Performance of Multiuser CDMA Receivers With Bursty ... - CiteSeerX

Performance of Multiuser CDMA Receivers With Bursty Traffic and Delay Constraints Kashif Mahmood∗ , Mikko Vehkaper¨a† , Yuming Jiang∗ ∗

†

Q2S, Norwegian University of Science and Technology, Norway School of Electrical Engineering and the ACCESS Linnaeus Center, KTH, Sweden

Abstract—In this work, we analyze the performance of linear multiuser CDMA receivers under bursty traffic and queuing delay constraints. An ON/OFF source is used to model the traffic burstiness and stochastic network calculus incorporates the queuing delay constraints in the analysis. At the physical layer, adaptive modulation and coding scheme is adopted and the channel service process is modeled by using a finite-state Markov chain. The following receivers are considered for performance evaluation: single-user matched filter, decorrelator, and linear minimum mean square error detector. We quantify the effect of these multiuser receivers on the traffic carrying capacity of uplink CDMA channels in the large system limit.

I. I NTRODUCTION The explosive growth of high speed services such as file transportation protocol, VOIP and IP video has resulted in a global demand for tetherless wireless access. Codedivision multiple-access (CDMA) is an efficient multiple access technique that allows the communication media to be shared among different users. In a direct-sequence CDMA (DS-CDMA), the signals on all the links are transmitted simultaneously but are separated by using spreading codes. These spreading codes become non-orthogonal, among others, because of the channel impairments such as multipath. As a result the intended signal for each user is contaminated by the signals from other users. This interference from other users is termed as multiple access interference (MAI). Multiuser detectors (MUDs) can be used for non-orthogonal CDMA which requires joint processing and decoding of users. Multiuser receivers mitigate, in addition to background noise, the MAI. Due to their relatively low computational complexity, practical MUDs are in general linear — the most common special cases being the single-user matched filter (SUMF), the decorrelator or zero-forcing (ZF) detector, and the linear minimum mean square error (LMMSE) detector. It is well established that the choice of detector has a significant effect on the system capacity (see, e.g., [1]–[3]) and information theoretic performance limits of linear MUDs, without traffic burstiness and queuing delay constraints, are known [1], [2]. Emerging wireless applications, such as VOIP and IP video, however require a bounded delay. In addition, one of the desirable features of CDMA is that they support bursty traffic sources [4]. It is therefore of paramount importance to study the effect of delay constraints and bursty traffic on the traffic carrying capacity of CDMA networks with linear MUDs. Such an attempt was made in [5] by using the effective bandwidth approach for cross-layer modeling of a wireless

CDMA networks with the LMMSE receiver. In [4], bursty data transmission in random access CDMA systems was considered wherein an ON/OFF source was used to model the burstiness. A performance comparison of SUMF, decorrelator and LMMSE receivers was carried out in [2] which mainly addresses physical layer issues and the modeling of link layer parameters such as delay and burstiness is not clear. Further, in [2], [4], [5] only memoryless channels are considered and the applicability of the proposed approach for quantifying the traffic carrying capacity of CDMA networks under delay constraints and burst traffic is not obvious. To the best of our knowledge there is no work which carries out the performance analysis of multiuser CDMA detectors with bursty traffic sources under delay constraints. In this paper, we carry out a performance analysis of multiuser CDMA receivers under delay and burstiness constraints. For the MUD, we consider the SUMF, decorrelator / ZF detector, and the LMMSE detector. We take a cross-layer approach to predict the traffic carrying capacity of multiuser CDMA networks under a given delay guarantee. To this end, we make use of stochastic network calculus (NetCal) [6] to model delay and burstiness. It is a bounding function based approach in which the input traffic and the service is characterized in terms of the envelope function and the service curve, respectively. A survey of different envelope functions and service curves can be found in [7], [8]. Stochastic NetCal enables the derivation of flow level performance bounds such as Prob[delay > D] ≤ ε, i.e. the probability of the delay exceeding a certain threshold D is upper bounded by ε. Specifically, we make use moment generating functions (MGF) based stochastic NetCal. MGF based stochastic NetCal was proposed in [9], and further established and developed in [10]. It uses bounds on the MGF of the arrival and service process to obtain the flow level performance measures such as delay and backlog bounds. In contrast to the previous work, for example in [5], here the multiuser CDMA channel is considered to have memory which is modeled by a finite-state Markov channel (FSMC) model. Furthermore, we consider a multiuser DS-CDMA system that employs adaptive modulation and coding (AMC) at the transmitter. Although we only consider ON/OFF traffic source in this work, the formulation is valid for any traffic arrival process for which the MGF or a bound on the MGF exists. In the numerical results, we show the impact of the delay guarantee, burstiness, SNR and the user load on the throughput under various conditions, such as fading speed and

ponoff

respectively. The average traffic arrival reads thus

1-poffon

1-ponoff

λ = πon Pr .

Off

On

We also define a burstiness parameter 1 1 B= + , ponoff poffon

poffon Fig. 1.

The ON/OFF Traffic Source

delay bound violation probability. II. S YSTEM M ODEL A. Preliminaries Let us begin by introducing some assumptions and results used later in the paper. At the network layer, we use a discrete time model indexed by the variable t ∈ N = {1, 2, . . .}. For each time index t, we shall later associate one data block or a code word that is transmitted over the wireless channel at the physical layer. We also define the moment generating function (MGF) of a discrete time stationary process X(t) indexed by   t as MX (θ, t) = EX eθX(t) , where E denotes expectation. Lemma 2.1: Consider a discrete-time homogenous Markov chain with state space {1, . . . , L0 }. The corresponding L0 × L0 transition matrix is denoted by PX while π X denotes the steady state vector. The MGF of a random process X(t) driven by such a Markov chain is given by [9, p.245] MX (θ, t) = π X (RX (θ)PX )t−1 RX (θ)1,

(1)

where θ > 0, t ≥ 0 and we denoted 1 for the column vector of ones. The rate matrix RX (θ) in (1) is defined as RX (θ) = diag(eθR1 , . . . , eθRL0 ),

(5)

(2)

where Rl0 is the workload processed in state l0 . The steady state vector π X = [π1 π2 · · · πL0 ] can be obtained by PL0 solving π X PX = π X with the condition l0 =1 πl0 = 1. B. Data Traffic Model A discrete time ON/OFF data source (depicted in Fig. 1) is frequently used to model arrivals (A) from bursty traffic sources such as a voice traffic source [4], [11]. The ON state corresponds to a talk spurt in which the data is generated at a rate of Pr while the OFF state corresponds to a speaker’s silence (zero arrival rate). Here we assume that the user stays in the ON state with probability 1 − ponoff and moves to the OFF state with probability ponoff . Similarly, 1 − poffon and poffon denote the respective probabilities for the OFF state. The transition matrix PA for the ON/OFF source reads   1 − ponoff ponoff PA = , (3) poffon 1 − poffon and the elements of the steady state vector πA = [πon πoff ] are poffon and πoff = 1 − πon , (4) πon = poffon + ponoff

(6)

which reflects the average time for the Markov process to change states twice. A small value of B, for a fixed πon , corresponds to low burstiness and vice-versa [12]. Consider the discrete time ON/OFF source discussed above and let A(0, t) be a stationary random process that represents the amount of data that has arrived during the time interval [0, t). The MGF of the ON/OFF arrival process MA (θ, t), described by a two state (L0 = 2) Markov chain, can then be obtained from (1) by replacing the matrix PX with (3), using (4) for the steady state vector πX , and values R1 = Pr and R2 = 0 for the workloads in (2). C. Wireless Channel Model The physical layer is considered to be a synchronous uplink CDMA channel with K users. The mobile station (MS) and the base station (BS) have a single transmit and receive antenna1 , respectively. Perfect channel state information (CSI) is assumed to be available at the BS, while the MSs have no access to CSI. For simplicity, we let the network layer time index t ∈ N correspond to physical layer code words that are transmitted during multiple symbol periods n = 1, 2, . . . The number of active users Kn ≤ K during the nth symbol period is a random variable, modeling the bursty information sources. The BS is assumed to know the set of active users at all times. Consider an arbitrary but fixed network layer time index t ∈ N, and let us omit t from the following presentation for notational simplicity. The discrete time received signal after matched filtering and sampling reads then [1] yn =

K X

hk sk bk,n + vn ∈ CM ,

n = 1, 2, . . . ,

(7)

k=1

where bk,n is the nth code symbol transmitted by the kth user. We let bk,n = 0 if the user k is not active and bk,n ∈ C \ {0} otherwise. The code symbols of the active users are assumed to be independent with zero mean and unit variance. The length of the signature sequences (spreading factor) {sk }K k=1 is M for all users. The random vector v ∼ CN(0, σ 2 IM ) represents samples of additive white Gaussian noise (AWGN), where IM denotes the identity matrix of size M × M , and CN(0, Ψ) stands for the circularly symmetric complex Gaussian (CSCG) distribution with covariance Ψ. For a given time index t (and code word), the wireless channels between the MSs and the BS are assumed to be fixed. The fading coefficients hk ∼ CN(0, 1) for all k = 1, . . . , K are considered to be independent and identically distributed (IID), but extension to unequal average received powers is straightforward. Random spreading [1]–[3], 1 The large system extension of the decoupling result in Section II-E for multiple receive antennas at the BS follows straightforwardly from [13], [14]. We concentrate on the single antenna case here for the ease of exposition.

[15], [16] is assumed to be employed at the MSs, so that the signature sequences {sk }K k=1 are IID CSCG random vectors 1 IM ). For future convenience, we denote the sk ∼ CN(0, M user load of the system at time instant n as αn = Kn /M ≤ α = K/M,

(8)

where α is the maximum system load when all users are simultaneously active. The average received signal to noise ratio (SNR) is defined as SNRavg = 1/σ 2 . Remark 1: Due to random spreading, the user load αn can be greater than one since the signature sequences between the users need not be orthogonal. Thus, there is no need for user scheduling at the BS when αn > 1. D. Linear Receivers for Multiuser CDMA In this paper we consider linear multiuser detectors that form the decision variables for the kth user at time instant n as zk,n = cH (9) k,n yn , The explicit form of ck ∈ CN depends on the choice of the MUD. Note also that given specific receiver type, the vector ck is a function of (a subset of the following variables): the noise variance σ 2 , the signature sequences {sk }K k=1 , and the channel coefficients {hk }K k=1 . Common special cases for ck are 1) the SUMF, 2) the decorrelator, or ZF detector, and 3) the LMMSE estimator [1]. The first one is the simplest non-trivial receiver. If the assigned signals to each user were orthogonal at the receiver front end then the SUMF followed by threshold yields the optimum detector [1]. However it is seldom the case–one of the reason being the channel impairments. Further the SUMF does not exploit the structure of the MAI. LMMSE receiver exploits the structure of the MAI while the decorrelator effectively nulls outs the interfering signals in an optimal way. SUMF has a computational complexity O(M Kn ) if all users at time instant n are detected. The ZF and the LMMSE detectors both have complexity O(Kn3 ). The latter requires, however, knowledge of the noise variance σ 2 that may be difficult to obtain in practice. Since the LMMSE receiver minimizes the MSE and uncoded bit error rate among all linear receivers, it works as a benchmark in this paper. E. Large System Limit To simplify the physical layer model, we consider throughout the paper the large system limit in which both the number users K and the spreading factor M grow without bound with a fixed ratio α = K/M . We also assume that the parameters of the ON/OFF source, described in Section II-B, stay fixed in the large system limit and πon > 0. Thus, in the large system limit, the instantaneous user load converges almost surely to αn → αon = πon α > 0,

∀n = 1, 2, . . .

(10)

where αon is the effective user load given each user is modeled via the ON/OFF traffic source. In this asymptotic region, the multiuser CDMA channel (7) also decouples to a set of singleuser channels [2], [3], [15], [16] that do not depend on the signature sequences {sk }K k=1 , as discussed below.

Consider the detection of code symbol bk,n of the kth user. For the linear receivers listed in Section II-D, the SNR after detection simplifies in the large system limit to [2], [15], [16] γk = pk /βk ,

(11)

where pk represents the instantaneous channel power in the decoupled single-user channel and is drawn according to the probability density function (PDF) f (pk ) = e−pk ,

pk ≥ 0.

(12)

Since the interference is Gaussian in the large system limit, the statistics of the decision variables (9) match the statistics of the received symbols of the equivalent single-user (su) channel √ su n = 1, 2, . . . (13) zk,n = pk bk,n + nsu k,n , where nsu k,n ∼ CN(0, βk ), n = 1, 2, . . . , are IID CSCG random variables independent of pk . The noise variance βk captures the multiple access interference suppression capability of the detector and depends on the selected receiver as [2], [15] SUMF βksumf = σ 2 + αon , Decorrelator/ZF σ2 , βkzf = 1 − αon σ 2 αon + (αon − 1)2 , βkzf = αon − 1 LMMSE Z ∞

βklmmse = σ 2 + αon

0

(14)

αon < 1, (15) αon > 1, (16)

p f (p)dp, 1 + p/β1lmmse

(17)

where the density f (p) is given in (12) and the fixed point equation (17) can be easily solved iteratively. For later use, we denote the average post-detection SNR Z γ k = γk f (pk )dpk = 1/βk , (18) and write the PDF of (11) as f (γk ) = βk e

βk γk

  1 γ = exp − . γ γ

(19)

In the following cross-layer analysis, we always consider the equivalent single-user system defined by (11) – (17) instead of the original system (7). The accuracy of our approach depends on the system size, K, M and the steady state probability πon . F. Adaptive Coded Modulation for Multiuser CDMA Consider an arbitrary but fixed time index t = 1, 2, . . . at the network layer. By the decoupling principle, the decision variable (9) of the kth user is statistically equivalent to the received symbol in the single-user channel (13). Let the code word of the kth user be bk = [bk,1 , . . . , bk,Nk ]T ∈ CNk , and let the received SNR over it be given by (11). If the elements

of bk belong to a discrete set M, the maximum rate that user k can reliably transmit over the wireless channel reads [17] IM (γk ) = log2 (|M|) − log2 (e) " # Z X √ 1 X −|v|2 −|v+ γ(b−˜ b)|2 e log2 e dv, (20) − π|M| ˜ b∈M

b∈M

where Nk is assumed to be sufficiently large and v ∈ C. To introduce AMC to the system, we assume that: 1) If the MS k is not active, it uses a code word bk = 0; 2) For active users, the code word is selected from a sequence Ck,0 , Ck,1 , . . . , Ck,L of random code books2 that have rates 0 = R0 < R1 < · · · < RL < ∞. All modulation sets Ml ⊂ C, l = 0, 1, . . . , L, satisfy E |bk,n |2 = 1, ∀bk,n ∈ Ml , where R0 = 0 denotes “outage”. In this paper, we assume that the BS has knowledge of γk for all MSs so that it can use (20) to find the suitable code books for the active users k lk∗ =

max {l : Rl ≤ IMl (γk )},

l=0,1,...,L

The wireless channel service process in the block Rayleigh fading channel (13) with AMC is modeled using an Lstate finite state Markov chain (FSMC). The block length (in seconds) is denoted by Tb and each transmission mode is mapped to one of the corresponding FSMC states. We let PS = [pl,l0 ] be the L × L transition matrix of the FSMC describing the channel. If the channel is slowly time varying and Tb relatively short, it is reasonable to assume that pl,l0 = 0 for all |l − l0 | > 1. The adjacent state transition probabilities for Rayleigh fading are then determined as [21] ( )Tb , l = 1, 2, . . . , L − 1, pl,l+1 ≈ N (Γl+1 πl (22) N (Γl )Tb pl,l−1 ≈ , l = 1, 2, . . . , L πl where N (Γ) denotes the level crossing rate (LCR) at SNR Γ, given a time varying Rayleigh fading channel with average SNR γ and PDF (19), i.e., s   2πΓ Γ fm exp − N (Γ) = , (23) γ γ

(21)

{lk∗ }

and feed back the indexes to the active MSs before the data transmission starts. Each active MS then uses Ck,lk∗ to transmit at rate Rl∗ over the CDMA channel with a vanishing probability of error as the code word lengths increase. In theory, we would like to have a large set of code books with finely spaced rates to accurately adjust to the channel conditions. However, designing arbitrary rate codes is difficult and as the number of code books L grows, a reliable and very high rate feedback channel is needed for code book selection. Thus, the number of transmission modes L is in practice small. Remark 2: The authors in, e.g., [18], [19] propose to simulate specific AMC schemes in AWGN channel and use parameter fitting to derive “analytical” error probabilities for them. Such results are, however, heavily dependent on the chosen error control codes. We consider instead the modulation constrained capacity (20) that gives an upper bound for all coding schemes with the same modulation sets and code rates. With the new code designs [20], we expect our results to provide a close description of modern wireless systems. III. D ELAY C ONSTRAINED T HROUGHPUT A NALYSIS In this section we formulate the traffic carrying capacity of CDMA networks under delay constraints which we call the delay constrained throughput. We concentrate on the decoupled single-user channel (13) of the kth user and omit both the user and the time indexes for notational convenience. Note that since all users are here statistically equivalent in the large system limit, it is sufficient to consider the delay constrained performance of a single user. Recall also that the impact of multiple access interference is captured by the equivalent noise variance given in (14) – (17) for the linear detectors investigated in this paper. 2 For simplicity of notation, we assume that code words of any desired lengths can be picked from all code books Ck,l .

where fm = v/ω is the maximum Doppler frequency of the channel, defined in terms of the vehicle speed v and the wavelength ω of the transmitted signal. The remaining transition probabilities are given as   p1,1 = 1 − p1,2 pL,L = 1 − pL,L−1 (24)  pl,l = 1 − pl,l−1 − pl,l+1 , l = 2, 3, . . . , L − 1, and the stationary probability πl of the FSMC being in state l reads     Z Γl+1 Γl+1 Γl − exp − , πl = f (γ)dγ = exp − γ γ Γl (25) where f (γ) is given in (19). The discrete time service of the channel3 is assumed to be an independent stationary random processes given by the cumulative processes S(s, t) with the corresponding MGF MS (θ, t). S(s, t) denotes the service offered by the time varying wireless channel in the interval [s, t). Let Nb denote the number of information bits per an upper layer data block and W the system bandwidth. Omitting again both the user and the time indexes for notational convenience, we can then write the number of data blocks transmitted in state l of the ˜ l = Rl Tb W/Nb . FSMC based channel service process as R b S (θ, t) = MS (−θ, t). The MGF of the Let us denote M random process S described by a homogeneous Markov chain (FSMC) with transition matrix PS can be obtained by (1) where PX is replaced with PS = [pl,l0 ] and using (22) – (24) as b S (θ, t) = π S (R ˜ S (θ)1, ˜ S (θ)(−θ)PS )t−1 R M (26) where

˜ S (θ) = diag(eθR˜ 1 , . . . , eθR˜ L ). R

(27)

3 It is assumed that the wireless channel described by the random process S is a dynamic server [9].

( dελ

= inf

θ>0

"

1 inf τ : τ ≥0 θ

ln

∞ X

! b S (θ, s) − ln ε MA (θ, s − τ )M

Ml BPSK BPSK QPSK QPSK 16-QAM 16-QAM 64-QAM

Rl (bps/Hz) 0 0.5 1 1.5 2.25 3 4.5

Γl (dB) -∞ -2.80 0.19 3.39 6.20 9.30 14.37

The steady state vector π S = [π1 π2 · · · πL ] can be obtained by making use of (25). The task is to find the maximum traffic which can be carried by the network for a given delay guarantee dg and violation probability ε. We call this the delay constrained throughput and is given by λd . Let dελ represent a bound with violation probability ε on the delay where λ denotes the average traffic arrival rate. Proposition 3.1: The delay constrained throughput λd of a DS-CDMA system under a delay guarantee dg is given as λd = α · max {λ | dελ ≤ dg },

≤0

,

(29)

s=τ

TABLE I T RANSMISSION MODES Modes 0 1 2 3 4 5 6

#)

(28)

dελ ,

where assuming FIFO scheduling, is given in (29) at the b S (θ, s) is given by (26). top of the page and M Proof: We refer the interested reader to [10]. Remark 3: The delay bound (29) is calculated using MGF based stochastic network calculus approach [10]. MGF for a variety of arrival models is available in the literature [9] b S (θ, t) is a but finding the MGF of the service process M challenging task for which we make use of (26). Having b S (θ, t), a stochastic bound on the delay dε in (29) obtained M λ can be obtained using Boole’s inequality, Chernoff’s bound, and applying the technique in [10]. IV. N UMERICAL R ESULTS We consider the AMC setup given in Table I. The transmission modes follow the HIPERLAN/2 specification (see, e.g., [18], [19]). The parameter Γl in Table I denotes the SNR point in decibels (dBs) where the lth mode is switched on. The physical layer of the system is parameterized by using a W = 20 MHz channel. The maximum size of the upper layer data blocks is fixed to Nb = 10 000 bits. The base time unit Tb is taken to be the time required to transmit a data block over a 90 Mbps channel, the maximum rate (4.5 × 20) ˜ S (θ) requires the entry Rl given in Table I. The rate matrix R which can be obtained from the Table I. The infinite sum in the delay bound formula (29) is computed for the first 4000 units of time. The steady state probability of being in the ON state πon is fixed to 0.1 while the probability of moving from ON to OFF state is ponoff = 0.5 unless stated otherwise. Fig. 2 depicts throughput as a function of the traffic burstiness for different receivers where the burstiness parameter is given by (6) and the delay guarantee requirement is 100 time

slots (TS). Note that the effective user load αon is fixed to 0.15 and, thus, the receiver experiences the same MAI in all cases. Clearly the LMMSE offers the highest throughput albeit it also deteriorates the most as a function of increasing burstiness. The reason for the ZF being worse than the SUMF is due to the low operating SNR where the noise amplification of the decorrelator becomes a major issue. Fig. 3 depicts throughput as a function of delay guarantee dg for different receivers. The general trend is very similar to Fig. 2. The throughput saturates after a given delay guarantee owing to the well understood concept that traffic arrival rate should never be greater than the capacity to ensure system stability. The impact of the average received SNR (SNRavg = 1/σ 2 ) on the throughput is studied in Fig. 4. As expected, increasing the average received SNR leads to an improvement in the throughput for all MUDs. For the considered system this is due to the fact that, the higher average received SNR increases the likelihood of a better channel at any given time instant and, thus, allows for higher rate code books to be used at the transmitters on average. One should observe that the ZF detector requires SNRavg > 3 dBs to be more efficient than the simple SUMF. Hence, the choice between the SUMF and ZF/decorrelator depends heavily on the operating conditions and QoS constraints. Finally, Fig. 5 depicts the delay constrained capacity as a function of the maximum user load α = K/M . With the delay constraint, the system throughput behaves similarly to the information theoretic case (see, e.g., [3, Fig. 1]), but the optimum user load shifts towards zero with the delay guarantee. Note that without delay constraints, the ZF detector has a zero rate only at a single point, whereas with delay constraints there is a region in α where the ZF/decorrelator does not yield a positive rate. This behavior is heavily dependent on SNRavg . V. C ONCLUSIONS AND F UTURE W ORK In this paper we study the interplay between physical layer interference suppression algorithms (multiuser detectors), the QoS constraints (delay guarantee and delay violation probability) and the network layer throughput. We show the impact of the delay guarantee, burstiness, SNR and the user load on the traffic carrying capacity of multiuser DS-CDMA system with linear multiuser detectors. The numerical results show that the behavior of the ZF detector deviates significantly form the information theoretic treatment, suggesting that it may not be good choice for the MUD unless the system is operating at high SNR. This work has its application in performance evaluation of wireless networks with bursty traffic sources where the maximum throughput for a given delay guarantee is of interest. While in this work, we only used ON/OFF source to illustrate the results, the same methodology can be applied to any traffic source with known MGF.

3.5 3 2.5 2 1.5 1 0.5 0 10

30

50 70 Burstiness (B)

Delay Constrained Throughput [Mbps] λd

Fig. 2. SNRavg = 2 dB, fm = 40 Hz, Delay Bound = 100 TS

Fig. 3.

9 8 7

90

8 7

α = K/M = 1.5, ε = 10−2 ,

5 4 3 2 1

Fig. 4. 3

LMMSE SUMF ZF

5 4 3 2 1 200 400 600 800 Delay Guarantee (dg) [time slots]

SNRavg = 2 dB, fm = 40 Hz,

1000

α = K/M = 1.5, ε = 10−2

R EFERENCES [1] S. Verd´u, Multiuser Detection. Cambridge, UK: Cambridge University Press, 1998. [2] D. N. C. Tse and S. V. Hanly, “Linear multiuser receivers: Effective interference, effective bandwidth and user capacity,” IEEE Trans. Inform. Theory, vol. 45, no. 2, pp. 641–657, March 1999. [3] S. Shamai and S. Verd´u, “The impact of frequency-flat fading on the spectral efficiency of CDMA,” IEEE Trans. Inform. Theory, vol. 47, no. 4, pp. 1302–1327, May 2001. [4] J. Zhang, M. Hu, and N. Shroff, “Bursty data over CDMA: MAI self similarity, rate control and admission control,” in IEEE INFOCOM, vol. 1, 2002, pp. 391–399. [5] F. Yu and V. Krishnamurthy, “Effective bandwidth of multimedia traffic in packet wireless CDMA networks with LMMSE receivers: A crosslayer perspective,” IEEE Trans. on Wireless Commun., vol. 5, pp. 525– 530, 2006. [6] Y. Jiang and Y. Liu, Stochastic Network Calculus. Springer, 2008. [7] S. Mao and S. Panwar, “A survey on envelope processes and their applications in quality of service provisioning,” IEEE Commun. Surveys and Tutorials, vol. 8, no. 3, pp. 2–19, Third Quarter 2006. [8] M. Fidler, “Survey of deterministic and stochastic service curve models in the network calculus,” IEEE Commun. Surveys & Tutorials, vol. 12, no. 1, pp. 59 –86, 2010. [9] C.-S. Chang, Performance Guarantees in Communication Networks. London, UK: Springer-Verlag, 2000. [10] M. Fidler, “An end-to-end probabilistic network calculus with moment generating functions,” in 14th IEEE International Workshop on Quality of Service (IWQoS), 2006, pp. 261–270. [11] V. Frost and B. Melamed, “Traffic modeling for telecommunications networks,” Communications Magazine, IEEE, vol. 32, no. 3, pp. 70–81, 1994.

LMMSE SUMF ZF

6

0 0

100

6

0.15 0

Delay Constrained Throughput [Mbps] λd

9 LMMSE SUMF ZF

Delay Constrained Throughput [Mbps] λd

Delay Constrained Throughput [Mbps] λd

4

2.5

2

4

SNR [dB]

6

8

10

α = K/M = 1.5, fm = 40 Hz, ε = 10−2 LMMSE SUMF ZF

2 1.5 1 0.5 0

Fig. 5.

0.5

1 User Load ( α )

1.5

2

SNRavg = 2 dB, fm = 40 Hz, ε = 10−2

[12] F. Ciucu, A. Burchard, and J. Liebeherr, “A network service curve approach for the stochastic analysis of networks,” SIGMETRICS Perform. Eval. Rev., vol. 33, no. 1, pp. 279–290, 2005. [13] S. V. Hanly and D. N. C. Tse, “Resource pooling and effective bandwidths in CDMA networks with multiuser receivers and spatial diversity,” IEEE Trans. Inform. Theory, vol. 47, no. 4, pp. 1328–1351, May 2001. [14] L. Cottatellucci and R. R. M¨uller, “CDMA systems with correlated spatial diversity: A generalized resource pooling result,” IEEE Trans. Inform. Theory, vol. 53, no. 3, pp. 1116–1136, Mar. 2007. [15] Y. Eldar and A. Chan, “On the asymptotic performance of the decorrelator,” IEEE Trans. Inform. Theory, vol. 49, no. 9, pp. 2309–2313, Sep. 2003. [16] J. Zhang, E. K. P. Chong, and D. N. C. Tse, “Output MAI distributions of linear MMSE multiuser receivers in DS-CDMA systems,” IEEE Trans. Inform. Theory, vol. 47, no. 3, pp. 1128–1144, March 2001. [17] A. Guill´en i F`abregas and G. Caire, “Coded modulation in the blockfading channel: Coding theorems and code construction,” IEEE Trans. Inform. Theory, vol. 52, no. 1, pp. 91–114, Mar. 2006. [18] Q. Liu, S. Zhou, and G. B. Giannakis, “Cross-layer combining of adaptive modulation and coding with truncated ARQ over wireless links,” IEEE Trans. on Wireless Commun., vol. 3, pp. 1746–1755, 2004. [19] J. Tang and X. Zhang, “Cross-layer modeling for quality of service guarantees over wireless links,” IEEE Trans. on Wireless Commun., vol. 6, no. 12, pp. 4504–4512, 2007. [20] T. Richardson and R. Urbanke, Modern Coding Theory. Cambridge, UK: Cambridge University Press, 2008. [21] H. Wang and N. Moayeri, “Finite-state Markov channel-a useful model for radio communication channels,” IEEE Transactions on Vehicular Technology, vol. 44, no. 1, pp. 163–171, 1995.