Analysis of Throughput and Fairness of WCDMA Networks with Downlink Scheduling

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Analysis of Throughput and Fairness of WCDMA Networks with Downlink Scheduling

Hua Fu1 and Dong In Kim2

1Dept. of Electrical and Computer Eng., National University of Singapore

E-mail: [email protected] of Engineering Science, Simon Fraser University

E-mail: [email protected]

Abstract In this paper we develop a theoretical frameworkwhich enables us to analyze the throughput and fairness perfor-mance of various downlink scheduling schemes in WCDMA net-works. For theoretical analysis, the following scheduling schemesare considered: (i) the Round Robin scheme, (ii) the maximumcarrier-to-interference ratio (C/I) scheme and (iii) the propor-tional fair scheme. By deriving a close approximation to the prob-ability density function of C/I, new, exact throughput and fair-ness expressions are formulated for the above scheduling schemes,

by which a rigorous quantitative comparison among them can beperformed. The results verify that throughput and fairness perfor-mance of each scheduling scheme depends largely on mobile userslocation in the active cell. In general, the Round Robin scheme hasthe worst throughput performance among them, while the propor-tional fair scheme provides a better trade-off between throughputand fairness.

I. INTRODUCTION

Recently the wideband code division multiple access

(WCDMA) has been chosen as the basic radio-access tech-

nology for the universal mobile telecommunication systems

(UMTS) [1]. These systems will provide high-speed packet

data services with different quality-of-service (QoS) support

[2], [3] and extend the services provided by the current cellular

networks, such as those based on the European GSM standard

and CDMA-based IS-95 networks. In order to support this high

traffic load a key component of a WCDMA downlink packet

network is the scheduling algorithm that decides which packets

should be transmitted and when. Several scheduling schemes

have been proposed [4][8] to achieve the trade-off betweenthroughput and fairness which are two key measures of the QoS

in the data network system. These scheduling schemes include

the Round Robin scheme, the maximum carrier-to-interference

ratio (C/I) scheme, and the proportional fair scheme.

Although studies in [4][8] demonstrated feasibility of theaforementioned scheduling schemes, they were mainly based

on computer simulations and experimental implementations.

They did not provide a theoretical analysis which is needed

before their results can be generalized. Moreover, even when

the problem of throughput and fairness analysis in downlink

WCDMA systems is best approached experimentally, appre-

ciation of the theoretical analysis will provide insights which

are vital to guiding the simulations and experiments to be per-

formed. The aim of this contribution is to develop a theoretical

framework, leading to such preliminary performance analysis.

The organization of the paper is as follows. In Section II,

the system model and assumptions are introduced. By using

This work was supported in part by the Natural Sciences and EngineeringResearch Council (NSERC) of Canada.

the well-known results in [9] and [10], we demonstrate that the

probability density function (PDF) of the C/I can be well ap-proximated by a lognormallydistributed function. The obtained

PDF of C/I is then applied for analyzing the throughput andfairness of various scheduling schemes in Section III. Section

IV presents numerical results to highlight the dependence of the

throughput and fairness performance on the system parameters.

II . SYSTEM MODEL AND ASSUMPTIONS

The network system model is similar to that in [11] and [12],

which consists of 19 regular hexagonal cells including the ac-

tive cell. No base station sectorization scheme or soft hand-

off is assumed. Both base stations and mobile users use omni-

directional antennas. We assume that there are M mobile usersin each cell. The transmitted powers are assumed to be equal

for all base stations. The channel model includes path loss, log-

normal shadowing and multipath short-term Rayleigh fading.

Following [11, eqn.(19)], [12, eqn.(6.79)] and assuming that

the packets are carried out in time-multiplexingmode and trans-

mitted at the full power available to the mobile user in an one-

by-one transmission fashion [7, Figure 1], the instantaneous

transmission rate Ri for user i (i = 1, 2, , M) in the activecell can be expressed as

Ri =W

i

P0i

(1 )(1 )P0i +18

j=1 Pji

=W

i

(1 )(1 ) + 18j=1 PjiP0i (1)where W is the system bandwidth, i = (Eb/I0)i is the ra-

tio of bit energy-to-interference-plus-noise density for the ithuser, is the fraction of the total base station power devotedto mobile users (1 is devoted to the pilot signal and othercommon channels) and represents the orthogonality factor toaccount for the self-interference 1 ( = 1 represents perfect or-thogonality). Note that since the backgroud noise is generally

well below the total signal power receivedfrom any base station

[11], [12], we have dropped the noise terms in (1). The quantity

Pji (j = 0, 1, , 18) is the total power received by user i fromthe jth base station (j = 0 represents the active cell base sta-tion). Since we assume equal transmitted power from all base

stations, the powers Pji are proportional to the path power at-tenuation due to path loss, lognormal shadowing and multipath

1It is caused by the multipath-delayed signals even with synchronous down-link transmission.

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Rayleigh fading. Thus, the termPjiP0i

in (1) can be expressed as

PjiP0i

=d4ji 10

(ji/10) ji

d40i 10(0i/10) 0i

. (2)

Here, dji is the distance between base stationj and mobile useri. The inverse fourth-law propagation is assumed. For eachi = 1, 2, , M and j = 0, 1, , 18, {ji} is a set of inde-pendent, zero-mean Gaussian random variables with variance

2, so that

10(ji/10)

is set of lognormally distributed ran-

dom variables. The set {ji} due to Rayleigh fading is a set ofindependent, exponentially distributed random variables with

variance 2. For each i and j , dji , ji , and ji are mutuallyindependent. For j = p and i = q, {ji , ji} are independentof {pq, pq}. The channels are identical since the variances2 and 2 are independent of i and j. Note that in practice,the lognormal shadow fading gains for two different base sta-

tions from a mobile user usually have some degree of correla-

tion [12]. However, the case of lognormal shadow fading withinter-channel correlation is more complicated in the throughput

analysis and will only be dealt with in future research.

To further simplify (1), we define the C/I of the ith user as

i =P0iPi

=P0i18j=1 Pji

=10(0i/10) 0i18

j=1

djid0i

410(ji/10) ji

(3)

where Pi =18

j=1 Pji represents the total interference. Us-ing the above definition, the instantaneous transmission rate R igiven in (1) can be rewritten as

Ri =

W

i

(1 )(1 ) + 1i W

i i. (4)

Here, for the purpose of analysis simplicity, we assume the or-

thogonality factor is near 1 (almost perfectly orthogonal).Before we proceed to analyze throughput and fairness perfor-

mance, the definitions of throughput and fairness are in order.

Following [3] and [4], the throughput is defined as the aver-

age data rate which is equal to the number of information bits

divided by the total transmission time. For example, suppose

user i can receive packet frames at the instantaneous data rateRi bits/s, where i = 1, 2, , M, and the corresponding prob-ability of system access for user i being served by the base sta-tion 0 is Qi. In time multiplexing mode, the throughput will begiven by [3], [4]

=Mi=1

Ri Qi bits/s. (5)

Note that in (5), Ri and Qi are deterministic variables. If theyare random variables, we should useE[Ri Qi] to replace Ri Qi,whereE[x] denotes the ensemble average of random variable x.

The definition of the fairness metric follows [13], [14], and

is given by

F =(M

i=1 Qi)2

M(Mi=1 Q

2i )

(6)

The value of this fairness metric ranges from 1/M to 1. Thevalue of 1 corresponds to a totally fair allocation and a totallyunfair allocation has a fairness of 1/M. Note that the fairness

definition given in (6) is slightly different from that given in

[13] and [14, (33)], where a totally fair allocation would give

the same average transmission rates rather than access prob-

abilities to all mobile users. To account for the average data

rate achieved by each user the fairness metric can be defined

as F =[Mi=1 E(RiPi)]2

M{Mi=1[E(RiPi)]2} . However, since definition (6) canprovide a unified metric which can judge the fairness of both

WCDMA high-speed downlink packet access system [4] and

cdma2000-based high data rate system [8], we use definition

(6) as the fairness metric in this paper.

III. THROUGHPUT AND FAIRNESS ANALYSIS

The key to throughput and fairness analysis is to find a way

to evaluate C/I i given in (3), for which we can compute in(5) and F in (6). Therefore, we first derive the PDF ofi.

We recall from (3) that the distribution ofi depends on threestatistically independent sets of random variables. Due to its

complexity the exact PDF of i is usually hard to analyze. Al-though the Monte Carlo simulation can serve a useful engineer-

ing tool for obtaining the histogram ofi [11, Fig. 4], [12, Fig-ure 6.9], it is more desirable to have a mathematical expression,

such as a reasonable approximation, from which the PDF of ican easily, accurately be estimated, and from which the behav-

ior of the throughput as a function of various system parameters

can easily be deduced.

First, denote the random variables Xji = 10(ji/10) ji as

the product of Rayleigh fading and lognormal shadowing fad-

ing, and denote Xi =

18j=1 (dji/d0i)

4 Xji. The C/I in (3)can then be written as

i = X0i/Xi (7)

The exact PDF of Xji exists in the integral expression whichhas no closed-form solution [15, eqn.(2.219]and, thus, presents

computational difficulties. It would be very useful to have an

approximation, and some researchers used a purely lognormal

distribution to approximate Xji and showed that the approx-imation is accurate at higher values of ( > 6 dB) [9],[15]. This follows from the fact that the composite variability

of Xji due to Rayleigh fading and lognormal shadow fading,at these values, is mainly dominated by the lognormal compo-

nent 10(ji/10). Since the experimental data has suggested the

choice of = 8 dB for standard deviation of ji [12], the ap-proximation is justified in our case. This approximation is very

convenient since such a distribution can be completely specified

in terms of the mean and standard deviation. Following [9], the

PDF ofXji is given by

f(Xji) =10/ ln10

22 Xjiexp

(10 log10 Xji )

2

22

(8)

where = 10 log10(22) 2.5 dB and = 2 + 5.572 dB.

Next, we note that the relative distance (dji/d0i) dependsonly on the users position within the active cell. Since the mo-

bile users periodically report their locations to the base stationfor the hard handoff, we can view the users position as known

to the base station. Actually, the user location can be detected

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by the network or mobile user. Release-99 of UMTS has speci-

fied three kinds of mobile positioning methods [1]. Condition-

ing on the ith user position, Xi in (7) will be a linear combi-nation of Xji . It is well known that the distribution of a sumof independent lognormal random variables has no closed-form

solution but can be well approximated by another lognormal

random variable [10]. Using the Fenton-Wilkinson approach

and noting that each component(dji/d0i)4 Xji is lognormally

distributed with logarithmic mean 40 log10 (dji/d0i) dB andstandard deviation dB, the PDF ofXi can be obtained as [15]

f(Xi) =10/ ln10

biXi

2exp

(10log10 Xi i)

2

2b2i

(9)

where i = 10log10

1.12472

102/10

18j=1

djid0i

4 b2i

2

and b2i = 10log10

1 +

(102/101)

18

j=1(dji/d0i)8

[

18

j=1(dji/d0i)4]

2

.

Finally, the distribution ofi in (7), which is the ratio of twoindependent lognormal random variables, can be evaluated as

f(i) =10/ ln10

22i

1

iexp

(10 log10 i i)

2

22i

(10)

where i = i and 2i = 2 + b2i . The correspondingcumulative distribution function (CDF) is given by

(i) = 1 12

erfc

10log10 i i

i

2

(11)

where erfc(x) denotes the complementary error function. ThePDF of i is plotted in Fig. 1 along with the simulation resultwhen the mobile user lies on the vertex of the active cell. The

figure shows that the PDF ofi can reasonably be approximatedby a lognormal distribution given by (9).

A. Throughput and fairness of Round Robin scheme

In the Round Robin scheme, the packet frame is equally di-

vided into M non-overlapping slots. Then the M slots are as-signed one at a time successively to each mobile user. Clearly,

this scheme provides a simple and fair sharing of transmission

time, but sacrifices loss in the throughput because the data rate

is much lower for users having very poor channel conditions.

Since the relative frequency for ith user to transmit is 1/M, thethroughput is given by

1 = E

Mi=1

1

MRi

=

1

M

Mi=1

0

Ri f(i) di. (12)

Substituting (4) and (10) into (12), we obtain, after manipula-

tion and simplification

1 =1

M

Mi=1

W

iexp

ln10

10i +

1

2

ln10

10

22i

. (13)

Using (6) and noting that the probability of system access for

each of users in the active cell is 1/M, the fairness F1 of RoundRobin scheme can be obtained as

F1 =(M

i=1 1/M)2

M(M

i=1 1/M2)

= 1. (14)

Equation (14) states that the Round Robin scheme is a totally

fair allocation scheme.

B. Throughput and fairness of maximum C/I scheme

In the maximum C/I scheme, a packet is transmitted to

the mobile user that experiences the largest C/I at any givenslot. That is, the base station measures i for each user, butonly selects the user m that is currently having the largest C/Im = maxi{i}. We can see that the maximum C/I schemeprovides maximum throughput at the expense of fairness, be-

cause all transmission time can be allocated to a single user

with good channel conditions.

To derive the throughput, first, conditioning on i, the rel-ative frequency for ith user to transmit is the probability thateach ofl (l = 1, 2, , M and l = i) is simultaneously belowi. But since {l} is a set of independent random variables, thisprobability will be the product of the independent probabilities

that each l,l=i < i. Using (11), we have

P[maxl=i

{l} < i] =M

l=1,l=i

1 1

2erfc

10 log10 i l

l

2

.

(15)

Then, by definition given in (5), the throughput for the maxi-

mum C/I scheme can be evaluated as

2 = E

Mi=1

Ri P[maxl=i

{l} < i]

=M

i=1

0

Ri P[maxl=i

{l} < i]f(i) di. (16)

Using (4), (10) and (15) in (16), one finally has

2 =

Mi=1

W

i

10/ ln1022i

0

exp

(10log10 i i)

2

22i

Ml=1,l=i

1 1

2erfc

10 log10 i l

l

2

di (17)

Since the releative frequency for ith user to transmit is givenby

0

P [maxl=i{l} < i] f(i) di, the fairness F2 of themaximum C/I scheme can be obtained as

F2 =

Mi=1

0

P [maxl=i{l} < i] f(i) di2

MM

i=1

0

P [maxl=i{l} < i] f(i) di2 .

(18)

Expressions (17) and (18) can be evaluated numerically. Note

that although the actual evaluation becomes cubersome when

the number of users is large, it still may be more favorable than

a complete simulation of a downlink WCDMA network. Fur-

ther, the numerical evaluation is not a difficult task on a com-

puter nowadays.

C. Throughput and fairness of proportional fair scheme

In the proportional fair scheme, the throughput monitoring

algorithm is included in a way that a user which does not get

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transmitted for too long gets its priority raised. The essential

feature of the scheme is that in each transmission slot the packet

is only transmitted to the user that experiences the largest C/I-to-mean-C/I ratio. In other words, the criterion of packet al-location will be based on the users relative C/I. Therefore,the proportional fair scheme can provide a good compromise

between throughput and fairness among the users.To derive the throughput, denote i and i = i/i as the

mean C/I and the C/I-to-mean-C/I ratio, respectively. Then,the proportional fair scheme is implemented by computing ifor each use, but selecting the user m that is currently havingthe largest C/I-to-mean-C/I ratio m = maxi{i}. The pro-cedure for obtaining the throughput is parallel to that followed

in the case of the maximum C/I scheme. First, using (10) themean C/I i can be evaluated as

i = E[i] = exp

ln10

10i +

1

2

ln10

10

22i

. (19)

Then, the PDF ofi which is a linear transformation of i (i.e.,the product ofi and constant 1/i) can be evaluated as

f(i) =10/ ln10

22i

1

iexp

[10 log10(i i) i]

2

22i

(20)

The corresponding CDF is given by

(i) = 1 12

erfc

10 log10 i i + 10 log10 i

i

2

. (21)

Finally, the relative frequency for user i to transmit is given by

P[maxl=i

{l} < i] = (22)M

l=1,l=i

1 1

2erfc

10log10 i l + 10 log10 l

l

2

.

Therefore, the throughput can be evaluated as

3 = E

Mi=1

Ri P[maxl=i

{l} < i]

=Mi=1

0

Ri P[maxl=i

{l} < i]f(i) di. (23)

Putting (4), (20) and (22) into (23) yields

3 =Mi=1

Wii

10/ ln1022i

0

exp (10 log10 i i)2

22i

Ml=1,l=i

1 1

2erfc

10log10 i l

l

2

di (24)

where i = i 10 log10 i.In the proportional fair scheme the probability of system ac-

cess ofith user is given by0

P [maxl=i{l} < i] f(i) di .Therefore, the fairness F3 can be obtained as

F3 =

Mi=1

0P [maxl=i{l} < i] f(i) di

2

MM

i=1

0P [maxl=i{l} < i] f(i) di

2

.

(25)

IV. CONCLUDING DISCUSSION

This section presents some numerical results to illustrate the

behavior of throughput and fairness as a function of the various

system parameters.

We consider a two-user system which is sufficient to help

illustrate our discussion. Fig. 2 shows two kinds of user dis-tribution, namely, scenario 1 and scenario 2. Points A and C

represent the location of user one. Points B and D represent the

location of user two. The system parameters used in our evalua-

tion are as follows: the system bandwidth W = 3.84 MHz, thevariance 2 = 1, the standard deviation = 8 and the powerfraction = 80%. We also assume both users have the same bitenergy-to-interference-plus-noise ratio, i.e., = i, i = 1, 2.

Figs. 3 and 4 plot the normalized throughput gains against

Eb/I0 for user distribution scenario 1 and scenario 2, respec-tively. We first compute the throughputs 1, 2 and 3 by us-ing equations (13), (17) and (24), respectively. The throughput

gains are then obtained by normalizing each throughput when

using one of the three scheduling schemes for Eb/I0 =5 dB, 6dB and 7 dB with that throughput when using the Round Robin

scheme for Eb/I0 = 5 dB. From these plots several conclusionscan be drawn. First, the Round Robin scheme gives the low-

est throughput of the three schemes. This is expected since in

the Round Robin scheme slot allocations are fixed in time and

implemented in an one-by-one fashion which completely ig-

nores the channel conditions. Therefore, when the user suffers

from deep fading due to lognormal shadowing and Rayleigh

fading, the throughput will decrease tremendously. Second,

the throughput is highly dependent on the users location. For

example, in scenario 1, when both users have the same dis-

tance from the base station, the mean 1 in (10) is equal to2 and the variance 21 is equal to 22 . This will cause the mean

C/I 1 = 2. Therefore, in this case choosing the user withthe maximum C/I is equivalent to choosing the user with thelargest relative C/I. This is why in Fig. 3 the throughput gainof the maximum C/I scheme is equal to that of the propor-tional fair scheme. However, when the two users have differ-

ent distance from the base station, the maximum C/I schemehas higher throughput gain than that of the proportional fair

scheme, as can be seen in Fig. 4 for scenario 2.

In Table 1, we provide the fairness comparison for differ-

ent scheduling schemes. The fairness metric F1, F2 and F3are obtained by using equations (14), (18) and (25), respec-

tively. It can be seen that in scenario 1, all three scheduling

schemes provide a totally fair allocation with a fairness of 1.

This is mainly due to the fact that if two users have equal dis-

tance from the base station 0, it can be shown that the proba-

bilities of system access0

P [maxl=i{l} < i] f(i) di in(18) and

0

P [maxl=i{l} < i] f(i) di in (25) are equal to1/M. Therefore, the fairness expressions F2 in (18) and F3 in(25) can be evaluated to yield the value of 1. As the distance

difference increases (see Fig.2 for scenario 2), the fairness of

maximum C/I scheme goes down faster than that of propor-tional fair scheme, as can be seen in Table 1. This indicates

that the proportional fair scheme can provide better trade-off

between throughput and fairness.

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Viterbi, CDMA/HDR: A bandwidth-efficient high-speed wireless dataservice for nomadic users, IEEE Communications Magazine, vol. 38,pp. 70-77, July 2000.

[4] 3GPP Technical Specification, Physical layer aspects of UTRA highspeed downlink packet access, 3GPP TR 25.848 v4.0.0, Mar. 2001.

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Table 1. Fairness comparison

Scheduling Scenario 1 Scenario 2

Round Robin Scheme 1 1

Maximum C/I Scheme 1 0.5410Proportional Fair Scheme 1 0.9977

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 510

3

102

101

100

101

102

i

probability

density

function

simulation result

theoretical result

Fig. 1. The PDF ofC/I i when the user lies on the vertex of the active cell.

Fig. 2. User distribution scenario in the active cell.

4.5 5 5.5 6 6.5 7 7.5

0.8

1

1.2

1.4

1.6

1.8

2

Eb/I

0(dB)

NormalizedThroughput

Round Robin scheme

Maximum C/I scheme

Proportional fair scheme

Fig. 3. Normalized Throughput versus Eb/I0 for user distribution scenario 1.

4.5 5 5.5 6 6.5 7 7.5

0.8

1

1.2

1.4

1.6

1.8

2

Eb/I

0(dB)

NormalizedThroughput

Round Robin scheme

Maximum C/I scheme

Proportional fair scheme

Fig. 4. Normalized Throughput versus Eb/I0 for user distribution scenario 2.

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Analysis of Throughput and Fairness of WCDMA Networks with Downlink Scheduling