Multiuser Scheduling over Rayleigh Fading Channels Fredrik Berggren
Riku J¨antti
Radio Communication Systems Dept. of Signals, Sensors and Systems Royal Institute of Technology S-164 40, Kista, SWEDEN Email:
[email protected]
Faculty of Technology Dept. of Computer Science University of Vaasa FIN-65101, Vaasa, FINLAND Email:
[email protected]
Abstract— Forthcoming wideband wireless systems are supposed to support services with loose delay requirements, which allows for scheduling of data. In this work we investigate fast transmission scheduling for downlink CDMA over a Rayleigh fading channel. For a logarithmic relation between the channel quality and the effective transmission rate, merely scheduling one user at a time may not result in maximum channel utilization. Therefore, we analyze the benefits of scheduling several users at a time with a suggested scheduler, containing regular CDMA as a special case. We consider a resource fair scheduler with the objective to asymptotically allocate the same amount of energy to all users. We derive the scheduling gain on closed-form and determine its limit values for both high and low average channel quality. The results show that the better the average channel quality becomes, the more users should be allowed to transmit simultaneously.
I. I NTRODUCTION To provide high-speed packet data services, emerging standards for next-generation DS-CDMA systems are currently extended to support higher data rates in the downlink. For this, time division along with fast channel adaptation techniques have been suggested. Both the outlined HDR mode [1] and HSDPA [2] consider a time divided downlink, in some cases offering peak rates around 10 Mbps. As a larger delay tolerance provides larger freedom to adapt to the channel conditions, transmission scheduling is an important means toward efficient channel utilization. The effect of multiuser diversity allows for transmitting data when the channel conditions are favorable. To reap the benefits of the asynchronous time variations, some form of channel estimation is needed as input to the scheduler. Rapid estimation and feedback of channel states have been suggested for HDR and HSDPA, for supporting fast scheduling, ARQ and link adaptation. Given the estimated channel quality, the task for the scheduler is to determine which user gives the “best” utilization, subject to some fairness constraint. Transmitting the users’ data in a oneby-one fashion can be regarded as providing an average data rate, as opposed to a continuous stream of data. In addition to multiuser diversity gains, time division benefits from less F. Berggren was supported by grant 18554-2 from VINNOVA. R. J¨antti was supported by grant 51521 from the Academy of Finland.
intracell interference. It has been shown that when the effective transmission rate is proportional to channel quality, this multiple access concept has merits in terms of substantially increased average throughput for static and fading channels respectively [3, 4]. Several fairness criteria and scheduler designs for downlink CDMA have previously been considered [4–11]. In particular it has been found that as a consequence of the multiuser diversity effect, the more users to choose from the larger gain from the scheduler. Thereto, large channel variations are not a drawback and could instead be preferred [4, 9, 11]. The above work has considered scheduling only one user at a time, i.e., time division. This is well motivated if there is a linear relation between effective transmission rate and channel quality, e.g., by adapting the processing gain by the symbol duration. Then the reduced intracell interference directly translates into a higher instantaneous transmission rate, which is sufficiently high to compensate for the periods transmission is not performed [3, 4]. However, if there is a diminishing return in rate from channel quality, the most efficient use of the channel is not necessarily given by single transmissions. For example, if the transmission rates are upper bounded or even discrete, there may be channel resources left for more than one user. In practice, a fully linear relation can be too optimistic if adaptive coding and modulation techniques are deployed. In this work, we will generalize the singleuser scheduling principle from [4] in order to allocate more than one user at a time to the channel. We will consider an asymptotic analysis, where fairness accounts for providing certain channel access time fractions among the users. That is, equal expected throughput is not necessarily guaranteed, rather the access to the channel. This will mean that the scheduler is resource fair in the sense that each user will asymptotically be allocated the same amount of energy. As special cases we have the previous scheduler [4] (one user on the channel) and the regular CDMA (all users on the channel). We will assume a logarithmic relation between effective transmission rate and average channel quality. It will be shown that the gain from scheduling a limited number of users at a time, is largest for channels with poor quality. That is if the signal-to-interference (SIR) is low, which could be caused by
poor orthogonality among the spreading codes or poor radio link conditions. For large SIRs, the average throughput with scheduling may in some cases become worse than regular CDMA, with all users on the channel. If the spreading codes exhibit good orthogonality, there is little to gain from deploying transmission scheduling. The performance of the scheduler is evaluated by numerical integration and derived limit values on closed-form. The rest of this paper is organized as follows. In Section II, the assumptions and system model are given. The scheduler is proposed and analyzed in Section III. In Section IV, the average gain is determined. The conclusions end the paper in Section V.
where the limit is stochastic and denotes convergence almost surely. The channel states ξi will be assumed mutually independent. Throughout the paper a Rayleigh fading channel is assumed. Consequently, the channel state ξi follows the exponential distribution given by 1 −x fξi (x) = ¯ e ξ¯i , ξi
x ≥ 0.
(3)
Thus the variations are due to small-scale fading and ξ¯i can be regarded as being comprised by shadowing and distance attenuation, varying on a much longer time scale. In Section IV, such channel variations will be included. III. M ULTIUSER S CHEDULING
II. S YSTEM M ODEL Consider a single cell in a DS-CDMA system with N users sharing a spreading bandwidth W . The data buffers are of infinite length and considered to be full all the time, i.e., queuing aspects are not considered. The total downlink transmission power is fixed at P . If one-by-one transmission is employed, all the transmission power in the cell can be given to a user when it transmits. If there is more than one user on the channel, we assume that P is shared equally among the users. Hence, we do not adapt the transmit power to the channel conditions, only to the number of users on the channel. Let for user i the constant Ii + νi denote the intercell plus background interference level and gi (t) the time variant link gain. We define the instantaneous channel state as the maximum SIR, ξi = gi P/(Ii + νi ). Since ξi is independent of any intracell interference, it serves as a relevant quantity of comparison between different schedulers. Consider the case where there are k users allocated to the channel giving each user a power of P/k. In the absence of multiuser detection, the instantaneous transmission rate of user i can be assumed to follow a logarithmic relation ξi (1) ri (ξi , k) = W log2 1 + θ(k − 1)ξi + k
where θi ∈ [0, 1] denotes the orthogonality factor, which is assumed to be a constant1 . We assume that ξi (t) can be accurately estimated and that it is a wide sense stationary and ergodic stochastic process for every user i. The channel state of each user has a mean E[ξi (t)|t] = ξ¯i for all t. From the these assumptions, we find the asymptotic data rate Z ∞ Z 1 T def ri (ξi , k)fξi (ξi ) dξi ri (ξi (t), k)dt = Ri (k) = lim T →∞ T 0 0 (2) 1 For large bandwidth systems and an AWGN channel, a linear relation between rate and quality is reasonable since prx prx lim W log2 1 + = log2 e bps, W →∞ I0 W I0
but we shall in this work refrain from adopting such an assumption.
A. Scheduling Algorithm Assume that the channel states of all users ξ = (ξi ) are error-free and available in the base station for scheduling decisions. The underlying scheduling principle is to schedule users when their channel quality is good relative to its expected quality. This will provide efficient channel utilization while ensuring fairness. The scheduler in [4], guarantees every user an asymptotic time fraction 1/N to the channel by the following rule. i∗ (ξ) = argmax1≤i≤N
ξi − ξ¯i ξ¯i
(4)
Now, let us generalize this principle by in any instant ordering the users such that their decision variables follow ξN − ξ¯N ξ1 − ξ¯1 ≥ ... ≥ ¯ ξ1 ξ¯N and schedule the L + 1 first users. Ties are broken arbitrarily. The values L = 0, . . . , N − 1 span one-by-one transmission to regular CDMA with N users on the channel. For a Rayleigh fading channel, we find that the conditional probability that user i is scheduled will equal L ξ ξ X N −1 − ij − i ∗ Pr[i (ξ) = i] = (1 − e ξ¯i )N −1−j e ξ¯i . (5) j j=0 Averaging (5) over (3), it can after some manipulations be found that any user will access the channel a fraction (L + 1)/N of the time. Hence, with a fixed power P/(L + 1), all users will asymptotically be allocated the same amount of energy. For a given L, the asymptotic throughput is defined as def Ri (L + 1) = E[ri (ξi , L + 1)1{i∗ (ξ)=i} ], (6) with the event indicator 1A =
(
1, 0,
if A occurs otherwise.
After some further manipulations, the asymptotic throughput
4.5
3 θ=1
4 2.5
3.5
Scheduling gain G
Scheduling gain G
3
2.5
2
2
θ=1 1.5
1.5
1
1 θ=0
0.5 θ=0 0
0
2
4
6
8
10 Average SIR
12
14
16
18
20
0.5
0
2
4
6
8
10 Average SIR
12
14
16
18
20
Fig. 1. The scheduling gain as function of average channel quality ξ¯i for different orthogonality factors θ = 0, 0.1, . . . , 1. The number of scheduled users equals 1 (L = 0) and N = 10.
Fig. 2. The scheduling gain as function of average channel quality ξ¯i for two different orthogonality factors θ = 0, 0.1, . . . , 1. The number of scheduled users equals 2 (L = 1) and N = 10.
of (1), (5) and (6) can be obtained on closed-form Z ∞ Ri (L + 1) = ri (ξi , L + 1)Pr[i∗ (ξ) = i]fξi (ξi ) dξi
In Fig. 1 we plot the scheduling gain for L = 0, i.e., one user is scheduled at a time from a total of N = 10. We observe that the larger θ, the more gain from scheduling. It is also noteworthy that for perfectly orthogonal channels, θ = 0, there exists a region (ξ¯i & 2.2) where regular CDMA transmission offers higher average throughput than the scheduled system. In that case, it is more favorable to let a user transmit all the time with 1/N :th of the power than 1/N :th of the time with full power. This is mainly due to that (1) increases less for large ξi , not making compensation for the silent periods possible. To fully assess performance, we study the limiting cases of high and low average channel quality. Using relations from Appendix, the scheduling gain (8) reduces to the form 1 L + 1 loge 1 + θL , (10) lim G = 1 N log 1 + ξ¯i →∞ e θ(N −1)
=
0 +1−j L+1 X NX j=1
N j
k=1 N +1−j−k
N +1−j k
×
j k (−1) × N loge 2 (N + 1 − j) (N + 1 − k) (L + 1)(N + 1 − k) − V ξ¯i (θL + 1) ! (L + 1)(N + 1 − k) (7) V ξ¯i θL def def R ∞ where V (x) = ex Γ(x) and Γ(x) = x t−1 e−t dt is the incomplete gamma function. B. Multiuser Scheduling Gain To compare with regular CDMA, L = N − 1, we define the scheduling gain as the normalized throughput def
G =
Ri (L + 1) , Ri (N )
(8)
where we can express Z ∞ Ri (N ) = ri (ξi , N )fξi (ξi ) dξi 0
=
1 N V ¯ − loge 2 ξi (θ(N − 1) + 1) ! N V ¯ . ξi θ(N − 1)
which could take values either below or above unity. In particular, it can be shown that for any L, lim G|θ=0 =
ξ¯i →∞
L+1 ≤ 1. N
Thus, if the channels are perfectly orthogonal and ξ¯i is large, regular CDMA transmission is always better. That is, for high ξ¯i , reducing the interference by scheduling results in an average throughput loss, since the effective transmission rate increases very little from the intracell interference reduction but the channel utilization reduces to (L + 1)/N . For θ > 0 and L = 0, the gain approaches lim G|θ>0,L=0 = ∞.
ξ¯i →∞
(9)
This is expected since Ri is for L = 0 independent of θ and an increasing function of ξ¯i . Ri (N ) on the other hand, can be
2.4
6 L=0 L=1 L=2
2.2 5 2 4 Average gain G
Scheduling gain G
1.8
1.6 θ=1
3
1.4 2 1.2
0.8
1
θ=0
1
0
2
4
6
8
10 Average SIR
12
14
16
18
20
Fig. 3. The scheduling gain as function of average channel quality ξ¯i for two different orthogonality factors θ = 0, 0.1, . . . , 1. The number of scheduled users equals 3 (L = 2) and N = 10.
bounded by Jensen’s inequality to Ri (N ) ≤ W log2 1 +
1 θ(N − 1)
,
k=1
k j (11) (N + 1 − j) (N + 1 − k)2
In particular for L = 0, it can after some algebra be shown that N X N k lim G|L=0 = (−1)N −k (N + 1 − k)2 k ξ¯i →0 =
k=1 N X
k=1
0
1
2
3
4
5 6 Average SIR C [dB]
7
8
9
10
Fig. 4. The average scheduling gain as function of channel quality C for two different orthogonality factors. The three upper curves have θ = 1 and the three lower θ = 0 The number of scheduled users equals 1-3 (L = 0, 1, 2).
IV. M ULTIUSER S YSTEM G AIN
which gives an increasing scheduling gain. Therefore, the curves in Fig. 1 for θ > 0, will eventually exceed unity as ξ¯i increases. As the channel quality degrades, we find that the limit value takes the following form: +1−j L+1 X NX N N + 1 − j (−1)N +1−j−k lim G = L+1 j k ξ¯i →0 j=1 ×
0
1 k
As opposed to the case of large ξ¯i , here, the scheduler gains over regular CDMA when N increases. It is noteworthy that G is in this case exactly equal to the scheduling gain that was found in [4] when the throughput was compared to that of a round robin scheduler, under a linear rate relation. This result is expected since log(1 + ξi ) = ξi + O(ξi2 ), i.e., a linear rate is well approximated in this region. In Figs. 2-3, we plot G when two and three users are scheduled, L = 1, 2. The region where the scheduling gain is larger than unity, is here broadened but the peak value at ξ¯i = 0 is lower. Here the gain always decreases with increasing ξ¯i .
As was found, the scheduling gain depends on the actual value of ξ¯i . To determine the gain of a system, the variations in ξ¯i must be accounted for. By regarding the scheduling gain as a conditional result, the system gain can be obtained by averaging over ξ¯i . If we let g be lognormally distributed with zero mean and standard deviation σ dB, then we can define ξ¯i as a compound random variable P c , ξ¯i = g · α · r Ii + ν
c>0
(12)
where r is the distance to the transmitter, also being a random variable. If we consider a circular cell with uniformly dispersed mobiles within the radius [rmin , rmax ], the density function becomes for x ≥ 0 10 1 × fξ¯i (x) = √ 2π σx loge 10 Z 1 1 10 log10 x+α10 log10 r−C 2 2r ) σ e− 2 ( dr (13) r2 rmin 1 − r2min rmax max
−α 10 log10 (cP rmax /(Ii
where C = + ν)) is the SIR at the cell border. The system scheduling gain can be obtained by averaging (7) and (9) over (13) and taking their ratio. In Fig. 4, we plot results obtained from numerical integration for different values of L and θ. Here rmax = 1000 · rmin , α = 4 and σ = 6 dB. It can be verified that, except for the case L = 0 and θ > 0, a higher average quality C provides less system gain from scheduling. This parallels the behavior in Figs. 1-3. If however, the orthogonality factor θ is known to take large values, single-user transmissions are preferable. It is observed that the larger L becomes, the less the span in system gain becomes between θ = 0 and 1.
V. C ONCLUDING R EMARKS If adaptive coding and modulation techniques are used, there may not necessarily exist a linear relation between the channel quality and the effective transmission rate. This may cause the channel to be underutilized had only one user been scheduled. Here, we have studied the effect of scheduling several users simultaneously, which was found to have merits when the average channel quality was good. In the limit of high average channel quality and almost orthogonal codes (θ small), the usual CDMA provides the best channel utilization. If the the orthogonality among the codes is small (θ large), one user should be allocated to the channel at a time. In practice, code and modulations levels are discrete, which suggests that these characteristics could be even more emphasized. A simple channel independent power control policy was adapted in this work. Other power control strategies may also be investigated. The analysis performed in this work considered a Shannon type of rate mapping. This implicitly assumes the interference to be Gaussian, which may be less valid when few users are scheduled. Further work is required to validate this assumption more rigorously. A PPENDIX The following expansions and identities are helpful in determining the scheduling gain. Z ∞ bc bc 1 −cξ a e Γ + log2 b log2 (b + aξ)e dξ = c loge 2 a 0 +1−j L+1 X NX j=1
k=1
×
N j
N + 1 − j (−1)N +1−j−k N k
L+1 j k = (N + 1 − j) (N + 1 − k) N lim V (k/x) = 0
x→0
lim V (k/x) = ∞
x→∞
For large x, V (k/x)
Z
= −γ − loge k − loge 1/x + (k + k(−γ − loge k − loge 1/x)) + O(x−2 ). x
∞ z
ta−1 e−t dt ∝ z a−1 e−z (1 + O(1/z)),
Γ(z) ∝
z→∞
(−1)n (1 + O(z + n)) , z → −n, n ∈ N + n!(z + n) R EFERENCES
[1] P. Bender, P. Black, M. Grob, R. Padovani, N. Sindhushayana, and A. Viterbi, “CDMA/HDR: A bandwidth-efficient high-speed wireless data service for nomadic users,” IEEE Commun. Mag., vol. 38, no. 7, pp. 70-77, 2000. [2] S. Parkvall, E. Dahlman, P. Frenger, P. Beming, and M. Persson “The high speed packet data evolution of WCDMA,” in Proc. IEEE VTC Spring, vol. 3, pp. 2287-2291, 2001. [3] F. Berggren, S.-L. Kim, R. J¨antti, and J. Zander, “Joint power control and intracell scheduling of DS-CDMA nonreal time data,” IEEE J. Sel. Areas Commun., vol. 19, no. 10, pp. 1860-1870, 2001. [4] F. Berggren, and R. J¨antti “Asymptotically fair scheduling on fading channels” in Proc. IEEE VTC Fall, vol. 4, pp. 1934-1938, 2002. Also to appear in IEEE Trans. Wireless Commun.. [5] N. Joshi, S. R. Kadaba, S. Patel, and G. S. Sundaram, “Downlink scheduling in CDMA data networks,” in Proc. ACM MobiCom, pp. 179190, 2000. [6] M. Andrews, K. Kumaran, K. Ramanan, A. Stoylar, and P. Whiting, “Providing quality of service over a wireless link,” IEEE Commun. Mag., vol. 39, no. 2, pp. 150-154, 2001. [7] X. Liu, E. K. P. Chong, and N. B. Shroff, “Opportunistic transmission scheduling with resource-sharing constraints in wireless networks,” IEEE J. Sel. Areas Commun., vol. 19, no. 10, pp. 2053-2064, 2001. [8] S. Borst, and P. Whiting, “Dynamic rate control algorithms for HDR throughput optimization,” in Proc. IEEE INFOCOM, vol. 2, pp. 976985, 2001. [9] J. M. Holtzman, “CDMA forward link waterfilling power control,” in Proc. IEEE VTC Spring, vol. 3, pp. 1663-1667, 2000. [10] J. M. Holtzman, “Asymptotic analysis of proportional fair algorithm,” in Proc. IEEE PIMRC, vol. 2, pp. 33-37, 2001. [11] P. Viswanath, D. N. C. Tse, and R. Laroia, “Opportunistic beamforming using dumb antennas,” IEEE. Trans. Inf. Theory, vol. 48, no. 6, pp. 1277-1294, 2002.
