Queuing Analysis on MIMO Systems with Adaptive Modulation and Coding

Transcription

1 Queuing Analysis on MIMO Systems with Adaptive Modulation and Coding Sheng Zhou, Kai Zhang, Zhisheng Niu and Yang Yang Tsinghua National Laboratory for Information Science and Technology Dept. of Electronic Engineering, Tsinghua University, Beijing , China Dept. of Electronic and Electrical Engineering, University College London (UCL) Torrington Place, London, WC1E 7JE, United Kingdom Abstract The combined MIMO with Adaptive Modulation and Coding (AMC) technology can provide high spectral efficiency and link robustness. Moreover, the diversity-multiplexing tradeoff of MIMO systems motivates the design of adaptive algorithms that switch between the two approaches to enhance performance. While most existing adaptive algorithms focus on physical layer, the QoS requirements of multimedia services are mainly parameters in link layer. Hence cross-layer analysis on the queuing behavior of MIMO-AMC systems is necessary, but remains open. In this paper, under the conditions of unsaturated traffic and finite-length buffer, we investigate the queuing characters of two representative categories of MIMO systems, namely the BLAST system and the space-time block coding (STBC) system. We successfully model the service processes of both STBC and BLAST coupled with AMC, which is the most challenging part of the queuing analysis. We observe a new tradeoff between diversity and multiplexing in terms of link layer packet loss rate and queuing delay, based on which we propose a cross-layer design of diversity-multiplexing switching scheme to optimize the QoS satisfaction of the MIMO-AMC systems. 1 I. INTRODUCTION Multiple-Input Multiple-Output (MIMO) technology has been shown to be able to improve transmission data rate through spatial multiplexing [1] or to enhance reliability through space-time coding techniques [2]. Together with MIMO technology, Adaptive Modulation and Coding (AMC) scheme can further enhance spectral efficiency by adjusting transmission parameters to the channel condition while satisfying a target error performance [3]. Consequently, the combined MIMO-AMC technology is a promising solution to offer high throughput for next generation wireless systems. For MIMO systems, there is a fundamental tradeoff between diversity and spatial multiplexing: higher spatial multiplexing gain leads to lower diversity and vice versa. In order to optimize throughput performance and robustness by making use of both the two approaches, adaptive algorithms that switch between multiplexing and diversity are proposed [4] [5]. Most of the existing adaptive schemes concentrate on physical layer to minimize symbol error rate and the switching 1 This research work was partially supported by the China Scholarship Council under the postgraduate student exchange program. criteria is basically the channel conditions. However, the QoS requirements of multimedia services are mainly packet loss rate, end-to-end delay, etc., which are the parameters in link layer. Pure physical layer analysis simply considers link layer as a saturate traffic provider, thus design problems rise when queuing at link layer is taken into account with practical conditions: unsaturated random arrival packets and finitelength buffer. Because of the dynamic property of the packet arrival, the queues may be empty even though the channel can offer high rate, and packet loss happens not only due to transmission error, but also queuing overflow with finitelength buffer. In addition, with AMC, the service process of the queue is no longer deterministic [6], especially for MIMO transmission schemes, the modeling of the service process with AMC is rather complex. Hence, to optimize link layer performance with various QoS requirement, cross-layer analysis on the queuing behavior of MIMO-AMC system is a necessary but challenging issue. Previous work [6] has investigated the queuing with AMC over single antenna wireless systems, in which the authors characterized the queuing service process dictated by AMC and constructed finite state Markov chain to solve the queue state recursion. Ref. [7] analyzed the queuing model of MIMO-AMC, especially on the delay constrained traffic. However, only diversity MIMO scheme (STBC: space-time block code) was investigated and infinite-length buffer was assumed. In [8], the authors provided some insight to the effect of MIMO wireless channels on TCP, then multiplexing and diversity MIMO approaches were compared. In the cross-layer analysis they adopted, the MIMO channel was simplified as an on-off Gilbert model, and the property of multi-rate offered by AMC was not reflected. In this paper, we analyze the queuing behavior of MIMO- AMC systems under the conditions of unsaturated traffic and finite-length queue. Two representative categories of MIMO systems are investigated and compared: BLAST [1] for multiplexing approach and STBC [2] for diversity approach. System model is described in Section II. In Section III, we model the service processes of BLAST and STBC with AMC, and the

2 accuracy of the model is later validated by simulations. Then we adopt queuing analysis on the two systems to evaluate their QoS performance: packet loss rate and queuing delay. In section IV, numerical and simulation results are given, and we also observe a new diversity-multiplexing tradeoff related to link layer performance. Motivated by the observation, we propose a cross-layer design of diversity-multiplexing switching algorithm to optimize the system QoS performance. A. System Overview II. SYSTEM MODEL We consider an end-to-end wireless transmission system with multiple antennas on both the transmitter side and the receiver side. On the transmitter side, link layer packets are buffered in a queue with finite length K, which operates in a first-in-first-out (FIFO) manner. The queue is served by merging packets into physical layer frames with constant duration T f, and we consider two different MIMO transmission modes: spatial diversity mode (STBC) and spatial multiplexing mode (BLAST). On the receiver side, according to the channel state information (CSI) estimated, an AMC selector determines the modulation-coding pairs (modes), and sends back them to the transmitter through an error free feedback channel. For STBC, one AMC mode is selected for all the transmit antennas, while for BLAST, AMC modes vary for different transmit antennas, i.e. each equivalent spatial stream has its own AMC mode. Due to different AMC modes, each frame may contain one or more link layer packets. B. MIMO Channel model The channel is assumed to be frequency flat, and remains invariant per frame, but varies independently from frame to frame. This corresponds to an i.i.d block fading model. There are transmit antennas and N r receive antennas, with N r, and perfect CSI at the receiver side is assumed. The received N r 1 signal vector y can be written as Es y = Hx + n, (1) where H is the N r channel matrix with i.i.d unit variance complex Gaussian elements. x is the 1 transmitted signal vector with normalized unit energy. E s is the total power transmitted. The receiver noise is the N r 1 vector n with i.i.d. complex Gaussian elements, and the noise power is N 0 = 1 for simplicity. Channel quality is represented by γ c = Es N 0 = E s as the average received signal-to-noise. For STBC, spatial diversity is provided by mapping each R T complex input symbols into orthogonal sequences of length T, that are simultaneously transmitted by transmit antennas. The information code rate R c = R/T is assumed to be 1. The diversity order of the STBC MIMO system is m s = N r. Using STBC SISO equivalence [10], the SNR at the output of the STBC decoder is Gamma distributed with parameter m s and mean m s γ c / : p γ STBC(γ) = γms 1 Γ(m s ) ( Nt γ c ) ms ( exp N ) tγ, (2) γ c TABLE I TRANSMISSION MODES AND CORRESPONDING PARAMETERS Mode (n) Modulation BPSK QPSK QPSK 16-QAM 64-QAM Coding rate 1/2 1/2 3/4 3/4 3/4 R n(bits/sym.) a n g n Γ n(db) where Γ(x) = t x 1 exp( t)dt is the Gamma function. 0 For BLAST, transmitted symbols are separated into substreams (one sub-stream per transmit antenna), and we allow different AMC modes used on sub-streams according to the post-detection SNRs of the sub-streams. Hence the information rate of the spatial multiplexing scheme is the sum of the substream rates. At the receiver side, zero forcing (ZF) detector is employed to sperate the sub-streams with equalizer G = E s H, where H is the pseudo inverse of H. The postdetection SNR γk BLAST of the kth sub-stream is γ BLAST k = γ c 1 [(H H) 1 ] k,k. (3) It has been shown that γk BLAST is Gamma distributed with parameter m b = N r + 1 and mean m b γ c / [10]. Equivalently, each of the sub-streams has diversity order m b and average SNR m b γ c /. Since the noise in the separated sub-streams is correlated, the identically distributed SNRs are not independent [10]. C. AMC With the target packet error rate constraint P 0, AMC maximizes the data rate by adjusting transmission parameters to the available CSI. The entire SNR range is divided into N +1 nonoverlapping consecutive intervals, with boundary points denoted by {γ n } N+1 n=0. When γstbc [γ n,γ n+1 ), mode n is selected for the STBC system, while γk BLAST [γ n,γ n+1 ), mode n is selected for the kth sub-stream in the BLAST system. Especially, we set mode n = 0 with rate R 0 = 0. In order to determine the boundary points {γ n } n=0 N+1, we fisrt approximate the packet error rate (PER) as [6] { 1, if 0 < γ < Γn PER n (γ) (4) a n exp( g n γ), if γ Γ n where n is the mode index, γ is the received/post-detection SNR (γ STBC for STBC, γ BLAST for BLAST). Parameter sets {a n,g n,γ n } are mode and packet-size dependent constants. Lsted in Table I, the transmission modes are adopted from the HIPERLAN/2 or the IEEE a standards [9], and the parameter values are obtained by simulation fitting with packet length N b = 1,080 [6]. Then we can get the boundary points with prescribed target PER P 0 by setting them to the minimum SNR required to achieve P 0 over a non-fading AWGN channel.

3 TABLE II VALUES OF PARAMETER ξ FOR DIFFERENT BLAST CONFIGURATIONS N r ξ Fig. 1. System time diagram of the discrete-time queuing model Inverting the PER expression in (4), we have γ 0 = 0 γ n = 1 g n ln( an P 0 ). γ N+1 = + n = 1,2,...N Note the boundaries are chosen ones P 0 is defined. We next calculate the average packet error rate. The average PER of AMC can be approximated as the ratio of the average number of packets in error over the total average number of transmitted packets [6] N n=1 PER R np n PER n N n=1 R. (6) np n where R n is the rate of mode n, and P n = Pr(n,m) is the probability with which mode n is chosen, so Pr(n,m s ) is the probability that STBC systems is in mode n, while Pr(n,m b ) is the probability that the kth sub-stream of BLAST system is in mode n; PER n denotes the average PER corresponding to mode n, especially for BLAST, it is for one sub-stream which is in mode n. Close-form expressions for P n and PER n can be found in [6]. One more comment is that using (6) for BLAST is not straightforward. However, as an approximation, we can assume packets are not interleaved among sub-streams, i.e. all symbols belonging to a packet will be transmitted in a fixed sub-stream during a frame time. Then the packet error of substreams can be considered independent, and the correlation between the SNRs of sub-streams does not affect the average PER result. The overall PER of BLAST system is the same with the PER of one sub-stream because the SNR distributions of sub-streams are identical. III. QUEUING ANALYSIS In this section, we will build a queuing model for the MIMO-AMC system, in which the arrival process, the service process (for both STBC and BLAST) and the queue state transition is detailed. The system time diagram is shown in Fig. 1, where we define each frame as a time-unit t, and the system state Q t is described as the number of packets in the queue at the end of time unit t. During time unit t, A t packets arrive from the upper layer. We also denote C t as the number of packets that can be transmitted at most (decided by AMC controller according to CSI) during time unit t. A. Arrival Process The arrival process A t is stationary with E{A t } = λt f, and it is independent of the queue state and the service process. We assume that A t is Poisson distributed P(A t = a) = (λt f) a exp( λt f ), a = 0,1,2,... (7) a! (5) where the value region of A t is defined as A t A = {0,1,2,..., }. B. Service Process We assume that the frame structure and AMC modes are designed to have integer number of packets per frame. Hence C t is a random variable, and it takes non-negative integer values from a set C with finite number of elements (because we have N +1 AMC modes). The value set C and the probability distribution of C t varies in STBC and BLAST systems. 1) Service Process in STBC System: For STBC, we denote the value set of C t as C STBC, and because of the STBC SISO equivalence, each element in C STBC corresponds to an AMC mode. Hence for STBC we have C t C STBC, C STBC = {c 0,c 1,...,c N } (8) where each c n = br n is an non-negative integer, and b depends on the system resource allocated per user in real systems [6]. We also have the probability relationship for STBC system Pr(C t = c n ) = Pn STBC. (9) 2) Service Process in BLAST System: For BLAST, at time unit t, we have C t = b R nk,t, (10) where R nk,t is the rate of kth sub-stream when it is in mode n k at time t. There will be (N +1) Nt rate combinations, each of them is denoted as (R n1,...,r nnt ). The set containing all these combinations is denoted as R BLAST. With the AMC rate set denoted as R = {R n } N n=0, we have R BLAST = R R R, (11) }{{} ones where denotes the direct product of two sets. Unfortunately, because of the SNR correlation among sub-streams, it is not possible to get the probabilities of each rate combination in R BLAST directly from the marginal distribution } N n=0. The joint distribution of the SNRs of substreams in the ZF BLAST system is hard to be characterized, and AMC makes the problem even more complex. On the other hand, the variance coefficient of the service process is one of the dominate factors when we consider the queue performance as shown later, so we choose to approximate the behavior of the service process of BLAST by keeping the variance of C t the same with the actual statistic of the service process. From the simulation results, we find that the SNR correlation coefficient between arbitrary two sub-streams is positive, which means that the SNRs of sub-streams tend to { n

4 be the same. Equivalently, the AMC rates of sub-streams have the same tendency, so we adopt this positive correlation to build the probability of each rate combination in R BLAST as Pr approx (R n1,...,r nnt ) =. n n k ξ Nt 1, if i j, n i n j n k ξ Nt 2,!(i j), s.t. n i = n j n k ξ Nt 3,!(i j k), s.t. n i = n j = n k {(n,i 2,...,i Nt )} \(n,n,...,n) Pr(R n,r i2,...,r int ), n 1 = n 2 = = n Nt = n (12) where!( ) means there only exists one such index group, and ξ is a parameter to be tuned, and the last probability description is to keep the marginal distribution {Pn BLAST } N n=0. With R BLAST and Pr approx (R n1,...,r nnt ), we can have C t value set C BLAST with no more than br N + 1 elements 2 by merging rate combinations in R BLAST with the same sumrate to one element, of which the probability is the sum of the probabilities of the corresponding rate combinations. The last thing is to determine ξ, and our criterion is θ(ξ, γ c ) := Var(C t) Var(C t ) real 1, (13) where Var( ) is the variance of a random variable, and Var(C t ) real represents the real statistic of C t (can be catched by simulation). Actually, ratio θ not only depends on ξ, but is also a function of γ c (channel quality). However, it is not necessary to determine ξ for every γ c realization, because from simulation we find that the ratio θ is not sensitive to γ c for a fixed ξ. It is also not sensitive to the target PER P 0, which affects the AMC thresholds. Thus we search for ξ that satisfies 25dB 10dB E(θ) := θ(ξ, γ c)d γ c 1, (14) 15dB where the γ c range [10dB,25dB] is chosen according to the discussion interests. In practise, the expectation calculation can be done over some chosen γ c values in the range. The ξ values for different BLAST configurations are listed in Table II. C. Queue State Transition Recall the description in the beginning of this section, the system state Q t is described as the number of packets in the queue at the end of time unit t, or equivalently, at the beginning of time unit t + 1, and its value region is denoted as Q t 2 Since each br n is an integer, so every element in C BLAST is an integer ranging from 0 to br N. Q = {0,1,2,...,K}. According to the channel state and the AMC mode, the transmitter moves at most C t packets out of the queue to the physical layer at the beginning of time unit t, after which the free slots in the queue at the beginning of time t is F t = K max{0,q t 1 C t }, where K is the queue length. Then arriving packets are placed in the queue throughout time t, so packets arrival can be equivalently considered as a bulk arrival at the end of time unit t as shown in Fig. 1. However, packets will be dropped when the queue is full, i.e., if A t > F t, A t F t packets will be dropped, otherwise all A t packets will be accepted by the queue. So the queue transition relation is Q t = min{k, max{0,q t 1 C t } + A t }. (15) Then the state transition probability matrix is P = {q i,j } (K+1) (K+1), 0 i,j K (16) where each element q ij is the transition probability from state Q t 1 = i to Q t = j. The value of q ij can be determined as q ij = P(Q t = j Q t 1 = i) = c C P(Q t = j Q t 1 = i,c t = c), (17) where we have P(Q t = j Q t 1 = i,c t = c) = { P(Qt = j max{0,i c}), if 0 j < K 1 P(Q t = j Q t 1 = i,c t = c). if j = K 0 j<k (18) It can be proved (similar to the Appendix of [6]) that the stationary distribution π = (π(0),π(1),...,π(k)) exists by solving π = πp and πe = 1, where π(i) = lim P(Q t = i) t and e is a (K +1) 1 vector with all elements equaling to 1. D. System Performance Based on the stationary distribution computed we can calculate the average delay and the average packet loss rate. First, we need to get the average packet dropping rate P d. From the expression of F t in Section III-C, we can get the number of packets dropped at time t as D t = max{0,a t K + max{0,q t 1 C t }}. Since A t and C t are independent of t, and we have the stationary distribution of Q, the ensembleaverage number of packets dropped per time unit D := E(lim t D t ) can be calculated as D = max{0,a K + max{0,q c}} a A,q Q,c C Based on (19), P d is given by T t=1 P d = lim D t T T t=1 A t P(A = a) P(Q = q, C = c). (19) = D λt f. (20) With P d, λt f and average queue length E(Q), we can get the average queuing delay W according to Little s Law [11] W = E(Q) (1 P d )λt f. (21)

5 Saturate Throughput E(C t ) (packets/frame) x2 STBC 4x4 STBC 2x2 BLAST 4x4 BLAST 2x3 BLAST Average SNR γ 0 (db) Fig. 2. Saturate throughput of different MIMO configurations The average packet loss rate P l can be computed as P l = 1 (1 P d )(1 PER), (22) where PER is the average packet transmission error rate (expressed in II-C). The average throughput can then be evaluated as η = λt f (1 P l ), hence smaller P l leads to larger average throughput. IV. NUMERICAL RESULTS AND DISCUSSIONS In this section, we provide numerical results based on our analytical expressions. We assume that the frame length T f = 2 (ms) and b = 2; queue length K = 30 (packets), the target packet error rate 3 P 0 = We also adopte Monte- Carlo simulations to validate our analysis. Brief discussions are offered related to applications of the proposed analysis in MIMO system design. A. Diversity-Multiplexing tradeoff First, in Fig. 2, we have shown E(C t ) for different MIMO configurations. This corresponds to the situation that there are always packets for transmission and the queue length is infinite. The comparison is to show the basic diversitymultiplexing tradeoff in terms of saturate throughput related to average SNR. At low SNR, multiplexing schemes (STBC) provide higher saturate throughput, while at high SNR, multiplexing schemes (BLAST) are better. Moreover, higher diversity gain can greatly improve the performance of BLAST as the curve of 2 3 BLAST shown in Fig. 2. Traditionally, when we choose from the diversity and multiplexing schemes, the one who can provide higher rate (under given error constraint) is preferred, for example, STBC for low SNR, BLAST for high SNR. However, when we have finitelength buffer and unsaturated traffic, another kind of tradeoff related to link layer performance is observed. The tradeoff mentioned is detailed in Fig. 3, in which we have set the average SNR γ c = 18dB, and we define ρ = λt f /E(C t ) as the traffic intensity. There is a good agreement between 3 Note that because SNR thresholds are the minimum SNRs required to achieve P 0, PER is much less than P 0 TABLE III VALUES OF E(C t) AND µ FOR DIFFERENT MIMO CONFIGURATIONS MIMO Config. 2 2 S 4 4 S 2 2 B 2 3 B 4 4 B E(C t) µ our analytical results and simulation results, and we have the following observations: (1) Packet loss P l is mainly caused by queuing overflow, i.e. it mostly depends on packet dropping rate P d. Only when ρ is very small, the effect of packet error shows up, and acts like a loss rate floor. (2) Though we choose γ c = 18dB where all the multiplexing schemes are with higher saturate throughput than corresponding diversity schemes (same antenna configuration) do, the comparison of packet loss rate depends on the arrival rate. Since P l mainly depends on P d, the dependance is tightly related to the queuing behavior of the two kinds of MIMO schemes. The packet dropping rate of a queuing system depends on two main factors: traffic intensity ρ and the randomness of the service process 4 represented by variance coefficient µ = Var(C t )/E(C t ) 2. Small ρ or small µ either leads to low P d. When ρ is small, µ is the crucial factor while ρ becomes more important when it increases. Given the average SNR when the multiplexing scheme can provide higher rates, i.e. larger E(c t ), it has smaller ρ comparison to the corresponding diversity scheme, but its µ is larger because of the higher randomness of the equivalent multiplexing channels (because of smaller diversity gain). Hence when arrival rate is low, which leads to small ρ for both STBC and BLAST, and since STBC has smaller µ, its P d performance is better. When arrival rate increases, as ρ acts more significantly, BLAST can offer better P d performance due to its smaller ρ. We can observe this tradeoff in the 2 2 STBC-BLAST pair and the 4 4 pair in Fig. 3, and we list the related parameters in Table III for comparison, where S is short for STBC and B is short for BLAST. We make two comments on the observed tradeoff: a) The tradeoff only exists when BLAST can offer higher saturate throughput; b) The tradeoff is more obvious when the rate gap between STBC and BLAST is relatively small. Thus, the tradeoff relation is related to the average SNR γ c. (3) Multiplexing schemes with higher diversity gain can greatly improve the P d performance. In Fig. 3 the performance of 2 3 BLAST is better than four other modes for most of the arrival rate values. The reason is that it has relatively high E(C t ) as a multiplexing scheme, together with smaller µ due to its diversity order 2. However, since the tradeoff depends on γ c, the advantage of 2 3 BLAST is weakened when γ c is lower, and for example, the cross-point between 2 3 BLAST and 4 4 STBC will be with larger P l. In Fig. 4, the delay performance is shown, in which the 4 The randomness of the arrival process does matters, but we don t consider it since we assume Poisson arrival for all situations.

6 Packet Loss Rate P l x2 STBC analy. 4x4 STBC analy. 2x2 BLAST analy. 2x3 BLAST analy. 4x4 BLAST analy. 2x2 STBC sim. 4x4 STBC sim. 2x2 BLAST sim. 2x3 BLAST sim. 4x4 BLAST sim Arrival Rate λt f (packets/frame) Average Delay W (frame durations) x2 STBC analy. 4x4 STBC analy. 2x2 BLAST analy. 2x3 BLAST analy. 4x4 BLAST analy Arrival Rate λt (packets/frame) f Fig. 3. Packet loss rate of different MIMO configurations Fig. 4. Average Delay of different MIMO configurations similar tradeoff trend is depicted. However, the multiplexing schemes outperform the diversity schemes even when their P d performance is still worse (comparison to Fig. 3). It implies that multiplexing schemes have advantages in decreasing queuing delay, and since queuing delay is the main part of end-toend delay, multiplexing schemes are more preferable for delay sensitive services with loose requirements for packet loss rate. B. Cross-Layer Design The diversity-multiplexing tradeoff with unsaturated traffic and finite-length buffer suggests cross-layer design of diversity-multiplexing switching algorithms. The framework proposed in this paper can be used in various cross-layer optimization schemes, we list two simple examples: (1) A MIMO system adaptively switch between BLAST and STBC according to QoS requirements, for example, minimizing P l or minimizing W. The system should not only consider the channel condition, but also the link layer packet arrival property. We take minimizing P l for instance. STEP 1 Collect average SNR γ c and average arrival rate λ periodically. Compute E(C t ) of STBC and BLAST, if STBC is better, choose STBC, otherwise, go to STEP 2. STEP 2 Compute P l using our framework for both STBC and BLAST, choose the one with smaller P l. (2) As the equivalent channel properties of STBC and BLAST are different, the AMC thresholds can be selected differently for the two schemes in order to minimize P l of each scheme by tuning P 0 respectively. Ref. [6] and [7] discussed the tuning detail, and our frame work can further provide a method to find the optimal P 0 for each MIMO transmission scheme, based on which the switching algorithm mentioned previously can be improved. The usage of our framework is straightforward, and it is not detailed regarding the space. V. CONCLUSION In this paper, we provided a cross-layer analysis to the performance of MIMO-AMC systems with finite-length buffer. We successfully modeled the service processes of both STBC and BLAST coupled with AMC, of which the accuracy is validated by simulation. Based on the performance analysis, we observed a new tradeoff between diversity and multiplexing related to link layer performance under the condition of unsaturated traffic and finite-length buffer. Tackling the tradeoff, a cross-layer design of diversity-multiplexing switching algorithm is proposed to optimize link layer performance with various QoS requirements. REFERENCES [1] G. J. Foschini, Layered space-time architecture for wireless communication in a fading environment when using multiple antennas, Bell Labs. Tech. J., vol. 1, pp , [2] V. Tarokh, H. Jafarkhani and A. R. Caldebank, Space-time codes from orthogonal designs, IEEE Trans. Inform. Therory, vol. 45, pp , July [3] M. S. Alouini and A. J. Goldsmith, Adaptive modulation over Nakagami fading channels, J. Wireless Commun., vol. 13, no. 1-2, pp , May [4] R. W. Heath Jr. and A. J. Paulraj, Switching between spatial multiplexing and transmit diversity based on constellation distance, Proc. of Allerton Conf. on Comm. Cont. and Comp., [5] S. Catreux, V. Erceg, D. Gesbert and R. W. Heath Jr., Adaptive modulation and MIMO coding for broadband wireless data networks, IEEE Comm. Mag., vol. 2, pp , June [6] Q. Liu, S. Zhou, and G. B. Giannakis, Queuing with adaptive modulation and coding over wireless links: cross-layer analysis and design, IEEE Trans. on Wireless Commun.,, vol. 4, no. 3, pp , May [7] J. S. Harsini and F. Lahouti, Queuing with adaptive modulation over MIMO wireless links for deadline contrained traffic: cross-layer analysis and design, Proc. of IEEE ICC 07., Glasgow, Scotland, Jun , [8] A. L. Toledo, X. Wang and B. Lu, A cross-layer TCP modelling framework for MIMO wireless systems, IEEE Trans. Wireless Commun., vol. 5, no. 4 pp , April [9] A. Doufexi, S. Armour, M. Butler, A. Nix, D. Bull, J. McGeehan and P. Karlsson, A comparison of the HIPERLAN/2 and ieee a wireless LAN standards, IEEE Commun. Mag., vol. 40, no. 5, pp , May [10] A. Paulraj, R. Nabar, D. Gore, Introduction to Space-Time Wireless Communications. Cambridge Univ. Press, Cambridge, UK, [11] L. Kleinrock, Queueing Systmes. Vol.1: Theory. John Wiley & Sons, New York, 1975.