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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.

Novel Batch Dependant Cross-Layer Scheduling for Multiuser OFDM Systems *

Nan Zhou1*, Xu Zhu*, Yi Huang* and Hai Lin+ Department of Electrical Engineering and Electronics, The University of Liverpool, Liverpool, UK + Department of Electrical and Information Systems, Osaka Prefecture University, Osaka, Japan

Abstract—We propose a novel packet batch dependant cross-layer scheduling scheme for the downlink multiuser orthogonal frequency division multiplexing (OFDM) system, which significantly outperforms the existing queue dependant scheduling scheme over a wide range of signal-to-noise (SNR) in terms of the system bandwidth efficiency, best-effort (BE) traffic throughput and quality of service (QoS) traffic delay, while requiring a lower complexity. Our batch dependant scheduling also provides a better tradeoff between the QoS traffic and BE traffic. It is also shown to guarantee the system stability. Index Terms—cross-layer optimization, orthogonal frequency division multiplexing (OFDM), quality of service (QoS), scheduling, resource allocation, stability.

I. INTRODUCTION Conventional network architectures, where each layer is designed to operate independently, do not utilize resources effectively. With the rapid increase of demands for high speed multi-media services of wireless networks, which are confronted with fading channels, limited bandwidth and competition of limited air resources among multiple users, a cross-layer design to meet the required quality of service (QoS) is desirable [1]. In [2], a cross-layer scheduling scheme for the medium access control (MAC) layer was proposed, which assigns priorities of connections according to the channel quality, QoS satisfaction and service priority across layers. In [3], a so-called modified largest weighted delay first (M-LWDF) scheduling scheme was proposed, based on the head-of-line (HOL) packet delay, relative data rate and QoS requirement. However, [2] and [3] assumed single carrier systems. Orthogonal frequency division multiplexing (OFDM) [1] is effective to combat frequency selective fading channels and support high data rate services, which has been widely used in WLAN (IEEE 802.11a & 11g), WiMAX (IEEE 802.16) and 3GPP LTE downlink systems [4]. In [5], a scheduling scheme was proposed for OFDM systems, which serves the QoS traffic and best-effort (BE) traffic together if the delays of the QoS packets do not approach the maximum allowable delay. In [6], an urgency and efficiency based packet scheduling (UEPS) scheme, which utilizes the HOL delay and channel quality, was proposed for OFDM systems. However, [5] and [6] only considered scheduling for the MAC layer, but did not consider adaptive resource allocation for the physical (PHY) layer. To exploit the synergy between scheduling and resource allocation at the PHY layer, a so-called maximum delay utility (MDU) cross-layer resource allocation and scheduling scheme was proposed in [7], which maximizes the utility function of the delay. Most previous work on scheduling was queue based, where in each slot the selected queues are served until they are empty or the PHY resources exhaust. However, this 1

leads to inefficiency if not all packets in the selected queues are urgent, while some packets in the unselected queues are more urgent. In this paper, we propose a novel packet batch dependant scheduling scheme for the downlink multiuser OFDM system. To the best of our knowledge, this is the first work to investigate scheduling based on packet batches, which is more flexible and efficient than the conventional queue based scheduling. The proposed scheduling scheme is combined with adaptive resource allocation based on the maximum weighted capacity (MWC) criterion [8], which is to maximize the sum of weighted capacities and allows a wide range of combinations of resource allocation and scheduling algorithms. The weights for MWC are determined by scheduling, as a function of a delay satisfaction (DS) indicator, the data amount and the traffic coefficient. We also consider multiple queues per user, which is a more practical case, but was not investigated in the literature. Simulation results show that the DS based scheduling scheme significantly outperforms the M-LWDF [3] and MDU [7] based scheduling schemes over a wide range of signal-to-noise (SNR), in terms of the system bandwidth efficiency, BE traffic throughput and QoS traffic delay, while requiring a much lower complexity. DS also provides a better tradeoff between the QoS and BE traffic than M-LWDF and MDU. Moreover, it is proven that the DS based scheduling guarantees the system stability. In Section II, we describe the MWC based cross-layer design. The DS based scheduling scheme is proposed in Sections III. Resource allocation and subcarrier allocation are described in Section IV. Stability analysis and complexity analysis are provided in Sections V and VI, respectively. Simulation results are shown in Section VII, and the conclusion is drawn in Section VIII. Throughout the paper, we use k to denote the user index, and i to denote the queue index. II. MAXIMUM WEIGHTED CAPACITY BASED CROSS-LAYER DESIGN We consider a downlink OFDM system with K users. Without loss of generality and for simplicity, we assume that each subcarrier is occupied by only one user [9]. With cross-layer optimization, the QoS information is transferred from the traffic controller to the subcarrier and power controller for resource allocation, and the resource allocation results are fed back to the traffic controller for scheduling. We assume a total bandwidth of B shared by N subcarriers, and the OFDM signaling is time slotted where the duration of each slot is Tslot. Let Ωk denote the index set of subcarriers allocated to user k (k=1,…,K). Let pk,n be the power allocated to user k on subcarrier n∈Ωk, hk,n the corresponding channel gain, and N0 the power spectral density of additive white Gaussian

Supported by the Overseas Research Students Award Scheme (ORSAS), UK, and the University of Liverpool, UK.

978-1-4244-2075-9/08/$25.00 ©2008 IEEE

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.

noise (AWGN). Assuming perfect channel estimation, the achievable instantaneous data rate of user k on subcarrier n is expressed as: B Rk , n = log 2 (1 + pk , nγ k ,n ) (1) N where γ k , n = | hk , n |2 ( N0 B N ) is the channel-to-noise power ratio for user k on subcarrier n. Thus, the total instantaneous data rate of user k is given by: Rk = ∑ Rk , n (2)

Substituting (5) into (4), it can be deduced that with M-LWDF, the queue which has a larger HOL packet delay, a larger Ri Ri (i.e., a higher instantaneous data rate relative to the average data rate for queue i), and a higher QoS requirement, is given a higher chance to be served first.

We employ a generalized maximum weighted capacity (MWC) based cross-layer optimization scheme. Letting Wk denote the weight for user k which contains the QoS information, our scheme is to maximize the sum of weighted capacities [8], i.e., to maximize

for queue i [10], where Qi denotes the average length of queue i calculated in the current slot, and λi denotes the arrival rate of queue i in the current slot. Similar to M-LWDF, MDU is also queue based, and the weight for queue i is expressed as: −1 ∂ Wi = ⋅ U (S ) (6) λi ∂Si i i The utility functions are defined in [7].

n∈Ωk

K

J = ∑ Wk Rk

(3)

k =1

subject to pk ,n ≥ 0 ,

K

∑∑ p k =1 n∈Ω k

k ,n

≤ p T , Ω i ∩ Ω j = ∅ (i ≠ j ) ,

Ω1 ∪ Ω2 ∪ ...Ω K ⊆ {1, 2,..., N } and Rk Tslot ≤ Qk , where pT is the total power, Qk denotes the total length of all queues for user k, and Tslot is the slot duration. The constraint Rk Tslot ≤ Qk is an additional constraint compared to the work in [8], which is to guarantee that no more resource is allocated to user k if the user has already obtains sufficient resources, to allow all data to be transmitted in current slot. The above user based cross-layer optimization can be easily extended to the queue based cross-layer optimization by using the following cost function: J = ∑ Wi Ri

(4)

i

where Ri denotes the total instantaneous data rate of queue i, and Wi denotes the weight for queue i. III. DELAY SATISFACTION BASED SCHEDULING The weights of the MWC based cross-layer optimization contain the QoS information and are obtained from scheduling at the MAC layer. We first review two queue based scheduling schemes: M-LWDF [3] and MDU [7], which can be combined with the queue based MWC cross-layer design with the cost function given by (4). Then a batch based scheduling scheme is proposed, which is referred to as DS. A. Modified Largest Weighted Delay First Based Scheduling The M-LWDF [3] based scheduling can provide delay sensitive services by keeping the delay of most packets below a threshold. We extend the work in [3] to the case of OFDM. Letting Li and Si denote the delay bound and HOL packet delay for queue i, define δ i as the maximum allowed probability of Si > Li . The weight of queue i is given by

B. Maximum Delay Utility Based Scheduling The scheduling scheme in [7] maximizes the utility functions with respect to the delay. Define U i ( Si ) as a decreasing utility function of the average HOL packet delay Si = Qi λi

C. Delay Satisfaction Based Scheduling The queue based M-LWDF and MDU scheduling schemes may lead to inefficiency if some packets in the selected queues are not urgent while some other packets in the unselected queues will be time out very soon. We now propose a scheduling scheme based on packet batches rather than queues, where a packet batch is defined as the data packets in the same queue which arrive in the same slot. Therefore, the batch based scheduling is much more flexible and efficient than the queue based scheduling. Our proposed scheduling scheme assigns a higher weight to the packet batch with a less DS, i.e., the data with the least DS should be sent out first. We define a DS indicator C k ,i ,l for the batch in queue i of user k, which arrives in slot l (l ∈ [ Lc − Li , Lc ]) , where Lc denotes the current slot index, Gi denote the guard time and Li is the delay bound for the queue i respectively. Let S k ,i ,l denote the delay for the batch of queue i for user k arriving in slot l. Note that Lc , Li , Gi and S k ,i ,l are all in the unit of

slot. The DS indicator C k ,i ,l is expressed as: Ck ,i ,l = Li − Gi − S k ,i ,l (7) which implies that the longer the data’s delay, the less the DS. Let Wk ,i ,l denote the weight of the batch corresponding to queue i of user k arriving in slot l, which is given by: (Ck ,i,l > 0) ( Dk ,i ,l + 1) βi (Ck ,i ,l + 1) Wk ,i ,l =  (8) βi ( Dk ,i ,l + 1) (−Gi ≤ Ck ,i ,l ≤ 0) 

(5)

where β i is the QoS coefficient or say priority of queue i, and Dk,i,l is the data amount of the batch belonging to queue i user k and arriving in slot l. 1) If Ck,i,l > 0, i.e., Sk,i,l < Li - Gi, the weight Wk,i,l increases with the decrease of the DS indicator. In particular, we have ∂Wk ,i ,l ∂Ck ,i ,l =  − β i (Ck ,i,l + 1) 2  ( Dk ,i ,l + 1) (9)

where Ri is the data rate of queue i averaged over slots.

which implies that the smaller the DS indicator is, the faster the weight increases.

Wi = −

log(δ i )Si Li Ri

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.

2)

If -Gi ≤ Ck,i,l ≤ 0 , i.e., Li - Gi ≤ Sk,i,l ≤ Li, Wk,i,l in (8) reaches the maximum value, implying that the batch arriving in slot l will be time out very soon, and should be sent out with no further delay. In weight calculation of the MWC based cross-layer optimization, only the first (the most urgent) Λi batches in queue i are used. The weight Wk in (3) is expressed as: Λi

Wk = ∑∑ Wk ,i ,l i

(10)

l =1

It is important to determine the value of Λi to make a tradeoff between the performance and complexity. With a large Λi, the complexity increases, while the performance on the queues with lower QoS classes improves due to obtaining more resources. With a small Λi, the complexity decreases, while more resources are allocated to the queues with higher QoS classes as they obtain higher weights. In practice, a reasonable value of Λi is below 50, and we set Λi=25 in our simulations. IV. SUBCARRIER ALLOCATION AND POWER ALLOCATION As the constraints for (3) are nondifferentiable, joint subcarrier and power allocation to maximize (3) can not be performed by using the Karush-Kuhn-Tucker conditions [11], especially in a real-time system [9]. Hence, we separate the solution into two steps—subcarrier allocation and power allocation. The following user based resource allocation can be easily extended to the queue based resource allocation by replacing all the index k by index i. A. Optimal Subcarrier Allocation We assume uniform power allocation across all subcarriers, i.e., each subcarrier is allocated a power pT/N. Letting Φ denote the user set, the dynamic algorithm to implement the MWC based subcarrier allocation is described as follows: 1) Initialization: Set Φ = {1,..., K } , Rk = 0 for ∀k ∈ Φ . 2) For n=1 to N Find k = arg max{Wk Rk , n } , assign subcarrier n to user k. k ∈Φ

Update Rk . If Rk Tslot > Qk , remove k from set Φ. B. Optimal Power Allocation Following the subcarrier allocation, the optimal power allocation can be obtained by using the Lagrange multiplier [12], and the optimal solution for pk,n is given by:   Wk pk , n =  K  (Wm Ω m  ∑ m =1 x x > 0 + where [ x ] =  , 0 x ≤ 0

 K   1 1    PT + ∑ ∑ −   γ k ,n  m =1 q∈Ω j γ m , q   ) 

+

(11)

and |Ωm| denotes the number of

subcarriers in set Ωm. V. STABILITY ANALYSIS In this section, we analyze the stability of the proposed cross-layer optimization scheme with the DS based scheduling. In a real system, the buffer size is limited, and each traffic stream has particular QoS requirements such as the delay and delay outage tolerance. Therefore, stability is highly related to

the QoS satisfaction. Letting Qk , L be the queue length of user k during slot L, a system is stable if [7] : 1 P −1 K lim sup ∑ ∑ E{Qk , L } < ∞ (12) P →∞ P L = 0 k =1 It can be shown that the proposed cross-layer optimization scheme guarantees the system stability. Since it is not straightforward to use (12), we introduce Lemma 1. Lemma 1: Define QL = [Q1, L , Q2, L ,..., QK , L ]T and K

Y ( QL ) = ∑ y ( Qk , L ) , where y (⋅) is a Lyapunov function k =1

[13]. A system is stable if there exist ∀a ∈ ( 0, ∞ ) and ∀b ∈ ( 0, ∞ ) , and for each slot L we have: K

L

E {Y ( QL +1 ) − Y ( QL )} ≤ a − b∑∑∑ Wk ,i ,l k =1

i

(13)

l =0

Assuming that the channel state is ergodic, a rate expectation exists for each resource allocation. Define the capacity region C which consists of all the rate expectations. Letting S denote the stability region, the system is stable if the average T

arrival rate vector λ = λ1 ,..., λK  ∈ S . It is easy to show S ⊆ C . However, the arrival rates locating within the capacity region is a necessary but insufficient condition for the system stability. Theorem 1: If the average arrival rates are within the capacity region, i.e., λ ∈ C , the DS based scheduling scheme guarantees the system stability defined by lemma 1. Due to space limitation, proof of the above is not given in this paper. VI.

COMPLEXITY ANALYSIS

In this section, we analyze the system complexity of the proposed cross-layer design of MWC+DS, compared to that of MWC+M-LWDF [3] and MWC+MDU [7]. The analysis is demonstrated in TABLE I, where we assume ω queues for each of the K users in each slot. Thus, there are totally ωK queues in the system. In TABLE I, the overall complexity of each cross-layer optimization scheme includes only the higher order complexity of resource allocation and scheduling, as the higher order complexity dominates with the increase of parameters [14]. When M-LWDF is combined with MWC cross-layer design, N iterations are required for subcarrier allocation, each containing ω K comparisons as MWC assigns N subcarriers to the total ω K queues. Power allocation is performed subsequently using the closed-form result in (11), and its complexity can be ignored compared to the complexity of subcarrier allocation [15]. Therefore, the complexity of resource allocation is O (ω KN ) in this case. The M-LWDF based scheduling uses (5) once to calculate the weight for each of the ω K queues in each slot. And therefore its complexity is O (ω K ) . Thus, the overall complexity of MWC+M-LWDF is O (ω KN ) , which increases linearly with the number of subcarriers, number of users and number of queues per user. Similarly, the overall complexity of MWC+MDU is O(ωKN).

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.

TABLE I. COMPLEXITY OF CROSS -LAYER OPTIMIZATION SCHEMES Complexity MWC + MWC + MWC + DS M-LWDF[3] MDU [7] Resource O(ω KN ) O(ω KN ) O( KN ) allocation Scheduling O(ω K ) O(ω K ) O (ω K Λ i )

Overall

O (ω KN )

O (ω KN )

O ( KN ) ( ω ≤ N Λ i ) O (ω K Λ i ) ( ω > N Λ i )

Combined with the DS based scheduling, the MWC based resource allocation requires K comparisons in each of the N iteration. Thus, the complexity of MWC is only O ( KN ) . The DS based scheduling needs to calculate weight Wk ,i ,l according to (10), for Λi batches in each of the ω K queues. Therefore, the complexity of the DS based scheduling is O ( ωΛi K ) . The overall complexity of MWC+DS is discussed as follows. Note that in general Λi < N . 1) If ω=1, i.e., each user has only one queue, all the crosslayer schemes have the same overall complexity of O(KN). 2) If 1 < ω ≤ N Λi , the overall complexity is O(KN), which is almost independent of the number of queues per user, benefiting from the batch based scheduling. Thus, MWC+DS achieves a complexity reduction of approximately ω times over MWC+M-LWDF and MWC+MDU. 3) If ω > N Λi , The overall complexity is O (ω K Λi ) . Therefore, the overall complexity of MWC+DS is around N/Λi times lower than that of MWC+M-LWDF and MWC+MDU. It can be concluded from the above that, with more than one queue for each user, which is true in practice, MWC+DS requires a lower complexity than MWC+M-LWDF and MWC+MDU, due to the batch based scheduling. For instance, with ω=3 queues per user, Λi = 25 and N=512 subcarriers, the complexity reduction achieved by MWC+DS is around 3 times. VII. SIMULATION RESULTS We use simulation results to show the performance of the proposed cross-layer optimization scheme, compared to that of M-LWDF [3] and MDU [7]. The total bandwidth of the downlink OFDM system is B=5 MHz, which is divided into N=512 subcarriers for K=32 users. The total power is PT=1 W. The channel has six independent Rayleigh fading paths with an exponential delay profile. The slot duration is Tslot=4 ms and we set Λi = 25 in (12). We assume that each user has three types of traffic queues: voice, variable bit rate (VBR) [16] video and best effort (BE) traffic, for which the maximum delay tolerances are 100 msec, 400 msec and 1 s, respectively. The voice and BE traffic have constant data rate of 64 Kbps and 500Kbps, respectively. The VBR video traffic uses a multiple-state model [16], where the data rate for each state is obtained from a truncated exponential distribution with a minimum data rate of 120 Kbps, a maximum data rate of 420 Kbps, and a mean data rate of 239 Kbps. The state duration is assumed to be independent and exponentially distributed with a mean of 160 ms. The QoS coefficients for voice, VBR video

and BE traffic are 1024, 512 and 1, respectively. SNR is defined as the average received signal power to noise power for each user. Figure 1 demonstrates the system bandwidth efficiency versus the average SNR for each user. DS significantly outperforms M-LWDF and MDU when the SNR is below 30 dB, while achieving a complexity reduction of around 3 times as discussed in Section VI. At SNR=25 dB, the bandwidth efficiency achieved by PD is around 295% and 125% of those achieved by M-LWDF and MDU, respectively. MDU provides a similar bandwidth efficiency to DS only at a very high SNR. Figure 2 shows the total throughput of the BE traffic. Similar trends to Fig.1 can be observed. DS provides a much higher BE throughput than M-LWDF and MDU when the SNR is below 30dB. MDU provides a similar BE throughput to DS only at a very high SNR. At SNR=30 dB, DS and MDU can afford all BE loads. In Fig. 3, the average delay of voice traffic is demonstrated. With a moderate to high SNR, the voice delay of DS is much lower than that of M-LWDF and MDU. At SNR=15 dB, the voice delay is only 10 ms for DS, while it is around 75 ms and 60 ms for M-LWDF and MDU, respectively. The average delay of video traffic is shown in Fig. 4, where DS outperforms the other two scheduling schemes over the moderate to high SNR range. The higher the SNR, the more advantages of DS over M-LWDF. MDU achieves the same video delay as DS only at SNR=30 dB. It can be observed from Figs. 2 to 4 that when the SNR is between 15 dB and 25 dB, DS achieves a much higher throughput of BE traffic than M-LWDF and MDU, while maintaining lower delays of voice and video traffic. This is because DS is based on packet batches and also considers the data amount for scheduling, which is more flexile and efficient than the queue based scheduling schemes. While M-LWDF and MDU assign a higher priority to voice and video traffic, and can afford BE traffic loads only if all voice and video traffic obtain adequate resources. Therefore, DS provides a better tradeoff between QoS and BE traffic than M-LWDF and MDU at moderate to high SNR, with a complexity reduction of 3 around times. VIII. CONCLUSION We have proposed a novel packet batch based cross-layer scheduling scheme, which considers the differences between the batches in the same queue, and therefore is more flexible and efficient than conventional queue based scheduling. The batch with a lower level of delay satisfaction, a higher QoS coefficient and more data amount, is assigned a higher weight for the MWC based cross-layer design. The proposed DS scheduling scheme significantly outperforms M-LWDF [3] and MDU [7] scheduling schemes with a moderate to high SNR, in terms of bandwidth efficiency, BE traffic throughputs, and QoS traffic delays. With a large number of subcarriers, the complexity required by MWC+DS is almost independent of the number of queues per user, while the complexities of MWC+M-LWDF and MWC+MDU are proportional to the number of queues per user.

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the ICC 2008 proceedings.

[3] [4]

[5] [6] [7]

[8] [9] [10] [11] [12] [13] [14] [15] [16]

16 DS M−LWDF MDU

14 Total throughput for BE traffic (Mbps)

[2]

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5

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15 20 Average SNR (dB)

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Fig. 2. BE throughput versus the average SNR 100 DS M−LWDF MDU

90 80 Average packet delay (ms)

[1]

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Fig. 3. Average voice packet delay versus the average SNR 5.5

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4 3.5 3 2.5 2

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1.5 50

1 0.5

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15 20 Average SNR (dB)

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Fig. 1. System bandwidth efficiency versus the average SNR

0

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Fig. 4. Average video packet delay versus the average SNR