TRADITIONALLY, wireless access networks, especially

Transcription

1 766 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 27 Cell-Throughput Analysis of the Proportional Fair Scheduler in the Single-Cell Environent Jin-Ghoo Choi and Saewoong Bah, Meber, IEEE Abstract The fairness concept has been widely studied in the area of data networs. The ost well-nown fairness criterion ax in fairness) gives priority to the iniu-rate session. Kelly questioned its appropriateness in his wors on the bandwidth sharing aong the end-to-end flows and proposed another fairness criterion preferring short-distance flows to enhance the overall throughput, which is called the proportional fairness PF). A siple scheduler achieving this objective was introduced in wireless access networs and revealed that it can achieve a good coproise between cell throughput and user fairness. Although it has received uch attention for soe tie, research on its perforance ainly depended on coputer siulations. In this paper, we analyze the PF scheduler to obtain the cell throughput, which is a priary-perforance etric, and extend the result to analyze the capacity of ultiple-input ultiple-output systes. We evaluate the effect of various paraeters on the throughput of the PF scheduler through the nuerical analysis. Index Ters Cell throughput, opportunistic scheduler, proportional fair scheduler. I. INTRODUCTION TRADITIONALLY, wireless access networs, especially cellular networs, have provided only voice service and were considered as an extension of wire-line telecounication systes. However, as deands for ubiquitous data services arose with the rapid growth of the Internet, wireless access networs and requireents placed upon the are evolving to support various high-data-rate HDR) services, including ultiedia. To accoodate these requireents, third-generation cellular systes such as wideband-code-division ultiple access W-CDMA) and cda2 have standardized enhancedradio access and their core networs, which currently consist of circuit switches and obility-supporting devices, are expected to be replaced by IP-enabled coponents including routers [1]. In wireless data networs, an iportant tool for resource anageent is the pacet scheduler. Several wors have shown that the scheduling policy can significantly affect syste perforances such as throughput, delay, fairness, and loss rate in wire-line and wireless doains [2]. Schedulers in wireless networs need to consider the unique characteristics of tie- Manuscript received April 3, 26. This wor was supported in part by the University IT Research Center Project and in part by the NRL Progra of KISTEP, Korea. The review of this paper was coordinated by Prof. Y.-B. Lin. J.-G. Choi was with the School of Electrical Engineering and INMC, Seoul National University, Seoul, Korea. He is now with Sasung Electronics, Seoul 1-742, Korea e-ail: cj@netlab.snu.ac.r). S. Bah is with the School of Electrical Engineering and INMC, Seoul National University, Seoul , Korea e-ail: sbah@netlab.snu.ac.r). Color versions of one or ore of the figures in this paper are available online at Digital Object Identifier 1.119/TVT varying and location-dependent channel conditions when contrasted with wired networs. Most research in wireless pacet scheduling used the twostate channel odel, where the channel fro the base-station BS) to a user is assued to be in a good or bad state [3], [4]. In the good state, pacet transission always succeeds without error while it fails in the bad state. Therefore, assigning the transission turn to users in the bad state results in throughput degradation vis-à-vis giving the turn to users with the good state. Schedulers are assued to have a reference-service odel describing the service aount that each session should be given in an error-free environent. Soe sessions are, due to yielded turns, behind their reference-service aount, and others receiving the turns first are ahead of it. The forer is said to be lagging, and the latter are leading. The schedulers are distinguished by the adopted reference odel and the copensation ethod for lagging sessions. For exaple, idealized wireless fair queuing IWFQ) uses WFQ as a reference odel and lagging sessions have absolute priority when their channel conditions return to good state. This abrupt service change causes perforance degradation of leading sessions [3]. The reference odel of channel-condition independent FQ C-IFQ) is start-tie FQ, which enables graceful service degradation of the leading sessions [4]. Other schees include the server-based fair approach SBFA) and wireless fair service WFS). The aforeentioned wireless pacet schedulers have the coon wea point of assuing the channel to have only two states. To now the channel state exactly, a BS requires fast feedbac fro obile terinals about ore detailed channel inforation, which increases the signaling burden. For instance, the channel state can be divided into 2 levels by using bits per feedbac. This structure of wireless lin-state has been realized in the IS-856 standard for the downlin direction, which is nown as the HDR syste of Qualco [5]. With the aid of increased signaling overhead, we can develop a better scheduler in ters of the average throughput. Several schedulers for this architecture have been proposed. Borst and Whiting [6] axiize cell throughput under the constraint that the noralized average throughput for each user is equal, where the noralization factor reflects the users required quality of service QoS). The schee in [7] achieves the sae objective with a different constraint, where the allocated tie portion of each user is assued given. This scheduler axiizes the objective function for exaple, based on utility), which depends only on the signal-to-noise ratio SNR). Liu et al. [8] consider various QoS constraints, such as absolute or relative) iniu-utility requireent, and uses /$ IEEE

2 CHOI AND BAHK: CELL-THROUGHPUT ANALYSIS OF THE PROPORTIONAL FAIR SCHEDULER 767 soe adaptive schees with tie-varying paraeters. It has a convergence proble when ipleented in real systes. Several wors dealt with scheduling policy in ultichannel systes [9], and any opportunistic schees that exploit tievarying channels have been presented in the context of ultiple access and ultiple antennas [1], [11]. We consider schedulers exploiting the tie-varying channel condition without explicit constraints. The axiu cell throughput can be obtained by serving the user with the best channel condition; however, it raises a serious fairness proble. Therefore, a pacet scheduler should eet a reasonable tradeoff between throughput and fairness. A frequently used fairness concept is the ax in fairness, which axiizes the iniu rate of a session only subject to the lin-capacity constraint. Kelly questioned the appropriateness of the ax in fair allocation of the bandwidth aong the sessions, proposing proportional fairness PF) [12]. Assuing elastic traffic, he derived a rate-allocation vector axiizing the su of all users satisfaction or utility) that is nown as the PF allocation. In PF, there is a tendency that short-distance sessions have priority over long distance ones when copared with ax in fairness [13]. In wireless access networs, in a related but different context, sessions with good channels are preferred over sessions with bad channels. A scheduler that achieves PF was introduced in [5]. The PF scheduler has low coplexity, so it has received uch attention [11], [14]. Kushner and Whiting [15] investigated the asyptotic behavior or convergence property of the algorith. Borst [16] showed that its perforance can be degraded in a dynaic situation with rando service deands and a variable nuber of users. However, the cell throughput obtainable by the PF scheduling policy has yet to be analyzed. Cell throughput is defined as the su of each user s average throughput and considered to be a priary-perforance etric for a scheduler in a wireless networ. In this paper, we analyze the PF scheduler and derive an approxiate expression for cell throughput. We apply the analysis results to calculate the capacity of ultiple-input ultipleoutput MIMO) systes, which have received significant attention recently. Through siulations, we confir that our analysis achieves accurate results, and we evaluate the various paraeters affecting scheduler perforance using nuerical evaluation of our analytic expression. This paper is organized as follows. In Section II, we introduce the syste and channel odel. Section III analyzes the PF scheduler by adopting the linear and logarithic odels for relating the feasible rate and the SNR. In Section IV, we obtain the approxiate cell capacity for MIMO systes under the PF scheduling policy. Then, we copare the siulation results with nuerical analysis in Section V, followed by concluding rears in Section VI. II. MODEL DESCRIPTION A. Syste and Channel We consider the downlin channel of a single-cell wireless access networ, where a BS serves N obile terinals or users). The downlin structure is very siilar to that of the IS-856 syste. It is a single broadband channel shared by all users in the tie-division-ultiplexing anner. The BS exploits the pilot signal, which is predefined by the protocol, in the specified position of each tie slot, and every obile easures it to obtain the channel gain. Each receiver feed bacs the inforation about the transission rate accurately and quicly. The BS receives the fed bac signal fro all the users to collect the current channel status. Based on the channel inforation, the radio-frequency scheduler selects a user aong the active 1 ones to be served in the next slot. Although various schedulers are proposed in this context, aong the, we concentrate on the PF scheduler. A pacet fro the selected user is transitted with the BS s full power, which is assued fixed for all users. The BS uses only the transission rate control. Integration with power control is outside the scope of this paper. Rate control can be realized by various techniques such as adaptive odulation and/or adaptive channel coding. When the transission power of the BS is P t, the receiving power of obile user is given by P = h 2 P t, where h is the channel gain. The channel gain reflects the effects of various physical phenoena such as scattering and absorption of radio waves, shadowing by terrestrial obstacles, and ultipath propagation. The channel gain fro the BS to user can be written as h = cd α s 1) where c is a constant incorporating the transission and receiving antenna gains, d is the distance fro the BS to user, α is the path-loss exponent estiated to be about 4. in typical urban environents), s is a rando variable for the shadow-fading effect, and represents the phasor su of the ultipath coponents. The shadow-fading effect s is nown to follow the log noral distribution with zero-ean and the variance σs 2 in decibels) in the log-scale. The ultipath-fading effect is odeled as the second-order chi-square or exponential rando variable with a ean of 1., which represents the Rayleigh fading channel. Since we consider the single-cell scenario, there is no intercell interference. Therefore, we can represent the SNR of user as Z = P /P n, where P n is the bacground-noise power, which includes the theral noise and other Gaussian interferences. Catreux et al. [17] defined the edian SNR at the cell edge ρ to represent the noise level of the wireless environent considered. Since ρ = cd α P t /P n, where D is the radius of the cell, we obtain the average SNR of user as Z = ρd/d ) α s. Note that the received-signal level follows the exponential distribution and that the noise power is constant so that the SNR can also be odeled by the exponential rando variable. 1 It eans that its downlin queue is already baclogged.

3 768 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 27 B. PF Scheduler A nuber of previous wors on wireless pacet schedulers have used the two-state channel odel and assued that the transission rate is fixed regardless of the channel state. In the recently proposed IS-856 syste, the channel state has ultiple levels, and it supports various transission rates by adopting lin-adaptation techniques. 2 Considering the scarcity of the wireless bandwidth, cell throughput is an iportant etric to evaluate wireless pacet schedulers. It is trivial to design a scheduler achieving axial cell throughput. In this case, when the channel gain of each user is given, the BS serves user first, whose agnitude of channel gain is larger than that of any other user. However, the absolute preference for users with good channel conditions ay bring about significant unfairness. The PF scheduler stries a reasonable coproise by allocating an equitable portion of tie slots to each user while giving preference to users in the good channel state [14], i.e., achieves PF [11]. Let us exaine the operation of the PF scheduler. The average throughput of each user is traced by an exponential oving average. At the beginning of each tie slot, each user feeds bac the channel state or the feasible rate) to the BS. The BS calculates the ratio of the feasible rate to the average throughput for each user, which is defined as the preference etric and is the ey selection criterion. The user with the axiu preference etric will be selected for transission at the next coing slot. This is described forally as follows. In tie slot n, the feasible rate of user is R [n], and its oving average is denoted by R [n]. Then, user = arg ax R [n]/ R [n] is served in tie slot n, and the average throughput of each user is updated by ) 1 1 t R c [n]+ 1 t c R [n], = R [n +1]= ) 2) 1 1tc R [n], where t c is the tie constant for the oving average. It is clear that the PF scheduler affects relative preference to users with good channels as opposed to absolute preference. In fast-fading environents, the channel state of a user fluctuates around an average level in a rando anner. When the nuber of users is sall and all the users happen to be in the bad state, then throughput in that slot will be low, regardless of whoever is scheduled. On the other hand, when the nuber of users is large, soe users will be in the good state with high probability. Therefore, we can obtain additional cell-throughput gain by scheduling the first to utilize the characteristics of fast-fading channels, which is called ultiuser diversity. Unlie the PF scheduler, the round-robin RR) scheduler decides scheduling order in advance. As each user taes the sae portion of tie slots, it is effort fair. It can adjust the transission rate of the selected user considering its channel 2 We use the ters channel state, channel condition, SNR, and feasible rate interchangeably. state but not the service order. Therefore, it cannot tae advantage of ultiuser diversity, which results in lower throughput than the PF scheduler. Most wireless fair schedulers, such as IWFQ, C-IFQ, SBFA, and WFS, reflect channel condition in selecting a user to be served. However, they usually have very liited inforation on the channel state and, accordingly, do not utilize the wireless channel efficiently. It ay cause a user to experience frequent pacet drops. They defer the service of a user with the bad channel teporarily and give its turn to a lagging user rather than a good user. They show lower throughput than an opportunistic schee lie the PF scheduler, which aes the latter attractive. The PF scheduler is practical because of its siplicity. It requires per-flow queue anageent, which is coon to every scheduling schee that reflects user s channel state. In general, per-flow queueing iposes a huge aount of burden on routers in a core networ since there are at least tens of thousands of flows. Fortunately, in an access networ, there are just tens of flows, and the cost becoes reasonable. Next, the PF scheduler needs the channel inforation of each user. To do so, BS broadcasts the pilot sybol periodically and gathers the easureent results fro each user. Using the pilot channel is coon aong cellular systes nowadays, and so, it does not incur extra burden. On the other hand, we need an additional control channel for the channel state feedbac, which cost increases with the nuber of users. There exists a tradeoff between the overhead and the accuracy of feedbac inforation. The coputational coplexity of the PF scheduler is just 3N 1) ultiplications and 2N 2) additions for given N users. Finally, we address the QoS issue of the PF scheduler, since it does not guarantee any inial perforance at all. If we change the preference etric slightly, the scheduler guarantees a certain aount of tie portion for user under the scheduling policy R [n]/ R [n]+λ and the iniu rate under R [n]/ R [n]+µ R [n], where λ and µ are the nonnegative constants interpreted as an offset that copensates for the user s channel state. If a user s channel is not good and, accordingly, its tie-share is less than the required, we need to assign a larger offset to the user [8]. III. ANALYSIS In this section, we derive an approxiate expression for the cell throughput under the PF scheduling policy. We ae the following assuptions. 1) Users are distributed uniforly throughout the entire cell area. 2) Every session or user) is always active in the downlin direction. We ignore the throughput loss due to the lac of data to be transitted. 3) The distribution of the channel gain of user h [n]) does not depend on slot n, and it is constant for the slot duration. As indicated in [11], the cell throughput of the PF scheduler decreases when the dynaic range of the channel variation is sall. Viswanath et al. [11] copared the scheduler throughput in the Rayleigh channel with

4 CHOI AND BAHK: CELL-THROUGHPUT ANALYSIS OF THE PROPORTIONAL FAIR SCHEDULER 769 that in the Rician channel through siulations, showing that the forer gives higher throughput. 4) Our odel uses the ratio of the SNR to the average SNR as the preference etric instead of the original PF etric of the ratio of the feasible rate to the average rate or throughput). That is, in tie slot n, the user is served with arg ax Z [n]/ Z [n], where Z [n] denotes the exponential oving averaged SNR. Although this criterion is not exactly the sae as that of the PF scheduler, they share the iportant characteristic of allocating alost the sae portion of tie slots to each user and allowing relative preference to users with good channels. 5) The feasible rate is a strictly onotonic increasing function of the SNR. It is also continuous while, in real systes, it taes values fro a discrete set of supported rates. For exaple, the IS-856 syste defines 11 feasible rates [5]. 6) The average throughput and the average SNR are obtained by the tie average rather than the oving average. By introducing the indicator function I [], which is one if user is scheduled in slot and zero otherwise, we obtain the throughput by R [n +1]= 1/n) n =1 R []I []. The procedures to obtain the average SNR are the sae. We add soe rears on assuptions 4) and 5). The feasible transission rate depends on various factors such as the SNR, the degree of signal distortion during the transission, the odulation schee, the channel-coding schee, the hardware structure of the receiver, etc. Aong these, the ost iportant factor is the SNR, so we express the feasible rate as a function of the SNR, aggregating other factors into a specific for. We consider two popular odels in this paper. They are the linear odel and the logarithic odel. The linear odel was widely adopted to describe CDMA systes, although it is not accurate in very high-snr regions. In this odel, our odified preference etric copletely atches that of the PF scheduler. In the case of the logarithic odel, it does not coincide with the genuine preference etric of the PF scheduler and introduces errors in the analysis. This is evaluated by coparing the analysis with siulations. A. Cell Throughput of the PF Scheduler Suppose that the average rate of user R [n]) gets stable and stationary as tie goes by because the feasible rate process R 1 [n],...,r N [n]) is stationary. Therefore, we can write the user s long-ter average throughput, which is an interediate result to obtain the cell throughput, as T = li n R [n] = li n E{R [n]i [n]} = li n E{R I }, assuing ergodicity and stationarity of R [n]. Note that T = E{R I = 1} E{I } is the product of the expected feasible rate when scheduled and the probability of choosing the user for scheduling. Now, we find the probability that the user is scheduled first. The redefined preference etric for the user is Γ [n] = Z [n]/ Z [n] at slot n. The average SNR Z [n] approaches a certain value Z, if the user s average rate converges. Since Z [n] is independent of tie, we can oit the tie index n and rewrite the preference etric as Z [n] Γ = li n Z [n] = Z 3) Z which follows an exponential distribution. The selected user sees that its preference etric is larger than that of any other user. Denoting the axiu preference etric of all the users except as Γ, we can express the probability that the user is scheduled at each slot as Pr{Γ > Γ }. Since Γ = ax{γ 1,...,Γ 1, Γ +1,...,Γ n } for N users within the cell, its cuulative density function cdf) F t) is given as F t) =Pr{Γ t} = F t) 4) where F t) is the cdf of Γ. The feasible rate strongly depends on the SNR, although the concrete relationship varies, depending on the underlying technologies for the physical and lin layers used. Since the SNR Z is given as Z Γ, the feasible rate R is represented as a function of Γ, that is, R = ξγ ). Since ξ ) is a strictly onotonic increasing function of Γ, it always has the inverse function ζ ), i.e., Γ = ξ 1 R )=ζr ). Then, the longter average throughput of user is T =Pr{Γ > Γ } E{R Γ > Γ } = = ˆR ˆR r d dr Pr{R r and Γ > Γ }dr r d dr Pr {Γ ζr) and Γ > Γ } dr 5) where ˆR indicates the supported axiu feasible rate. After soe anipulation, we obtain T = ζ ) ζ) ξt)f Γ t)f t)dt 6) where f Γ t) denotes the probability density function pdf) of Γ. Consider the case where the SNR of user in tie slot n can be represented as the product of the constant c and the rando variable C [n]. That is, Z [n] =c C [n]. Z /c is nown to be the sae for all s when the following conditions are satisfied [14]: 1) rando variable C [n]s are independent identically distributed, regardless of users and tie slots; 2) C [n]s are independent of the transission power; and 3) the preference etric is a linear function of the SNR. Since our odel corresponds to this case with c = Z and 3 C [n] = 2, Z /c = Z /Z = Γ is the sae for all s. Deleting the subscript, we can see that, fro 3), the pdf of 3 Note that 2 is the exponential rando variable with the ean 1., regardless of.

5 77 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 27 given. Now, it needs to be averaged for all possible shadow fading and for the entire cell area and then ultiplied by the nuber of users to obtain the cell throughput. If we denote the expectation of the shadowing by E s { } E s {T } = βw N MN)E s{z } = βw ) α D N MN)E s{s }ρ. 1) Representing the average over the entire area by E A { }, we can express the cell throughput ˆT cell as d Fig. 1. Multiuser diversity factor in the linear rate odel plotted fro 9). ˆT cell = N E A {E s {T }} = βwmn)e s {s } Ω 1 A A ) α D ρ da 11) d Γ is given as f Γ t) =1/Γ) exp t/γ)) for t, and fro 4), F t) =1 exp t/γ))) for t. Then, 6) can be rewritten as T = ζ ) ζ) ξt) 1 Γ exp t ) 1 exp t dt. 7) Γ Γ)) Next, we consider the function R = ξγ ) in the linear and logarithic fors. 1)Linear Model: In this odel, the feasible rate is linearly proportional to the SNR. Past wors adopted this odel to describe CDMA systes and used it as an approxiation to represent relations containing the sall variation of SNR. When the relation is given as R = βwz, where β is a constant and W is syste bandwidth, we obtain the average throughput of user fro 7) as follows: T = βwz N N te t 1 e t ) dt 8) where βwz )/N) =1/N )E{R } represents the average throughput of user when the RR scheduler is used. The second ter is the auxiliary gain by using the PF scheduling, and we define it as the ultiuser diversity factor MN). Aftersoe anipulation, we have MN) =N = ) N 1 1) +1) 2 9) which is plotted in Fig. 1. The ultiuser diversity factor increases rapidly with the increase in the nuber of users. This is owing to the very high SNR occasionally experienced during fast fading. Recognizing that the linear odel overestiates the feasible rate in this region, we can see that the result has liited eaning in practical environents. We derived a user s long-ter average throughput, assuing that the user s location is fixed or that the shadowing effect is where A and Ω A represent the cell and its area, respectively. In this case, we can see that the cell throughput does not have a finite value. This eans that the linear odel is not accurate for very high SNR and produces unreasonably HDRs. Consequently, the extreely high throughput of users close to the BS gives rise to an infinite cell throughput. We can obtain the finite cell throughput by enforcing the distance between the BS and users to be larger than D = ηdη <1). This reflects the fact that the transission antenna of BS is, typically, apart fro the ground. With this constraint, we can express the cell throughput as ˆT cell = W 2ρβ 1 η 2 α 2 α 1 η 2 exp ) ) 2 ln σ s MN) 12) by using E s {s } = expln 1)/1 2)σ s ) 2 ). Henceforth, we use the noralized cell throughput T cell = ˆT cell /W to ae it independent of syste bandwidth, which is the spectral efficiency of the cell in unit of bits per second per Hertz. 2)Logarithic Model: Adaptive odulation schees have been introduced to perfor lin adaptation. Aong those, M-level quadrature aplitude odulation M-QAM) is proising, where M is typically set to the power of two. With the increase of M, both the inforation bit rate and the bit error rate for the given SNR increase. Therefore, the large M is used in the good channel condition, while the sall M is used in the bad condition. It was verified that, with M-QAM, the feasible transission rate is related to the SNR in the logarithic anner [17], [18]. In this paper, we adopt the for proposed in [17] R = W log 2 1+ Z ) 13) K where K is a constant depending on the syste design and the target bit error rate required for reliable transission. We call K the syste-efficiency factor, since the feasible rate for the sae SNR varies according to K. When K =1., 13) becoes the well-nown Shannon-capacity forula.

6 CHOI AND BAHK: CELL-THROUGHPUT ANALYSIS OF THE PROPORTIONAL FAIR SCHEDULER 771 Fig. 2. Accuracy of the approxiation for the product of the exponential and the exponential integral. Fig. 3. Accuracy of the approxiation for the average of the ln1 + µs). Substituting R = ξγ )=W )/ln 2) ln1 + T /K)Γ ) into 7), we obtain the long-ter average throughput of user as follows: T = W ln 2 = W ln 2 = ln 1+ Z ) K t e t 1 e t ) dt N 1 ) 1) ln 1+ Z ) K t e +1)t dt. 14) As ln+µt)e κt dt =1/κ)expκ/µ)expintκ/µ) in [19], we 4 can rewrite 14) as T = W ln 2 ) N 1 1) +1) = ) K exp +1) expint Z ) K +1). 15) Z Although expt)expintt) is nuerically unstable for large t owing to the exponential ter, it is closely approxiated by ν 1 ln 2 ln1 + ν 2 t 1 ) for a wide range of t in [17], where ν 1 and ν 2 are set to 1.4 and.82, respectively. This aes 15) atheatically tractable, and Fig. 2 shows its accuracy. Applying this approxiation to 15), we obtain T Wν 1 = N 1 ) 1) +1) ln 1+ ) ν 2 K +1) Z. 16) 4 expintt) is defined as t ln x e tx dx and called the exponential 1 integral function. Taing the expectation of the shadow fading, we can approxiate the above as ) N 1 1) E s {T } Wν 1 +1) = E s {ln 1+ ν 2 K +1) Z )}. 17) Fro Z = ρd/d ) α s, we need to calculate the for E s {ln1 + µs)}. We oit the subscript here for notational siplicity. In [17], Pr{Y =ln1+µs) y} is shown to be close to 1 1/2)erfcy ln µ)/ln 1/1) 2σ s )), where erfcx) is the copleentary error function defined as 2/ π) x dt, and σ e t2 s is the standard deviation of shadowing s. This indicates that Y =ln1+µs) is well approxiated by the Gaussian rando variable with the ean ln µ and the standard deviation ln 1/1)σ s ). Catreux et al. [17] also pointed out that it is valid only for y 1. Since we are concerned with the ean rather than the cdf itself, we can use this approxiation. However, this gives an unsatisfactory result for sall µ. Especially, when µ is less than 1., we get a negative ean by the approxiation, while Y is always larger than zero. It otivates us to derive another approxiation E s {ln1 + µs)} =ln1+µ)+ν 3 18) where ν 3 = ln 1/2) σ s / 2π). The accuracy of this approxiation is justified by Fig. 3, and the detailed derivation is given in the Appendix. The approxiation has a tendency of overestiating the exact value, and the gap increases with the increase of µ, which causes the analysis error. Using this approxiation, we obtain E s {T } Wν 1 = ln ) N 1 1) +1) D 1+ ν 2 ρ K +1) d ) α ) + ν 3 ). 19)

7 772 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 27 Finally, taing the average for the entire cell area, we can write the noralized cell throughput as T cell Nν 1 = where B is defined as B =Ω 1 A A = 2 D 2 D ) N 1 1) +1) B + ν 3 ) 2) ) α ) ln 1+b da Dd ) α ) D r ln 1+b dr. 21) r We define b =ν 2 ρ)/k + 1)) for frequent references below and changed the duy variable d to r for readability. Now, we integrate the for x ln1 + µx α )dx to obtain B, but this is difficult to do, except in soe special cases. B can be exactly calculated for the integer of α by referring to [19]. We present the result for α =4, which is the ost interesting case B =ln1+b )+2b 1 2 arctan b ) In extending our analysis to a ulticell environent, we face the obstacle that the SNR of each user no longer follows the exponential distribution because the interference fro adjacent cells is tie-varying rather than constant. Therefore, the scheduling etrics of users are not independent identically distributed rando variables, and the long-ter average throughput of each user is too hard to obtain. This eans that we cannot siplify 5) as 7). We can proceed with the analysis by representing the interferences fro other cells as their average value. In this case, we will have a larger analysis gap in cell throughput copared to the single-cell case. B. Cell Throughput of the RR Scheduler When the RR scheduler is used, every user has the sae portion of tie slots, regardless of the channel condition. Then, the long-ter average throughput of the user is given as T = 1 N E{R } = 1 N ζ ) ζ) ξt)f Γ t)dt. 23) As F t) =1 exp t/γ)),wehavef t) 1 for N =1. Then, coparing 23) with 6), we can see that the average throughput and the cell throughput) of RR scheduler is equal to that of PF scheduler when N =1. The RR scheduler has no ultiuser-diversity gain, and therefore, its cell throughput is constant, regardless of the nuber of users. IV. EXTENSION TO MIMO SYSTEMS A MIMO syste has architecture with ultiple transit and receive antennas, which can give a draatically iproved capacity and/or diversity gain. It is one of the active ongoing research areas and a proising technology due to its advantage of spatial ultiplexing. As the PF scheduler can run on top of this antenna structure without difficulty, we extend our analytic expression of the cell throughput to the MIMO case. A. Spatial Multiplexing MIMO Systes Suppose that the BS has n T transit antennas and that each obile has n R receive antennas. n R should be larger than or equal to n T for this antenna structure to be used as a spatial ultiplexing syste. In this paper, we assue that n T = n R = n A. Henceforth, we represent the ith transit antenna by TA i and the jth receive antenna by RA j. In the single-user case, the user s data strea is split into n T substreas that are concurrently transitted via the transit antennas odulating carriers independently with the sae frequency. Let the sybol transitted by TA i be x i and the channel gain fro TA i to RA j be h ij. Then, the signal received by RA j is given as y j = n T i=1 h ij x i + n j 24) where n j denotes the bacground noise. If the total transit power of the BS is equally allocated to all the transit antennas and all the channels follow the independent and flat Rayleigh fading odel, the SNR of user at RA j, which is Z j), follows an exponential distribution under the assuption that the zero-forcing receiver is used [17]. 5 Its ean is given as Z /n A, where Z is the average SNR of the corresponding single-input single-output SISO) systes and equal to ρd/d) α s, as shown in Section II-A. 6 The receiver does not detect the substreas jointly. The channel atrix and its pseudoinverse should be easured and calculated accurately at the receiver. In the ultiuser case, all the substreas ay not be for one user s data. That is, several users can be selected for transission in a slot, and a user can be allocated to ultipletransit antennas. For exaple, if a user is selected and allocated for L n A ) transit antennas, L pacets fro the user can be siultaneously transitted via the allocated L antennas. The user receives and decodes all pacets transitted, but it accepts only the pacets fro its allocated antennas). B. Deployent of the PF Scheduler In spatial ultiplexing systes, n A subchannels are created between the BS and a user. Therefore, to apply PF scheduler, we should obtain the channel-state inforation for all subchannels and all users. User feeds bac, at each slot, the axiu feasible rate R i), assuing that the user is allocated for TA i. The odulation and coding schee deterines the feasible rate according to the subchannel i s condition. As a consequence, the scheduler nows all the feasible rates. Table I shows a 5 When the SNR is high, its perforance is close to the iniu-eansquare-error receiver. 6 We oit the subscript, representing a user, for the brevity of the notation.

8 CHOI AND BAHK: CELL-THROUGHPUT ANALYSIS OF THE PROPORTIONAL FAIR SCHEDULER 773 TABLE I EXAMPLE OF THE FEASIBLE RATE MATRIX independently chooses a user to be served, a axiu of n A users can be scheduled in a slot. Coparing these two, we can find that the ultiuser style always gives higher throughput than the single-user one but requires the larger signaling and coputational burden. 7 The feasible rate of the single-user style is given as R s = n A i=1 R i), and that of the ultiuser style is R = n A i=1 R i), where i represents the selected user in the single-user style, and i is the selected user for TA i in the ultiuser style. Considering R i) R i) i for every i, wehaver R s. The total feasible rate, for the exaple in Table I, is = 9 by selecting user 2 in the single-user style and = 12 by selecting users 2 and 3 for the ultiuser style. Let us focus on the ultiuser style, since it fully utilizes the virtue of spatial ultiplexing. We obtain n A parallel scheduling planes, and the total cell throughput is given as the su of the throughput of each plane. Each plane is the sae as for the SISO case, except that the average SNR is reduced by using ultiple-transit antennas. We can see that the expression for the cell throughput is easily generalized for the spatial ultiplexing syste. Especially, for the logarithic-rate odel, the cell throughput is given, fro 2) and 21), as Tcell M Nn A ν 1 = ) N 1 1) +1) B + ν 3 ) 25) where B includes the ter b, which is redefined as b = ν 2 ρ)/n A K +1)). Fig. 4. Two deployent styles of the PF scheduler in MIMO systes. a) Single-user style. b) Multiuser style. case that the nubers of users and antennas are four and three, respectively. With this atrix, we can consider two different styles of deploying the PF scheduler. To facilitate explanation, we logically divide the PF scheduler into two parts. They are the channel estiator and the user selector. The channel estiator gives the feasible rate of each user to the user selector, which calculates the preference etric and selects a user to be served by updating the oving average of each user s throughput. We depict the conceptually in Fig. 4. The first style is to serve only one user in a slot, which is called the single-user style. The estiator gives out the aggregate feasible rate of each user to the user selector that selects the best user with the axial preference etric. This is the sae as the SISO case, except that the feasible rate of user is given as the su of R i) s. The second is with a user-selector per-transit antenna, and it is called the ultiuser style. There are n A instances of the user selector, with each taing charge of a transit antenna. The channel estiator gives the ith selector the rate inforation of each user, which is feasible when allocated for TA i. That is, the ith selector has R i) sfor1 N. Since each selector V. S IMULATION RESULTS In this section, we present siulation results for the cell throughput under the PF scheduling policy and copare the with our analysis results. The throughput gap between the genuine PF scheduler and our eulated one is also anifested. Then, we investigate the effect of various paraeters on the throughput through nuerical analysis and siulations. A single-cell scenario is considered in siulations, and the cell radius is set to 1. The slot length for the downlin channel is 1.67 s. The transission power of BS is fixed at 1 W, the path-loss exponent α is four, the standard deviation of shadow fading σ s is 8 db, and the edian SNR at the cell edge ρ is set to db. The bacground noise power and antenna gains are adjusted according to ρ. The syste-efficiency factor K, which deterines the feasible transission rate for the given SNR, is 8 db. These paraeters are used as default if not stated otherwise. Before studying the PF scheduler, we scrutinize the eulated PF scheduler that was derived fro the genuine PF by iposing soe assuptions, which are justified by analysis and siulation. We used tie average rather than oving average to obtain the average throughput of each user. The tie average can be written as a recursive for R [n +1]= R [n]+1/n)r [n]i [n] R [n]) and the oving average 7 It is enough, in the single-user style, to feed bac the aggregated feasible rate rather than the respective feasible rates, which alleviates the signaling overhead.

9 774 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 27 Fig. 5. Coparison of the oving average with the tie average. Fig. 6. Aount of tie-share taen by each user in the PF scheduler. as R [n +1]= R [n]+1/t c )R [n]i [n] R [n]). The stochastic-approxiation theory asserts that the average rate converges as tie goes by, if the weighting 1/t c ) is sall enough. For siulations, we located two users user 1 and user 2) at the distance of 1 and 2, respectively, away fro a BS, and neglected the shadowing for siplicity. Fig. 5 shows that the oving average approaches the tie average as t c goes up. The eulated PF scheduler uses the ratio of SNR to average SNR as the preference etric instead of the ratio of feasible rate to average rate. The two etrics are siilar in ters of their preference of users with the good channel, but they are not always coincident. To see the ipact of discrepancy, we study the tie-share of each user with various channel conditions. We aligned 2 users fro a cell center toward the edge to ae the channel condition of each user diverse. User is positioned at D/2) in eters) away fro BS, where D represents the cell radius. In the eulated PF, each user receives the sae aount of tie slots because the preference etrics are independent identically distributed rando variables. In contrast, Fig. 6 shows that the PF scheduler soewhat favors users with the good channel. The dotted line indicates the tie-share of each user in the eulated PF scheduler. Fro these, we anticipate that the PF scheduler will show larger throughput than the eulated one, and the gap will get larger with the variation of users channel conditions. Then, we copare the cell throughput of PF scheduler with that of the eulated PF scheduler and our analysis result. Each siulation runs for 1 slots, and 1 outcoes of the sae siulation runs are averaged to represent a single point of the graph. We only consider the logarithic odel for the relation between the feasible rate and the SNR, since the linear odel has only liited eaning in practical situations. Fig. 7 presents the noralized cell throughput, varying the nuber of users fro 1 to 5. We observe that the siulation result for the eulated PF scheduler is close to that for the genuine PF scheduler. As expected, there is a tendency that the throughput of the PF scheduler is slightly higher than that of the eulated one. With single-user, PF, eulated PF, and RR Fig. 7. Noralized cell throughput of PF scheduler, eulated PF scheduler, RR scheduler, and analysis. a) Coparison of analysis and siulation: α =4, σ s =8 db, ρ = db, and K =8 db. b) Relative error to the genuine PF scheduler.

10 CHOI AND BAHK: CELL-THROUGHPUT ANALYSIS OF THE PROPORTIONAL FAIR SCHEDULER 775 Fig. 8. Effect of the path-loss exponent on the cell throughput: σ s =8dB, ρ =db, K =8dB, and N =3. Fig. 1. Effect of the edian SNR at the cell edge on the cell throughput: α =4, σ s =8dB, K =8dB, and N =3. Fig. 9. Effect of the standard deviation of shadowing on the cell throughput: α =4, ρ =db, K =8dB, and N =3. schedulers, all have siilar throughput. However, the throughput of the RR scheduler is constant, regardless of the nuber of users, while those of the PF and the eulated PF schedulers increase logarithically with the increase of the nuber of users. Their gap coes fro the effect of the ultiuser diversity for N>1. For a sall nuber of users, the PF scheduler shows any various tie shares copared to the eulated one. As the nuber of users increases, the channel variation aong users lessens, and the eulated PF scheduler goes close to the genuine PF one gradually. Their throughputs in analysis are coparable with soe positive offset. The approxiation 18) causes the error, because it gives a little larger value than the real one. The gap can be saller by letting ν 3 saller. Next, we investigate the variation of cell throughput according to the path-loss exponent α, the standard deviation of shadowing σ s, the edian SNR at the cell edge ρ, and the syste-efficiency factor K, which are the ajor paraeters affecting the perforance of the PF which scheduler [see 2)]. In Figs. 8 11, the nuber of users is fixed to 3. Figs. 8 and 9 Fig. 11. Effect of the syste-efficiency factor on the cell throughput: α =4., σ s =8dB, ρ =db, and N =3. show that the throughput increases with the increase of α and σ s. The PF scheduler obtains larger throughput when soe users are with the good state by scheduling the first. In Fig. 9, the analysis underestiates the cell-throughput increase with respect to large σ s. This eans that, even though approxiation 18) is generally accurate, the effect of shadowing is not counted appropriately. Fig. 1 shows that the throughput rapidly increases with ρ. However, the large ρ requires BS to increase the power exponentially to cancel out the effect of path loss. Fig. 11 says that the throughput is inversely proportional to the systeefficiency factor. Therefore, the syste with efficient coding and/or odulation schees can obtain higher throughput gain. Recollecting approxiation 18), we get the overestiated throughput for large µ, which is ν 2 ρ)/k + 1))D/d ) α [see 19)]. This explains the analysis error with respect to large α, large ρ, and sall K in Figs Fro these results, we can conclude that the effects of the paraeters on cell

11 776 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 27 Fig. 12. Coparison of opportunistic schedulers: Noralize throughput of each group. Fig. 13. Coparison of opportunistic schedulers: Total noralized throughput. throughput are siilar to that of B + ν 3, although they are represented in the fors of alternating series. We copare the PF scheduler with other opportunistic schedulers by choosing the schees proposed in [6] and [7] as the copetitors. They axiize the total average throughput while satisfying soe QoS constraints. In [7], called Liu s here, each user requires a certain aount of tie slots to be served. We assue that all users deand an equal portion of slots. The scheduler in [6], called Borst s, equalizes the noralized throughput of each user, where the weight is allocated apriori by reflecting both user s channel state and QoS requireent. We set it to one for all users so that they have the sae average throughput. In siulations, we consider 2 users and divide the into two groups Group1 and Group2), each with ten users. Group1 is located at 2 apart fro BS, and the distance of Group2 fro BS varies fro 2 to 8. We do not consider the shadowing in this case. Liu s scheduler gives the sae portion of tie slots to each user so that the throughput of Group1 does not change, even if Group2 goes away fro the BS, while that of Group2 drops rapidly according to the channel condition, as shown in Figs. 12 and 13. The throughput of the PF scheduler is very siilar to that of Liu s, that is, actually equivalent to the eulated PF scheduler in ters of tie-slot allocation. The throughput of Group1 increases slowly with the distance of Group2, because average channel conditions between the groups get ore different. Borst s scheduler gives the equal average throughput to each user. Therefore, as Group2 gets away fro the center of the cell, Group2 users need to be assigned ore slots to copensate for the reduced transission rate, which leads to throughput decrease. Borst s scheduler shows the sallest throughput even when every user is located at the sae distance of 2. In Fig. 14, we plot the noralized cell throughput for the MIMO syste with PF scheduling. The nuber of antennas is set to one, two, and four, respectively. The cell throughput increases with the nuber of antennas, as expected. Fig. 15 shows the variation of the ultiuser diversity gain as a function of Fig. 14. Noralized cell throughput with ultiple antennas: α =4, σ s =8dB, K =8dB, and ρ =db. the nuber of antennas. As the nuber of antennas increases, the ean and variance) of the SNR at each receive antenna decreases. When ρ is 2 db, ost users are in the high-snr region. In this region, the feasible rate hardly increases, even if the SNR becoes excessively high because of fast fading. Therefore, the ultiuser diversity gain is very sall. The ultiple antennas bring the SNR at each receive antenna down to the relatively low region, and then, opportunistic scheduling can give a large throughput gain. On the other hand, when ρ is db, ost users are in the sall SNR region where the ultiuser diversity gain is very sensitive to the variance of the SNR. In this region, the reduced variance of the SNR can deteriorate the ultiuser diversity gain. VI. CONCLUSION In legacy wireless access networs, user fairness was not a pressing consideration due to reservation-based allocation of

12 CHOI AND BAHK: CELL-THROUGHPUT ANALYSIS OF THE PROPORTIONAL FAIR SCHEDULER 777 Then, we obtain the upper bound by Fig. 15. Multiuser diversity gain as a function of the nuber of antennas: α =4, σ s =8dB, K =8dB, and N =3. E s {ln1 + µs)} = < 1 ln1 + µx)f X x)dx ln1 + µ)f X x)dx + =ln1+µ) =ln1+µ)+ and the lower bound by f X x)dx ln 1 + µ)x) f X x)dx ln x f X x)dx ln 1 1 σ s 27) 2π cellular channels and copetitive bandwidth consuption of IEEE systes. Several newly proposed systes, however, seriously consider user fairness and operate with pacet schedulers that treat users differently according to the channel condition to achieve enhanced perforance goals. The PF scheduler is one of the strong candidates targeted for balancing the cell throughput against user fairness by considering the channel condition. In this paper, we analyzed the PF scheduler to obtain an analytic expression for the cell throughput. To do this, we adopted a slightly odified etric, which could be explicitly supported by the relationship between the transission rate and the SNR, which is linear or logarithic. We verified the accuracy of our analysis by coparing it with siulation results. The paraeters affecting the throughput were specified by the expression, and their effects were also investigated. We extended our analysis results for the case of MIMO systes. This paper is expected to facilitate understanding the perforance of the PF scheduler in wireless environents for networ design and resource diensioning, enabling perforance evaluation using analytic tools that copleent siulation-driven evaluation. APPENDIX We show that E s {ln1 + µs)} ln1 + µ)+ν 3, ν 3 = ln 1/2) σ s )/ 2π), which is, in fact, the average of the upper and lower bound. Let X be the log noral rando variable with the ean db and the variance σ 2 s in decibels) in log-scale. Since Y = 1 log X follows the Gaussian distribution with the ean zero and the variance σ 2 s,frof X x) = f Y y)dy/dx), the pdf of X is given as f X x) = c 1 π x exp c ln x) 2) 1, c = ln 1. 2σ s 26) E s {ln1 + µs)} = > ln1 + µx)f X x)dx ) ln 1 + µ)x µ 1+µ f X x)dx =ln1+µ) f X x)dx + µ 1+µ ln x f X x)dx =ln1+µ). 28) Therefore, we approxiate E s {ln1 + µs)} by ln1 + µ)+ ln 1/2) σ s )/ 2π). REFERENCES [1] L. Bos and S. Leroy, Toward an all-ip-based UMTS syste architecture, IEEE Netw., vol. 15, no. 1, pp , Jan./Feb. 21. [2] A. K. Pareh and R. G. Gallager, A generalized processor sharing approach to flow control in integrated services networs: The single-node case, IEEE/ACM Trans. Netw., vol. 1, no. 3, pp , Jun [3] S. Liu and V. Bharghavan, Fair scheduling in wireless pacet networs, IEEE/ACM Trans. Netw., vol. 7, no. 4, pp , Aug [4] T. S. E. Ng, I. Stoica, and H. Zhang, Pacet fair queuing algoriths for wireless networs with location-dependent errors, in Proc. IEEE INFOCOM, San Francisco, CA, 1998, pp [5] A. Jalali, R. Padovani, and R. Panaj, Data throughput of CDMA-HDR a high efficiency-high data rate personal counication wireless syste, in Proc. IEEE VTC, Toyo, Japan, 2, pp [6] S. Borst and P. Whiting, Dynaic rate control algoriths for HDR throughput optiization, in Proc. IEEE INFOCOM, Anchorage, AK, 21, pp [7] X. Liu, E. K. P. Chong, and N. B. Shroff, Opportunistic transission scheduling with resource-sharing constraints in wireless networs, IEEE J. Sel. Areas Coun., vol. 19, no. 1, pp , Oct. 21. [8], A fraewor for opportunistic scheduling in wireless networs, Coput. Netw., vol. 41, no. 4, pp , Mar. 23. [9] Y. Liu and E. Knightly, Opportunistic fair scheduling over ultiple wireless channels, in Proc. IEEE INFOCOM, San Francisco, CA, 23, pp

13 778 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 56, NO. 2, MARCH 27 [1] X. Qin and R. Berry, Exploiting ultiuser diversity for ediu access control in wireless networs, in Proc. IEEE INFOCOM, San Francisco, CA, 23, pp [11] P. Viswanath, D. N. C. Tse, and R. Laroia, Opportunistic beaforing using dub antennas, IEEE Trans. Inf. Theory, vol. 48, no. 6, pp , Jun. 22. [12] F. Kelly, Charging and rate control for elastic traffic, Eur. Trans. Telecoun., vol. 8, no. 1, pp , Jan [13] L. Massoulié and J. Roberts, Bandwidth sharing: Objectives and algoriths, IEEE/ACM Trans. Netw., vol. 1, no. 3, pp , Jun. 22. [14] J. M. Holtzan, Asyptotic analysis of proportional fair algorith, in Proc. IEEE PIMRC, San Diego, CA, 21, pp [15] H. J. Kushner and P. A. Whiting, Convergence of proportional-fair sharing algoriths under general conditions, IEEE Trans. Wireless Coun., vol. 3, no. 4, pp , Jul. 24. [16] S. Borst, User-level perforance of channel-aware scheduling algoriths in wireless data networs, in Proc. IEEE INFOCOM, San Francisco, CA, 23, pp [17] S. Catreux, P. F. Driessen, and L. J. Greenstein, Data throughputs using ultiple-input ultiple-output MIMO) techniques in a noiseliited cellular environent, IEEE Trans. Wireless Coun., vol. 1, no. 2, pp , Apr. 22. [18] A. J. Goldsith and S.-G. Chua, Variable-rate variable-power MQAM for fading channels, IEEE Trans. Coun., vol. 45, no. 1, pp , Oct [19] I. S. Gradshteyn and I. M. Ryzhi, Tables of Integrals; Series and Products, 4th ed. New Yor: Acadeic, Jin-Ghoo Choi received the B.S. and Ph.D. degrees in electrical engineering fro Seoul National University, Seoul, Korea, in 1998 and 25, respectively. In 25, he wored for KT, Korea, as a cell planning engineer. He is currently a Senior Engineer with Sasung Electronics, Seoul. His research interests include resource anageent and pacet scheduling in wireless networs. Saewoong Bah M 94) received the B.S. and M.S. degrees in electrical engineering fro Seoul National University, Seoul, Korea, in 1984 and 1986, respectively, and the Ph.D. degree fro the University of Pennsylvania, Philadelphia, in Fro 1991 to 1994, he was with AT&T Bell Laboratories as a eber of technical staff, where he wored for AT&T networ anageent. In 1994, he joined the School of Electrical Engineering, Seoul National University, where he is currently a Professor. His areas of interests include perforance analysis of counication networs and networ security. Dr. Bah is a eber of Who s Who in Science and Engineering.