Cross layer adaptive transmission optima

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 dat

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

time-varying 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 INTRODUCTION

IN modern and future wireless communications,

maximiz-ing throughput under limited available energy and

band-width 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 1 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

Engineer-ing, National University of Singapore (e-mail: athoang@i2r.a-star.edu.sg).

M Motani is with the Department of Electrical and Computer

En-gineering, National University of Singapore, Singapore 119260 (e-mail:

motani@nus.edu.sg).

Digital Object Identifier 10.1109/TCOMM.2008.060214.

Transmitter Wireless Channel Receiver

Control Signals Buffer

Packets

Fig 1 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

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

au-thors 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 do 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

transmis-sion 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

subop-timal schemes We also confirm via numerical

compu-tations 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 PROBLEMDEFINITION

A System Model

We consider a single-user system depicted in Fig 1 Time

is divided into frames of equal length of T f seconds each and

frame i refers to the time period [iT f , (i+1)T f) The number

of packets arriving to the buffer during frame i is denoted

by A i We assume that these A i packets are only added to

the buffer at the end of frame i We consider the case when

distribution of the number of packets arriving during each time

frame is assumed known and denoted by p A (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

addi-tive white Gaussian noise (AWGN) W and N o /2 respectively

denote the channel bandwidth and noise power density The

fading process is represented by a stationary and ergodic

K-state Markov chain, with the channel K-states numbered from 0

is denoted by γ g During each frame, the channel is assumed

to remain in a single state Letting G i denote the channel state

during time frame i, the channel state transition probability is

defined as

P G (g, g  ) = Pr{G i = g  | G i−1 = g}. (1)

We assume that P G (g, g  ) are known for all g and g  The

stationary distribution of each channel state g is denoted by

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 p G (g) as well

as the channel state transition probabilities P G (g, g ) can be derived [13], [14]

B Adaptive Transmission

We denote the system state in frame i by S i = (B i , G i),

where B i is the number of packets in the buffer at the

beginning of frame i while G i is the channel state during

frame i At the beginning of frame i, we assume that the

transmitter and the receiver have complete knowledge ofS i

We assume that, based on S i, the transmitter can vary

its transmit power and rate For frame i, let P i (Watts)

and U i (packets/frame) denote the transmit power and rate, respectively We must have 0 ≤ U i ≤ B i In addition, we

assume that P i ∈ P where P is the set of all power levels

that the transmitter can operate at We call a pair (U i , P i)

a control action for frame i Note that the transmitter can change its transmission rate U i by changing the coding and/or modulation schemes [4], [15]–[17]

Let P b (g, u, P ) be the function that gives the BER when the channel state is g and the transmit power and rate are

coding, modulation, and detection schemes used We further

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

P p (g, u, P )

=

L



j=l

L

j



P b (g, u, P ) j

1 − P b (g, u, P )(L−j)

As an example, let us change the transmission rate by

varying the constellation size of an M-ary quadrature

ampli-tude 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 b (g, u, P ) ≤ 0.2 exp



o(2u − 1)



C Throughput Maximization Problem

We adopt the following definition of the system throughput

Definition 1: The system throughput is the long-term

aver-age rate at which packets are successfully transmitted For an

average packet arrival rate λ, a buffer overflow probability

be calculated by

We consider the following optimization problem:

Throughput Maximization Problem: At the beginning of

the transmitter and the receiver, select the transmission

subject to an average transmit power constraint P

D Satisfying a BER Constraint

From this point on until the end of Section IV, we adopt

an extra constraint that the control action (U i , P i) 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, P b) be the minimum power needed to transmit

state is g and the BER constraint is P b P (u, g, P b) depends on

the specific coding, modulation, and detection schemes being

used Furthermore, we must have P (u, g, P b ) ∈ P In case

there is no power level inP 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, P b) by:

= arg min

P ∈P



g



1.5



.

(5)

In general, we assume that P (u, g, P b ) is non-decreasing in u and non-increasing in g.

As the BER performance is always kept at P b, from (2), the

packet error probability P p is always kept unchanged When

both λ and P p are fixed, from (4), it is clear that maximizing

the system throughput is equivalent to minimizing P of 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

L o (b, u) = E max{0, A + b − u − B} (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 min

U0, ,U T −1 lim sup

T →∞

1

T E

T −1



i=0

(L o (B i , U i))

(7)

subject to:

lim sup

T →∞

1

T E

T −1



i=0

III PARETOOPTIMALPOLICIES 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

J avr = lim sup

T →∞

1

T E

T −1



i=0

where C I (b, g, u) is the immediate cost incurred in state (b, g) when the control action (u, P ) is taken, i.e.,

C I (b, g, u) = P (u, g, P b ) + βL o (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 L β are the average power and packet loss rate (due to buffer overflow)

obtained when minimizing J avr for a particular value of β, then L β is also the minimum achievable loss rate given a

power constraint of P β In other words, for each value of β, minimizing J avr gives us a Pareto optimal point (L β , P β) in

the Loss Rate versus Power Constraint curve.

The problem of minimizing J avr is an infinite horizon average cost Markov decision process (MDP) with system state S i = (B i , G i ), control action U i, and immediate cost

function C I (B i , G i , U i) 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

action u is taken, the probability of the system being in state

s  = (b  , g ) in the next time frame is

P S (s, s  , u) = Pr{S i+1 = s  | S i = s, U i = u}

= P G (g, g  )P B (b, b  , u), (12)

where

P B (b, b  , u) = Pr{B i+1 = b  |B i = b, U i = u}. (13)

Furthermore,

B i+1 = min{B i + A i − U i , 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., U i =

μ i (B i , G i) We have the optimization problem

π ∗= arg min

π J avr (π)

= arg min

π

lim sup

T →∞

1

T E

T −1



i=0

In our system, as all states are connected, there exists a

stationary policy π ∗ , i.e μ i ≡ μ for all i, which is a

solution to (15) To simplify the notation, we just write

U i = π ∗ (B i , G i) As it is shown in [19], using a simple policy

iteration algorithm, an optimal policyπ ∗can be reached in a

finite number of steps

Finally, it is also useful to consider the discounted cost

problem defined as:

arg min

π J α (b,g, π) = arg min

T →∞E T −1

i=0

C I (B i , G i , U i ) |B0= b, G0= g, π

(16)

where 0 < α < 1 is the discounting factor As the immediate

cost function C I is bounded, the limit in (16) always exists

As shown in [19], when α → 1 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

α (b, g) = min

u C I (b, g, u) + α

K−1

g =0

∞



a=0



P G (g, g )

α (min{b + a − u, B}, g )

.

(17)

Equation (17) is particularly useful for analyzing the structure

of the optimal policy

IV STRUCTURE OF THEOPTIMALPOLICY

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]

We will show the above effect for a simple FSMC

model which has three possible states, i.e., K = 3.

In particular, we assume that 0 = γ0 < γ1 < γ2 Moreover, in the channel model, transitions can only

hap-pen between adjacent channel states, i.e., P G (0, 2) =

P G (2, 0) = 0 while P G (0, 0), P G (0, 1), P G (1, 1), P G (1, 0),

P G (1, 2), P G (2, 2), P G (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 C I (b, g, u) Second, transmitting at rate u in state (b, g) also reduces the

future cost

C F (b, g, u) = α K−1

g =0

∞



a=0



P G (g, g  )p A (a)

α (min{b + a − u, B}, g )

.

(18)

For state (b, g) with b > 0, g > 0, let 0 ≤ u1 < u2 ≤ b be

two possible transmission rates Let

ΔI (b, g, u1, u2) = C I (b, g, u2) − C I (b, g, u1) (19) and

ΔF (b, g, u1, u2) = C F (b, g, u1) − C F (b, g, u2). (20)

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 u1to u2 Clearly, action

u2 is more favorable than u1 in state (b, g) if and only if

ΔI (b, g, u1, u2) < Δ F (b, g, u1, u2) From (11) and (19), we have

ΔI (b, 1, u1, u2) − Δ I (b, 2, u1, u2) = P (u2, 1, P b)

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 β o such that for every β > β oand 0≤ u1< u2≤ b,

the following inequality holds:

ΔI (b, 1, u1, u2) − Δ I (b, 2, u1, u2)

Theorem 1: For each buffer state b > 0, let β obe defined

as in Lemma 1 and β > β o, then the optimal transmission

rate for each state (b, g), g > 0, is non-increasing in g.

u ∗ be the optimal transmission rate at states (b, 1) and (b, 2)

respectively Suppose 0≤ u ∗ < u ∗ ≤ b From (17) we have

C I (b, 1, u ∗

1) + C F (b, 1,u ∗

1)

≤ C I (b, 1, u ∗

2) + C F (b, 1, u ∗

2). (23)

C I (b, 2, u ∗

2) + C F (b, 2,u ∗

2)

≤ C I (b, 2, u ∗

1) + C F (b, 2, u ∗

1). (24)

Inequalities (23) and (24) respectively imply (25) and (26)

ΔI (b, 1, u ∗

1, u ∗

2) = C I (b, 1, u ∗

2) − C I (b, 1, u ∗

1)

≥ C F (b, 1, u ∗

1) − C F (b, 1, u ∗

2) = ΔF (b, 1, u ∗

1, u ∗

2), (25)

ΔI (b, 2, u ∗ , u ∗ ) ≤ Δ F (b, 2, u ∗ , u ∗ ). (26)

From (25) and (26) we have:

ΔI (b, 1, u ∗

1, u ∗

2) − Δ I (b, 2, u ∗

1, u ∗

2)

≥ Δ F (b, 1, u ∗

1, u ∗

2) − Δ F (b, 2, u ∗

1, u ∗

2), (27) which contradicts Lemma 1 and therefore, u ∗

1≥ u ∗

2

Comment: Theorem 1 shows that for a certain correlated

fading channel model and average transmission power

con-straint, 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.

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

A Taking Packet Loss Due to Transmission Error into

Ac-count

Given the transmission rate u, power P , channel state g,

and the packet error probability of P p (g, u, P ), the expected

number of packets lost due to transmission error is

L e (g, u, P ) = uP p (g, u, P ) (28) For fixed packet arrival rate, maximizing the system

through-put is equivalent to minimizing total packet loss rate due to

both buffer overflow and transmission error So we have the

optimization problem:

arg min

U i ,P i lim sup

T →∞

1

T E

T −1



i=0



L o (B i , U i ) + L e (G i , U i , P i)

(29)

subject to:

lim sup

T →∞

1

T E

T −1



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 C I Now we have:

C I (b, g, u, P ) = P + β

L o (b, u) + L e (g, u, P )

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

and sufficient condition for a power level to be optimal is that

it must satisfy

P = arg min

P ∈P C I (b, g, u, P )

= arg min

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)

Letπ be a stationary policy which maps system states into

transmission rate for each frame i, i.e., U i = μ i (B i , G i) Define

I (b, g, u) = min

P ∈P C I (b, g, u, P ) (35) and

J avr (π) = lim sup

T →∞

1

T E

T −1



i=0

I (B i , G i , U i )|π

We have to solve the optimization problem

π ∗ = arg min

Again, this problem can be solved efficiently using dynamic programming techniques [19]

VI NUMERICAL RESULTS ANDDISCUSSION

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.

A System Parameters

Packets arrive to the buffer according to a Poisson

distribu-tion with average rate λ = 103 and 3× 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 N o /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

f D = 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 rate,

variable-power MQAM scheme similar to that described in [4] Let T s

be the symbol period of the MQAM modulator and assume a

Nyquist signaling pulse, sinc(t/T s), is used so that the value

of T s is fixed at 1/W seconds When the symbol period T s

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

so that when a signal constellation of size M = 2 u is used,

exactly u packets are transmitted during each time frame.

TABLE I

C HANNEL STATES AND TRANSITION PROBABILITIES

P k,k+1 0.0641 0.0807 0.0859 0.0835 0.0745 0.0590 0.0361 0

P k,k−1 0 0.0641 0.0807 0.0859 0.0835 0.0745 0.0590 0.0361

0

1

2

3

4

5

6

7

Channel states

1 packet in buffer

5 packets in buffer

10 packets in buffer

14 packets in buffer

Fig 2 Structure of optimal policies, i.e., transmission rates for different

channel states when the buffer occupancy is fixed at1, 5, 10, 14 packets.

Power constraintP = 16dB Channel is correlated over time (Tab I).

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 γ k and the BER constraint is P b 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.

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,

f D = 10 Hz, P b= 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 3 As can be

seen, for each buffer occupancy, the optimal transmission rate

increases when the channel gain increases

0 1 2 3 4 5 6

Channel states

1 packet in buffer

5 packets in buffer

10 packets in buffer

14 packets in buffer

Fig 3 Structure of optimal policies, i.e., transmission rates for different channel states when the buffer occupancy is fixed at1, 5, 10, 14 packets.

Power constraintP = 16dB Channel is i.i.d over time.

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

Power (dB)

Ch_Adpt Ch_Ivn

Fig 4 Normalized packet loss rate (due to buffer overflow only) versus average transmission power. B = 15, λ = 3 packets/frame, P b = 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 P b = 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

14 16 18 20 22 24 26

0.1

0.15

0.2

0.25

0.3

0.4

0.5

0.6

Power (dB)

MDP_I, BER = 10−3 MDP_I, BER = 10−4 MDP_I, BER = 10−5 MDP_I, BER = 10−6 MDP_II, 20 power levels

Fig 5 Normalized packet loss rate (due to buffer overflow and transmission

errors) versus average transmission power. B = 15, λ = 3 packets/frame.

Channel model is correlated over time and is given in Table I.

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 4

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 5 is for correlated channel model We plot the

perfor-mances 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 4 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

0.1 0.15 0.2 0.25 0.3 0.4 0.5 0.6

Power (dB)

MDP_I, BER = 10−3 MDP_II, 5 power levels MDP_II, 10 power levels MDP_II, 20 power levels

Fig 6 Normalized packet loss rate (due to buffer overflow and transmission errors) versus average transmission power.B = 15, λ = 3 packets/frame.

Channel model is correlated over time and is given in Table I.

0.01

0.05 0.1 0.2 0.3 0.4

Power (dB)

MDP_I, BER = 10−3 MDP_I, BER = 10−4 MDP_I, BER = 10−5 MDP_I, BER = 10−6 MDP_II, 20 power levels

Fig 7 Normalized packet loss rate (due to buffer overflow and transmission errors) versus average transmission power.B = 15, λ = 3 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 4 to 40dB) As can be seen, even with only 5 different power levels, the MDP II scheme can perform much better than MDP I schemes Figs 7 and 8 show result for i.i.d channel models and similar effects can be observed

14 16 18 20 22 24

0.01

0.05

0.1

0.2

0.3

0.4

Power (dB)

Normalized Packet Loss Rate MDP_I, BER = 10

−3

MDP_II, 5 power levels

MDP_II, 10 power levels

MDP_II, 20 power levels

Fig 8 Normalized packet loss rate (due to buffer overflow and transmission

errors) versus average transmission power. B = 15, λ = 3 packets/frame.

Channel model is i.i.d over time.

VII CONCLUSION

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

APPENDIX

PROOF OFLEMMA1 Let us first prove the following Lemmas 2, 3, and 4

Lemma 2: For all 0 ≤ g < K, J ∗

α (b, g) is increasing in the buffer occupancy b.

be a bounded and increasing function on the state space (b, g).

For i = 1, 2, , let

J i (b, g) = min

u C I (b, g, u)

+ α K−1

g =0

∞



a=0

P G (g, g  )p A (a)J i−1

where q(b, a) = min {b + a, B} Note that from the value

iteration algorithm for solving discounted cost problem (17),

we have J ∗

α (b, g) = lim i→∞ J i (b, g) for all 0 ≤ b ≤ B and

0 ≤ g < K Now assuming J i−1 (b, g) is increasing in b for

all g, we will show that J i (b, g) is also increasing in b for

all g For 0 < b ≤ B, let u ∗ be a value that achieve the minimization in (38)

a) If u ∗= 0, then from (38) we have:

J i (b − 1, g) ≤ C I (b − 1, g, 0) + α K−1

g =0

∞



a=0

P G (g, g  )p A (a)J i−1

g =0

∞



a=0



P G (g, g  )p A (a)

= J i (b, g).

(39)

b) If u ∗ > 0, then from (38) we have:

J i (b − 1, g) ≤ C I (b − 1, g, u ∗ − 1) + α K−1

g =0

∞



a=0

P G (g, g  )p A (a)J i−1

K−1

g =0

∞



a=0



P G (g, g  )p A (a)

= J i (b, g).

(40)

We have proved that if J i−1 (b, g) is increasing in b for all

α (b, g) = lim i→∞ J i (b, g) is increasing in b for all g.

Lemma 3: For all 0 ≤ b1 < b2 ≤ B and all 0 < g < K,

1) − C F (b1, g, u ∗

1) = C I (b2, g, u2)

1) + C F (b2, g, u2) − C F (b1, g, u ∗

1).

(41)

1)

= P (u ∗

Therefore

(42)

It is clear that the left hand side of (42) is bounded when β

increases, so the proof is completed

Lemma 3 is for situation in which the channel state g > 0, when g = 0, we have the following lemma.

Lemma 4: For all 0 ≤ b1< b2≤ B, J ∗

α (b1, 0) increases without bound when β increases.

Proof: When the channel is in state 0, no transmission is

possible, therefore

+ α

K−1

g=0

∞



a=0

P G (0, g)p A (a)J ∗

α



,

+ α K−1

g=0

∞



a=0

P G (0, g)p A (a)J ∗

α



.

(43)

+ α K−1

g=0

∞



a=0

P G (0, g)p A (a)

α



α



= β

.

(44)

The inequality in (44) is due to Lemma 2 From (44), it is

clear that J ∗

β increases and the proof is completed.

Proof of Lemma 1: First of all, we have

ΔI (b, 1, u1, u2) − Δ I (b, 2, u1, u2)

= W N o

f(u2, P b ) − f(u1, P b)  1

2



Therefore, the left hand side of (22) does not depend on β.

For the right hand side of (22), we have:

ΔF (b, g, u1, u2) = α K−1

g =0

∞



a=0

P G (g, g )

p A (a) (J ∗

α (q(b − u1, a), g  ) − J ∗

α (q(b − u2, a), g ))

(46)

Now

ΔF (b, 1, u1, u2) − Δ F (b, 2, u1, u2)

= α

K−1

g =0

∞



a=0



P G (1, g  ) − P G (2, g )



α



α



(47)

= α K−1

g =1

∞



a=0

P

G (1, g  ) − P G (2, g )



α



α



+ α∞

a=0

P

G (1, 0) − P G (2, 0)



α



α



.

(48)

As β increases, the first term in (48) is bounded from below

(from Lemmas 2 and 3) while the second term increases

without bound (from Lemma 4) This combined with (45)

completes the proof

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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 cross-layer 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 in-formation 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.

rate for each state (b, g), g > 0, is non-increasing in g.

u ∗ be the optimal transmission rate at states... 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. ..

IEEE WCNC 2004, Atlanta, Mar 2004, pp 1891–1896.

[2] ——, ? ?Cross- layer adaptive transmission: coping with incomplete

sys-tem state information,”