Cross layer adaptive transmission with i

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL XX, NO Y, MONTH 2007 Cross-layer Adaptive Transmission with Incomplete System State Information Anh Tuan Hoang, Member, IEEE, and Mehul Motani, Member, IEEE Abstract— We consider a point-to-point communication system in which data packets randomly arrive to a finite-length buffer and are subsequently transmitted to a receiver over a timevarying wireless channel Data packets are subject to loss due to buffer overflow and transmission errors We study the problem of adapting the transmit power and rate based on the buffer and channel conditions so that the system throughput is maximized, subject to an average transmit power constraint Here, the system throughput is defined as the rate at which packets are successfully transmitted to the receiver We consider this buffer/channel adaptive transmission when only incomplete system state information is available for making control decisions Incomplete system state information includes delayed and/or imperfectly estimated channel gain and quantized buffer occupancy We show that, when some delayed but error-free channel state information is available, optimal buffer/channel adaptive transmission policies can be obtained using Markov decision theories When the channel state information is subject to errors and when the buffer occupancy is quantized, we discuss various buffer/channel adaptive heuristics that achieve good performance In this paper, we also consider the tradeoff between packet loss due to buffer overflow and packet loss due to transmission errors We show by simulation that exploiting this tradeoff leads to a significant gain in the system throughput Index Terms— Cross-layer design, adaptive transmission, throughput maximization, partially observable Markov decision processes I I NTRODUCTION In this paper, we study the problem of buffer and channel adaptive transmission in a point-to-point wireless communication scenario with the objective of maximizing the system throughput, subject to an average transmit power constraint We term our adaptive transmission schemes cross-layer since transmission decisions at the physical layer take into account not only the channel condition but also the data arrival statistics and buffer occupancy, which are the parameters of higher network layers Our system model is depicted in Fig Time is divided into frames of equal length and during each frame, data packets arrive at the transmitter buffer according to some known stochastic distribution The buffer has a finite length and when there is no space left, arriving packets are dropped Manuscript received November 08, 2006; revised May 31, 2007, and August 22, 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 Data packets in the buffer are transmitted to a receiver over a discrete-time block-fading channel The fading process is represented by a finite state Markov chain (FSMC) ( [1], [2]) We define the system state during each time frame as the combination of the buffer occupancy and the channel state and assume that there is a signaling mechanism for the transmitter and receiver to exchange some system state information (SSI) In our system model, data packets are subject to loss due to buffer overflow and transmission errors We define the system throughput as the rate at which packets are successfully transmitted to the receiver The control problem is to adapt the transmit power and rate according to some SSI so that the system throughput is maximized, subject to an average transmit power constraint We are interested in scenarios in which only an incomplete observation of the instantaneous system state is available for making control decisions Incomplete SSI includes delayed and/or imperfectly estimated channel state and quantized buffer occupancy The case when control decisions can be made based on complete SSI is considered in our related work [3], where interesting structural properties of optimal adaptive transmission policies are studied In the context of adaptive transmission, our paper is related to the well-known works of Goldsmith in [4] and [5] In these works, it is shown that when the channel state information (CSI) is available at both the transmitter and receiver, the optimal power allocation scheme that achieves the capacity of a time-varying wireless channel, subject to an average transmit power constraint, exhibits a water-filling structure over time The insight is that the transmitter should transmit at a higher power and rate when the channel is good while reducing the transmit power in poorer channel conditions However, data arrival statistics and buffer conditions are not of concern in [4] and [5] In the context of cross-layer design, our paper is closely related to the works in [6]–[16], which consider similar problems of buffer/channel adaptive transmission An early work of Collins and Cruz adapts transmit power and rate based on the queue length and channel condition in order to minimize the average transmit power, subject to an average delay constraint [6] In [7] Berry and Gallager quantify the behavior of the power-delay tradeoff in the regime of asymptotically large delay The same model is further studied in [8], [9], with some structural properties of the optimal policies identified In [10], Rajan et al consider a more generalized queueing model where packets can be dropped They propose transmission policies that are near-optimal, in terms of minimizing packet loss subject to an average delay and an average power constraint In [11], Karmokar et al further extend the IEEE TRANSACTIONS ON COMMUNICATIONS, VOL XX, NO Y, MONTH 2007 Transmitter Packets Wireless Channel Receiver transmission given imperfect SSI We exploit the tradeoff between packet loss due to buffer overflow and packet loss due to transmission errors This tradeoff results in a performance gain in the overall system throughput • We show how buffer and channel adaptive transmission can be carried out given incomplete SSI In particular, we show that optimal adaptive policies can be obtained for the cases when some delayed but error-free channel state information is available When the channel state information is subject to errors and when the buffer occupancy is quantized, we present various buffer/channel adaptive heuristics that achieve good performance The rest of this paper is organized as follows In Section II, we present our system model and discuss the approach that can be used to obtain optimal adaptive transmission policies when the transmit power and rate can be chosen based on a perfect knowledge of the instantaneous system state Next, in Section III, we discuss the situations in which the transmitter and receiver only have partial information about the current buffer and channel states In Section IV, we show that optimal control policies can be obtained when some delayed but error-free channel states are available for making decision When this is not possible, we propose various heuristics to obtain policies with good performance in Section V Numerical results and discussion are given in Section VI Finally, we conclude the paper in Section VII • Buffer Control Signals Fig System model of a point-to-point wireless communication scenario Data packets arrive to the buffer according to some stochastic distribution The packets are then transmitted over a time-varying wireless channel There are control signals for the transmitter and receiver to exchange buffer and channel state information tradeoff to include average packet delay, average transmit power, and average packet dropping probability They also propose a suboptimal policy that approximates the behaviors of the optimal policies In [12]–[16] the problem of cross-layer adaptive transmission is considered from a different angle in which transmission is carried out given a fixed amount of energy and a limited amount of time The authors adapt the transmit power and rate according to the amount of data remaining, the present time relative to the deadline, and the present channel state, in order to maximize the achievable throughput ( [12]–[14]) or to maximize the probability of a data file being successfully transmitted ( [15], [16]) We note that the works in [6]–[16] assume perfect knowledge of the instantaneous buffer occupancy and channel state In [17], Karmokar et al consider the problem of adapting the error control coding scheme base on some imperfect observations of traffic statistics and channel condition In particular, the channel observations are in the form of NACK/ACK that are fed back from the receiver to the transmitter Similar to our paper, the problem in [17] is formulated as a partially observable Markov decision process (POMDP) Even though the problem setup in [17] differs from that of our paper in several points, the authors come to a similar conclusion that, given partial observations, a heuristic called QMDP ( [18]) achieves good performance An important contribution that differentiates our work from [6]–[16] is that we exploit the tradeoff between packet loss due to buffer overflow and packet loss due to transmission errors Our results show that, by balancing these sources of packet loss, significant gain in the system throughput can be achieved From the implementation point of view, when imperfect channel state information is considered, it is not possible to calculate transmit power to guarantee a target packet error rate We note that the problem formulation in [10] and [16] allows for optimizing over both packet losses due to transmission failure and buffer overflow However, 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 ( [19], [20]), Liu at 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 [19], [20] adapt to the channel state information only, not to the buffer and data arrival statistics The main contributions of this paper can be summarized as follows • We present tractable models of buffer/channel adaptive II T HROUGHPUT M AXIMIZATION P ROBLEM A System Model The system model considered in this paper is depicted in Fig Time is divided into frames of equal length of Tf seconds During frame i, Ai packets arrive at the transmitter buffer We assume that Ai is independent and identically distributed (i.i.d.) over time and follows a stationary distribution pA (a) Each data packet contains L bits, the buffer can store up to B packets and when the buffer is full, all arriving packets are dropped We further assume that arriving packets are only added to the buffer at the end of each time frame We consider a discrete-time block-fading channel with additive white Gaussian noise (AWGN) The fading process is represented by a stationary and ergodic K-state Markov chain, with the channel states numbered from to K − The power gain of channel state g, g ∈ {0, K − 1}, is denoted by γg During each time frame, we assume that the channel remains in a single state, between two consecutive frames, the probability of transitioning from channel state g to channel state g ′ is denoted by PG (g, g ′ ) The stationary distribution of each channel state is denoted by pG (g) In general, a finite state Markov channel model (FSMC) is suitable for modeling a slowly varying flat-fading channel [1], [21]–[23] A FSMC is constructed for a particular fading distribution, e.g., log-normal shadowing or Rayleigh fading, by first partitioning the range of the fading gain into a finite number of sections Then each section of the gain value corresponds to a state in the Markov chain Given knowledge of the fading process, the stationary distribution pG (g) as well IEEE TRANSACTIONS ON COMMUNICATIONS, VOL XX, NO Y, MONTH 2007 as the channel state transition probabilities PG (g, g ′ ) can be derived For more details, the reader is referred to [1], [21]– [23] Let Bi denote the number of packets in the buffer at the beginning of frame i and Gi denote the channel state throughout frame i, the system state at frame i is S i (Bi , Gi ) For time frame i, let Pi (Watts) and Ui (packets/frame) denote the transmit power and rate, respectively We have ≤ Ui ≤ Bi and Pi ∈ P , where P is the set of all power levels at which the transmitter can operate buffer/channel adaptive transmission policies can be obtained for the case of complete SSI With complete SSI, the throughput maximization problem can be reformulated as the problem of minimizing the weighted sum of the long-term packet loss rate and the average transmission power In particular, consider the following problem of selecting transmission rate and power (Ui , Pi ): B Buffer and Channel Adaptive Transmission where Given a particular system state (b, g), where b is the buffer occupancy and g is the channel state (0 ≤ b ≤ B, ≤ g < K), each chosen pair of transmission rate and power (u, P ) results in some expected number of packets lost due to buffer overflow and transmission errors We characterize these losses by two functions: Lo (b, u) is the expected number of packets lost due to buffer overflow and Le (g, u, P ) is the expected number of packets discarded due to transmission error Note that in this paper, we not consider retransmission of erroneous packets For our system model, when the data arrival process is fixed, maximizing the system throughput is equivalent to minimizing total packet loss due to buffer overflow and transmission errors This is achieved by varying the transmission rate and power (Ui , Pi ) according to some knowledge of S i Note that there are various ways for the transmitter to change its transmission rate Ui It can be done by changing the channel coding scheme [24], i.e by encoding data bits in the buffer using different code rates while keeping the transmission rate for the coded bits fixed Ui can also be varied by keeping the symbol rate fixed and changing the signal constellation size of a modulator [5], [8], [25] In existing communication standards such as IEEE.802.11 and IEEE.802.16, different transmission rates are achieved by combinations of different coding and modulation schemes C Buffer Overflow and Transmission Error Tradeoff At this point, let us point out an interesting tradeoff between the two sources of packet loss, i.e., buffer overflow and transmission errors Consider a particular system state (b, g) and a fixed transmit power P If we increase the transmission rate u, the amount of buffer overflow is reduced However, increasing u when P is fixed results in a greater number of packet transmission errors The reverse is also true, for fixed P , the amount of packet transmission errors can be reduced by lowering the transmission rate u, but that will be at the cost of increasing the buffer overflow rate This argument highlights the need to find a good tradeoff between packet transmission errors and buffer overflow when choosing transmit power and rate In this paper, our control decision strives for an optimal tradeoff between these two sources of packet loss D Throughput Maximization with Complete SSI Before considering buffer/channel adaptive transmission with incomplete SSI, let us briefly discuss how optimal E T T −1 , (1) C(b, g, u, P ) = P + β (Lo (b, u) + Le (g, u, P )) (2) arg lim sup Ui ,Pi T →∞ C(Bi , Gi , Ui , Pi ) i=0 Here β is a positive weighting factor that gives the priority of reducing packet loss over conserving power When β is increased, we tend to transmit at a higher rate in order to lower the packet loss rate at the expense of using higher transmit power On the other hand, for smaller values of β, the average transmission power will be reduced at the cost of increasing the packet loss rate If P β and Lβ are the average power and packet loss rate (due to buffer overflow and transmission errors) obtained when solving (1) for a particular value of β, then Lβ is also the minimum achievable loss rate given a power constraint of P β For our system model in which the channel state Gi evolves according to a stationary, ergodic Markov process, the optimization problem in (1) can be classified as an infinitehorizon, average-cost Markov decision process [26] For such a problem, given complete system SSI, there exists a stationary control policy that is optimal Let π be a stationary policy which maps system states into transmission rate and power for each frame i, i.e., π(Bi , Gi ) (Ui , Pi ) Defining Javr (π) = lim sup E T T →∞ T −1 i=0 C(Bi , Gi , Ui , Pi ) | π , (3) the optimization problem in (1) becomes π ∗ = arg Javr (π) π (4) The above infinite-horizon, average-cost Markov decision process (MDP) can be solved effectively using dynamic programming techniques such as policy iteration and value iteration [26, Chapter 6] It is also useful to consider the discounted cost of using policy π with initial system state (b, g), i.e., Jα (b, g, π) T −1 = lim E T →∞ i=0 αi C (Bi , Gi , Ui , Pi ) |B0 = b, G0 = g, π , (5) where < α < is the discounting factor As the immediate cost function C(b, g, u, P ) is bounded, the limit in (5) always exists Correspondingly, we have the problem of finding a control policy that minimizes the discounted cost, i.e., π ∗α = arg Jα (b, g, π) π (6) IEEE TRANSACTIONS ON COMMUNICATIONS, VOL XX, NO Y, MONTH 2007 It can be shown that π∗α converges to π ∗ which is the solution of (4) as α → ( [26, Chapter 6]) Moreover, let Jα∗ (b, g) be the minimum discounted cost when starting with initial state (b, g), the solution of the discounted cost problem satisfies the simple Bellman equation ( [26, Chapter 6]): K−1 ∞ Jα∗ (b, g) = C (b, g, u, P ) + α (u,P ) pA (a)Jα∗ PG (g, g ′ ) g′ =0 (7) a=0 min{b − u + a, B}, g ′ The physical interpretation of (7) is that, for the discounted cost problem, at each stage of control, the optimal control action should minimize the sum of the immediate cost C(.) and the α-weighted future cost, provided that in the subsequent future stages, optimal control actions are selected This elegant Bellman equation is useful for analyzing the structural properties of optimal control policies It is also the inspiration behind the effective QMDP heuristic ( [18]) when only incomplete system state information is available for making control decisions This is discussed in Section VB.3 III I NCOMPLETE S YSTEM S TATE I NFORMATION Let us now consider the cases when only imperfect knowledge of the instantaneous system state is available for making control decisions Rather, the transmit power and rate are adapted based on a partially observed system state which includes quantized buffer occupancy and delayed and/or imperfectly estimated channel state A Quantized Buffer State Information Although the transmitter usually knows the exact buffer occupancy, we may not want to adapt the transmission parameters to this exact value Firstly, the buffer occupancy can change frequently, therefore, adapting to its exact value may require a significant amount of signaling from the transmitter to the receiver Secondly, apart from the signaling issue, we may want to quantize the buffer occupancy in order to reduce the complexity in obtaining and implementing the buffer/channel adaptive policies Given that the buffer capacity is B and the number of channel states is K, using the exact buffer occupancy results in the total number of system states of (B + 1)K When B and K are large, by quantizing B using a small number of levels, we can significantly reduce the number of system states and consequently reduce the complexity of obtaining and implementing the adaptive transmission policies We can quantize the buffer occupancy using a small number of thresholds and only update the transmit power and rate when there is a threshold crossing In this paper, the buffer occupancy is quantized using M + thresholds, i.e., = b0 < b1 < < bM = B+1 The buffer is said to be in state k, ≤ k < M , if the number of packets currently queueing satisfies bk ≤ b < bk+1 Denoting the quantized buffer occupancy at time i by B i , we have B i = bk , where k satisfies bk ≤ Bi < bk+1 (8) B Delayed Imperfect Channel Estimates We assume that the channel gain is first estimated at the receiver, then quantized into one of the possible values {γ0 , γ1 , γK−1 }, and finally the estimated channel index is fed back to the transmitter This process introduces both delay and errors in the transmitter knowledge of the channel state If we take into account the effects of both delay and errors, then at time i, what available at the transmitter is a sequence of delayed imperfect estimates of the channel states up to time i − m, i.e., {G0 , Gi−m }, i ≥ m ≥ Note that mTf is the total estimation and feedback delay We account for the fact that Gi can be erroneous by the following function: PE (g, g) = Pr(Gi = g | Gi = g), (9) which gives the probability of wrongly estimating channel state g as channel state g Note that PE (g, g) depends on the specific channel estimation technique employed at the receiver In this paper, we assume that the channel estimation error does not depend on the chosen transmission parameters and is i.i.d over time We also assume that PE (g, g) is known at the transmitter for all pairs (g, g) As an example, let us assume that if the actual channel state is g, then the estimated channel gain prior to quantization is of the form: γ = γg + v, (10) where v is a Gaussian random variable with zero mean and variance σ Quantizing γ to the closest value in the set {γ0 , γ1 , γK−1 } to obtain the estimated channel index g, we have: γg + γg+1 − 2γg √ PE (g, g) = erf 2 2σ γg + γg−1 − 2γg √ − erf , < g < K − 1, 2σ (11) and PE (g, 0) = + erf γ0 + γ1 − 2γg √ 2σ PE (g, K − 1) = − erf γK−2 + γK−1 − 2γg √ 2σ , (12) , (13) where erf(.) is the standard error function IV O PTIMAL A DAPTIVE T RANSMISSION P OLICIES G IVEN D ELAYED E RROR - FREE C HANNEL S TATES In this section, we consider a special case in which the channel information for choosing the transmit power and rate at time frame i is of the form {G0 , Gi−m−n , Gi−m−n+1 , Gi−m }, i ≥ m + n, m ≥ 0, n ≥ This means that, at time i, in addition to the imperfect channel estimates {Gi−m−n+1 , Gi−m }, the transmitter knows all the exact channel states up to time i − m − n This assumption can be justified by the fact that the accuracy of channel estimation process may be improved if the receiver is given extra time and information to processing [5] For example, when a certain estimation delay is permitted, the receiver can interpolate between past IEEE TRANSACTIONS ON COMMUNICATIONS, VOL XX, NO Y, MONTH 2007 and future estimates to obtain more accurate predictions Therefore, our assumption corresponds to the case when the delay (m + n)Tf is long enough so that the receiver can obtain a near perfect channel estimate Due to the Markov property of the channel model, it is enough to only maintain a truncated sequence of the channel observation history which can be represented by the following channel observation vector: At time i, given that the buffer occupancy is Bi = b and the channel observation vector is H i = (g, g), if the transmission rate and power are set to u and P respectively, the average number of packets lost due to buffer overflow is still given by Lo (b, u) while the expected number of packets lost due to transmission error is H i = (Gi−m−n , Gi−m−n+1 Gi−m ) A When H i = (Gi−1 , Gi ) To simplify the derivations, we consider the case when H i = (Gi−1 , Gi ) Physically, this means that at time i, the transmitter knows the exact previous channel state Gi−1 and has an estimate of the current channel state Gi This corresponds to setting m = and n = in (14) We note that the subsequent derivations can be extended for general values of m and n At time i, given the channel observation vector H i = (Gi−1 , Gi ), we can derive the conditional probability distribution of the channel state Gi as: Pr(Gi = g|H i = (g, g)) = Pr(Gi = g|Gi−1 = g, Gi = g) = = = Pr(Gi = g, Gi−1 = g, Gi = g) Pr(Gi−1 = g, Gi = g) Pr(Gi = g, Gi = g|Gi−1 = g)Pr(Gi−1 = g) Pr(Gi = g|Gi−1 = g)Pr(Gi−1 = g) Pr(Gi = g, Gi = g|Gi−1 = g) = Pr(Gi = g|Gi−1 = g) PG (g, g)PE (g, g) K−1 ′ ′ g′ =0 PG (g, g )PE (g , g) Pr H i+1 = (g ′ , g ′ )|H i = (g, g) = Pr Gi = g′ , Gi+1 = g ′ |H i = (g, g) = Pr Gi = g′ |H i = (g, g) Pr(Gi+1 = g ′ |Gi = g ′ ) (16) K−1 PG (g ′ , k)PE (k, g ′ ) k=0 Knowing the dynamics of H i together with the cost of a transmission action in each state (Bi , H i ), an MDP can be readily formulated, i.e., similar to that given in Section II-D, to minimize the weighted sum of the long term packet loss rate and average transmit power B When H i = Gi−m In the special case when H i = Gi−m , i.e., the transmission decisions at time i can be made based on the perfect knowledge of channel state at time i − m, the number of possible values for H i is K As the result, the size of the newly form MDP is the same as the size of the MDP for the case of complete channel state information V A DAPTIVE T RANSMISSION P OLICIES W HEN NO E RROR - FREE C HANNEL S TATE I S AVAILABLE Now, we consider the situation when no delayed error-free channel estimate is available for choosing transmit power and rate At time i, the transmitter knows a sequence of imperfect channel estimates which can be represented by the following channel observation vector: I i = (G0 Gi−m ) (18) A Optimal Control Policy Given Delayed Imperfect Channel Estimates With i.i.d Channel Model In the special case when the channel states are i.i.d over time, there is no extra information gained by keeping estimates of past channel states We suppose that during frame i, the transmitter knows the estimates of channel state i, i.e., Gi , then the channel observation vector I i in (18) is simplified to defined as I i = Gi (19) PI (g, g ′ ) Based on (15), the dynamics of H i can be written as: = ρG (g ′ , g, g) × (17) The dynamics of I i can be derived as: (15) PH (g, g, g′ , g ′ ) ρG g, g, g Le (g, u, P ) g=0 (14) As there are K possible channel states, the number of all possible channel observation vectors H i is K n+1 The important point to note is that even though the channel state information is incomplete, the number of possible values for H i is still finite This allows the problem of minimizing a weighted sum of the long term packet loss rate and average transmit power to be formulated as a finite-state MDP, with the actual channel state Gi being replaced by the channel observation vector H i In order to fully specify the MDP, we need to derive the dynamics of H i , together with the cost functions associated with choosing transmission rate and power (u, P ) in state (Bi , H i ) ρG g, g, g K−1 LH e (g, g, u, P ) = Pr(I i+1 = g ′ |I i = g) K−1 = Pr(Gi+1 = g ′ |Gi = g) = PE (g, g ′ )pG (g) (20) g=0 Also, during time frame i, given that the channel estimate is I i = g, we can derive the probability distribution of the current channel states as φG (g, g) Pr(Gi = g|I i = g) = Pr(Gi = g|Gi = g) PE (g, g)pG (g) (21) = K−1 ′ ′ g′ =0 PE (g , g)pG (g ) IEEE TRANSACTIONS ON COMMUNICATIONS, VOL XX, NO Y, MONTH 2007 At time i, given that the buffer occupancy is Bi = b and the channel observation vector is I i = (g), if the transmission rate and power are set to u and P respectively, the average number of packets lost due to buffer overflow is still given by Lo (b, u) while the expected number of packets lost due to transmission error is Due to this it is essentially impossible to obtain an optimal adaptive policy based on either I i or Ψi as doing so may require infinite time and memory Therefore, instead of aiming for an optimal control policy, let us look at some approaches that can be used to approximate it All of these approximations start with the assumption that we have already obtained the MDP policy π ∗ , i.e., an optimal policy when the system state is fully observable 1) Employing the MDP Policy π ∗ : The most straightforward approach is to ignore the partial observability of the channel states and just employ policy π ∗ In other words, at time i, given the channel estimate Gi and buffer occupancy Bi , the transmission parameters are set as: K−1 LIe (g, u, P ) = φG g, g Le (g, u, P ) (22) g=0 Note that the number of possible values for I i is K Knowing the dynamics of I i together with the cost of a transmission action in each state (Bi , I i ), an MDP can be readily formulated, i.e., similar to that given in Section II-D, to minimize the weighted sum of the long term packet loss rate and average transmit power B Suboptimal Control Policies Given Imperfect Channel Estimates Now let us consider the case when the channel states are correlated over time and at time i, the transmitter knows only a sequence of delayed imperfect channel estimates I i = (G0 Gi−m ) To simplify the notations, we further assume that m = 0, however, when m > the analysis is similar The control problem in this situation can be modeled as a partially observable Markov decision process (POMDP) For a POMDP in which the system states are correlated over time, in order to make an optimal control decision, the controller needs to keep track of the entire observation history That means for our control problem, the transmitter needs to record the entire channel estimation history, i.e., I i , in order to select optimal transmit power and rate Instead of remembering the entire observation history, the controller in a POMDP can keep track of the so called belief state, which is the probability distribution of the system state, conditioned on the observation history For our particular problem, we can define Ψi as the belief channel state at time i, i.e., then Ψi (g) = Pr(Gi = g | Ψ0 , G0 , Gi ), (23) where the initial probability distribution Ψ0 is assumed known In case Ψ0 is not given, it can be set to Ψ0 (g) = pG (g), i.e., the stationary distribution of the channel states The advantage of keeping a belief state for every time frame is that it contains all relevant information for making control actions [26] Furthermore, in the next time frame, given a new channel estimation Gi+1 = g, the new belief state can be readily derived from (Ui , Pi ) = π ∗ (Bi , Gi ) (25) 2) The Most Likely State Heuristic: In this approach, we first determine the state that the channel is most likely in, i.e., GMLS = arg max {Ψi (g)} i (26) g∈{0, K−1} Note that Ψi is the belief channel state at time i and is calculated using (24) Then the transmission parameters are set as: (Ui , Pi ) = π ∗ (Bi , GMLS ) i (27) This approach, which is usually termed the Most Likely State (MLS) approach, was proposed in [27] 3) The QMDP Heuristic: This approach relates to the discounted cost problem defined in (6) Let the Q function be defined as: Q(b, g, u, P ) = C(b, g, u, P ) K−1 ∞ +α g′ =0 a=0 PG (g, g ′ )pA (a)Jα∗ min{b − u + a, B}, g ′ , (28) from the Bellman equation (7), when the system state is fully observed, Q(b, g, u, P ) represents the cost of taking action (u, P ) in state (b, g) and then acting optimally afterward Based on this, the popular QMDP heuristic takes into account the belief state for one step and then assumes that the state is entirely known [18] Applying to our control problem, at time i, given the buffer occupancy Bi and the belief channel state ΨI , the transmission rate and power are chosen according to: K−1 Ψi+1 (g) = Pr(Gi+1 = g | Ψ0 , G0 , , Gi , Gi+1 = g) = Pr(Gi+1 = g|Ψi , Gi+1 = g) = = Pr(Gi+1 = g, Gi+1 = g|Ψi ) Pr(Gi+1 = g|Ψi ) K−1 ′ ′ g′ =0 Ψi (g )PG (g , g) K−1 K−1 ′ ′′ ′′ ′ g′ =0 PE (g , g) g′′ =0 Ψi (g )PG (g , g ) PE (g, g) (24) Unfortunately, the number of possible channel observation vectors I i and possible belief channel states Ψi are infinite (Ui , Pi ) = arg u∈{0, Bi }, P ∈P Ψi (g)Q(Bi , g, u, P ) g=0 (29) For a deeper discussion on different approaches to approximate an optimal solution for POMDP, please refer to [28] 4) The Minimum Immediate Cost Heuristic: Finally, to assess the effectiveness of the MDP, MLS, and QMDP approaches, which are all MDP-based, we introduce a nonMDP heuristic called the Minimum Immediate Cost (MIC) approach In the MIC approach, at time frame i, given the belief state Ψi , the transmission parameters are selected so IEEE TRANSACTIONS ON COMMUNICATIONS, VOL XX, NO Y, MONTH 2007 Correlated Channel Model that the expected immediate cost is minimized, i.e., 0.6 K−1 arg Ψi (g)C(Bi , g, u, P ) u∈{0, Bi }, p∈P g=0 0.4 (30) VI N UMERICAL RESULTS AND D ISCUSSION A System Parameters The system for our numerical study is as follows Packets arrive to the buffer according to a Poisson distribution with rate λ = × 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 AWGN noise power density is No /2 = 10−5 Watt/Hz We consider two 8state FSMCs as described in Table I, where the channel model in Scenario is obtained by quantizing the fading range of a Rayleigh fading channel that has average gain γ = 0.8 and Doppler frequency fD = 10 Hz and the channel model in Scenario corresponds to fD = 20 Hz Adaptive transmission is based on a variable-rate, variablepower M-ary quadrature amplitude modulation (MQAM) scheme similar to that described in [5] 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 contains F modulated symbols and therefore, Tf = F Ts Here we set F = L = 100 so that when a signal constellation of size M = 2u is used, exactly u packets are transmitted from the buffer during each time frame Given a particular system state (b, g), a control action (u, P ), and a Poisson arrival with rate λ, the expected number of packets lost due to buffer overflow is B−b+u−1 Lo (b, u) = (λTf ) − pA (a) a=0 (31) B−b+u − (B − b + u) − pA (a) , a=0 where exp(−λTf )(λTf )a (32) a! We assume that a transmitted packet is in error if at least V out of the L bits in the packet are in error The expected number of packets discarded due to transmission errors can be calculated by pA (a) = L Le (g, u, P ) =u j=V L j (Pb (g, u, P )) j (L−j) (1 − Pb (g, u, P )) (33) , where Pb (g, u, P ) is the (uncoded) bit error rate when using transmit power P and rate u on channel state g Pb (g, u, P ) Normalized Packet Loss Rate (Ui , Pi ) = OCPI, fixed BER = 10−3 −4 OCPI, fixed BER = 10 −5 OCPI, fixed BER = 10 −6 OCPI, fixed BER = 10 OCPI without BER constraint 0.5 0.3 0.25 0.2 0.15 0.1 14 16 18 20 22 24 26 Power (dB) Fig Performance of optimal buffer/channel adaptive scheme with and without a BER constraint Channel model is given in Table I, Scenario can be approximated by ( [5]): Pb (g, u, P ) = 0.2 exp −1.5 P γg W N o(2u − 1) (34) We consider the performance of different approaches discussed in Sections IV and V When the packet arrival rate is fixed, maximizing the system throughput is equivalent to minimizing total packet loss due to buffer overflow and transmission error Therefore, for each scheme, the long-term packet loss rate versus average transmit power is plotted B Performance with Buffer Overflow and Transmission Error Tradeoff In Fig 2, we plot the performance of the optimal buffer/channel adaptive transmission policies with and without a BER constraint Here, we assume that the system state information is perfect and consider optimal control policies (termed OCPI) We also assume that a packet is in error if any bit in the packet is corrupted, this means V = in (33), this is also assumed for the results plotted in Figs and The OCPI policies without any BER constraint are obtained by solving the MDP in (4) The OCPI policies with a BER constraint are obtained by solving some similar MDP described in [7]–[9] As can be seen, when the BER constraint is relaxed, significant gain can be achieved When the fixed BER is set to relatively high values, i.e 10−3 and 10−4 , adaptive policies perform well in low range of transmission power but become much worse than the policies without BER constraint when the power is high On the other hand, when the fixed BER is set to a relatively low value, i.e 10−6 , the performance of adaptive policies is much worse than that of the policies without BER constrant in the low power range To further understand the tradeoff between buffer overflow and transmission errors, in Fig 3, we separately plot the packet loss due to buffer overflow and packet loss due to transmission errors for optimal buffer/channel adaptive policies with and IEEE TRANSACTIONS ON COMMUNICATIONS, VOL XX, NO Y, MONTH 2007 TABLE I C HANNEL STATES AND TRANSITION PROBABILITIES ( AN 8- STATE FSMC OBTAINED BY QUANTIZING A R AYLEIGH FADING CHANNEL WITH AVERAGE GAIN 0.8 AND D OPPLER FREQUENCY 10 H Z IN S CENARIO AND 20 H Z IN S CENARIO 2) Channel states k γk Pkk Pk,k+1 Pk,k−1 γk Pkk Pk,k+1 Pk,k−1 Scenario Scenario 0 0.9359 0.0641 0 0.8718 0.1282 0.1068 0.8552 0.0807 0.0641 0.1068 0.7104 0.1613 0.1282 0.2301 0.8334 0.0859 0.0807 0.2301 0.6668 0.1718 0.1613 0.3760 0.8306 0.0835 0.0859 0.3760 0.6612 0.1670 0.1718 0.5545 0.8420 0.0745 0.0835 0.5545 0.6841 0.1489 0.1670 0.7847 0.8665 0.0590 0.0745 0.7847 0.7330 0.1181 0.1489 1.1090 0.9048 0.0361 0.0590 1.1090 0.8097 0.0723 0.1181 1.6636 0.9639 0.0361 1.6636 0.9277 0.0723 10 10 −3 Overflow/Errors (BER = 10 ) −6 Overflow/Errors (BER = 10 ) Overflow/Errors (no BER constraint) −1 10 Overflow_ Rate/Error_Rate Normalized Packet Loss Rate 10 −2 10 −3 Overflow Rate (BER = 10 Error Rate (BER = 10−3) Overflow Rate (BER = 10−6) −3 10 −6 Error Rate (BER = 10 ) Overflow Rate (no BER constraint) Error Rate (no BER constraint) −4 10 −5 10 10 10 10 10 12 14 16 18 20 22 24 26 Power (dB) Fig Packet loss due to buffer overflow and transmission errors of optimal buffer/channel adaptive scheme with and without a BER constraint Channel model is given in Table I, Scenario without a BER constraint It is clear that, without a BER constraint, an optimal policy varies the transmission error rate dynamically according to the available transmit power In particular, at low power, a greater number of transmission errors can be tolerated in order to reduce buffer overflow On the other hand, when plenty of transmit power is available, a good adaptive policy should transmit at a high rate and high power to minimize both transmission errors and buffer overflow This argument can be further illustrated in Fig 4, where we plot the ratio between packet loss due to buffer overflow and packet loss due to transmission errors C Performance Under Quantized Buffer Occupancy First, let us look at the performance of the buffer/channel adaptive transmission approach when the buffer occupancy is quantized When the buffer occupancy is quantized, the performance of policy π ∗ (obtained by solving (4)) depends on two factors, i.e., the number of quantized buffer states, and the selected quantization thresholds Clearly, the greater the number of quantized states, the closer the performance to the optimal At the same time, given a fixed number of quantized states, the performance depends on the set of selected thresholds An intuitive way to select good quantization thresholds is to divide the range of buffer occupancy more finely at the 10 14 16 18 20 22 24 26 Power (dB) Fig Ratio between packet loss due to buffer overflow and packet loss due to transmission errors of optimal adaptive scheme with and without a BER constraint Channel model is given in Table I, Scenario range of high probability distribution For example, if we know that most of the time, the buffer occupancy is low, then a greater number of thresholds should be set at low values In Fig 5, we plot the performance of π ∗ , in terms of total long term packet loss rate versus average transmit power, for different buffer quantization schemes The number of quantized buffer states is increased from two to four In particular, in the first quantization scheme, we set a single threshold at When the buffer occupancy is less than 7, it is quantized to 0, otherwise, it is quantized to Similarly, for the case of three quantized buffer states, we set the two thresholds at and 9, and for the case of four quantized buffer states, we set the three thresholds at 3, 6, and 10 For the results in Fig 5, as well as in Figs 6-9, we assume that a packet is in error if more than ten out of 100 bits in the packet are corrupted, this means V = 11 in (33) As can be seen, when only two quantized states are used, there is a significant loss compared to the case of adapting to the exact buffer occupancy However, the performance loss is reduced significantly when the number of quantized buffer states is increased to three and four When four quantized buffer states are used, the performance is very near optimal This suggests that we can often quantize the buffer occupancy in order to reduce the complexity of the adaptive transmission policy IEEE TRANSACTIONS ON COMMUNICATIONS, VOL XX, NO Y, MONTH 2007 0.3 BCDI OCDI_1 OCDI_2 (σ = 0.1) OCDI_2 (σ = 0.05) OCPI 0.3 quantized buffer states (threshold = 7) quantized buffer states (thresholds = 4, 9) quantized buffer states (thresholds = 3, 6, 10) Using exact buffer occupancy (16 states) 0.2 Normalized Packet Loss Rate Normalized Packet Loss Rate 0.25 0.25 0.15 0.1 0.05 10 12 14 16 18 20 22 0.2 0.15 0.1 24 Power (dB) 0.05 10 12 14 16 18 20 22 24 Power (dB) Fig Performance of π∗ under quantized buffer state information The performance is in terms of normalized packet loss rate versus average transmit power System parameters are given in Section VI-A Channel model is given in Table I, Scenario Fig Performance, i.e., normalized packet loss rate versus average transmit power, for different adaptive transmission schemes given delayed error-free channel state information System parameters are given in Section VI-A Channel model is in Tab I, Scenario 0.3 without suffering significant performance degradation Let us look at the performance of different buffer/channel adaptive transmission schemes when a delayed error-free channel state and an accurate buffer occupancy are available for making control decisions We consider two scenarios In the first scenario, at time frame i, the transmitter knows the exact channel state at time i−1, i.e., Gi−1 In the second scenario, in addition to knowing Gi−1 , the transmitter also has an estimate of the channel state at time i, i.e., Gi Note that both of these scenarios have been discussed in Section IV In both cases, we have shown that optimal transmission policies, which maximize the system throughput given incomplete channel state information, can be obtained To facilitate the discussion, we term the optimal adaptive policies under the first and second scenarios OCDI and OCDI (Optimal Control under Delay Information and 2) In addition to this, we also look at the approach of blindly employing policy π ∗ with delayed information This approach is termed BCDI (Blind Control under Delay Information) We plot the packet loss rate versus average transmit power for each scheme Here, the packet loss rate is normalized by the average packet arrival rate Clearly, the packet loss rates of all schemes are lower-bounded by the packet loss rate when optimal adaptive policies are employed with perfect system state information, that is, the OCPI curve The performance of OCDI 1, OCDI 2, BCDI, and OCPI schemes are given in Figs and Fig corresponds to channel model in Table I Scenario while Fig is for the channel model in Scenario In Figs and 7, we observe, as expected, that the performance of all schemes under delayed channel state information is lower-bounded by the performance of optimal transmission scheme with perfect channel knowledge More importantly, the 0.25 Normalized Packet Loss Rate D Performance of Different Approaches Given Delayed Error-free Channel State BCDI OCDI_1 OCDI_2 (σ = 0.1) OCPI 0.2 0.15 0.1 10 12 14 16 18 20 22 24 Power (dB) Fig Performance, i.e., normalized packet loss rate versus average transmit power, for different adaptive transmission schemes given delayed channel state information System parameters are given in Section VI-A Channel model is in Tab I, Scenario performance degradation increases when the channel changes faster (Fig 6) This is expected because when the channel changes faster, the delayed channel state contains less information about the current channel state The second observation that we can make from Figs and is that the greater amount of information an adaptive scheme has, the better its performance is In particular, the OCDI scheme performs better than BCDI scheme and OCDI scheme performs better than OCDI The performance of scheme OCDI improves when the quality of the channel estimate Gi is improved For example, when σ = 0.05, the performance of OCDI is quite close to that of the optimal scheme under perfect SSI When the channel estimate Gi has high error probability (σ = 0.1), the performance of OCDI approaches that of OCDI However, the performance gain of IEEE TRANSACTIONS ON COMMUNICATIONS, VOL XX, NO Y, MONTH 2007 10 0.3 0.4 MIC BCEI MLS QMDP OCPI 0.2 0.15 0.1 0.05 10 12 14 16 18 20 MIC BCEI MLS QMDP OCPI 0.3 Normalized Packet Loss Rate Normalized Packet Loss Rate 0.25 0.35 22 Power (dB) Fig Performance, i.e., normalized packet loss rate versus average transmit power, for different adaptive transmission schemes given imperfect channel estimate System parameters are given in Section VI-A Channel model is in Tab I, Scenario The standard deviation of channel estimating noise is σ = 0.05 OCDI comes at a cost of higher complexity In particular, the number of internal channel states for OCDI is K while it is K for OCDI E Performance of Different Approaches Given Imperfect Channel Estimates Now let us look at the performance of different buffer and channel adaptive transmission schemes when no errorfree channel state information is available at the transmitter In particular, during time slot i, the transmitter only has an estimate of the channel state, i.e., Gi For this numerical study, we assume that the estimation error for the channel gain has a Gaussian distribution with zero mean and variation of σ The estimation statistics can be computed using equation (11) - (13) As has been discussed in Section V-B, for the general case of correlated channel model, when no perfect channel estimate is available at the transmitter, it is not practical to look for optimal adaptive transmission policies Instead, there are various approaches that can approximate optimal control policies at lower complexity These approaches are: BCEI, MLS, QMDP and they have been discussed in Section V-B Note that BCEI is the approach that blindly employs policy π ∗ with erroneous channel state information Again, we plot the performance of different adaptive schemes in terms of normalized packet loss rate versus average transmit power The performance of all schemes are compared to the case when an optimal scheme is employed under perfect SSI, that is, the OCPI curve The performance of different classes of adaptive policies is given in Figs and Fig is obtained for the case when σ = 0.05 and Fig is for the case when σ = 0.1 In both Figs and 9, the channel model in Table I, Scenario 2, is used As can be seen, the MIC approach, which only tries to minimize the immediate cost during each time frame and does 0.25 0.2 0.15 0.1 0.05 10 12 14 16 18 20 22 Power (dB) Fig Performance, i.e., normalized packet loss rate versus average transmit power, for different adaptive transmission schemes given imperfect channel estimate System parameters are given in Section VI-A Channel model is in Tab I, Scenario The standard deviation of channel estimating noise is σ = 0.1 not take the dynamics of the system into account has the worst performance Significant performance gain can be achieved by using BCEI, MLS, and QMDP approaches This shows the important of structuring the problem as a partially observable Markov decision process Among the three approaches BCEI, MLS, and QMDP, it seems that QMDP performs best We note that there is no significant extra complexity when using QMDP instead of BCEI or MLS, therefore, QMDP is a good choice to cope with imperfect estimated channel state information Between BCEI and MLS, MLS tends to perform better at low power range, while at higher power range, BCEI achieves better results However, we note that the difference in the performance of BCEI and MLS is not significant, therefore, the simpler approach, i.e., BCEI, is preferable VII C ONCLUSION In this paper, we consider the problem of buffer and channel adaptive transmission for maximizing the throughput of a transmission over a wireless fading channel, subject to an average transmit power constraint We consider scenarios in which the system state information for making control decisions is incomplete This includes delayed and/or imperfectly estimated channel state and quantized buffer occupancy We also allow for a tradeoff due to the loss from both transmission errors and buffer overflow and obtain significant throughput improvement This paper shows the importance of cross-layer design in achieving good performance for wireless data communication system This paper also demonstrates that, even when the system state is not fully observable, buffer and channel adaptive transmission can still be implemented in an effective manner IEEE TRANSACTIONS ON COMMUNICATIONS, VOL XX, NO Y, MONTH 2007 R EFERENCES [1] H S Wang and N Moayeri, “Finite-state markov channel - a useful model for radio communication channels,” IEEE Trans Veh Tech., vol 44, pp 473–479, Feb 1995 [2] C C Tan and N C Beaulieu, “On first-order markov modeling for the rayleigh fading channel,” IEEE Trans Comm., vol 48, no 12, pp 2032–2040, Dec 2000 [3] A T Hoang and M Motani, “Cross-layer adaptive transmission: Optimal strategies 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Lovejoy, “A survey of algorithmic methods for partially observable markov decision processes,” Annals of Operation Research, vol 28, no 1, pp 47–65, 1991 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 PLACE in electrical engineering at the National University PHOTO of Singapore in 2005 HERE 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 PLACE information theory and coding for CDMA systems PHOTO Prior to his Ph.D., he was a member of technical HERE 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