Abstract — The focus of this paper is on joint channel-aware and queue-aware scheduling schemes for the downlink of Orthogonal Frequency Division Multiple Access (OFDMA) networks. The objective is to enhance the stability of the queues while improving the network throughput. For this purpose, a Multi-Carrier Opportunistic Stable Queue (MC-OSQ) scheduling scheme is proposed, and a multi-carrier Modified Largest Weighted Delay First (M-LWDF) scheduling scheme is extended. The proposed stability metric used by MC-OSQ makes the network queues stable, while improving the network aggregate throughout.The comparable performance of MC-OSQ with that of M-LWDF reveals that the proposed stability metric is effective in reflecting the queue status in queue-aware scheduling schemes.

I. INTRODUCTION Resource (namely power and spectrum) allocation is crucial in wireless communication networks to compensate users’ service degradation due to channel impairments. Multipath propagation, shadowing, and path loss cause random variations of wireless channel. To combat some fading channel impairments, such as, intersymbol interference, Orthogonal Frequency Division Multiplexing (OFDM) technique has been developed. However, more performance improvement can be achieved when a multiple access mechanism based on OFDM, i.e., OFDMA, is in place. The granularity and flexibility of OFDMA in subcarrier and power allocation can improve multiuser diversity gain [1] while mitigating users’ service degradations. To gain from multiuser diversity, an opportunistic scheduling scheme assigns each subcarrier to a user that has the highest channel gain on that subcarrier. Despite throughput maximization, opportunistic scheduling schemes cause users with poor channel quality to fail accessing the channel most of the time and to suffer from backlogged traffic in their queues. So, an applicable scheduling scheme for wireless networks should be both Channel-Aware and Queue-Aware (CAQA) to take advantage of users’ channel diversity while being fair to unfortunate users with long backlogged queues.

Opportunistic Stable Queue (OSQ) and M-LWDF are CAQA scheduling schemes which have been recently proposed for single-carrier networks in [2, 3], and [4], respectively. OSQ schedules users’ traffic based on channel quality and stability status of each queue defined as the ratio of average departure rate to average arrival rate of the queue. In OSQ scheme, users with low queue stability or good channel gain have higher priority to be allocated resources. Similarly, M-LWDF privileges users with good channel quality. But, the queue status metric which is considered for scheduling is the queue length. The success of OSQ and M-LWDF in scheduling single-carrier networks traffic motivated us to propose a multi-carrier OSQ denoted as MC-OSQ and extend M-LWDF for scheduling traffic in multicarrier OFDMA networks. The contribution of this paper is twofold. First, we propose a mathematical model for subcarrier assignment and power allocation in OFMDA networks where the traffic is scheduled by MC-OSQ scheme. Second, we extend M-LWDF scheduling scheme for resource allocation in OFDMA. Due to the high computational complexity of the mathematical models, a suboptimal approach based on decoupling techniques is proposed. Assuming equal power allocation, subcarriers are assigned; then, a greedy bit loading algorithm is deployed for power allocation. The performance of both scheduling schemes is evaluated in scenarios where users are homogenous or heterogeneous in terms of traffic arrival rate or channel quality. Numerical results demonstrate the comparable performance of the proposed scheduling schemes. The remaining of the paper is organized as follows. The system model, including the network architecture, bit loading procedure, and MC-OSQ and M-LWDF problem formulations, is described in section II. A literature review of some relevant works is presented in section III, which is followed by the heuristic approaches for MC-OSQ and M-LWDF scheduling schemes. Performance evaluation of these schemes in terms of

A Stability-Based Scheduling Scheme for

OFDMA Networks

*Mohammad Fathi, *Hassan Taheri, **Mehri Mehrjoo *Department of Electrical Engineering, Amirkabir University of Technology, Tehran, Iran

**Department of Electrical and Computer Engineering, University of Sistan and Baluchestan, Iran email:{mfathi, htaheri}@aut.ac.ir, mehrjoo@ece.usb.ac.ir

2010 5th International Symposium on Wireless Pervasive Computing (ISWPC)

978-1-4244-6857-7/10/$26.00 ©2010 IEEE 477

throughput and stability is presented in section IV. The paper is concluded in section V.

II. SYSTEM MODEL

A. Network Architecture We consider downlink scheduling in a network with point to multipoint infrastructure. Data traffic is admitted to the network and buffered in separate queues in a base station (BS). The scheduler located in BS assigns OFDM subcarriers and allocates a portion of BS power to each subcarrier based on channel state information (CSI) and queue state information (QSI), which are assumed to be available at the scheduler and remain unchanged during each scheduling interval. The scheduling decision is made periodically at the beginning of each OFDM symbol, noted as time slot. Fig.1 illustrates the system architecture along with CSI and QSI signaling.

Fig. 1. Network Architecture

B. Bit Allocation Channel gain for a given user m on subcarrier k during a time slot is defined as αmk=|Hmk|2/N, where N represents noise power and Hmk depends on path loss, shadowing, and fading. Accordingly, Received Signal to Noise Ratio (SNR) is obtained by γmk=αmk pk, where pk is allocated power to subcarrier k. The achievable rate for user m on subcarrier k with bandwidth Δf (in Hz) can be expressed as 2log (1 )mk mkc f βγ= Δ +⎢ ⎥⎣ ⎦ , where β=-1.5 / (ln (5.BER)) is a function of the link bit error rate (BER) [5]. An adaptive modulation (AM) takes channel SNR into account to adopt the type and order of modulation of each subcarrier [6]. We consider M-QAM modulation with M=2c, c = 1,…, C, where C determines the highest modulation level allowed. The set of possible received SNR is partitioned into C+1 disjoint regions by boundary points b0<b1<…<bC<bC+1, where b0=0, bC+1=∞, and bc=(2c-1) (ln(5BER)) /(-1.5). Therefore, discrete rate transmission (in bps) of user m is obtained as follows:

2log (1 )mk mk Cmk

mk C

f bc

f C bβγ γ

γ⎧Δ + <⎢ ⎥⎪ ⎣ ⎦= ⎨ Δ ≥⎪⎩

(1)

C. MC-OSQ and M-LWDF Problems Formulations

MC-OSQ and M-LWDF schemes can be formulated as optimization problems where the objective functions and constraints are determined by the scheduling goals and users’ requirements.

The main objective of MC-OSQ is maximizing long term throughput subject to stabilizing queues in the system. Let the total number of users and subcarriers be M and K, respectively. Long term throughput maximization is obtained by

max1

M

mm

U η=

= ∑ , (2)

where 1

1lim min( ( ), ( ))t

m m mt nc n Q n

tη

→∞ == ∑ is the long term average

throughput of user m [7] and ( ) ( )m mkk m

c n c n∈Φ

= ∑ is the

achievable rate of user m upon the assignment of subcarrier set Φm. The throughput of user m at time slot n is bounded by its queue length Qm(n).

The objective function is restricted with the set of constraints (3) which maintain the same degree of stability among users:

1 2( ) ( ) ... ( )MS n S n S n= = = (3)

At time instant n, the degree of stability Sm(n) of queue m, with instantaneous arrival rate ( )mr n , is defined as

( )( )

( )m

mm

c nS n

r n= , (4)

where ( )mc n and ( )mr n are the average departure rate and the average arrival rate of queue m, respectively. An exponentially weighted moving average (EWMA) technique is adopted to estimate average rates. At the beginning of time slot n+1 average departure rate ( )mc n and average arrival rate ( )mr n are updated as follows:

( 1) (1 1/ ) ( ) (1/ ) ( )m c m c mc n T c n T c n+ = − + , (5)

( 1) (1 1/ ) ( ) (1/ ) ( )m c m c mr n T r n T r n+ = − + . (6)

Tc represents the average window length. Timing representation of arrival and departure rates of queue m is shown in Fig.2.

The instantaneous queue length is updated by

( 1) min(( ( ) ( )) ( ), )m m m mQ n Q n c n r n B++ = − + (7)

where B is the buffer size.

BS

Scheduler

λ1 Q1

.

.

.

User 1

User 2

User M

.

.

.

Channel Information Queue Information

λ2

λM QM

d1

d2

dM

Q2

478

In OFDMA networks, each subcarrier is assigned exclusively to one user, so the sets of subcarriers assigned to users should be disjoint:

m m m m′ ′Φ ∩ Φ = ∅ ∀ ≠ . (8)

We allow all subcarriers to be assigned to users, i.e.,

1| |

M

mm

K=

Φ =∪ . (9)

Fig. 2. Timing representation of arrival and departure rates

The objective function along with the network constraints form the optimization problem (P1), which is a Mixed Integer Nonlinear Programming (MINLP) problem with integer variable

set { }1

Mm m

Φ= and continuous variable set { }

1K

pk k = .

{ } { } 1,1 1

1

1

1 2

( 1) : max (10)

: (11)

| | (12)

( ) (13)

( ) ( ) ... ( ) (14)

M

mM K mpm km k

m mM

mmK

k BSk

M

P U

subject to m m

K

p n P

S n S n S n

η=Φ

= =

′

=

=

= ∑

′Φ ∩Φ = ∅ ∀ ≠

Φ =

≤∑

= = =

∪

Similarly, M-LWDF scheme can be represented mathematically by an optimization programming problem (P2) as follows.

{ } { } 1,1 1

1

1

( ) ( )( 2): max (15)

( )

: (16)

| | (17)

( ) (18)

Mm m

M K mp mm km k

m mM

mmK

k BSk

Q n c nP U

c n

subjectto m m

K

p n P

=Φ= =

′

=

=

= ∑

′Φ ∩Φ =∅ ∀ ≠

Φ =

≤∑

∪

The objective of M-LWDF is to maximize( ) ( )

( )m m

m

Q n c nc n

at time

slot n subject to the network constraints (16), (17) and (18). The optimal solution of (P1) and (P2) identify a subcarrier set assigned to each user and a portion of PBS allocated to each subcarrier.

III.HEURISTIC ALGORITHMS The MINLP formulation is a common approach for joint subcarrier assignment and power allocation problems in OFDMA networks [7-12]. However, due to the computational complexity of finding the optimal solutions of MINLP problems, some heuristic approaches have been proposed for practical implementation of the resource allocation problems. In subsection III.A, we review some of these approaches briefly. Due to the high complexity of finding optimal solution of (P1) and (P2) in real-time, in subsections III.B and III.C, we propose a two-stage heuristic solution for (P1) and (P2) that can be performed in practical time. In the first stage, subcarrier assignment is performed based on joint CSI and QSI of users, assuming uniform power allocation to users. Then, having subcarriers assignment, we modify the power allocations in the next stage.

A. Literature Review Assuming uniform power allocation in [7], first subcarriers are assigned, and then allocated power to assigned subcarriers is optimized. In [8], a subcarrier assignment scheme is proposed to minimize transmit power and satisfy minimum rate constraints of users. The assignment problem is formulated as an integer programming, and a suboptimal solution is proposed by continuous relaxation of integer variables. Subcarrier assignment problem with fix modulation level is solved in [9] using Hungarian method [10]. Hungarian is a solution for assignment problems with high computational complexity of O(K4), where K is the number of subcarriers. A dynamic subcarrier assignment based on users’ traffic and channel state information has been derived in [11] by modeling each user’s buffer as an M/D/1 queue with finite space. However, the BS power constraint has not been considered in this work. A utility-based theoretical framework for radio resource allocation in OFDMA networks is modeled in [12], and some heuristic approaches are proposed for the cases that the utility functions are concave.

B. Subcarrier Assignment Subcarriers are assigned using a priority matrix

( ) [ ( )]mkn nψΨ = , at time slot n, where ( )mk nψ is the priority value of user m on subcarrier k. Each subcarrier is assigned to a user who has the highest priority value. For MC-OSQ scheduling scheme, we define the priority value of user m on subcarrier k as

Time

Queue m

rm(n)

cm(n) n n+1

rm cm

( )mr n( )mc n

( 1)mr n +( 1)mc n +

479

( )( ) ( )10 S nmmk mkn c nψ −= , (19)

where ( )mkc n is obtained under the assumption that BS power is distributed equally among subcarriers. For M-LWDF scheduling scheme, the priority value ( )mk nψ is defined as

( )( ) ( ).

( )mk

mk mm

c nn Q n

c nψ = . (20)

C. Power Allocation Having subcarriers assignment, problem (P1) and (P2) are converted to a Non-Linear Programming (NLP) optimization problem, which can be solved optimally by water-filling approach. However, as water-filling is based on continuous rate adaptation, we adopt a greedy bit loading algorithm where bits are loaded on subcarriers progressively [12]. According to (1), user m with channel gain mkα requires

2 1bk

kmk

pβα

−= to transmit kb bits on subcarrier k. Therefore, the

additional required power to transmit one more bit on this

subcarrier is12 2b bk k

kmk

pβα

+ −Δ = . To maximize overall transmitted

bits in a time slot, subcarriers with smaller required power must be given higher priority in the power allocation algorithm. As a result, the greedy bit loading algorithm, shown in TABLE 2, chooses a subcarrier with the smallest kpΔ value to increase

kb to 1kb + iteratively. This is repeated until the overall allocated power pt exceeds the BS power, PBS.

TABLE 2: POWER ALLOCATION ALGORITHM

IV. PERFORMANCE EVALUATION We evaluate the performance of MC-OSQ and multi-carrier M-LWDF in terms of throughput, queue stability and fairness. Also, to show the advantages of CAQA scheduling schemes over queue-aware (only) scheduling schemes, we develop a multi-carrier stable queue (MC-SQ) scheme. Irrespective to

users’ channel quality, MC-SQ assigns all subcarriers to a user with minimum degree of stability at each time slot. At the beginning of time slot n, available packets in queues are scheduled for transmissions. The achievable packet rate of each subcarrier is determined assuming that the channel condition is static during that time slot. The packet arrival process distribution is Poisson, and the duplexing mode is frequency division. Table 3 lists the parameters used in the simulation. Two network scenarios, denoted as net-1 and net-2, are simulated. In net-1, traffic arrival rates are heterogeneous, and users are located at the same distance from the BS. In net-2, the users have been distributed randomly in a cell with 500 meters radio range, and their arrival rates are homogenous.

TABLE 3: SIMULATION PARAMETERS

Description Notation value total bandwidth 3.2MHz number of OFDM subcarriers K 256 number of users M 8 base station power PBS 2Watt noise power per subcarrier -110dB bit error rate 10-3 packet length 66*8 bit buffer size for each user B 25 packets time slot length 10 ms path loss exponent 3 shadowing standard deviation 10dB mean of Rayleigh fading channel 0dB maximum modulation level C 6 time constant Tc 50

The average throughput (packet/time slot) and the average degree of stability for a network with 16 users are shown in fig. 3 for net-1, where the users are sorted in an ascending order according to their arrival rates. As shown, MC-OSQ performs slightly better than M-LWDF, especially when the arrival rates increase. In other words, the degree of stability is more effective than the queue length in reflecting the queue status when the arrival rate increases. MC-SQ scheme has the lowest performance in this figure as it does not consider the users’ CSI. Fig. 4 demonstrates per user throughput and stability performance versus variable number of users. The scheduling schemes performance degrades by increasing the number of users; MC-OSQ and M-LWDF have almost the same performance which is much better than the ones of MC-SQ. Fairness is another scheduling performance metric which is investigated. In Fig. 5, Jain fairness indices [13] of examined schemes are drawn for variable number of users. Jain fairness index is defined as:

1

*

*

* * *

: 0, 0: {

2 2 ;

arg min ;

( ) ;

1; ;

}

k

b bk k

kmk

kk

BSk

k k k

Initialization pt b for all kdo

p for all k

k p

if pt p P break

else b b pt pt p

βα

+

= =

−Δ =

= Δ

+ Δ >

= + = + Δ

480

2

1

2

1

( )M

mm

M

mm

SF

M S

=

=

=∑

∑ , (21)

which represents the fairness of a scheme in maintaining the same degree of stability among the users. As shown in Fig. 5,

fairness of M-LWDF drops more rapidly than the ones of MC-OSQ when the number of users increases and the resources are scarce to serve them efficiently. Also, it is observed from fig. 4 and 5 that MC-SQ provides high fairness but low degree of stability with variable number of users.

2 4 6 8 10 12 14 162

3

4

5

6

7

8

User ID

Ave

rage

thr

oghp

ut

MC-OSQM-LWDFMC-SQ

2 4 6 8 10 12 14 16

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

User ID

Ave

rage

sta

bilit

y

MC-OSQM-LWDFMC-OQ

Fig. 3. Performance comparison of the scheduling schemes for net-1

12 13 14 15 16 17 18 19 202

3

4

5

6

7

8

9

Number of users

Thr

ough

put

per

user

MC-OSQM-LWDFMCSQ

12 13 14 15 16 17 18 19 20

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of users

Ave

rage

sta

bilit

y

MC-OSQ

M-LWDFMCSQ

Fig. 4. Performance comparison of the scheduling schemes for net-2

481

12 13 14 15 16 17 18 19 200.92

0.94

0.96

0.98

1

Number of users

Fai

rnes

s

MC-OSQ

M-LWDF

MC-SQ

Fig. 5. Jain fairness index

V. CONCLUSION The degree of stability is proposed as a metric for queue-aware scheduling in OFDMA wireless networks. Numerical results show that the proposed metric is successful in reflecting a queue status, in terms of backlogged traffic, comparing with the queue length metric used in M-LWDF scheduling scheme. Specifically, it maintains almost the same degree of stability and fairness among users with heterogeneous arrival rates, while improves the throughput with respect to M-LWDF scheduling scheme.

REFERENCES [1] P. Viswanath, D.N.C. Tse and Laroia, “Opportunistic beam forming using

dumb antennas,” IEEE Transaction on Information Theory, vol. 48, pp. 1277–1294, June 2002.

[2] M. Mehrjoo, X. Shen, and K. Naik, “A Joint Channel and Queue-Aware Scheduling for IEEE 802.16 Wireless Metropolitan Area Networks,” WCNC Conference, 2007.

[3] M. Mehrjoo and X. Shen, “An Efficient Scheduling Scheme for eterogeneous Traffic in IEEE 802.16 Wireless Metropolitan Area Networks,” in Proc. International Symposium on Telecommunications (IST), 2008, pp. 263–267.

[4] M. Andrews, M. Andrews, K. Kumaran, K. Ramanan, A. Stolyar, P. Whiting, and R. Vijayakumar, “Providing quality of service over a shared wireless link,” IEEE Communication. Magazine, pp. 150–154, Feb. 2001.

[5] A. J. Goldsmith and S. G. Chua, “Variable-rate variable-power MQAM for fading channel,” IEEE Transaction on Communication., vol. 45, no. 10, pp. 1218–1230, Oct. 1997.

[6] X. Qiu, and K. Chawla, “On the performance of adaptive modulation in cellular systems,” IEEE Transaction on Communication, vol. 47, pp. 884–895, June 1999.

[7] C. Mohanram, and S. Bhashyam, “Joint subcarrier and power allocation in channel-aware queue-aware scheduling for multiuser OFDM,” IEEE Transaction on Wireless Communication, vol. 6, no. 9, pp. 3208-3213, Sep. 2007.

[8] C. Y. Wong, R. S. Cheng, K. B. Lataief, and R. D. Murch, “Multiuser OFDM with adaptive subcarrier, bit, and power allocation,” IEEE J. Sel. Areas Communication., vol. 17, no. 10, pp. 1747–1758, Oct. 1999.

[9] S. Pietrzyk, and G. J. M. Janssen, “Multiuser subcarrier allocation for QoS provision in the OFDMA systems,” in Proceeding of IEEE Veh. Technol. Conf., vol. 2, Sep. 2002, pp. 1077–1081.

[10] H.W. Kuhn, “The Hungarian method for the assignment problem,” Naval Research Logistic Quarterly, Vol. 2, pp. 83-97, 1955.

[11] G. Li and H. Liu, “Dynamic resource allocation with finite buffer constraint in broadband OFDMA networks,” IEEE Wireless Comm. and Networking, v. 2, pp. 1037-1042, March 2003.

[12] G. Song and Y. Li, “Cross-layer optimization for OFDM wireless networks-Part I: Theoretical framework”, IEEE Transactions on Wireless Communications, vol. 4, no. 2, 2005.

[13] R. Jain, The art of computer systems performance analysis: techniques for experimental design, measurement, simulation and modeling. New York: Wiley, 1991.

482