Robust cross - Layer scheduling design in multi - User multi - Antenna wireless systems

us er th ro ug hp ut (b its /H z/s ) number of nT the contribution of nT to total data user throughput at high and low SNR Homogeneous scheduling, SNR =8dB Homogeneous scheduling, SNR =0dB Heterogeneous scheduling, SNR =8dB Heterogeneous scheduling, SNR =0dB high SNR low SNR (a) Network Capacity of data users (bits/Hz/s): 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 0 5 10 15 20 25 30 n u m be r o f v oi ce u se rs number of nT contribution of nT to voice capacity (number of voice users with mean time−delay ≤ 50 ms) Heterogeneous scheduling, SNR =8dB Homogeneous scheduling, SNR =8dB Heterogeneous scheduling, SNR =0dB Homogeneous scheduling, SNR =0dB high SNR low SNR (b) Voice Capacity (number of supported voice users) Figure 5.6 Spatial multiplexing gains of voice and data users (BW = 20kHz, Kdata =20, Kvoice =2, T0 = 20ms). 82 0 20 40 60 80 0 2 4 6 8 10 12 14 16 18 20 D at a: a ve ra ge th ro up ut (bi t/H z/s ); V oic e: me an tim e d ela y(m s) simulation time ( each point 2000 time slots) Homogeneous scheduling with bursty data traffic Data user throughput Voice user mean time delay λdata = 0.2 λdata = 0.7 λdata = 0.1 λdata = 0.8 (a) Homogeneous scheduling: 0 20 40 60 80 0 2 4 6 8 10 12 14 16 18 D at a: a ve ra ge th ro up ut (b it/H z/s ); V oic e: me an tim e d ela y ( ms ) simulation time ( each point 2000 time slots) the effect of bursty data traffic to voice with heterogeneous scheduling Data user throughput Voice user mean time delay λdata = 0.2 λdata = 0.7 λdata = 0.1 λdata = 0.8 (b) Heterogeneous scheduling: Figure 5.7 Transient performance of voice users in the presence of bursty data loading. BW = 20kHz, nT =4, Kdata =20, Kvoice =2, T0 = 20ms. 83 where the summation term is neglected due to the high SNR region. Hence, from (5.6), the instan- taneous capacity utility with |A| = Q is given by: Ccap(Q) = Q∑ k=1 log2 ( P0|hkwk|2 Q ) = Q∑ k=1 log2 ( P0λ (Q) k Q ) , where λ(Q)k is the largest eigenvalue of ( B∗A,kBA,k )−1 B∗A,kh ∗ khkBA,k. Consider Ccap(Q)− Ccap(Q−m) = m log2(P0) + ∑ k log2(λ (Q) k ) − ∑ k log2(λ (Q−m) k ) +Q log2(Q)− (Q−m) log2(Q−m) > 0 for large P0. Hence, we have Ccap(Q) ≥ Ccap(Q−m) for all Q ≤ nT and m ≥ 1. Therefore, the optimizing A has cardinality of nT . 84 Chapter 6 Cross-layer Downlink Scheduling with Het- erogeneous Delay Constraints 6.1 Overview In Chapter 5, we have shown that it is important to consider the effect of the source statistics, queueing delays and application level requirements into the cross-layer scheduling design. How- ever, the proposed heterogeneous scheduling design is based on a heuristic way by naively giving priority to voice users. It ensures stable delay performance but cannot guarantee any specific delay requirement for Voice users. It is also not clear how well the proposed allocation policy performs compared with the optimal performance. In this chapter, we shall focus on the design and performance analysis of cross-layer schedulers for multi-antenna systems with specific delay requirements. Similar to previous chapters, we con- sider a wireless multimedia system with a base station (with nT transmit antennas), K1 client users (with a single antenna) running delay sensitive application and K2 background data users (with a single antenna) running delay insensitive applications such as emails or FTP. In fact, different ap- plications can have very different QoS requirements. We shall propose an analytical framework to address the heterogeneous cross-layer scheduling problem, which exploits the spatial multiplexing gain and the multi-user selection diversity gains to maximize the system throughput, and at the same time fulfil the QoS requirements for heterogenous users. The physical layer performance is 85 modeled based on information theory and the delay (buffer) dynamics is modeled based on queue- ing theory. Based on the analytical framework, a novel cross-layer scheduler, which dynamically adjusts the rate, power and user selection, is obtained for the mixed user application. 6.2 System Model 6.2.1 Multi-user Physical Layer Model We consider a communication system with a base station (with nT transmit antennas), K1 delay sensitive users and K2 bursty delay insensitive users (K = K1 + K2 users with single receive antenna), which is the same as the system model in Chapter 5. Referring to the linear transmit- receive processing model introduced in Section 2.3 and Fig. 2.4, the channel fading between the base station and the k-th mobile user is characterized by the 1 × nT dimension channel vector, hk, which is modeled as a circularly symmetric complex Gaussian vector with zero mean and covariance InT . Furthermore, it is assumed that the duration of the scheduling slot (2 ms) is much shorter than the coherence time of the fading channel1. In other words, the channel fading is quasi-static within an encoding frame. We shall consider zero-forcing (ZF) approach at the base station which only has a linear order of complexity. With ZF transmit scheme, a number of orthogonal beams are formed at the base station with each beam carrying independent streams of information corresponding to the selected set of mobiles. In general, K streams of information data to K individual users at the base station transmitter are encoded independently. The vector (K × 1) of encoded symbols, U = [u1, ..., uK ], are processed by the power control diagonal matrix (K×K) √P = diag(√p1, ...,√pK) followed by the beamforming matrix (nT ×K), W = [w1, ...,wK ]. pk ≥ 0 is the average transmit power for user k and wk is the nT ×1 complex beamforming weight for user k. Hence, the nT ×1 transmitted signal X from the base station is given by: X = K∑ k=1 Xk = K∑ k=1 uk √ pkwk. (6.1) 1We are targeting for pedestrian users where the channel coherence time is of the order of 40ms. 86 The received signal of the k−th user yk is given by: yk = √ pkhkwkuk︸ ︷︷ ︸ Information + ∑ j 6=k √ pjhkwjuj︸ ︷︷ ︸ Multi-beam Interference +Zk, (6.2) where the first term contains the desired signal and the middle term represents the multi-beam interference due to simultaneous transmission of independent information streams. Using the ZF approach for the weight selections (wk), we have hkwj = 0 and hence, the multi-beam interference term is zero-out. The ZF beamforming (ZFBF) weights calculation can be found in Section 2.3. The received signal for mobile user k after ZF processing is given by yk = √ pkhkwkuk + Zk. (6.3) Hence, the maximum achievable date rate of the k-th user during the fading block is given by the maximum mutual information between uk and yk and is given by rk = log2 ( 1 + pk|hkwk|2 σ2z ) . (6.4) Let A = {k ∈ [1, K] : pk > 0} be the admissible user set (set of user indices with positive power allocation). Due to the ZF processing, we have cardinality constraint on A, i.e. |A| ≤ nT . Furthermore, we assume the total transmit power from the base station is constrained by P0, i.e.∑ k∈A pk ≤ P0. 6.2.2 Source Model - Delay Sensitive and Delay Insensitive Consider a system with K users consisting of two classes of users (delay sensitive and delay insensitive), as shown in Fig. 6.1. Packets of each user k arrive at the user’s queue (at random arrival time) according to a Poisson process with independent rate λk, with each packet of fixed size F2. Each user k has a mean packet delay requirement τk. The nature of k-th user’s application 2In order to avoid partial delivery problem in the subsequent analysis (i.e. for simplicity of analysis, we assume a packet will not be digested over multiple time slots), the file size F defined here are assumed to be small enough compared to service rate. 87 p1 p1 p2 p1 p2 p3 p4 p1 p2 data queues Heterogeneous Scheduler p1 update data buffer p1 p1 p2 p2 p3 p3 User 1 User 2 User K User K-1 K 1 Class-1 users power constraint delay constraint queueing state Q channel state H rate allocation power allocation active user set K 2 Class-2 users Figure 6.1 Queueing model and scheduling model is thus characterized by the tuple (λk, τk), representing the average incoming traffic rate and delay requirement. We further assume that the same class users have the same tuple (λk, τk) setting. The transmission process for each user can be modeled as an M/G/1 queue [67] with non- selection vacation (If the user is not selected in current time slot). Define the average system time of user k (queueing time and transmission time) as E[Tk]: E[Tk] = E[Wk] + E[Xk], (6.5) where E[Wk] is the expected waiting time in queue before the packets start being served; E[Xk] is the average transmission time for k-th user’s packets. 6.3 Formulation of the Cross-layer Design for Heterogeneous Users The scheduling design can be modeled as an optimization problem in the resource space to maximize the system level performance under some physical layer constraints (such as the transmit power and multiplexing scheme) and network layer constraints (such as delay requirements). We 88 shall consider the average system throughput as the optimization objective. Uthp(r1, . . . , rK) = E [Gthp(r1, .., rK)] = E [∑ k∈A rk ] , (6.6) where the expectation E[·] is taken over all channel state and queue state realization; Gthp de- notes the instantaneous system goodput. The scheduling problem can be cast into the following optimization problem: Problem 6.1 Cross-Layer Formulation Choose the admissible user set policy {A(H,Q)}, the power allocation policy {pk(H,Q)} as well as the rate allocation policy {rk(H,Q)}3 so as to maximize the average system throughput in (6.6) subject to average power constraint, delay constraint and degree of freedom constraint, i.e.: max sk∈[0,1],pk≥0 E [∑ k sk log2 ( 1 + pk|hkwk|2 σ2z )] (6.7) s.t. C1 : ∑ k sk ≤ nT C2 : E[ ∑ k skpk] ≤ P0 C3 : E[Tk] ≤ τk,∀k ∈ [1 : K], where H and Q represent the channel state and queueing state respectively; sk is an indicator which has sk = 1 if user k is selected and sk = 0 otherwise. C1 is degree of freedom constraint due to the ZFBF transmit scheme; C2 is the average total power constraint; And C3 is the delay requirement of each user on average system time E[Tk]. The solution of above optimization problem (6.7) is not explicit due to the fact the average delay constraint in C3 is not in the form of optimizing parameters ({rk},{pk} and {sk}). Thus we shall first establish the relationship between system time (Tk) and transmission rate rk in the following lemma. 3Assuming perfect channel state information is available and powerful coding such as LDPC is used, the optimal rate is approximated by Shannon Capacity (6.4) 89 Lemma 5 (Equivalent Delay Constraint) A necessary and sufficient condition of delay constraint C3 in (6.7) is E[Xk] + λkE[X 2 k ] + λE[Xk](E[s¯k]/E[sk])(ts) 2 (1− λkE[s¯k]/E[sk]) ≤ τk, (6.8) where Xk is the transmission time of the packet of user k; λk and τk are respectively the incoming rate and delay requirement of user k; E[sk] denotes the selection probability of user k and E[s¯k](= 1− E[sk]) is the probability that user k is not selected. Corollory 1 In terms of transmission rate rk, a necessary condition for delay constraint in (6.8) is: E[skrk] ≥ ρk (6.9) where ρk = (2− λkts)E[sk] + λk(2τk + ts) + √ c− b2/a 4τk F, where sk is the selection indicator and a, b, c is respectively given by a = (2 − λkts)2 + 8λkτk, b = (4λkts − 8λkτk − 2λk(λkts)(2τk + ts)) and c = (λk(2τk + ts))2; F is the packet size; ts is the time slot duration (2ms). Proof 5 Please refer to Appendix 6A. With the result from Corollary 1, the cross-layer scheduling problem in Eqn. (6.7) can be reformulated as: max sk∈{0,1},pk≥0 E [∑ k sk log2 ( 1 + pk|hkwk|2 σ2z )] (6.10) s.t. C˜1 : ∑ k sk ≤ nT C˜2 : E [∑ k skpk ] ≤ P0 C˜3 : E [ sk log2(1 + pk|hkwk|2 σ2z ) ] ≥ ρk. Notice that the ρk (given by Eqn.(6.9)) is a function of selection probability, incoming rate and delay requirement. The solution of optimization problem in Eqn.(6.10) can be found by Lagrange method described in the next section. 90 6.4 Solution of the Cross-Layer Optimization Problem Note that the problem of cross-layer scheduler design in Eqn.(6.10) is a mixed convex opti- mization (on pk) and combinatorial search problem (on active set A). The optimal solution can be obtained through two steps by separating the binary variable (s1, ..., sK) from the continuous variable (p1, .., pK). 6.4.1 Convex Optimization on (p1, .., pK) Given any active user set {s1, ..., sK}, the optimizing power vector (p1, .., pK) of the optimiza- tion problem in Eqn.(6.10) is determined by the following Lagrangian: L(pk) = E [∑ k∈A log2(1 + pk|hkwk|2 σ2z ) ] −γk ( E [ log2(1 + pk|hkwk|2 σ2z ) ] − ρk ) +µ ( E[ ∑ k∈A pk]− P0 ) . (6.11) Thus for any given channel realization H, the optimal power allocation can be determined by: L˜(pk) = ∑ k∈A log2 ( 1 + pk|hkwk|2 σ2z ) − γk ( log2 ( 1 + pk|hkwk|2 σ2z ) − ρk ) + µ (∑ k pk − P0 ) . (6.12) The first order derivative on pk from above equation gives the optimal power vector, which is a result of multi-level water-filling p∗k = ( 1− γk µ − σ 2 z |hkwk|2 )+ (6.13) for all k ∈ A, where {γk} and µ are lagrangian parameters determined by KKT condition: γk ( E [ log2 ( 1 + pk|hkwk| 2 σ2z )] − ρk ) = 0 E [∑ k∈A pk ]− P0 = 0. (6.14) Since there is no closed-form solution for {γk} and µ. We shall devise a Lagrangian multiplier finder algorithm described in Appendix 6B to obtain the numerical solution of {γk} and µ. 91 6.4.2 Combinatorial Search on Admissible Set This step is to find the optimal binary vector (s1, .., sK) satisfying the constraint ∑ k sk ≤ nT . The sum rate can be maximized by exploiting the multi-user diversity gain while at the same time maintaining the delay constraints for all users. The optimal active set can be obtained by performing exhaustive search among all possible ac- tive user sets and the total search space is given by ∑nT m=1  K m . However, at moderate K and nT , the computational complexity of the optimal search algorithm is huge and is beyond imple- mentation limit. Hence, we shall apply the low-complexity genetic selection algorithm introduced in Appendix 3A (Chapter 3) for the search of A. 6.5 Numerical Results and Discussions For illustration, we consider two types of users in the system, namely the K1 delay-sensitive users and the K2 delay-insensitive users. The system bandwidth is 20kHz and the packet size is assumed to be F = 80 bits. Each simulation point is obtained by 5000 channel fading realizations. 6.5.1 Delay Performance of the Proposed Scheduler Figure 6.2 illustrates the mean packet delay (given by average system time E[Tk]) of two classes users versus delay insensitive background traffic. It is observed that for the proposed optimal heterogeneous scheduler, delay constrained users maintain the target delay requirement in a stable way with respect to the increasing traffic load of class-2 delay insensitive users (For example the delay constraint for class-1 users is 5 time slots in Fig. 6.2). Hence the delay sensitive application is almost not affected by the increasing traffic loading of delay insensitive application, which is highly desirable in the wireless system where heterogeneous applications coexist. On the other hand, the mean packet delay of class-2 users increases rapidly with the increasing data traffic load. 92 0.1 0.15 0.2 0.25 0.3 0 50 100 150 200 250 300 350 The incoming rate for delay insensitive user, Power SNR = 10 dB Pa ck et m ea n de la y tim e class−1 user 1 (delay constraint 5) class−1 user 2 class−2 user (delay constraint 1000) class−2 user 2 Figure 6.2 Mean packet delay (number of time slots) of two class users versus background data traffic λ (packets/time slot) (BW = 20kHz, nT =4, ts = 2ms) 93 6.5.2 System Throughput Performance Figure 6.3 compares the proposed optimal heterogeneous scheduler with the heuristic hetero- geneous scheduler proposed in Chapter 4 (also in [68]) and the baseline Round-robin scheduler (RRS). The heuristic heterogeneous scheduling design in Chapter 4 gives priority to the delay sen- sitive users in a heuristic way regardless of their channel condition. In RRS, the users take turns to transmit regardless of their channel condition and delay requirements. With the delay requirement (τ1 = 10, τ2 = 1000 time slots), the optimal heterogeneous sched- uler has a significant capacity gain over the heuristic heterogeneous scheduler for both high SNR (solid lines, 16dB) and low SNR (dashed lines, 6dB) cases at different nT s. Notice that the pro- posed scheduler exhibits higher capacity gain over nT (by more effectively exploiting multiplexing gain and multi-user diversity gain) than the heuristic scheduler. It can be also observed that, with- out incorporating the channel state and queue state into the scheduling design, the baseline RRS scheduler suffers a significant capacity loss. 6.5.3 Delay and Power Tradeoff The delay and power tradeoff has been already analytically studied in [37]. We shall present some discussion on the delay and power tradeoff exhibited in the proposed scheduling design. For the proposed scheduling design, there exists a minimum transmit power required to meet all users’ delay constraints. This minimum power is obtained by assigning just enough power to satisfy the inequality constraint in C˜3 of Eqn. (6.10), which is given by:  E [ log2 ( (1−γk) µ |hkwk|2 σ2z )] = ρk Pmin = E ∑ k [( 1+γk µ − σ2z|hkwk|2 )+] , (6.15) where ρk is given in Eqn.(6.9). E[·] is the ensemble average in terms of channel variation. The value of minimum power can be numerically obtained in the step-2 of the iterative Lagrangian multiplier finder algorithm elaborated in Appendix 6B. The behavior of the minimum power-delay tradeoff is characterized in Fig. 6.4, which outlines the feasible power operation region. As is shown, a more stringent delay constraint requires a 94 higher total transmit power. Increasing the number of users and the incoming traffic loading (of delay sensitive users) will both result in a higher minimum power requirement. For example, increasing the user numbers from {K1 = 2, K2 = 2} to {K1 = 4, K2 = 6} doubles the required minimum power at delay constraint τ2 = 5 time slots. Appendix 6A: Proof the Lemma 5 and Corollary 1 - Proof of Lemma 5 In the proposed scheduling framework, the buffer for each user can be modeled as an M/G/1 queue [67]. Due to the fact there will be the non-section time slot when the user is not selected, it can be further modeled as an M/G/1 queue with non-selection vacation. We shall the derive delay performance of each user with the model of M/G/1 queue with non-selection based on queueing theory. Consider the j-th data packet of user k arriving its queue (For notation simplicity, we shall drop the user index k in the derivation), the average system time of this packet E[T ] (including the time spent in queue and in transmission) is given by: E[T ] = E[W ] + E[X], (6.16) where E[X] is the average transmission time for this packet; E[W ] is the expected waiting time in queue before being served. As shown in Fig. 6.5, the waiting time W consists of three parts [67]:(1) the residual time R which means the remaining time for digesting the current packet being served when j-th data packet arrives; (2) the total transmission time for the NQ packets (the sum of NQ packets’ trans- mission time ∑NQ k=1 E[Xk]) currently in the queue when j-th packet arrives ∑j−1 n=j−NQ Xn; (3) the average vacation time due to non-selection (E[ZT ]). Thus the expected waiting time E[W ] is given by: E[W ] = E[R] +NQE[X] + E[ZT ]. (6.17) The mean transmission time E[X], average residual time E[R] and the average non-selection vacation time ZT are respectively elaborated in the following. 95 2 4 6 8 10 12 0 5 10 15 Number of transmit antenna nT, K1 = 8, K2=8 Su m ra te o f a ll u se rs (b its /H z/s ) optimal sch, SNR=16dB optimal sch, SNR=6dB heuristic sch, SNR=16dB heuristic sch, SNR=6dB RRS sch, SNR=16dB RRS sch, SNR=6dB Figure 6.3 Comparison of optimal, heuristic heterogeneous schedulers and Round-Robin scheduler on Spatial multiplexing gains over nT (BW = 20kHz, K1 = 8, K2 = 8, ts = 2ms, SNR = 6 dB and 16 dB) 96 0 5 10 15 20 25 30 35 40 0 2 4 6 8 10 12 14 16 18 20 Delay constraint for 1st−class user τ1 (number of time slots) M in im um to ta l p ow er delay and minimum power tradeoff nT=4, K1=4, K2=6, λ1 = 0.4 nT=4, K1=4, K2=6, λ1 = 0.2 nT=4, K1=2, K2=2, λ1 = 0.2 Figure 6.4 Minimum transmission power versus first class users delay requirement the packet being served j-th arriving packet waiting time due to the existing N Q packets and non-selection time slot: residual time R : remaining service time of the packet being served when j-th packet arrives queue buffer for user k : with N Q packets in buffer when j-th packet arrives j-1 j-N Q X j-1 (t)+... +X j-N Q (t) + Z T (t) Figure 6.5 Waiting time model consisting of three parts 97 Mean packet transmission time E[X]: Considering i-th time slot for user k, si denotes the se- lection indicator which has si = 1 when the user is selected and si = 0 otherwise. ri is the transmission rate for the user (unvarying within the time slot (ts = 2ms)). ni refers to the total number of transmitted packets (of user k) within time slot i which is given by ni = ritsF . The average packet transmission time of the investigated user can be approximated as the total transmission time divided by the total number of packets digested, i.e: E[X] = lim N−>∞ ∑N i=1 sits∑N i=1 sini = lim N−>∞ 1 N ∑N i=1 sits 1 N ∑N i=1 si rits F = E[sj]F E[sjrj] . (6.18) Calculation of mean residual time E[R]: The mean residual time E[R] is calculated in a similar way to that of M/G/1 queues with vacations [67]. which is given by: E[R] = lim t→+∞ 1 t ∫ t 0 r(τ)dτ = lim t→+∞ ( M(t) t ∑M(t) i=1 1 2 X2i M(t) + L(t) t ∑L(t) i=1 1 2 Z2i L(t) ) = λE[X2] 2 + λE[X] 2 E[s¯] E[s] ts. (6.19) Expected waiting time modeling in terms of E[R] and E[s]: By expressing the selection prob- ability as E[s] = NQE[X] NQE[X]+E[ZT ] , the mean waiting time in (6.17) is also given by E[W ] = E[R] + NQE[X] E[s] . By applying NQ = λE[W ] (Little’s theorem) and ρ = λE[X], we obtain the following modified Pollaczek-Khinchin formula: E[W ] = E[R] 1− ρ/E[s] . (6.20) By substituting Eqn.(6.16) ∼ (6.20)to the delay constraint for user k on system time E[Tk] = E[Wk] + E[Xk] ≤ τk, the delay constraint is equivalently given by: E[Xk] + λE[X2k ] + ρ(E[s¯k]/E[sk])ts 2(1− ρ/E[sk]) ≤ τk, (6.21) which is the expression in Eqn.(6.8). - Proof of Corollary 1 98 With the fact that E[X2k ] = V ar[Xk] + (E[Xk]) 2 ≥ (E[Xk])2, a necessary condition for the delay requirement on system time in Eqn.(6.21) is achieved by: E[Xk] + λ(E[Xk]) 2 + ρ(E[s¯k]/E[sk])ts 2(1− ρ/E[sk]) ≤ E[Xk] + λE[X2k ] + ρ(E[s¯k]/E[sk])ts 2(1− ρ/E[sk]) ≤ τj. (6.22) From Eqn.(6.18), we have E[sk]F E[skrk] + λ( E[sk]F E[skrk] )2 + ρ(E[s¯k] E[sk] )ts 2(1− ρ E[sk] ) ≤ τk. (6.23) Solving the standard quadratic inequality, a necessary condition for delay constraint C3 in Eqn.(6.7) is established in terms of rate rk: E[skrk] ≥ (2− λkts)E[sk] + λj(2τk + ts) + √ c− b2 4a 4τk F , ρk, where a = (2−λkts)2+8λkτk, b = (4λkts−8λkτk−2λk(λkts)(2τk+ts)) and c = (λk(2τk+ts))2. Appendix 6B: Iterative Lagrange Multiplier Search Algorithm The optimal rate allocation and power allocation strategies in (6.4) and (6.13) are obtained in terms of the Lagrange multipliers. By substituting the power allocation (6.13) into (6.14), the Lagrange multipliers (γk and µ) is given by the solutions of the following equations: γk ( E [( log2 ( (1− γk) µ |hkwk|2 σ2z ))+] − ρk ) = 0 (6.24) and E [∑ k∈A ( 1− γk µ − σ 2 z |hkwk|2 )+] = P0. (6.25) There is no closed-form solution for the Lagrange multipliers µ and γk from the above system of equations ((6.24) and (6.25)). Thus we shall resort to the iterative numerical searching method to determine the µ and γk. A Lagrangian multiplier finder algorithm is devised based on bisection searching method. 99 According to the homogeneous characteristic among the same class of users, we only need to determine two distinct γk for all users (γ1 for class-1 user and γ2 for class-2 user). The flow chart of Lagrangian multiplier finder algorithm is illustrated in Fig. 6.6. Without loss of generality, let index j = 1 represent a particular user from class 1 and index j = 2 a particular user from class-2. The γ1 for class-1 user and γ2 for class-2 user can be respectively determined by the representative user (j = 1, 2) from each class. To find the Lagrange multiplier µ and γ = {γ1, γ2}, we define the following bisection functions. fj(µ, γ) = γj [ 1 N N∑ n=1 ( log2 [ (1 + γj) µ |hj(n)wj(n)|2 σ2z ])+ − ρj ] (6.26) and P (µ, γ) = P0 − 1 N N∑ n=1 ∑ k∈A [( 1 + γk µ − σ 2 z |hk(n)wk(n)|2 )+] , (6.27) where N is the number of time slots that is to be averaged for approximating the ensemble average in (6.24). Eqn.(6.26) is the bisection function used to determine the optimal vector γ∗ for given µ, which is based on the delay constraint equation (Eqn.(6.24)). Eqn.(6.27) is the bisection function to determine the optimal µ∗ based on the average total power constraint (Eqn.(6.25)). Notice that the ensemble average in the system of equations Eqn.(6.24) and Eqn.(6.25) is approximated by time average in the bisection functions. The Lagrangian multiplier finder algorithm consists of the following three major steps: Step 1 - Initialization: Choose an arbitrary µ, initialize a feasible search region for γ1 and γ2 denoted as [γj,0, γj,0], such that  fj(µ, γj,0) 0 for j = [1, 2] . Step 2 - bisection search on γ1 and γ2: (a) For user j = [1, 2], we shall update γj,n and {γj,n, γj,n} based on Eqn. (6.28) until |fj(µ, γ)|2 ≤ ε/2 . γj,n = γj,n + γj,n 2 , 100 γj,n+1 =  γj,n if fj(µ, γj,n) > 0γj,n if fj(µ, γj,n) < 0 and γj,n+1 =  γj,n if fj(µ, γj,n) > 0γj,n if fj(µ, γj,n) < 0 , where n is the iteration index; ε is the error tolerance for stopping criteria. (b) Repeat bisection algorithm in (a) until we find a γ∗ such that (|f1(µ, γj,n)|2+f2(µ, γj,n)|2) ≤ ε where fj(µ, γ) is given by Eqn. (6.26) for j = [1, 2]. At this moment, the total consumed power reaches the minimum power level Pmin required to satisfy all users rate constraint (constraint C˜3 in Eqn. 6.10). Step 3 - Redistribution of the remaining power by adjusting µ using bisection method: Given γ∗(µ) obtained in step 2, determine the remaining power (P (µ, γ) = P0 − Pmin) from Eqn. (6.27). As illustrated in Fig. 6.6, if P (µ, γ) < 0 , the problem is infeasible because the pro- vided power is insufficient to meet all the delay requirements. If P (µ, γ) = 0 , then {µ, γ∗} obtained in step 2 is the solution. If P (µ, γ) > 0, it means there exists remaining power to be allocated and the solution is obtained as follow. Given {µ, γ∗} obtained in Step 2, we first initialize a feasible search region of µ , denoted as [µ0, µ0] such that  P (µ, γ(µ0)) 0 . The search for the correct µ∗ is based on the following bisection iterative procedure: µn = µn + µn 2 , µn+1 =  µn if P (µn, γ∗(µn)) > 0µn if P (µn, γ∗(µn)) < 0 and µn+1 =  µn if P (µn, γ∗(µn)) > 0µn if P (µn, γ∗(µn)) < 0 . For each µn obtained from Eqn.6.28, repeat Step 1 and Step 2 to update γ∗(µn) . The iteration on µn in Eqn.6.28 terminates until |P (µn, γn)|2 < ε. The final solution for Lagrange multiplier is given by {µ∗, γ∗(µ∗)}. 101 Exit: Infeasible 0 min Is ? (Check once only) P P≥ No Exit: Feasible Yes * * 1 2 2 * Given , find { , } with bisection method till ( , ) (Step 2) µ γ γ µ ε<f γ 0 1 2 Initialize with and feasible search region of { , } (Step 1) µ µ γ γ 2 *Is ( , ( )) ?n nP µ µ ε<γ Update based on bisection method using Eqn. (17) (Step 3) nµ No Yes Figure 6.6 Flow chart of iterative lagrange multiplier algorithm. 102 Chapter 7 Conclusions and Future Work 7.1 Conclusions In this thesis, we first investigate the performance of cross-layer scheduling with imperfect CSIT (due to estimation error or outdatedness) in downlink multi-user MISO and uplink multi- user SIMO systems. We found that while the system performance (average system throughput) improves quickly as the number of antennas in the base station (nT or nR) increases (due to spatial multiplexing) with perfect CSI, the system performance suffers a significant loss on the system throughput gain in the presence of outdated CSI or imperfect CSI. This is due to the mis-scheduling and packet outage problems. To capture the effect of potential packet transmission outage due to the imperfect CSIT, we de- fine the average system goodput, which measures the average b/s/Hz successfully delivered to the K mobiles, as the performance objective. Followed by two heuristic schemes targeted for realiz- ing the potential spatial multiplexing gains, a systematic and optimal framework for the cross-layer scheduling design with imperfect CSI constraint and discrete rate set constraint are proposed. We introduce some special structures on the scheduling algorithm in order to simplify the optimization. Specifically, the problem is decomposed into two parts: the optimal inner scheduling based on im- perfect CSIT and the optimal transmission modes design based on inner scheduling output. We formulate the cross-layer design as a mixed convex and combinatorial optimization problem with respect to the imperfect CSIT statistics. By applying a Chi-square approximation on the outage probability, we obtain a closed-form solution for the rate and power adaptation for any given target 103 packet error probability. The offline transmission mode design for rate adaptation is formulated as an optimization problem equivalent to the conventional scalar quantization problem, which can be effectively solved by Lloyd algorithm. By considering the statistic of CSIT errors into the design, we have shown that the proposed schemes provide significant goodput enhancement. Robust scheduling and rate adaptation with respect to imperfect CSIT assume delay-tolerant applications (homogeneous users), in which the objective is to maximize the system goodput re- gardless of how much delay a user’s packet may experience. This is not realistic for wireless multimedia applications. In Chapters 5 and 6, cross-layer scheduling for heterogenous users appli- cation are investigated. We first propose a heuristic heterogeneous scheduling in which scheduling priority is given to delay stringent users (voice users) in a naive way. If there is any remaining resource (both power and spatial channel) after the voice user scheduling, it will be distributed to the data users in an optimal way. The proposed heterogeneous scheduler ensures a stable quality of service for voice users. Yet it cannot guarantee any explicit delay requirement due to the heuristic design. In Chapter 6, we formulate the cross-layer heterogeneous scheduling as an optimization problem with a specific delay constraint for each user. By considering the statistic of channel state and queue state into the design, the proposed scheme obtains significant throughput gain and desired delay performance. 7.2 Future Work The work presented in this thesis can be extended in various ways. Here we give some examples for potential future research. • Cross-layer scheduling with heterogeneous delay constraints and imperfect CSIT In the thesis, the cross-layer scheduling design for heterogenous users application assumed perfect CSIT for simplicity. It is straightforward to extend this work to study robust schedul- ing design with heterogeneous delay constraints for imperfect CSIT. Instead of maximizing 104 the average system throughput subject to delay requirements, the objective could be maxi- mizing the average system goodput (due to the transmission outage) or proportional fairness subject to heterogenous delay constraints. • Robust beamforming with imperfect CSIT for the cross-layer scheduling We are assuming linear Zero-forcing processing and MMSE beamforming at the base station throughout the thesis. The ZFBF and MMSE beamforming weights are obtained based on the estimated CSI as if they were perfect. This actually simplifies the design and leads to sub- optimal beamforming configuration. An interesting extension is to regard the beamforming weight as the optimization parameters in the design framework. It would be interesting to jointly optimize the beamforming weights, power allocation, rate allocation and user selec- tion to maximize the average system goodput based on the error statistic of CSIT. • Cross-layer scheduling design in multi-cell wireless systems In this thesis, we are considering cross-layer scheduling for a single cell wireless system with a centralized scheduler. Basically the base station determines the scheduling output and co- ordinates the transmission among users to maximize the system capacity (with perfect CSIT) or goodput (with imperfect CSIT). In a practical multiple cells wireless system (with multi- antenna base stations and single-antenna users), there exists multi-cell interference besides multi-user interference within the cell. The multi-cell cooperation is expected to be able to effectively reduce the interference throughout the network. 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Meilong Jiang Dept. of Electrical and Electronic Engineering Phone: (852) 2857-8410 Room 807, CYC Building mljiang@ieee.org The University of Hong Kong, Hong Kong mljiang Research Interest 1. Cross layer design in multi-user OFDM/MIMO systems, Ad Hoc wireless systems, and relay-based wireless systems 2. Physical layer algorithm design in OFDM/MIMO-based wireless systems such as channel estima- tion, timing and frequency synchronization, space time coding, multi-user detection, interference cancellation 3. Digital baseband ASIC design, digital hardware design/prototyping for future wireless systems such as WLAN, UWB, and WiMAX Education 1. PhD, Electrical and Electronic Engineering, Jan. 2007, The University of Hong Kong, Hong Kong 2. Master of Science, Electrical and Electronic Engineering, Apr. 2002, Beijing University of Posts and Telecommunications, China 3. Bachelor of Science, Physics, 1st Honor, Jul. 1999, Nanchang University, China Research Experience Research Assistant Sep. 2005 - Dec. 2006 Cross-layer scheduling for downlink multi-user MIMO systems with heterogeneous delay constraints Research Assistant Sep. 2004 - Aug. 2005 Cross-layer scheduling and rate adaptation for downlink multi-user MIMO systems with imperfect channel state information (CSI) Research Assistant Sep. 2003 - Aug. 2004 Space-time uplink scheduling in multi-user multi-antenna systems; space-time downlink scheduling for voice and data application Technical Experience Research Assistant Feb. 2005 - Dec. 2006 Project on MBOA Ultra-wideband (UWB) Wireless Communications Chipset Design Research Assistant Feb. 2004 - Jan. 2005 Project on High Capacity Wireless LAN Access Point Chipset Design Hardware Design Engineer Mar. 2002 - Aug. 2003 Digital IF Design in CDMA2000-1X Base Station, Eastern Communications Ltd./Datang Mobile Ltd., Beijing, PRC Research Assistant Sep. 2000 - Feb. 2002 Project on FPGA implementation of RFID Reader/Writer List of Publications 1) K.N. Lau, M. L. Jiang and Y.J. Liu, ”Analysis of Space Time uplink Scheduling with Channel Estimation Error in Multiple Antennas System,” IEEE Transactions on Wireless Communications, Vol. 5, No. 6, Jun. 2006, pp. 1250-1253. 2) K.N. Lau and M. L. Jiang, ” Performance Analysis of Multi-user Downlink Space-time Scheduling for TDD Systems with Imperfect CSIT,” IEEE Transactions on Vehicular Technologies, Vol. 25, No. 1, Jan. 2006, pp. 296-305. 3) K.N Lau, M. L. Jiang, ” On the Rate Adaptation and Scheduling Design of Downlink Multi-user, Multiple-antenna Base Station with Imperfect CSIT,” under second revision, IEEE Transactions on Wireless Communications, 2006. 4) M.L. Jiang, S.W Hui, K.N. Lau and W.H. Lam, ”Cross-layer Downlink Scheduling of Multi- user Multi-Antenna Systems for Wireless Multimedia Applications with Heterogeneous Delay Con- straints,” to be submitted to IEEE Transactions on Wireless Communications, 2006. 5) M.L. Jiang, S.W Hui, K.N. Lau and W.H. Lam, ” Cross-layer Downlink Scheduling of Multi- user Multi-Antenna Systems for Wireless Multimedia Applications with Heterogeneous Delay Con- straints,” IEEE ISIT2006, Jul. 2006, Seattle, Washington, USA. 6) K.N Lau and M.L. Jiang, ” On the Rate Adaptation and Scheduling Design of Downlink Multi- user, Multiple-antenna Base Station with Imperfect CSIT,” IEEE GLOBECOM 2005, Nov. 2005, St. Louis, MO, USA. 7) M.L. Jiang and K.N. Lau, ” On the design of Uplink Multi-Antenna Space Time Scheduling with CSI error,” IEEE PIMRC 2004, Sep. 2004.. Barcelona, Spain. 8) K.N.Lau, M.L.Jiang, S. Liew and O.C. Yue, ” Performance Analysis of Downlink Multi-Antenna Scheduling for Voice and Data Applications,” Allerton Conference 2004, Sep. 2004, the Allerton House, Monticello, IL, USA.