IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 4, APRIL 2013 733

On Multiple Users SchedulingUsing Superposition Coding over Rayleigh Fading Channels

Ammar Zafar, Student Member, IEEE, Mohammad Shaqfeh, Member, IEEE,Mohamed-Slim Alouini, Fellow, IEEE, and Hussein Alnuweiri, Member, IEEE

Abstract—In this letter, numerical results are provided toanalyze the gains of multiple users scheduling via superpositioncoding with successive interference cancellation in comparisonwith the conventional single user scheduling in Rayleigh block-fading broadcast channels. The information-theoretic optimalpower, rate and decoding order allocation for the superpositioncoding scheme are considered and the corresponding histogramfor the optimal number of scheduled users is evaluated. Resultsshow that at optimality there is a high probability that only twoor three users are scheduled per channel transmission block.Numerical results for the gains of multiple users scheduling interms of the long term throughput under hard and proportionalfairness as well as for fixed merit weights for the users arealso provided. These results show that the performance gain ofmultiple users scheduling over single user scheduling increaseswhen the total number of users in the network increases, and itcan exceed 10% for high number of users.

Index Terms—Rayleigh block-fading; broadcast channels; mul-tiple users scheduling; successive interference cancellation.

I. INTRODUCTION

CHANNEL-AWARE opportunistic scheduling is com-monly applied in current wireless communication sys-

tems in order to exploit the fading channels of the users whenthey are at their peak conditions. This is often referred to asmultiuser diversity (MUD) gain [1]. The common schedulingscheme is to select only a single user per channel transmissionblock. Single user scheduling, in its generic form, is basedon selecting the user with the best weighted instantaneouschannel capacity, where the users’ weights control the trade-off between total throughput and fairness among the network’susers [2].

From information-theoretic perspective, single user (SU)selection is optimal in the special case of maximizing the sum(of all users) throughput of the system, where the user with thebest channel quality gets selected [3]. However, for all otheroperating points of the system in which the objective is tomaximize a weighted sum of the users’ rates, SU selection issuboptimal. The optimal scheme in this case is superpositioncoding (SC) with successive interference cancellation (SIC)at the receivers. This is the capacity-achieving scheme for

Manuscript received November 5, 2012. The associate editor coordinatingthe review of this letter and approving it for publication was K. K. Wong.

A. Zafar and M.-S. Alouini are with the Computer, Electrical and Mathe-matical Sciences and Engineering (CEMSE) Division, King Abdullah Univer-sity of Science and Technology (KAUST), Thuwal, Makkah Province, SaudiArabia (e-mail: {ammar.zafar, slim.alouini}@kaust.edu.sa).

M. Shaqfeh and H. Alnuweiri are with the Department of Electricaland Computer Engineering, Texas A&M University at Qatar, C/o QatarFoundation, PO Box 23874, Doha, Qatar (e-mail: {Mohammad.Shaqfeh,Hussein.Alnuweiri}@qatar.tamu.edu).

This paper was made possible by YSREP grant # 2-011-2-002 from theQatar National Research Fund (a member of Qatar Foundation). Furthermore,KAUST funded the efforts of A. Zafar partially and M.-S. Alouini fully. Thestatements made herein are solely the responsibility of the authors.

Digital Object Identifier 10.1109/LCOMM.2013.021213.122465

the degraded broadcast channel [4]. With SC, multiple usersare scheduled simultaneously over the same channel block,and hence the codewords of the different users interfere witheach other. So, this is a non-orthogonal transmission scheme.Therefore, the rates of the individual users become less thanwhen they are scheduled alone due to the interfering code-words of the other users. However, each user gets scheduledmore frequently and hence on the long term the individualand the aggregate (i.e. sum) rates get enhanced given thatthe allocation of the rate, power and decoding order over theusers is optimized. The optimal rate and power allocation forSC with SIC at the receivers was studied in the literaturein [5] and [6]. It is not that all users get scheduled overall channel blocks, but rather the number of scheduled usersvaries dynamically based on the channel conditions of all userssuch that only the users with good channel conditions getselected. The statistics of the optimal number of scheduledusers using SC depend on the users’ channels statistics. Inthis letter, Rayleigh fading channels are considered.

Due to the prospected gains of multiple users schedulingover SU scheduling, there has been some recent work onthe performance of multiple users scheduling via SC withSIC at the receivers and its application in the next genera-tion communication standards such as Long Term Evolution(LTE) systems. Two examples of such works are [7] and[8]. However, in these works, the optimal power and rateallocation for SC as studied in [5], [6] was not taken intoconsideration. Moreover, these works considered the uplink orthe multiple access channel (MAC). The optimized SC schemewas considered in [9]. However, the comparison between SUscheduling and SC was done based on the achievable rateregion of a two-user scenario. This was insightful, but it cannotbe extended into higher number of users exceeding three.Therefore, to compare the two scheduling schemes for highernumber of users, other suitable comparison criteria need to befound, which is done in this letter.

The contribution of this letter is in providing new insightfulnumerical simulations of the optimal SC scheme. In particular,two aspects of the system are considered. The first one isthe statistics of the optimal number of scheduled users overRayleigh fading channels and the statistics of the averagechannel access ratio of each user. The second one is comparingthe long-term average sum rate (i.e. throughput) achieved bythe optimal SC scheme and the conventional SU schedulingscheme. The gain of the optimal SC scheme over the SUscheduling scheme is obtained versus the total number ofusers in the system. To have consistent comparisons, the twoscheduling schemes are compared at three different properlydefined operating points as explained in more detail in Sec-tion III.

1089-7798/13$31.00 © 2013 IEEE

734 IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 4, APRIL 2013

II. PRELIMINARIES

A. System Model

Consider a system with one source node (S) and M usersor destination nodes (D) yielding a broadcast (i.e. downlink)channel. The information sent for each user is independentof the information sent to other users. It is assumed thatall nodes are equipped with a single antenna. The channelsbetween the source and the users are block-fading. Therefore,the time and frequency grid is divided into channel blocks,called resource units (RUs), with a certain time duration andbandwidth. The channel is assumed to be constant over oneRU and changes randomly and independently from one RU toanother. Therefore, within one RU, the channel between thesource and any user is modeled as additive white Gaussiannoise (AWGN) channel. Furthermore, the channels of the usersare assumed to be independent. Without loss of generality,it is assumed that each RU has the same time duration andbandwidth. Constant power, denoted by P̄ , per RU is alsoassumed.

For notation, the index k is used for the RUs. hi[k] is thechannel response between the source and the ith user overthe kth RU. γi[k] = |hi[k]|2P̄

N0denotes the Signal-to-Noise-

Ratio (SNR), where N0 is the additive noise power spectraldensity. Ri[k] and R̄i denote respectively the achievable rate(in bits/sec/Hz) by the ith user in the kth RU and the long-term average (over all RUs) achievable rate (throughput) ofthat user. Furthermore, the channels of all users are assumedto be Rayleigh block-faded.

It is assumed that the source has full channel state infor-mation (CSI) of all channels through feedback. The sourceincorporates the CSI in the scheduling task in order to obtainMUD gains. The scheduling schemes are optimized to be thesolution of the optimization problem

max

M∑i=1

μiR̄i =1

K

M∑i=1

μi

K∑k=1

Ri[k] (1)

where the total number of RUs, i.e. K , approaches infinity andμi is the merit weighting factor assigned to the ith user. Forthe SU scheduling scheme, there is an additional constraintthat only a single user can have a non-zero rate per RU. Thisadded constraint makes SU scheduling suboptimal.

B. Single User Selection

With the SU selection constraint, the solution of (1) yieldsselecting the scheduled user according to

ζ[k] = arg maxi

μiRi[k] i = 1, . . . ,M, (2)

where ζ[k] is the index of the selected user in the kth RU.Therefore, the achievable rate by SU scheduling is given as

Ri[k] =

{log2

(1 + P̄ |hi[k]|2

N0

): i = ζ[k]

0 otherwise.(3)

The average throughput of a user is given by (e.g. [2])

R̄i =

∫ ∞

0

rfRi(r)∏l �=i

FRl

(μir

μl

)dr. (4)

where fR(r) and FR(r) are respectively the probabilitydensity function (PDF) and cumulative distribution function(CDF) of the achievable rate.

The ratio of the number RUs in which a user is scheduledto the total number of RUs, denoted by τi, is given by

τi =

∫ ∞

0

fRi(r)∏l �=i

FRl

(μir

μl

)dr. (5)

C. Superposition Coding with Successive Interference Can-cellation For Decoding

A procedure to optimize the power and rate allocation forthe SC based on a greedy algorithm that involves marginalutility functions for every user was suggested in [6] for thecase of variable power allocation over the channel blocks(i.e. RUs). The straightforward extension to the constant powerper RU case was outlined in [9]. The latter case is summarizedas follows; The marginal utility functions in the kth RU aredefined for every user as

ui(k, z) ≡ μi

1|hi[k]|2 + z

, 0 ≤ z ≤ P̄

N0, (6)

where z refers to the interference levels. The rate (inbits/sec/Hz) and power (in Joul/sec/Hz) allocated to each userdepends on the marginal utility functions according to

Ri[k] =1

ln 2

∫Ai[k]

11

|hi[k]|2 + zdz, Pi[k] = N0

∫Ai[k]

dz,

(7)where the period Ai[k] is defined as

Ai[k] ≡{z ∈

[0,

P̄

N0

]: ui(k, z) > uj(k, z) ∀j �= i

}. (8)

Notice that if Ai[k] is an empty set, then the ith user isnot scheduled in the corresponding RU. The users’ order inthe SIC process is in the order of decreasing μ, meaning thateach user decodes the codewords of users who have higherμ’s before decoding its own codeword.

The long term throughput of the ith user is given by1 [9]

R̄i =

∫ 1

0

∫ ∞

0

11γ + z

fγi(γ) .

∏j �=i

Fγj

([μiγ

μj + μjγz − μiγz

]∗)dγdz,

(9)

where the notation [x]∗ is defined as [10]

[x]∗ =

{∞ : x < 0

x otherwise.(10)

The ratio of the number of RUs in which the ith user isscheduled (channel access ratio) can be found by

τi =

∫ 1

0

∫ ∞

0

fγi(γ)∏j �=i

Fγj

([μiγ

μj + μjγz − μiγz

]∗)dγdz

(11)

1An equivalent expression to the one in [9] is used since here the PDF andCDF of SNR (γ = |h|2P̄ /N0) are required, while in [9] the PDF of |h|2 isused.

ZAFAR et al.: ON MULTIPLE USERS SCHEDULING USING SUPERPOSITION CODING OVER RAYLEIGH FADING CHANNELS 735

III. CRITERIA FOR THE COMPARISONS BETWEEN

SCHEDULING SCHEMES

For the simulation framework, the following model for thesystem is used. It is assumed that the users are uniformlydistributed in a circular region around the base station betweentwo bounds dmin and dmax for the minimum and maximumdistance from the base station. Therefore, the PDF and CDFfor the distance of a user from the base station is given by

fD(d) =2d

d2max − d2min, FD(d) =

d2 − d2min

d2max − d2min(12)

Let’s assume that there are M users in the cell. Thedistances of the users are obtained randomly using (12)and then they are arranged in descending order, denoted byd1, d2, · · · , dM from the maximum to the minimum distance.The expected values of the ordered distances are obtained by

d̄i = i

(M

i

)∫ dmax

dmin

xfD(x)FD(x)M−i (1− FD(x))i−1

dx

(13)From the expected values for the distances, we obtain

the expected values for SNR, arranged in ascending order,according to the model (γ̄i = β (d̄i)

−α), where α is the pathloss exponent and β is a constant related to the transmissionpower and antenna gains. In our simulations, we use α = 3,β = 103, dmin = 1 and dmax = 10. Therefore, the averageSNR ranges between 0dB at dmax and 30dB at dmin.

We compare between SC and SU selection for variablenumber of users. For a given number of users, we generatethe expected SNR values of the users using the describedmodel for uniform distribution in a circular cell. Furthermore,we compare the scheduling schemes for three different andwell-defined optimization functions in order to have a broaderpicture of the gains of SC at different operating points of thesystem. The challenging task is to obtain the users’ weightingfactors (i.e. μ) for every case.

A. Hard Fairness – Equal Rates

In this case, the optimization objective function is

max min(R̄i, i ∈ {1, 2, · · · ,M}) (14)

The optimal solution is achieved when all users have equalthroughput on the long-term (R̄i = R̄j ∀i, j). The problemformulation in (14) is equivalent to (1) with the users’ weightsproperly selected to make the average rates of the users equal.A numerical method is applied to obtain the users’ weightswithin a small tolerance value from the equal rates point, asdescribed in Algorithm 1. δ controls the speed of convergence.Notice that the weights to achieve hard fairness in SU selectionis different from the SC case.

B. Proportional Fairness

In this case, the optimization objective function is

max

(M∏i=1

R̄i

)1/M

(15)

It can be shown that (15) is equivalent to (1) with theweight of each user selected to be the reciprocal of its averagethroughput (μi = 1/R̄i) [11]. Since the average throughputs

Algorithm 1 Weight Update for Equal Rate

Initialize R̄i = 0 ∀ i = 1, · · · ,MInitialize μM = 1for k = 1 to K do

Obtain Ri[k] using (3) for SU or (7) for SC.

R̄i(k + 1) = R̄i(k) +1

k+1

(Ri − R̄i(k)

) ∀ i

μi(k + 1) = μi(k) + δ(R̄M (k + 1)− R̄i(k + 1)

)end forreturn μi(K) ∀ i

2 3 4 5 6 7 8 9 102

2.5

3

3.5

4

4.5

Number of Users (M)S

um o

f Lon

g T

erm

Thr

ough

puts

(bi

ts/s

ec/H

z)

SC−PFSC−Fixed WeightsSU−PFSC−Equal RateSU−Fixed WeightsSU−Equal Rate

Fig. 1. Comparison of SC and SU in term of sum of long term throughputsof all the users.

cannot be known in advance, they are obtained using arecursive algorithm such that the weights and throughputs arecontinuously updated until they converge into their steady statevalues. Such an algorithm was applied in [12].

C. Fixed Pre-determined Users’ Weighting Factors

In this case, we apply (1) directly and use the same users’weights for both SU selection and SC. We selected the weightsarbitrary to be the reciprocal of the achievable rate at theaverage SNR for each user (μi = 1/ log2 (1 + γ̄i)).

IV. NUMERICAL RESULTS AND CONCLUSIONS

Fig. 1 shows the comparison between SU selection and SCin terms of the long term sum throughput plotted versus thetotal number of users in the system for the system modeldescribed in Section III. As expected, for both SC and SUselection, the sum throughput increases as the number of usersincreases due to the MUD gains, and SC achieves higher sumthroughput than SU selection. To quantify the gain of SC, weshow in Fig. 2 the ratio between SC and SU selection in termsof the maximum achievable values for the objective functions(14), (15) and (1), for hard fairness, proportional fairnessand equal weights, respectively using three different networkmodels2. A new learned conclusion is that the gain of SC overSU selection increases as the number of users increases.

2To give a more comprehensive analysis, two additional models to obtainthe average SNR of the users are used for Fig. 2. In the first model, theSNRs of the best and worst channels are similar to the original model whilethe SNRs in between are distributed uniformly between these two limits. Inthe second model, we use our original model with a smaller range betweenminimum and maximum SNR values, which are changed to 5 dB and 25 dB,respectively.

736 IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 4, APRIL 2013

2 3 4 5 6 7 8 9 101

1.05

1.1

1.15

Number of Users (M)

Gai

n

Equal Rate−Different SNR distributionFixed Weights−Different SNR distributionFixed Weights−Smaller SNR rangeProp. FairnessFixed WeightsEqual Rate−Smaller SNR rangeProp. FairnessF−Smaller SNR rangeProp. Fairness−Different SNR distributionEqual Rate

Fig. 2. Gain in performance of SC over SU as a function of number ofusers, M .

This is true for the three different operating points which areinvestigated in this work. Another important conclusion is thatthe gain of SC can exceed 10% for high number of users.Furthermore, the gain of SC, as a function of the numberof users, is to a big extent similar for the three optimizationfunctions although they reflect three different operating pointsin terms of the trade-off between total throughput and fairness.

Fig. 3 shows the channel access ratio for the individual usersin a ten-user scenario. As expected, the channel access ratiowith SC is greater for each user than SU selection. This is dueto the fact that multiple users can be scheduled in the sameRU. Another important observation is that with SC, the userswith worst and best channel conditions are scheduled morefrequently than the remaining users. Notice that this is notthe case with SU selection.

The statistics of the number of scheduled users per RU forSC at proportional fairness are shown in Fig. 4 for numberof users M = 10, M = 6 and M = 3. We found that thestatistics have a very similar pattern for hard fairness as wellas fixed weights case. Therefore, their results are not shown.An important learned conclusion from Fig. 4 is that even if thenumber of users is high, e.g. 10, we have high probabilitythat a limited number of users are scheduled in each RU.We could even have a single user scheduling, but it has lowprobability of occurring. The expected number of scheduledusers using SC is important when we consider the practical im-plementation of the scheme due to the complexity constraints.Therefore, in our opinion, 10% increase in throughput for twoor three layers of superposition coding is a good compromisebetween prospected gains and added complexity. Therefore,we believe that superposition coding has the potential tobe a good option for practical implementation in nextgeneration systems.

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1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

User

τ i

SC−PFSC−Fixed WeightsSC−Equal RateSU−PFSU−Fixed WeightsSU−Equal Rate

Fig. 3. Channel access ratio of each user for a system with 10 users. Theaverage SNR, γ̄, increases with the increase in user index. Therefore, user 10has the highest SNR.

1 2 3 4 5 6 7 8 90

0.2

0.4

0.6

Users selected in one blockfr

eque

ncy

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freq

uenc

y

1 2 30

0.2

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0.6

Users selected in one block

freq

uenc

yFig. 4. Bar plot of number of users selected in one RU for a system with10, 6 and 3 users respectively.

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