Quality of service constrained throughput optimisation for joint adaptive transmission with...

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Published in IET CommunicationsReceived on 18th May 2008Revised on 12th September 2008doi: 10.1049/iet-com.2008.0305

ISSN 1751-8628

Quality of service constrained throughputoptimisation for joint adaptive transmissionwith automatic repeat request overblock-fading channelsJ.S. Harsini F. LahoutiWireless Multimedia Communications Laboratory, Center of Excellence in Applied Electromagnetic Systems, School ofElectrical and Computer Engineering, University of Tehran, IranE-mail: [email protected]

Abstract: An optimisation framework for wireless link adaptation which maximises the system throughput subjectto a packet-level quality of service (QoS) constraint is presented. The authors consider joint adaptive variation of thetransmitted power level, transmission data rate and packet-error-rate (PER) at the physical layer to improve thethroughput performance of the selective-repeat automatic repeat request (SR-ARQ) protocol over block-fadingchannels. Specifically, for discrete-rate coded M-QAM schemes, the authors present a power and rate adaptationalgorithm that guarantees a target PER constraint. The proposed framework also facilitates optimising thethroughput performance for delay constrained wireless applications, which imposes a limit on the number ofretransmissions for the ARQ. In particular, a link adaptation scheme is presented, which guarantees a targetpacket-loss rate (PLR) when a truncated SR ARQ protocol at the data-link layer is employed. Numerical resultsindicate that the proposed adaptation schemes compared with other adaptive schemes noticeably enhance thesystem throughput. Also, in the case of truncated ARQ, the results illustrate a fundamental trade-off betweenthe delay and PLR QoS metrics achieved at the maximum throughput performance.

1 IntroductionTo respond to increasing demand for modern data services,the wireless networks are expected to facilitate high datarate applications with diverse quality of service (QoS)requirements. However, guaranteeing QoS over wirelesslinks is challenging as they are subject to variousimpairments such as channel fading. Link adaptation at thetransmitter, in particular adaptive modulation and coding(AMC) and automatic repeat request (ARQ), arepromising approaches towards high throughput wirelesscommunications with sufficient reliability. Today, AMCand ARQ are already included in recent wireless standardssuch as IEEE 802.11a/n [1, 2], and IEEE 802.16e [3].An AMC scheme improves the system performance byselecting a suitable channel code and modulationconstellation based on the time-varying channel conditions[2, 4]. Also, the ARQ protocol increases the reliability by

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retransmissions of erroneously received packets, while ithelps increase the system throughput compared with thescenario of using forward-error correction (FEC) codes atthe physical layer alone [5].

Recently, aiming at achieving a better system performance,combining the AMC scheme at the physical layer with theARQ protocol at the data-link layer has been studied inthe literature [6–12]. Yun et al. first proposed an analyticalexpression for the throughput performance of a selective-repeat (SR) ARQ protocol combined with adaptivemodulation over block-fading channels [6]. They showedthat the considered scheme improves the throughput, whencompared with a single-rate ARQ scheme. In [7], using amultistate Markov packet-error model, the throughputperformance of different ARQ strategies in adaptivemodulation systems is analysed for correlated fadingchannels. For this channel model, in [8] analytical

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expressions are presented for the throughput of jointARQ-AMC scheme under a general multiuser schedulingrule and different arrival traffic loads. The throughput ofthe hybrid type-II ARQ protocol in the presence of AMCat the physical layer, is also investigated in [9, 10], whereincremental redundancy is transmitted for combineddecoding. Using a cross-layer approach, the effect of atruncated-ARQ protocol on the spectral efficiencyperformance of an AMC scheme is studied in [11] and,subsequently, an optimal design perspective is adopted by[12]. Also, in [13] a joint AMC-truncated-ARQadaptation scheme is used to provide a desirable delay-reliability trade-off over fading channels.

A common result of all these studies is that a joint AMC-ARQ scheme yields a better system performance comparedwith the case of single-rate ARQ scheme, and improves theQoS provisioning over fading channels. As validated in[14], when the performance measure is the systemthroughput, designing the AMC scheme based on anobjective function that includes both the effect of channelerror and possible packet retransmissions is more relevant.Therefore it is of special interest to provide optimisedsolutions for AMC design, including power and rateadaptation, which directly maximises the throughputperformance of a joint ARQ-AMC system in the presenceof packet-level QoS constraints. This design problem is themain focus of the current article. Naturally, an accuratepacket-level error model facilitates the analysis.

The assumed system model employs adaptive transmissionat the physical layer in conjunction with the SR ARQprotocol at the data-link layer. The main contributions arepresented in two parts. First, we consider optimising thethroughput performance for data-packet traffic. We obtainoptimal solutions for power adaptation and AMC modeswitching levels that maximise the system throughputsubject to a target packet-error-rate (PER) and an averagetransmit power constraint. Next, we consider adaptivetransmission policy design in the presence of the truncated-ARQ protocol, for service provisioning of delay-constrainedtraffic. For this scenario, we propose an analyticalexpression for the throughput performance of the system.Then, aiming at maximising this objective function, apower and rate adaptation policy is derived, whichguarantees a target packet-loss rate (PLR) in the presenceof a delay constraint imposed on the number of possiblepacket retransmissions. The results demonstrate thatcompared with other adaptive schemes, the proposed cross-layer design considerably improves the throughputperformance of the system, while guaranteeing the QoSrequirements. The proposed adaptation scheme also allowsfor quantifying a fundamental trade-off between the delayand PLR QoS metrics achieved at the optimum point ofthroughput performance.

The rest of this paper is organised as follows. Section 2presents the system and channel model. We derive the

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optimised solutions for power, data rate and PERadaptation in Section 3. Link adaptation in the presence ofa truncated-ARQ protocol is investigated in Section 4.Numerical results are provided in Section 5, and theconcluding remarks are presented in Section 6.

2 System modelWe consider a point-to-point wireless packet communicationlink, which employs AMC and power control at the physicallayer, and an SR ARQ protocol at the data-link layer (Fig. 1).Input packets that have arrived from higher layers of stack arequeued at the transmitter buffer, grouped into frames andtransmitted over a wireless channel. At the transmitter,AMC provides multiple transmission modes, each of whichis specified by a modulation and an FEC code pair. Thetransmitter selects an AMC mode for transmission andadapts transmit power on a frame-by-frame basis, based onthe channel state information (CSI) fed back from thereceiver. At the receiver, in case a packet is detectederroneous, the ARQ protocol initiates a retransmission. Weadopt the packet and frame structures as in [7], where adata frame has a fixed time duration and includes a fixednumber of symbols. Since the number of symbols thatrepresents a data packet depends on the employed AMCmode, each frame carries a variable number of packets. Thefollowing assumptions are made:

i. There are always sufficient packets available at thetransmitter queue (heavy traffic) and, thus, we onlyconsider the effect of the channel process on thethroughput. This assumption is reasonable, for example, ata wireless access point (AP) providing a mobile terminalwith a service such as TCP file transfer or UDP-basedvideo/audio continuous data streaming, which is fed to theAP from an Internet server through a wired link. Ofcourse, in the case of TCP file transfer, an ARQ-basederror recovery control at the link-layer will prevent thethroughput degradation of the TCP as elaborated in [15].

ii. It is assumed that a strong cyclic-redundancy-check codeis used, so that packet error detection at the receiver is perfect.

iii. We assume perfect channel estimates at the receiver. TheCSI, here channel signal to noise ratio (SNR), is fed backreliably to the transmitter without delay.

Figure 1 System and channel model

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2.1 Channel model and AMC modes

We assume a frequency flat-fading channel with stationary andergodic time-varying gain h with unit average power gain, andadditive white Gaussian noise (AWGN) n with zero meanand variance s2. A block-fading model is adopted, where thechannel gain remains constant during a block-length (here aframe), and independently changes from one block to anotheraccording to the distribution of the channel fading. This suitswireless links with slowly moving terminals [5, 16].

Let �S denote the average transmit power. Transmittingwith power �S, the pre-adaptation received SNR intransmitting the kth frame is g(k) ¼ �S h(k)j j

2=s2, whereh(k)j j

2 denotes the instantaneous channel power gain.Because of stationary assumption, the distribution of g(k) isindependent of k and is denoted by pg(g). To simplify thenotation, we also omit the frame index k in SNR g.

We use an AMC scheme with Nþ 1 transmission modes,each corresponding to a distinct range of the channel SNRdenoted by [gn, gnþ1), n ¼ 0, 1, . . . , N, where g0 ¼ 0,gNþ1 ¼ 1. The channel is said to be in state n if the channelSNR fed back from the receiver falls within the interval [gn,gnþ1). When the channel is in state n, the mode n of AMCis chosen and data are transmitted with rate Rn(bits/symbol)and power Sn(g). No data is sent when g [ [g0, g1)corresponding to the outage mode with R0 ¼ 0.

2.2 PER modelling

To optimise the performance, we need an expressionfor channel PER that is invertible and differentiable interms of the received SNR. To approximate the PER of acoded discrete-rate M-QAM scheme over an AWGNchannel, two fitting expressions, min 1, a exp(�gg)

� �and

1= 1þ (ag)gð Þ are suggested by [11] and [17], respectively,

where a and g are constants that depend on the channelcoding and modulation. In this paper, we use the followingexpression as a function of post-adaptation received SNRg Sn(g)= �S� �

to approximate PER in mode n

PERn(g) ¼

1, 0 � g , Gn

an exp �gn

Sn(g)�S

g

� �, g � Gn

8<: (1)

The parameters {an, gn, Gn} are mode and packet-sizedependent constants, and can be obtained by least squarefitting the expression (1) to the exact PER. In Section 5,the accuracy of the PER model in (1) is verified.

3 Throughput optimisation ofjoint SR ARQ-AMC schemeIn this section, we focus on the throughput performance of ajoint SR ARQ-AMC scheme over a fading channel, whereno limit is imposed on the number of retransmissions.Aiming at maximising the throughput, an optimisation


problem is formulated in the presence of a packet-levelQoS constraint at the physical layer.

3.1 Throughput performance of jointSR ARQ-AMC scheme

When AMC is employed, a variable number of packets aretransmitted in each frame. The number of packetstransmitted in frame t is denoted by Ct. Let b denote thenumber of packets that can be accommodated per frame intransmission mode 1. If cn denotes the number of packetstransmitted per frame, in mode n, we have cn ¼ bRn/R1

and Ct [ {c0, c1, . . . , cN }. The parameter b may be relatedto the number of time-slots allocated to the assumedconnection per frame using a resource allocation strategy.Therefore the AMC scheme yields a queue service processwith a total of Nþ 1 states {cn}N

n¼0, where the service statecn corresponds to the channel state n, and the probabilityPr(Ct ¼ cn) ¼ Pr(gn � g , gnþ1).

During the transmission of a frame using any of the AMCmodes, two assumptions are made: (i) we assume that thepower adaptation policy guarantees a fixed PER P0 forpackets of that frame over the wireless channel, as is thedesign subject in the next section; (ii) it is assumed thatAWGN affects each transmitted packet of the frameidentically (with the same noise power) and independently.The latter assumption is valid when the frame transmissiontime is short compared with the time scale of channelvariations. In SR ARQ, only erroneously received packetsin a frame are retransmitted. Therefore we define thethroughput performance as follows. Let rn denote thenumber of correctly received packets per frame, whenthe queue service process is in state n. Thus, rn follows abinomial distribution with parameters (cn, p ¼ 1 2 P0)

Pr(rn ¼ m) ¼cn

m

� �1� P0

� �mP

cn�m0 ,

m ¼ 1, 2, . . . , cn (2)

with mean hn ¼ E{rn} ¼ cn(1� P0) ¼ bRn(1� P0)=R1.

Accordingly, the average number of correctly receivedpackets per frame over the fading channel, is given by

�h ¼ E rk(t)(t)� �

¼XN

n¼0

E rn

� �:Pr(gn � g , gnþ1)

¼ 1� P0

� �:XN

n¼1

cn:

ðgnþ1

gn

pg(g) dg (3)

Here rk(t)(t) is a random process, which indicates the numberof correctly received packets in frame t, when the queueservice process is in state k(t) [ {0, 1, 2, . . . , N }, that is, ifk(t) ¼ n, then rk(t)(t) ¼ rn. We define the averagenormalised throughput as the ratio of the average numberof correctly received information bits per frame to the total


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number of bits transmitted per frame with the mode 1 of theAMC scheme, as in [6, 7], that is

T (P0) ¼ND � h

b(ND þ NH)þ NC

¼ K (1� P0)XN

n¼1cn:

ðgnþ1

gn

pg(g) dg (4)

where ND and NH are the number of payload and header bitsin a packet, respectively, NC is the number of control bits in aframe, and K ¼ ND=(b(ND þ NH)þ NC).

3.2 Adaptive transmission policy designunder target PER constraint

We now maximise the throughput performance as presentedin (4) subject to a prescribed PER and an averagetransmission power constraint. Consider an instantaneousPER constraint, that is, PERn(g) ¼ IPER � Ptgt, gn �

g � gnþ1; 8n ¼ 1, . . . , N , where Ptgt and IPER denote thetarget PER and the achievable instantaneous PER,respectively. From (1) and the assumed PER constraint,power adaptation in mode n is given by

Sn(g)�S¼

1

gngln

an

IPER

� , Gn � gn � g � gnþ1 (5)

Using (5), the optimisation problem of interest is

min{gn}N

n¼1,IPER�T (IPER) s. t.

C1 :XN

n¼1

1

gn

lnan

IPER

� :

ðgnþ1

gn

1

gpg(g) dg � 1

C2 : IPER � Ptgt

C3:(Nþ2) : gn � Gn, n ¼ 1, 2, . . . , N

(6)

In (6), C1 is the average power constraint over AMC modesin the form of E{Sn(g)} � �S; C2 is the instantaneous PERconstraint; and the last N conditions ensure that for eachmode n, the power is allocated based on (5). To solve (6),we first assume equality in C2, and search for an optimalsolution. Applying the Karush–Kuhn–Tucker (KKT)conditions [18] to determine the optimal solution, weconstruct the Lagrangian of (6) as follows

L(g1, . . . ,gN , l, b1, . . . , bN ) ¼ �K (1� Ptgt)

�XN

n¼1

cn

ðgnþ1

gn

pg(g) dgþ lXN

n¼1

dn:

ðgnþ1

gn

1

gpg(g) dg� 1

!

þXN

n¼1

bn(gn � Gn) (7)

where (b1, . . . , bN ), l are the Lagrange multipliers, anddn ¼ 1=gn ln(an=Ptgt). Based on the KKT conditions, the

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optimal solution (g�1, g�2, . . . , g�N ) and the correspondingLagrange multipliers (b�1, . . . , b�N ) and l�, must satisfy thefollowing conditions

@L

@gn

(g�1, . . . , g�N , l�, b�1, . . . , b�N )¼ 0, n¼ 1, 2, . . . , N (8)

XN

n¼1

dn �

ðg�nþ1

g�n

1

gpg(g) dg� 1

l� � 0 (9)

g�n � Gn, b�n � 0, n¼ 1, 2, . . . , N

b�n(g�n �Gn)¼ 0, n¼ 1, 2, . . . , N

With (8), we have

@L

@gn

g�1, g�2, . . . , g�N , l�, b�1, b�2, . . . , b�N� �

¼

K (1� Ptgt)c1pg(g�1)� l�d1

pg(g�1)

g�1þb�1 ¼ 0, n¼ 1

�K (1� Ptgt)cn�1pg(g�n)þK (1� Ptgt)cnpg(g�n)

þl�dn�1

pg(g�n)

g�n� l�dn

pg(g�n)

g�nþb�n ¼ 0, n� 2

8>>>>><>>>>>:

(10)

Using the complementary slackness conditions [18] in (9), ifg�n . Gn, then b�n ¼ 0. Following (10), if g�n . Gn we obtain

g�1 ¼d1

c1

m�

g�n ¼dn � dn�1

cn � cn�1

m�, n¼ 2, . . . , N

(11)

where m� ¼ l�=K (1� Ptgt)� 0 is a constant. As a result, thegeneral form of the optimal mode switching levels can bewritten as

g1 ¼maxd1

c1

m, G1

� �

gn ¼maxdn � dn�1

cn � cn�1

m, Gn

� �, n¼ 2, . . . , N

(12)

To find the constant m in (12), we use a numerical searchbased on a bisection root-finding method [19], such thatthe switching levels in (12) satisfy the power constraint C1

in (6) with equality.

As our experiments show, for a large value of Ptgt, or a verysmall value of the average channel SNR, the power constraintin (6) may not be met with equality, that is, a solution for m isnot available. In fact, in such cases the use of full averagetransmit power to improve the throughput by increasingthe summation term in (4) is not possible and, hence, theremaining transmit power is used to reduce the IPERvalue, which in turn further increases the throughput. The

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solution in this case is simple; we first set g�n ¼ Gn, 8n, andthen obtain the achievable IPER� ,Ptgt value, by findingthe root of the following equation

XN

n¼1

1

gn

lnan

IPER*

� :

ðGnþ1

Gn

1

gpg(g) dg ¼ 1 (13)

Obviously, for such cases a fixed throughput performance isobtained for any required target PER above the thresholdIPER�, and the algorithm leads to an achievable PERsmaller than Ptgt. We refer to the adaptation policy in thissection as AP-IPER. Note that for the practical range ofPtgt and the selected AMC modes, we have dn . dn21, and(dn 2 dn21)/(cn 2 cn21) . (dn21 2 dn22)/(cn21 2 cn22). Asa result, the sequence of fg�ng in (11) is increasing in n, thatis, we have g�1 , g�2 , � � � , g�N , where m� is a positivevalue.

The following theorem states a sufficient condition for theoptimality of the proposed solution in (12).

Theorem 1: For a Rayleigh fading channel model,the proposed solution in (12) is an extreme point of thefeasible set of problem (6), and globally minimises theobjective function in (6).

Proof: The proof is provided in Appendix 1. A

3.3 Adaptive transmission policy designunder target BER constraint

In [10, 13], the PER for an AMC transmission mode, isobtained based on the BER, assuming that the channelbit-errors in a packet are uniformly distributed and areindependent. In this case, if the transmission policy isrequired to provide a target PER (Ptgt) constraint, one mayimpose a target BER (BERtgt) constraint on the design oftransmission policy obtained through the followingequation [13]

BERtgt ¼ 1� (1� Ptgt)1=NB (14)

where NB is the number of bits in a packet. However, thestated underlying assumptions are not longer valid forcoded M-QAM-based transmission and, hence, thisapproximation leads to a large overestimation of the PER,as demonstrated in [11, 20].

Aiming at a performance comparison, here we also presenta link adaptation design which maximises the packetthroughput in (4) subject to a target PER constraint in anindirect manner. We first use (14) to obtain a targetBERtgt from the target PER, Ptgt. We then construct thedesired optimisation problem subject to a target BERconstraint BERtgt. As in the case of PER modelling, weapproximate the BER performance of the AMC mode n


over the AWGN channel using the following expression

BERn(g) ¼ en exp �qn

Sn(g)�S

g

� �(15)

The parameters {en, qn} are AMC mode dependentconstants, and can be obtained by least square fitting theexpression (15) to the exact BER. In Section 5, we verifythe accuracy of this BER model.

Consider the instantaneous BER constraint, BERn(g) ¼BERtgt, gn � g � gnþ1, 8n. From (15) and the BERconstraint, the required power adaptation in mode n isobtained as

Sn(g)�S¼

1

qngln

en

BERtgt

!, gn � g � gnþ1 (16)

Using (16), the desired optimisation problem is as follows

min{gn�0}N

n¼1,�T (BERtgt) s. t.

C1 :XN

n¼1

1

qn

lnen

BERtgt

!:

ðgnþ1

gn

1

gpg(g) dg � 1

(17)

where C1 condition is the average power constraint. Theproblem in (17) can be considered as a special case of theoptimisation problem (6) with IPER! BERtgt, an ! en,gn ! qn, and G1 ¼ G2 ¼ � � � ¼ GN ¼ 0. Therefore the

optimal solution for (17) can be written as

g�1 ¼ln (e1=BERtgt)

q1c1

l�

g�n ¼qn�1 ln (en=BERtgt)� qn ln (en�1=BERtgt)

qnqn�1(cn � cn�1)l�,

n ¼ 2, . . . , N

(18)

where the constant l� . 0 is found such that the modeswitching levels in (18) satisfy the power constraint C1 in(17) with equality. We refer to this policy as AP-IBER.

4 Joint truncated SR ARQ-AMCfor delay-constrained trafficThe analysis proposed in the previous section is adequate forwireless applications with no delay constraint. To providedelay guarantees for real-time traffic, here, we impose amaximum number of retransmissions per packet (Nr) forthe ARQ protocol. The number Nr is obtained as themaximum packet delay that can be tolerated by theapplication divided by the round-trip time of the wirelesslink. Since the proposed optimisation framework is set upat the packet level, it can be used in a cross-layer approach,to maximise the average throughput subject to a prescribed


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PLR constraint at the data-link layer, while satisfyingthe packet delay constraint. The key assumptions used inthe analysis are as follows:

i. As a consequence of a block-fading channel model, thechannel packet error process is i.i.d. on the multipletransmissions of one packet because of ARQ [5, 21].

ii. If the receiver detects a packet in error after Nr

retransmissions, the packet is considered lost. We refer tothe rate (percentage) of such packet losses as PLR.

According to (ii), a finite Nr leads to potential packet lossand, hence, we consider a prescribed target PLR at the data-link layer, denoted by Ploss. Using an AP-IPER transmissionpolicy, a fixed target PER can be guaranteed over thefading channel. Therefore according to (i), Ploss can berelated to the target PER at the physical layer, denoted byPtgt, Nr

, as follows

Ploss ¼ PNrþ1tgt,Nr

(19)

4.1 Throughput optimisation for jointtruncated SR ARQ-AMC scheme

As previously stated a limited number of retransmissions maylead to packet loss. Therefore it is necessary to consider theeffect of such losses in the calculation of the throughputperformance. The following proposition provides the firststep for throughput calculation in this case.

Proposition 1: For a joint truncated SR ARQ-AMCscheme which guarantees a target PLR Ploss, the averagenumber of correctly received packets per frame over theblock-fading channel is as follows

�h(Nr) ¼ 1� Ptgt,Nr

� �XN

n¼1

cn �

ðgnþ1

gn

pg(g) dg (20)

where the target PER Ptgt,Nris related to Ploss through (19),

that is, Ptgt,Nr¼ P

1=(Nrþ1)loss .

Proof: The proof is provided in Appendix 2. A

Obviously, for Nr ¼ 1, a zero target PLR (Ploss ¼ 0) isachieved and the expression (20) reduces to (3) withPtgt,1 ¼ P0. Using (20), the average normalised throughputis expressed as

T (Nr, Ptgt,Nr) ¼

ND:h(Nr)

b(ND þ NH)þ NC

¼ K (1� Ptgt,Nr)XN

n¼1

cn:

ðgnþ1

gn

pg(g) dg (21)


where K is defined in Section 3.1. To maximise the averagethroughput in (21) subject to a prescribed PLR constraintPloss, we consider a target PER constraint Ptgt,Nr

for thetransmission policy design. Given fixed Nr and Ploss, thedesired optimisation problem is then formulated as

min{gn}N

n¼1,IPER�T Nr, IPER

� �s. t.

C1 :XN

n¼1

1

gn

lnan

IPER

� :

ðgnþ1

gn

1

gpg(g) dg � 1

C2 : IPER � Ptgt,Nr

C3 : (Nþ2) : gn � Gn, n ¼ 1, 2, . . . , N

(22)

The optimisation problem in (22) has the same structure asthe optimisation problem in (6). Therefore the sameprocedure proposed in Section 3 can be used to solve (22).Also, the optimality of the derived solution can beestablished as presented in Theorem 1.

5 Numerical resultsIn this section, we present numerical results for the throughputand PLR performance of the proposed link adaptation schemes.We also verify the BER and PER exponential error models.For numerical results, we use the seven modes of the AMCscheme in the IEEE802.11a standard [1]. Each mode isdescribed by a convolutionally coded Mn-QAM scheme.Table 1 presents these AMC transmission modes and theircorresponding BER and PER fitting parameters according to(1) and (15), respectively. The PER fitting parameters inTable 1 are obtained for the packet length NB ¼ 1080 bits.In the calculation of throughput performance, the effects ofheader and control bits are ignored, that is, we simply setK ¼ 1/b in (4) and (21). Although our derivations arefor general fading distributions, in the numerical results, aRayleigh fading channel model is used, that is, pg(g) ¼1=g exp(�g=g), g � 0, where g is the average received SNR.In the following experiments, a default value g ¼ 12 dB is used.

5.1 Methods for comparison: constant-power AMC designs

In the previous studies, constant-power AMC schemes arewidely used, for example, in [6–13]. A design, whichdirectly satisfies a target IPER constraint PERn(g) � Ptgt,is proposed in [11]. We refer to this policy as CP-IPER.Also, a design that satisfies a target BER constraintBERn(g) � BERtgt is presented in [13], which is referredto as the CP-IBER policy in this paper.

5.2 Validation of error models

To validate the PER and BER models in (1) and (15), theerror performance of the AMC modes in Table 1 issimulated over an AWGN channel. Because of limitedchannel bandwidth, in simulations Nyquist data pulses with

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Table 1 Transmission modes for AMC scheme and their corresponding fitting parameters for PER and BER modelling

Mode (n) 1 2 3 4 5 6 7

modulation BPSK BPSK QPSK QPSK 16-QAM 16-QAM 64-QAM

coding rate 1/2 3/4 1/2 3/4 1/2 3/4 2/3

rate: Rn bits/sym. 0.5 0.75 1 1.5 2 3 4

an 1042.107 311.0644 706.9783 264.0134 264.5420 171.0575 228.3775

gn 11.5300 4.9410 2.8090 1.2220 0.6165 0.2500 0.09596

Gn, dB 22.1991 0.6510 3.6842 6.5925 9.5655 13.1319 17.5279

en 22.9886 9.2258 21.0521 9.8552 4.9234 5.3228 3.5895

qn 13.1500 6.0920 3.3110 1.5420 0.7027 0.3162 0.1105

zero ISI are used. With a frame length of 20 ms and b ¼ 2,the lowest and highest system rates corresponding to mode1 and mode 7 of the AMC scheme are 108 and 864 kb/s,respectively.

In Fig. 2, we depict the BER performance of the AMCtransmission modes and their corresponding fitting curvesbased on (15). We also plot in Fig. 3 the PERperformance of the AMC modes, fitting curves based on(1), and PER derived from the BER based on (14). Figs. 2and 3 show that the BER and PER error models in (15)and (1) well approximate the exact BER and PER,respectively. Fig. 3 also illustrates that the PER derivedfrom the BER leads to an overestimated PER.

5.3 Throughput performance of joint SRARQ-AMC scheme (Nr ¼ 1)

Fig. 4 depicts the average normalised throughput against thetarget PER for different adaptation policies. Note that for

Figure 2 Bit error rate for AMC transmission modes: starsdenote (simulated) exact BER, and solid lines are fittingcurves to the exact BER


constant-power transmission policies the instantaneous PERdepends on the channel SNR and the AMC mode, and isonly guaranteed to be smaller than the target PER. Thereforein Fig. 4, the results for constant-power schemes are obtainedby setting P0 in equation (4) to the average PER achieved bythese schemes, similar to [8]. Note that at the end of thethroughput curve for the AP-IPER policy in Fig. 4, wherethe target PER is large, the optimal solution is obtainedbased on (13), that is, a fixed throughput is achieved (seeSection 3.2 for more details). In this figure, we observe thefollowing: (i) The proposed AP-IPER policy considerablyoutperforms the other adaptation schemes. In particular, atPtgt ¼ 0.03, which is near the peak of the throughput curvefor the AP-IPER policy, this policy yields about 19 and 25%throughput gain compared with the CP-IPER and CP-IBER policies, respectively. (ii) The AP-IPER policy whencompared with the AP-IBER yields a better throughputperformance, especially for large values of Ptgt. This is because

Figure 3 PER for AMC transmission modes: stars denote(simulated) exact PER, solid lines are fitting curves to theexact PER and dashed lines show PER obtained from BERfitted curves in Fig. 2


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of the fact that using (14) to derive the target BER from thetarget PER (Ptgt) leads to a large underestimation of theBER, especially for Ptgt . 0.001. This in turn imposes morerestriction on the adaptation policy, limiting the throughputenhancement in problem (17). (iii) We note that there is anoptimal target PER that yields the highest throughputperformance over the fading channel. This value can be foundusing an appropriate search algorithm. (iv) The peak of theAP-IPER throughput curve occurs at a lower target PER,compared with the CP-IPER policy (Ptgt ¼ 0.03 against0.3). This in turn requires a smaller buffer size and also leadsto a smaller queuing delay for the packet traffic.

To show the effect of the channel SNR, in Fig. 5 we depictthe normalised throughput against the average received SNRfor different adaptation policies at Ptgt ¼ 0.03. It is evidentthat the AP-IPER policy outperforms other adaptationpolicies over a large range of channel SNR. However, this

Figure 4 Average normalised throughput against targetPER for different adaptation policies

Figure 5 Average normalised throughput against averageSNR for different adaptation policies, Ptgt ¼ 0.03

Commun., 2009, Vol. 3, Iss. 6, pp. 1030–104010.1049/iet-com.2008.0305

is not the case for small SNRs (below 3 dB). This isbecause the solution for the AP-IPER policy in this SNRregion is determined based on (13).

5.4 Throughput performance of jointtruncated SR ARQ-AMC scheme

In Figs. 6 and 7, we plot the average throughput, and theachievable PLR against the target PLR for differentnumber of retransmissions using the AP-IPER policy,respectively. As observed, for each value of Nr, when thetarget PLR is larger than a threshold level, the solution ofproblem (6) is obtained based on (13) and a fixed PLR isachieved, for example, Ploss ¼ 4 � 1023 for Nr ¼ 1. Thisissue is further elaborated in Section 3.2. However, inpractice, a system design around the peak of thethroughput curve in Fig. 6 is desirable. Another twoimportant observations from Fig. 6 are as follows: (i) there

Figure 6 Average normalised throughput against the targetPLR for different Nr

Figure 7 Achievable PLR against the target PLR for differentNr

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is a fundamental trade-off between the delay and PLR QoSmetrics determined based on the peak of the throughputcurves in Fig. 6, that is, if we accept a larger delay (largerNr), the maximum throughput occurs at a lower targetPLR; (ii) according to IEEE 802.11a standard, a PLR of1–10% indicates a reasonable point of operation for packetservices without the delay constraint, when the packetlength is 1500 bytes [1]. However, based on (14), for apacket length of 1080 bits, this translates to a PLRrequirement of about 0.09–0.94%. Now, assume that aspecific application can tolerate a packet delaycorresponding to Nr , 4. For such an application, ourresults in Fig. 6 indicate that Nr ¼ 1 captures thethroughput gain and no more retransmissions are required.

To compare the results, in Fig. 8, we depict the throughputperformance against the target PLR, for the CP-IPERpolicy. As evident, the AP-IPER policy outperforms theCP-IPER policy for any pair of delay and PLR QoSconstraints. An important observation is that, when thesystem throughput reaches its maximum level, the AP-IPER policy exhibits a stronger delay-PLR trade-offcompared with the CP-IPER scheme. In fact, in contrastto AP-IPER, for CP-IPER and the PLR range of 0.09–0.94%, accepting a larger packet delay (larger Nr) leads toimproved throughput performance.

Comparing the results in Figs. 6 and 8 indicates that theAP-IPER is a superior transmission policy, achieving animproved throughput while satisfying the PLR QoSconstraint.

6 ConclusionsWe proposed an optimisation framework for wireless linkadaptation, which maximises the throughput performanceof an SR ARQ protocol at the data-link layer, using

Figure 8 Average normalised throughput against the targetPLR for different number of Nr

Results are obtained using CP-IPER policy


adaptive transmission at the physical layer. Specifically, forcoded M-QAM schemes, we derived rate and poweradaptation policies, which maximise the average throughputsubject to a target PER constraint. We also consider theapplication of the proposed adaptation schemes for real-time wireless services, where a limit is imposed on thenumber of ARQ retransmissions. We optimised a linkadaptation scheme that guarantees a target PLR and apacket-delay constraint in the presence of a truncated SRARQ protocol. The results demonstrate the superiorperformance of the proposed link adaptation schemes,achieving enhanced throughput performance under QoSconstraints.

7 AcknowledgmentsThe authors would like to thank Prof R. Faraji-Dana forhelpful discussions on the optimality analysis in Section3. This work is supported in part by IranTelecommunication Research Center (ITRC) undercontract T/500/9235.

8 Rererences

[1] DOUFEXI A., ARMOUR S., BUTLER M., ET AL.: ‘A comparison ofthe HIPERLAN/2 & IEEE 802.11a wireless LAN standards’,IEEE Commun. Mag., 2002, 5, pp. 172–180

[2] PENG F., ZHANG J., RYAN W.E.: ‘Adaptive modulation andcoding for IEEE 802.11n’. Proc. IEEE WirelessCommunications and Networking Conf., Hong Kong,March 2007, pp. 656–661

[3] ‘Mobile WiMAX: ‘Part II a comparative analysis’.WiMAX Forum, May 2006

[4] ALOUINI M.S., GOLDSMITH A.J.: ‘Adaptive modulation overNakagami fading channels’, J. Wirel. Commun., 2000, 13,(1–2), pp. 119–143

[5] MALKAMAKI E., LEIB H.: ‘Performance of truncated type-IIhybrid ARQ schemes with noisy feedback over blockfading channels’, IEEE Trans. Commun., 2000, 48, (9),pp. 1477–1487

[6] YUN J., JEONG W., KAVEHRAD M.: ‘Throughput analysis ofselective repeat ARQ combined with adaptive modulationfor fading channels’. Proc. IEEE Conf. MILCOM, Anaheim,USA, October 2002, pp. 710–714

[7] YUN J., KAVEHRAD M.: ‘Markov error structure forthroughput analysis of adaptive modulation systemscombined with ARQ over correlated fading channels’, IEEETrans. Veh. Technol., 2005, 54, (1), pp. 235–245

[8] LE L.B., HOSSAIN E., ALFA A.S.: ‘Delay statistics andthroughput performance for multi-rate wireless networks


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under multiuser diversity’, IEEE Trans. Wirel. Commun.,2006, 5, (11), pp. 3418–3428

[9] FALAHATI S., SVENSSON A.: ‘Hybrid type-II ARQ schemeswith adaptive modulation systems for wireless channels’.Proc. IEEE Conf. VTC, Amsterdam, Netherlands, September1999, pp. 2691–2695

[10] WU D., CI S.: ‘Cross-layer combination of hybrid ARQ andadaptive modulation and coding for QoS provisioning inwireless data networks’. Proc. IEEE/ACM QShine,Waterloo, Canada, August 2006

[11] LIU Q., ZHOU S., GIANNAKIS G.B.: ‘Cross-layer combining ofadaptive modulation and coding with truncated ARQ overwireless links’, IEEE Trans. Wirel. Commun., 2004, 3, (5),pp. 1746–1755

[12] HARSINI J.S., LAHOUTI F.: ‘Joint optimization of adaptivemodulation and coding, power control and packetretransmission over block fading channels’. Proc. IEEEITW, Bergen, Norway, July 2007, pp. 67–71

[13] WU D., CI S., WANG H.: ‘Cross-layer optimization for videosummary transmission over wireless networks’, IEEE J. Sel.Areas Commun., 2007, 25, (4), pp. 841–850

[14] ZHENG H., VISWANATHAN H.: ‘Optimizing the ARQperformance in downlink packet data systems withscheduling’, IEEE Trans. Wirel. Commun., 2005, 4, (2),pp. 495–506

[15] CHASKAR H.M., LAKSHMAN T., MADHOW U.: ‘TCP over wirelesswith link level error control: analysis and designmethodology’, IEEE/ACM Trans. Netw., 1999, 7, (5),pp. 605–615

[16] OZAROW L.H., SHAMAI S., WYNER A.D.: ‘Information theoreticconsiderations for cellular mobile radio’, IEEE Trans. Veh.Technol., 1994, 43, (2), pp. 359–378

[17] SONG K.B., MUJTABA S.A.: ‘On the code-diversityperformance of bit-interleaved coded OFDM in frequency-selective fading channels’. Proc. IEEE Conf. VTC, Orlando,USA, October 2003, pp. 572–576

[18] BOYD S., VANDENBERGHE L.: ‘Convex optimization’(Cambridge University Press, UK, 2004)

[19] PRESS W.H., TEUKOLSKY S.A., VETTERLING W.T., FLANNERY B.P.:‘Numerical recipes in C’ (Cambridge University Press, UK,1997, 2nd edn.)

[20] KHALILI R., SALAMATIAN K.: ‘A new analytic approachto evaluation of packet error rate in wireless networks’.Proc. Int. Conf. Communication Networks andServices Research, Nova Scotia, Canada, May 2005,pp. 333–338


[21] ROSBERG Z., SIDI M.: ‘Selective-repeat ARQ: the jointdistribution of the transmitter and the receiverresequencing buffer occupancies’, IEEE Trans. Commun.,1990, 38, (9), pp. 1430–1438

[22] LUENBERGER D.G.: ‘Linear and nonlinear programming’(Addison-Wesley, Massachusetts, 1984, 2nd edn.)

9 Appendix 1Before presenting a proof for theorem 1, we introduce thefollowing lemma. Let f1 g1, . . . , gN

� �denote the objective

function in (6). Instead of minimising the function f1 inproblem (6), consider a new problem that maximises thefunction f2 ¼ �f1 g1, . . . , gN

� �over constraints of problem (6).

Lemma 1: For a Rayleigh fading distribution, that is,pg(g) ¼ 1=g exp(�g=g), both the function f2 andconstraint functions in (6) are convex functions of{gi . 0}N

i¼1.

Proof of Lemma 1: First, consider the f2 function. Since@2f2=@gi@gj ¼ 0; i, j ¼ 1, . . . , N, i = j, for the function f2to be convex, it is sufficient to have @2f2=@g

2i � 0 [18].

We have

@2f2@g2

i

¼ �K (1� Ptgt) ci � ci�1

� �@pg gð Þ

@g

g¼gi

¼K (1� Ptgt)

g2ci � ci�1

� �exp �gi=g� �

(23)

Since ci . ci�1 and Ptgt , 1, we have @2f2=@g2i � 0, 8i.

Now, consider the C1 constraint in (6), denotedby h(g1, . . . , gN ) ¼

PNn¼1 dn:

Ð gnþ1

gn1=gð Þpg(g) dg� 1 � 0,

where dn ¼ 1=gn ln(an=Ptgt). It is easy to checkthat @2 h=@gi@gj ¼ 0, i = j, and

@2 h

@g2i

¼di � di�1

� �g2

i

pg gi

� ��

di � di�1

� �gi

@pg gð Þ

@g

g¼gi

(24)

Since di . di�1 (this is a property of AMC modes), the firstterm in the RHS of (24) is positive. Also,

@pg gð Þ=@gjg¼gi¼ �1= �g2 exp(�gi= �g) � 0, and therefore the

second term in the RHS of (24) is also positive;consequently, @2 h=@g2

i � 0 and, hence, h is convex. Wefinally note that because of linearity, the last N constraintsin (6) are convex. A

We denote the maximum channel SNR by Gmax from apractical perspective and define it such that Pr g � Gmax

� �is

negligible. Therefore we assume that the feasible region forthe solution of (6) is a bounded set denoted by V, whichincludes all solutions in the form g1, . . . , gN

� �.

Corollary: 1: According to Lemma 1, the function f2 andconstraint functions in (6) are convex functions of {gi}

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variables over the feasible set V. Therefore the set V is aconvex set [18, Section 4.2.1].

Proof of theorem 1: From corollary 1, we know that thefeasible set V is convex. Therefore we are dealing withmaximising a convex function ( f2) over a feasible convexset V, which is shaped based on the constraints of problem(6). It is known that for a convex function defined on abounded, closed convex set, if the function has a (global)maximum over the set, then it is achieved at an extremepoint of this set [22, pp.181, Theorem 3].

Note that f2 is a finite convex function and@f2=@gi ¼ �K (1� Ptgt) ci � ci�1

� �pg gi

� �, 0, 8i. Therefore

we immediately infer that f2 is a convex monotonicdecreasing function over the set V, which has a maximumover this set. Based on this fact and theorem 3 from [22],we infer that the proposed solution in (12) specifies anextreme point of the set V, which globally maximises thefunction f2. A

10 Appendix 2Here, we obtain an expression for the throughputperformance of a joint Nr-truncated ARQ-AMC scheme,when a target PLR Ploss is provisioned. To simplify thenotation, we denote the target PER at the physical layer byp0 ¼ Ptgt,Nr

¼ P1=(Nrþ1)loss .

Consider the transmission of T frames of data, whereT� 1. We denote the number of packets that aretransmitted within these frames by Np(T ). If T issufficiently large, this number will be (almost) equal totheir average, that is

limT!1

Np(T ) ¼ T � En Np(1)n o

¼ T :X

n

cn Pr(gn � g , gnþ1) (25)

For an Nr-truncated ARQ system, the limit in (25) includesboth successfully received packets and dropped (discarded)packets because of a limited number of retransmissions.

Assume that there are a total of A packets in thetransmitter queue to be transmitted during T frames. We

0he Institution of Engineering and Technology 2009

denote the number of successfully received packets duringthese T frames by a, and we assume that there are �apackets that are ultimately not correctly received (discarded)in this time, following all possible retransmissions. We have

a ¼ (1� pNrþ10 ) � A (26)

�a ¼ pNrþ10 � A ¼

pNrþ10

1� pNrþ10

a (27)

a � �nts þ �a � (Nr þ 1) ¼ Np(T ) ¼ T :X

n

cn Pr(gn � g , gnþ1)

(28)

where �nts denotes the average number of transmissions for asuccessfully received packet. To obtain �nts, we define therandom variable nts as the number of transmissions for asuccessfully received packet. This random variable has thefollowing normalised probability mass function

Pr(nts ¼ i) ¼(1� p0)pi�1

0

1� pNrþ10

, i ¼ 1, 2, . . . , Nr þ 1 (29)

The average number of transmitted packets for successfulreception of one packet can be computed as follows

�nts ¼ E{nts}¼1

1� pNrþ10

� 1(1� p0)þ 2(1� p0)p0þ � � � þ (Nrþ 1)(1� p0)pNr0

h i

¼1þ (Nr þ 1)p

Nrþ20 � (Nr þ 2)p

Nrþ10

(1� pNrþ10 )(1� p0)

(30)

Substituting (27) and (30) into (28), and solving for a, weobtain

a ¼ (1� p0) �T �X

n

cnPr(gn � g , gnþ1) (31)

which leads to the following equation for the average numberof correctly received packets in each frame

�h(Nr)¼ limT!1

a

T¼ (1� Ptgt,Nr

)X

n

cn:

ðgnþ1

gn

pg(g) dg (32)


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