QoS Aware Power Allocation for Combined Guaranteed Performance and Best Effort Users in OFDMA Systems Mohmmad Anas∗ , Kanghee Kim∗ , Seokjoo Shin† and Kiseon Kim∗ ∗ Department
of Information and Communications Gwangju Institute of Science and Technology (GIST) 1 Oryong-dong, Buk-gu, Gwangju, 500-712, Republic of Korea Tel: +82-62-970-2264, Fax: +82-62-970-2274 Email:
[email protected] † Department of Internet Software Engineering Chosun University, Gwangju, 501-759, Republic of Korea
Abstract— One of the key issues in design of an orthogonal frequency division multiple access (OFDMA) based system is subcarrier and power allocation. Assuming knowledge of the instantaneous channel gains for all users, we propose a resource allocation scheme for an OFDMA based system to simultaneously provide two services, Guaranteed Performance (GP) and Best Effort (BE), with different quality-of-service (QoS). Subcarrier and power allocation are carried out sequentially to reduce the complexity, and an optimal power allocation procedure is derived. Furthermore, we present a reduced complexity suboptimal power allocation algorithm.
I. I NTRODUCTION Broadband wireless access (BWA) based on orthogonal frequency division multiple access (OFDMA) is considered as a standard for next generation (NextG) communication systems, to provide high-rate data communication over a wireless channel [1]. One of the goals of NextG is to provide heterogeneous services with diverse quality-of-service (QoS) requirements [2]. In this paper, we consider two types of users, Guaranteed Performance (GP) and Best Effort (BE), differentiated on the basis of required data rate and bit error rate (BER) criteria. Applications that require guaranteed QoS, such as bounded BER, and a guarantee on the throughput, are called GP services. On the other hand, applications those are less sensitive to instantaneous variations in available bandwidth and which do not require guarantees on throughput, are called BE services. In the context of Asynchronous Transfer Mode (ATM), this corresponds to Available Bit Rate (ABR) service category [3], which can adapt to the bandwidth unused by the GP service classes. In particular, we suppose that the BE users share the remaining bandwidth, that is left unused by the GP users. So far, several papers [4]-[7] have dealt with the problem of resource allocation for the multiuser OFDM system in a This work was supported (in part) by the Ministry of Information & Communication, Republic of Korea, under the Information Technology Research Center (ITRC) Support Program
downlink transmission. In [4], dynamic subchannel allocation is performed to maximize the minimum capacity of all users under the total transmit power constraint. In [5], authors attempted to minimize the total transmit power under fixed performance requirements (i.e., Rate and BER) and a given set of user data rates. They focus on the practical algorithms that can support real-time multimedia data whose data rates are generally fixed and BER requirements are same. In [6], an analytical proof for optimal subcarrier allocation, for transmit power adaptation is given to maximize the sum capacity of the users. In [7], an optimal power allocation is proposed to satisfy each users data rate proportionally. None of them [5]-[7], however, has considered the differentiation among the users on the basis of rate and BER requirements simultaneously. In this paper we generalize the resource allocation to applications where we are interested in simultaneously providing services with different QoS characterized by rate and BER performance. Without the restriction on how to assign the services to the subcarriers, we thus have Γ, SNR gap, or target BER as an additional variable to optimize. Hence, we propose a resource allocation algorithm, considering a practical scenario with users of dual service class differentiated on the basis of rate and BER constraints in an OFDMA system. Ideally, subcarriers and power should be allocated jointly to achieve the optimal solution. However this poses an extreme computational burden on the Base Station (BS) in order to reach the optimal allocation. Separating the subcarrier and power allocation is a way to reduce the complexity since the number of variables in the objective function is almost reduced by half [7]. Here, we assume a modified subcarrier allocation algorithm given in [4] to service combined GP and BE users. In this context we derive an optimal power allocation solution following analysis in [7] and propose a reduced complexity power allocation algorithm, the simulation results are compared with [4]. The rest of this paper is organized as follows. Section II contains system model and the problem formulation. In
Base Station Transmitter Subcarrier and bit/power allocation algorithm
Channel condition from user k
Subcarrier 1
User 1, R1
Subcarrier 2
User 2, R2
.….
.….
Encoder
IFFT and Parallel to Serial
Add cyclic prefix and D/A
Subcarrier N
User K, RK
Receiver for User 1
Channel for User 1
Receiver for User 2
Channel for User 2 …
Receiver for User K
Channel for User K
Subcarrier 1 FFT and Serial to Parallel
Subcarrier 2 Subcarrier Selector
.….
A/D and remove cyclic prefix
User K decoder
User K, RK
Subcarrier N
Subcarrier and bit/power allocation information
Receiver for User K
Fig. 1.
System Model of a downlink OFDMA System
Section III the optimal solution is derived and a suboptimal algorithm is proposed in Section IV. In Section V, we give simulation results of the proposed algorithm.Section VI contains the concluding remarks. II. SYSTEM M ODEL AND P ROBLEM F ORMULATION A schematic diagram of an OFDMA system used in this paper is shown in Fig. 1. In the figure, K denotes the total number of users and N denotes the total number of subcarriers. At the transmitter, the serial data stream from the K users are fed into the encoder block. Using the channel information from all K users, the subcarrier and bit/power allocation algorithm is applied to assign different subcarriers to different users. Here, we assume that a subcarrier at a particular time is not being shared among users. The number of bits and power allocated to each subcarrier is also determined in the process. This information is used to configure the encoder and the input data is encoded and transmitted accordingly. At the receiver, the subcarrier and bit/power allocation information is used to configure the subcarrier selector and decoder to extract the data from the subcarriers assigned to the k th user. Let’s assume that bk denotes a set of data symbols for the k th user and pk,n is the power allocated to the k th user’s nth subcarrier. Under the assumptions above, is the transmitted signal from the base station is detected by the k th user’s receiver, the decision statistic zk,n for the k th user’s and nth
subcarrier data symbol can be written as zk,n = bk,n pk,n hk,n + ηn
(1)
where, hk,n is a random variable representing the fading for the nth subcarrier between the base station and k th user’s receiver. ηn denotes the additive white Gaussian noise B . B is (AWGN) with mean zero and variance σ 2 = N0 N assumed to be total available bandwidth, hence signal-to-noise ratio (SNR) for the k th user’s nth subcarrier signal is, SNRk,n =
pk,n | hk,n |2 = pk,n Hk,n B N0 N
(2)
where, N0 is the noise power spectral density and Hk,n is carrier-to-noise ratio (CNR) for k th user’s nth subcarrier. Assuming that QAM modulation and ideal phase detection are used as in [6], the BER for the k th user’s nth subcarrier signal is bounded by −1.5SNRk,n 1 BER ≤ exp (3) 5 (2qk,n − 1) where, qk,n is the number of bits in each data symbol. Note that the BER bound (3) is valid for qk,n ≥ 2 and 0 ≤ SNRk,n ≤ 30 dB. For a given BER rearranging (3) yields the maximum number of bits in a symbol to be transmitted for the k th user’s nth subcarrier as SNRk,n qk,n = log2 1 + bits/symbol (4) Γ
where Γ = − ln(5BER)/1.5. Since the data rate of user k is viewed as the sum of the user’s subcarrier’s data rate, the data rate of user k in the OFDMA system is represented by qk,n B pk,n Hk,n = log2 1 + Rk = bps (5) T N Γ n∈Ωk
n∈Ωk
where, Ωk is the set of subcarriers allocated to user k. T is the OFDMA symbol duration i.e., T = N B seconds. In this paper, users are classified as either GP or BE users, out of total K users, first K1 are assumed to be GP users and, the next K − K1 are assumed to be BE users. Since BE users have no strict data rate requirements, we formulate our problem to maximize the sum-capacity of BE users for a given BER while satisfying the data rate requirements of all the GP users for a given BER under the total power constraint. Hence, the optimization problem is formulated as K B pk,n Hk,n log2 1 + (6) max N Γ2 pk,n k=K1 +1 n∈Ωk B pk,n Hk,n log2 1 + subject to: = Rk N Γ1
where, λk and σk are positive constants. We differentiate (7) with respect to pk,n and set each derivative to 0 to obtain, K
1 ∂L H1,n B = −λ1 + λk + σ1 = 0 (8) ∂p1,n N ln2 Γ1 + H1,n p1,n
k=2
Hk,n γ1 B ∂L |k=2,...,K1 = −λ1 − λk + σk = 0 ∂pk,n γk N ln2 Γ1 + Hk,n pk,n (9) ∂L Hk,n B |k=K1 +1,...,K = − λ1 + σ k = 0 ∂pk,n N ln2 Γ2 + Hk,n pk,n (10) A. Power Distribution for GP Users The optimal power distribution for single GP user is derived using (8) and (9). Here, we assume that each user’s subcarriers are arranged according to the ascending order of CNR i.e., Hk,1 ≤ Hk,2 ≤ . . . ≤ Hk,Nk [7].
n∈Ωk
pk,n = pk,1 +
K
pk,n ≤ Ptotal
pk,n ≥ 0 for all k, n R1 : R2 : . . . : RK1 = γ1 : γ2 : . . . : γK1 Ω1 ∪ Ω2 ∪ . . . ∪ ΩK ⊆ {1, 2, . . . , N } where, B is the total bandwidth; N is the total number of subcarriers; Ptotal is the total available power; Γ1 = − ln(5BER1 )/1.5 and Γ2 = − ln(5BER2 )/1.5 are the SNR gap for GP and BE users respectively; Ωk is the set of subcar1 riers for user k and they are mutually exclusive; {γk }K k=1 is a set of values proportional to the GP users rate. III. O PTIMAL P OWER A LLOCATION The optimization problem in (6) is a concave function of power hence is equivalent of maximizing the lagrangian of the aforementioned optimization problem [8]. B pk,n Hk,n log2 1 + L(pk,n , λk , σk ) = N Γ2 k=K1 +1 n∈Ωk K pk,n + λ1 Ptotal − K
k=1 n∈Ωk
σk pk,n .
where, Pk |k=1,2,...,K1 =
Nk
(12)
pk,n
(13)
Nk Hk,n − Hk,1 Γ1 Hk,n Hk,1 n=2
(14)
n=1
Vk =
(15)
and, Nk is the number of subcarriers in Ωk .
k=1 n∈Ωk
n∈Ωk
K
H1,1 P1 − V1 1 N1 + log2 W1 log2 1 + γ1 N Γ1 N1 1 Nk Hk,1 Pk − Vk = + log2 Wk log2 1 + γk N Γ1 Nk
N N1 k k Hk,n Wk = Hk,1 n=2
B p1,n H1,n + log2 1 + λk N Γ1 n∈Ωk k=2 γ1 B pk,n Hk,n log2 1 + − γk N Γ1 +
(11)
Equation (11) shows the optimal power distribution for a single user. More power will be put into the subcarriers with high CNR gain. This is waterfilling in frequency domain [7]. Using (11) and the rate requirements for GP users given in (6) we get,
k=1 n∈Ωk
K1
Hk,n − Hk,1 Γ1 Hk,n Hk,1
(7)
B. Power Distribution for BE Users Similarly, using (10), the optimal power distribution for a single BE user is derived. pk,n = pk,1 +
Hk,n − Hk,1 Γ2 Hk,n Hk,1
(16)
Subsequently, power distribution among the BE users is derived using (16) and assuming the data rate requirements for each BE user as equal, we get
NK HK,1 PK − UK + log2 WK log2 1 + N Γ2 NK Hk,1 Pk − Uk Nk + log2 Wk = log2 1 + N Γ2 Nk where, Nk Pk |k=K1 +1,K1 +2,...,K = pk,n
start K1
(17)
(18)
n=1
Uk =
Nk Hk,n − Hk,1 Γ2 Hk,n Hk,1 n=2
IV. P ROPOSED S UBOPTIMAL A LGORITHM In the proposed algorithm we subdivide power allocation procedure for GP and BE users. This approach is said to be suboptimal as we use an equal-power allocation scheme for BE users, and waterfilling solution to allocate power to GP users. Since, the equal-power distribution among subcarriers is shown to be near optimal in [6] for the sum capacity maximization problem under power constraints, the proposed scheme is said to be suboptimal. Fig. (2) summarizes the proposed suboptimal algorithm. Details of the proposed suboptimal power allocation algorithm are described as follows: Step 1. In the initial step, in order to quantify the amount of combined resources (power and subcarriers), we assume that the amount of power assigned to the users should be proportional to the number of subcarriers allocated. This initial step is based on the reasonable assumption made in [9]. Hence, estimate the total power allocated to GP and BE users respectively in proportion to the number of subcarrier allocated to GP and BE user class i.e., the power allocated to GP users Nk
and BE users is P = Ptotal k=1N respectively.
and, P = Ptotal − P
Step 2. Allocate power to GP user’s under the data rate constraints so as to minimize the use of total available power, P , using waterfilling solution. This problem is similar to [7] with the exception that [7] didn’t consider BER differentiation. Step 3. Calculate power margin from step 2 (i.e., γm = P − K
Pk ). Sum of margin from step 2 and power allocated to
k=1
Power allocated to GP users, P = Ptotal
∑N
k
k =1
N
Power allocated to BE users, P " = Ptotal − P '
Allocate power to GP user’s subcarrier under data rate constraint so as to minimize the use of total available power, P’, to GP users according to the proposed optimal solution K1
(19)
Set of nonlinear equations in (12), (17), along with total power constraint can be solved iteratively for {Pk }K k=1 using Newton’s Method. Consequently (11), (13), (16), (18) are solved for the power distribution among each subcarrier of each user. We notice from (11) and (16) that the only difference between power allocation to GP users and BE users is the SNR gap (i.e., Γ1 and Γ2 ). Set of nonlinear equations in (12), (17) can be approximated to a single equation for special case of high CNR. The approximations follow exactly as in [7] and are not outlined here.
K
1
Distribute power among GP users and BE users proportional to the number of subcarriers allotted to each service
'
Calculate power margin
Power margin, γm = P ' − ∑ Pk k =1
Capacity of each BE user, Above found margin is then distributed equally among the subcarriers alloted to BE users
Rk k =K +1,K +2,…,K = 1
1
∑
n ∈Ωk
stop
Fig. 2.
⎛ ⎞⎟ ⎟ ⎜⎜⎜ 1 P '' + γm H k ,n ⎟⎟⎟ ⎜ log 2 ⎜⎜1 + K ⎟ N Γ ⎟⎟ ⎜⎜ N k 2 ⎟⎟⎟ ∑ ⎜⎜ ⎝ ⎠⎟ k =K 1 +1
Flowchart of the proposed suboptimal algorithm
BE users from step 1 (i.e., γm +P ) is then equally distributed among the subcarriers allocated to the BE users. Hence, the capacity of each BE user is represented as, ⎛ ⎞ Rk =
⎜ B γm + P Hk,n ⎟ ⎜ ⎟ log2 ⎜1 + ⎟ K
N Γ2 ⎠ ⎝ n∈Ωk Nk
(20)
k=K1 +1
V. S IMULATION R ESULTS To investigate the performance of the proposed algorithm simulations has been performed with the following parameters: number of subcarriers N = 64; the number of users, K, was in between 4 and 16; the required BER is 10−5 and 10−3 for GP and BE users respectively; rate requirements of GP users are considered to be equal. The channel is considered to be frequency selective multipath channel consisting of six independent Rayleigh multipaths, with an exponential decaying profile. The maximum delay spread is 5 microsecond. The maximum doppler frequency spread is 30 Hz. The total power available at the base station is 64 W. The power spectrum density of additive white Gaussian noise is -80 dBW/Hz. The overall bandwidth is 1 MHz. The user locations are assumed to be equally distributed. Fig. (3) shows the plot of the minimum capacity of user vs. user number in the OFDMA system at different BER. We can see from fig.(3) that dynamic resource allocation achieve significantly higher capacity gain over fixed time division multiple access (TDMA) i.e., a fixed time slot is allotted to each user in TDMA. Also the equal-power allocation is shown to give near similar performance as that of optimal power allocation. Fig. (4) shows the capacity gain vs. the user number at different BER. We can see that capacity gain of optimal power
110
3.5
Proposed Optimal at Pe = 1e-3
Proposed Optimal at Pe = 1e-3 Proposed Optimal at Pe = 1e-5
min(Rk ) [bits/s/Hz]
2.5
Method in [4] at Pe = 1e-3
Method in [4] at Pe = 1e-5
Method in [4] at Pe = 1e-5
Fixed-TDMA at Pe = 1e-3 Fixed-TDMA at Pe = 1e-5
2
1.5
1
0.5
0
Proposed Optimal at Pe = 1e-5
100
Method in [4] at Pe = 1e-3
Capacity gain over TDMA [%]
3
90
80
70
60
50
4
6
Fig. 3.
8
10 12 Number of users (K)
14
16
Capacity in OFDMA system vs. user number
allocation and equal power allocation over TDMA increases as the number of user increases. This phenomenon is also known as multiuser diversity. Also we can see that in a system of 16 users with the proposed optimal power allocation solution achieves 18.67% and 19.64% more capacity gain than the scheme with equal power, when compared to fixed TDMA at BER of 10−3 and 10−5 respectively. VI. C ONCLUSION AND F UTURE WORKS In this paper, we present an analytical solution for optimal power allocation to provide services to heterogeneous users differentiated on the basis of required QoS for determined subcarrier allocation. Furthermore, we propose a reducedcomplexity suboptimal power allocation algorithm, where we allocate power to GP users using optimal waterfilling solution and BE users according to the equal-power allocation method. Result shows that the equal-power allocation method has near same performance as optimal waterfilling solution. At the same time we see a proportional reduction in computational complexity with the increase in the number of BE users. This follows from the fact that for the equal-power allocation we need to equally divide the power among the subcarriers allocated to that of BE users, and only have to calculate the optimal power allocation for tight QoS GP users. As is seen that to obtain an optimal solution, to provide heterogeneous service, we need to solve a set of multivariate nonlinear equations which adds to system complexity. Hence,
40
4
6
Fig. 4.
8
10 12 Number of users (K)
14
16
Capacity gain over TDMA vs. user number
as a future work, we recognize the need to implement the proposed reduced-complexity suboptimal power allocation algorithm and to compare its performance with the expected performance. R EFERENCES [1] IEEE Standard for Local and Metropolitan Area Networks Part 16: Air Interface for Fixed Broadband Wireless Access Systems, IEEE Std 802.16-2001. [2] K. Kim, I.S. Koo, S. Sung, and K. Kim, ”Multiple QoS support using MLWDF in OFDMA adaptive resource allocation,” Proc. of IEEE LANMAN Workshop 2004, pp. 217-222, March 2004. [3] E. Altman, and H.J. Kushener, “Admission Control for Combined Guaranteed Performance and Best Effort Communications Systems under Heavy Traffic,” SIAM J. Control and Optimization, vol. 37, no. 6, pp. 1780-1807, 1999. [4] W. Rhee, and J.M. Cioffi, “Increase in Capacity of Multiuser OFDM system Using Dynamic Subchannel Allocation,” Proc. of IEEE VTC 2000, vol. 2, May 2000. [5] C.Y. Wong, R.S. Cheng, K.B. Letaief, and R.D. Murch, “Multiuser OFDM with Adaptive Subcarrier, Bit, and Power Allocation,” IEEE Journal on Selected Areas in Communications, vol. 17, no. 10, October 1999. [6] J. Jang, and K.B. Lee, “Transmit Power Adaptation for Multiuser OFDM Systems,” IEEE Journal on Selected Areas in Communications, vol. 21, no. 2, February 2003. [7] Z. Shen, J.G. Andrews, and B.L. Evans, “Optimal Power Allocation in Multiuser OFDM Systems,” Proc. of IEEE GLOBECOM 2003, vol. 1, December 2003. [8] W. Yu, and J.M. Cioffi, “On Constant Power Water-filling,” Proc. of IEEE ICC 2001, pp. 1665-1669, June 2001. [9] H. Yin, and H. Liu, “An Efficient Multiuser Loading Algorithm for OFDM-based Broadband Wireless Systems ,” Proc. of IEEE Globecom 2000, vol. 1, pp. 103-107, November 2000.
