On the Throughput of Multi-Source Multi-Destination Relay Networks with Queueing Constraints

# On the Throughput of Multi-Source Multi-Destination Relay Networks with Queueing Constraints

Yi Li, M. Cenk Gursoy and Senem Velipasalar The authors are with the Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY, 13244 (e-mail: yli33@syr.edu, mcgursoy@syr.edu, svelipas@syr.edu).The material in this paper was presented in part at the IEEE Wireless Communications and Networking Conference (WCNC), New Orleans, LA, March 2015.
###### Abstract

In this paper, the throughput of relay networks with multiple source-destination pairs under queueing constraints has been investigated for both variable-rate and fixed-rate schemes. When channel side information (CSI) is available at the transmitter side, transmitters can adapt their transmission rates according to the channel conditions, and achieve the instantaneous channel capacities. In this case, the departure rates at each node have been characterized for different system parameters, which control the power allocation, time allocation and decoding order. In the other case of no CSI at the transmitters, a simple automatic repeat request (ARQ) protocol with fixed rate transmission is used to provide reliable communication. Under this ARQ assumption, the instantaneous departure rates at each node can be modeled as an ON-OFF process, and the probabilities of ON and OFF states are identified. With the characterization of the arrival and departure rates at each buffer, stability conditions are identified and effective capacity analysis is conducted for both cases to determine the system throughput under statistical queueing constraints. In addition, for the variable-rate scheme, the concavity of the sum rate is shown for certain parameters, helping to improve the efficiency of parameter optimization. Finally, via numerical results, the influence of system parameters and the behavior of the system throughput are identified.

broadcast channel, buffer overflow, decode-and-forward relaying, effective capacity, fixed-rate transmissions, multiple-access channel, statistical queueing constraints, throughput, variable-rate transmissions.

## I Introduction

Increasing transmission rates, improving energy efficiency and enhancing reliability are important considerations in wireless communications. Various advanced schemes have been proposed to address these concerns. One such strategy is cooperative communications. In particular, relay networks can greatly enhance the performance for long distance transmissions among users and improve resource efficiency. For instance, the throughput of relay networks have been analyzed by several studies. In [1], the achievable rates of Gaussian orthogonal multi-access relay channels in which multiple sources communicate with one destination with the help of one relay were investigated, which were also proved to have a max-flow min-cut interpretation. The throughput region of the same system model was also given in [2] with superposition block Markov encoding and multiple access encoding. Further analysis was also provided in [3], in which the optimal resource allocation strategy was studied to achieve the maximum sum rate. In [4], the system throughput region of the generalized multiple access relay network , which includes multiple transmitters, multiple relays, and a single destination, was studied. In all cases, with the help of relay nodes, the channel conditions effectively improve for long distance wireless communication, and performance enhancements are realized.

A further generalization of multiple-access relay channels is to introduce multiple destination nodes. These models are referred to as multi-source multi-destination relay networks. Multi-source multi-destination relay network model can be seen as a combination of multiple-access, broadcast, and two-hop relay channels, and it can be used to address scenarios in which multiple pairs of users simultaneously communicate with the help of a relay node. A basic practical example of these models is cellular operation in which multiple mobile users within a cell communicate with each other through a base station, which essentially acts as a relay unit between the source and destination nodes111Moreover, in LTE-Advanced cellular standards, relaying and coordinated multi point (CoMP) operation are introduced to provide enhanced coverage and capacity at cell edges, and multi-user relay models can be realized in these operation modes as well.. Such networks have been analyzed in several recent studies. In [5], the throughput of the amplify-and-forward multi-source multi-destination relay network was studied, when the relay was equipped with multiple antennas. Based on this work, the same authors studied the impact of imperfect CSI in [6], and proposed an antenna selection algorithm to improve the performance. In [7], the joint power optimization was investigated for the multi-source multi-destination relay network, and in [8], network coding was applied to this type of network, and the system performance was evaluated.

In addition to cooperative operation, due to the critical delay/buffer requirements in real-time data transmissions, such as in live video streaming, quality of service (QoS) guarantees should be provided for acceptable performance in wireless systems supporting multimedia traffic. With this motivation, we consider the throughput of the multi-source multi-destination relay networks under statistical queueing constraints, imposed as limitations on the decay rate of buffer overflow probabilities at all nodes in the system. In [9], effective bandwidth was introduced as a measure of the system throughput under such statistical queueing or QoS constraints. More specifically, effective bandwidth has been defined as the minimum constant transmission rate required to support time-varying arrivals while the buffer overflow probability decays exponentially with increasing overflow threshold. In [10], effective bandwidths of departure processes with time-varying service rates were investigated, and the theory of effective bandwidth was employed to analyze the performance of high speed networks in [11]. Later, effective capacity was defined in [12] as a dual concept to characterize the maximum constant arrival rates that can be supported by time-varying wireless transmission rates again under statistical queueing constraints.

As noted before, beside QoS requirements, reliability and robustness are important concerns in wireless systems, especially when CSI is not available at the transmitter. In this situation, data transmission can be performed at fixed rates, and reliability can be ensured via automatic repeat request (ARQ) protocols, which trigger retransmissions in cases of decoding failure. This effectively enables the transmitter to adapt to the channel conditions with only limited feedback from the receiver. Queueing analysis has also been performed when ARQ is employed in the communication system. For instance, in [19], queueing models were formulated and performance analysis was conducted for go-back-N and selective repeat ARQ protocols, and the energy efficiency of ARQ with fixed transmission rates was analyzed in [20] under statistical queueing constraints.

In this paper, the system throughput of multi-source multi-destination relay network is investigated under statistical queueing constraints primarily for a network model with two source-destination pairs and one intermediate relay. The following are our main contributions:

1. We characterize the throughput of the multi-source multi-destination relay network under queueing constraints by using the stochastic network calculus framework and effective capacity formulations. We identify the impact of resource allocation policies and decoding strategies on the performance.

2. We extend our analysis to the network with more than two source-destination pairs and also to the model with full-duplex relay operation.

3. We perform an effective capacity analysis for the case in which CSI is not available at the transmitter nodes, all transmitters are sending the data at fixed rates, and an ARQ protocol is employed.

The rest of this paper is organized as follows. We describe our system model in Section II, and introduce preliminary concepts regarding statistical queueing constraints and arrival rates in Section III. In Section IV, system throughput is characterized when CSI is available at the transmitters, and several properties of the throughput are identified. In Section V, extensions of the system model treated in Section III are addressed. In Section VI, we analyze the system throughput when there is no CSI at the transmitters. Finally, we draw conclusions in Section VII.

## Ii System Model

In this paper, we consider a multi-source multi-destination relay network model with two pairs of sources and destinations, as depicted in Figure 1. In this system, two sources and send information to their corresponding destinations and with the help of an intermediate relay node, and there is no direct link between the source nodes and their destinations. This assumption is accurate if the source and destination nodes are sufficiently far apart in distance. We assume that only needs the packets coming from source , where . Each source node has a buffer, keeping the packets to be transmitted to the relay node. The arrival rates at source nodes and are assumed to be constant, and are denoted as and respectively. At the relay node, there are two buffers222In practice, only one physical buffer is sufficient at the relay node to store the received packets from and . In the analysis, we essentially decompose this physical buffer into two equivalent virtual buffers, in each of which data for only one destination is stored and first-in first-out policy is employed., one for keeping the decoded information coming from source , and the other for the decoded data of .

In our setup, relay node performs decode-and-forward relaying and works in half-duplex mode, and hence it cannot transmit and receive at the same time. The entire transmission process can be divided into two phases, namely multiple-access phase and broadcast phase. In the multiple-access phase, both and transmit to the relay node simultaneously through a multiple-access channel. Relay node attempts to decode their messages by using certain decoding orders, and the decoded information bits are stored in their corresponding buffers at the relay. We assume that if fixed-rate transmissions are employed, transmission fails if the rate is greater than the instantaneous capacity of the link for a given decoding strategy at the relay333It is assumed that errors are detected reliably at the receivers, and when the system employs ARQ protocol, acknowledgement (ACK) and retransmission request (RQ) packets are assumed to be received with no errors..

The received discrete-time signal at the relay node can be expressed as

 Yr[i] =g1[i]X1[i]+g2[i]X2[i]+nr[i], (1)

where for represents the transmitted signal from source node , is the fading coefficient of the link, and is the additive Gaussian noise at the relay.

In the broadcast phase, relay node forwards information bits to their destinations through a broadcast channel. The received signal at is

 Yj[i] =hj[i]Xr[i]+nj[i],j=1,2 (2)

where stands for the transmitted signal from , is the additive Gaussian noise at , and represents the channel fading coefficient of the link. Magnitude-squares of the fading coefficients in both phases are denoted by and , for . In our analysis, we consider block fading and assume that fading coefficients stay constant in one time block, and change independently from block to block. While our analysis is general and applicable to any fading distribution with finite variances, we assume Rayleigh fading in all channels in our numerical analysis.

The transmitted signals are subject to energy constraints given by for and , where is the system bandwidth and for is the transmit power constraint for the corresponding node. The additive noise terms for are independent, zero-mean, circularly symmetric, complex Gaussian random variables with variances . Then, signal-to-noise ratios are defined as

 \footnotesize{SNR}k=¯PkN0B (3)

where .

Finally, there are three important system parameters: , and . denotes the fraction of time allocated to the multiple-access phase, and hence the fraction of time allocated to the broadcast phase is . represents the fraction of power allocated by the relay to the transmission of the message intended for , and therefore the fraction of power allocated to the transmission to is . In the multiple-access phase, relay node decodes the received signal using different decoding orders, and the fraction of time allocated to decoding order and at the relay node are denoted by and , respectively. This time sharing strategy between different decoding orders is used only for the case of variable-rate transmissions, performed when CSI is available at all transmitters. For fixed-rate transmission schemes, decoding order is part of the decoding strategy, which is fixed for each node.

## Iii Preliminaries on Statistical Queueing Constraints and Arrival Rates

In our work, we assume that the queueing constraints are imposed so that buffer overflow probability decays exponentially fast, i.e., we have

 Pr{Q≥qmax}≈γe−θqmax (4)

where is the stationary queue length, is a sufficiently large buffer overflow threshold, is the probability that buffer is non-empty, and is called the QoS exponent. This QoS exponent can more precisely be formulated as

 θ=limqmax→∞−logPr{Q≥qmax}qmax. (5)

It is obvious that a larger value implies stricter constraints on the buffer overflows.

In our analysis, the departure process from each buffer is assumed to be a stationary process. We first define the asymptotic logarithmic moment generating function (LMGF) of an arrival or service process as a function of the QoS parameter as 444Throughout the text, logarithm expressed without a base, i.e., , refers to the natural logarithm .

 ΛA(θ)=limn→∞logE{eθ∑ni=1a[i]}n. (6)

It can be easily verified that the asymptotic LMGF of a constant-rate arrival process, , is .

In our multi-source multi-destination relay system, we assume that the QoS exponents at the source nodes and and the relay node are denoted by , and , respectively. Hence, we wish to have the buffer overflow probability at node to behave approximately as for large .

From the theory of effective bandwidth and effective capacity [9], [10], [12], the buffer overflow probability decays exponentially as or faster at if the constant arrival rate at satisfies

 Rj=−ΛSj,R(−~θj)~θj,j=1,2 (7)

for some . Above is the asymptotic LMGF of service process at .

The above arrival rate formulation considers only the queueing constraints at the source nodes. However, we need to address the constraints at the relay buffers as well. With the characterization of the effective bandwidth of departure processes in queues with time-varying service rates, it was shown in [10] that the buffer overflow probabilities at the relay decay as or faster for large if we have

 ΛR(^θj)+ΛR,Dj(−^θj)=0 (8)

for some , . Above, is the LMGF of the service rate at for the transmission of the message to .

In (8), is the asymptotic LGMF of the arrival process to (or equivalently the departure process from ) and is formulated as [10, equation (18)]

 (9)

for .

Hence, in order to comply with both the source and relay queueing constraints, the arrival rate at should satisfy (7) and (8) simultaneously.

In this paper, system throughput is characterized by the pair of maximum constant arrival rates and that can be supported by the relay network with two pairs of source-destination nodes in the presence of statistical queueing constraints.

Finally, we provide a list of notations together with their descriptions in Table I.

## Iv Throughput of the Two-Source Two-Destination Relay Network With Variable Transmission Rates

In this section, we study the throughput of the two-source two-destination relay network with variable-rate transmissions. Under the assumption that CSI is available at each transmitter, transmitters adapt their transmission rate to the instantaneous channel conditions, and the departure rates at each buffer are given by the corresponding instantaneous channel capacities. To perform an effective capacity analysis at each node with a buffer, we have to first identify the instantaneous transmission rates as functions of the fading coefficients.

### Iv-a Instantaneous Transmission Rates in Multiple User Relay Networks

We initially describe the instantaneous transmission rates of four links. Let us first consider the multiple-access phase in which links and are active simultaneously. When the decoding order at the relay is given by , i.e., the information sent from node is decoded first, and the information sent from node is decoded after interference cancelation, then the maximum instantaneous achievable rates at and are given, respectively, by [16]

 ⎧⎨⎩RS1,R{1,2}=Blog2(1+\footnotesize{SNR}1z11+\footnotesize{SNR% }2z2),RS2,R{1,2}=Blog2(1+% \footnotesize{SNR}2z2). (10)

If the decoding order at the relay node is , then we have

 ⎧⎨⎩RS1,R{2,1}=Blog2(1+\footnotesize{SNR}1z1),RS2,R{2,1}=Blog2(1+% \footnotesize{SNR}2z21+\footnotesize{SNR}1z1). (11)

If we perform time-sharing between two decoding orders with parameter , then the rates of links and are characterized by (10) in fraction of the time, and the rates are characterized by (11) rest of the time. Overall, the transmission rates between the source nodes and the relay node can be expressed as

 RSj,R=δRSj,R{1,2}+(1−δ)RSj,R{2,1}, (12)

for .

In the broadcast phase, relay node forwards packets to their corresponding destinations. In this phase, only links and are active. When the channel conditions are available at the relay node and destinations, the instantaneous transmission rates are given by

 ⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩RR,D1=Blog2(1+ρ\footnotesize{SNR}rω11+(1−ρ)% \footnotesize{SNR}rω1\mathds1{ω1<ω2}),RR,D2=Blog2(1+(1−ρ)% \footnotesize{SNR}rω21+ρ\footnotesize{SNR}rω2\mathds1{ω2<ω1}) (13)

where is indicator function.

### Iv-B Stability Conditions

With the expressions of the instantaneous rates for both the multiple-access channel and broadcast channel described above, we can characterize the stability region in the space. Stability at the source buffers is ensured by requiring the arrival rates to satisfy (7), which actually leads to compliance with the stricter condition that the tail distribution of the buffer length decays exponentially fast. The stability conditions at the relay node requires the average arrival rate to be less than or equal to the average departure rate at each buffer in the relay. Hence, the stability conditions can be formulated as

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩τ(δE{RS1,R{1,2}}+(1−δ)E{RS1,R{2,1}})≤(1−τ)E{RR,D1},τ(δE{RS2,R{1,2}}+(1−δ)E{RS2,R{2,1}})≤(1−τ)E{RR,D2}. (14)

Plugging (10), (11), and (13) into (14), we obtain

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩(1−τ)E{Blog2(1+ρ\footnotesize{SNR}rω11+(1−ρ)\footnotesize{SNR}% rω1\mathds1{ω1<ω2})}≥τ(δE{Blog2(1+% \footnotesize{SNR}1z11+\footnotesize{SNR}2z2)}+(1−δ)E{Blog2(1+\footnotesize% {SNR}1z1)}),(1−τ)E{Blog2(1+(1−ρ)\footnotesize{% SNR}rω21+ρ\footnotesize{SNR}rω2\mathds1{ω2<ω1})}≥τ(δE{Blog2(1+% \footnotesize{SNR}2z2)}+(1−δ)E{Blog2(1+% \footnotesize{SNR}2z21+\footnotesize{SNR}1z1)}). (15)

All feasible (,,)-tuples satisfying the inequalities in (15) form the stability region in the space. Hence, we formally define the the stability region in the space as

 Ξ={(ρ,τ,δ)|ρ,τ,andδthat % satisfy (???)}. (16)

For a certain time sharing scheme at the relay node with fixed , since is the time fraction allocated to the multiple-access phase, lower value is more likely to satisfy the stability condition, and the two inequalities in (15) provide two upper bounds on as functions of . The power allocation parameter has a different influence on these two phases. With more power allocated to transmission to in the broadcast phase, the corresponding buffer in the relay can support a higher value while satisfying the stability constraint.

### Iv-C Throughput Region under Statistical Queueing Constraints

As noted before, for a certain parameter setting, the system throughput is defined as the pair of constant arrival rates and , which can be supported by two-hop links and , respectively, under queueing constraints. Since stability is a prerequisite for effective capacity analysis, our system throughput is only defined with parameter values included in the stability region. For those parameter settings outside the stability region, at least one of the queueing constraints cannot be satisfied, and the system throughput is set to zero. Using the results in the previous section, to comply with queueing constraints at all nodes, for has to satisfy (7) and (8) simultaneously, which leads to the following characterization of the system throughput.

###### Theorem 1

For any parameter setting that satisfies the stability conditions, the maximum constant arrival rate , which can be supported at source node for in the presence of all queueing constraints, is given by

 (17)
###### Proof:

We know that both and links are restricted by two queueing constraints, one at the corresponding source node, and the other one at the relay node. We consider these two constraints separately, and then combine the results. First, we only consider the constraints at the source nodes. According to (7), the maximum arrival rate that can be supported under queueing constraints at a source node is given by

 Rj=−ΛSj,R(−θj)θj, (18)

for . Similarly, when we only consider the queueing constraint at the relay node, the maximum arrival rates should satisfy

 Rj=⎧⎨⎩−1θrΛR,Dj(−θr)θr≤θj−1θj(ΛR,Dj(−θr)+ΛSj,R(θr−θj))θr>θj, (19)

which is obtained from (8) and (9). Combining these results, the overall maximum arrival rates that can be supported by the system should be the minimum of (18) and (19), i.e.,

 Rj=⎧⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪⎩min{−1θjΛSj,R(−θj),−1θrΛR,Dj(−θr)}θr≤θjmin{−1θjΛSj,R(−θj),−1θj(ΛR,Dj(−θr)+ΛSj,R(θr−θj))}θr>θj, (20)

for . Using the definition of LMGF in (6), (20) can be expressed in terms of the instantaneous rates, which is given by (17).

\qed

Following this characterization, some properties of the system throughput are shown in the next subsection.

### Iv-D Properties of the System Throughput under Queueing Constraints

In the previous subsection, we have characterized the throughput of the two-source two-destination relay network. Based on (17), we next analyze the behavior of the throughput in the parameter space, and establish several convexity properties, which can lead to simplifications in parameter optimization.

###### Theorem 2

In the stability region, for a given pair, the maximum arrival rates , and the sum rate are concave over the time sharing parameter between different decoding orders at the relay.

###### Proof:

Depending on the relationship between and for , there are two possible cases identified by (17).

.
In this case, the throughput is given by

 Rj=min{Rj,1,Rj,2} (21)

where and are defined as

 ⎧⎪⎨⎪⎩Rj,1=−1θjlog(E{e−θjτ(δRSj,R{1,2}+(1−δ)RSj,R{2,1})}),Rj,2=−1θrlog(E{e−θr(1−τ)RR,Dj}). (22)

By taking the second order derivative with respect to , we can easily show the concavity of and . The second order derivative of is given by (23) on the next page.

According to the Cauchy-Schwarz inequality, two random variables and should satisfy . Assuming that

 U=e−12θjτ(δRSj,R{1,2}+(1−δ)RSj,R{2,1}), (24)

and

 V=(RSj,R{1,2}−RSj,R{2,1})U, (25)

we can easily determine that the part inside the large curly brackets in (23) can be written as and hence is nonnegative. Then, we can readily determine that , which indicates that is a concave function of . From (22), we notice that the expression of does not contain . In other words, is a constant function in terms of , and . Hence, we can still regard as a concave function of .

Since the pointwise minimum of concave functions is concave [21], the concavity of and with respect to the time sharing parameter follows immediately when .

.
In this case, the throughput is given by

 Rj=min{Rj,1,Rj,3} (26)

where is defined as

 Rj,3=− 1θj(log(E{e−θr(1−τ)RR,Dj}) (27) +log(E{e(θr−θj)τ(δRSj,R{1,2}+(1−δ)RSj,R{2,1})})).

We have already shown the concavity of in the previous case, and we can show the concavity of following the same approach. The second order derivative of is given by (28) on the next page.

Again using the Cauchy-Schwarz inequality, we have , and the concavity follows. Since is the pointwise minimum of and , is a concave function of . Now, we have shown in both cases that and are concave functions of .

Finally, since the sum of two concave functions is also a concave function, the sum rate is concave as well. \qed

Theorem 2 indicates that there exists a globally optimal time sharing parameter for the two possible decoding orders at the relay, which can be determined via convex optimization methods. Similarly, the system throughput functions are also concave functions of , which is the parameter for time allocation between the multiple-access and broadcast phases.

###### Theorem 3

In the stability region, for given power allocation parameter and time-sharing parameter , the maximum arrival rates , and the sum rate are concave over the time allocation parameter .

###### Proof:

Similar to the proof of Theorem 2, Theorem 3 can be proved easily by evaluating the derivatives with respect to . The second order derivatives of , and with respect to are given, respectively, by (29)-(31) on the next page.

Using the Cauchy-Schwarz inequality and concavity-preserving property of pointwise minimum, the concavity of , and the sum rate follow readily. \qed

Using these results, we can maximize the system throughput over and under stability constraints by employing efficient convex optimization methods.

### Iv-E Numerical Results

In this subsection, numerical results are provided to further analyze the throughput of the two-source two-destination relay network with variable transmission rates. Our numerical results are based on (17).

In order to verify our analysis, we have conducted Monte Carlo simulations in which we have generated arrivals to the buffer at constant rates determined by our theoretical characterization in (17) and also generated random (Rayleigh) fading coefficients to simulate the wireless channel and random transmission rates. We have tracked the buffer occupancy and overflows for different threshold levels. We plot the simulated logarithmic buffer overflow probabilities as functions of the overflow threshold in Figs. 2 and 3. In each simulation, we generate time blocks to estimate the buffer overflow probability, and repeat each simulation times to evaluate the averages. We set the queueing constraints as , and the constant arrival rates at nodes and are determined from (17). In both figures, , , dB. In Fig. 2, we set dB. Note from (4) that . Therefore, the slope of the logarithmic overflow probability is expected to be proportional to . Although (4) requires large , our simulation results show that can be approximated as a linear function of starting from relatively small . In Fig. 2, the slopes of the logarithmic overflow probabilities at buffers in and are and , respectively. This implies that simulation results demonstrate perfect agreement with the analysis and the arrival rates given by (17) fit the queueing constraints at and exactly. We also observe that the logarithmic overflow probabilities of the two relay buffers decay faster with steeper slopes than our requirement of . In this specific example, due to relay having a relatively large transmit power, the system performance is mainly decided by the multiple-access phase, which is the bottleneck of the system. Although the relay can potentially support higher and , this is not allowed by the multiple-access phase. As we reduce the transmission power of the relay node, the system bottleneck shifts to the broadcast phase and the situation is reversed. In Fig. 3, we reduce to dB. Now, the arrival rates given by (17) fit the queueing constraints at the relay exactly, and the corresponding slopes are and , respectively. On the other hand, the decays of the overflow probabilities at the source nodes are faster, meaning that sources can potentially support higher arrival rates but this leads to the violation of the overflow constraints at the relay buffers and is therefore not allowed. Overall, these simulation results, while confirming the analysis, also interestingly unveil the critical interactions between the queues and buffer constraints.

For the rest numerical results in this subsection, we consider Rayleigh fading and we set dB and dB. Fig. 4 shows the influence of the position of the relay node for different values. We assume a symmetric model, in which , and and , where stands for the distance between and . The overall distance , and the position parameter . Obviously, , and the smaller value of indicates that relay is closer to the source. Path loss as a function of distance is incorporated into the statistics of fading powers, and hence, we have and for . In the figure, we see that the maximum sum rate is achieved when is close to , which means that it is better to place the relay in the middle between the source and destination in this symmetric setting. When the relay is close to the source nodes, the channels between the relay and destinations deteriorate and the overall throughput is limited by the broadcast links. Similarly, the multiple-access links become the bottleneck when is close to . Also, we observe that the system throughput decreases when increases due to tighter queueing constraints. This occurs because when is small, the effective capacity is closer to the Shannon capacity, and as increases, effective capacity diminishes and approaches the zero-outage capacity (which is, for instance, zero in Rayleigh fading).

In Fig. 5, we consider an asymmetric scenario in terms of QoS exponents, and again plot sum rate vs. relay location parameter . We fix and determine the optimal value of for each given . When , , the maximum sum rate is achieved at . In this case, relay should be placed closer to the destinations to support more stringent queueing constraints at the relay. On the other hand, when , , the optimal position for the relay is at . Hence, the relay needs to be closer to the source nodes to support their stricter queueing constraints. These observations indicate the sensitivity of optimal relay placement to different QoS requirements.

Figs. 6 and 7 demonstrate the concavity555These concavity results can simplify the search for the optimal parameter setting with the use of convex optimization tools. of the sum rate with respect to and , respectively, when the parameter values are in the stability region. In these two figures, , and . In Fig. 6, the sum rate curves first increase with , and then decrease very fast after reaching the maximum sum rate. As exceeds a threshold, the sum rates drop to , because stability conditions are violated beyond this threshold. In Fig. 7, the sum rate curves are concave with respect to the decoding parameter , and the optimal values which maximize the sum rate are all close to . In this case, relay allocates time to two decoding orders equally. However, note that these results are again for a symmetric scenario in which all QoS exponents are the same. In Fig. 8, we address a heterogeneous setting in terms of QoS exponents. For instance, when and , the optimal value of is 1. Hence, sum rate is maximized when the decoding order at the relay is always fixed as , i.e., relay initially decodes data arriving from source in the presence of interfering signal of . The underlying reason for this result is the following. Source operates under stricter QoS constraints with respect to and consequently can support smaller arrival rates and needs, in turn, smaller transmission rates which can be sustained even in the presence of interference. If the roles are switched (i.e., if we have and ), then the optimal value of is zero. If the QoS exponents are more comparable (e.g., and or and ), we notice that optimal values of start to slightly deviate from the two extremes of 0 and 1.

Fig. 9 shows the throughput regions of the two-source two-destination relay network under different queueing constraints. The boundary of the throughput region is obtained by searching over the three-dimensional parameter space. When achieves its maximum value, is close to , and is slightly greater than , because decoding order and more power in the link can help link to support higher arrival rates. Similar results are also obtained for the maximum value of the arrival rate .

## V Extension to Multiple Source-Destination and Full-Duplex Models

In this section, we study the extensions of the two-source two-destination relay model, addressed in Section IV. The first extension is to generalize the results to the multi-source multi-destination relay model with more than two source-destination pairs. The second extension is to full-duplex operation of the relay.

### V-a Multi-Source Multi-Destination Relay Network

In this subsection, we consider a multiple-user model in which sources send information to their corresponding destinations with the help of a relay node. The magnitude-squares of the fading coefficients of links and are represented by and , respectively. We assume that there are separate buffers in the relay node, each one for the information arriving from a different source. Again, all buffers in the relay node are assumed to share the same QoS exponent , while source node may have its own QoS exponent . Compared with the two-user model, adding more users only increases the dimension of the parameter space while the analytical methods and results essentially remain the same.

In this multi-user setting, system parameters and become vectors, while the time allocation parameter is still a scalar. The definition of is kept the same as the fraction of time allocated to the multiple-access phase. The power allocation parameter becomes an -dimensional vector , and the component represents the fraction of power allocated to the data transmission to , . The elements of should be between and , and satisfy . Since there are different decoding orders at the relay in the multiple-access phase, the time-sharing parameter becomes an -dimensional vector , and the component represents the fraction of time allocated to the decoding order at the relay node. Similarly, all the elements of should be between and , and satisfy .

In the multiple-access phase, we denote the decoding order at the relay as , which is a permutation of . With this decoding order, the instantaneous rate of the link is characterized by

 RSki,R,πk=Blog2⎛⎜⎝1+\footnotesize{SNR}kizki1+∑Nj=i+1\footnotesize{SNR}kjzkj⎞⎟⎠. (32)

Given a time sharing vector , the rate of the link is given by

 (33)

for . For the broadcast channel, the instantaneous rate is given by

 RR,Dj=Blog2⎛⎝1+ρj% \footnotesize{SNR}rωj1+∑Nl=1,l≠jρl% \footnotesize{SNR}rωj\mathds1{ωj<ωl}⎞⎠, (34)

for . Similarly, the stability region in the parameter space is defined as

 Ξ={(τ, ρ1,⋯,ρN,δ1,⋯,δN!)|τ,ρandδthat satisfy τE{RSj,R}≤(1−τ)E{RR,Dj}, N∑i=1ρi=1andN!∑i=1δi=1,for allj=1,2,⋯,N}. (35)

In this multiple-user setting, the dimension of the parameter space becomes much higher than that in the two-user model. For a set of parameters that guarantee the stability conditions, the throughput of the link under queueing constraints satisfies (7) and (8) simultaneously, and hence is given by (17), for , with the instantaneous rate expressions provided above.

### V-B Full-Duplex Two-Source Two-Destination Relay Network

In this subsection, we extend our analysis on half-duplex system to full-duplex two-source two-destination relay network. In full-duplex mode, nodes , and relay transmit all the time, and thus there is no parameter in this case. In full-duplex mode, relay node experiences self-interference due to its transmission to and . The received signal at relay is given by

 Yr[i]=g1[i]X1[i]+g2[i]X2[i]+Is[i]+nr[i] (36)

where is the self-interference term caused by simultaneous transmission of the signal that relay sends to and . Since relay knows the signal it sends to the destination nodes, it can perform self-interference cancelation. Here, we assume that relay can perfectly eliminate its self-interference, and the received signal at relay after self-interference cancelation is given by (1). In order to satisfy the stability conditions at the relay, two source nodes need to reduce their transmission power, when the arrival rates are larger than departure rates. Therefore, we introduce two new parameters, and , which represent the fraction of power that and use for transmission, respectively. Obviously, we have . Parameters and are defined in the same way as in the half-duplex mode. Then, for a given pair, the instantaneous transmission rates in the multiple-access phase become

 ⎧⎨⎩RS1,R{1,2}=Blog2(1+α1\footnotesize{SNR}1z11+α2\footnotesize{SNR}2z2),RS2,R{1,2}=Blog2(1+α2% \footnotesize{SNR}2z2), (37)

if the decoding order at the relay node is . On the other hand, we have

 ⎧⎨⎩RS1,R{2,1}=Blog2(1+α1\footnotesize{SNR}1z1),RS2,R{2,1}=Blog2<