On the Performance of the RelayARQ Networks
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 spatiallycorrelated 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 powerlimited individually. With adaptive power allocation, the results demonstrate the efficiency of relayARQ techniques in different conditions.
I Introduction
Relayassisted 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 longterm evolution advanced (LTEA) 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 wellestablished 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 dataintensive 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 powerlimited performance of the relayARQ setups.
The basic principles of different ARQ protocols are derived in [2, 3, 4, 5, 6, 7, 8]. Power allocation in ARQbased singleuser (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 relayARQ 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., spacetime 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. longterm average transmission rate) of STCbased relayARQ systems. The energy and spectrum efficiency of the basic and hybrid relayARQ networks are verified in [32, 33, 34] as well. Also, [35] designs a multirelayARQ 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 delaylimited throughput and deriving a closedform 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 relayARQ setups in fastfading 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 outagelimited energy minimization in singleuser and relayARQ networks. Opportunistic relaying, rate adaptation and analyzing the energydelay tradeoff curve are considered by [43], [44] and [45, 46], respectively, where the direct sourcedestination link is ignored. Also, the throughput, the packet error rate and the effective capacity of different ARQassisted relay networks are studied in [47, 48, 49], respectively. Power scaling in MIMO and cognitive radio relayARQ networks is addressed in [50] and [51], respectively. Finally, [52] studies a relayARQ 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 movingrelay systems [54, 55]. As a result, there might be considerable correlation between the sourcerelay and the sourcedestination 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 sourcedestination and the relaydestination links may be spatiallycorrelated. 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 relayARQ networks in cases where there is either a longrun sum power constraint on the source and the relay or when each of the source and the relay are powerlimited individually. Adaptive power allocation between the retransmissions is used to improve the system performance. We derive closedform expressions for the average power, the throughput and the outage probability of different relayARQ protocols in the cases with independent or spatiallycorrelated fading channels. Moreover, we investigate the effect of fading temporal variations on the data transmission efficiency of the relayARQ 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 ARQbased 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 spatiallycorrelated fading conditions, with noisy/noisefree feedback signals. Finally, our discussions on the users’ message decoding probabilities (Theorems 13) have not been presented before.
The results show that there is a structural procedure to study different performance metrics of relayARQ 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 relayARQ 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) signaltonoise ratios (SNRs). Compared to the fixedlength coding scheme, the throughput of the relayARQ network increases when variablelength coding is utilized. With the practical range of spatial correlations, the performance of the relayARQ network is not sensitive to the spatial correlation. However, the data transmission efficiency of the network is reduced at highlycorrelated conditions.
Ii System model
We consider a relayassisted communication setup consisting of a source, a relay and a destination. The channel coefficients in the sourcerelay, the sourcedestination and the relaydestination 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 ARQbased 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 blockfading conditions:

Quasistatic. 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).

Fastfading. Here, the channel coefficients are supposed to change in each retransmission round.
The quasistatic model, studied in Subsections IV.AB, represents the scenarios with slowmoving or stationary users, e.g., [12, 11, 56]. On the other hand, the fastfading, studied in Subsection IV.C, is an appropriate model for the high speed users and frequencyhopping 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 blockfading 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 errorfree, but we later investigate the effect of erroneous feedback bits as well (Section IV).
RelayARQ model: The considered relayARQ 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 negativeacknowledgment (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 errorfree 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 sourcedestination 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 relaydestination link experiences better average characteristics than the sourcedestination 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 Rayleighfading conditions the proposed scheme outperforms the stateoftheart approaches, in terms of outage probability/throughput.
Iii Problem formulation
In this paper, we study the problem of
(1) 
In words, we investigate the longterm 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 powerlimited scenarios:

Scenario 1. The total power for data transmission in the relayARQ 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 electricitybill 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 batterylimited [11, 9, 10, 13, 12].
To study (1), the following procedure is considered (please see Fig. 1 as well). First, we derive closedform 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 closedform expressions.
In three steps we obtain the closedform 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.
Iiia 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]
(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
(3) 
(4) 
(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.
IiiB 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)(IIIA) as functions of and The details are explained as follows.
According to the definitions, the outage probability is found as
(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
(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 (IIIA), is obtained by
(8) 
From (7)(8) and , the throughput (2) is found as
(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
(10) 
and, using we have
(11) 
With the same arguments, the expected activation period of the source, i.e., in (3), is found as
(12) 
which, along with (IIIB), leads to
(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
(14) 
(15) 
(16) 
Note that the summations in (IIIB)(IIIB) are on all possible activation conditions of the relay. Finally, from (IIIA), (8), (IIIB), (IIIB), the total transmission power is obtained by
(17) 
From (6)(IIIB), it follows that the only difference between different ARQ protocols is in the probability terms and . Also, to derive closedform 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 spatiallyindependent Rayleighfading conditions
For the spatiallyindependent Rayleighfading 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 relayARQ setup in the presence of RTD and INR protocols is as follows.
Iva RTD Protocol in QuasiStatic 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
(18) 
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 (IVA). Note that as a maximum of (re)transmissions is considered. For independent Rayleighfading channels, (IVA) is rephrased as
(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
(20) 
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, (IVA) gives the message decoding probability of destination for all possible activation conditions of the relay. For independent Rayleighfading condition, is found as
(21) 
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
IvB INR Protocol in QuasiStatic Conditions
Considering a maximum of INRbased 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
(23) 
and
(24) 
respectively, where (IVB) is based on the assumption that the relay is active in rounds Also,
(25) 
denotes the maximum decodable rate of the destination at the th round, given that the relay is inactive.