Frequency-Time Scheduling Algorithm for OFDMA Systems

Rabie Almatarneh, Mohamed Ahmed, Octavia Dobre Faculty of Engineering and Applied Science, Memorial University of Newfoundland and Labrador

St. John’s, Newfoundland, Canada, {rabie, mhahmed, odobre} @ mun.ca

Abstract— Frequency-time scheduling is an essential radio resource management (RRM) function in Orthogonal Frequency Division Multiple Access (OFDMA) wireless systems. In the literature, there are several OFDMA scheduling algorithms such as the Hungarian and Max-Max algorithm. However, such algorithms do not consider the multiuser diversity into account. In this paper, we propose a scheduling algorithm that exploits the multiuser diversity in both time and frequency domains. Also, the proposed algorithm utilizes the Proportional Fairness (PF) criterion to achieve fairness among users in the system. In order to support multimedia bursty traffic, our algorithm allows more than one user to share a subband in each time frame. The proposed algorithm iteratively assigns the available subbands to be shared among different users concurrently. We compare the performance of the proposed algorithm with other OFDMA scheduling algorithms in the literature. Results show that the proposed algorithm outperforms other algorithms in terms of the throughput with comparable fairness performance.

Index Terms— Adaptive modulation and coding, Channel state information, Jain’s fairness index, Max-Max algorithm, OFDMA, Proportional fair scheduling, RRM.

I. INTRODUCTION OFDMA is an appealing multiple access technique specified for wireless networks, such as IEEE 802.16 [1]. This allows dynamic scheduling of the OFDM subcarriers to different users with two degrees of freedom, in the frequency and time domains. OFDMA efficiently utilizes the OFDM subcarriers; a pool of orthogonal narrowband subcarriers are assigned to carry data streams for different users simultaneously. In OFDMA systems, a group of adjacent subcarriers are usually grouped to represent a frequency virtual subband. Moreover, Adaptive Modulation and Coding (AMC) mode is employed to enhance the subbands spectral efficiency.

In [2]-[7], the downlink dynamic subcarrier scheduling is studied in the point-to-point networks over OFDM environment and the subcarrier assignment problem is formulated as a maximal bipartite matching problem. The Hungarian algorithm which maps subbands to users with the maximum possible throughput is implemented as an optimal solution. However, this algorithm is computationally expensive, and an iterative matrix based algorithm is implemented as a suboptimal solution to reduce the complexity with small decrease in the performance. In [8]-[9], the same subcarrier assignment

problem is solved as in [2] for mesh networks, by using integer linear programming. The downlink dynamic subcarrier scheduling problem is approached in [10] by using a PF scheduling scheme. Three algorithms are proposed, with different historical rate updating schemes. However, the distribution of subcarriers over different users is not fully utilized. OFDM PF scheduling solutions based on the PF scheme is presented in [11], and three algorithms are developed. However, the OFDM PF algorithms do not consider the frequency domain scheduling of subbands to meet the OFDMA capabilities; only one user accesses all the available subbands at each scheduling time.

In this paper, we present the performance of the Hungarian, Max-Max [7], and RR algorithms as candidate solutions for the subband scheduling problem. The Hungarian algorithm solves the assignment problem between users and subbands in order to achieve the maximum possible throughput. It allocates one subband to one user in every scheduling time. The Hungarian algorithm is computationally expensive and does not exploit multiuser diversity.

The Max-Max algorithm provides close performance to the Hungarian algorithm with lower complexity. The Max-Max algorithm establishes a cost matrix which contains the instantaneous data rate for all users on all subbands. The matrix dimension is U*N, with U represents the number of users, and N the number of subbands. A subband is allocated to the user with the highest instantaneous data rate, after which the user’s row and subband’s column are deleted. This operation is repeated until all subbands are allocated to all users.

RR algorithm admits users to access the subbands in a cyclic fashion; each user access one subband regardless of the channel state information (CSI). The three aforementioned algorithms allow only one user to access a subband in a fame duration. Also, these algorithms do not compensate for the service starvation over time. Moreover, these algorithms are not able to support multimedia bursty traffic.

We propose a modified Max-Max scheduling algorithm in order to exploit multiuser diversity, improve fairness, and meet the bursty traffic requirements. The multiuser diversity is exploited by allowing a user to access more than one subband, and also not granting a user on any subband

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during a frame time based on the channel conditions and the service satisfaction level. Fairness between users is maintained by applying the PF criterion over the time domain. Furthermore, the bursty traffic requirements are guaranteed by allowing subband sharing within the frame duration.

The performance of the proposed modified Max-Max and OFDM OF scheduling algorithm in terms of throughput and fairness are presented. Simulations show that our proposed algorithm provides higher throughput than the OFDM OF algorithm with close fairness behavior. The rest of this paper is organized as follows. Section II provides the system model. The assignment problem formulation and the proposed modified Max-Max scheduling algorithm are described in Section III. Simulation parameters and results are given in Section IV. Finally, conclusions are drawn in Section V.

II. SYSTEM MODEL In this paper, we consider a single cell downlink scenario for an OFDMA wireless system. The bursty traffic of the users is modeled based on the exponential distribution. In this model, the smallest data entity which the base-station can handle is a data packet. We consider fixed size packets. The inter-arrival time between packet requests is modeled as a random variable with an exponential distribution. The user’s location in the cell is a random variable uniformly distributed.

Each subband contains a number of adjacent subcarriers, which are highly correlated in frequency domain. In time domain, the frame duration is divided into time symbols. Fig. 1 shows the frequency-time resources of the OFDMA system [11]. As shown in Fig. 1, the minimum allocable resource unit is represented by the intersection between a symbol in time domain and a subband in frequency domain. The goal is to allocate the resource units to users with time and frequency diversity in order to maximize the total system throughput with reasonable fairness among users. Moreover, AMC is used to enhance the subbands efficiency. The suitable modulation level and coding rate are decided depending on the CSI of each subband.

Path loss, shadowing, and small scale fading are considered in the signal-to-noise ratio (SNR) calculation. Correlated Rayleigh fading is assumed between subcarriers. In order to meet the target bit error rate (BER) in the system, we consider the worst case subcarrier fading in each subband for the SNR and link budget calculations. Although the worst case subcarrier fading is considered in a subband, the overall SNR calculation does not significantly change because the fading difference between subcarriers within a subband is insignificant because the fading is highly correlated.

Fig. 1. Frequency-time OFDMA resources [11].

III. SCHEDULING ALGORITHMS In this section, the scheduling problem and the proposed modified Max-Max scheduling algorithm are introduced.

A. The Subband Scheduling Proplem Formulation

The subband scheduling for an OFDMA system with U users and N subbands can be described as follows [2], [7]-[8],

∑ ∑= =

U

u

N

nnunu sr

1 1,, max (1)

subject to

}, ..., ,3 ,2 ,1{ ,1:11

, NnsCU

unu ∈∀=∑

=

(2)

}, ..., ,3 ,2 ,1{ ,1:21

, UusCN

nnu ∈∀=∑

=

(3)

:3C QoS requirements, (4)

where ru,n is the data rate of user u on subband n, and su,n is a selection indicator, su,n={0, 1}, with su,n=1 meaning that subband n is allocated to the user u. Conditions C1 and C2 enforce a one-to-one mapping between users and subbands, and the optimal solution can be achieved using the maximal bipartite matching assignment [2].

Different versions of the Hungarian and heuristic solutions are proposed for this problem [7]-[9]. However, the throughput achieved using those solutions is low compared to the expected throughput of the OFDMA systems that support bursty traffic from multiple users. Furthermore, this one-to-one solution does not exploit multiuser diversity as it guarantee a subband for each user all the time. Also, this solution cannot guarantee a fair

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treatment among users, even if it allocates one subband to each user. The reason is that some users are assigned subbands with bad CSI which prevents those users from utilizing the assigned subbands.

Considering the efficiency and the bursty traffic requirements of the OFDMA systems, it is more suitable to allow subband sharing among users. In our solution, more than one user can share the same subband. In order to exploit multiuser diversity, users with bad conditions are prevented from accessing subbands until their CSI are improved. However, queuing some users within a frame duration does not affect the long-term fairness because the queued users already experience bad channel conditions and cannot benefit from the available subbands. With the PF criterion, the scheduler compensates for service starvation of a user when either the user is queued for a specific time or its channel conditions are improved in the next time frames.

In order to overcome the throughput reduction and guarantee fair treatment between users, the conditions (C1 and C2) are dropped. Hence, the subcarrier scheduling problem is no longer a maximal bipartite matching assignment, and a user can access more than one subband. In order to guarantee fairness among users, we use a PF scheduling criterion, expressed as [6], [11]

),1(/)( max arg , 1−=

≤≤tRtDj unuUu

(5)

where j is the selected user, and Du,n(t) and Ru(t) are the instantaneous data rate of user u on subband n and historical average data rate of user u respectively. Ru(t) is updated for all users on each subband as follows, [10]

⎪⎪⎩

⎪⎪⎨

⎧

=+−−

≠−−=

, , )(1)1() 11 (

, , )1() 11 ()(

, jutDT

tRT

jutRT

tR

nuc

uc

uc

u (6)

where Tc represents the observation window in time frames. Tc can be chosen as a tradeoff between efficiency and fairness. Larger values of Tc imply that the scheduling algorithm selects users in a more greedy approach to maximize the system’s throughput. On the other hand, smaller Tc values indicate more fairness among users in penalty of throughput degradation as more users with lower data rates are granted. However, Tc =0 means that the PF ranking does not consider any historical data rate information.

This PF criterion is integrated into the proposed modified Max-Max scheduling algorithm in order to guarantee fair treatment between users. The PF criterion compensates for the service shortage of users; also it gives more priority to users who have better instantaneous data rates in order to maximize the throughput.

B. Proposed Modified Max-Max Scheduling Algorithm

We modify the Max-Max algorithm as follows. 1) A granted user will not excluded in next scheduling iteration unless its queue is empty 2) A subband will not excluded in the next scheduling iteration unless it is fully utilized 3) A PF criterion is implemented to evaluate ranking of users on the subbands based on the instantaneous and historical data rates.

The proposed algorithm is divided into two stages. In the first stage, the instantaneous data rates, Du,n(t), are evaluated for each user on each subband. A cost matrix that reflects the ranking of all users on all subbands is generated based on the PF rule given in (5). In the second stage, the modified Max-Max scheduling algorithm is applied on the cost matrix as follows. In every iteration, the algorithm selects the user with the highest rank among all users on all subbands. After the user and subband are selected, the remaining part of the subband is reserved for that user depending on its traffic loads and channel conditions. The scheduled user requirements and the selected subband capacity are updated. A user will be removed from the cost matrix if its queue is empty. A subband will be removed from the cost matrix if the remaining part cannot support at least one packet for any user. Finally, at the end of each iteration, the historical data rate, Ru(t), for each user is updated based on (6). The algorithm terminates the iterations if all the subbands are consumed or all the users are satisfied. Moreover, at the beginning of every frame duration, the new instantaneous data rates, Du,n(t), are evaluated, and combined with the historical data rate, Ru(t), to start new cost matrix. The flow chart of the modified Max-Max scheduling algorithm is presented in Fig. 2.

Start a new frame duration

Estimate the CSIDu,n(t)

Build the cost matrixDu,n(t)/Ru(t-1)

Iterations end?Yes

Schedule the best userj=Du,n(t)/Ru(t-1)

No

Update Ru(t)Update statistics

Update user's queue

Update subband's capacity

Fig. 2. Modified Max-Max algorithm flow chart.

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IV. SIMULATION RESULTS A. Simulation Setup

The signal bandwidth is set to 20 MHz, and the carrier frequency to 2 GHz. The number of subcarriers equals 256, and each 8 subcarriers are grouped together into a subband. Moreover, the subcarrier with the worst SNR is selected to represent other subcarriers to guarantee a target BER probability within each subband. The transmit power is set to 0 dBW, while the noise power is assumed -130 dBW. The path loss exponent equals 4, and the standard deviation of the lognormal shadowing equals 10 dB. Also, we set the cell radius to 1500 m, and the number of users in the cell to 32.

In the time domain, the frame duration is 2 ms, and the symbol duration 16 μs. The AMC scheme parameters are given in Table 1, while the complete set of simulation parameters in Table 2. The proposed algorithm is examined by using four values of the observation window, Tc=5000, Tc =3000, Tc=1000, and Tc=0. When Tc=0, the scheduling algorithm ignores the previous history, Ru(t), and establish the cost matrix based only on the instantaneous data rate, Du,n(t). Ignoring the historical achievable data rates of users dramatically increases the system’s throughput at the cost of low fairness among users. The fairness among users is measured using the Jain’s fairness index [12],

, )(

1

2

2

1

∑

∑

=

=⎥⎦

⎤⎢⎣

⎡

= N

ii

N

ii

x

xxf (7)

Where xi is the average traffic achieved by user i.

Table 1. AMC Scheme [10].

Modulation Scheme

Code rate

Bits/symbol

SNR (dB)

BPSK 1/4 1/4 -2.9

BPSK 1/2 1/2 -0.2

QPSK 1/2 1 2.2

8PSK 1/2 3/2 5.2

8PSK 2/3 2 8.4

64QAM 1/2 3 11.8

64QAM 2/3 4 15.1

Table 2. Simulation Parameters.

Parameter Value

Bandwidth 20 MHz

Carrier frequency 2 GHz

Number of subcarriers 256

Number of subbands 32

Transmit power 0 dB

Noise power -130 dBW

Path loss exponent 4

Shadowing standard deviation 10 dB

Number of users 32

Packet size 180 bits

Frame duration 2 ms

Symbol duration 16 μs

Cell radius 1500 m B. Simulation Results

In this section, the performance of the proposed modified Max-Max scheduling algorithm is analyzed. The efficiency of the three one-to-one algorithms (Hungarian, Max-Max, and RR) in terms of throughput is presented in Fig. 3. The Hungarian solution outperforms the Max-Max and RR algorithms, as it guarantees the optimal user-subband assignment under the C1 and C2 constrains. The RR algorithm shows low throughput, as it does not consider the CSI while assigning subbands to users.

Jain’s fairness index is presented in Fig. 4 for the Hungarian, Max-Max, and RR algorithms. RR is the fairest algorithm as it tries to equally allocate resources to users in the system. The Hungarian and the Max-Max algorithms shows lower fairness because they admit users based on the instantaneous data rates. Although these two algorithms guarantee a subband for each user within the frame duration, the fairness is not high because some users access subbands without the ability to transmit due to bad channel conditions. The throughput of the proposed modified Max-Max and the OFDM PF scheduling algorithms is presented in Fig. 5. As expected, results in Fig. 5 show that the larger the observation window, Tc, the higher the throughput for both algorithms because the algorithms try to admit users with better CSI. Both algorithms show dramatical throughput increase compared to the one-to-one algorithms as these two algorithms exploit multiuser diversity. However, the proposed modified Max-Max algorithm shows better performance compared to the OFDM PF algorithm as it

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efficiently utilizes the resources in the frequency domain, and can handle efficiently the bursty traffic because of the subband sharing.

The fairness of the proposed modified Max-Max and OFDM PF scheduling algorithm is presented in Fig. 6 and Fig. 7, respectively. Both algorithms show same fairness among users. Also, we notice that as the observation window increases, the fairness decreases among users because the PF criterion gives more priority to users with higher data rates which cause other users to wait for longer time. Meanwhile, these two algorithms show slightly better fairness compared to the Hungarian and Max-Max algorithms due to the PF criterion implementation. It is evident that pursuing short-term fairness between users in the OFDMA systems by applying the one-to-one scheduling algorithms decreases the system throughput dramatically, without improving the fairness.

V. CONCLUSION In this paper, we propose a modified Max-Max scheduling algorithm that allows multiple users to share a subband concurrently, in order to exploit multiuser diversity and guarantee effective scheduling of the multimedia traffic from multiple users. Simulation results prove the efficiency of this scheduling algorithm in wireless systems that involve multiple users with multimedia traffic requirements. The one-to-one solution, represented by the Hungarian, Max-Max, and RR scheduling algorithms, does not exploit multiuser diversity. Also, this cannot support the bursty traffic requirements from multiple users, as it pursues a direct fair treatment, which severely affects the throughput, without improving the long-term fairness among users.

REFERENCES [1] “IEEE standard for local and metropolitan area networks part 16:

Air interface for fixed and mobile broadband wireless access systems, ” IEEE, Tech. rep. 802.16, Oct. 2004.

[2] S. H. Ali, K. D. Lee, and V. Leung, “Dynamic resource allocation in OFDMA wireless metropilitan area networks,“ IEEE Wireless Communications, vol. 14, pp. 6-13, Feb. 2007.

[3] K. D. Lee and V. C. Leung, “Fair allocation of subcarrier and power in an OFDMA wireless mesh network,“ IEEE Jornal of Selected Areas in Communications, vol. 24, pp. 2051-2060, Nov. 2006.

[4] C. Y. Wong and K. B. Letaief, “Multiuser OFDM with adaptive subcarrier, bit, and power allocation,“ IEEE Jornal ofSelected Areas in Communications, vol. 17, pp. 1747-1765, Oct.1999.

[5] G. Song and Y. Lee, “Cross-layer optimization for OFDM wireless networks—Part II: Algorithm development,“ IEEE Transactions on Wireless Communications, vol. 4, pp. 625-634, Mar. 2005.

[6] G. Huang, H. Juan, M. S. Lin, and C. Chang, “Radio resource management of heterogeneous services in mobile WiMAX systems,“ IEEE Wireless Communications, vol. 14, pp. 20-26, Feb. 2007.

[7] Z. Zhang , Y. He, and E. Chong, “Opportunistic downlink scheduling for multiuser OFDM systems,“ in Proc. Wireless Communications and Networking Conference, vol. 2, Mar. 2005, pp. 1206-1212.

[8] D. Niyato and E. Hossain, “A radio resource management framework for IEEE 802.16-based OFDM/TDD wireless mesh networks,“ in Proc. IEEE International Conference, vol. 9, June 2006, pp. 3911-3916.

[9] Y. J. Zhang and K. B. Letaief, “Multiuser adaptive subcarrier-and-bit allocation with adaptive cell selection for OFDM systems,“ IEEE Transactions on Wireless Communications, vol. 3, pp. 1566-1575, Sep. 2004.

[10] W. Anchun, X. Liang, Z. S. Xibin, and Y. Yan, ”Dynamic resource management in the fourth generation wireless systems,” in Proc. International Conference on Communication Technology, vol. 2, Apr. 2003, pp. 1095-1098.

[11] H. J. Zhu and R. H. Hafez, “Scheduling schemes for multimedia service in wireless OFDM systems,“ IEEE Wireless Communications, vol. 14, pp. 99-105, Oct. 2007.

[12] R. Jain, D. Chiu, and W. Hawe, “A quantitative measure of fairness and discrimination for resource allocation in shared computer systems,” Digital Equip. Corp., Littleton, MA, DEC Rep., DEC-TR-301, Sep. 1984.

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Fig. 4. Jain’s fairness index of the Hungarian, Max-Max, and RR algorithms.

Fig. 3. Throughput of the Hungarian, Max-Max, and RR algorithms.

Fig. 5. Throughput of the proposed modified Max-Max and OFMD PF algorithms.

Fig. 6. Jain’s fairness index of the proposed modified Max-Max algorithm.

Fig. 7. Jain’s fairness index of the OFDM PF algorithm.

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