throughput estimation of downlink packet access systems based on a point mass approximation concept

3356 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 59, NO. 7, SEPTEMBER 2010

Throughput Estimation of Downlink PacketAccess Systems Based on a Point Mass

Approximation ConceptJunsu Kim, Member, IEEE, Sung Ho Moon, Member, IEEE, and Dan Keun Sung, Senior Member, IEEE

Abstract—Although the system throughput of a wireless com-munication system considering multiple users’ geographical dis-tribution on a 2-D plane and various traffic characteristics is animportant performance metric, its mathematical analysis is verydifficult due to the nonindependent identically distributed distrib-ution of the received signal-to-noise ratio (SNR) of each mobile ter-minal and the complex dynamics of the traffic patterns. Therefore,complicated system-level simulations have been the only approachto evaluate the system throughput of downlink packet-access sys-tems. In this paper, we propose an efficient analytical approach toestimate the system throughput of downlink packet-access systemsby using point mass approximation, which is widely used in physicsand mechanics. The proposed analysis framework enables themathematical analysis of the system throughput performance byconsidering various positions of mobile terminals and traffic pat-terns without the complicated and heavy system-level simulations.Although the proposed approach has a tradeoff relation betweencomputation complexity and accuracy, we show that it is possibleto obtain an effective accuracy with moderate complexity using theproposed analytical approach.

Index Terms—High-speed downlink packet-access (HSDPA),packet-access system, point mass, scheduling, system throughput,throughput analysis.

I. INTRODUCTION

P ERFORMANCE analysis of a wireless communicationsystem is very important to predict problems and enhance

the system. Generally, two-step analysis procedures are used toevaluate the performance of a wireless communication system.The first step is the link-level analysis, which evaluates thephysical-layer performance, including coding and modulation,of a single link between one transmitter and one receiver.The results of the link-level analysis are the bit error rate,

Manuscript received August 12, 2009; revised January 25, 2010 andMay 8, 2010; accepted June 15, 2010. Date of publication July 26, 2010;date of current version September 17, 2010. This work was supported in partby the National Research Foundation of Korea Grant funded by the KoreanGovernment under Grant NRF-2009-352-D00121 and in part by the Ministryof Knowledge Economy under the Information Technology Research Centersupport program supervised by the National IT Industry Promotion Agency(NIPA) under Grant NIPA-2010-(C1090-1011-0011). The review of this paperwas coordinated by Dr. P. Lin.

J. Kim is with the Department of Electrical and Computer Engineering,University of British Columbia, Vancouver, BC V6T 1Z4, Canada (e-mail:[email protected]).

S. H. Moon is with LG Electronics, Anyang 431-749, Korea (e-mail:[email protected]).

D. K. Sung is with the Department of Electrical Engineering, Korea Ad-vanced Institute of Science and Technology, Daejeon 305-701, Korea (e-mail:[email protected]).

Digital Object Identifier 10.1109/TVT.2010.2056942

the block error rate, or the frame error rate (FER) along thesignal-to-noise ratio (SNR). Since the link-level analysis onlyfocuses on a single air link, any system-level features, includingresource allocation, channel activity of a traffic pattern, anduser distribution, are not considered. The second step is asystem-level analysis. It considers multiple transmitters andmultiple receivers, and each air link between the transmitter andthe receiver is modeled by the link-level analysis results. Thepurpose of the system-level analysis is to evaluate the overallperformance of a wireless communication system in variousscenarios in terms of system throughput, packet delay, delayvariation, and so on.

The link-level analysis can be performed using simulationand mathematical analysis for some modulation and codingschemes under additive white Gaussian noise (AWGN) orRayleigh fading channel environments. Moreover, there is alsoan open source library, which is called a coded modulationlibrary, to simulate various combinations of modulation andcoding schemes [1]. If we obtain the link-level performanceunder the AWGN channel, then we can also easily obtain thelink-level results under various fading channel environmentsusing effective signal-to-interference-plus-noise ratio mapping[2]. Therefore, the link-level performance analysis is no longera bottleneck for the evaluation of the wireless communicationsystem.

Unlike the link-level performance analysis, the system-levelanalysis for a general case requires intensive simulation work.If we consider that multiple users are randomly distributed ona 2-D plane, then the distribution of each user’s received SNRbecomes nonindependent identically distributed (non-i.i.d.) dueto the different path loss. The mathematical analysis of wirelessscheduling in a non-i.i.d. environment is extremely complexand impossible for many cases in which we are interested.Therefore, most analytical approaches for throughput analysisassume an i.i.d. environment [3]–[5]. On the other hand, asystem-level simulation approach has been a practical methodto obtain the exact performance of packet-access systems in var-ious environments [6]. However, the development of a system-level simulator (SLS) is also complex, and a large amount ofcomputing power is required to achieve accurate and reliablesimulation results.

The objective of this paper is to propose a practicalapproach to analyze the system throughput of wireless com-munication systems. We consider downlink packet-access sys-tems such as high-speed downlink packet access (HSDPA)of Third-Generation Partnership Project (3GPP) [7] and

0018-9545/$26.00 © 2010 IEEE

KIM et al.: ESTIMATION OF DOWNLINK PACKET ACCESS SYSTEMS BASED ON A POINT MASS CONCEPT 3357

1xEV-DO (HDR) of 3GPP2 [8]. Our proposed analytical ap-proach is based on the concept of point mass approximation,which is used in physics and mechanics, to alleviate the com-plexity of geographical randomness of multiple users whilemaintaining the characteristics of downlink packet-access sys-tems. We will show that it is possible to analyze the systemthroughput with low complexity by using our proposed analyt-ical approach.

The rest of this paper is organized as follows: InSection II, we introduce the concept of point mass approx-imation into the downlink packet-access system to analyzethe system throughput. Then, we analyze the throughput ofthe downlink packet access system using the point mass ap-proximation in Section III. To evaluate the performance ofour proposed analysis framework, we compared the analyticalthroughput with the simulation-based throughput of HSDPAin Section IV. The computational complexity and applicabilityof the proposed scheme are discussed in Section V. Finally,conclusions and future work are drawn in Section VI.

II. ANALYSIS FRAMEWORK USING

POINT MASS APPROXIMATION

To analyze the system throughput, it is required to con-sider a situation in which there are multiple users with ran-dom locations and traffic sources. In particular, multiple usersand random locations significantly increase the computationalcomplexity in the analysis of schedulers [3]. As an alterna-tive approach, we propose an analysis framework based onpoint mass approximation. Point mass approximation, whichis widely used in physics and mechanics, approximates thecontinuously distributed mass in a certain surface to a pointmass that represents a lumped mass located at the center ofmass. As the number of point masses increases, the approxi-mation becomes more accurate although the complexity alsoincreases.

Multiple users randomly distributed in a cell area can be seenas a mass distributed in a certain material. Since each user hasits own traffic volume, as more users are located close together,more traffic to be handled is generated in a certain area. There-fore, mobile users in wireless communication systems exhibitsimilar characteristics to the mass in the physical field. In thepoint mass approximation, a continuously distributed mass isdivided into several point masses. Several point masses areinterpreted as several groups of users. If a group of users ismapped into a mass, then the average distance of the group ofusers in the group from a base station (BS) corresponds to thecenter of mass. Service for the traffic of a certain user is mainlydetermined by the wireless channel quality between the BS andthe user, which depends on the distance if we only considerpath loss and Rayleigh fading. In this situation, it is reasonableto regard the average distance from the BS to a group of usersas the center of mass in terms of SNR.

In our analysis framework, we divide a cell area into multiplering-shaped regions, which are called rings, and each ring isbounded by two radii (ri−1, ri], where i is the ring number. Agroup of active users in each ring area can be mapped into avirtual user in the corresponding ring. Then, the virtual user

TABLE IMAPPING OF POINT MASS APPROXIMATION INTO

CELLULAR COMMUNICATION SYSTEMS

corresponds to the point mass, and its distance from the BS isthe center of mass. Note that even if we consider a 2-D cellarea, the distance of a virtual user can be a center of mass interms of received SNR because its large-scale component onlydepends on distance. The distance of a virtual user from the BSis the average distance of the actual users in the correspondinggroup from the BS. Then, it is a constant determined bythe geographical distribution of the actual users and the ringboundary (ri−1, ri].

Table I summarizes the mapping of point mass approxima-tion into cellular communication systems. The continuouslydistributed mass in the physical field is associated with ran-domly located users with traffic sources in a 2-D cell. Then,the point mass and the center of mass are interpreted as thevirtual users (rings) and the average distance of the actual usersfrom the BS, respectively. Using the concept of the point massapproximation, the randomly distributed multiple users aregrouped into several virtual users with deterministic distances.Hence, the proposed analysis framework approximates the 2-Dcell with multiple users into the 1-D model with a smallernumber of virtual users at deterministic locations. Then, thescheduler is modeled to select a virtual user instead of the actualuser. Therefore, the analysis complexity can significantly bereduced regardless of the number of users and the randomnessof the users’ locations. We will describe the procedure forsystem throughput analysis using the proposed framework inthe following section. Note that we assume a max.C/I scheduler[9], which provides maximum achievable system throughputby scheduling a user with the best SNR at each time, in thefollowing analysis.

III. SYSTEM THROUGHPUT ANALYSIS OF DOWNLINK

PACKET-ACCESS SYSTEMS

A. Geometry and Received SNR Analysis

Suppose that there exist Nr virtual users in a cell, where Nr

denotes the number of rings. The boundaries of the ring i are(ri−1, ri], where i = 1, . . . , Nr, and r0 = 0. The deterministicring boundary can be formulated as ri = i(rc/Nr), where rc

denotes the cell radius. If the actual users are uniformly distrib-uted, then the probability density function (pdf) of the distancesfrom the BS to the users can be expressed as fR(r) = (2r/r2

c ),0 < r ≤ rc, where r is the distance from the BS [10]. Theprobability that an actual user is located in the ring i can becalculated as

pi =1

N2r

(2i − 1). (1)


The average number of actual users in the ring i, that is, m̄i,is expressed as

m̄i =Nue

N2r

(2i − 1) (2)

where Nue is the total number of actual users. The averagedistance from the BS to the actual users in the ring i can bederived as

r̄i =23

rc

Nr

i3 − (i − 1)3

i2 − (i − 1)2. (3)

According to our proposed framework, (3) becomes thedistance from the BS to the virtual user i. For other randomdistributions, including the Gaussian distribution, a similargeometry analysis is also possible. If we consider path lossand Rayleigh fading, then the received SNR of the virtualuser i can be derived as zi = (α2Ptx/r̄

ni PN ) = (α2/r̄n

i )ρ0,where n is the path loss exponent, ρ0 = (Ptx/PN ), Ptx isthe transmitter power, PN is the noise power, and α is thefast fading coefficient, which is a Rayleigh-distributed randomvariable. Then, the received SNR of the virtual user i at adistance of r̄i becomes the exponentially distributed randomvariable of which the distribution is expressed as

fZi(z) =

r̄ni

ρ0exp

(− r̄n

i

ρ0z

). (4)

B. System Throughput Analysis

The scheduler in a BS selects a user to serve according to thereported channel-state information (CSI) of the users. However,every user is not an eligible candidate for scheduling sincesome users may not have any traffic to transmit. Therefore, thescheduler should choose a user with the best SNR among thebacklogged users. Backlogging probability of a virtual user isdefined as the probability that there exists data traffic to transmitto the virtual user, and selection probability is the probabilitythat a virtual user is selected by a scheduler.

1) Backlogging Probability of Virtual User i: Since the vir-tual user i can be considered as a single queue, the backloggingprobability is the same as the utilization factor of the queuingsystem [11]. The utilization factor is defined as the ratio of thearrival rate to the service rate of a queue.

Let the traffic generation rate of a single traffic source be gbits per second. Since a single virtual user aggregates the trafficof multiple actual users, the mean traffic arrival rate of the ringi, that is, λi, can be defined as

λi = m̄ig (5)

where m̄i is the average number of actual users in the ring i, asformulated in (2).

Before analyzing the service rate of the virtual user i, wedefine the link capacity of the virtual user i, which is denotedas Clink,i. Link capacity is defined as the maximum limit ofdata transmission rate of a physical air link. Therefore, it isdetermined by the coding and decoding performance of a sys-tem. Generally, downlink packet-access systems have several

Fig. 1. Link capacity for HSDPA in each ring Nr = 10.

combinations of modulation orders and code rates, which arecalled adaptive modulation and coding (AMC) levels. EachAMC level has its own transmission block size and decodingperformance for a given SNR. Therefore, the link capacity isthe maximum value of successful transmission rate among allthe AMC levels for a given SNR. It can be formulated as

Clink,i = arg maxl

Ll {1 − FERl(zi)}TTI

(6)

where Ll is the transmission block size in bits for the AMClevel l, FERl(zi) is the FER of the AMC level l for a givenSNR value zi, and TTI is the transmit time interval in seconds.The transmission block size and the transmit time interval aregiven in the specification of each system. The FER can beobtained by a link-level simulation for each AMC level. Fig. 1shows the link capacity of the HSDPA system at each ring forvarying ρ0(= Ptx/PN ) values when there are ten rings in a cell,i.e., Nr = 10.

The service rate is defined as the dequeuing rate whenthe queue is not empty [11]. A virtual user can be servedwith channel service rate when the virtual user is selected.Therefore, the service rate of the virtual user i is the same asthe channel service rate scaled by the ratio of scheduled periodover backlogged period by the definition of service rate. Theservice rate of the virtual user i, that is, μi, can be derived as

μi =Psel,i

νiClink,i (7)

for a given selection probability of Psel,i.


Using (5) and (7), we can formulate the backlogging proba-bility as

νi = min(

λi

μi, 1

)

= min(

νi

Psel,i

m̄ig

Clink,i, 1

). (8)

As shown in (8), the backlogging probability formula hasa recursive form. The backlogging probability is a functionof the selection probability, and the selection probability isalso a function of the backlogging probability. Therefore, wehave to obtain the selection probability and the backloggingprobability using an iterative approach from (10) and (8). Amore detailed description of the iterative approach is presentedin the Appendix.

2) Selection Probability of Virtual User i: Let νi denote thebacklogging probability of the virtual user i. If we let a set ofbacklogged virtual users be denoted by Br and i ∈ Br, then theprobability that the virtual user i has the best SNR in Br can beformulated as

Cs(iBr) =

∞∫0

fZi(z)

∏j∈Brj �=i

FZj(z) dz (9)

where fZi(z) and FZj

(z) are the pdf in (4) and the cumulativedistribution function (cdf) of the received SNR of the virtualusers i and j, respectively. The selection probability of thevirtual user i, which is denoted by Psel,i, is derived as

Psel,i =

⎛⎜⎝νi

Nr∏j=1j �=i

(1−νj)

⎞⎟⎠ Cs (iBr ={i})

+Nr∑j=1j �=i

⎛⎜⎜⎝νiνj

Nr∏k=1k �=ik �=j

(1−νk)

⎞⎟⎟⎠ Cs (iBr ={i, j})

+Nr∑j=1j �=i

Nr∑k=1k �=ik �=j

⎛⎜⎜⎜⎜⎝νiνjνk

Nr∏l=1l �=il �=jl �=k

(1−νl)

⎞⎟⎟⎟⎟⎠ Cs (iBr ={i, j, k})

+· · ·+

⎛⎜⎝νi

Nr∏j=1j �=i

νj

⎞⎟⎠ Cs (iBr ={1, 2, . . . , Nr}) (10)

for a given backlogging probability of each virtual user. Eachterm in (10) represents the probability that the virtual user ihas the best channel quality among the backlogged virtual usersin Br. The number of summation terms is dependent on thenumber of rings calculated as

∑Nr−1n=0

(Nr−1

n

). Although a more

accurate analysis is possible as the number of rings increases,the number of summation terms also increases.

3) System Throughput: Once a virtual user i is scheduled,the virtual user i is served with a rate of Clink,i bits/s. Therefore,

TABLE IISIMULATION PARAMETERS

the effective service rate allocated to the virtual user i becomesPsel,i × Clink,i bits/s in the average sense. Then, the entireservice rate is

∑Nr

i=1 Psel,iClink,i. If we assume that the ratecontrol is perfectly performed to meet the required FER, andthe required FER is denoted by β, then the system throughputcan be expressed as

Rsys = (1 − β)Nr∑i=1

Psel,iClink,i. (11)

Generally, the BS selects an appropriate AMC level toachieve an FER lower than the target FER β. Consequently,(11) becomes the achievable lower bound of the throughput.

IV. NUMERICAL EVALUATION

A. Performance Evaluation

We need to compare the analytical results of the proposedframework with the simulation results to evaluate the perfor-mance of the proposed analysis framework. We develop anSLS of the HSDPA system [7]. The SLS consists of a singleBS and multiple users with random locations in a cell [6].The BS has a traffic queue for each user and transmits thedata from the queue to a scheduled user. Each mobile stationcalculates the received SNR value using an SNR mappingfunction and determines the feedback values (channel qualityindicator value, power control command, and ACK/NACK)from the received SNR values and link-level simulation results.We use the same link-level simulation results as those used forthe proposed approximation and in Fig. 1. The traffic modelused in the SLS is file transfer protocol (FTP) traffic [2], [12].We obtained our simulation results for a different number ofusers. For each case, we performed about 100 iterations and500 s of simulation time for each iteration to obtain a steady-state simulation result. Considering a 2 ms radio frame ofHSDPA, we simulated 250 000 frames for each iteration. Otherdetailed parameters for the SLS are shown in Table II.

Fig. 2 compares the throughput of the HSDPA system withthe FTP traffic between the analytical results using the proposedanalysis framework with five rings and simulated results. Notethat σ in the Gaussian case is the standard deviation of the


Fig. 2. Throughput comparison in the HSDPA system with FTP traffic.

distances from the BS to the users. Since more users are locatedin the cell-center area when users are Gaussian distributed, thesystem throughput becomes higher than for the uniformly dis-tributed case. We can observe that the analytical approximationresults are close to the exact values in the figure. Since theproposed analysis framework transforms the actual users withrandom locations to virtual users with fixed locations, there maybe an approximation error. If we increase the number of ringsin the ring queue model, more precise analysis is possible, butthe computation complexity in (10) also increases.

Fig. 3 compares the analytical results with the exact valueswhen we consider a Gaussian user distribution with σ = 0.5. Inthis figure, we show the two analytical results: One is the resultwith five rings, and the other is the result with eight rings. Asshown in this figure, the approximation error becomes smalleras the number of rings increases.

B. Improving the Approximation Accuracy

The proposed analysis framework approximates the actual2-D cell into the 1-D model with a smaller number of virtualusers compared with the number of actual users. Hence, theactual multiuser diversity (MUD) gain achieved by the compe-tition among the actual users is approximated by the MUD gainachieved by the competition among the virtual users (rings).Generally, since Nue > Nr, the MUD gain achieved by theproposed scheme is smaller than the actual MUD gain. Theapproximation error shown in the previous numerical resultspartially comes from this underestimation of the achievableMUD gain. To recover the MUD gain, we take into accountcompetition among the users in the same ring. While the ap-proximation that every user in the same ring has the same SNRdistribution as in (4) may reduce the computational complexity,it is not possible to consider an additional MUD gain achieved

Fig. 3. Throughput comparison in the HSDPA system with FTP traffic for aGaussian distribution with σ = 0.5.

by the competition among the users in the same ring. Instead of(4), we can consider the distribution of the user with the bestSNR in ring i as [13]

fZi(z) =

m̄∗i r̄

ni

ρ0exp

(− r̄n

i

ρ0z

){1 − exp

(− r̄n

i

ρ0z

)}m̄∗i −1

(12)

FZi(z) =

{1 − exp

(− r̄n

i

ρ0z

)}m̄∗i −1

(13)

where m̄∗i = max(1, m̄i), and m̄i is the average number of

users in ring i.Fig. 4 shows the improved result in terms of approxima-

tion accuracy by considering the MUD gain in the ring in auniform user-distribution case. Black circle markers representthe throughput obtained from simulations, star markers withdotted line indicate the case when the MUD gain in the ringis not considered, and triangle markers with dotted line indicatethe case when the MUD gain in the ring is considered. It isobserved that if we consider the competition of users in thesame ring, then it is possible to improve the approximationaccuracy. On the other hand, if we use (12) and (13) instead of(4), the computational complexity increases because (12) and(13) should be calculated for every different Nue, whereas (4)does not need to be calculated because it is independent of Nue.

V. DISCUSSIONS

A. Computational Complexity

While the proposed throughput approximation frameworkadopting a point mass approximation approach may replacecomplex system-level simulations by simplifying a 2-D cell


Fig. 4. Improvement of approximation accuracy by taking into account MUDgain in each ring for HSDPA system with FTP traffic in a uniform distributioncase.

with randomly located multiple users, it still requires somecomputational complexity if we improve the approximationaccuracy. The computational complexity mainly comes fromthe integral in (9) and the selection-probability calculationin (10).

Equation (9) is an integral of fZi(z)

∏j∈Brj �=i

FZj(z), which

is the multiplication of a pdf and multiple cdfs. The num-ber of multiplication is the same as the number of ringswhen all rings are backlogged. For example, if there are fiverings and letting i = 1, then the multiplication term becomesfZ1(z)FZ2(z)FZ3(z)FZ4(z)FZ5(z), which is too complex tohave a closed-form expression for its integral. Thus, we have toobtain the results of (9) using a numerical analysis algorithm.If our proposed model is not used, then the number of multipli-cations of the pdf and cdfs is the same as the number of usersNue. For example, if there are 20 users in a cell, then we haveto calculate the integral of multiplication of the pdf and cdfs for20 users. Its computational complexity increases as the numberof users increases, whereas the computational complexity of theproposed scheme is maintained as a constant once the numberof rings is determined. The proposed approximation frameworkis able to analyze the throughput for a large number of userswith a much smaller number of rings.

Equation (10) induces an additional computational complex-ity. The number of summation terms in (10) exponentiallyincreases according to the number of rings. Since the num-ber of summation terms is

∑Nr−1n=0

(Nr−1

n

), we have 16 and

128 summation terms for Nr = 5 and Nr = 8, respectively.If we numerically calculate the integral function in (9) by

dividing the integration interval into NI subintervals, then thecomputational complexity of (9) is expressed as nNI , where

n denotes the cardinality of Br [14]. Hence, for a given set ofbacklogged queue Br, the number of operations required for Cs

is(Nr−1

n

)(n + 1)NI , and the entire number of operations for

selection probability calculation is∑Nr−1

n=0

(Nr−1

n

)(n + 1)NI .

Since each term in (10) includes Nr times of multiplicationsand one multiplication with Cs, the overall number of opera-tions required to calculate (10) Nop is obtained as

Nop =Nr−1∑n=0

(Nr − 1

n

){Nr + 1 + (n + 1)NI} . (14)

Equation (14) provides the computational complexity ofthe proposed approximation. Nop rapidly increases as Nr in-creases. On the other hand, while Nr generally has a valuesmaller than 10, NI is much greater than Nr to obtain theaccurate integration, which depends on a specific numerical-analysis algorithm. Therefore, the computational complex-ity additionally depends on NI rather than Nr, and wecan relatively achieve accurate approximations with moderatecomplexity.

B. Applicability

Since the point mass approximation provides a flexible andconvenient technique to approximate a certain system withcontinuous or granular distribution of mass or user, it can beapplied for various purposes.

One important performance metric of wireless communica-tion systems is the maximum capacity. This can be achieved bytaking into account full-queue traffic, which assumes that everyuser always has sufficient data to send. Since the backloggingprobability of the virtual user i is always 1, i.e., νi = 1, (7) isfixed once Psel,i is determined. Accordingly, (8) is no longerneeded in the iteration. Hence, the capacity can easily beanalyzed.

The proposed analysis framework can be applied for varioustypes of multiuser-scheduling systems, including the downlinkpacket-access system. Since the point mass approximationconverts a real system with infinite degrees of freedom intoa simple model with finite degrees of freedom, it is obviousthat there is a tradeoff between computational complexity andaccuracy. Hence, for some applications that require highlyaccurate throughput measure, the proposed scheme may notbe suitable. On the other hand, in the case when we needoverall throughput performance without complex system-levelsimulations, the proposed scheme provides quite an accurateestimation with moderate complexity.

VI. CONCLUSION

Since it is very complicated to analyze the system throughputof downlink packet-access systems in randomly distributed userlocations and various traffic source environments, we haveproposed a practical analysis framework. The proposed analysisframework introduces point mass approximation, which ap-proximates the actual users with random locations into vir-tual users with deterministic locations. Although the proposedframework induces approximation error, we showed that the


Fig. 5. Convergence of the selection and backlogging probabilities in theHSDPA system with a Gaussian user distribution with σ = 0.5, Nue = 20,and Nr = 5.

analytical results of the ring queue model are closely approxi-mate to the exact system throughput of the HSDPA system withFTP traffic.

We formulated recursive equations to find the selection andbacklogging probabilities and presented an iterative method tocalculate the probabilities. We showed that the probabilitiesconverge within a few iterations. Finally, the proposed analy-

sis framework enables the numerical analysis of the systemthroughput considering geographical user distribution, trafficsource, and wireless channel simultaneously with low complex-ity and reasonable approximation error.

APPENDIX

ITERATIVE APPROACH

If we consider a discrete-time domain, the current back-logging status of the virtual user i may affect the selectionprobability the next time, and the selection probability of thenext time also changes the backlogging probability. Therefore,we can rewrite the discrete-time version of (8) as

νi[t] = min(

νi[t − 1]Psel,i[t − 1]

m̄ig

Clink,i, 1

)(15)

where t = 1, 2, . . ..It is possible to solve the backlogging probability and the

selection probability using (10) and (15) with an initial condi-tion νi[0] = 1. This initial condition implies that every trafficsource starts to generate traffic simultaneously. With this initialcondition and (10), we can calculate the initial selection prob-ability. Then, the next backlogging probability can be derivedusing (15). This is one iteration cycle. After several iterations,the backlogging and selection probabilities converge to certainvalues. Since a wireless communication system always has asystem throughput in a certain circumstance, the iteration in(15) always converges. Fig. 5 shows an example of convergenceof the selection probability and the backlogging probability ofthe HSDPA system with FTP traffic.

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Junsu Kim (S’04–A’09–M’10) received the B.S.,M.S., and Ph.D. degrees in electric engineering andcomputer science from the Korea Advanced Insti-tute of Science and Technology (KAIST), Daejeon,Korea, in 2001, 2003, and 2009, respectively.

From March 2009 to August 2009, he was a Post-doctoral Research Fellow sponsored by Brain Korea21 with KAIST. Since October 2009, he has beenwith the University of British Columbia, Vancouver,BC, Canada, where he is currently a PostdoctoralResearch Fellow. His research interests include radio

resource management, wireless scheduling algorithms, cognitive radio systems,and cooperative diversity techniques.

Sung Ho Moon (S’01–M’07) received the B.S.,M.S., and Ph.D. degrees in electrical engineeringfrom the Korea Advanced Institute of Science andTechnology (KAIST), Daejeon, Korea, in 1999,2001, and 2006, respectively.

From March 1999 to August 2006, he was aTeaching and Research Assistant with the De-partment of Electrical Engineering and ComputerScience, KAIST. He researched multiple-input–multiple-output (MIMO) and hybrid automatic re-peat request (HARQ) schemes as a visiting scholar

with Stanford University, Stanford, CA, from 2006 to 2007. Since 2007, hehas been a Senior Research Engineer with Mobile Communication TechnologyResearch Laboratory, LG Electronic, Anyang, Korea, where he worked forIEEE 802.16m, WiMAX, and Third-Generation Partnership Project–Long-Term Evaluation-A standards and development. His research interests includefrequency and code-hopping systems for packet data transmission, wirelessscheduling algorithms, link and system-level simulations for third- and fourth-generation wireless communication systems, synchronization channel, relay,carrier aggregation, HARQ, and constellation rearrangement in MIMO.

Dan Keun Sung (S’80–M’86–SM’00) received theB.S. degree in electronics engineering from SeoulNational University, Seoul, Korea, in 1975 and theM.S. and Ph.D. degrees in electrical and computerengineering from the University of Texas at Austinin 1982 and 1986, respectively.

Since 1986, he has been with the faculty of theKorea Advanced Institute of Science and Technology(KAIST), Daejeon, Korea, where he is currently aProfessor with the Department of Electrical Engi-neering. From 1996 to 1999, he was the Director

of the Satellite Technology Research Center, KAIST. He was the DivisionEditor of the Journal of Communications and Networks. He has publishedapproximately 450 papers in journals and conferences and has approximately210 patents (or pending patents). His research interests include mobile commu-nication systems and networks, with special interest in resource management,wireless local area networks, wireless personal area networks, high-speednetworks, next-generation internet protocol-based networks, traffic controlin wireless and wireline networks, signaling networks, intelligent networks,performance and reliability of communication systems, and microsatellites.

Dr. Sung is a member of the National Academy of Engineering of Korea. Heis the Editor of the IEEE Communications Magazine. He was the recipient ofthe 1992 National Order of Merits, the Dongbaek Medal, the 1997 ResearchAchievement Award, the 1997 MoMuc Paper Award, the 2000 Academic Ex-cellence Award, the Best Paper Award from the 2000 Asia-Pacific Conferenceon Communications, the 2004 This Month’s Scientist Award from the Ministryof Science and Technology and the Korea Science and Engineering Foundation,the 2005 Paper Award from the Next Generation PC International Conference,and the 2010 Patent Award from the Korea Intellectual Property Office.

throughput estimation of downlink packet access systems based on a point mass approximation concept

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