nm =⌈

bpayload + bL1L2

Rm · �(1 − ϕ) · nsymb�⌉

, (3)

where bpayload is the size of an IP packet payload with oneVoIP frame, bL1L2 is the Layer 1 and Layer 2 control signallingoverhead (as number of bits), Rm is the normalized throughput

1525-3511/08/$25.00 ©2008 IEEE

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2008 proceedings.

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for MCS m and a given BLER target, ϕ is the physical layeroverhead (the portion of symbols in a chunk that are reservedfor physical signals, e.g., reference signals), and nsymb is thetotal number of symbols per chunk. Normalized throughput fora given MCS and BLER target can be obtained, for example,from link-level simulations. Similar to [3], we use the followingapproximation,

Rm ≈ RCm · RM

m · (1 − BLER) , (4)

where RMm is the modulation rate (the number of bits per trans-

mitted symbol), RCm is the coding rate for an Additive White

Gaussian Noise (AWGN) channel, and BLER is the targetBLER. Note that the total number of chunks for transmittingone VoIP frame may exceed nm since the frame may need to beretransmitted a number of times before it is correctly receivedat the destination.

To incorporate MIMO antenna systems into our systemmodel, we utilize the fact that spatial multiplexing with MIMOyields a linear (in the minimum of the number of transmitand receive antennas) increase in capacity for no additionalpower or bandwidth expenditure [5], [7]. For instance, theShannon channel capacity values for SISO and 2 × 2 MIMOantenna systems are typically found as W · log2 (1 + SINR)and 2 · W · log2 (1 + SINR), respectively, where W is thebandwidth, and SINR is the SINR per receive antenna withone transmit antenna. Therefore, to account for 2×2 MIMO inour model, we double the normalized throughput Rm.

Error control. Hybrid Automatic Repeat-reQuest (HARQ)with chase combining is one of the possible error control mech-anisms, also considered for LTE [4], and the one assumed inour model. This implies that each retransmission is an identicalcopy of the original transmission and all incorrectly receivedcoded data blocks are stored at the receiver and combined (usingmaximum-ratio combining for each bit) before being fed to thedecoder. Therefore, given a maximum power ρmax per chunk,we compute the number of retransmissions by

⌈ρjm

ρmax

⌉. Then,

the minimum number of chunks needed to transmit the entireVoIP frame with MCS m is as follows,

ηjm =⌈

ρjm

ρmax

⌉· nm . (5)

The accumulated SINR after all retransmissions is then com-pared to the target SINR value. This means that the minimumamount of power needed to deliver the entire VoIP frame touser j does not depend on the number of retransmissions andis given by ρjm ·nm, provided that MCS m remains unchangedfor all HARQ transmissions to one user.

To this end, we have considered two resources in the network:the power resource and the frequency spectrum resource. Thetotal amount of power allocated for all DL transmissions in acell at any time shall not exceed the cell maximum limit Pmax,and the number of chunks at any time shall not exceed NSB .The goal is to find the maximum number of users in cell i∗

that can receive their VoIP frames within a given time interval,given the amount of the available resources in the cell.

III. AN OPTIMIZATION MODEL

A. Problem Statement

Observe that the problem described in the end of Section IIrequires the knowledge of the scheduler implemented in thenetwork because we need to know the users served at each

particular time point, the selected MCS for each of the users,and the size of each transmitted block. Our goal, however, isto enable evaluating the technology potential in terms of VoIPcapacity. Therefore, we are interested in finding the maximumachievable capacity in cell i∗ without restrictions imposed byany particular scheduling algorithm, except that we want thesystem to be long-term fair to VoIP users. We aggregate thepower resource and the frequency resource over some timeinterval (we assume it to be equal to the average transmissioninterval of VoIP frames). To ensure network fairness withrespect to users, we require scheduling at most one VoIPframe per user during the considered time interval. Furthermore,we assume that all users within the cell coverage area havethe same chance of being served, which is equivalent to therequirement of uniform distribution of served users [1]. Thus,the optimization problem we tackle can be stated as follows.

Given the minimum required cell coverage degree, find themaximum number of VoIP users within the cell coverage areathat can be scheduled in DL during T TTIs such that

a) the total amount of power consumed by all transmissionsduring the entire time interval is at most T · Pmax,

b) the total number of utilized chunks is at most T · NSB ,c) the served users are uniformly drawn from a given distri-

bution.

Requirements a) and b) are further referred to as the aggregatedpower and the aggregated RB constraints, respectively.

B. A Mathematical FormulationTo formulate the problem mathematically, we use the follow-

ing two sets of variables.

xjm ={

1 if a user in bin j is served and uses MCS m,0 otherwise.

sj ={

1 if bin j experiences coverage loss in cell i∗ ,0 otherwise.

We assume one user per bin and uniform user distribution(adapting the model to a non-uniform distribution is straight-forward), i.e., set J can be used to denote the set of users.A complete problem formulation (denoted by VoIPDL-1) ispresented below.∑

j∈J ′

∑m∈M

xjm −→ max (7a)

s. t. sj +∑

m∈Mxjm ≤ 1, j ∈ J ′ (7b)

∑j∈J ′

∑m∈M

ηjm xjm ≤ T · NSB (7c)

∑j∈J ′

∑m∈M

(ρjm nm) xjm ≤ T · Pmax (7d)

∑j∈J ′

sj ≤ α

1 − α

∑j∈J ′

∑m∈M

xjm (7e)

sj−sj+1 ≥∑

m∈M

(x(j+1)m−xjm

), j ∈ J ′ \ {|J ′|} (7f)

xjm ∈ {0, 1}, j ∈ J ′,m ∈ M (7g)

sj ∈ {0, 1}, j ∈ J ′ (7h)

In VoIPDL-1, we use J ′ to denote sequence {1, 2, . . . , |J |}such that J ′ maps onto a random permutation of elements inset J . Set J ′ has been introduced to model random sam-pling of users within a given service area of cell i∗ where


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served users are selected following a given distribution. Con-straints (7b) enforce selection of at most one MCS for eachuser. Constraints (7c) and (7d) are the aggregated RB and powerconstraints, respectively. Constraint (7e) ensures the maximumallowed coverage loss α (0 ≤ α < 1), and constraints (7f)enforce that the served users must be selected sequentially fromset J ′ within the cell coverage area.

Proposition 1: Given the system model described in Sec-tion II, the network capacity is always RB-limited when thetransmit power per chunk is limited by ρmax ≤ P max

NSB .Proof: Observe that after dividing constraint (7d) by ρmax,

ρjmnm

ρmax ≤ ηjm holds for all j ∈ J ′,m ∈ M. In other words,for any modified coefficient in the left-hand side of (7d), thecorresponding coefficient in (7c) is at least as large. Further-more, the right-hand side of (7d) is either equal to or larger thanthat of (7c), which makes constraint (7d) redundant. Hence theconclusion.

The optimal solution to VoIPDL-1 gives the maximum num-ber of VoIP users for a given user distribution snapshot (orpermutation J ′ of J ) that can be scheduled in DL withinthe given time interval. However, to obtain statistically reliableresults to the problem described in Section III-A, it is notsufficient to solve VoIPDL-1 for set J ′ generated only once.One way to address this issue is to apply Monte-Carlo method,i.e., to solve a sequence of VoIPDL-1 problems, one for eachgenerated set J ′, and then find a statistical result, e.g., theaverage optimal capacity over all instances. In Section IV, wepresent an alternative approach where we exploit the resultof Proposition 1 and the observation that the optimal solutionto VoIPDL-1 can be easily constructed in polynomial time ifMCS is known for each user.

IV. A SOLUTION APPROACH BASED ON STATISTICAL

SAMPLING FROM FITTED DISTRIBUTIONS

A. General DescriptionOur solution approach is, instead of generating J ′, to con-

sider the original set J and work with distributions of the η-parameters and values of ρjmj

nmj, where mj is the MCS se-

lected for user j according to the link adaption strategy adoptedin the network. More specifically, given MCS mj for eachuser j, we propose to fit a discrete probability distribution toset η = {ηjmj

− 1, j ∈ J } of retransmissions and a continuousprobability distribution to set ρ = {ρjmj

nmj, j ∈ J }, where

J = {j ∈ J : ρjmjnmj

≤ ρ′}, ρ′ is the 100(1 − α)-thpercentile of set ρ, and α is the maximum allowed cell coverageloss. The main idea is to exploit the Central Limit Theorem1 toderive the properties of the sum of random variables drawn fromthe two distributions and thus to enable statistical estimation ofthe left-hand sides of the aggregated power and RB constraintsfor a certain number of users without generating user distribu-tion snapshots. Further, we describe our link adaption strategyand in the next section we discuss how we deal with each ofthe two resource constraints and put the things together.

Link adaptation strategy. Provided that the network per-formance is typically RB-limited, we adopt a link adaptionalgorithm which consists of the following two steps.a) Select set Mj ⊂ M of MCSs that minimize the number

of chunks for transmitting a VoIP frame to user j, i.e.,

Mj = arg minm∈M

ηjm . (8)

1By the Central Limit Theorem, if the sum of variables has a finite variance,then it will be approximately normally distributed.

b) If |Mj | > 1, select MCS mj from Mj that minimizestransmit power ρjm, i.e.,

mj = arg minm∈Mj

ρjm . (9)

Observe that mj defined by (9) minimizes the left-hand side ofconstraint (7c) and thus ensures that the solution obtained bya simple greedy packing heuristic is optimal if the power re-source remains underutilized. Note, however, that the presentedlink adaption strategy is not always optimal from the powerconsumption point of view due to the round-up operator in (5).In fact, it is typically optimal for users with a bad channelquality because the minimum power ρjm grows exponentiallywith the MCS order resulting in many chunks for transmittingthe entire VoIP frame at the required quality. For users with agood channel quality, higher-order MCSs are typically selected,which implies increasing target SINRs and thus higher transmitpower levels. On the other hand, these users consume a smalleramount of cell power in total compared to those that suffer frombad channel conditions. Therefore, our link adaption strategyallows us to get near-optimal VoIP capacity solutions.

B. An Algorithm for Solving the Problem for Fixed ρmax

1) Aggregated RB Constraint: Parameters ηjmjmodel the

minimum number of chunks required to transmit a VoIP frameto user j with modulation mj . To approximate the probabilitydistribution for the parameters, we fit the negative binomialdistribution2 NegBin(r, p) to set η, i.e., find real-valued pa-rameters r and p. Once we know parameters r and p of thefitted negative binomial distribution, we can compute, withoutgenerating different user distribution snapshots, the left-handside of the aggregated RB constraint that occurs with some prob-ability P for N users. We exploit the fact that the probabilitydistribution of the minimum total number of chunks needed toserve N users decreased by N is given by NegBin(N ·r, p) andcompute the minimum total number of chunks (with probabilityP ) for N random users as N + NegBinCDF−1(P,N ·r, p),where NegBinCDF−1(·) is the inverse of the negative bino-mial CDF. Note that negative binomial distribution is typicallyused for over-dispersed data sets, i.e., when variance exceedsthe sample mean. Otherwise, Poisson distribution can be usedinstead.

2) Aggregated Power Constraint: To model probability dis-tribution of real-valued coefficients in the aggregated powerconstraint, we fit gamma distribution Γ(k, θ) to set ρ, i.e.,find a shape parameter k and a scale parameter θ. By theCentral Limit Theorem, the probability distribution of theminimum total amount of power consumed by all transmissionsto N users is given by Γ(N ·k, θ), and the minimum totalpower consumption for N users is equal with probability Pto GammaCDF−1(P,N ·k, θ), where GammaCDF−1(·) isthe inverse of the gamma CDF.

3) The Maximum Number of Served Users and the CellCapacity: With the fitted distributions NegBin(N ·r, p) (orPoiss(N ·λ)) and Γ(N ·k, θ), our goal is to maximize N , i.e.,the number of users that can be scheduled in DL such that theaggregated power and RB constraints are met. A mathematical

2The negative binomial distribution is a discrete probability distribution thatgives a probability of the number of failures before a given number of successesis achieved in a series of independent and identically distributed Bernoulli trials.


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formulation (denoted by VoIPDL-2) of this optimization prob-lem is presented below.

N −→ max (10a)

N + NegBinCDF−1(P,N ·r, p) ≤ T · NSB (10b)

GammaCDF−1(P,N ·k, θ) ≤ T · Pmax (10c)

N ∈ Z+ (10d)

To solve VoIPDL-2, we first find separately the maximumnumber N1 of users that constraint (10b) can admit, and then themaximum number N2 of users that the power constraint (10c)can admit. This approach is justified by the fact that the left-hand sides of constraints (10b) and (10c) are monotonously non-decreasing functions of N . N1 and N2 can be found numerically(e.g., by a simple binary search). Once N1 and N2 are known,the maximum number of users that can be served in cell i∗, isfound as N∗ = min{N1, N2}, and the cell capacity is computedas N∗

v , where v is VoIP activity factor (a relative part of aconversation session during which one user is talking).

Note that neither VoIPDL-1 nor VoIPDL-2 takes into accountSilence Descriptor (SID) frames that are sent during silenceperiods of discontinuous transmission (DTX). Usually, SIDframes are sent periodically (e.g., once every 160 ms) suchthat the receiver could recreate the sender’s background noisein order to avoid a “dead channel” that causes an unnaturalsounding audio signal. Due to relatively small frame sizes [6]and larger transmission intervals (in [6], for example, SIDframes are sent eight times less frequent than VoIP frames),power and RB consumption by SID frames is not significantand therefore usually ignored in capacity studies. It is, however,possible to account for them in VoIPDL-2 by either increasingthe amount of transmitted data to each user or applying the samedistribution fitting approach as for VoIP frames and changingcorrespondingly the left-hand sides of (10b) and (10c).

C. An Algorithm for Solving the Problem for Variable ρmax

So far, we have only been considering a scenario withfixed ρmax. One might also want to optimize this parameterwithin some given range. We solve this optimization problemnumerically by a binary search, where the algorithm describedin Section IV-B is used to find N1(ρmax) and N2(ρmax) foreach ρmax. Our algorithm is based on the observation thatthe maximum number of users admitted by the aggregatedRB constraint is a monotonously non-decreasing function ofρmax, whilst the maximum number of users admitted by theaggregated power constraint is a monotonously non-increasingfunction of ρmax.

The effect of changing ρmax is two-fold. First, the numberof retransmissions decreases with ρmax (see equation (5)),which, however, does not apply to the power resource constraintbecause of chase combining. Second, increasing ρmax may leadto changing MCS mj for some users (see the definition ofmj), i.e., it may become optimal to use a higher-order MCS,which leads to a smaller nmj

and larger ρjmj. Furthermore,

because ρjm increases faster than nm decreases, the total powerconsumption for user j is likely to also increase, resulting insmaller N2.

We define resource utilization ratio µ = N1(ρmax)

N2(ρmax) . The ratiois greater than one when the limiting resource is power, smallerthan one when the number of available chunks is limiting thecell capacity, and equals one when utilization of both resources

is balanced. Observe that if the equilibrium point exists, i.e.,µ = 1.0 for some value of ρmax within a given range, thenthe maximum cell capacity is achieved at this point. Otherwise,the maximum cell capacity with respect to ρmax is achieved ateither the lower or the upper bound of the given range.

V. NUMERICAL EXPERIMENTS

A. Test Network

We have experimented with networks having regular hexag-onal layout, 19 sites with three sectors per site, and regulargeographically defined cell coverage areas. Each site is equippedwith three directional antennas installed 15 m above the averagerooftop level and having azimuth of 0◦, 120◦ and 240◦. Thehorizontal pattern of antennas is specified in [1]. Three inter-site distances (ISDs) have been considered: 500 m, 1000 m,and 1732 m, corresponding to cell radii of 333.3 m, 666.7 m,and 1154.7 m, respectively. Radio propagation has been mod-eled by a distance-dependent received signal model with addedlog-normal shadowing having standard deviation of 8 dB [1]limited by ±12 dB (shadowing gain/loss values outside thisinterval are zero). Assuming an LTE-like parameter setting,three modulation schemes (QPSK, 16QAM, 64QAM) and 19coding schemes have been considered (31 MCSs in total). Otherparameters are specified in Table I. SID frame transmissions andsession initiation and session description protocol (SIP/SDP)messages were not considered.

Figure 1 demonstrates the accuracy of fitted distributions for100 randomly sampled users and two values of ρmax, or tobe more exact, for ratio ρmax

ρ equal 1.0 and 1.5. (The ratio isfurther referred to as the link power ratio, or LPR.) The plottedCDFs are for a network with ISD=500 m, frequency reuse (FR)of one, SISO antenna systems, and maximum coverage lossα = 0.1. Note that the CDF representing the power resource isfor the scaled aggregated power constraint, i.e., both sides ofthe constraint were divided by ρmax. In this case, the right-handside of the power constraint is 500/LPR, whilst the right-handside of the RB constraint remains 500. With this, as can be

TABLE I

PARAMETER SETTING

Frequency 2000 MHzBandwidth B 5 MHzBandwidth utilization factor k 0.9Sub-carrier spacing ∆f 15 kHzNumber of sub-carriers per RB nf 12Frequency reuse (FR) factor 1 and 3

TTI 1 msTime interval 20 ms (T = 20)OFDM symbols per TTI 7 × 2 (short CP)

Distance-dependent path loss [1] 128.1 + 37.6 · log10(d[km])Bin size 10 m × 10 mInter-site distance (ISD) 500 m, 1000 m, 1732 mShadowing Log-normal, 8 dB stand. deviationAntenna type directional; SISO, 2×2 MIMOAntenna gain 16 dBi gainNoise power νj −120 dBmMax. cell transmit power P max 43 dBmBlock error rate (BLER) 0.1

Codec rate 12.2 kbpsIP packet payload bpayload 32 BytesPHY overhead ϕ 32 % of symbols, +5 % (2×2 MIMO)L1/L2 overhead bL1L2 40 bits per VoIP frameModulation rates RM

m 2 (QPSK), 4 (16QAM), 6 (64QAM)VoIP activity factor v 0.5


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100 200 300 400 500 600 7000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Left−hand side of the aggregated constraints

CD

F

RB: Empirical CDF, LPR=1.0RB: NegBinCDF, LPR=1.0Power: Empirical CDF, LPR=1.0Power: Gamma CDF, LPR=1.0RB: Empirical CDF, LPR=1.5RB: NegBinCDF, LPR=1.5Power: Empirical CDF, LPR=1.5Power: Gamma CDF, LPR=1.5

Fig. 1. Empirical and fitted CDFs for ISD=500 m, α = 0.1, FR=1, SISO.

seen in the figure, the number of available RBs is the limitingfactor when LPR = 1.0, and the available resources allow forscheduling 100 VoIP users in DL if we admit P ≤ 38 %.For LPR = 1.5, the limiting factor is power and the availablecell resources admit 100 users with P ≤ 83 %. The DL VoIPcapacity is thus larger when LPR = 1.5.

B. Numerical Results

Applying the methodology described in Section IV, we firststudy how each of the two resources is utilized depending onLPR. We experiment with LPRs in the interval [1, 6] with astep of 0.1. Note that equal power allocation among sub-bandswith fully utilized cell power resource, a strategy commonlyassumed so far for LTE networks, is a special case when LPRequals one. For each value of LPR, we applied the approachdescribed in Section IV-B to find the maximum number of usersthat the aggregated power and RB constraints admit separately.Probability P = 0.95 has been assumed. Figures 2(a)-2(d) showthe obtained results for SISO, and Figures 3(a)-3(d) demonstrateresults for networks with 2 × 2 MIMO.

Figures 2(a) and 2(b) show the results obtained for thethree test networks for FR = 1 and two different α-values.A circle marker denotes the equilibrium point where the twocurves intersect and where the maximum capacity in cell i∗ isachieved. (Note, however, that the y-axis shows the number ofusers scheduled in DL but not the cell capacity.) Comparingthe number of scheduled users at the equilibrium point to thenumber of scheduled users when LPR is one, we observe animprovement of more than 10 %. Note also that for each α, themaximum number of scheduled users occurs at approximatelythe same LPR. The trade-off between coverage and capacity canbe observed when comparing the two figures — the maximumnumber of users drops by more than 30 % when α decreasesfrom 0.1 to 0.05, which corresponds to 90 % and 95 % coveragedegree, respectively.

Figures 2(c) and 2(d) show results when FR = 3, i.e.,when the total number of available chunks is one third ofthat when FR = 1, the set of interfering cells consists ofonly those with antennas having zero azimuth, and the averagetransmit power ρ per sub-band in interfering cells is threetimes of that for FR = 1. The maximum number of users inboth figures is achieved at a higher LPR compared to whenFR = 1. This is because the network performance becomes lesslimited by interference and is mostly limited by bad channel

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RB constraint (500m)Power constraint (500m)RB constraint (1000m)Power constraint (1000m)RB constraint (1732m)Power constraint (1732m)

(a) α = 0.1, fr=1

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Fig. 2. Resource utilization and the equilibrium points, SISO.

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Fig. 3. Resource utilization and the equilibrium points, 2×2 MIMO.

quality experienced by a few users. Optimizing LPR gives animprovement over 30 % when FR = 3.

Comparing the results for SISO and MIMO when FR = 1, weobserve that the number of scheduled users in DL increases byapproximately 70 % and 80 % for ISD=500 m and higher ISDs,respectively. For FR = 3, DL VoIP capacity is strongly RB-limited and at some ρmax value the maximum number of usersachieves its maximum (166 users, i.e., one chunk is sufficientfor transmitting a VoIP frame to any user such that the requiredcell coverage degree is achieved).

So far, we have only considered probability P fixed at 0.95.Next, we study the relation between the maximum DL VoIPcapacity (taking into account VoIP activity factor v) and prob-ability P which varies in the range [0.8, 1]. Figures 4 and 5present numerical results for SISO and MIMO, respectively.

For SISO and FR = 1, Figures 4(a) and 4(b) show theresults for LPR = 1.0 and optimized LPR, respectively. Wemake the following observations from the two figures. First,the maximum capacity decreases with probability P , but the


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decrease is moderate unless the degree of certainty is very high(e.g., P is above 98 %). Thus, for ISD=1000 m and maximumcoverage loss α = 0.1, DL VoIP capacity of 200 users withP = 80 % and 175 users with P = 98 %. However, with 99.9 %certainty no more than 145 users can be guaranteed. Also, wenote that in both figures the difference between the maximumcapacity for ISD=1000 m and ISD=1732 m is relatively smallcompared to that for ISD=500 m. The maximum capacity gainachieved by optimizing LPR when FR = 1 is about 12–18 %for various P (compare Figures 4(a) and 4(b)).

Figures 4(c) and 4(d) illustrate the relationship between themaximum capacity and probability P for the networks withFR = 3. First, for a given maximum coverage loss α andprobability P , in most cases we observe a smaller differencein the maximum capacity among the three ISDs than whenFR = 1. Second, the effect of optimizing LPR is significantlylarger when FR = 3. Third, for the same ISD and the same α,the maximum capacity is in average 20 % and 50 % higher for afixed and optimized LPR, respectively, when FR = 3 than thatwhen FR = 1.

Next, we compare the results obtained for MIMO to those wehave analyzed for SISO. First, similar to the SISO scenarios, DLVoIP capacity is more robust with respect to probability P whenFR = 3. Note also that when FR = 3 and LPR is optimized, DLVoIP capacity is at its maximum (332 users) for all values ofP and is therefore not shown in a figure. Second, for FR = 1,similar to SISO, the capacity is the highest for ISD=1000 m.With FR = 3, the capacity increases with ISD, although itis almost the same for ISD=1000 m and ISD=1732 m. Third,introducing MIMO increases the capacity by 70–80 % whenFR = 1, both for non-optimized and optimized LPR. However,the increase is much smaller (25–30 %) when FR = 3 and LPRis non-optimized, and it is even smaller, due to RB limitation,when LPR is optimized. The latter demonstrates inefficiency ofstatically applying the frequency reuse of three strategy with5 MHz bandwidth and 2×2 MIMO antennas.

Disclaimer: The presented results are reasonable only forrelative comparisons, but are optimistic in absolute values dueto omitting many system- and link-level details.

VI. CONCLUSIONS

We have presented an approach for statistical estimation ofDL VoIP capacity in OFDM-based FDD networks and somenumerical results for test networks with LTE-like parametersettings. The approach can also be applied on realistic data setsand adopted for networks with an irregular topology. For thestudied networks, we observed that interference is the factor thatsignificantly limits the network capacity, which results in lowcapacity for any fixed link power ratio and small capacity gainachieved by optimizing the link power ratio in the networkswith frequency reuse of one. In the networks with frequencyreuse of three, the reduced amount of interference allows fora large capacity improvement (up to 30 %) by means of powercontrol. With MIMO antennas and 5 MHz bandwidth, frequencyreuse of three may limit the gain from power adjustment dueto a smaller frequency range per cell. Remember also that foraccurate results, more accurate modeling and detailed studiessupported by network simulations are needed.

VII. ACKNOWLEDGMENT

The author wishes to thank Anders Furuskar, Fredrik Persson,and Stefan Wanstedt from Ericsson Research, Sweden, for their

0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 150

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(a) fr=1, ρmax = P max

NSB

0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 150

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(b) fr=1, optimized ρmax

0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 150

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0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 150

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(d) fr=3, optimized ρmax

Fig. 4. DL VoIP capacity versus probability P , SISO.

0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 150

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0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 150

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(c) fr=3, ρmax = P max

NSB

Fig. 5. DL VoIP capacity versus probability P , 2×2 MIMO.

valuable comments and practical suggestions.

REFERENCES

[1] 3GPP TR 25.814, v7.1.0, Physical layer aspects for evolved UniversalTerrestrial Radio Access (UTRA), Sep. 2006.

[2] 3GPP TS 36.211, v1.2.0, Physical Channels and Modulation (Release 8),June 2007.

[3] E. Dahlman, H. Ekstrom, A. Furuskar, Y. Jading, J. Karlsson, M.Lundevall, and S. Parkvall. The 3G Long-Term Evolution — RadioInterface Concepts and Performance Evaluation. In Proc. of the 63rd IEEEVehicular Technology Conference, pp. 137–141, May 2006.

[4] E. Dahlman, S. Parkvall, J. Skold, and P. Beming. 3G Evolution: HSPAand LTE for Mobile Broadband. Academic Press, 2007.

[5] G. J. Foschini and M. J. Gans, On limits of wireless communications ina fading environment when using multiple antennas, Wireless PersonalCommunications, 6(3), pp. 311-335, March 1998.

[6] F. Persson. Voice over IP Realized for the 3GPP Long Term Evolution.In Proc. of the 66th IEEE Vehicular Technology Conference, Sep. 2007.

[7] E. Telatar. Capacity of Multi-Antenna Gaussian Channels. In EuropeanTransactions on Telecommunications, 10(6), pp. 585–596, Nov. 1999.


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