On the Performance of the Relay-ARQ Networks

# On the Performance of the Relay-ARQ Networks

Behrooz Makki, Thomas Eriksson, Tommy Svensson, Senior Member, IEEE
The authors are with Department of Signals and Systems, Chalmers University of Technology, Gothenburg, Sweden, Email: {behrooz.makki, thomase, tommy.svensson}@chalmers.seThis work was supported in part by the Swedish Governmental Agency for Innovation Systems (VINNOVA) within the VINN Excellence Center Chase.
###### Abstract

This paper investigates the performance of relay networks in the presence of hybrid automatic repeat request (ARQ) feedback and adaptive power allocation. The throughput and the outage probability of different hybrid ARQ protocols are studied for independent and spatially-correlated fading channels. The results are obtained for the cases where there is a sum power constraint on the source and the relay or when each of the source and the relay are power-limited individually. With adaptive power allocation, the results demonstrate the efficiency of relay-ARQ techniques in different conditions.

## I Introduction

Relay-assisted communication is one of the promising techniques that have been proposed for the wireless networks. The main idea of a relay network is to improve the data transmission efficiency by implementation of intermediate relay nodes which support the data transmission from a source to a destination. The relay networks have been adopted in the 3GPP long-term evolution advanced (LTE-A) standardization [1] and are expected to be one of the core technologies for the next generation cellular systems.

From another perspective, hybrid automatic repeat request (ARQ) is a well-established approach for wireless networks [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. The ARQ systems can be viewed as channels with sequential feedback where, utilizing both forward error correction and error detection, the system performance is improved by retransmitting data that has experienced bad channel conditions. Thus, the combination of relay and ARQ improves the performance of wireless systems, because the ARQ makes it possible to use the relay only when it is needed.

Due to the fast growth of wireless networks and data-intensive applications in smart phones, green communication via improving the power efficiency is becoming increasingly important for wireless communication. The network data volume is expected to increase by a factor of every year, associated with increase of energy consumption, which contributes about of global emissions [14]. Hence, from an environmental point of view, minimizing the power consumption is a very important design consideration, and green data transmission schemes must be taken into account for the wireless networks [15, 16, 17, 18, 19, 20]. Moreover, as most wireless devices operate with limited battery power, it is very important to find ways of maximizing the device lifetime by efficiently utilizing the limited power. These are the main motivations for this paper, in which we analyze the power-limited performance of the relay-ARQ setups.

The basic principles of different ARQ protocols are derived in [2, 3, 4, 5, 6, 7, 8]. Power allocation in ARQ-based single-user (without relay) networks is addressed by, e.g., [9, 10, 11, 12, 13]. Also, [21, 22, 23, 24, 25, 26, 27, 28] study the problem in relay networks. There are a number of papers dealing with energy efficiency and power allocation in relay-ARQ setups. These works can be divided into two categories, as stated in the following.

In [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41], the source and the relay use, e.g., space-time codes (STCs) to make a distributed cooperative antenna and retransmit the data simultaneously in rounds when the relay is active; With an outage probability constraint, [29, 30] (resp. [31]) study the energy efficiency (resp. long-term average transmission rate) of STC-based relay-ARQ systems. The energy and spectrum efficiency of the basic and hybrid relay-ARQ networks are verified in [32, 33, 34] as well. Also, [35] designs a multi-relay-ARQ network using Alamouti codes. Assuming the source and the relay to be close, [36, 37] investigate the throughput of relay networks using different ARQ protocols. Optimizing the delay-limited throughput and deriving a closed-form expression for the average power of the source are addressed by [38] and [39], respectively. Considering the incremental redundancy (INR) protocol, [40] studies the performance of the relay-ARQ setups in fast-fading conditions. Finally, the results of [40] are extended in [41], where the system performance is compared with cases having only one of the source or the relay active in the retransmissions. Implementation of STCs in these works is based on the assumption that there is perfect synchronization between the source and the relay.

In [42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52], only one terminal (either the source or the relay) is active in the retransmission rounds, as opposed to [29, 30, 31, 32, 33, 34, 35, 36, 37, 39, 38]. For instance, [42] studies the outage-limited energy minimization in single-user and relay-ARQ networks. Opportunistic relaying, rate adaptation and analyzing the energy-delay tradeoff curve are considered by [43], [44] and [45, 46], respectively, where the direct source-destination link is ignored. Also, the throughput, the packet error rate and the effective capacity of different ARQ-assisted relay networks are studied in [47, 48, 49], respectively. Power scaling in MIMO and cognitive radio relay-ARQ networks is addressed in [50] and [51], respectively. Finally, [52] studies a relay-ARQ network using superposition coding. References [40, 41, 51, 33, 34, 52, 29, 30, 45, 46, 31, 32, 35, 36, 50, 37, 38, 43, 44, 47, 48, 49] are based on the assumption that there is a fixed transmission power for the source and the relay. Meanwhile [31, 24, 45, 46] optimize the power allocation between the source and the relay under a sum power constraint, while they use the same powers in all retransmissions. Also, [42] investigates the power allocation between the retransmissions for basic ARQ schemes and [39] studies the average power of the source with repetition time diversity (RTD) ARQ and a fixed power for the relay.

In theoretical investigations, the communication links between the source, the relay and the destination are normally assumed to be independent [42, 51, 33, 41, 40, 34, 52, 29, 30, 45, 46, 31, 32, 35, 36, 50, 37, 39, 38, 43, 44, 47, 48, 49, 53]. This is an appropriate model for many practical scenarios [42, 51, 33, 40, 41, 34, 52, 29, 30, 45, 46, 31, 32, 35, 36, 50, 37, 39, 38, 43, 44, 47, 48, 49, 53] and makes it possible to analyze the system performance analytically. However, the independent fading channel is not always a realistic model. For instance, the relay is normally located close to the destination in moving-relay systems [54, 55]. As a result, there might be considerable correlation between the source-relay and the source-destination fading coefficients. Also, e.g., [52] demonstrates the cases where the source is connected to the destination through a relay which is close to the source. In this case, the source-destination and the relay-destination links may be spatially-correlated. For these reasons, it is interesting to extend the independent fading model to the case where there is spatial correlation between the channels.

In this paper, we study the throughput and the outage probability of the relay-ARQ networks in cases where there is either a long-run sum power constraint on the source and the relay or when each of the source and the relay are power-limited individually. Adaptive power allocation between the retransmissions is used to improve the system performance. We derive closed-form expressions for the average power, the throughput and the outage probability of different relay-ARQ protocols in the cases with independent or spatially-correlated fading channels. Moreover, we investigate the effect of fading temporal variations on the data transmission efficiency of the relay-ARQ systems.

As opposed to [29, 30, 40, 31, 32, 33, 34, 35, 36, 37, 39, 38], we study the scenario where only one of the source or the relay is active in each ARQ-based retransmission round. Also, the problem setup of the paper is different from the ones in [29, 30, 31, 40, 41, 32, 33, 34, 35, 36, 37, 39, 38, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52] because 1) we consider adaptive power allocation between retransmissions of hybrid ARQ protocols, 2) the results are obtained with different sum and individual power constraints on the source and the relay and 3) we investigate the system performance in both independent and spatially-correlated fading conditions, with noisy/noise-free feedback signals. Finally, our discussions on the users’ message decoding probabilities (Theorems 1-3) have not been presented before.

The results show that there is a structural procedure to study different performance metrics of relay-ARQ networks experiencing different fading models. Optimal power allocation is shown to be very useful in terms of outage probability, throughput and coverage region of the relay-ARQ network, when there is a sum power constraint on the source and the relay. With individual power constraints on the source and the relay, however, optimal power allocation increases the throughput (resp. reduces the outage probability) only at low (resp. high) signal-to-noise ratios (SNRs). Compared to the fixed-length coding scheme, the throughput of the relay-ARQ network increases when variable-length coding is utilized. With the practical range of spatial correlations, the performance of the relay-ARQ network is not sensitive to the spatial correlation. However, the data transmission efficiency of the network is reduced at highly-correlated conditions.

## Ii System model

We consider a relay-assisted communication setup consisting of a source, a relay and a destination. The channel coefficients in the source-relay, the source-destination and the relay-destination links are denoted by and , respectively. Also, we define and which are referred to as the channel gains in the following. A maximum number of ARQ-based retransmission rounds is considered, i.e., the data is (re)transmitted a maximum of times. Moreover, we define a packet as the transmission of a codeword along with all its possible retransmission rounds. In each packet, information nats are sent to the destination and the length of the subcodeword used in the -th round of the ARQ is denoted by . Thus, the equivalent data rate, i.e., the code rate of the ARQ, at the end of the -th round is given by

We study the system performance for two different block-fading conditions:

• Quasi-static. In this model, the channel coefficients are assumed to remain fixed within a packet period, and then change to other values based on their probability density functions (pdf).

• Fast-fading. Here, the channel coefficients are supposed to change in each retransmission round.

The quasi-static model, studied in Subsections IV.A-B, represents the scenarios with slow-moving or stationary users, e.g., [12, 11, 56]. On the other hand, the fast-fading, studied in Subsection IV.C, is an appropriate model for the high speed users and frequency-hopping setups where the channel quality changes in the retransmissions independently, e.g., [41, 56, 13].

In each link, the channel coefficient is assumed to be known by the receiver, which is an acceptable assumption in block-fading channels [9, 10, 11, 12, 13]. However, there is no instantaneous channel state information available at the transmitters except the ARQ feedback bits. The ARQ feedback signals are initially assumed to be received error-free, but we later investigate the effect of erroneous feedback bits as well (Section IV).

Relay-ARQ model: The considered relay-ARQ protocol works as follows. In each packet period, the data transmission starts from the source. If the data is decoded by the destination, an acknowledgement (ACK) is fed back by the destination to the source and the relay, and the retransmissions stop. Otherwise, the destination transmits a negative-acknowledgment (NACK). Only one terminal (either the source or the relay) is active in each retransmission round; the relay becomes active and the source turns off, as soon as the data is decoded by the relay. That is, if the relay successfully decodes the message, it sends an ACK to the source and starts retransmission until the destination decodes the data or the maximum number of retransmissions is reached. In other words, with error-free feedback bits, the following cases may occur during a packet transmission period: 1) receiving an ACK from the relay and a NACK from the destination, the source turns off and the relay starts retransmission. 2) With NACKs from the relay and the destination, the data is retransmitted by the source. 3) Receiving an ACK from the destination, the source ignores the ARQ feedback of the relay and the retransmissions stop (Performance analysis in the cases with noisy feedback bits is studied in Section IV.D.).

The motivations for considering the proposed data transmission model are as follows. Letting the relay retransmit instead of the source when the source-destination link experiences bad condition makes it possible to exploit the potential diversity gain through the relay channel. Also, in practice, the relay is located such that the relay-destination link experiences better average characteristics than the source-destination link. Therefore, it is more beneficial to use the power resources for the relay, instead of dividing the power between the source and the relay, if the relay decodes the message. Finally, as seen in the following, for Rayleigh-fading conditions the proposed scheme outperforms the state-of-the-art approaches, in terms of outage probability/throughput.

## Iii Problem formulation

In this paper, we study the problem of

 max∀R(m),Psm,PrmΩ,Ω={η,−Pr(% Outage)}subjecttoΔ,Δ={Φ% total≤ϕtotal,(Φs≤ϕs)&(Φr≤ϕr)}. (1)

In words, we investigate the long-term throughput and the outage probability as the evaluation yardsticks. The optimization parameters are the equivalent data transmission rates as well as and which denote the source and the relay power used in the -th retransmission round, respectively (because the noise variances are set to 1, and , in dB and , represent the transmission SNR as well). Finally, the throughput and the outage probability are optimized under two different power-limited scenarios:

• Scenario 1. The total power for data transmission in the relay-ARQ setup is limited, which is represented by . Here, is the total power in the source and the relay, averaged over many packet transmissions, and denotes the total power constraint. This scenario is of interest in the green communication concept, where the goal is to minimize the total average power required for data transmission [15, 16, 17, 18, 19, 20], and also for electricity-bill minimization.

• Scenario 2. There are individual power constraints on the source and the relay, which is represented by in (1). Here, and are the average power in the source and the relay, respectively, and and denote their corresponding thresholds. This scenario models the case where the source and the relay are battery-limited [11, 9, 10, 13, 12].

To study (1), the following procedure is considered (please see Fig. 1 as well). First, we derive closed-form expressions for the functions , , , and which are involved in (1). Then, since (1) is a nonconvex problem, iterative optimization algorithms are used to optimize the parameters based on the closed-form expressions.

In three steps we obtain the closed-form expressions of , , , and . The first step is to define the metrics and the constraints as functions of a few expected values. Then, in the second step, we derive the expectations as functions of predefined probability terms. The last step is to represent the considered probabilities in terms of i.e., the optimization parameters of (1). Interestingly, the two first steps are independent of the considered ARQ protocol and the fading channel model. Thus, they are explained in the two following subsections. The third step, however, depends on the characteristics of the ARQ schemes and the fading channel model. For this reason, we specify the results for different ARQ protocols and fading channel models in Sections IV and V.

### Iii-a Step 1: Definitions

The outage probability is defined as the probability of the event that the data can not be decoded by the destination when the data (re)transmission is stopped. Also, the throughput (in nats per channel use (npcu)) is given by [4, 11, 12, 7]

 η=limK→∞∑Kk=1Qk∑Kk=1ttotalk=limK→∞1K∑Kk=1Qk1K∑Kk=1ttotalk(a)=E{Q}E{Ttotal}. (2)

Here, is the number of information nats successfully decoded by the destination in the -th packet transmission. Also, denotes the total number of channel uses in the -th packet transmission, i.e., if the message retransmission of the -th packet continues for rounds (see (8)). Note that in each packet part of the data may be (re)transmitted by the source or the relay and where and are the source and the relay activation periods in the -th packet transmission, respectively. In general, and are random values which follow the random variables and , respectively, as functions of the channel realizations. Also, in (2) is based on the law of large numbers, with representing the expectation operator, and the fact that the limits, e.g., , , exist [11, 12, 7].

With the same arguments, the average power terms , and are obtained by

 Φs=limK→∞∑Kk=1ξsk∑Kk=1tsk=E{Ξs}E{Ts}, (3)
 Φr=limK→∞∑Kk=1ξrk∑Kk=1trk=E{Ξr}E{Tr}, (4)
 Φtotal =limK→∞∑Kk=1ξtotalk∑Kk=1ttotalk=limK→∞∑Kk=1ξsk+∑Kk=1ξrk∑Kk=1ttotalk =E{Ξs}+E{Ξr}E{Ttotal}. (5)

Here, and are the source, the relay and the total transmission energy in the -th packet transmission, respectively, with . Also, and denote the random variables corresponding to and , respectively. Note that the metrics and constraints are functions of a few expected values.

### Iii-B Step 2: Deriving the Expected Values

Let us define the following events:

• is the event that the data is successfully decoded by the destination in the -th (re)transmission round while it was not decodable before. In this case, the codeword may have been sent by the source or relay.

• represents the event that the relay is active in rounds with In this case, the source message has been decoded by the relay in the -th round and, consequently, the source turns off in the successive retransmissions. The relay data retransmission is stopped in the -th round if the destination can decode the data or the maximum number of retransmissions is reached.

• is the event that the source stops data retransmission in the -th round. In this case, either the maximum number of retransmissions is reached or the data has been decoded by the relay or the destination.

The defined events are used to express (2)-(III-A) as functions of and The details are explained as follows.

According to the definitions, the outage probability is found as

 Pr(Outage)=1−M+1∑m=1Pr(Am). (6)

If the data is decoded by the destination at any (re)transmission round, all information nats of the packet are received. Hence, the expected number of received information nats in each packet is

 E{Q}=Q(1−Pr(Outage))=QM+1∑m=1Pr(Am). (7)

If the data is decoded at the end of the -th round, the total number of channel uses is . Also, the total number of channel uses is if an outage occurs, where all possible retransmission rounds are used. Thus, the expected number of total channel uses, i.e., in (2) and (III-A), is obtained by

 E{Ttotal}=M+1∑m=1(m∑i=1li)Pr(Am)+(M+1∑i=1li)Pr(% Outage). (8)

From (7)-(8) and , the throughput (2) is found as

 η=∑M+1m=1Pr(Am)∑M+1m=1Pr(Am)R(m)+1−∑M+1m=1Pr(Am)R(M+1). (9)

If the source stops data (re)transmission at the end of the -th round, the total energy consumed by the source is . Therefore, the source consumed energy is a random variable given by

 Ξs=m∑i=1Psili,if Sm,m=1,…,M+1, (10)

and, using we have

 E{Ξs} =M+1∑m=1((m∑i=1Psili)Pr(Sm)) =QM+1∑m=1((m∑i=1Psi(1R(i)−1R(i−1)))Pr(Sm)). (11)

With the same arguments, the expected activation period of the source, i.e., in (3), is found as

 E{Ts}=M+1∑m=1((m∑i=1li)Pr(Sm))=QM+1∑m=1Pr(Sm)R(m) (12)

which, along with (III-B), leads to

 Φs=∑M+1m=1((∑mi=1Psi(1R(i)−1R(i−1)))Pr(Sm))∑M+1m=1Pr(Sm)R(m). (13)

Given that the data is retransmitted by the relay in the rounds, its consumed energy is which is consumed during channel uses. Thus, we can use the definition of the same procedure as in (10)-(13) and to write

 E{Ξr}=∑∀n
 E{Tr} =∑∀n
 Φr=E{Ξr}E{T% r}= ∑∀n

Note that the summations in (III-B)-(III-B) are on all possible activation conditions of the relay. Finally, from (III-A), (8), (III-B), (III-B), the total transmission power is obtained by

 Φtotal=ϖ∑M+1m=1Pr(Am)R(m)+1−∑M+1m=1Pr(Am)R(M+1), ϖ≐M+1∑m=1(m∑i=1Psi(1R(i)−1R(i−1)))Pr(Sm) +∑∀n

From (6)-(III-B), it follows that the only difference between different ARQ protocols is in the probability terms and . Also, to derive closed-form expressions for the outage probability, the throughput and the average power terms, the final step is to represent the probabilities and as functions of i.e., the optimization parameters of (1). Sections IV and V are devoted to obtain the probability terms for different ARQ protocols and fading channel models.

## Iv Performance analysis in spatially-independent Rayleigh-fading conditions

For the spatially-independent Rayleigh-fading channels the fading coefficients follow and Thus, the pdf of the channel gains are given by and Here, and are the fading parameters determined by the path loss and shadowing between the corresponding terminals. Performance analysis of the relay-ARQ setup in the presence of RTD and INR protocols is as follows.

### Iv-a RTD Protocol in Quasi-Static Conditions

Using the RTD protocol, the same codeword is (re)transmitted in each round and the receiver performs maximum ratio combining (MRC) of the received signals. Thus, the equivalent data rate at the end of the -th round is with and representing the initial data rate and the length of the codeword, respectively. Moreover, with MRC, the received SNR of, e.g., the relay at the end of the -th retransmission round increases to Thus, the data is decoded by the relay at the end of the -th round (and not before) if which is based on the fact that with SNR the maximum decodable rate is if a codeword is repeated times [12, 7]. In this way, , i.e., the probability that the source stops retransmission at round , is found as

 Pr(Sm)={αm+βmifm=1,…,M,γMifm=M+1, αm=Pr(log(1+gsrm−1∑i=1P% si)

Here, is the probability that the relay decodes the data at round , before the destination. Moreover, denotes the probability that the destination decodes the data at round while the relay had not decoded the message up to the end of the -th round (the message may be decodable by the relay at the -th round). Also, the source retransmits the codeword times, if none of the relay and the destination have decoded the data until the -th round, leading to in (IV-A). Note that as a maximum of (re)transmissions is considered. For independent Rayleigh-fading channels, (IV-A) is rephrased as

 αm=⎛⎝Fgsr(eR−1∑m−1i=1Psi)−Fgsr(eR−1∑mi=1Psi)⎞⎠Fgsd(eR−1∑mi=1Psi) =⎛⎜⎝e−λsreR−1∑mi=1Psi−e−λsreR−1∑m−1i=1Psi⎞⎟⎠(1−e−λsdeR−1∑mi=1Psi), βm=⎛⎝Fgsd(eR−1∑m−1i=1Psi)−Fgsd(eR−1∑mi=1Psi)⎞⎠Fgsr(eR−1∑m−1i=1Psi) =⎛⎜⎝e−λsdeR−1∑mi=1Psi−e−λsdeR−1∑m−1i=1Psi⎞⎟⎠(1−e−λsreR−1∑m−1i=1Psi), γM=⎛⎜⎝1−e−λsdeR−1∑Mi=1Psi⎞⎟⎠(1−e−λsreR−1∑Mi=1Psi). (19)

With the same procedure, the probability that the destination decodes the codeword at the -th round (and not before), i.e., is obtained by

 Pr(Am)=βm+m−1∑j=1εj,m, εj,m=Pr(log(1+gsrj−1∑i=1Psi)

Here, is the probability that the relay decodes the codeword at the -th round and helps the destination until it decodes the message at round . Thus, (IV-A) gives the message decoding probability of destination for all possible activation conditions of the relay. For independent Rayleigh-fading condition, is found as

 εj,m=ωjθj,m,ωj=Fgsr(eR−1∑j−1i=1Psi)−Fgsr(eR−1∑ji=1Psi)=e−λsreR−1∑ji=1Psi−e−λsreR−1∑j−1i=1Psi,θj,m=∫eR−1∑ji=1Psi0fgsd(x)×Pr(grd∑m−1i=j+1Pri

where is the decoding probability of the relay at round (and not before). Also, represents the decoding probability of the destination at the -th round, given that the relay is active in rounds

Finally, i.e., the probability that the relay is active in rounds is determined as

 Pr(Bn,m)={εn,mifm≠M+1,ϑnifm=M+1, ϑn=Pr(log(1+gsrn−1∑i=1Psi)

where and are defined in (IV-A) and (21), respectively, and is obtained with the same procedure as in (21).

Using (IV-A)-(IV-A), we can express the outage probability, the throughput and the average power functions of the relay-RTD scheme in terms of and investigate the system performance, as stated in the following.

### Iv-B INR Protocol in Quasi-Static Conditions

Considering a maximum of INR-based retransmission rounds, information nats is encoded into a parent codeword of length Then, the codeword is punctured into subcodewords of lengths which are sent by the source/relay in the successive retransmission rounds. In each round, all received subcodewords are combined by the receivers (relay and destination), to decode the message. In this case, the results of [57, 58, chapter 15], [59, chapter 7] can be used to show that the maximum data rates which are decodable by the relay and the destination at the -th round are obtained by

 Urm(gsr) =m∑i=1lilog(1+gsrPsi)m∑k=1lk =R(m)m∑i=1(1R(i)−1R(i−1))log(1+gsrPsi), (23)

and

 Udj,m(gsd,grd) =∑ji=1lilog(1+gsdPsi)+∑mi=j+1lilog(1+grdPri)∑mk=1lk =R(m)(j∑i=1(1R(i)−1R(i−1))log(1+gsdPsi) +m∑i=j+1(1R(i)−1R(i−1))log(1+grdPri)),j

respectively, where (IV-B) is based on the assumption that the relay is active in rounds Also,

 Udm,m(gsd)≐ ∑mi=1lilog(1+gsdPsi)m∑k=1lk =R(m)m∑i=1(1R(i)−1R(i−1))log(1+gsdPsi) (25)

denotes the maximum decodable rate of the destination at the -th round, given that the relay is inactive.

Although having high throughput and low outage probability, variable-length coding INR results in high packeting complexity [13, 12]. In order to reduce the complexity, fixed-length coding INR scheme is normally considered where setting in (IV-B)-(IV-B) leads to and

 Ur, fixed-lengthm(gsr)=1mm∑i=1log(1+gsrPsi), (26)
 Ud, fixed-lengthj,m(gsd,grd) =1m(j∑i=1log(1+gsdPsi)+m∑i=j+1log(1+grdPri)),j≤m. (27)

From (IV-B)-(IV-B), we can find the probabilities for the INR protocol; Replacing the terms, e.g., of the RTD by for the INR, one can use the same procedure as in (IV-A)-(IV-A) to recalculate the parameters