Cell throughput analysis of the Proportional Fair scheduler in the single cell environment

Transcription

1 Cell throughput analysis of the Proportional Fair scheduler in the single cell environment Jin-Ghoo Choi and Seawoong Bahk IEEE Trans on Vehicular Tech, Mar 2007 *** Presented by: Anh H. Nguyen February 21, 2013

2 Outline Introduction 1 Introduction

3 Outline Introduction 1 Introduction

4 Schedulers Introduction Round Robin: sequentially allocates resource to users. Loss in multiuser diversity. Min-max: maximize the minimum rate. Proportional Fair [1, 2]: allocates reasonable portion of the resource to all users while giving preference to the users with good channel condition.

5 Outline Introduction 1 Introduction

6 A downlink multiuser system where the BS serves N users The received signal at user k is P k = h k 2 P t. h k = cd α k s k m k, (1) where c is constant, d k is distant BS-user k, random variable s k is for shadowing effect (log-normal with variance σ 2 s db), m k represents Rayleigh fading. The average received SNR of user k Z k = ρ(d/d k ) α s k, (2) where D is the radius of the cell, ρ = cd α P t /P n the average SNR at the cell edge.

7 Proportional Fair scheduler PF select user k k = arg max k R k [n] R k [n], (3) where R k [n] the instantaneous rate, R k [n] is the average throughput of user k { (1 R 1 t k [n + 1] = c ) R k [n] + 1 t c R k [n] k = k (1 1 t c ) R k [n] k k (4), where t c is the time constant for the moving average.

8 Outline Introduction 1 Introduction

9 Assumptions Introduction Users are distributed uniformly throughout the entire cell area. Every session is always active in the downlink direction. The distribution of channel gain of user k does not depend on time slot n and is constant for the slot duration. In this model, the ratio of the SNR to the average SNR is used. The feasible rate is a strictly monotonic increasing function of the SNR. Average throughput and average SNR are obtained by the time average.

10 Cell throughput of the PF scheduler Suppose the average rate of user k, R k [n], gets stable and stationary as time goes by T k = lim n Rk [n] = lim n E{R k [n]i k }, (5) with I k is the indicator which equal 1 when the user is allocated. The preference metric is Z k [n] Γ k = lim n Z k [n] = Z k, (6) Z k where Z k, Z k are the instantaneous and the average SNR.

11 Cell throughput of the PF scheduler The longterm average throughput of user k is T k = Pr{Γ k > Γ k }E{R k Γ k > Γ k } = ξ( ) ξ(0) ξ(t)f Γk (t)f Γk (t)dt, (7) where the instantaneous rate R k = ξ(γ k ), f Γk (t) is the distribution of Γ k = Z k Z k, and F Γk (t) is the distribution of the maximum Γ j with j = 1,..., K and j k.

12 Cell throughput of the PF scheduler Under Rayleigh fading, throughput of user k is T k = ξ( ) ξ(0) where ξ(t) is the rate function ξ(t) 1 ( Γ exp t ) ( ( 1 exp t N 1 dt, (8) Γ Γ))

13 PF - linear model Introduction The feasible rate is linearly proportional to the SNR R k = βwz k. The average throughput T k = βw Z k te t (1 e t ) N 1 dt N 0 ( ) βw = N M(N) E s ( Z k ) ( ) ( ) α βw D = N M(N) E s (s k )ρ, (9) d k with Z k = ρ(d/d k ) α s k and M(N) = N N 1 m=0 ( N 1 m ) ( 1)m (m+1) 2.

14 PF - linear model Introduction Taking average over the entire cell E A {.}, the cell throughput is ˆT cell = NE A {E s {T k }} = βwn(m)e s {s k }Ω 1 A = W 2ρβ 1 η 2 α 2 α 1 η 2 exp A ( D ρ d k ) α da ( ( ln σ s by using E s {s k } = exp(((ln10)/10 2σ s ) 2 ). ) 2 ) M(N), (10)

15 PF - logarithmic model ) The rate to user k is R k = W log 2 (1 + Z k K, where K is a constant depending on the system design and the target BER. Similarly, T k = W ln2 0 ( ln 1 + Z ) k K t e t (1 e t ) N 1 dt = W N 1 ( ( ln2 N 1 m ) ( 1)m (m + 1) K Z exp (m + 1) k m=0 N 1 W ν 1 m=0 ( ( N 1 m ) ( 1)m (m + 1) ln 1 + ) Ei ( ) (m + 1) K Zk ) ν 2 K (m + 1) Z k. (11) where ln(1 + at)e bt dt = 1 0 a exp(b/a)ei(b/a), the parameters ν 1 = 1.4 and ν 2 = 0.82.

16 PF - logarithmic model Taking expectation over shadowing fading and average over the entire cell area N 1 T cell Nν 1 ( N 1 m ) ( 1)m (m + 1) (B m + ν 3 ), (12) where B m is defined as m=0 B m = 2 D 2 D 0 ( ( ) α ) D r ln 1 + b m dr, (13) r with b m = (ν 2 ρ)/(k (m + 1)). Note B m can be exactly calculate for α integer. When α = 4, B m = ln(1 + b m ) + 2bm 0.5 arctan bm 0.5

17 Outline Introduction 1 Introduction

18 MIMO systems n T transmit antennas, n R receive antennas, n T = n R = n A. The signal received by RA j is y j = n T i=1 h ij x i + n j, (14) where n j denotes noise. Then, Z (j) k has exponential distribution. The cell throughput in logarithmic rate model is given N 1 T cell = Nn A ν 1 ( N 1 m ) ( 1)m (m + 1) (B m + ν 3 ). (15) m=0

19 PF in MIMO systems

20 Simulations results Single cell D = 1km. Transmit power P t = 10W. pathloss exponent α = 4, shadow fading σ s = 8dB. The median SNR at the cell edge ρ = 0dB. System efficiency factor K = 8dB. Two user 100, 200m from the BS.

21 Time average vs. moving average

22 Time average vs. moving average

23 Time average vs. moving average

24 Time average vs. moving average

25 Outline Introduction 1 Introduction

26 Conclusion Introduction Question: Is PF the best? We look for an alternative/complementary algorithm. Guarantee fairness. Have good performance. Be practical.

27 References Introduction F. P. Kelly, A. K. Maulloo, and D. K. Tan, Rate control for communication networks: shadow prices, proportional fairness and stability, Journal of the Operational Research society, vol. 49, no. 3, pp , J.-G. Choi and S. Bahk, Cell-throughput analysis of the proportional fair scheduler in the single-cell environment, Vehicular Technology, IEEE Transactions on, vol. 56, pp , march Thank you! Questions?