Effective Capacity of Buffer-Aided Full-Duplex Relay Systems with Selection Relaying

# Effective Capacity of Buffer-Aided Full-Duplex Relay Systems with Selection Relaying

\authorblockNDeli Qiao The author is with the School of Information Science and Technology, East China Normal University, Shanghai, China, 200241 (e-mail: dlqiao@ce.ecnu.edu.cn).This work was supported in part by the National Natural Science Foundation of China under Grant 61172085. This paper will be presented in part at the IEEE Global Communications Conference (Globecom), San Diego, CA, Dec. 2015.
###### Abstract

In this work, the achievable rate of three-node relay systems with selection relaying under statistical delay constraints, imposed on the limitations of the maximum end-to-end delay violation probabilities, is investigated. It is assumed that there are queues of infinite size at both the source and relay node, and the source can select the relay or destination for data reception. Given selection relaying policy, the effective bandwidth of the arrival processes of the queue at the relay is derived. Then, the maximum constant arrival rate can be identified as the maximum effective capacity as a function of the statistical end-to-end queueing delay constraints, signal-to-noise ratios (SNR) at the source and relay, the fading distributions of the links, and the relay policy. Subsequently, a relay policy that incorporates the statistical delay constraints is proposed. It is shown that the proposed relay policy can achieve better performance than existing protocols. Moreover, it is demonstrated that buffering relay model can still help improve the throughput of relay systems in the presence of statistical delay constraints and source-destination link.

Buffer-aided relay, statistical delay constraints, selection relaying, effective capacity, intree-network.

## I Introduction

Relay channels can help improve the system coverage and throughput, and hence information-theoretic analysis of relay channels has been the research forefront for decades (see, e.g., [1]-[3]). For instance, Laneman et al. in [2] considered different relaying strategies, such as decode-and-forward (DF) and selection relaying, and showed that considerable cooperative diversity can be achieved with the relaying schemes. Among the relaying protocols, selection relaying schemes are attractive due to their potential to improve bandwidth utilization and cooperation diversity [3]. While providing powerful results, information-theoretic studies generally assume no buffer at the relay.

Recently, it has been shown that the achievable throughput can be further improved with the introduction of buffering relay model [4]. This is generally due to the information storage at the relay such that the shortcomings of existing relaying schemes caused by bad channel conditions can be overcome. The analysis of the buffer-aided relay systems has attracted much attention recently (see, e.g., [5]-[13] and references therein). For instance, the authors in [5] analyzed the two-hop relay system with buffer-aided relaying for adaptive and fixed rate transmission, and proposed the throughput-optimal buffer-aided relaying protocols with significant performance improvements. In [6], the authors proposed a max-max relay selection protocol, which chooses the source-relay and relay-destination link with the strongest channel gain. They found that this policy can achieve better performance than systems without buffering relay [3]. In [13], the authors proposed a relay policy based on the relay link with strongest channel gain and the direct link that help improve the system performance. However, these works on buffer-aided relaying systems rarely consider the buffer at the source. In the presence of the buffer at the source, the delay experienced at the source buffer will be taken into account as well for the end-to-end delay, and the queue dynamics of the interacted queues are generally difficult to analyze. Note that in [14], the authors investigated the buffer-aided relay systems with buffer at the source and energy-harvesting capability at each node, although the analysis is based on the average arrival and service rate, and only stability regions of different strategies are considered.

In this paper, we follow a different approach to analyze the buffer-aided relay systems. We consider the statistical delay constraints, imposed on the limitations of the maximum end-to-end delay violation probabilities. We assume that there are buffers of infinite size at both the source and relay node, each subject to the statistical queueing constraints imposed on the limitations of buffer overflow probabilities. We consider the end-to-end delay composed of the queueing delay at the source and relay buffer. To handle the queueing dynamics of the relay networks, we employ the concept of effective bandwidth, which defines the bandwidth usage of given processes [33]. More recently, Wu and Negi in [15] defined the dual concept of effective capacity, which provides the maximum constant arrival rate that can be supported by a given departure process while satisfying statistical delay constraints. The analysis and application of effective capacity in various settings have attracted much interest recently (see e.g. [16]-[29] and references therein). For instance, Tang and Zhang in [16] analyzed the power allocation policies of relay networks with only one relay, where the relay node is assumed to have no buffer, i.e., the packets arriving to the relay node are forwarded immediately. In [18], Efazati and Azmi considered a multirelay network, and proposed a novel transmission scheme that selects different strategies such as best relay selection and distributed space-time coding based on the criterion that maximizes the obtained effective capacity. Still, there is no buffer at the relays in the considered system model. Parag and Chamberland in [24] provided a queueing analysis of a butterfly network with constant rate for each link, while assuming that there is no congestion at the intermediate nodes. For the buffer-aided relay networks with buffer at the source, Du and Zhang in [25] analyzed the throughput and power allocation policy in two-hop links under statistical end-to-end delay constraints, while imposing symmetric statistical delay constraints to the queues at the source and relay. The effective capacity of the two-hop link in the presence of statistical queueing constraints is given in [27], and the performance of multi-relay links is analyzed in [28]. As a further step, we derived the maximum effective capacity of the two-hop links under statistical delay constraints with asymmetric delay constraints to the queues in [29]. However, to the best knowledge of the author, there is no related work considering the buffer-aided relay systems with source-destination link.

In this work, we consider the buffer-aided relay systems with source-destination link. We assume that the channel state information (CSI) of all links is available at the source and relay, while the destination has no CSI of the source-relay link. We assume that the source employs selection relaying strategy, i.e., the source selects the relay or destination for data reception based on the CSI available. Since the CSI of the source-relay link is not available at the destination, we assume that the relay policy is informed to the destination via an one-bit acknowledge (ACK) signal such that the destination can perform successive decoding of the received signals when destination is selected for data reception. Given the relay protocol and statistical queueing constraints of the queues, we characterize the effective bandwidth of the arrival processes to the relay, which is one of the major findings of this work since it can be extended to various relay networks and constitutes an important basis for the ensuing effective capacity analysis. Then, based on the statistical delay tradeoff established in [29], we characterize the maximum effective capacity under the statistical delay constraints with fixed relay strategy. Also, we propose a relay scheme taking into account the statistical delay constraints. We show that the maximum effective capacity of the proposed variable relay schemes can be characterized similarly. Through numerical evaluation, this study reveals the benefits of buffering relay model in the presence of delay constraints and source-destination link. The main contributions of this paper are summarized as follows:

1. We provide a framework for analyzing the throughput of the relay systems with selection relaying under statistical delay constraints.

2. We determine the maximum effective capacity of the relay system with arbitrary selection relaying policy, and we obtain the limiting performance as the delay constraints vanish.

3. We propose a relay scheme based on statistical delay constraints that can further improve the achievable throughput compared with the existing best relay selection policy of buffer-aided relay systems.

The rest of this paper is organized as follows. Section II discusses the necessary preliminaries on the system model, the statistical delay constraints and effective capacity. In Section III, we present our main results on the effective capacity and relay policy, with numerical results given in Section IV. Finally, Section V concludes this paper, while some lengthy proofs are relegated to Appendix.

## Ii Preliminaries

### Ii-a System Model

The three-node relay communication link is depicted in Fig. 1. The source can select the relay or destination for data reception. In this model, there are buffers of infinite size at both the source and relay node. In this work, we assume full-duplex relay such that transmission and reception can be performed simultaneously. Note that full-duplex relaying can be achieved through some form of analog self-interference cancellation followed by digital self-interference cancellation in the baseband domain [31], [32].

The discrete-time input and output relationships in the th symbol duration are given by

 Yr[i] =gsr[i]Xs[i]+I[i]+nr[i] (1) Yd[i] =gsd[i]Xs[i]+grd[i]Xr[i]+nd[i] (2)

where for denote the input signal from the source S and the relay R, respectively. The inputs are subject to individual average energy constraints where is the bandwidth. represent the received signal at the relay R and the destination D, respectively. We assume that the fading coefficients are jointly stationary and ergodic discrete-time processes, and we denote the magnitude-square of the fading coefficients by , , and . Note that denotes the self-interference incurred by the full-duplex operation at the relay, which may include linear and nonlinear components of the transmitted signal of the relay [32]. In [31, Appendix A.1], the received signal’s SNR can be modeled as with relative SNR loss due to self-interference. Hence, we normalize the channel gain of the link by to take into account the self-interference, i.e., . Denote . Assuming that there are complex symbols per second, we can easily see that the symbol energy constraint of implies that the channel input has a power constraint of . Above, in the channel input-output relationships, the noise component is a zero-mean, circularly symmetric, complex Gaussian random variable with variance for . The additive Gaussian noise samples are assumed to form an independent and identically distributed (i.i.d.) sequence. We denote the signal-to-noise ratio at source as , and at relay as .

### Ii-B Statistical Delay Constraits

With the above mentioned settings, we first need the following result from [33].

###### Theorem 1 ([33])

Suppose that the queue is stable and that both the arrival process and service process satisfy the Gärtner-Ellis limit, i.e., for all , there exists a differentiable logarithmic moment generating function (LMGF) such that111Throughout the text, logarithm expressed without a base, i.e., , refers to the natural logarithm .

 limn→∞logE{eθ∑ni=1a[n]}n=ΛA(θ), (3)

and a differentiable LMGF such that

 limn→∞logE{eθ∑ni=1c[n]}n=ΛC(θ). (4)

If there exists a unique such that

 ΛA(θ∗)+ΛC(−θ∗)=0, (5)

then

 limQ{max}→∞logPr{Q>Q{max}}Q{max}=−θ∗. (6)

where is the stationary queue length.

This theorem tells us that the tail distribution of the stationary queue length decays exponentially. The proof of the theorem takes advantage of the large deviations principles (see e.g. [34] and [35]) to characterize the exponential decay rate of buffer overflow probability. In particular, for large , we have the approximation for the buffer violation probability: . Hence, while larger corresponds to more strict queueing constraints, smaller implies looser queueing constraints.

Assume that the first-in first-out (FIFO) queues are saturated, and hence they always attempt to transmit [14]. Then the queueing delay violation probability can be written equivalently as [17], [19]

 Pr{D>D{max}}≐e−J(θ)D{max} (7)

where we defined when , and

 J(θ)=θδ=−ΛC(−θ) (8)

is the statistical delay exponent associated with the queue, with the LMGF of the service rate, and is decided by the arrival and departure processes jointly. Now, we can express the probability density function of random variable as

 pD(x)=∂∂x(1−Pr{D>x})≐J(θ)e−J(θ)x. (9)

In this work, we seek to identify the maximum constant arrival rate to the source that can be supported by the relay system while satisfying the statistical delay constraints. Therefore, we need to guarantee that the data transmission of the flow with the largest end-to-end delay should satisfy the statistical delay constraints, i.e., information flow over two queues. Consider two concatenated queues with statistical queueing constraints specified by and , for queue 1 and queue 2, respectively. Given the queueing constraints specified by and with (6) satisfied for each queue, we define

 J1(θ1)=−ΛC,1(−θ1),andJ2(θ2)=−ΛC,2(−θ2), (10)

where and are the LMGF functions of the service rate of queue 1, 2, respectively. For data going through both queues, the end-to-end queueing delay violation probability can be characterized as222Note that the end-to-end delay consists of the queueing and transmission delays. As indicated in [30, Section IV], the flow of data bits are treated as the flow of a fluid in the theory of effective bandwidth, in which case the transmission delay can be negligible if . The end-to-end delay can be approximated by the queueing end-to-end delay [17], [19].

 Pr{D1+D2>D{max}}≐1−∫D{% max}0∫D{max}−D10pD(D1)pD(D2)dD2dD1 =⎧⎪⎨⎪⎩J1(θ1)e−J2(θ2)D{max}−J2(θ2)e−J1(θ1)D{% max}J1(θ1)−J2(θ2),J1(θ1)≠J2(θ2)(1+J1(θ1)D{max})e−J1(θ1)D% {max},J1(θ1)=J2(θ2). (11)

Thereby, we need to guarantee that

 Pr{D1+D2>D{max}}≤ε. (12)

In this way, we can guarantee that the data transmissions through the relay, i.e., information flow over two queues, satisfy the statistical delay constraints. Then, the delay constraints of the whole system can be satisfied. Note that characterizes the statistical delay constraints with maximum delay violation probability and maximum delay .

To facilitate the following analysis, we need the following tradeoff between and .

###### Lemma 1 ([29])

Consider the following function

 ϑ(J1(θ1),J2(θ2)) =J2(θ2)e−J1(θ1)D{max}−J1(θ1)e−J2(θ2)D{max}J2(θ2)−J1(θ1) =e−J0D{max}=ε,for0≤ε≤1, (13)

where is defined as the statistical delay exponent associated with . Denoting as a function of , we have

1. is continuous. For , we have

 Φ(J1(θ1))=Jth(ε) (14)

where

 Jth(ε)=−1D{max}(1+W−1(−εe)) (15)

where is the Lambert W function, which is the inverse function of in the range .

2. is strictly decreasing in .

3. is convex in .

4. , and .

This lemma indicates that to have the end-to-end delay constraints satisfied, we must increase if is decreased; vice versa.

### Ii-C Effective Capacity

Denote the queue at source S as queue 1, and the queue at relay R as queue 2. Denote as the set of pairs such that (12) can be satisfied. Assume and at the source and the relay node with . Assume that the constant arrival rate at the source is , and the channels operate at their capacities. Then, the effective capacity with statistical queueing constraints is defined as the maximum constant arrival rate such that both the queueing constraints can be satisfied. More specifically, to satisfy the queueing constraint at the source, we should have

 ~θ≥θ1 (16)

where is the solution to

 R=−ΛC,1(−~θ)~θ (17)

and is the LGMF of the service rate for queue 1, i.e., queue at the source.

Also, in order to satisfy the queueing constraint of the relay node R, we must have

 ^θ≥θ2 (18)

where is the solution to

 ΛA,2(θ)+ΛC,2(−θ)=0. (19)

where is the LGMF of the arrival process to queue 2, is the LGMF of the service process of queue 2, i.e., queue at the relay.

Note that we can derive the effective capacity with following the method provided in [27, Theorem 2]. After these characterizations, effective capacity of the buffer-aided relay system under statistical delay constraints can be formulated as follows.

###### Definition 1

The effective capacity of the buffer-aided relay system with statistical delay constraints specified by is given by

 Rε(ε,D{max})=sup(θ1,θ2)∈ΩRE(θ1,θ2). (20)

Hence, effective capacity is now the maximum constant arrival rate that can be supported by the relay system under statistical delay constraints.

## Iii Effective Capacity with Selection Relaying Protocol in Block-Fading Channel

In the following, we first discuss the transmission strategy in detail to obtain the associated channel rate of the links. Then, assuming that are given, we obtain the effective bandwidth of the arrival processes of the queue at the relay given selection relaying policy. Next, we derive the effective capacity for arbitrary relay policy in a general form. Afterwards, we propose a relay policy taking into account the statistical delay constraints.

### Iii-a Transmission Strategy and Channel Rate

We assume that perfect CSI of all links is available at S and R, while only the CSI of the links and is available at D. The transmission power levels at the source and relay are fixed and hence no power control is employed (i.e., nodes are subject to short-term power constraints). We further assume that the channel capacity for each link can be achieved, i.e., the service processes are equal to the instantaneous Shannon capacities of the links. We consider a block fading scenario in which the fading stays constant for a block of seconds and changes independently from one block to another.

We consider selection relaying protocols, i.e., the source selects the relay for data reception when the channel condition at the relay is larger than certain threshold [2]. Denote as the region such that when , the source S selects the relay R for data reception. Therefore, stands for the relay strategy employed in the system. The relay strategy is forwarded by the source to the destination through an one-bit acknowledge (ACK) signal such that the destination can perform successive decoding of the received signals when .

When , due to the buffer at the relay, the transmitted messages from the source and the relay node are different. So we cannot exploit spatial diversity as we can when there is no buffer at the relay. In this case, we have a two-hop channel while the transmitted signal of link forms interference to the link. See Fig. 2 for the illustration of the information flow. Then the instantaneous service rates for the queues at the source and relay node are given by

 Csr =TBlog2(1+\footnotesize{SNR}~zsr), (21) Crd =TBlog2(1+\footnotesize{SNR}rzrd1+\footnotesize{SNR}zsd). (22)

When , S selects the destination D for data reception. In this case, we have a two-user multiple access channel. See Fig. 3 for the illustration of the information flow. Note that the destination can perform successive decoding of the received signal from the source and relay node. Define as the region depending on and such that when , the destination D decodes the received signal in the order of , i.e., the sent signal from the source sees no interference, and the instantaneous rates are given by [36]

 Csd =TBlog2(1+\footnotesize{SNR}zsd), (23) Crd =TBlog2(1+\footnotesize{SNR}rzrd1+\footnotesize{SNR}zsd). (24)

On the other hand, when , the decoding order at D is , i.e., the sent signal from the relay sees no interference, and the instantaneous rates are given by

 Csd =TBlog2(1+\footnotesize{SNR}zsd1+\footnotesize{SNR}rzrd), (25) Crd =TBlog2(1+\footnotesize{SNR}rzrd). (26)

To summarize, we have the service rates of the queues at the source and relay node as

 Cs=⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩TBlog2(1+% \footnotesize{SNR}~zsr),z∈ZTBlog2(1+\footnotesize{SNR}zsd),z∈Zc∩Z0TBlog2(1+\footnotesize{SNR}zsd1+% \footnotesize{SNR}rzrd),z∈Zc∩Zc0 (27)

and

 Cr=⎧⎪⎨⎪⎩TBlog2(1+% \footnotesize{SNR}rzrd1+\footnotesize{SNR}zsd),z∈Z∪{Zc∩Z0}TBlog2(1+\footnotesize{SNR}rzrd),z∈Zc∩Zc0 (28)

respectively. Above, the rates are in the units of bits per block or equivalently bits per seconds. These can be regarded as the service processes of the queues at the source and relay, respectively.

To ensure the stability of the queue at the relay, we need to enforce the following condition [33]

 Ez∈Z{TBlog2(1+% \footnotesize{SNR}~zsr)}

where denotes the expectation over the region , i.e., with the probability density function of . The above equation implies that the average arrival rate of the queue at the relay should be less than the average service rate. We assume that the above condition is satisfied for the relay policies in consideration, otherwise the effective capacity is deemed as zero.

### Iii-B Effective Bandwidth of the S-R Link

Note that and can be easily derived with the relay policy after we obtain the service rate of queue 1 and 2 in (27) and (28), respectively. Then, from the discussions in Section II-C, we must obtain the effective bandwidth of the arrival processes of the queue at the relay to derive the effective capacity.

This can be achieved by borrowing the idea of intree-network [33]. Note that queue 1 is the predecessor of queue 2, or equivalently, queue 2 is the successor of queue 1. Now the three-node relay network can be viewed as an intree network. The routing depends on the relay protocol designed. Following the definitions in [33, Section 9.4], we define the routing variable

 p1[i]={1,z∈Z,0,z∈Zc. (30)

That is, if the departure from queue 1 at th symbol is routed to the successor, i.e., queue 2, and if the departure from queue 1 at th symbol leaves the intree network, i.e., goes to the destination node. Now, we have the log-moment generating function for the routing process as

 Λp1(θ,Z) =limn→∞logE{eθ∑ni=1p1[i]}n. (31)

Recalling the definitions in Section II-C, we have the following result.

###### Proposition 1

Given and the routing process specified by , the LMGF of the departure process from the source to the relay, or equivalently the arrival process to the relay node, is given by

 ΛA,2(θ)=⎧⎨⎩RΛp1(θ,Z),0≤Λp1(θ,Z)≤~θ,R~θ+ΛC,1(Λp1(θ,Z)−~θ),Λp1(θ,Z)>~θ. (32)

where is the constant arrival rate to queue 1, and is defined in (17).

Proof: Note that the only arrival to the relay node is from the source. Following the procedures described in [33, Section 9.4], we know that

 ΛA,2(θ)=Ξ1(Λp1(θ,Z)) (33)

where is the effective bandwidth of the departure process from queue 1. Since the arrival to the source is constant, we have

 Ξ1(θ)={θR,0≤θ≤~θ~θR+ΛC,1(θ−~θ),θ>~θ. (34)

Substituting the value of to the above equation yields the result directly.

### Iii-C Effective Capacity

Under the block fading assumption, the logarithmic moment generating functions for the service processes of queues at the source S and the relay R as functions of are given by [16]

 ΛC,1(θ) =logE{eθCs}andΛC,2(θ)=logE{eθCr} (35)

where and are given by (27) and (28), respectively. Now, due to the assumption that the fading changes independently from one block to another, we can, for instance, simplify the logarithmic moment generating function in (4) as . If fading is correlated, such simplifications are in general not possible and analysis needs to be based on the limit forms of the asymptotic logarithmic moment generating functions. However, if the service rates can be regarded as Markov modulated processes, then it is shown in [33, Section 7.2] that where denotes the spectral radius or equivalently the maximum of the absolute values of the eigenvalues of the matrix , and is a matrix which depends on the transition probabilities of the Markov process. Similarly, we can derive the limit forms of the asymptotic logarithmic moment generating function of , , if the routing process can be viewed as Markov modulated processes with representing the transition matrix of the associated Markov process. In such cases, an analysis similar to the one given in this paper can be pursued to identify the maximal effective capacity.

Combining (35) with (10) gives us and as follows

 J1(θ1) =−log(Ez∈Z{e−θ1TBlog2(1+\footnotesize{SNR}~zsr)}+Ez∈Zc∩Z0{e−θ1TBlog2(1+\footnotesize{SNR}zsd)} +Ez∈Zc∩Zc0{e−θ1TBlog2(1+% \footnotesize{SNR}zsd1+\footnotesize{SNR}rzrd)}), (36) J2(θ2) (37)

Now, the LMGF for the routing process (31) can be written as

 Λp1(θ,Z) =log(Pr{z∈Zc}+eθPr{z∈Z}). (38)

According to (32), the LMGF for the arrival process of the queue at the relay is

 ΛA,2(θ)=⎧⎪⎨⎪⎩RΛp1(θ,Z),0≤Λp1(θ,Z)≤~θ,R~θ+logE{e(Λp1(θ,Z)−~θ)Cs},Λp1(θ,Z)>~θ. (39)

For the following analysis, we need to characterize the relationship between and . We have the following result.

###### Lemma 2

Consider the function

 Λp1(θ,Z)=log(Pr{z∈Zc}+eθPr{z∈Z}). (40)

This function has the following properties:

1. , and .

2. is increasing in .

3. is a convex function of .

Proof:

1. This property can be readily seen by evaluating the function at , and noting that .

2. The first derivative of with respect to can be evaluated as

 ˙Λp1(θ)=eθPr{z∈Z}Pr{z∈Zc}+eθPr{z∈Z}>0. (41)

Hence, it is increasing in .

3. This property follows immediately since is log-convex, and non-negative multiplication does not alter the convexity.

###### Remark 1

Therefore, when , we have . As increases, increases. Also, increases at least linearly with . Note that is decreasing in , since the is increasing in [33]. Considering (41), we have

 limθ→0θΛp1(θ,Z)=1˙Λp1(0)=1Pr{z∈Z} (42)

where L’Hospital’s rule is used.

From Lemma 2, we can show that is still a decreasing function of similar to [27, Lemma 1], which is fundamental to this article.

With the selection relaying strategy , we can establish an upperbound on the arrival rates supported by the relay system with any specific .

###### Proposition 2

The constant arrival rates, which can be supported by the buffering relay system with statistical queueing constraints specified by at the source and relay, respectively, are upperbounded by

 R ≤min{−1θ1logE{e−θ1Cs},−1Λp1(θ2,Z)logE{e−θ2Cr}} (43) =min{J1(θ1)θ1,J2(θ2)Λp1(θ2,Z)} (44)

Proof: See Appendix -A.

Define

 Ωε={(θ1,θ2): J1(θ1) and J2(θ2) are solutions to (???)}.

Similar to the discussions in [29] regarding two-hop channels, we can iterate over satisfying (1) and obtain the following result. Note that and represent the largest or smallest channel gain of link , respectively.

###### Theorem 2

The effective capacity of the buffer-aided relay systems with selection relaying strategy subject to statistical delay constraints specified by is given by the following:

Case I: If ,

 Rε(ε,D{max})=Jth(ε)θ1,th, (45)

where (,) is the unique solution pair to , and .

Case II: If ,

 (46)

where is the solution to , and is the smallest value of with satisfying

 −1θ1logE{e−θ1Cs}=−1θ1(logE{e−θ2Cr} +logE{e(Λp1(θ2,Z)−θ1)Cs}). (47)

Moreover, if , where is given by with

 θ1=Λp1(θ2,Z), (48)

then the solution to (2) with is unique.

Case III: If ,

 (49)

where is the solution to , and (,) is the unique solution to

 J1(θ1)θ1=J2(θ2)Λp1(θ2,Z) (50)

with .

Proof: See Appendix -B.

###### Remark 2

The above theorem covers all the possibilities that symmetric or asymmetric delay constraints on the queues at the source and relay node can be optimal for achieving the maximum effective capacity of the relay system. Case I refers to the case that the maximum throughput can be achieved with symmetric delay constraints at the queues of the source and relay node. Case II represents the case when the statistical delay constraints at the relay can be more stringent, while Case III shows the scenario for more strict delay constraints at the source. Recalling [27, Theorem 2], we know that as , and , and hence

 limε→1Rε(ε,D{max}% ) =min{limθ1→0J1(θ1)θ1,limθ1→0J2(θ2)Λp1(θ2,Z)} (51) =min{E{Cs},E{Cr}Pr{z∈Z}}. (52)

### Iii-D Selection Relaying Protocols

Considering the expression of the effective capacity and the associated conditions in Theorem 2, we note that finding the optimal relaying protocol in closed-form analytical expressions seems intractable for a general scenario. With this in mind, we consider a simplified case in which the relaying protocol is decided by a function of and , and is denoted as . The channel state region is given by

 Z={z:~zsr>g(zsd)}. (53)

Also, assume that the decoding strategy at the destination is given by , such that

 Z0={z:zrd>f(zsd)}. (54)

#### Iii-D1 Max Channel Gain (MCG)

A typical relay policy with buffer-aided relay system is to select the link with stronger channel condition or received SNR [6]. Then, we have

 g(zsd)=zsd. (55)

We know that . With this relaying scheme , we can obtain the effective capacity according to Theorem 2.

#### Iii-D2 Max Delay Exponent (MDE)

In this part, we propose a relay policy that takes into account the statistical delay constraints as well. Assume that the optimal statistical queueing constraints that maximize the effective capacity are given. With this parameter set , we consider the relay strategy that maximizes the statistical delay exponent at the source, in which case the effective capacity can be potentially improved. Assume that the channel of the link is independent of the links and . Combining (36), (53) and (54), we can express the statistical delay exponent at the source as

 J1(θ1,g(zsd)) =−log(∫∞0∫∞g(zsd)(1+%\footnotesizeSNR~zsr)−β1p(zsd,~zsr)d~zsrdzsd +∫∞0∫g(zsd)0∫∞f(zsd)(1+\footnotesize{SNR}zsd)−β1p(zsd,~zsr)p(zrd)dzrdd~zsrdzsd +∫∞0∫g(zsd)0∫f(zsd)0(1+\footnotesize{SNR}zsd1+\footnotesize{SNR}rzrd)−β1p(zsd,~zsr)p(zrd)dzrdd~zsrdzsd) (56)

Then, the associated relay strategy should be the solution to the following optimization problem

 maxgJ1(θ1,g(zsd)). (57)

We can obtain the relaying strategy specified as below.

###### Theorem 3

Given , the relay strategy as a function of that maximizes the statistical delay exponent at the source is given by

 g(zsd,θ1)=1\footnotesize{SNR}(e−1β1logEzrd{e−θ1Csd,%MAC}−1) (58)

where , and

 Csd,MAC=⎧⎪⎨⎪⎩TBlog2(1+% \footnotesize{SNR}zsd),zrd≥f(zsd),TBlog2(1+\footnotesize{SNR}zsd1+% \footnotesize{SNR}rzrd),zrd

Proof: Define , where is the optimal function, is any constant, and represents arbitrary perturbation. Now, a necessary condition that needs to be satisfied for the solution to (57) is [37]

 (60)

By noting that , and from (III-D2) and (60), we can obtain

 ∫∞0(−(1+\footnotesize{SNR}g(zsd,θ1))−β1+∫∞f(zsd)(1+% \footnotesize{SNR}zsd)−β1p(zrd)dzrd +∫f(zsd)0(1+\footnotesize{SNR}zsd1+\footnotesize{SNR}rzrd)−β1p(zrd)dzrd)p(zsd,g(zsd,θ1))η(zsd)dzsd=0 (61)

Since the above equation holds for any , it follows that

 −(1+\footnotesize{SNR}g(zsd,θ1))−β1+∫∞f(zsd)(1+\footnotesize{SNR}zsd)−β1p(zrd)dzrd +∫f(zsd)0(1+\footnotesize{SNR}zsd1+\footnotesize{SNR}rzrd)−β1p(zrd)dzrd=0 (62)

which after rearranging and defining shown in (59) yields (58).

###### Theorem 4

With the optimal , we can show the following:

1. is increasing in .

2. is decreasing in .

Proof: See Appendix -C.

With the above properties, we know that given , there is a unique with achieving , and as increases, increases, and decreases. Therefore, as we iterate over , we know that the achievable rate of the relay system decreases as , or equivalently , increases. Then, we can still apply the method in Theorem 2 to obtain the maximum effective capacity associated with the proposed relay policy. Instead of fixed relaying policy , we now have different