I Introduction
Over the past few decades, wireless communications and networking have witnessed an unprecedented growth. The growing demands require high data rates, considerably large coverage areas and high reliability. Relayassisted wireless networks have been proposed as a candidate solution ot fulfill these requirements [1], since relays can decrease the delay and can also provide increased reliability and higher energy efficiency [2, 3]. A relaybased cooperative wireless system operates as follows: There is a number of source users that transmit packets to a common destination node, and a number of relay nodes that assist the source users by storing and retransmitting the packets that failed to reach the destination; e.g., [4, 5, 6, 7]. A cooperation strategy among sources and relays specifies which of the relays will cooperate with the sources. This problem gives rise to the usage of a cooperative space diversity protocol [8], where each user has a number of “partners” (i.e., relays) that are responsible for retransmitting its failed packets.
Ia Related work
Cooperative relaying is mostly considered at the physical layer, and is based on informationtheoretic considerations. The classical relay channel was first examined in [9] and later in [2]. Recently cooperative communications have received renewed attention, as a powerful technique to combat fading and attenuation in wireless networks; e.g., [8, 10]. Most of the research has concentrated on informationtheoretic studies. Recent works [4, 5, 7, 11] shown that similar gains can be achieved by networklayer cooperation. By networklayer cooperation, relaying is assumed to take place at the protocol level avoiding physical layer considerations.
In addition, random access recently regained interest due to the increased number of communicating devices in 5G networks, and the need for massive uncoordinated access [12]. Random access and alternatives schemes and their effect on the operation of LTE and LTEA are presented in [12], [13], [14]. In [15], the effect of random access in CloudRadio Access Network is considered.
The characterization of the stable throughput region, i.e. the stability region, which gives the set of arrival rates such that there exist transmission probabilities under which the system is stable, is a meaningful metric to measure the impact of bursty traffic and the concept of interacting nodes in a network; e.g., [16, 17, 18].
Except throughput, delay is another important metric, which recently received considerable attention due to the rapid growth on supporting realtime applications, which in turn require delaybased guarantees. However, due to the interdependence among queues, the characterization of the delay even in small networks with random access is a rather difficult task, even for the noncooperative collision channel model [19]. In [19], queueing delay was studied with the aid of the theory of boundary value problems. The traditional collision model is accurate for modeling wireline communication, however, it is not an appropriate model for probabilistic reception in wireless multiple access networks. For the noncooperative multipacket reception (MPR) model, delay analysis was performed in [20], based on the assumption of a symmetric network. Recently, the authors in [21] generalized the model in [20, 19] by considering timevarying links between nodes where the channel state information was modeled according to a GilbertElliot model. The study of queueing systems using the theory of boundary value problems was initiated in [22], and a concrete methodological approach was given in [23, 24]. The vast majority of queueing models are analyzed with the aid of the theory of boundary value problems referring to continuous time models, e.g., [25, 26, 27, 28, 29, 30, 31, 32]. On the contrary, there are very few works on the analysis of discrete time models [19, 21, 18, 33, 34]. This is mainly due to the complex boundary behavior of the underlying random walk, which reflects the interdependence and coupling among the queues.
IB Contribution
Our contribution is summarized as follows. We consider a cooperative wireless network with saturated source users, two relay nodes with adaptive transmission control, and a common destination. Our primary interest is to investigate the stability conditions, throughput performance, and provide expressions for the queueing delay experienced at the buffers of relay nodes. The time is slotted, corresponding to the duration of a transmission of a packet, and the sources/relay nodes access the medium in a random access manner. The sources transmit packets to the destination with the cooperation of the two relays. If a transmission of a user’s packet to the destination fails, the relays store it in their queues and try to forward it to the destination at a subsequent time slot. Moreover, the relays have also external bursty arrivals that are stored in their infinite capacity queues. We consider MPR capabilities at the destination node.
We assume that there is no coordination among relays and sources, but the destination node can sense both of them. Here, we assume that the destination node gives “priority” to the sources when it senses that they will transmit. If it senses that all of the sources will remain silent then it switches to the relays. The relays are accessing the wireless channel randomly and employ a statedependent transmission protocol. More precisely, a relay adapts its transmission characteristics based on the status of the other relay in order to exploit its idle slots, and to increase its transmission efficiency, which in turn leads towards selfaware networks [35]. More specifically, we assume that each relay node is aware of the state of the other one. Note that this feature is common in cognitive radios [7, 35]. To the best of our knowledge this variation of random access has not been reported in the literature. The contribution of this work has two main parts focused on the stable throughput region, and the detailed analysis of the queueing delay at relay nodes.
IB1 Stability analysis and throughput performance
We provide the throughput analysis of the general twouser network with MPR capabilities and the symmetric user network under random access. The performance characterization for symmetric users can provide insights on scalability of the network. In addition, we provide the stability conditions for the queues at the relays.
IB2 Delay Analysis
The second part of the contribution of this work is the delay analysis. Except its practical implications, our work is also theoretically oriented. To the best of our knowledge there is no other work in the related literature that deals with the detailed delay analysis of an asymmetric random access cooperative wireless system with adaptive transmissions and MPR capabilities.
To enhance the readability of our work we consider the case of source users, and focus on a subclass of MPR models, called the “capture” channel, under which at most one packet can be successfully decoded by the receiver of the node , even if more than one nodes transmit. We need to mention, that the assumption of two users is not restrictive, and our analysis can be extended to the general case of users. Moreover, our analysis remains valid even for the case of general MPR model. However, in both cases some important technical requirements must be further taken into account, which in turn will worse the readability of the paper. Besides, our aim here is to focus on the fundamental problem of characterizing the delay in a cooperative wireless network with two relay nodes, and our model and analysis can serve as a building block for the more general case.
Our system is modeled as a twodimensional discrete time Markov chain, and we show that the generating function of the stationary joint relay queue length distribution by solving a fundamental functional equation with the aid of a RiemannHilbert boundary value problem. Furthermore, each relay node employs an adaptive transmission policy, under which it adapts its transmission probabilities based on the status of the queue of the other relay. Moreover, the kernel of this functional equation has never been treated in the related literature. More precisely,

Based on a relation among the values of the transmission probabilities we distinguish the analysis in two cases, which are different both in the modeling, and in the technical point of view. In particular, the analysis leads to the formulation of two boundary value problems [36]
(i.e., a Dirichlet, and a RiemannHilbert problem), the solution of which will provide the generating function of the stationary joint distribution of the queue size for the relays. This is the key element for obtaining expressions for the average delay at each user node. To our best knowledge, it is the first time in the related literature on cooperative networks with MPR capabilities, where such an analysis is performed.

Furthermore, for the twouser, tworelay symmetric system, we provide explicit expression for the average queueing delay, without the need of solving a boundary value problem.
Concluding, the analytical results in this work, to the best of our knowledge, have not been reported in the literature.
The rest of the paper is organized as follows. In Section II we describe the system model in detail. Section III is devoted to the investigation of the throughput and the stability conditions for the asymmetric MPR model of , while in Section IV we generalize our previous results for the general case of users with MPR capabilities. In Section V we focus on the delay analysis for the general asymmetric twouser with tworelays network. The fundamental functional equation is derived, and some preparatory results in view of the resolution of the functional equation are obtained. We formulate and solve two boundary value problems, the solution of which provide the generating function of the stationary joint queue length distribution of relay nodes. The basic performance metrics are obtained, and important hints regarding their numerical evaluation are also given. In Section VI we obtain explicit expressions for the average delay at each relay node for the symmetrical system without solving a boundary value problem. Finally, numerical examples that shows insights in the system performance are given in Section VII.
Ii Model description and notation
In this work, we consider a network consisting of saturated userssources, two relays, and one destination node. In this section, we will describe the case of saturated users assisted by two relays as depicted in Fig. 1. We focus on the twouser scenario in order to facilitate the presentation and the description of the cooperation protocol.
Iia Network Model
We consider a network of saturated source users, i.e. sources and , two relay nodes, denoted by and , and a common destination node depicted in Fig. 1.^{1}^{1}1In this section we will present the system model for the case of two users. However, in Section IV, we consider the case where there are symmetric users The sources transmit packets to the node with the cooperation of the relays. The packets have equal length and the time is divided into slots corresponding to the transmission time of a packet.
We assume that the relays and the destination have multipacket reception (MPR) capabilities and the success probabilities for the transmissions will be provided in Section IIC. MPR is a more suitable model than the collision channel since it can capture better the wireless transmissions. The sourceusers have random access to the medium with no coordination among them. At the beginning of a slot, the source user attempts to transmit a packet with a probability , , i.e., with probability remains silent. The nodes are assume to have priority over the relays. More specifically, the sources and the relays transmit in different channels, however, the destination node can overhear both of the channels. However, the destination node gives “priority” to the sources if it senses that they will transmit. If the destination senses no activity from the sources, it switches to the relays. We assume that this sensing time is negligible. If a packet transmission from a source to the destination fails and at the same time if at least one of the relays will be able to decode this packet, then will store it in its queue with a probability, and it will forward it to the destination at a subsequent time slot. The queues at the relays are assumed to have infinite size.
IiB Description of Relay Cooperation
If a transmission of a user’s packet to the destination fails, the relays overhear the wireless transmission, they can store it in their queues with a probability, and try to forward it to the destination at a subsequent time slot. In case that both relays receive the same packet from a user, they choose randomly which will store the packet. In particular, we define the probability that a transmitted packet from the th source will be stored at the queue of th relay if the relay is able to decode it. This probability captures two scenarios, (i) the partial cooperation of a relay, which was introduced in [37] and (ii) when both relays receive the failed packet from node , then the first one will keep in its queue with probability and with probability will be stored in the queue of the second relay. We would like to emphasize that in case that only one relay, i.e the first relay, will receive correctly a failed packet, then it will store it in its queue with probability . This probability, controls the amount of the cooperation that this relay provides. However, in this work we assume that if only one relay receives successfully a packet that fails to reach the destination, then this packet will be stored in its queue. When both relays decode correctly a failed packet, if we assume that , then the packet will enter one queue only, either the first or the second one. If we will assume that , then there is a probability that the failed packet will not be accepted in the queues of the relays and it has to be retransmitted in a future timeslot by its source.
Let be the number of packets in the buffer of relay node , , at the beginning of the th slot. Moreover, during the time interval (i.e., during a time slot) the relay , generates also packets of its own (i.e., exogenous traffic). Let
be a sequence of i.i.d. random variables where
represents the number of packets which arrive at in the interval , with . The network with pure relays can be obtained by replacing .In case node senses no activity from the source users at the beginning of a slot, it switches to the channel of relay nodes. If there are stored packets in the buffers of the relays, they will also attempt to transmit a packet to the node with a probability.
Due to the interference among the relays, we consider the following opportunistic access policy: If both relays are non empty, , transmits a packet with probability . If (resp. ) is the only nonempty, it adapts its transmission probability. More specifically, it transmits a packet with a probability , in order to utilize the idle slot of the neighbor relay node.^{2}^{2}2We consider the general case for instead of assuming directly . This can handle cases where the node cannot transmit with probability one even if the other node is silent, e.g., when the nodes are subject to energy limitations. It is outside of the scope of this work to consider specific cases and we intent to keep the proposed analysis general. Note that in such a case, a relay node is aware about the state of its neighbor.^{3}^{3}3In such a shared access network, it is practical to assume a minimum exchanging information of one bit between the nodes.
IiC Physical Layer Model
The MPR channel model used in this work is a generalized form of the packet erasure model. In wireless networks, a transmission over a link is successful with a probability. We denote the success probability of the link between nodes and when the set of active transmitters are in . For example, denotes the success probability for the link between the first source and the first relay when both sources are transmitting. The probability that the transmission fails is denoted by . In order to take also into account the interference among the relays, we have to distinguish the success probabilities when a relay transmits and the other is active or inactive (i.e., it is empty). Thus, when , the success probability of the link between relay node and node when relay node is the only non empty is denoted by . In this work we distinguish this case, in order to have more general results that can capture scenarios that one relay can increase its transmission power when the other relay is empty, thus silent, in order to achieve a higher success probability. Thus, . The probabilities of successful packet reception can be obtained using the common assumption in wireless networks that a packet can be decoded correctly by the receiver if the received SINR (SignaltoInterferenceplusNoiseRatio) exceeds a certain threshold. The SINR depends on the modulation scheme, the target bit error rate and the number of bits in the packet [38] and the expressions for the success probabilities can be found in several papers, i.e for the case of Raleigh fading refer in [4]. On the other hand, if source user , is the only that transmits, denotes the probability that its packet is successfully decoded by the destination, while with probability this transmission fails.
We now provide the service rates , seen at relay nodes. For the first relay we have
(1) 
Similarly we have the service rate at the second relay
(2) 
Note that the success probability (resp. ) refers to the case where a submitted packet from relay (resp. ) is successfully decoded by node , and includes both the case where only a packet from (resp. ) is decoded, both the case where both relays have successful transmissions (i.e. MPR case).
We define the following two variables in order to simplify the presentation in the analysis
(3) 
and
(4) 
These variables can be seen as an indication regarding the MPR capability for each user. If , then the interference caused by the other user is negligible.
Symbol  Explanation 

The number of packets in relay node at the beginning of slot  
The number of packets arriving during in relay node ,  
The expected number of external arrivals in relay node , , during a slot  
Transmission probability of source ,  
Transmission probability of relay node , , when both users are active (i.e., nonempty)  
Transmission probability of relay node , , when it is the only active (i.e., nonempty) node  
Success probability of the link between node and when the set of the transmitted nodes are in  
Success probability of relay node , , when it is the only active (i.e., nonempty) node  
Success probability of the link between source and when both sources transmit,  
but source fails to directly reach node , , 
Iii Throughput and Stability Analysis for the twouser case – General MPR case
In this section, we provide the analysis for the twouser case under the MPR channel model. More specifically, we provide the throughput analysis for the two users and in addition we derive the stability conditions for the queues at the relays.
Based on the definition in [18], a queue is said to be stable if and . 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. If the average arrival rate is greater than the average service rate, then the queue is unstable and the value of
approaches infinity almost surely. The stability region of the system is defined as the set of arrival rate vectors
, for which the queues in the system are stable.We start the analysis by deriving the throughput per user which allow us to calculate the endogenous arrivals at the relays.
Iiia Throughput per user
Here we will consider the throughput per (source) user when both queues of the relays are stable. Conditions for stability are given in a subsequent subsection. When the queues at the relays are not stable the throughput per user can be obtained using the approach in [4].
The throughput per user , , is the direct throughput when the transmission to the destination is successful plus the throughput contributed by the relays (if they can decode the transmission) in case of a failed transmission to the destination. Thus, the throughput seen by the first user is given by
(5) 
where
(6) 
and
Similarly we can obtain the throughout for the second user.
The aggregate or networkwide throughput of the network when the queues at the relays are both stable is
(7) 
IiiB Endogenous arrivals at the relays
Here, we will derive the internal (or endogenous) arrival rate from the users to each relay. We would like to mention that the relays have also their own traffic (exogenous) denoted by for the th relay.
A packet from a user can enter a queue at one relay if the transmission to the destination fails and at the same time at least one relay decodes correctly the packet. In the twouser case with MPR capabilities, in a relay up to two packets can enter the queue.
Here we will derive the endogenous arrival from user to the first relay denoted by . The is also the probability that a transmitted packet by the first user will enter the queue at the first relay. So, the term denotes the endogenous arrival probability from th user, , to the queue at the th relay, .
The endogenous arrival rate is given by
The endogenous arrival rate is given by
Similarly we can define , and . Note that , which is the relayed throughput for the first user defined in the previous subsection.
The average arrival rate at the relay is given by
(8) 
Recall that denotes the probability that the transmitted by the first source packet which is correctly received by both relays and failed to reach the destination will enter the queue at the first relay. The term denotes the probability that the packet will enter the queue at the second relay. Thus, a packet can enter only one queue so we avoid wasting resources by transmitting the same packet twice.
IiiC Stability conditions for the queues at the relays
We now proceed with the investigation of the stability conditions, based on the concept of stochastic dominant systems developed in [17, 18]. The stability region of the system is defined as the set of arrival rate vectors , for which the queues of the relay nodes are stable. Here, we will derive the stability analysis for the total average arrival rate at each relay, .
The next theorem provides the stability criteria for the twouser general MPR case.
Theorem III.1.
The stability region is given by where
(9) 
and
(10) 
Proof.
Since the average service rate of each relay depends on the queue size of the other relay, the stability region cannot be computed directly. Thus, we apply the stochastic dominance technique introduced in [17], i.e. we construct hypothetical dominant systems, in which the relay with the empty queue transmits dummy packets, while the nonempty relay transmits according to its traffic.
In the first dominant system, the first relay transmit dummy packets and the second relay behaves as in the original system. All the rest operational aspects remain unaltered in the dominant system. Thus, in this dominant system, the first queue never empties, hence the service rate for the second relay is
(11) 
Which can be rewritten as
(12) 
Then, we can obtain stability conditions for the second relay by applying Loynes’ criterion [39]. The queue at the second source is stable if and only if , that is . Then we can obtain the probability that the second relay is empty by applying Little’s theorem, i.e.
(13) 
Thus, after applying Loynes’ criterion, the stability condition for the first relay in the first dominant system is
(15) 
The stability region obtained from the first dominant system is given by
(16) 
Similarly, we construct a second dominant system where the second relay transmits a dummy packet when it is empty and the first relay behaves as in the original system. All other operational aspects remain unaltered in the dominant system. Following the same steps as in the first dominant system, we obtain the stability region, , of the second dominant system.
(17) 
An important observation made in [17] is that the stability conditions obtained by the stochastic dominance technique are not only sufficient but also necessary for the stability of the original system.
The indistinguishability argument [17] applies to our problem as well. Based on the construction of the dominant system, it is easy to see that the queue sizes in the dominant system are always greater than those in the original system, provided they are both initialized to the same value and the arrivals are identical in both systems. Therefore, given , if for some , the queue at the first relay is stable in the dominant system, then the corresponding queue in the original system must be stable. Conversely, if for some in the dominant system, the queue at the first relay saturates, then it will not transmit dummy packets, and as long as the first relay has a packet to transmit, the behavior of the dominant system is identical to that of the original system since dummy packet transmissions are eliminated as we approach the stability boundary. Therefore, the original and the dominant systems are indistinguishable at the boundary points. ∎
The stability region obtained in Theorem III.1 is depicted in Fig. 2. To simplify presentation, we denote the points with the following expressions
Remark 1.
The stability region is a convex polyhedron when the following condition holds . In the previous condition, when equality holds, the region becomes a triangle and coincides with the case of timesharing of the channel between the relays. Convexity is an important property since it corresponds to the case when parallel concurrent transmissions are preferable to a timesharing scheme. Additionally, convexity of the stability region implies that if two rate pairs are stable, then any rate pair lying on the line segment joining those two rate pairs is also stable.
Remark 2.
The case of pure relays can be obtained easily by replacing .
Remark 3.
The network without relay’s assistance can be obtained by . In this case, we have a network with saturated users and also two users with bursty traffic that transmit packets only when the saturated users are silent.
Remark 4.
One can connect the endogenous arrivals from the users to the relays with the stability conditions, obtained in Theorem III.1, by replacing the relevant expressions of and into and .
Remark 5.
A slightly different scenario is captured by the case where the relays can transmit in a different channel than the users and the destination can hear both channels at the same time. The receivers at the relays are operating at the same channels where the users are transmitting. In this case, we can have a full duplex operation at the relays on different bands. Thus, we have the following average service rate for the first relay
(18) 
Similarly we have the service rate at the second relay
(19) 
The stability analysis for this case can be trivially obtained by the presented analysis thus, it is omitted. However, this scenario has applicability in nowadays relayassisted networks.
Iv Throughput and Stability Analysis – The symmetric user case for the General MPR case
Here we will generalize the analysis provided in the previous section for the user case. However, due to presentation clarity we will focus on the symmetric user case. The users attempt to transmit with probability . The success probability from a user to the destination is the same for all the users, thus, in order to characterize it we just need the number of active users, i.e. the interference. This probability is denoted by to capture the case that users are attempting transmission (including the user we intend to study its performance), similarly we define .
Iva Endogenous arrivals at the relays and throughput performance
The direct throughput of a user to the destination in the case of symmetric users is given by
(20) 
We will derive the endogenous arrivals at the first and the second relay respectively in order to calculate the relayed throughput in the network. For the symmetric user case we denote the endogenous arrivals from the users at the first (second) relay as ().
We need to characterize the average number of packet arrivals from the users at each relay. Thus, we define as the probability that packets will arrive in a timeslot at the first relay. Similarly, we define . Then, the average endogenous arrival rate at the th relay is given by
(21) 
The probability where is given by
(22) 
Similarly, we obtain for the second relay. Note that the networkwide relayed throughput when both relays are stable is given by . Thus, the aggregate or networkwide throughput of the network when both relays are stable is given by
(23) 
Recall that the total arrival rate at relay is , consisting of the endogenous arrivals from the users and the external traffic.
Below provide the stability conditions at the relays.
IvB Stability conditions for the queues at the relays
The service rates at the at the first relay is given by
(24) 
Similarly, we have the service rate at the second relay is given by
(25) 
Following the same methodology as in the proof of Theorem III.1, we obtain the stability conditions for the symmetric user case. The stability conditions are given by where
(26) 
and
(27) 
V Delay analysis: The twouser case
This section is devoted to the analysis of the queueing delay experienced at the relays. Our aim is to obtain the generating function of the joint stationary distribution of the number of packets at relay nodes. In the following we consider the case of users, and focus on a subclass of MPR models, called the “capture” channel, under which at most one packet can be successfully decoded by the receiver of the node , even if more than one nodes transmit.
In order to proceed, we have to provide some more information regarding the success probabilities of a transmission between nodes that were defined in subsection IIC. More precisely, we have to take into account the number as well as the type of nodes that transmit (i.e., source or relay node). This is due to several reasons, such as the fact that generally the channel quality between relay nodes and destination node is usually better than between sources and destination, as well as due to the wireless interference, since the channel quality is severely affected by the the number of nodes that attempt a transmission. Moreover, it is crucial to take into account the possibility that a failed packet can be successfully decoded by both relays, as well as the ability of the “smart” relay nodes to be aware of the status of the others, which in turn leads to selfaware networks. With that in mind we consider the following cases:

Both sources transmit

When both sources failed to transmit directly to the node , the failed packet of source is successfully decoded by relay with probability , , , where with probability , the relay failed to decode both packets. Note also that , , . Due to the total probability law we have

When source 1 (resp. source 2) is the only that succeeds to transmit a packet at node , i.e., its transmission was successfully decoded by node , then with probability (resp. ), , the failed packet of source (resp. source ) is successfully decoded by the relay . On the contrary, with probability (resp. ), the relay failed to decode the packet from source (resp. source ), and thus, it is considered lost. Due to the total probability law we have,


Only one source transmit, say source , and the other remains silent. When source fails to transmit directly to node , its failed packet is successfully decoded by relay with probability , , , where with probability , the relay fails to decode the packet. Due to the total probability law we have
Note that the cases and refer to the case where only one source cooperate with a relay. However, we have to distinguish it in two cases because in the former one, there is an interaction among sources since both of them transmit, while in the latter one, only one source transmit and the other remains silent (i.e., there is no interaction). Such an interaction, plays a crucial role on the values of the success probabilities. In wireless systems, the feature of interference and interaction among transmitting nodes is of great importance and have to be taken into account.
If both relays transmit simultaneously, with probability , the packet transmitted from is successfully received by node , while with probability , both of them failed to be received by the node , and have to be retransmitted in a later time slot. Recall also the success probabilities ,
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