IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 56, NO 5, MAY 2008 799 Cross-layer Adaptive Transmission: Optimal Strategies in Fading Channels Anh Tuan Hoang, Member, IEEE, and Mehul Motani, Member, IEEE Abstract—We consider cross-layer adaptive transmission for a single-user system with stochastic data traffic and a timevarying wireless channel The objective is to vary the transmit power and rate according to the buffer and channel conditions so that the system throughput, defined as the long-term average rate of successful data transmission, is maximized, subject to an average transmit power constraint When adaptation is subject to a fixed bit error rate (BER) requirement, maximizing the system throughput is equivalent to minimizing packet loss due to buffer overflow When the BER requirement is relaxed, maximizing the system throughput is equivalent to minimizing total packet loss due to buffer overflow and transmission errors In both cases, we obtain optimal transmission policies through dynamic programming We identify an interesting structural property of these optimal policies, i.e., for certain correlated fading channel models, the optimal transmit power and rate can increase when the channel gain decreases toward outage This is in sharp contrast to the water-filling structure of policies that maximize the rate of transmission over fading channels Numerical results are provided to support the theoretical development Index Terms—Cross-layer design, adaptive transmission, throughput maximization, Markov decision process I I I NTRODUCTION N modern and future wireless communications, maximizing throughput under limited available energy and bandwidth is and will be a challenging task The task becomes even harder in scenarios when the data arrival processes are stochastic, the buffer space is limited, and the transmission medium is time-varying In this paper, we study a problem of adapting the transmission parameters of a single-user system according to the data arrival statistics, buffer occupancy, and channel condition in order to maximize the system throughput We consider the single-user system depicted in Fig Time is divided into frames of equal length During each frame, data packets arrive to the buffer according to some known stochastic distribution The buffer has a finite length and when there is no space left, arriving packets are dropped and considered lost The transmitter transmits data in the buffer over a discrete-time block-fading channel The fading process is represented by a finite state Markov chain (FSMC) We define the system state during each time frame as the combination of the buffer occupancy and the channel state Paper approved by R Fantacci, the Editor for Wireless Networks and Systems of the IEEE Communications Society Manuscript received April 4, 2006; revised August 28, 2006, November 8, 2006, and February 19, 2007 A T Hoang is with the Department of Networking Protocols, Institute for Infocomm Research (I2R), 21 Heng Mui Keng Terrace, Singapore 119613 Previously, he was with the Department of Electrical and Computer Engineering, National University of Singapore (e-mail: athoang@i2r.a-star.edu.sg) M Motani is with the Department of Electrical and Computer Engineering, National University of Singapore, Singapore 119260 (e-mail: motani@nus.edu.sg) Digital Object Identifier 10.1109/TCOMM.2008.060214 Transmitter Packets Wireless Channel Receiver Buffer Control Signals Fig A single-user system with stochastic data arrival, limited buffer, and time-varying channel In this paper, we assume that the transmitter and receiver have complete knowledge of the instantaneous system state information (SSI) We deal with imperfect SSI (e.g., delayed, erroneous, and quantized SSI) in [1], [2] In general, for the SSI to be available at the transmitter and receiver, some processing and signaling is required Assuming that both the transmitter and receiver have complete knowledge of the current system state, we consider the problem of adapting the transmit power and rate during each time frame according to the SSI so that the system throughput is maximized, subject to an average transmit power constraint The system throughput is defined as the rate at which packets are successfully transmitted We first consider the case when the adaptation is subject to a fixed bit error rate (BER) requirement This may be appropriate when a certain quality of service is mandated by communication standards or specific user applications In this case, maximizing the system throughput is equivalent to minimizing the packet loss rate due to buffer overflow When the BER requirement is relaxed, we take into account the tradeoff between packet loss due to buffer overflow and packet loss due to transmission errors Our work is closely related to the work by Goldsmith in [3] and [4] The objective of our work and that of [3], [4] are similar, i.e., to maximize the throughput of transmission over a time-varying channel subject to an average transmit power constraint However, we take into account the effects of stochastic data arrival, finite-length buffer, and transmission errors, and adapt the transmit power and rate to both the channel gain and buffer occupancy With this formulation, we point out an interesting structural property of the optimal policies, i.e., for certain correlated fading channel models, the optimal transmit power and rate can increase as the channel gain decreases This is in sharp contrast to the water-filling structure of the capacity achieving policy in [3], [4] Taking a broader view, our work follows the cross-layer design approach, which aims to take the system variations and statistics at multiple layers of the protocol stack into account In our work, the transmission decisions, which are part of the physical layer, take into account the data arrival statistics and the buffer condition, which are the parameters of higher layers In this context, our paper is closely related to the works 0090-6778/08$25.00 c 2008 IEEE 800 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 56, NO 5, MAY 2008 in [5]–[8], in which a similar system model with stochastic data arrival, a finite-length buffer, and a time-varying channel is considered However, our work is different from [5]–[8] in several significant ways First, while the objective of our work is to maximize the system throughput, [5]–[8] concentrate more on achieving good quality of service (QoS), which is defined as the average delay experienced by each data packet Second, the objective of maximizing the throughput motivates us to consider the effects of transmission errors, which are not considered in [5]–[8] Third, in [5]–[8], the authors characterize how the optimal transmission rate depends on the channel condition; however, their characterization is only for the case when the fading process is independent and identically distributed (i.i.d.) over time In that case, the structure of the optimal policies is similar to water-filling In our work, we look at the dependency when the fading process is time correlated and make an interesting observation The works in [9] and [10] also take both data arrival and channel statistics into account when carrying out adaptive transmission While their formulation allows for optimizing over both packet losses due to transmission failure and buffer overflow, their assumptions result in no packet losses due to transmission errors Specifically, their policies never transmit above the Shannon capacity and they assume no transmission errors at rates below capacity In their recent works ( [11], [12]), Liu et al take into account both packet losses due to transmission errors and buffer overflows Their definition of system throughput is also similar to ours However, the policies considered in [11], [12] adapt to the channel state information only, not to the buffer and data arrival statistics With respect to the existing literature, the main contributions of this paper can be summarized as follows • • • • We obtain, via dynamic programming, optimal policies which maximize the system throughput for two scenarios, i.e., with and without a fixed BER requirement When the BER requirement is relaxed, we show that there is a tradeoff between packet loss due to transmission errors and packet loss due to buffer overflow We show that for certain correlated channel models and relatively large power constraints, the optimal transmission power and rate can increase as the channel gain decreases This effect is in contrast to policies which operate in the spirit of water-filling We present numerical results to support the theoretical development Specifically, we compare via simulation the performance of our optimal policies to various suboptimal schemes We also confirm via numerical computations the structure of the optimal policies mentioned above The rest of this paper is organized as follows In Section II, we define our throughput maximization problem In Section III, we describe our approach to solve the problem, via dynamic programming Section IV deals with the structure of the optimal policies In Section V, we relax the BER constraint and consider both buffer overflow and transmission error In Section VI, we present numerical results and discussion We end with some concluding remarks in Section VII II P ROBLEM D EFINITION A System Model We consider a single-user system depicted in Fig Time is divided into frames of equal length of Tf seconds each and frame i refers to the time period [iTf , (i+1)Tf ) The number of packets arriving to the buffer during frame i is denoted by Ai We assume that these Ai packets are only added to the buffer at the end of frame i We consider the case when {Ai } is i.i.d over time so that the index i can be omitted The distribution of the number of packets arriving during each time frame is assumed known and denoted by pA (a) The average packet arrival rate is λ All packets have the same length of L bits The buffer can store up to B packets and if a packet arrives when the buffer is full, it is dropped and considered lost We consider a discrete-time block-fading channel with additive white Gaussian noise (AWGN) W and No /2 respectively denote the channel bandwidth and noise power density The fading process is represented by a stationary and ergodic Kstate Markov chain, with the channel states numbered from to K−1 The channel power gain of state g, g ∈ {0, K−1}, is denoted by γg During each frame, the channel is assumed to remain in a single state Letting Gi denote the channel state during time frame i, the channel state transition probability is defined as PG (g, g ) = Pr{Gi = g | Gi−1 = g} (1) We assume that PG (g, g ) are known for all g and g The stationary distribution of each channel state g is denoted by pG (g) In general, a finite state Markov channel (FSMC) is suitable for modeling a slowly varying flat-fading channel [13], [14] A FSMC is constructed by first partitioning the range of the fading gain into a finite number of sections Then, each section corresponds to a state in the Markov chain Given knowledge of the fading process, the stationary distribution pG (g) as well as the channel state transition probabilities PG (g, g ) can be derived [13], [14] B Adaptive Transmission We denote the system state in frame i by S i = (Bi , Gi ), where Bi is the number of packets in the buffer at the beginning of frame i while Gi is the channel state during frame i At the beginning of frame i, we assume that the transmitter and the receiver have complete knowledge of S i We assume that, based on S i , the transmitter can vary its transmit power and rate For frame i, let Pi (Watts) and Ui (packets/frame) denote the transmit power and rate, respectively We must have ≤ Ui ≤ Bi In addition, we assume that Pi ∈ P where P is the set of all power levels that the transmitter can operate at We call a pair (Ui , Pi ) a control action for frame i Note that the transmitter can change its transmission rate Ui by changing the coding and/or modulation schemes [4], [15]–[17] Let Pb (g, u, P ) be the function that gives the BER when the channel state is g and the transmit power and rate are P and u respectively Pb (g, u, P ) depends on the specific coding, modulation, and detection schemes used We further HOANG and MOTANI: CROSS-LAYER ADAPTIVE TRANSMISSION: OPTIMAL STRATEGIES IN FADING CHANNELS assume that a packet is in error if at least l out of its L bits are corrupted Then we can characterize the packet error probability in terms of u, g, P as Pp (g, u, P ) L = j=l L Pb (g, u, P )j − Pb (g, u, P ) j (L−j) (2) As an example, let us change the transmission rate by varying the constellation size of an M-ary quadrature amplitude modulator (MQAM) while fixing its symbol rate From [18], assuming ideal coherent phase detection, the BER for a particular transmit power P and rate u bits per QAM symbol can be upper-bounded by P γg Pb (g, u, P ) ≤ 0.2 exp −1.5 W No (2u − 1) (3) C Throughput Maximization Problem We adopt the following definition of the system throughput Definition 1: The system throughput is the long-term average rate at which packets are successfully transmitted For an average packet arrival rate λ, a buffer overflow probability Pof , and a packet error rate Pp , the system throughput can be calculated by throughput = λ − λPof − Pp (λ − λPof ) = λ(1 − Pof )(1 − Pp ) P (u, g, Pb ) P ∈P P ≥ W No γg − log(5Pb )(2u − 1) 1.5 Lo (b, u) = E max{0, A + b − u − B} (5) (6) where the expectation is with respect to the distribution of A, i.e., the number of packets arriving in the frame Our optimization problem can be written as: arg lim sup U0 , ,UT −1 T →∞ E T T −1 (Lo (Bi , Ui )) (7) i=0 subject to: Ui ∈ {0, 1, Bi } lim sup E T (4) D Satisfying a BER Constraint From this point on until the end of Section IV, we adopt an extra constraint that the control action (Ui , Pi ) must be selected so that a fixed BER is satisfied From a practical point of view, many existing communication protocols require a fixed BER Furthermore, enforcing a BER requirement enables us to have a good comparison between our optimal adaptive transmission policies and those obtained in [3]–[6], where a BER constraint is also enforced Let P (u, g, Pb ) be the minimum power needed to transmit u packets in a frame of length Tf seconds when the channel state is g and the BER constraint is Pb P (u, g, Pb ) depends on the specific coding, modulation, and detection schemes being used Furthermore, we must have P (u, g, Pb ) ∈ P In case there is no power level in P that satisfies both the transmission rate u and the BER constraint when the channel state is g, then it means that transmission rate u is not feasible in channel state g As an example, if an adaptive MQAM scheme as described at the end of Section II-B is employed, from (3), we can approximate P (u, g, Pb ) by: = arg In general, we assume that P (u, g, Pb ) is non-decreasing in u and non-increasing in g As the BER performance is always kept at Pb , from (2), the packet error probability Pp is always kept unchanged When both λ and Pp are fixed, from (4), it is clear that maximizing the system throughput is equivalent to minimizing Pof So from now on, we concentrate on minimizing the rate at which packets are dropped due to buffer overflow For frame i, given that there are b packets in the buffer and we decide to transmit at rate u packets/frame, the expected number of packets that are dropped due to buffer overflow is T →∞ We consider the following optimization problem: Throughput Maximization Problem: At the beginning of each time frame i, given that the system state S i is known to the transmitter and the receiver, select the transmission parameters (Ui , Pi ) so that the system throughput is maximized, subject to an average transmit power constraint P 801 ∀i = 0, 1, T − 1, (8) T −1 P (Ui , Gi , Pb ) ≤ P (9) i=0 III PARETO O PTIMAL P OLICIES Instead of directly solving the above optimization problem, we reformulate it as a problem of minimizing a weighted sum of the long-term packet drop rate and average transmit power In particular, we aim to minimize Javr = lim sup T →∞ E T T −1 CI (Bi , Gi , Ui ) , (10) i=0 where CI (b, g, u) is the immediate cost incurred in state (b, g) when the control action (u, P ) is taken, i.e., CI (b, g, u) = P (u, g, Pb ) + βLo (b, u) (11) In (11), β is a positive weighting factor that gives the priority to reducing packet loss over conserving power In particular, by increasing β, we tend to transmit at a higher rate in order to lower the packet loss rate at the expense of more transmit power On the other hand, for smaller values of β, the average transmission power will be reduced at the cost of increasing packet loss rate As pointed out in [6], if P β and Lβ are the average power and packet loss rate (due to buffer overflow) obtained when minimizing Javr for a particular value of β, then Lβ is also the minimum achievable loss rate given a power constraint of P β In other words, for each value of β, minimizing Javr gives us a Pareto optimal point (Lβ , P β ) in the Loss Rate versus Power Constraint curve The problem of minimizing Javr is an infinite horizon average cost Markov decision process (MDP) with system state S i = (Bi , Gi ), control action Ui , and immediate cost function CI (Bi , Gi , Ui ) For an MDP to be well defined, we also need to characterize the dynamics of the system given a control action in a particular system state Supposing the system state at time frame i is S i = s = (b, g) and a control 802 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 56, NO 5, MAY 2008 action u is taken, the probability of the system being in state s = (b , g ) in the next time frame is PS (s, s , u) = Pr{S i+1 = s | S i = s, Ui = u} = PG (g, g )PB (b, b , u), (12) where PB (b, b , u) = Pr{Bi+1 = b |Bi = b, Ui = u} (13) Furthermore, Bi+1 = min{Bi + Ai − Ui , B} (14) Based on (12), (13), (14), the system dynamics are well defined Let π = {μ0 , μ1 , μ2 , } be a policy which maps system states into transmission rates for each frame i, i.e., Ui = μi (Bi , Gi ) We have the optimization problem We will show the above effect for a simple FSMC model which has three possible states, i.e., K = In particular, we assume that = γ0 < γ1 < γ2 Moreover, in the channel model, transitions can only happen between adjacent channel states, i.e., PG (0, 2) = PG (2, 0) = while PG (0, 0), PG (0, 1), PG (1, 1), PG (1, 0), PG (1, 2), PG (2, 2), PG (2, 1) are all positive Let us look at the insight behind the dynamic programming equation (17) When the system is in state (b, g), b > 0, g > 0, there are two effects of taking a control action u First, transmitting at rate u incurs an immediate cost CI (b, g, u) Second, transmitting at rate u in state (b, g) also reduces the future cost K−1 ∞ CF (b, g, u) = α Jα∗ π ∗ = arg Javr (π) π = arg lim sup π T →∞ E T T −1 CI (Bi , Gi , Ui ) (15) T −1 arg Jα (b,g, π) = arg π π lim E T →∞ αi (16) i=0 CI (Bi , Gi , Ui ) |B0 = b, G0 = g, π Jα∗ (b, g) K−1 ∞ = CI (b, g, u) + α u ΔI (b, g, u1 , u2 ) = CI (b, g, u2 ) − CI (b, g, u1 ) (19) ΔF (b, g, u1 , u2 ) = CF (b, g, u1 ) − CF (b, g, u2 ) (20) and As can be seen, ΔI (b, g, u1 , u2 ) is the increase in immediate cost while ΔF (b, g, u1 , u2 ) is the reduction in future cost when the transmission rate is increased from u1 to u2 Clearly, action u2 is more favorable than u1 in state (b, g) if and only if ΔI (b, g, u1 , u2 ) < ΔF (b, g, u1 , u2 ) From (11) and (19), we have ΔI (b, 1, u1 , u2 ) − ΔI (b, 2, u1 , u2 ) = P (u2 , 1, Pb ) PG (g, g ) (17) Equation (17) is particularly useful for analyzing the structure of the optimal policy IV S TRUCTURE OF THE O PTIMAL P OLICY In this section, we will point out that, for certain FSMC models in which the fading process is correlated over time, when the transmission power constraint is relatively large, the optimal transmission power and rate are non-increasing in the channel gain This is counter to the well known water-filling structure of the capacity-achieving link adaptation policy, which allocates more transmission power to good channel states and less power to bad channel states [3] (21) We state the following lemma, the proof of which is given in the Appendix Lemma 1: For each buffer state b > 0, there exists a constant βo such that for every β > βo and ≤ u1 < u2 ≤ b, the following inequality holds: ΔI (b, 1, u1 , u2 ) − ΔI (b, 2, u1 , u2 ) < ΔF (b, 1, u1 , u2 ) − ΔF (b, 2, u1 , u2 ) g =0 a=0 pA (a)Jα∗ (min{b + a − u, B}, g ) (min{b + a − u, B}, g ) − P (u1 , 1, Pb ) − P (u2 , 2, Pb ) + P (u1 , 2, Pb ) where < α < is the discounting factor As the immediate cost function CI is bounded, the limit in (16) always exists As shown in [19], when α → the solution of the discounted cost problem converges to that of the average cost problem in (15) Moreover, the solution of the discounted cost problem satisfies a simple dynamic programming equation given by (18) For state (b, g) with b > 0, g > 0, let ≤ u1 < u2 ≤ b be two possible transmission rates Let i=0 In our system, as all states are connected, there exists a stationary policy π ∗ , i.e μi ≡ μ for all i, which is a solution to (15) To simplify the notation, we just write Ui = π ∗ (Bi , Gi ) As it is shown in [19], using a simple policy iteration algorithm, an optimal policy π ∗ can be reached in a finite number of steps Finally, it is also useful to consider the discounted cost problem defined as: PG (g, g )pA (a) g =0 a=0 (22) Theorem 1: For each buffer state b > 0, let βo be defined as in Lemma and β > βo , then the optimal transmission rate for each state (b, g), g > 0, is non-increasing in g Proof: We present a proof by contradiction Let u∗1 and ∗ u2 be the optimal transmission rate at states (b, 1) and (b, 2) respectively Suppose ≤ u∗1 < u∗2 ≤ b From (17) we have CI (b, 1, u∗1 ) + CF (b, 1,u∗1 ) (23) ≤ CI (b, 1, u∗2 ) + CF (b, 1, u∗2 ) CI (b, 2, u∗2 ) + CF (b, 2,u∗2 ) ≤ CI (b, 2, u∗1 ) + CF (b, 2, u∗1 ) (24) Inequalities (23) and (24) respectively imply (25) and (26) ΔI (b, 1, u∗1 , u∗2 ) = CI (b, 1, u∗2 ) − CI (b, 1, u∗1 ) ≥ CF (b, 1, u∗1 ) − CF (b, 1, u∗2 ) = ΔF (b, 1, u∗1 , u∗2 ), (25) HOANG and MOTANI: CROSS-LAYER ADAPTIVE TRANSMISSION: OPTIMAL STRATEGIES IN FADING CHANNELS ΔI (b, 2, u∗1 , u∗2 ) ≤ ΔF (b, 2, u∗1 , u∗2 ) (26) From (25) and (26) we have: and sufficient condition for a power level to be optimal is that it must satisfy P = arg CI (b, g, u, P ) ΔI (b, 1, u∗1 , u∗2 ) − ΔI (b, 2, u∗1 , u∗2 ) (27) ≥ ΔF (b, 1, u∗1 , u∗2 ) − ΔF (b, 2, u∗1 , u∗2 ), which contradicts Lemma and therefore, u∗1 ≥ u∗2 Comment: Theorem shows that for a certain correlated fading channel model and average transmission power constraint, the optimal transmission rate is non-increasing in the channel gain In fact, our numerical results (see Section VI) show an even stronger effect, i.e., in some cases, the optimal transmission rate decreases when the channel gain increases 803 P ∈P (34) = arg {P + βLe (g, u, P )} P ∈P In other words, in each system state, we only have to decide which rate the transmitter should use After that, the power level will follow directly by solving (34) Let π be a stationary policy which maps system states into transmission rate for each frame i, i.e., Ui = μi (Bi , Gi ) Define CI∗ (b, g, u) = CI (b, g, u, P ) (35) P ∈P V R EMOVING THE BER C ONSTRAINT and In this section, we relax the BER constraint and allow the trade off between packet loss due to buffer overflow and packet loss due to transmission errors Javr (π) = lim sup T →∞ E T T −1 CI∗ (Bi , Gi , Ui )|π (36) i=0 We have to solve the optimization problem A Taking Packet Loss Due to Transmission Error into Account π∗ = arg Javr (π) Given the transmission rate u, power P , channel state g, and the packet error probability of Pp (g, u, P ), the expected number of packets lost due to transmission error is Again, this problem can be solved efficiently using dynamic programming techniques [19] Le (g, u, P ) = uPp (g, u, P ) (28) For fixed packet arrival rate, maximizing the system throughput is equivalent to minimizing total packet loss rate due to both buffer overflow and transmission error So we have the optimization problem: arg lim sup Ui ,Pi T →∞ E T Lo (Bi , Ui ) + Le (Gi , Ui , Pi ) i=0 (29) Ui ∈ {0, 1, Bi } ∀i = 0, 1, T − 1, lim sup T →∞ (37) VI N UMERICAL RESULTS AND D ISCUSSION In this section, we present numerical results to illustrate the previous theoretical development We focus on the structure of the optimal buffer and channel adaptive transmission policies as well as the performance, in terms of the packet loss rate, normalized by the arrival rate λ T −1 subject to: Pi ∈ P π ∀i = 0, 1, T − 1, E T (30) (31) T −1 Pi ≤ P (32) i=0 B Optimal Policies Similar to the approach in Section III, we can reformulate the above optimization problem as a problem of minimizing a weighted sum of the total packet loss rate (due to buffer overflow and transmission error) and average transmission power The only modification needed here is for the immediate cost function CI Now we have: CI (b, g, u, P ) = P + β Lo (b, u) + Le (g, u, P ) (33) At time i, let the system state be S i = s = (b, g) and a control action (u, P ) is taken, the probability of the system being in state s = (b , g ) in the next time frame is still characterized by (12), (13), (14) An important point to note from (12), (13), (14) is that the chosen transmission power level P does not have any effect on the system dynamics Therefore, given a choice of transmission rate u, the necessary A System Parameters Packets arrive to the buffer according to a Poisson distribution with average rate λ = 103 and × 103 packets/second All packets have the same length of L = 100 bits The buffer length is B = 15 packets The channel bandwidth is W = 100 kHz and the power density of AWGN noise is No /2 = 10−5 Watt/Hz We consider both cases of correlated and i.i.d fading channels For the correlated channel model, we use an 8-state FSMC as described in Table I This channel model is obtained by quantizing the power gain of a Rayleigh fading channel that has average power gain γ = 0.8 and Doppler frequency fD = 10 Hz For the i.i.d channel model, the values of the channel gains are the same as in Table I; however, the channel evolves independently over time with all states being equiprobable Adaptive transmission is based on a variable-rate, variablepower MQAM scheme similar to that described in [4] Let Ts be the symbol period of the MQAM modulator and assume a Nyquist signaling pulse, sinc(t/Ts ), is used so that the value of Ts is fixed at 1/W seconds When the symbol period Ts is kept unchanged, varying the signal constellation size of the modulator gives us different data transmission rates As has been specified in Section II, the power and rate adaptation are carried out in a frame-by-frame basis Each frame consists of F modulated symbols, i.e., Tf = F Ts We set F = L = 100 so that when a signal constellation of size M = 2u is used, exactly u packets are transmitted during each time frame 804 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 56, NO 5, MAY 2008 TABLE I C HANNEL STATES AND TRANSITION PROBABILITIES Channel states k γk Pkk Pk,k+1 Pk,k−1 0 0.9359 0.0641 0.1068 0.8552 0.0807 0.0641 0.2301 0.8334 0.0859 0.0807 0.3760 0.8306 0.0835 0.0859 0.5545 0.8420 0.0745 0.0835 0.7847 0.8665 0.0590 0.0745 1.1090 0.9048 0.0361 0.0590 1.6636 0.9639 0.0361 packet in buffer packets in buffer 10 packets in buffer 14 packets in buffer packet in buffer packets in buffer 10 packets in buffer Packets transmitted/frame Packets transmitted/frame 14 packets in buffer 4 1 Worst state Channel states Best state Fig Structure of optimal policies, i.e., transmission rates for different channel states when the buffer occupancy is fixed at 1, 5, 10, 14 packets Power constraint P = 16dB Channel is correlated over time (Tab I) Worst state Channel states Best state Fig Structure of optimal policies, i.e., transmission rates for different channel states when the buffer occupancy is fixed at 1, 5, 10, 14 packets Power constraint P = 16dB Channel is i.i.d over time 0.5 0.4 Mormalized Packet Loss Rate As discussed in Sections III and V, we consider two classes of buffer and channel adaptive transmission policies In the first class, transmit power and rate are selected subject to a fixed BER requirement We use (5) to approximate the power needed to transmit u bits per QAM symbol when the channel gain is γk and the BER constraint is Pb This class of policies is called MDP I The other class of adaptive transmission policies is called MDP II In MDP II policies, the BER requirement is removed and packet loss due to transmission errors is taken into account Also, for MDP II policies, we assume that the set P of possible power levels is finite −3 MDP_I (BER = 10 ) Ch_Adpt Ch_Ivn 0.45 0.35 0.3 0.25 0.2 0.15 0.1 B Structure of Optimal Policies First, let us look at the structure of MDP I policies obtained by solving (15) for the correlated FSMC given in Table I In Fig.2, we plot the optimal transmission rates of an MDP I policy obtained when λ = 103 packets/sec, B = 15 packets, fD = 10 Hz, Pb = 10−3 and P = 16 dB for different values of the buffer occupancies As can be seen, for a particular value of the buffer occupancy, the optimal transmission rate increases when the channel gain decreases toward the outage point (state 0) This is consistent with our discussion in Section IV For comparison, we also obtain an optimal policy for the i.i.d channel model and plot its structure in Fig As can be seen, for each buffer occupancy, the optimal transmission rate increases when the channel gain increases 12 14 16 18 20 22 24 Power (dB) Fig Normalized packet loss rate (due to buffer overflow only) versus average transmission power B = 15, λ = packets/frame, Pb = 10−3 Channel model is given by Table I C Packet Loss due to Buffer Overflow Now we compare the performance of MDP I policies with those of other adaptive transmission schemes All transmission is subject to a BER constraint of Pb = 10−3 and we only care about packet loss due to buffer overflow We consider two other types of policies, channel inversion, i.e., C Inv, and channel adaptive, i.e., C Adpt In a C Inv policy, the transmission rate is always kept unchanged and given the HOANG and MOTANI: CROSS-LAYER ADAPTIVE TRANSMISSION: OPTIMAL STRATEGIES IN FADING CHANNELS 0.6 805 0.6 −3 −3 MDP_I, BER = 10 −5 MDP_I, BER = 10 0.4 −6 MDP_I, BER = 10 MDP_II, 20 power levels 0.3 0.25 0.2 0.15 0.1 14 MDP_I, BER = 10 MDP_II, power levels MDP_II, 10 power levels MDP_II, 20 power levels 0.5 MDP_I, BER = 10−4 Normalized Packet Loss Rate Normalized Packet Loss Rate 0.5 0.4 0.3 0.25 0.2 0.15 0.1 16 18 20 22 24 26 14 16 18 20 Power (dB) channel gain, the necessary power is calculated based on (5) to guarantee the target BER In a C Adpt policy, we use the optimal link-adaptive policy that maximizes the transmission rate for our channel model under some power constraint and with the assumption that there are always packets to transmit The performance of the three schemes, in terms of normalized packet loss rate (due to buffer overflow) versus consumed power are shown in Fig As expected, MDP I outperforms the other two classes of adaptive policies For low value of average transmit power, the performance of MDP I policies and C Adpt policies are very close while that of the C Inv policies is much worse This is expected, since at low power, the structure of an MDP I policy is similar to that of the C Adpt, and by focusing on conserving power, the system performance is improved At high power, the performance of MDP I and C Inv policies are close and it is interesting to see that the C Inv scheme results in less packet loss rate relative to the C Adpt scheme This means that at this high range of average transmission power, if we only adapt to the channel, the performance can be worse than not doing any rate adaptation at all D Packet Loss due to Buffer Overflow and Transmission Errors Now we take packet transmission errors into account and compare the performance, in terms of total normalized packet loss rate (due to buffer overflow and transmission errors) versus average transmit power, of the two classes of buffer and channel adaptive transmission policies, namely MDP I and MDP II Fig is for correlated channel model We plot the performances of MDP I policies corresponding to BER values of 10−3 , 10−4 , 10−5 , 10−6 and an MDP II scheme that has 20 different power levels, from to 40 dB As can be seen, among all the schemes, the MDP II scheme performs best For high values of BER, i.e 10−3 and 10−4 , MDP I policies perform well in low ranges of transmission power while become much worse than the MDP II policies when the power is high 24 26 Fig Normalized packet loss rate (due to buffer overflow and transmission errors) versus average transmission power B = 15, λ = packets/frame Channel model is correlated over time and is given in Table I 0.4 0.3 Normalized Packet Loss Rate Fig Normalized packet loss rate (due to buffer overflow and transmission errors) versus average transmission power B = 15, λ = packets/frame Channel model is correlated over time and is given in Table I 22 Power (dB) 0.2 0.1 0.05 0.01 MDP_I, BER = 10−3 −4 MDP_I, BER = 10 MDP_I, BER = 10−5 MDP_I, BER = 10−6 MDP_II, 20 power levels 14 16 18 20 22 24 Power (dB) Fig Normalized packet loss rate (due to buffer overflow and transmission errors) versus average transmission power B = 15, λ = packets/frame Channel model is i.i.d over time On the other hand, for low value of BER, i.e 10−6 , the performance of MDP I is much worse than MDP II in low power range This can be explained by looking at the structure of the MDP II As MDP II tries to balance between packet loss due to buffer overflow and transmission errors, when the power constraint is low, it tends to transmit at relatively high BER values and when the power constraint is high, it transmits at low BER levels In other words, at low power, the structure of a MDP II scheme is similar to those of MDP I schemes corresponding to high BER constraints On the other hand, when the power constraint is high, MDP II is closer to a MDP I with low value of BER constraint In Fig 6, we plot the performance of different MDP II policies that correspond to different numbers of possible power levels (from to 40dB) As can be seen, even with only different power levels, the MDP II scheme can perform much better than MDP I schemes Figs and show result for i.i.d channel models and similar effects can be observed 806 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL 56, NO 5, MAY 2008 all g For < b ≤ B, let u∗ be a value that achieve the minimization in (38) a) If u∗ = 0, then from (38) we have: Normalized Packet Loss Rate 0.4 0.3 0.2 Ji (b − 1, g) ≤ CI (b − 1, g, 0) 0.1 K−1 ∞ PG (g, g )pA (a)Ji−1 q(b − 1, a), g +α 0.05 g =0 a=0 (39) K−1 ∞ < CI (b, g, 0) + α PG (g, g )pA (a) g =0 a=0 0.01 Ji−1 q(b, a), g −3 MDP_I, BER = 10 MDP_II, power levels MDP_II, 10 power levels MDP_II, 20 power levels 14 16 18 b) If u∗ > 0, then from (38) we have: 20 22 Ji (b − 1, g) ≤ CI (b − 1, g, u∗ − 1) 24 Power (dB) K−1 ∞ Fig Normalized packet loss rate (due to buffer overflow and transmission errors) versus average transmission power B = 15, λ = packets/frame Channel model is i.i.d over time In this paper, we considered the problem of buffer and channel adaptive transmission for maximizing the system throughput subject to an average transmit power constraint Given that accurate buffer and channel states are available for making decisions, we show how optimal control policies can be obtained for transmission with and without a fixed BER requirement Our paper highlights some important issues in wireless data communications First, as nodes are only equipped with limited batteries and have to operate within a dynamic environment, cross-layer design is essential to achieve good performance and conserve energy Second, when statistics at multiple layers are taken into account, the popular intuition associated with layered design may no longer be true For example, this paper shows that, in a correlated fading channel, the structure of the optimal buffer and channel adaptive transmission policies can be in sharp contrast to the well known strategy of water-filling A PPENDIX P ROOF OF L EMMA Let us first prove the following Lemmas 2, 3, and Lemma 2: For all ≤ g < K, Jα∗ (b, g) is increasing in the buffer occupancy b Proof: This lemma can be proved by induction Let J0 be a bounded and increasing function on the state space (b, g) For i = 1, 2, , let Ji (b, g) = CI (b, g, u) u K−1 ∞ PG (g, g )pA (a)Ji−1 q(b − u, a), g PG (g, g )pA (a)Ji−1 q(b − u∗ , a), g +α g =0 a=0 ∗ (40) K−1 ∞ < CI (b, g, u ) + α PG (g, g )pA (a) g =0 a=0 VII C ONCLUSION +α = Ji (b, g) , (38) Ji−1 q(b − u∗ , a), g We have proved that if Ji−1 (b, g) is increasing in b for all g then the same is true for Ji (b, g) Therefore, by induction, Jα∗ (b, g) = limi→∞ Ji (b, g) is increasing in b for all g Lemma 3: For all ≤ b1 < b2 ≤ B and all < g < K, Jα∗ (b2 , g) − Jα∗ (b1 , g) is upper bounded when β increases Proof: Let u∗1 be the optimal transmission rate in state (b1 , g) and u2 = u∗1 + b2 − b1 , then Jα∗ (b2 , g) − Jα∗ (b1 , g) ≤ CI (b2 , g, u2 ) + CF (b2 , g, u2 ) − CI (b1 , g, u∗1 ) − CF (b1 , g, u∗1 ) = CI (b2 , g, u2 ) − CI (b1 , g, u∗1 ) + CF (b2 , g, u2 ) − CF (b1 , g, u∗1 ) (41) As u2 = u∗1 + b2 − b1 , CF (b2 , g, u2 ) = CF (b1 , g, u∗1 ) while CI (b2 , g, u2 ) − CI (b1 , g, u∗1 ) = P (u∗1 + b2 − b1 , g, P b ) − P (u∗1 , g, P b ) Therefore Jα∗ (b2 , g) − Jα∗ (b1 , g) ≤ P (u∗1 + b2 − b1 , g, P b ) − P (u∗1 , g, P b ) (42) It is clear that the left hand side of (42) is bounded when β increases, so the proof is completed Lemma is for situation in which the channel state g > 0, when g = 0, we have the following lemma Lemma 4: For all ≤ b1 < b2 ≤ B, Jα∗ (b2 , 0) − Jα∗ (b1 , 0) increases without bound when β increases Proof: When the channel is in state 0, no transmission is possible, therefore Jα∗ (b1 , 0) = CI (b1 , 0, 0) K−1 ∞ g =0 a=0 where q(b, a) = min{b + a, B} Note that from the value iteration algorithm for solving discounted cost problem (17), we have Jα∗ (b, g) = limi→∞ Ji (b, g) for all ≤ b ≤ B and ≤ g < K Now assuming Ji−1 (b, g) is increasing in b for all g, we will show that Ji (b, g) is also increasing in b for = Ji (b, g) +α PG (0, g)pA (a)Jα∗ q(b1 , a), g , g=0 a=0 Jα∗ (b2 , 0) = CI (b2 , 0, 0) K−1 ∞ +α g=0 a=0 PG (0, g)pA (a)Jα∗ q(b2 , a), g (43) HOANG and MOTANI: CROSS-LAYER ADAPTIVE TRANSMISSION: OPTIMAL STRATEGIES IN FADING CHANNELS Therefore Jα∗ (b2 , 0) − Jα∗ (b1 , 0) K−1 ∞ = CI (b2 , 0, 0) − CI (b1 , 0, 0) PG (0, g)pA (a) Jα∗ q(b2 , a), g +α (44) g=0 a=0 − Jα∗ q(b1 , a), g > CI (b2 , 0, 0) − CI (b1 , 0, 0) = β L(b2 , 0) − L(b1 , 0) The inequality in (44) is due to Lemma From (44), it is clear that Jα∗ (b2 , 0) − Jα∗ (b1 , 0) increases without bound when β increases and the proof is completed Proof of Lemma 1: First of all, we have ΔI (b, 1, u1 , u2 ) − ΔI (b, 2, u1 , u2 ) = W No f (u2 , Pb ) − f (u1 , Pb ) 1 − γ1 γ2 (45) Therefore, the left hand side of (22) does not depend on β For the right hand side of (22), we have: K−1 ∞ ΔF (b, g, u1 , u2 ) = α PG (g, g ) (46) g =0 a=0 pA (a) (Jα∗ (q(b − u1 , a), g ) − Jα∗ (q(b − u2 , a), g )) Now ΔF (b, 1, u1 , u2 ) − ΔF (b, 2, u1 , u2 ) K−1 ∞ PG (1, g ) − PG (2, g ) pA (a) =α (47) g =0 a=0 Jα∗ q(b − u1 , a), g − Jα∗ q(b − u2 , a), g 807 [6] R A Berry and R G Gallager, “Communication over fading channels with delay constraints,” IEEE Trans Inform Theory, vol 48, no 5, pp 1135–1149, May 2002 [7] M Goyal, A Kumar, and V Sharma, “Power constrained and delay optimal policies for scheduling transmission over a fading channel,” in Proc IEEE INFOCOM’03, Mar 2003, pp 311–320 [8] A Fu, E Modiano, and J Tsitsiklis, “Optimal energy allocation for delay-constrained data 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and K M Wasserman, “Analysis on markov modeling of packet transmission over wireless channels,” in Proc IEEE WCNC’02, Mar 2002, pp 876–880 [15] B Vucetic, “An adaptive coding scheme for time-varying channels,” IEEE Trans Commun., vol 39, pp 653–663, May 1991 [16] W T Webb and R Steele, “Variable rate qam for mobile radio,” IEEE Trans Commun., vol 43, pp 2223–2230, July 1995 [17] A T Hoang and M Motani, “Buffer and channel adaptive modulation for transmission over fading channels,” in Proc ICC’03, July 2003, pp 2748–2752 [18] G J Foschini and J Salz, “Digital communications over fading radio channels,” Bell Syst Tech J., pp 429–456, Feb 1983 [19] P R Kumar and P Varaiya, Stochastic Systems: estimation, identification, and adaptive control Englewood Cliffs, NJ: Prentice Hall, 1986 K−1 ∞ =α PG (1, g ) − PG (2, g ) pA (a) g =1 a=0 Jα∗ q(b − u1 , a), g − Jα∗ q(b − u2 , a), g (48) ∞ PG (1, 0) − PG (2, 0) pA (a) +α a=0 Jα∗ q(b − u1 , a), − Jα∗ q(b − u2 , a), As β increases, the first term in (48) is bounded from below (from Lemmas and 3) while the second term increases without bound (from Lemma 4) This combined with (45) completes the proof R EFERENCES [1] A T Hoang and M Motani, “Buffer and channel adaptive transmission over fading channels with imperfect channel state information,” in Proc IEEE WCNC 2004, Atlanta, Mar 2004, pp 1891–1896 [2] ——, “Cross-layer adaptive transmission: coping with incomplete system state information,” submitted to IEEE Trans Commun., 2006 [3] A J Goldsmith and P P Varaiya, “Capacity of fading channels with channel side information,” IEEE Trans Inform Theory, vol 43, pp 1986–1992, Nov 1997 [4] A J Goldsmith and S G Chua, “Variable-rate variable-power mqam for fading channels,” IEEE Trans Commun., vol 45, no 10, pp 1218– 1230, Oct 1997 [5] B Collins and R Cruz, “Transmission policy for time varying channel with average delay constraints,” in Proc 1999 Allerton Conf on Commun Control and Comp, 1999, pp 1–9 Anh Tuan Hoang (IEEE Member) received the Bachelor degree (with First Class Honours) in telecommunications engineering from the University of Sydney in 2000 He completed his Ph.D degree in electrical engineering at the National University of Singapore in 2005 Dr Hoang is currently a Research Fellow at the Department of Networking Protocols, Institute for Infocomm Research, Singapore His research focuses on design/optimization of wireless comm networks Specific areas of interest include crosslayer design, dynamic spectrum access, and cooperative communications Mehul Motani is an Assistant Professor in the Electrical and Computer Engineering Department at the National University of Singapore He graduated with a Ph.D from Cornell University, focusing on information theory and coding for CDMA systems Prior to his Ph.D., he was a member of technical staff at Lockheed Martin in Syracuse, New York for over four years Recently he has been working on research problems which sit at the boundary of information theory, communications and networking, including the design of wireless ad-hoc and sensor network systems He was awarded the Intel Foundation Fellowship for work related to his Ph.D in 2000 He is on the organizing committees for ISIT 2006 and 2007 and the technical program committees of MobiCom 2007 and Infocom 2008 and several other conferences He participates actively in IEEE and ACM and has served as the secretary of the IEEE Information Theory Society Board of Governors