Relayassisted Multiple Access with Fullduplex MultiPacket Reception
Abstract
The effect of fullduplex cooperative relaying in a random access multiuser network is investigated here. First, we model the selfinterference incurred due to fullduplex operation, assuming multipacket reception capabilities for both the relay and the destination node. Traffic at the source nodes is considered saturated and the cooperative relay, which does not have packets of its own, stores a source packet that it receives successfully in its queue when the transmission to the destination has failed. We obtain analytical expressions for key performance metrics at the relay, such as arrival and service rates, stability conditions, and average queue length, as functions of the transmission probabilities, the self interference coefficient, and the links’ outage probabilities. Furthermore, we study the impact of the relay node and the selfinterference coefficient on the peruser and aggregate throughput, and the average delay per packet. We show that perfect selfinterference cancelation plays a crucial role when the SINR threshold is small, since it may result to worse performance in throughput and delay comparing with the halfduplex case. This is because perfect selfinterference cancelation can cause an unstable queue at the relay under some conditions.
Index terms— Fullduplex, relay, cooperative communications, networklevel cooperation, multiple access, stability, random access networks.
I Introduction
Driven by the exponential traffic growth and the everincreasing demands for wider spectrum, the quest for higher spectral efficiency and enhanced reliability and coverage is creating a new impetus for cooperative communication systems. Cooperative communication aims at increasing the link data rates and the reliability of timevarying links, by overcoming fading and interference in wireless networks. Among the various cooperation techniques to increase throughput, fullduplex relaying has recently gained significant attention. The vast majority of research papers have considered halfduplex or outofband fullduplex systems, in which terminals cannot transmit and receive at the same time, or over the same frequency band. However, the use of nodes with inband fullduplex capability, i.e. terminals that transmit and receive simultaneously over the same frequency band, is constantly increasing in current wireless networks as they can potentially double the network spectral efficiency. Moreover, fullduplex relay systems open a whole new spectrum of capabilities, such as collision detection in contentionbased networks. In this work, we focus on a relayassisted random access network and we analyze the effect of fullduplex cooperative relaying in the network performance, namely arrival and service rates, stability conditions, and average queue length at the relay.
Ia Related Work
The classical relay channel was originally introduced by van der Meulen [2], and earlier work on the relay channel was based on informationtheoretic formulations, e.g. [3]. Most cooperative techniques that have been studied so far focus on the benefits of physical layer cooperation [4]. Nevertheless, there is evidence that the same gains can be achieved with network layer cooperation, which is plain relaying without any physical layer considerations [5, 6]. Recently several works have investigated relaying performance at the MAC layer [5, 6, 7, 8, 9, 10, 11, 12]. More specifically, in [5], the authors have studied the impact of cooperative communication at the medium access control layer with TDMA. They introduced a new cognitive multiple access protocol in the presence of a relay in the network. In [13] the notion of partial network level cooperation is introduced by adding a flow controller at the relay, which regulates the amount of provided cooperation depending on the conditions of the network. The classical analysis of random multiple access schemes, like slotted ALOHA [14], has focused on the socalled collision model. Random access with multipacket reception (MPR) has attracted attention recently [15, 16, 17, 18]. All the above approaches come together in the model that we consider.
In wireless networks, when a wireless node transmits and receives simultaneously in the same frequency, the problem of selfinterference arises. Selfinterference mitigation is a key challenge in inband fullduplex systems. Informationtheoretic aspects of this problem can be found in the pioneering work of Shannon [19], although the capacity region of the twoway channel is not known for the general case [20]. The informationtheoretic limits of inband fullduplex relaying have been studied focusing on the idealistic case of perfect selfinterference cancelation [21, 22]. There exist several techniques that allow the possibility of perfect selfinterference cancelation [20]. However, in practice, there are several technological limitations and challenges [23, 24], which may limit the accuracy and the effectiveness of selfinterference cancelation. Various methods for performing selfinterference cancelation at the receivers can be found in [25] and [26]. The main result therein is that there is a tradeoff between transceiver complexity and selfinterference cancelation accuracy. In [27, 28], it was demonstrated in practice real implementations of simultaneous transceivers, where the selfinterference problem has been mitigated through RF isolators and echo cancellers, coupled with baseband digital filtering. Furthermore, some recent results have also shown that full duplex is possible, proposing specific designs, e.g. [29, 30], which mainly focus on the physical and the medium access control (MAC) layer design. Choi et al. in [29] designed a practical singlechannel fullduplex wireless system, combining three selfinterference cancellation schemes, as well as RF and digital interference cancellation. Jain et al. [30] presented a fullduplex radio design using signal inversion and adaptive cancellation. Unlike [29], the authors in [30] consider wideband and high power systems. In theory, this new design has no limitation in terms of bandwidth or power. Therefore, building fullduplex wireless networks (such as fullduplex 802.11n wireless networks) has started becoming feasible. Fang et al. [31] proposed a collisionfree fullduplex broadcast MAC and studied crosslayer optimization of MAC and routing in fullduplex wireless networks under various resource and social constraints. In [32] the comparison of performance of half and fullduplex relay is studied at the physical layer, in [33] is investigated the effect of channel estimation errors on the ergodic capacities for bidirectional fullduplex transmission. An information theoretic study in [34] compares multiantenna half and fullduplex relaying from the perspective of achievable rates.
IB Contribution
In this work, we complement and extend the work in [1]. We study the operation of a cooperative node relaying packets from a number of users/sources to a destination node as shown in Fig. 1. We assume MPR capability for both the relay and the destination node. The relay node can transmit and receive at the same time over the same frequency band (inband full duplex). We assume random medium access, slotted time, and that each packet transmission takes one timeslot. The wireless channel is modeled as Rayleigh flatfading channel with additive white Gaussian noise. A user transmission is successful if the received signaltointerferenceplusnoise ratio (SINR) is above a certain threshold . We also assume that acknowledgements (ACKs) are instantaneous and error free. The relay does not have packets by itself and the source nodes are considered saturated with unlimited amount of traffic. The selfinterference cancellation at the relay is modeled as a variable power gain, mainly because we are studying the impact on the network layer^{1}^{1}1The selfinterference cancellation at the relay is modeled as a variable power gain that affects the success probability with which the relay will receive a packet and is described in Section II.. Studying in detail the physical layer implementation of selfinterference mitigation and considering specific selfinterference cancelation mechanisms is beyond the scope of this paper. We obtain analytical expressions for key performance characteristics of the relay queue, such as arrival and service rates, and we derive conditions for stability and the average queue length as functions of the transmission probabilities, the selfinterference coefficient, and the links’ outage probabilities. In particular, we study the impact of the relay node and the selfinterference coefficient on the peruser and the networkwide throughput, as well as the average delay per packet. Furthermore, we derive expressions for both the peruser and aggregate throughput when the queue at the relay is unstable, for which case we do not have though any guarantees for bounded delay.
The remainder of the paper is organized as follows: Section II describes the system model and in Section III we present the main characteristics of the relay queue, such as the average arrival and service rates. In Section IV, we provide expressions for the peruser and the aggregate throughput. The average delay per packet is obtained in Section V. Numerical results are presented in Section VI, and finally Section VII concludes the paper.
Ii System Model
Iia Network Model
We consider a network with sources, one relay node, and a single destination node. The sources transmit packets to the destination using a cooperative relay; the case of is depicted in Fig. 1. We assume that the queues of both sources are saturated, i.e., no external arrivals and unlimited buffer size, and that the relay does not have packets of its own but only forwards the packets it has received from the two users. The relay node stores a source packet that it receives successfully in its queue when the direct transmission to the destination node has failed. We assume a random access channel where is the transmit probability of the relay given that it has packets in its queue, and for is the transmit probability for the th user. The receivers at the relay and the destination nodes are equipped with multiuser detectors, hence they can decode packets from more than one transmitter at a time. Furthermore, the relay can simultaneously transmit and receive packets (full duplex).
IiB Physical Layer Model
The MPR channel model used in this paper is a generalized form of the packet erasure model. We assume that a packet transmitted by node is successfully received by node if and only if , where is a threshold characteristic of node . The wireless channel is subject to fading; let be the transmit power at node and be the distance between and . The received power at when transmits is where is a random variable representing smallscale fading. Under Rayleigh fading, is exponentially distributed [35]. The received power factor is given by where is the path loss exponent with typical values between and . We model the selfinterference by a scalar as in [36] and [37]. We refer to the as the selfinterference coefficient. When , no selfinterference cancelation technique is used, while models perfect selfinterference cancelation. The success probability in the link is given by
(1)  
where is the set of transmitting nodes at the same time, is the parameter of the Rayleigh fading random variable, is the receiver noise power at and when and else. The analytical derivation for this success probability can be found in [35].
Note: The selfinterference is modeled through and it affects the success probability when the relay transmits and receives simultaneously. The value of captures the accuracy of the selfinterference cancelation. As approaches it is closer to the pure full duplex operation. When is the operation is the half duplex operation since the success probabilities for the users in this case are very close to .
IiC Queue Stability
We adopt the definition of queue stability used in [38].
Definition 1.
Denote by the length of queue at the beginning of timeslot . The queue is said to be stable if
(2) 
If , the queue is substable. If a queue is stable, then it is also substable. If a queue is not substable, then we say it is unstable.
Loynes’ theorem [39] states that if the arrival and service processes of a queue are strictly jointly stationary and the average arrival rate is less than the average service rate, then the queue is stable.
Iii Performance Analysis for the Relay Queue
In this section, we derive expressions for key performance metrics for the relay queue, namely arrival and service rates, stability conditions, and average queue length. The analysis is provided for two cases: (i) when the network consists of two nonsymmetric in general users, (ii) for symmetric users.
This section is an intermediate step before investigating the impact of the relay node in the peruser throughput, the aggregate throughput, and the average per packet delay. In order to study those quantities, we need to first compute the average arrival and service rate of the relay, the average queue length, and the stability conditions. The stability of a queue is translated to bounded queue size, which implies finite queuing delay.
Iiia Twouser Case
We study first the relay queue characteristics for the twouser case. In this network, at each timeslot, the relay can receive at most two packets (one per user) and to transmit at most one.
The probability that the relay receives packets in a given timeslot when its queue is empty is denoted by , and otherwise (not empty). The expressions for the are rather lengthy and are presented in Appendix A. The average arrival rate at the relay when its queue is empty is denoted by , and by when it is not (derived in Appendix A). The probability that the relay queue increases by packets when is empty is denoted by , and when it is not; is the probability that the queue decreases by one packet. Note that and are in general different quantities, however in halfduplex relay systems^{2}^{2}2The case of halfduplex relay is studied in [18], for which the analysis is simpler compared to the fullduplex case..
The next theorem presents the main relay queue characteristics for the twouser case.
Theorem III.1.
The key performance measures for the relay queue in a twouser network are provided below.

The average service rate is
(3) where is the success probability between the relay and the destination when the transmitting nodes are the relay and nodes and . can be computed from (1).

The probability that the queue at the relay is empty is
(4) 
The average arrival rate is
(5) 
The average relay queue size is
(6)
Proof.
See Appendix A. ∎
IiiB Symmetric user Case
We now investigate the case of a symmetric user network^{3}^{3}3Our work could be generalized to the asymmetric case; nevertheless the expressions will be significantly involved without providing any meaningful or crisp insights.. Each user attempts to transmit in a slot with probability ; the success probability to the relay and the destination when nodes transmit are given by and , respectively. There are two cases for the , i.e., and , denoting the success probability when relay remains silent or transmits, respectively. Those success probabilities for the symmetric case are given by and . , , , . There are two cases for the , i.e., and , denoting the success probability when the relay remains silent or transmits respectively. The success probabilities are given by and , where is the distance between the users and the relay, is the parameter of the Rayleigh fading random variable at channel , is the path loss exponent and is the selfinterference coefficient.
The next theorem summarizes the results for the characteristics of the relay queue for the symmetric user case.
Theorem III.2.
The key performance measures for the relay queue in the symmetric user network are provided below.

The average service rate is
(7) 
The probability that the queue at the relay is empty is
(8) 
The average relay queue size is
(10)
Proof.
See Appendix B. ∎
Iv Throughput Analysis
In the previous section, we provided the main results on the relay queue characteristics, including the empty queue probability and the average queue length. Here, we derive the peruser throughput and the network aggregate throughput with one cooperative relay and users.
The peruser throughput, for the th user is given by , where denotes the direct throughput from user to the destination, i.e., the transmitted packet reaches the destination directly, without using the relay. When the transmission to the destination is not successful, and at the same time the relay node receives the packet correctly, then it stores it to its queue, and the contributed throughput by the relay for the user is denoted by . When the queue at the relay is stable, is the arrival rate from user to the queue.
The term can also be interpreted as the probability that a transmitted packet from user reaches the destination directly, and is the probability of unsuccessful transmission from user to the destination while the packet is received at the relay.
The percentage of th user’s traffic that is being relayed is .
In the following subsection, we provide expressions for and for the twouser and the symmetric user cases.
Iva Peruser and Aggregate Throughput: Twouser Case
The direct throughput to the destination for the th user, , is given by
(11)  
When the relay queue is stable, the contributed throughput to user , , is the arrival rate from user to the relay queue. Note that a packet from user enters the relay queue when the transmission to the destination is not successful and at the same time the relay is able to decode that packet. The relayed throughput of user is given by
(12)  
The throughput for the th user is given by
(13) 
In the above equations, the queue is assumed to be stable, hence the arrival rate from each user to the queue is a contribution to the overall throughput. The aggregate throughput is . Notice that the peruser throughput is independent of as long as it is in the stability region. This is due to the fact that the product is constant (does not depend on ). The proof is straightforward and thus is omitted.
IvB Peruser and Aggregate Throughput: Symmetric user Case
In this subsection, we provide expressions for the direct and the relayed peruser and aggregate throughput. The notation used in Section IIIB applies here as well. Furthermore, the peruser throughput is denoted by , the direct throughput to the destination by , and the relayed throughput by .
The direct throughput is given by
(14)  
The throughput contributed by the relay (when the queue at the relay is stable), , is given by
(15)  
The peruser throughput for the cooperative relay network when the relay queue is stable is given by
(16) 
The aggregate throughput is .
Remark 1.
When the queue is unstable, the aggregate throughput is the summation of the direct throughput among the users and the destination plus the service rate of the relay. However, when the queue is unstable, the queue size increases to infinity, thus there is no guarantee for finite queueing delay.
V Delay Analysis
In Section III, we studied the performance of the relay queue in terms of the probability of empty queue and the average queue length. That section was an intermediate step for our main goal, which is to study the impact of the relay node in the network in terms of throughput and the delay. In the previous section, we obtained the peruser and the aggregate throughput for a relay network with stable relay queue and commented on the case of unstable relay queue. In this section, we analyze an important network performance measure, the delay, and derive analytical expressions for the average delay required to deliver a packet from the source to the destination.
Theorem V.1.
The average delay for a packet received at the destination when it is in the head of the user queue is given by
(17) 
where and is the th user relayed and peruser throughput, respectively. and is the average arrival and service rate of the relay, respectively, and is the average queue length of the relay.
Proof.
See Appendix C. ∎
The expressions for and are given in Section IV. The expressions for , , are summarized in Theorem III.1 and III.2 for the twouser and the symmetric user case, respectively.
Note that the term in (17) is the queueing delay, which is the time a packet spends in queue, the time the packet is assigned to the queue for transmission and the time it starts being transmitted. In the meantime, the packet waits while other packets in the queue are transmitted.
Remark 2.
When the relay queue is unstable, the average queue length can be arbitrarily large, thus the average queueing delay tends to infinity. In (17), when the queue is unstable, then the average delay also tends to infinity. In the case of unstable queues, flow control policies could be applied for packet dropping, however this is beyond the scope of our paper.
Vi Numerical Results
In this section, we provide numerical results to validate the above theoretical performance analysis. For exposition convenience, we consider the case where all users have the same link characteristics and transmission probabilities. The parameters used in the numerical results are as follows: distances are , , and in meters, the path loss exponent is , and the receiver noise power . The transmit power for the relay is mW and for the th user is mW.
Via Per user and Aggregate Throughput
Figs. 2(a) and 4(a) present the peruser throughput versus the number of users in the network for different values of and , and for and , respectively. Figs. 2(b) and 4(b) show the aggregate throughput versus the number of users. When , we observe that for and (almost perfect selfinterference cancelation) the relay queue is unstable for relative small number of users. This is because for small values of , it is more likely to have more successful transmissions from the users to the relay, while at the same time the relay can transmit at most one packet per timeslot. For the queue is never unstable for the selected set of parameters, while for and , throughput gains are evident as compared to no selfinterference cancelation.
In Figs. 3 and 5, we plot the percentage of traffic that is being relayed in the network (cf. Section IV) for and respectively, for the case of a stable queue.
Figs. 6(a) and 8(a) present the peruser throughput versus the number of users in the network for different values of and , and for and , respectively. Figs. 6(b) and 8(b) show the aggregate throughput versus the number of users. Finally, Figs. 7 and 9 show the percentage of traffic that is being relayed.
Note that when the percentage tends to (or ), the contributed throughput by the relay tends to be the total network throughput.
The gains from the relay are more pronounced for large , whilst in the case of and quasi perfect selfinterference cancelation, we tend to have an unstable queue, which affects the delay per packet as we will see in the next subsection.
ViB Average Queue Length and Average Delay per Packet
In this subsection, we provide numerical results for two key performance metrics, namely the average relay queue size and the average delay per packet.
Figs. 10(a) and 11(a) present the average queue length of the relay for and . The average queue length is among the factors that affect the average delay per packet as presented in (17) of Theorem V.1. Figs. 10(b) and 11(b) illustrate the average delay per packet for and .
For we included the perpacket delay for the network without the relay for comparison reasons. We observe that in that case the cooperative relay node does not provide any gains for increasing number of users, as the delay for the relay network is larger than the delay without using a relay. For , the delay for the network without the relay is much larger, e.g., it starts with timeslots for users and goes up to for users. In that case, the use of a relay is beneficial in terms of throughput and perpacket delay.
Figs. 12(a) and 13(a) show the average queue length for and respectively. In Figs. 12(b) and 13(b) and 13(b), we illustrate the average delay per packet.
The delay for the network without the relay is significantly large, e.g., for the delay is greater than timeslots and for the delay is more than timeslots. In those cases, the existence of the relay offers significant gains not only in terms of throughput but also in the delay performance.
When we have almost perfect selfinterference cancelation (except the case of ), we observe significant gains in the delay performance compared to the case of the quasi halfduplex relay ().
Vii Conclusions
In this paper, we explored fullduplex communication in which a cooperative node relays packets from a number of sources to a common destination node in a random access network with multipacket reception capability for both the relay and the destination node. Considering a multiple capture model and the selfinterference due to fullduplex relay operation, a transmission is successful if the received SINR is above a certain threshold .
We provided analytical expressions for the performance of the relay queue, namely stability conditions, arrival and service rates, and average queue length. We studied the peruser and the aggregate throughput, and showed that the peruser throughput does not depend on the relay transmit probability under stability conditions. We also studied the impact of the selfinterference coefficient on the peruser throughput, the networkwide throughput, and the average perpacket delay. We showed that the selfinterference coefficient plays a crucial role when is small (and tends to zero) since it may result in an unstable queue. However, for large values and perfect selfinterference cancelation, the gains in terms of throughput and delay are more pronounced.
Future extensions of this work may include users with nonsaturated queues, i.e. sources with external random arrivals, as well as scenarios where the cooperative relay node has packets on its own and different service priorities.
Appendix A Proof of Theorem iii.1
We provide here the proof of Theorem III.1, which presents the main result for the relay queue characteristics for the twouser case.
Analysis of the average arrival and service rate:
The average service rate , is given by (3), where is the transmit probability of the relay given that it has packets in its queue, and for is the transmit probability for the th user. The term is the success probability of link when the transmitting nodes are and and can be calculated based on (1).
The average arrival rate of the queue is given by , where is the average arrival rate at the relay queue when the queue is empty and when it is not. , where is the probability of receiving packets given that the queue is empty. Accordingly, , where is the probability of receiving packets when the queue is not empty.
The expressions for are given by
(18)  
(19) 
In order to compute for instance (i.e., the relay receives one packet), we have to take into account all the possible combinations, which are either the received packet is transmitted by the first or the second user (with all the possible combinations of active/idle users). When the relay queue is not empty, the expressions for the are given by
(20)  
(21)  
In order to fully characterize the average arrival rate at the relay, we have to compute the probability the queue is empty. We model the queue at the relay as a discrete time Markov Chain (DTMC), which describes the queue evolution and is presented in Fig. 14. Each state is denoted by an integer and represents the queue size at the relay node. The transition matrix of the above DTMC is a lower Hessenberg matrix given by
(22) 
The elements of the matrix P, are , and . The quantity () is the probability that the queue size increases by packets when the queue is empty (not empty). Note that because when the queue is empty, the probability of packets arriving is the same with the probability that the queue size increases by packets; when the queue is not empty however, this is not true. For example the probability of two packets arriving is not the same with the probability of doubling the queue size; this is because both arrivals and departures can occur at the same time. The expressions for the are given by
(23)  
(24) 
(25)  
(26)  
Note that and