Tradeoff Power Control for Cellular Systems Changho Suh
Ali T. Koc and Shilpa Talwar
Wireless Foundations, Department of EECS University of California at Berkeley Email:
[email protected]
Communication Technology Group Intel Corporation Email: {ali.t.koc,shilpa.talwar}@intel.com
Abstract— We consider distributed uplink open-loop power control for orthogonal frequency division multiple access (OFDMA) cellular systems. To touch simultaneously on two main performance metrics in cellular systems, average throughput and celledge throughput, we focus on a metric of tradeoff performance. First we investigate two extreme cases to show that the wellknown SNR-based scheme and the INR-based scheme maximize cell-edge throughput and average throughput, respectively. Based on these two extremes, we propose a new power control scheme that trades off well both metrics. The idea is to combine the optimum power of two extremes in a linear fashion. System-level simulations show that the proposed scheme has better tradeoff performance as compared to two schemes: (1) a trivial timesharing scheme (where a tradeoff curve shows the straight line that connects two extreme points); (2) another alternative (that combines the optimum power of two extremes in an exponential way).
I. I NTRODUCTION Power control has been extensively explored in code division multiple access (CDMA) cellular systems to combat a near-far problem due to intra-cell interference [1], [2], [3]. Obsessed with solving this problem, researchers employed power control as a means of guaranteeing the same qualityof-service of all users by maintaining their target signal-tointerference-and-noise-ratio (SINR) levels simultaneously. To this end, Foschini and Miljanic [4] developed an iterative and distributed power control scheme under a closed-loop assumption that BS tracks its instantaneous SINR and feeds back power adjustment information to mobiles. Yates [1] developed a general framework that describes all power control problems of this type. As for the open-loop case, one may use signal-to-noise-ratio (SNR) instead of SINR since interference information is not fed back to each mobile. But the main idea is essentially the same, which is to maintain the received signal power level. However, recent communications standards such as 3GPP-LTE [5] and IEEE 802.16m (WiMAX) [6] employ a different multiple access scheme: orthogonal frequency division multiple access (OFDMA), since it has many advantages (as compared to CDMA) especially to broadband systems. In OFDMA systems, orthogonal subcarriers are assigned to different users without being overlapped, thereby not causing intra-cell interference. Hence, there is no near-far problem. A question remains: “How should CDMA power control (the SNR-based scheme that focused on the near-far problem) be changed in OFDMA systems?”. Since we do not need to consider the near-far problem, we instead consider other performance metrics: average throughput and cell-edge
throughput, both of which are crucial factors in cellular systems. Obviously, maximizing average throughput conflicts with maximizing cell-edge throughput, i.e., those two metrics have a tradeoff relationship. The goal of our paper is to develop a power control scheme that trades them off well. To gain insights into this, we first investigate two extreme optimization problems: (1) maximizing cell-edge throughput; (2) maximizing average throughput. As for constraints, we set maximum power limit and interference-level limit (to other cells). We assume independent resource allocations between cells. It turns out that the known SNR-based scheme is optimum for maximizing cell-edge throughput. On the other hand, maintaining interference-to-noise-power-ratio (INR) turns out to be optimum for the other extreme, which is called the INRbased scheme in standards contexts [7], [5], [6]. Next, based on these two extremes, we propose a power control scheme that trades off average throughput and celledge throughput well. The idea is to combine the optimum power of two extreme cases in a linear way. To characterize tradeoff performance, we introduce a tradeoff curve. Through system-level simulations, we show that the proposed scheme has better tradeoff performance than other schemes: a straight forward time-sharing scheme (which has a tradeoff curve connecting two extreme points with the straight line) and another alternative (which combines the optimum power of two extremes in an exponential manner). Earlier research has addressed tradeoff issues to develop power control schemes [8], [9]. Especially Jindal et.al. [8] developed fractional power control that captures tradeoff performance. The scheme therein tunes a parameter −s (0 ≤ s ≤ 1) (raised to an exponent of channel gain) to sweep intermediate policies between two extremes: the SNR-based scheme (the channel-inversion scheme, s = 1) and the constant power scheme (s = 0). However, tradeoff performance is not explicitly characterized there, i.e., it is hard to know how much overall spectral efficiency is improved by sacrificing celledge throughput. Also performance metrics therein (the outage probability and transmission capacity targeted for peer-to-peer wireless networks) are different from ours, which are targeted for cellular systems. II. S YSTEM M ODEL We consider distributed open-loop uplink OFDMA systems where resources (subcarrier/power) are independently allocated between cells. We assume full spectral sharing (fre-
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quency reuse = 1) and uplink-downlink channel reciprocity. We assume to treat interference as noise (no successive interference cancelation). Also we assume interference is Gaussian and the capacity is determined by the Gaussian channel Shannon capacity. This assumption is reasonable when Gaussianlike codebooks are used in the systems. Let g be channel gain from a mobile to its own base station (BS) and gk be channel gain from a mobile to kth interferee BS. Since we assume channel reciprocity, from downlink channel gains (measured from downlink preamble or pilot), we can estimate uplink channel gains. Specially we assume g = SIRDL , (1) ηk gk where SIRDL denotes downlink signal-to-interference-powerratio. Interference power is measured from downlink preamble of other BS. Here ηk is a calibration factor due to different power levels between BSs. If downlink preamble power is the same across all the BSs, then ηk = 1, ∀k. Throughout the paper, we will assume ηk = 1, ∀k for simplicity. Let gk , (2) gsum := which indicates the aggregation of interfering effects to the other cells. We assume slow fading channels, i.e., channels vary in a slower time scale as compared to a communication time scale. Then, when a mobile with g uses transmit power of SNR (normalized to noise power), it can transmit at a rate of gSNR bits/s/Hz log 1 + INRrec + 1 where INRrec indicates interference power received at BS. Finally we assume maximum power limit and interfering power limit: INRtar , SNR ≤ min SNRmax , gsum
optimization problem is simply formulated by considering one particular realization. Average Throughput Maximization: gSNR , max log 1 + SNR INRrec + 1 INRtar s.t. 0 ≤ SNR ≤ min SNRmax , . gsum
(3)
Notice that the received interference power INRrec depends on other-cell power control. Hence, SNR is independent of INRrec . Note that the objective function is concave over SNR and Slater’s condition holds. By strong duality, it suffices to solve the Karush-Kuhn-Tucker conditions [10]. Consider a Lagrange dual function: gSNR − λ1 (−SNR) L(SNR,λ) = log 1 + INRrec + 1 (4) INRtar − λ2 SNR − min SNRmax , . gsum Using this dual function, we can write the KKT conditions: dL(SNR∗ , λ∗ ) = 0; dSNR INRtar −SNR∗ ≤ 0, SNR∗ − min SNRmax , ≤ 0; gsum λ∗1 ≥ 0, λ∗2 ≥ 0; INRtar ∗ ∗ ∗ ∗ λ1 SNR = 0, λ2 SNR − min SNRmax , = 0, gsum (5) where the last condition is called complementary slackness. From these, we get INRtar . (6) SNR∗ = min SNRmax , gsum
where SNRmax indicates maximum transmission power normalized to noise power. INRtar indicates interfering power limit. In other words, aggregated interference created from one mobile, denoted by gsum SNR, is limited by INRtar . It turns out that the constraint of interfering-power-level controls the whole-network throughput performance. In other words, by restricting interfering power level, we can impose some centralized policy to the whole networks. In Section III-D, we will deal with this issue in more details.
The solution is quite intuitive. The objective is not the whole network throughput, but individual-cell throughput. Also SNR is independent of other-cell power allocations. Therefore, transmitting power as high as possible should provide satisfactory solution. Remarks: Interestingly, the solution is the same as the well known INR-based scheme [5], [6], [7]. Total interfering power gsum SNR∗ is constant for sufficiently large SNRmax :
III. O PTIMAL P OWER C ONTROL FOR T WO E XTREMES
gsum SNR∗ = INRtar .
Let us start by examining two extreme optimization problems: (1) maximizing average throughput; (2) maximizing celledge throughput. A. Maximizing Average Throughput Since uplink channel gain is assumed to be known, maximizing average throughput is equivalent to maximizing individual throughput for given channel realization. Hence, an
(7)
Notice that the solution is due to an interference-power-level constraint. Without the constraint, the optimum solution would be the max-power transmission scheme. As mentioned earlier, putting a constraint of INRtar can provide some centralized policy to control the whole-network performance. It turned out numerically that there exists the optimum INRtar to maximize the whole-network throughput. The detailed optimization will be discussed in Section III-D.
978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.
B. Maximizing Cell-Edge Throughput The other extreme optimization is to maximize cell-edge throughput (the minimum data rate). Given minimum channel gain gmin , an optimization is formulated as follows. Cell-Edge Throughput Maximization: gmin SNR max log 1 + , SNR INRrec + 1 INRtar . s.t. 0 ≤ SNR ≤ min SNRmax , gsum Note that by (1) and (2) gsum is a function of gmin : gmin gsum = . SIRedge DL
(8)
(9)
Similar to the previous optimization problem, solving the KKT conditions gives the following solution: INRtar ∗ SNRedge = min SNRmax , . (10) gsum The issue remains of how to determine power of cell-center users with general channel gain g ≥ gmin . Note that the worst user (cell-edge user) with gmin should have the minimum throughput, i.e., gSNR ≥ gmin SNR∗edge ,
∀g ≥ gmin .
(11)
Otherwise the objective in the above optimization violates the “max min” rule. Since the user with better channel gain g ≥ gmin is excluded in the objective, it is the optimum solution for this user to transmit the lowest power as possible. Therefore, for general channel gain g, the optimum transmission power is gmin SNR∗edge SNR∗ = g (12) gmin SNRmax gmin INRtar , = min . g g · gsum Remarks: Interestingly, the solution is the well-known SNR-based scheme. Notice that the received signal power gSNR∗ is constant for sufficiently large SNRmax : gmin INRtar gSNR∗ = = INRtar SIRedge (13) DL , gsum where the second equality follows from (9). C. Maximizing x% Cell-Edge Throughput To capture tradeoff performance between average throughput and cell-edge throughput, one can think of a parameterized objective, which is to maximize x% cell-edge throughput. Here x% indicates the cell-boundary portion to a total cell area. For example, in the case of circular cellular model with radius R, x% cell-edge users belong to the interval [Rthres , R] x where Rthres = R 1 − 100 so that the ring area is x%. For this case, users are classified into two groups: (1) x% cell-edge users; (2) (100 − x)% cell-center users. Let gthres denotes average channel gain at threshold boundary Rthres : gmin gthres = , (14) α/2 (1 − x/100)
where α is a path loss exponent. By the same arguments in the previous sections, it easily follows that ⎧
⎨ min SNRmax , INRtar , gmin ≤ g ≤ gthres gsum
SNR∗ = thres INR SIR g SNR tar max th DL ⎩ min , g ≥ gthres , , g g (15) denotes the SIR of a mobile located in the where SIRthres DL threshold boundary Rthres . Note that the solution has a hybrid form: cell-edge users follow the INR-based scheme and cellcenter users keep the SNR-based scheme. D. Impact of INRtar So far we have considered an individual-cell objective. However, in practical systems, we are interested in the whole network performance that depends on a design parameter INRtar . Therefore, INRtar needs to be optimized from a perspective of whole-network performance. In fact, the optimum INRtar is a function of many system parameters such as a pass loss exponent, a cellular structure (a linear cell or a hexagonal cell or a sectored cell), a cell radius, shadowing, Rayleigh fading, etc. Hence, it is challenging to obtain an explicit form of INR∗tar that captures all of system parameters. To simplify the optimization problem, only through this section, we consider a seven-cell circular cellular model (1st tier interference model) suffering from only path loss (no shadowing and no Rayleigh fading). Also assume that there is only one dominant interferer nearest to the desired BS. Let g11 be channel gain from a mobile to the desired BS. Let g22 be channel gain from a dominantly interfering mobile to its corresponding BS. Then, we have g11 = rGα , g22 = rGα , 1 2 where r1 and r2 are distances from a mobile to its own BS respectively; and G is constant due to large-scale path loss. Assume that users are uniformly distributed and independent, 2r i.e., r1 , r2 ∼ f (r) = R 2 , where R denotes a cell radius. Due to symmetry, it is sufficient to consider throughput of one cell. So an optimization problem can be formulated as: R R Rthres 4 max fe r2 dr2 + fc r2 dr2 r1 dr 2 INRtar (R2 − Rthres )R2 Rthres Rthres 0 where
⎛
fe = log ⎝1 + ⎛ ⎜ ⎜ fc = log ⎜1 + ⎝
min
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α 2R r2α INRtar Gr2α SNRmax Rthres −1 min (2R−r2 )α Rα , (2R−r2 )α thres
min
⎞
+1
⎟ ⎟ ⎟. ⎠
For simplicity we assume that a dominant interferer is lying along with the straight line between two BSs. Lemma 1 (Optimum INR∗tar ): For x = 0%, the optimum INRtar is GSNRmax INR∗tar = gmin SNRmax = . (16) Rα
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Fig. 1. INRtar optimization: x% cell-edge throughput as a function of INRtar .
Proof: For x = 0%, we have Rthres = R. So the objective function can be simplified into ⎞ ⎛ R max min GSNR , INR 2 α tar ⎠ dr2 .
α R r2 log ⎝1 + Gr2 SNRmax r2α INRtar R2 0 + 1 min (2R−r , α α (2R−r )α 2) R 2 Notice that the objective function is increasing over INRtar ; and INRtar is limited by gmin SNRmax due to a maximum power constraint. Therefore, the objective is maximized when max . Notice that at this optimum value, only INR∗tar = GSNR Rα cell-edge user transmits the maximum power while cell-center users transmit the lower power according to the SNR-based scheme. For x = 0%, we could not find the optimum INR∗tar . Instead, as shown in Fig. 1, we do the grid search by plotting the objective function (x% worst-user throughput) as a function of INRtar . Interestingly, the function is increasing for almost all cases of x%. On the other hand, the function is increasing and decreasing for x = 100%. However, it is almost flat. Therefore, we can say that the optimum INRtar is similar for all x’s. Here we used WiMAX system parameters for a cell radius, a path loss model, maximum transmit power and noise power [6] (see Table I). Using these, we get GSNRmax Rα (23−(−103.8861)) 10 (8.6696 × 10−14 ) · 10 = 3.76 0.5 7.58dB.
INR∗tar =
(17)
IV. P ROPOSED P OWER C ONTROL In the previous section, we show how we found the optimum power control for two extremes. We then investigated the optimization that maximizes x% cell-edge throughput as one of tradeoff schemes. The question arises: “How well does the x%-parameterized scheme capture tradeoff performance
Fig. 2. Tradeoff curves for different power control schemes: analysis results assuming path loss only (no shadowing and no Rayleigh fading).
between average throughput and cell-edge throughput?”. It turned out that it does not trade them off well. In this section, therefore, we propose a new power control scheme that has better tradeoff performance: Fig. 2 shows average/cell-edge throughput tradeoff curves for different power control schemes. We assumed a simple seven-cell circular cellular model which suffers only from large-scale path loss (no shadowing and no Rayleigh fading). We assumed WiMAX system parameters for a cell radius, a path loss model and SNRmax (see Table I). Also we used the INRtar = 7.58dB which was optimized in the previous section. We considered the whole network throughput performance. ⎞ ⎛ GSNR∗ (r1 ) R R 4 r1α ⎠ dr1 dr2 , r1 r2 log ⎝1 + GSNR∗ (r ) Tavg = 4 2 R 0 0 + 1 α (2R−r2 ) ⎞ ⎛ R GSNR∗ (R) 2 α ⎠ dr2 , r2 log ⎝1 + GSNR∗R(r ) Tedge = 2 2 R 0 α + 1 (2R−r2 )
∗
where SNR (r) depends on specific power control schemes. Note that by symmetry it is sufficient to consider one-cell throughput. Observe two extreme points: x = 0% and x = 100%. The straight forward tradeoff performance can be obtained via time sharing between these two extremes. Time sharing means that we use the SNR-based scheme (x = 0%) during some portion λ(0 ≤ λ ≤ 1) of time slots and INR-based scheme (x = 100%) for remaining time slots ((1 − λ) portion). So we get the straight line (the dashed curve) which connects these two points. Next notice that the x%-parameterized scheme (the cross-marked curve) is even worse than the straight forward time-sharing scheme. This implies that maximizing x% worstuser throughput is not an appropriate objective for obtaining good tradeoff performance, although it is optimal in a sense of giving higher priority to x% cell-edge users. This motivated
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us to come up with a new power control scheme that achieves better tradeoff performance. Characterizing fundamental limits of tradeoff performance is challenging because it is not easy to find the limit of improved average throughput given the decrease of cell-edge throughput. We therefore propose a specific power control scheme, which is definitely the lower bound of the limit. In order to gain insights into this scheme, let us re-interpret the INR-based scheme and SNR-based scheme in terms of target SNR. Target SNR means the target received signal-tonoise-power-ratio at the desired BS. From (6) and (12), for sufficiently large SNRmax , we get INR-based : SNR∗tar = INRtar SIRDL , SNR-based : SNR∗tar = INRtar SIRedge DL . We can think of two natural tradeoff schemes: one is to combine the optimum power of two extremes in a linear way; the other is to combine them in an exponential way. Specifically, the linear combination scheme has the form of SNR∗tar = (1 − γ)INRtar SIRedge DL + γINRtar SIRDL ,
Parameters Transmission Bandwith Cell Layout Inter-Base Station Distance # of Active Mobiles per Sector A Path Loss Model Log Normal Shadowing A Fading Channel Profile Penetration Loss Carrier Frequency A Resource Block (RB) Size Maximum Transmit Power Noise Power per RB Scheduling HARQ Operation Target PER A Traffic Model
Assumptions 10 MHz 19 Hexagonal Cells 3 Sectors, The Wrap-Around-Model [11] 1.0 km 10 The NGMN model: P L(dB) = 128.1 + 37.6 log10 rkm + 2.52 μ = 0 dB, σ = 8 dB Inter-site correlation = 0.5 ITU Pedestrian B 20 dB 2.5 GHz 175.04 kHz 23 dBm (= 200 mW) −103.8861 dBm (= 4.0869 × 10−11 mW) Proportional-fair scheduler Chase Combining, max of 4 re-tx, Retransmission interval = 1 ms 30% Full Buffer
(18)
where 0 ≤ γ ≤ 1. The exponential combination scheme is 1−β · (INRtar SIRDL )β , SNR∗tar = (INRtar SIRedge DL )
TABLE I S YSTEM -L EVEL -S IMULATION PARAMETERS FOR W I MAX S YSTEMS
(19)
where 0 ≤ β ≤ 1. Note that γ = 0 (or β = 1) corresponds to the SNR-based scheme and γ = 1 (or β = 1) to the INR-based scheme, respectively. Remarks: Intuitively, linear-combination scheme should have better tradeoff performance than an exponentialcombination scheme. Note that throughput has log relationship with power. This implies that the exponential combination of power gives roughly a linear combination of throughput. The approximation is very accurate especially in high SINR regime. We can see this in Fig. 2. Observe that the exponential combination scheme (the circle-marked curve) is similar to the time-sharing scheme (the dashed curve). Now note that log is a concave function. Using Jensen’s inequality, throughput at the average power is greater than the average value of throughput. Therefore, the linear combination of power gives better throughput than exponential one, resulting in better tradeoff performance. It is numerically verified in Fig. 2 as well. The curve of the linear-combination scheme is outer than the exponential-combination-scheme curve, which implies better tradeoff performance. In the next section, this conclusion will be also verified for more realistic WiMAX systems through system level simulations. V. S YSTEM -L EVEL -S IMULATION R ESULTS In this section, we provide realistic system-level-simulation results under practical OFDMA systems (WiMAX [6]). See Table I for detailed system parameters. Fig. 3 shows tradeoff curves of linear combination and exponential combination schemes. Here x-axis and y-axis indicate average throughput and cell-edge (5% user) throughput, respectively. For each scheme, we vary parameter γ (or β) to plot tradeoff curves.
Remarks: In system level simulations, the linear combination scheme is slightly better than exponential one, while the gap is quite large in the analysis (Fig. 2). The one possible reason is that in our analysis, we assume a seven-cell circular model (not sectored) and large-scale path loss only (no Shadowing and no Rayleigh fading). This may significantly affect the concavity level of a throughput function, which highly determines tradeoff gain. We did not evaluate how those factors (such as fast-fading effects and cellular structures) affect the concavity of the throughput function, but such evaluation could be the basis of future research. Fig. 4 shows the user throughput CDF for the proposed power control. The SNR-based scheme (γ = 0) has the best cell-edge throughput, while INR-based scheme has the best average throughput. For γ = 0.1, we obtain compromised tradeoff performance. VI. C ONCLUSION In CDMA systems, the role of power control is to combat a near-far problem due to severe intra-cell interference; hence, power control aimed to maintaining the same QoS of different users without considering overall spectral efficiency. In OFDMA systems, on the other hand, there is no nearfar problem, so we have room to consider other performance metrics which are important in cellular systems. In this paper, we emphasize that a power control scheme in OFDMA cellular systems can be designed to trade off average throughput and cell-edge throughput, rather than to focus only on one performance metric. For this, we provided a metric of tradeoff performance. We then proposed a new power control scheme which combines the optimum power of two extremes in a linear manner. Numerically we verified that the proposed scheme has a good tradeoff curve, i.e., the curve is concave. But the limit of tradeoff performance is still unknown. It is our
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[7] A. M. Rao, “Reverse link power control for managing inter-cell interference in orthogonal multiple access systems,” Proc. of IEEE VTC Fall, pp. 1837–1841, Oct. 2007. [8] N. Jindal, S. Weber, and J. G. Andrews, “Fractional power control for decentralized wireless networks,” IEEE Transactions on Wireless Communications, vol. 7, pp. 5482–5492, Dec. 2008. [9] W. Xiao, R. Ratasuk, A. Ghosh, R. Love, Y. Sun, and R. Nory, “Uplink power control, interference coordination and resource allocation for 3GPP E-UTRA,” IEEE Vehicular Technology Conference, pp. 1–5, Sept. 2006. [10] S. P. Boyd and L. Vanderberghe, Convenx Optimization. Cambridge University Press, 1st ed., 2004. [11] T. Hytonen, “Optimal wrap-around network simulation,” Helsinki University of Technology: Report A432, Oct. 2001.
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ultimate goal to find the limit, as well as address the closed loop power control. R EFERENCES [1] R. D. Yates, “A framework for uplink power control in cellular radio systems,” IEEE Journal of Selected Areas in Communications, vol. 13, pp. 1341–1347, Sept. 1995. [2] J. Zander, “Performance of optimum transmitter power control in cellular radio systems,” IEEE Transactions on Vehicular Technology, vol. 41, Feb. 1992. [3] J. Zander, “Distributed co-channel interference control in cellular radio systems,” IEEE Transactions on Vehicular Technology, vol. 41, Aug. 1992. [4] G. J. Foschini and Z. Miljanic, “A simple distributed autonomous power control algorithm and its convergence,” IEEE Transactions on Vehicular Technology, vol. 42, pp. 641–646, Nov. 1993. [5] 3GPP TR 25.814 V7.1.0., “Physical Layer Aspects for Evoloved Universal Terrestrial Radio Access”. Oct. 2007. [6] IEEE, 802.16m-08/0004r2, “IEEE802.16m Evaluation Methodology Document (EMD)”. July 2008.
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