Optimal downlink multi-user MIMO cross-layer ... - IEEE Xplore

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Optimal Downlink Multi-user MIMO Cross-layer Scheduling Using HOL Packet Waiting Time Cheng Wang, Student Member, IEEE, and Ross D. Murch, Senior Member, IEEE Abstract— Cross-layer design is a promising technique to achieve performance enhancement for wireless systems. In this paper, we seek to exploit the synergy between the PHY (physical) layer and the MAC (Multiple Access Control) layer when a multiple-input multiple-output (MIMO) wireless system is operating at the PHY layer. We propose two optimal downlink multi-user MIMO scheduling algorithms which consider both the queueing state information from the MAC layer and the channel state information from the PHY layer when making scheduling decision. Our approaches are optimal in the sense that they can achieve maximum system throughput while at the same time guaranteeing the stability of the system. That is, they guarantee stability of the system whenever it is feasible with any other scheduling algorithm. Computer simulations demonstrate that significant gains are achieved by our cross-layer design compared to traditional strict-layered designs. Additionally, by making use of the HOL (head of line) packet waiting time our optimal scheduling algorithms can provide a more balanced delay performance for non-uniform traffic compared to the existing optimal scheduling algorithm which does not consider it. Index Terms— cross-layer, scheduling, downlink, multi-user MIMO.

I. I NTRODUCTION

W

ITH the tremendous growth in the use of wireless communications, there is a need to exploit new ways to achieve performance enhancement. One promising prospect for the enhancement and improvement of wireless systems is known as cross-layer design. Currently communication systems are based on a layered approach, such as the 7-layer OSI model, where specific functions for communication are compartmentalized into independent layers with defined interfaces between them. While this simplifies the overall system design by keeping various components separate, it leads to sub-optimal system performance since global optimization is not performed. In this paper, we seek to exploit the synergy between the PHY layer and the MAC layer when a multiple-input multiple-output (MIMO) wireless system is operating at the PHY layer, with particular emphasis on packet scheduling. By jointly considering the queueing state information from the MAC layer and the channel state information from the PHY layer, our goal is to design optimal scheduling algorithms and thereby achieve maximum throughput and maintain stability of the system in presence of time varying wireless channel and Manuscript received September 10, 2004; revised April 26, 2005; accepted November 6, 2005. The editor coordinating the review of this paper and approving it for publication is X. Shen. This work was supported by the Hong Kong RGC grant HKUST 6164/04E. The authors are with the Department of Electrical and Electronic Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong (e-mail: {eeelva, eermurch}@ust.hk). Digital Object Identifier 10.1109/TWC.2006.04613.

random packet arrival process. We use the term optimal in the sense that it achieves maximum system throughput while at the same time guaranteeing the stability of the system. That is, it guarantees stability of the system whenever it is feasible with any other scheduling algorithm and where stability is defined as stability-in-the-mean [8] so that the mean queue length of the system is bounded. In previous work it has been recognized [1]-[6] that by employing multi-user MIMO (MU-MIMO) techniques in the downlink, several users can be served simultaneously in frequency and time, and the overall system capacity can be increased without the need for additional spectrum or power in ideal uncorrelated channels. However in reality the channels are not ideal and strong fading correlation often exists since usually the BS is unobstructed and there are only a few dominant scatterers. This will have an impact on the spatial separability of the users. To avoid severe cochannel interference, adaptive resource allocation is therefore considered in e.g. [7], where heuristic algorithms are proposed to allocate channels to users and adjust downlink beamforming vectors based on criteria such as minimum induced or received interference. However in these studies, the queueing state of the MAC layer is not considered and this will lead to degradation in overall system performance. On the other hand, due to the characteristics of the wireless channel such as user mobility and time-varying channel capacity, scheduling algorithms which only consider the queueing state will result in very poor system performance. Recently, optimal scheduling algorithms in the sense that the algorithm can guarantee stability of the system whenever it is feasible with any other scheduling algorithm for the wireless system have been studied in [9]-[11]. In [9], the throughput region of the system is established and a scheduling algorithm which considers the queue backlog and wireless channel state is proved to be optimal using the method of Lyapunov stability analysis. [10] basically follows a similar approach as in [9] and proposes several sub-optimal scheduling policies. Optimal scheduling policies which use HOL (head of line) packet waiting time can be found in [11], [12]. However in [11] only one user can be scheduled to transmit at a time, which is an unnecessary constraint in a MIMO broadcast channel; While in [12] an M × N input-queued packet switch is treated and the connection between input and output can only be on or off. Our investigation is different in that we analyze a MUMIMO scheduling algorithm which jointly considers the HOL packet waiting time and the time-varying physical channel state. We show that the algorithm we propose is optimal and can provide a more balanced delay performance for non-

c 2006 IEEE 1536-1276/06$20.00 

NAME et al.: OPTIMAL DOWNLINK MULTI-USER MIMO CROSS-LAYER SCHEDULING USING HOL PACKET WAITING TIME

uniform traffic. That is different users don’t share the same arrival rate. We further show that instead of using the HOL packet waiting time, a scheduling algorithm that uses a linear combination of the HOL packet waiting time and the queue length is also optimal, which provides additional freedom to achieve desired performance. The remainder of this paper is organized as follows. In the next section, we introduce the system model and prove the optimality of the two scheduling algorithms we propose. In section III, numerical simulations of delay-throughput performance of various scheduling algorithms are provided. Conclusions are drawn in section IV. II. O PTIMAL D OWNLINK M ULTI - USER MIMO S CHEDULING A. System Model We consider the downlink of a narrowband multi-user MIMO system with one base station (BS) and N users. The BS is equipped with M transmit antennas and the ith user is equipped with Li receive antennas. As a result several users can be served simultaneously by using downlink MU-MIMO transmit schemes [1]-[6]. Packets destined for each user are buffered in separate FIFO (first in first out) queues with infinite space at the BS. Time is divided into frames with the constraint that only integer number of packets can be served in a frame. We also adopt the convention that packets arriving during the current frame cannot be served until the next frame. The downlink wireless channels are assumed to be quasi-static for each frame, i.e. channels remain unchanged within the frame duration, but vary independent and identically from frame to frame. At the beginning of each frame the BS makes scheduling decision in reaction to the queueing state, e.g. the waiting time of the HOL packet, the queue length, as well as the physical channel state, where we assume that the channel state information is perfectly known to the BS at the beginning of each frame. The scheduling the form of∗a transmission rate vector  decision is in ∗ , where Ri ≥ 0 is the transmission R∗ = R1∗ , R2∗ , . . . , RN N rate for the ith user in packets per frame and i=1 Ri∗ > 0. Generally, the packet arrival process is described in terms of the probability distribution of the inter-arrival times of packets [13]. We assume that the packet arrival processes of all users are mutually independent. For each user the inter-arrival times are independent and identically distributed (i.i.d), therefore, the stream of arrivals forms a stationary renewal process. Denote ai (p), i = 1, 2, . . . , N as the number of packets that arrive for user i in frame p, we assume ai (p), p = 0, 1, 2, . . . to be i.i.d. random variables with a finite second moment, i.e. E a2i (p) < ∞. The vector of the expectation of the inter-arrival  time of each user is assumed to  be 1/A1 , 1/A2 , . . . , 1/AN , thus A  i represents the average arrival rate of user i, i.e. E a (p) = A i   i . Throughout the paper we denote A = A1 , A2, . . . , AN as an arrival rate  vector. And  we use W(p) = W1(p), W2 (p), . . . , WN (p) , Q(p) = Q1 (p), Q2 (p), . . . , QN (p) to represent the vector of HOL packet waiting time of each queue and the vector of queue length of each queue at the beginning of frame p respectively.

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Throughout this paper, we refer to stability as stability-inthe-mean [8] so that the mean queue length of the system is bounded. The following is the formal definition of stabilityin-the-mean. K−1 N 1   E Q (p) < ∞ Definition: A system is stable if K i p=0 i=1 for all K. B. Throughput Region Assume the channel states can be modeled by a Markov Chain with transitions occurring on the boundary of the frames. Let S denote the channel state space and πs be the steady state probability that the channel is in state S. Each channel state represents a set of feasible transmission rate vectors {R : R ∈ S}, which depend on the physical channel conditions, the MU-MIMO transmit schemes employed, etc. Proposition: For an arrival rate vector A, the necessary and sufficient condition for the system to be stabilizable is for A to strictly lie in the interior of the throughput region. Given the set of feasible transmission rate vectors of all channel states, the throughput region of the downlink system is given by     πs φSR R , (1) Λ= λ:λ= S∈S

R∈S

where  S ∈ S, R ∈ S, φSR are nonnegative numbers such that φSR ≤ 1 for all S ∈ S.

R∈S

The proof of the above proposition is fairly straightforward so we omit it here and recommend [9] for a proof. We claim that this throughput region is different from the information theoretic capacity region of the downlink multi-user MIMO broadcast channel (MIMO BC). Here the throughput region is related to the sets of feasible transmission rate vectors, which depend on the choice of MU-MIMO transmit scheme, coding, power constraint, etc. C. Optimal Downlink Multi-user MIMO Scheduling Algorithm – OMSW Theorem 1: The scheduling algorithm which selects the rate vector R∗ (p) such that R∗ (p) = arg max

R∈S(p)

N 

γi Wi (p)Ri ,

(2)

i=1

guarantees stability of the system whenever the arrival rate vector is strictly in the interior of the throughput region, where  γ1 , γ2 , . . . , γN is any arbitrary set of positive weights, which can be used to indicate the QoS requirement of each user. We call this scheduling algorithm optimal MU-MIMO scheduling based on waiting time (OMSW). Proof: The evolution of the HOL packet waiting time vector W(p) is given by the following equation: ⎧ d i (p) ⎪ ⎪ ⎪ Wi (p) + 1 − τim (p) di (p) < Qi (p) ⎪ ⎪ ⎪ m=1 ⎨ di (p) ≥ Qi (p), Wi (p+1) = 1 ⎪ ai (p) > 0 ⎪ ⎪ ⎪ ⎪ di (p) ≥ Qi (p), ⎪ ⎩ 0 ai (p) = 0 (3)

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where τim (p) is the inter-arrival time between the mth and (m + 1)th packet (counted from the HOL packet) of user i’s queue at the beginning of frame p, di (p) is the number of packets that can be served in frame p for user i. For any queue i whose average arrival rate is zero, the queue is always empty and the waiting time is always zero. It does not contribute to the summation in the selection criterion (2), so without loss of generality, we set di (p) = 0 for all p if Ai = 0. d i (p) τim (p) to simplify the notation, we can Let Ti (p) =   Wi (p + 1) ≤ max Wi (p) − Ti (p), 0 + 11 ,

Wi2 (p + 1) 2    ≤ Wi (p) − Ti (p) + 1 + 2 max Wi (p) − Ti (p), 0   (5) ≤ Wi2 (p) + Ti2 (p) + 1 − 2Wi (p) Ti (p) − 1 .

−2

N 

i=1

N 

   γi Ai E Ti2 (p)W(p)

N 

N 

N 

m=1

i=1

−2

   γi Ai E Ti2 (p)W(p)

     γi Wi (p) E di (p)W(p) − Ai

i=1

≤ B−2

N 

     γi Wi (p) E di (p)W(p) − Ai

(6)

i=1

where B =

N  i=1

γi Ai +

N  i=1

(7)

γi Wi (p)(Ai + ε) ≤

N 

   γi Wi (p)E di (p)W(p) .

(8)

i=1

N       γi Wi (p) E di (p)W(p) − Ai ≥ ε γi Wi (p), (9) i=1

N 

γi Wi (p).

(10) N  i=1

γmin Wi (p) ≤

γi Wi (p), so we get       E L W(p + 1) − L W(p) W(p) ≤ B − 2εγmin

N 

Wi (p).

(11)

i=1

Taking the unconditional expectation of the inequality above, we have       E L W(p + 1) − E L W(p)

i=1

N 

   γi Wi (p)E di (p)W(p)

Let γmin = min(γ1 , γ2 , . . . , γN ), then



N 

γi Wi (p)Ri

i=1

In addition, since A lies strictly in the interior of the throughput region, there exists a positive vector ε = [ε, ε, . . . , ε], such that A + ε still strictly lies in the interior of the throughput region and satisfies

  di (p)  i=1     γi Wi (p) Ai E E τim (p) di (p), W(p) − Ai ,

γi Ai +

N 

i=1

where the outer expectation is taken with respect to di (p) conditioned on W(p), =

R∈S

≤ B − 2ε

   γi Ai E Ti2 (p)W(p)

i=1

S∈S N 

γi Wi (p)Ri

i=1

R∈S

πS max

N 

and then we can find that       E L W(p + 1) − L W(p) W(p)

m=1

N 



φSR

i=1

  d   i (p)  γi Wi (p) Ai E τim (p)W(p) − Ai

γi Ai +

i=1

−2

=

i=1

i=1

=

≤



i=1

by using inequality (5) we obtain,       E L W(p + 1) − L W(p) W(p)

N 

πS

S∈S

N 

i=1

N 



Therefore we have

N    L W(p) = γi Ai Wi2 (p),

i=1

≤

i=1

Define the Lyapunov function

γi Ai +

γi Wi (p)Ai

i=1

(4)

where the inequality occurs when di (p) ≥ Qi (p), ai (p) = 0. Squaring both sides of the above inequality we get:

N 

N 

m=1

easily see that

≤

Since A lies strictly in the interior of the throughput region, we have

    γi Ai max E Ti2 (p)W(p) < ∞

since di (p) is bounded.

1 When d (p) ≥ Q (p), we include the effect of the inter-arrival time i i of the still-haven’t-arrived-packets, and in this case Wi (p) < Ti (p), then max(Wi (p) − Ti (p), 0)=0, Wi (p + 1) ≤ 1.

≤ B − 2εγmin

N    E Wi (p) .

(12)

i=1

By summing over p from 0 to p = K − 1, and telescoping, we obtain   1   1   E L W(K) − E L W(0) K K K−1 N     2εγmin ≤B− E Wi (p) , (13) K p=0 i=1

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then

Note that in [11], there is a statement of the optimality of a special case of Theorem 2, where only one user can be      1 B 1 scheduled at a time and αi βi = 0, αi + βi > 0 for all i. E L W(K) E Wi (p) ≤ − K p=0 i=1 2εγmin 2εγmin K Proof: The evolution of the queue length vector Q(p) is    given by the following recursive equation: 1 + E L W(0) (14) 2εγmin K Qi (p + 1) = max(Qi (p) − di (p), 0) + ai (p) (20) Using the non-negativity of the Lyapunov function yields, Squaring both sides of (20) we get: K−1 N  1   E Wi (p) Q2i (p + 1) K p=0 i=1 2    ≤ Qi (p) − di (p) + a2i (p)  B 1   ≤ E L W(0) for all K. (15) + +2ai (p) max Qi (p) − di (p), 0 2εγmin 2εγmin K   ≤ Q2i (p) + d2i (p) + a2i (p) − 2Qi (p) di (p) − ai (p) (21) Since the initial waiting times Wi (0), i = 1, 2, . . . , N are bounded, Define the Lyapunov function K−1 N     1 N   E Wi (p) ≤ M < ∞ for all K. (16)    K p=0 i=1 L W(p), Q(p) = αi Ai W 2 (p) + βi Q2 (p) , K−1 N 

i

where M is a finite constant. Let Amax max(A1 , A2 , . . . , AN ), then        E Qi (p) = E E Qi (p) Wi (p)     = Ai E Wi (p) ≤ Amax E Wi (p) .

=

By using inequality (5) and (21), similar to the derivation of (10) we get       E L W(p + 1), Q(p + 1) − L W(p), Q(p) W(p), Q(p) (17)

K−1 N N   Amax K−1   1   E Qi (p) ≤ E Wi (p) K p=0 i=1 K p=0 i=1

(18)

This result proves that a maximum weight type scheduling algorithm which takes into consideration both the HOL packet waiting time and the physical channel state is optimal. In the following sub-section we propose and prove another optimal stability-achieving scheduling algorithm. D. Optimal Downlink Multi-user MIMO Scheduling Algorithm - OMSWQ Instead of using the HOL packet waiting time, a scheduling algorithm namely optimal MU-MIMO scheduling based on waiting time and queue length (OMSWQ), which uses a linear combination of the HOL packet waiting time and the queue length can be shown to be optimal in a similar way to the optimality of OMSW. Theorem 2: The scheduling algorithm which selects the rate vector R∗ (p) such that R∗ (p) = arg max

R∈S(p)

N    αi Wi (p) + βi Qi (p) Ri ,

≤ B  − 2ε

N    αi Wi (p) + βi Qi (p) .

(22)

i=1

Therefore, we have stability-in-the-mean,

≤ Amax M < ∞ for all K.

i

i=1

  where B  < ∞ since di (p) and E a2i (p) are bounded. By taking the unconditional expectation of the above inequality, summing over p from 0 to p = K −1 and telescoping, we have   1   1   E L W(K), Q(K) − E L W(0), Q(0) K K K−1 N     2ε E αi Wi (p) + βi Qi (p) , (23) ≤ B − K p=0 i=1 Without loss of generality, we assume αi > 0, βi > 0 for i = 1, . . . , N1 , αi > 0, βi = 0 for i = N1 + 1, . . . , N1 + N2 , and αi = 0, βi > 0 for i = N1 + N2 + 1, . . . , N , where 0 ≤ N1 ≤  N1 + N2 ≤ N . Let γmin = min(α1 , . . . , αN1 , β1 , . . . , βN1 ),   = min(αN1 +1 , . . . , αN1 +N2 ) and βmin = αmin min(βN1 +N2 +1 , . . . , βN ), then we have   1   1   E L W(K), Q(K) − E L W(0), Q(0) K K N1 K−1      2εγmin ≤ B − E Wi (p) + Qi (p) K p=0 i=1 −

i=N1 +1

(19)

i=1

guarantees stability of the system whenever the arrival rate vector is strictly in the interior of the throughput region, where {α1 , α2 , . . . , αN }, {β1 , β2 , . . . , βN } are any arbitrary sets of non-negative weights such that αi + βi > 0 for all i.

K−1 N1 +N2   2εαmin   E Wi (p) K p=0

−

K−1   2εβmin

K

N 

  E Qi (p) ,

(24)

p=0 i=N1 +N2 +1

 Define  Amax1 = max(A  1 , . . . , AN1 +N2 ), we have E Wi (p) ≥  E Qi (p) , i = 1, . . . , N1 + N2 according Amax

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to (17). Then we further obtain   1   1   E L W(K), Q(K) − E L W(0), Q(0) K K N1 K−1     1 2εγmin (1 +  ) E Qi (p) ≤ B − K Amax p=0 i=1 −

K−1 N1 +N2   2εαmin   E Qi (p)  KAmax p=0 i=N1 +1

−

K−1   2εβmin

K

N 

  E Qi (p) ,

(25)

p=0 i=N1 +N2 +1

 1 ), αmin , β  ), then Amax Amax min   1   1   E L W(K), Q(K) − E L W(0), Q(0) K K K−1 N     2εγ min ≤ B − E Qi (p) , (26) K p=0 i=1

 Let γmin = min(γmin (1 +

Because the initial waiting times Wi (0), i = 1, 2, . . . , N and the initial queue lengths Qi (0), i = 1, 2, . . . , N are bounded and the Lyapunov function is non-negative, we get K−1 N  1   E Qi (p) < ∞ for all K. K p=0 i=1

At the PHY layer, the total transmit power Pt for each transmission is assumed to be 15dB and the noise at each user ni , i = 1, 2, . . . , N is assumed to be additive white Gaussian with ni ∼ N 0, I , where I represents the identity matrix. We use the spatial water-filling transmission strategy [14] to compute the transmission rate for single-user transmission. For simultaneous multi-user transmission, we use the MUMIMO decomposition scheme (please refer to [1], [2] for the details of this scheme) to decompose the multi-user MIMO system into parallel independent single-user MIMO systems and then use the spatial water-filling transmission strategy for each single-user MIMO system, where the total transmit power Pt = 15dB is equally allocated to these single-user systems. For simplicity we constrain that the feasible transmission rate vector has either 1 or 2 positive entries, i.e. no more than 2 users can be transmitted to simultaneously. B. Simulation Results We compare the performance of our proposed scheduling algorithm with that of the following four scheduling algorithms. • Optimal MU-MIMO scheduling based on queue length (OMSQ) The scheduling algorithm [9], [10] selects the rate vector R∗ (p) such that

(27)

∗

R (p) = arg max

R∈S(p)

That is OMSWQ is also optimal. •

III. N UMERICAL R ESULTS A. Simulation Setup We utilize a semi-correlated flat fading MIMO channel model as described in [15] in the simulations, where it is assumed that the BS is unobstructed while all the users are in rich scattering environment. Hence the channel between the BS and user i can be modeled as Hi = Gi Fi , where Gi is an Li × Di matrix with zero-mean unit-variance i.i.d complex Gaussian entries. Fi is the steering matrix of size Di ×M containing Di steering vectors of the transmit antenna array corresponding to Di DODs (directions of departure). In our simulations, a semi-correlated flat fading channel with 50 DODs randomly and independently distributed in an angle spread of 20◦ according to a uniform distribution is used. At the BS we use an 8-element ULA (uniform linear array) with equidistant spacing between antenna elements to be half a wavelength. There are 8 users in the system each of which has 2 receive antennas, and the users are assumed to be randomly and independently distributed in [−π, +π] according to a uniform distribution around the BS. The symbol and the frame duration are assumed to be 40μs and 2.4ms respectively, so each frame contains 60 symbols. The packet size is assumed to be 128 bits, and we constrain that only integer number of packets can be transmitted in a frame. We consider non-uniform traffic and model the packet arrivals for each user as Poisson processes with arrival rate vector A = TAR 16 [1, 2, 2, 2, 2, 2, 2, 3], where TAR is the total arrival rate in packets/frame.

N 

Maximum rate (MaxRate) The scheduling algorithm selects the rate vector R∗ (p) such that N  ∗ Ri . R (p) = arg max R∈S(p)

•

γi Qi (p)Ri .

i=1

i=1

Maximum waiting time (MaxW) The scheduling algorithm selects two users who have the largest sum HOL packet waiting time to transmit to simultaneously, i.e.   {i, j} = arg max Wi (p) + Wj (p) . i∈1,...,N j∈1,...,N i=j

•

Maximum queue length (MaxQ) The scheduling algorithm selects two users who have the largest sum queue length to transmit to simultaneously, i.e.   {i, j} = arg max Qi (p) + Qj (p) . i∈1,...,N j∈1,...,N i=j

We set γi = 1, i = 1, 2, . . . , N for OMSW and OMSQ in all the simulations below to indicate that all the users have the same delay QoS requirement. All these five algorithms compared in the simulations share the same set of feasible transmission rate vectors when making scheduling decision and this ensures fair comparison. Also note in our simulations when there are not enough packets to be transmitted for the selected 1 or 2 users, the remaining symbols of the frame can be used for the transmission of other groups of users in the descending order of priority according to the scheduling algorithm.

NAME et al.: OPTIMAL DOWNLINK MULTI-USER MIMO CROSS-LAYER SCHEDULING USING HOL PACKET WAITING TIME 100

10

OMSW OMSQ MaxRate MaxQ MaxW

OMSW OMSQ MaxRate MaxQ MaxW

90

difference of mean packet delay in frames

9

system throughput in packets/frame

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8

7

6

80

70

60

50

40

30

20

5 10

4

0

4

5

6

7

8

9

10

Fig. 1.

Fig. 3.

Comparison of system throughput.

4

5

6

7

8

9

10

total arrival rate in packets/frame

total arrival rate in packets/frame

Difference of mean packet delay between user 1 and user 8.

100

OMSW OMSQ MaxRate MaxQ MaxW

90

mean packet delay in frames

80

70

60

50

40

30

20

10

0

4

5

6

7

8

9

10

total arrival rate in packets/frame

Fig. 2.

Comparison of mean packet delay across all users.

Fig.1 provides performance comparison in terms of system throughput. When the system is stable, the system throughput is just the total arrival rate. As expected, OMSW and OMSQ perform the best and the system is stable when the total arrival rate is as large as 9 packets/frame, which means 9 [1, 2, 2, 2, 2, 2, 2, 3] is a point near the boundary of A = 16 the throughput region of this system. The maximum total arrival rate for MaxRate can successfully handle is around 7.5 packets/frame. MaxW and MaxQ perform extremely poor compared to the other three algorithms, and they render the system unstable when the total arrival rate is around 5.25 and 5.5 packets/frame respectively. This is because during the scheduling process, only the MAC layer queueing state is considered regardless of the PHY layer channel state. Although MU-MIMO transmit scheme is used at the physical layer, users’ spatial separability is not taken into consideration, which leads to significant degradation in system performance. In Fig.2 we investigate the performance of mean packet delay across all users for the five scheduling algorithms. The result is consistent with the system throughput performance we demonstrate in Fig.1. When the total arrival rate is above

the maximum we observed in Fig.1, the corresponding mean packet delay across all users increases quickly. A key reason for us to propose optimal scheduling algorithms using HOL packet waiting time is that although OMSQ is also optimal in the sense that it can guarantee stability of the system whenever it is feasible with any other scheduling algorithm, it can lead to starvation of a nonempty queue with low arrival rate when the traffic is non-uniform, i.e. users don’t share the same arrival rate. We investigate the impact of non-uniform traffic on the difference of mean packet delay performance among users in Fig.3. In the simulation setup we know that the arrival rate of user 8 is three times the arrival rate of user 1, however since we set γi = 1 for all i to indicate that all users share the same delay QoS requirement, a good scheduling algorithm should balance the delay of each user, i.e., keep the delay of different user close to each other. From the figure we can see that algorithms based on HOL packet waiting time have superior performance compared to their counterparts who only use queue length as the queueing state information in the scheduling algorithm and the one which does not consider the queueing state information. Especially when the system becomes unstable, the mean packet delay difference between user 1 and user 8 grows quickly unbounded for these three algorithms. While algorithms which consider the HOL packet waiting time overcome this limitation, the mean packet delay difference remains low even when the system becomes unstable. This is because at the beginning of each frame, the increase in the weight of the algorithms which only use queue length as the queueing state information is biased when the traffic is non-uniform, i.e. the queue length of the ith user is increased by ai (p), so users with high arrival rate nearly always get larger weights and this leads to starvation of nonempty queues with low arrival rate; While for algorithms using HOL packet waiting time, the increase in the weight is unbiased by the arrival rate. If a queue is not served its HOL packet waiting time keeps increasing and eventually the weight will increase to a value that ensures it is served. From the simulation results above we can see when HOL packet waiting time is used, the delay of different users are

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well balanced, and we can expect although haven’t shown in the simulation results here that when queue length is used, the queue lengths of different users will be balanced. Thus we can expect that OMSWQ has more freedom to achieve desired performance. For example, voice packets are delay sensitive and are labeled with deadlines, in order that the QoS of the voice users with low arrival rate is still acceptable, i.e. packet loss due to expiration has to be controlled under a certain limit, their weight on HOL packet waiting time should be much larger than the weight on queue length. Likewise, if the buffer size at the BS is limited for each user, then in order to avoid packet loss due to overflow of the buffer, users with high arrival rate should put larger weight on the queue length. The two scheduling algorithms we propose are shown to be optimal based on performing the scheduling on the queue level, i.e. at the beginning of each frame, the BS only decides once which queues to be served regardless of the states of the packets behind the HOL packet. However we can perform the scheduling on the packet level, i.e. scheduling is performed on a finer time scale, and it is not difficult to realize. Once the BS has decided which queues to be served, the HOL packets of these selected queues can be transmitted at once, no need to wait for the whole frame to be scheduled. At the same time update the priorities related to the queues which have been scheduled. Since the physical channels remain unchanged, the feasible transmission rates need not to be recalculated, thus updating the priorities is fairly easy. We can expect that the delay performance of scheduling on the packet level will be better than that of scheduling on the queue level. And we can also expect that this has greater impact on waiting time based scheduling algorithm, since the difference between the waiting time of the HOL packet and the packet behind can be large, while for queue length it is only decreased by one. IV. C ONCLUSION In this paper we investigate the cross MAC-PHY layer packet scheduling problem. In particular, two downlink MUMIMO scheduling algorithms using HOL packet waiting time are proposed and are shown to be optimal. When making scheduling decision, our algorithms jointly consider the queueing state information from the MAC layer and the channel state information from the PHY layer. As a result, the variations of both the queueing state and the channel state are effectively exploited and significant gain is achieved compared to traditional strict-layered design as we demonstrate in the simulation results. Furthermore, when compared to the existing optimal scheduling algorithm which only use the queue length as the queueing state information we find that the existing optimal scheduling algorithm can lead to starvation of nonempty queues with low arrival rate when the traffic is non-uniform, while our optimal scheduling algorithms using HOL packet waiting time overcome this limitation and provide a balanced delay performance. R EFERENCES [1] R. L. -U Choi and R. D. Murch, “A downlink decomposition transmit pre-processing technique for multi-user MIMO systems,” in Proc. IST Mobile & Wireless Telecommunications Summit, June 2002.

[2] Lai-U Choi and R. D. Murch, “A transmit preprocessing technique for multiuser MIMO systems using a decomposition approach,” IEEE Trans. Wireless Commun., vol. 3, no. 1, pp. 20-24, Jan. 2004. [3] Q. H. Spencer and M. Haardt, “Capacity and downlink transmission algorithms for a multi-user MIMO channel,” in Proc. 36th Asilomar Conf. Signals, Systems, and Computers, vol. 2, pp. 1384-1388, Nov. 2002. [4] Q. H. Spencer, A. L. Swindlehurst, and M. Haardt, “Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels,” IEEE Trans. Signal Processing, vol. 52, no. 2, pp. 461-471, Feb. 2004. [5] A. Bourdoux and N. Khaled, “Joint TX-RX optimisation for MIMOSDMA based on a null-space constraint,” in IEEE Proc. VTC’02, vol. 1, pp. 171-174, Sept. 2002. [6] M. Schubert and H. Boche, “A unifiying theory for uplink and downlink multiuser beamforming,” in Proc. Inter. Zurich Sem. On Broadband Communications, Access, Transmission, Networking, pp. 27.1-27.6, Feb. 2002. [7] I. Koutsopoulos, T. Ren, and L. Tassiulas, “The impact of space division multiplexing on resource allocation: a unified approach,” in IEEE Proc. INFOCOM’03, vol. 1, pp. 533-543, Apr. 2003. [8] P. R. Kumar and S. P. Meyn, “Stablilty of queueing networks and scheduling policies,” IEEE Trans. Automat. Contr., vol. 40, no. 2, pp. 251-260, Feb. 1995. [9] M. J. Neely, E. Modiano, and C. E. Rohrs, “Power and server allocation in a multi-beam satellite with time varying channels,” in IEEE Proc. INFOCOM’02, vol. 3, pp.1451-1460, June 2002. [10] T. Ren, R. J. La, and L. Tassiulas, “Optimal transmission scheduling with base station antenna array in cellular networks,” in IEEE Proc. INFOCOM’04, vol. 3, pp. 1684-1693, Mar. 2004. [11] M. Andrews et al., “Providing quality of service over a shared wireless link,” IEEE Commun. Mag., vol. 39, no. 2, pp. 150-154, Feb. 2001. [12] N. McKeown et al., “Achieving 100% throughput in an input-queued switch,” IEEE Trans. Commun., vol. 47, No. 8, pp. 1260-1267, Aug. 1999. [13] L. Kleinrock, Queueing Systems, Volume I: Theory. New York: Wiley, 1975. [14] E. Telatar, “Capacity of multi-antenna Gaussian channels,” European Trans. Telecommun., vol. 10, no. 6, pp. 585-595, 1999. [15] M. T. Ivrlac, W. Utschick, and J. A. Nossek, “Fading correlation in wireless MIMO communication systems,” IEEE J. Select. Areas Commun., vol. 21, no. 5, pp. 819-828, June 2003.

Cheng Wang (S’03) received the Bachelor’s degree in Electronic Science and Engineering from the NanJing University, NanJing, JiangSu, China, where she graduated in 2002 and was ranked first in the department. She is currently working toward the Ph.D. degree in the Department of Electrical and Electronic Engineering, the Hong Kong University of Science and Technology, Kowloon, Hong Kong. Her research interests include multi-user MIMO wireless communication systems, adaptive resource allocation and cross-layer design and optimization.

Ross D. Murch (S’85−M’87−SM’98) is a Professor of Electrical and Electronic Engineering at the Hong Kong University of Science and Technology. His current research interests include multiple antenna systems, compact antenna design, MIMO, WLAN, B3G and Ultra-Wide-Band (UWB) systems for wireless communications. He has several US patents related to wireless communication, over 150 published papers and acts as a consultant for industry and government. In addition he is an editor for the IEEE Transactions on Wireless Communications and was the Chair of the Advanced Wireless Communications Systems Symposium at ICC 2002. He is also the founding Director of the Center for Wireless Information Technology at the Hong Kong University of Science and Technology which was begun in August 1997. He is the program Director for the MSc in Telecommunications at the Hong Kong University of Science and Technology. From August-December 1998 he was on sabbatical leave at Allgon Mobile Communications (manufactured 1 million antennas per week), Sweden and AT&T Research Labs, NJ, USA.