IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 11, NOVEMBER

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1 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 11, NOVEMBER Buffer-Aided Diamond Relay Network With Block Fading and Inter-Relay Interference Renato Simoni, Vahid Jamali, Student Member, IEEE, Nikola Zlatanov, Robert Schober, Fellow, IEEE, Laura Pierucci, Senior Member, IEEE, and Romano Fantacci, Fellow, IEEE Abstract A simple diamond half-duplex relay network composed of a source, two decode-and-forward half-duplex relays, and a destination is considered, where a direct link between the source and the destination does not exist. For this network, we study the case of buffer-aided relays, where the relays are equipped with buffers. Each relay can receive data from the source, store it in the buffer, and forward it to the destination, when the channel conditions are advantageous. Thereby, buffering enables adaptive scheduling of the transmissions and receptions over time, which allows the network to exploit the diversity offered by the fading channels. For the considered halfduplex network, four transmission modes are defined based on whether the relay nodes receive or transmit. In this paper, we derive the locally optimal scheduling of the transmission modes over time and investigate the achievable average rate, when the relays are affected by inter-relay interference. Since the proposed buffer-aided transmission policies introduce unbounded delay, we provide a sub-optimal buffer-aided transmission policy with limited delay. Moreover, for inter-relay interference cancellation, we consider two coding schemes with different complexities. In the first scheme, we employ dirty paper coding, which entails a high complexity, whereas in the second scheme, we adopt a low-complexity technique based on successive interference cancellation at the receiving relay nodes and optimal power allocation at the transmitting nodes. Our numerical results show that the proposed protocols, with and without delay constraints, outperform existing protocols for the considered network from the literature. Index Terms Buffer-aided relaying, diamond relay channel, block fading, delay, inter-relay interference, dirty paper coding, and interference cancellation. I. INTRODUCTION THE cooperative communications paradigm has recently gained particular interest due to its inherent capability to exploit spatial diversity, thus increasing reliability and/or Manuscript received July 27, 2015; revised February 2, 2016 and June 13, 2016; accepted August 5, Date of publication August 18, 2016; date of current version November 9, This paper was presented at the IEEE International Conference on Communications, London, U.K., June The associate editor coordinating the review of this paper and approving it for publication was J. M. Romero-Jerez. R. Simoni, L. Pierucci, and R. Fantacci are with the Department of Information Engineering, University of Florence, Florence, Italy ( renato.simoni@unifi.it; laura.pierucci@unifi.it; romano.fantacci@unifi.it). V. Jamali and R. Schober are with the Institute for Digital Communication, University of Erlangen Nuremberg, Erlangen 91058, Germany ( vahid.jamali@fau.de; robert.schober@fau.de). N. Zlatanov is with the Department of Electrical and Computer Systems Engineering, Monash University, Melbourne, VIC 3800, Australia ( nikola.zlatanov@monash.edu). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TWC information rate [1] [6]. In a cooperative communication network, each node is assumed to receive/transmit data as well as to act as a relay for the other nodes by helping them to forward the data to their respective destinations. A key example is the parallel relay channel, also known as the diamond relay network, which consists of a source, a destination, and a set of relay nodes. This network was first considered in [7] for the case of two relays, which employed ideal full-duplex transmission/reception without self-interference. However, ideal full-duplexing is challenging in practice due to severe self-interference [4]. In particular, the large difference between the powers of the transmitted and received signals leads to the saturation of the analog amplifiers of the receiving chain and makes self-interference cancellation extremely difficult. For this reason, half-duplex relays, where transmission and reception in the same time slot and the same frequency band is not allowed, have been widely adopted in the literature [4], [5]. One of the simplest transmission protocols for the two-relay diamond network is relay selection where, in each time slot, only one node is selected to receive from the source and retransmit the received information to the destination [8]. However, the average data rate can be further improved by allowing both relays to transmit or receive simultaneously and exploiting efficient coding strategies, which have been proposed for the broadcast, multiple access, and interference channels [3]. Successive relaying, introduced in [4] and [9], can significantly increase the data rate by employing two parallel transmissions in each time slot. In fact, during the same time slot, one relay receives data from the source while the other relay transmits to the destination, and in each time slot, the relays change their roles in transmission and reception. In this context, the authors of [6] have recently considered the case where all links in the two-relay diamond network are characterized by additive white Gaussian noise (AWGN) channels and interrelay interference is neglected. Thereby, the authors introduced the Multi-hopping Decode-and-Forward (MDF) protocol, in which four transmission modes are defined depending on whether the two relays transmit or receive. Each time slot was divided into four sub-slots, where in each sub-slot one of the aforementioned transmission modes was adopted, and the sub-slot lengths were optimized for rate maximization. An achievable data rate and an upper bound on the capacity for the half-duplex two-relay diamond network were provided in [5], where again the interference between the relays was neglected IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

2 7358 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 11, NOVEMBER 2016 The protocols in [5], [6], [8], and [9] assumed a fixed and predefined schedule which established when the relays of the diamond relay network transmitted and received. However, if the links are affected by fading, the fixed scheduling of the relays for reception and transmission may not allow the network to fully exploit the diversity offered by the channel. To overcome this limitation, the authors in [10] [12] proposed link selection protocols for the diamond relay network, where only one of the source-to-relay or relay-to-destination links was selected according to their respective channel qualities. For this to be possible, the relay nodes must be equipped with buffers for storage of the information received from the source [13]. The protocol in [10] chooses the source-to-relay link with the highest signal-to-noise ratio (SNR) in the odd time slots and the relay-to-destination link with the highest SNR in the even time slots, where the relay selected for transmission can be different from the relay selected for reception. In contrast, the protocol in [11] selects the link with the highest SNR among all source-to-relay and relay-to-destination links in each time slot. However, the protocols in [10] and [11] are heuristic as far as rate maximization is concerned. A bufferaided protocol for relay selection maximizing the average data rate for relay selection was proposed in [12]. However, the buffer-aided protocols in [10] [12] assume single relay selection, i.e., only one relay is selected for transmission or reception in each time slot, which guarantees a low complexity but limits the achievable rate. In this work, we propose a mode selection protocol for the two-relay Buffer-aided Diamond (BaD) channel, which further improves the data rates by combining the use of buffers at the relays [12] with the advanced coding strategies for simultaneous transmission in [6]. Thereby, we assume that all nodes are equipped with a single antenna, and both relays are half-duplex. Assuming availability of channel state information (CSI), the proposed protocol, which we refer to as BaD protocol throughout the paper, specifies the optimal selection of the transmission mode as well as the corresponding coding scheme based on the fading state in each time slot, such that the achievable average rate of the BaD network is maximized. Furthermore, we consider practical issues arising in the implementation of the proposed protocol such as limiting the delay, interference cancellation at low computational cost, and adaptive computation of the protocol parameters. In particular, the proposed optimal BaD protocol introduces an unbounded delay and constitutes a performance upper bound for any delay-limited buffer-aided relaying protocol for the considered diamond relay channel. However, unfortunately, incorporating an average delay constraint in the throughput maximization problem yields an untractable optimization problem. Hence, in this paper, based on the intuition that the optimal delayunlimited BaD protocol provides and by limiting the maximum buffer size at the relays, we introduce a modified BaD mode selection protocol which guarantees a limited average delay. Moreover, in order to cancel the inter-relay interference, dirty paper coding (DPC) is employed. Specifically, by using DPC, the source can encode its signal such that the receiving relay can completely remove the interference caused by the transmitting relay [14], [15]. Since the encoding and decoding processes for DPC have an exponential complexity in the block length [16], practical implementation might be challenging. Hence, we propose an alternative strategy for interrelay interference cancellation which has a low computational cost. For the considered BaD network, inter-relay interference occurs when one relay transmits to the destination and the other relay receives from the source simultaneously, which is a particular instance of the Z interference channel, whose achievable rates have been studied in [17] and [18]. Therefore, depending on its severity, two different approaches for coping with the interference are employed. If the interference between the relays is weak, the receiving relay decodes the source s signal by treating the interference as noise, otherwise, the relay node removes the interference through successive cancellation without reducing the data rate [19], [20]. Indeed, for the first approach, the SNR of the source s signal degrades with increasing strength of the interference channel, which leads to a data rate reduction. Thus, the highest loss in performance for these two methods compared to DPC occurs when the interference level is not sufficiently high to be removed through successive cancellation but considerably reduces the data rate compared to the case without interference. For this reason, for the proposed low-complexity BaD protocol, we introduce power control variables that optimize the transmission power such that the data rate is maximized when the interference cannot be entirely removed. The last practical consideration for the BaD protocol concerns the computation of the protocol parameters that are needed for the selection of the transmission modes in each time slot. In particular, instead of performing a numerical search which also requires knowledge of the channel statistics, we propose a low complexity algorithm that adaptively estimates the parameters required for the BaD protocol without knowledge of the channel statistics. As benchmark schemes, we compare the proposed BaD protocol with the protocols introduced in [6] and [12], respectively. We show through numerical simulations that the BaD protocol exploits the full diversity gain of the BaD channel and achieves higher data rates than the existing protocols. Moreover, we derive a performance upper bound based on the max-flow min-cut theorem [21]. Our numerical results show that the gap between the rate achieved with the proposed protocol and the considered upper bound is within 1 bit/symb for the scenarios considered in Section V. Furthermore, we consider the limited delay case and show that even if the average delay is limited to only a few time slots, the delaylimited BaD protocol achieves data rates that are close to the case with infinite delay and outperforms the protocols in [6] and [12]. Finally, we show that depending on the level of the inter-relay interference, the proposed low-complexity BaD protocol can achieve data rates close to those of the BaD protocol with DPC. The remainder of the paper is organized as follows. In Section II, the channel model and the four transmission modes are presented. In Section III, the rate maximization problem is formulated and the proposed optimal mode selection policy for unbounded delay is presented. In Section IV, practical constraints such as limited delay, low-complexity implementation, and adaptive estimation of the protocol

3 SIMONI et al.: BUFFER-AIDED DIAMOND RELAY NETWORK 7359 parameters are considered. In Section V, the proposed protocols are numerically evaluated and compared to benchmark schemes. Finally, Section VI concludes the paper. II. SYSTEM MODEL In this section, we introduce the channel model and review the four transmission modes for the two-relay diamond network. A. Channel Model We consider a diamond half-duplex decode-and-forward relay network consisting of a source node S, two halfduplex relays R1 and R2, and a destination node D without a direct link between source and destination. We assume that the relays are equipped with unlimited-size buffers and that time is divided into slots of equal duration, indexed by t = 1, 2,..., N, where N. Furthermore, we assume flat block fading on all links, such that the channel coefficients remain constant during one time slot and change from one time slot to the next. We also assume that all fading processes are stationary and ergodic. Let P s, P 1, and P 2 denote the transmit powers of nodes S, R1, and R2, respectively, and assume that they are fixed for all time slots. Let h (t), h (t), h 1d (t), andh 2d (t) denote the complex-valued channel coefficients of the S-R1, S-R2, R1-D, and R2-D channels, respectively. Moreover, the noises at the receivers of R1, R2, and D are assumed to be AWGNs with variances σr1 2, σ r2 2,andσ d 2, respectively. Let γ (t) and γ (t) denote the SNRs of the S-R1 and S-R2 links in the t-th time slot, respectively, where γ si (t) = P s h si (t) 2 /σri 2, i = 1, 2. Similarly, let γ 1d (t) and γ 2d (t) denote the SNRs of the R1-D and R2-D links in the t-th time slot, respectively, where γ id (t) = P i h id (t) 2 /σd 2, i = 1, 2. We assume that the CSI of all links is available at all nodes, such that each node can determine the transmission strategy according to Theorems 1 and 2 provided in Sections III and IV, respectively. This assumption is justified if the coherence time of the fading channels is much larger than the time required for CSI acquisition. We also assume that all transmitting nodes employ capacity-achieving codes. For notational simplicity, we define functions C(x) log(1 + x) and [x] b a min{b, max{x, a}}, a b. B. Transmission Modes For the considered two-relay BaD network, the amount of data stored in the buffers of the relays may change in each time slot. To keep track of the queues at the buffers, we denote the amount of normalized information in bits/symbol in the buffer of the i-th relay at the end of time slot t by Q i (t), i {1, 2}. We assume that the relays forward independent data streams for simplicity of analysis and due to practical considerations. For the case when the relays make use of potential correlations between the information stored in their respective buffers, the problem formulation is more involved and the resulting optimal protocol does not lend itself to a simple form. For instance, the relay nodes would have to employ Slepian-Wolf coding to transmit correlated information Fig. 1. The four possible transmission modes in the half-duplex diamond relay network are illustrated. to the destination [22]. This coding scheme requires synchronous coherent transmission by the relay nodes which is difficult to implement in practice. Here, depending on whether the two half-duplex relays are transmitting or receiving, four transmission modes are considered in each time slot, cf. Fig. 1. The transmission modes, the respective coding schemes, and the corresponding dynamics of the queues are introduced in following. Mode 1 (Broadcast channel (BC) mode): In this mode, S transmits to R1 and R2 (Fig. 1a). Thereby, we assume that the source broadcasts two superimposed codewords, where each codeword is intended for one of the relays. The two codewords are transmitted with different powers, different rates, and carry independent messages intended for the relays. Let α(t)p s and [1 α(t)]p s denote the powers of the codewords intended for R1 and R2, respectively, where 0 α(t) 1. Let R (1) si (t) denote the data rate transmitted from source to relay i in Mode 1 in time slot t. According to the capacity region of the degraded Gaussian broadcast channel [3], the transmission rates R (1) (t) and R(1) (t) have to satisfy R (1) R (1) (t) = C(1) (t) C(α(t)γ ( (t)), = C α(t)γ (t) [1 α(t)]γ (t) + 1 if γ ) (t) >γ (t), otherwise, (1a) (1b) (t) = C(1) (t) (2) C ( [1 α(t)]γ (t) ) (, ) if γ (t) <γ (t) = [1 α(t)]γ (t) (2a) C, otherwise. α(t)γ (t) + 1 Thereby, the intended codewords for R1 and R2 can be successfully decoded at each of the relays. For this mode, the amount of information in the queue of the buffer at relay i, at the end of time slot t, increases to Q i (t) = Q i (t 1) + R (1) si (t), i = 1, 2. Mode 2 (Multiple Access (MA) Mode): In this mode, the relays simultaneously transmit their codewords to the destination forming an MA channel (Fig. 1b). In this case, the transmission rates are limited not only by the capacity of the MA channel, but also by the amount of data available in the buffers. As a result, the transmission rates of the relays have to satisfy R (2) 1d (t) = min{c(2) R (2) 2d 1d (t), Q 1(t 1)} (t) = min{c(2) 2d (t), Q 2(t 1)} (3a) (3b) where C (2) 1d (t) and C(2) 2d (t) denote the maximum possible rates of the MA channel [3] from R1 and R2 to D in time slot t,

4 7360 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 11, NOVEMBER 2016 respectively, and are given by C (2) 1d (t) = β(t)c (γ 1d(t)) +[1 β(t)]c ( 2d (t) = β(t)c γ2d (t) 1 + γ 1d (t) C (2) ( γ1d (t) ) 1 + γ 2d (t) (4a) ) +[1 β(t)]c (γ 2d (t)). (4b) Here, 0 β(t) 1 is a time sharing variable which divides time slot t into two sub-slots, where the length of the first sub-slot is the β(t) fraction of time slot t and the length of the second sub-slot is the 1 β(t) fraction of time slot t. In the first sub-slot, the maximum ( rates of ) the R1-D and R2-D channels are C (γ 1d (t)) and C γ2d (t) 1+γ 1d (t), respectively. Similarly, in the second ( sub-slot, ) the maximum rates of R1 and R2 to D are C γ1d (t) 1+γ 2d (t) and C (γ 2d (t)), respectively. D employs successive decoding [3], where in the first subslot, D decodes first the codeword of R2 while treating the codeword of R1 as noise. After the codeword of R2 has been decoded, D subtracts it from the received signal, and decodes the codeword of R1. In the second sub-slot, D uses the same decoding procedure but the decoding order is reversed. For this transmission mode, the amount of information at relay i, atthe end of time slot t, decreases to Q i (t) = Q i (t 1) R (2) id (t), i = 1, 2. Mode 3 (Successive Relaying Mode): S and R2 transmit codewords in the same time slot to R1 and D, respectively (Fig. 1c). During the simultaneous transmission, R2 causes interference to R1. The particularity of this case is that node S already knows the codeword that R2 is transmitting, since the information in this codeword was previously transmitted to R2 by S itself. Thereby, using the CSI of the network, DPC [14, Lemma 1] can be employed to mitigate the interrelay interference since node S has full knowledge of the interference signal transmitted by R2. Using DPC, the total interference at R1 can be cancelled and yet R1 can reliably decode codewords transmitted by S with a rate equal to the capacity of the S-R1 channel. Hence, for Mode 3, in time slot t, the transmission rate from S to R1, denoted by R (3) (t), and the transmission rate from R2 to D, denoted by R (3) 2d (t), have to satisfy (t) = C(γ (t)) (5a) (t) = min{c(3) 2d (t), Q 2(t 1)}, (5b) where C (3) 2d (t) C(γ 2d(t)), and the queue sizes are updated as Q 1 (t) = Q 1 (t 1)+ R (3) (t) and Q 2(t) = Q 2 (t 1) R (3) 2d (t). Mode 4 (Successive Relaying Mode): This mode is identical to Mode 3 but R1 and R2 switch roles (Fig. 1d). Therefore, the corresponding transmission rates from S to R2 and from R (3) R (3) 2d R1 to D, denoted by R (4) (t) and R(4) 1d (t), respectively, have to satisfy (t) = C(γ (t)) (6a) (t) = min{c(4) 1d (t), Q 1(t 1)}, (6b) where C (4) 1d (t) C(γ 1d(t)), and the corresponding queue sizes are updated as Q 1 (t) = Q 1 (t 1) R (4) 1d (t) and Q 2(t) = Q 2 (t 1) + R (4) (t). R (4) R (4) 1d III. OPTIMAL MODE SELECTION POLICY In this section, we formulate an optimization problem for maximization of the rate of the two-relay BaD network, and provide the optimal mode selection policy as the solution to this problem. A. Problem Formulation In order to formulate the mode selection problem, we define mode selection variables q k (t) {0, 1}, k ={1, 2, 3, 4}, where k denotes the index of the mode. In particular, if q k (t) = 1, then mode k is selected, and if q k (t) = 0, then mode k is not selected in time slot t. Since in each time slot only one mode can be selected, 4 k=1 q k (t) = 1, t, must hold. For the problem formulation, we assume that the source has always data to transmit (i.e., it is fully backlogged) and that the number of time slots N satisfies N.Usingq k (t), k, t, the average rates from S to R1, S to R2, R1 to D, and R2 to D denoted by R, R, R 1d,and R 2d, respectively, can be expressed as R = R = R 1d = R 2d = lim N 1 N [q 1 (t)r (1) (t) + q 3(t)R (3) (t)] (a) = E{q 1 (t)r (1) (t) + q 3(t)R (3) (t)} lim N 1 N [q 1 (t)r (1) (t) + q 4(t)R (4) (t)] (a) = E{q 1 (t)r (1) (t) + q 4(t)R (4) (t)} lim N 1 N [q 2 (t)r (2) 1d (t) + q 4(t)R (4) 1d (t)] (a) = E{q 2 (t)r (2) 1d (t) + q 4(t)R (4) 1d (t)} lim N 1 N [q 2 (t)r (2) 2d (t) + q 3(t)R (3) 2d (t)] (7a) (7b) (7c) (a) = E{q 2 (t)r (2) 2d (t) + q 3(t)R (3) 2d (t)}, (7d) where R (1) (t), R(1) (t), R(2) 1d (t), R(2) 2d (t), R(3) (t), R(3) 2d (t), (t), andr(4) (t) are the transmission rates given in (1b), R (4) 1d (2a), (3b), (3a), (5a), (5b), (6a), and (6b), respectively, and the equalities (a) = in (7) hold due to the assumed ergodicity of the fading. Since there is no direct link from the source to the destination, the average rate received by the destination, denoted by R d, is given by the sum of the average rates transmitted by the relays to the destination, i.e., R d = R 1d + R 2d. (8) Our goal is to maximize R d by optimally selecting the transmission modes in each time slot, i.e., by optimal selection of the value of q k (t), k, t. In the following, we provide a useful condition which simplifies the formulation of the rate maximization problem. Lemma 1 (Optimal Queue Condition): The average rate of the considered two-relay BaD network is maximized when the

5 SIMONI et al.: BUFFER-AIDED DIAMOND RELAY NETWORK 7361 following conditions hold R = C 1d and R = C 2d, (9) where C 1d = E{q 2 (t)c (2) 1d (t) + q 4(t)C (4) 1d (t)} (10a) C 2d = E{q 2 (t)c (2) 2d (t) + q 3(t)C (3) 2d (t)}. (10b) Furthermore, when, q k (t), k, t are chosen such that (9) holds, we obtain R 1d = C 1d and R 2d = C 2d. (11) Proof: Please refer to Appendix A. Remark 1: From Lemma 1, we observe that average rates R 1d and R 2d do not depend on the queue memory Q i (t) when (9) holds, cf. (11). In fact, if (9) holds, the arrival rates at the relays are equal to the departure rates, therefore, the min functions in (3a), (3b), (5b), and (6b), and in the average transmission rates R 1d and R 2d become negligible as N. Now, we are ready to formulate the rate maximization problem as follows maximize q k (t),α(t),β(t) C 1d + C 2d subject to C1: R = C 1d C2: R = C 2d C3: 4 k(t) = 1, t k=1 C4: q k (t) {0, 1}, t, k C5: 0 α(t) 1, t C6: 0 β(t) 1, t, (12) where constraints C1 and C2 correspond to the two conditions in (9), respectively, constraint C3 guarantees that only one mode is selected in each time slot, constraints C4, C5, and C6 specify the feasible domains of q k (t), α(t), and β(t), respectively. Furthermore, since constraints C1 and C2 are satisfied, according to Lemma 1, the identities in (11) hold, thereby R d in (8) is replaced with the objective function in (12). B. Proposed Mode Selection Protocol We note that the optimization problem in (12) is not jointly convex in all optimization variables q k (t), α(t), and β(t). In particular, the optimization problem in (12) is a mixed integer-continuous problem [23]. Therefore, in Appendix B, we first relax the binary condition q k (t) {0, 1} to q k (t) [0, 1] and find the solution of the relaxed problem using the Lagrange dual formulation [23]. Thereby, we show that the solution is always at the boundaries of interval [0, 1]. Hence, the binary relaxation does not affect the optimal solution. Nevertheless, the relaxed problem is still non-convex due to e.g. the multiplication of q k (t) with β(t) and C (1) si (t) which is a function of α(t). Therefore, the proposed mode selection protocol corresponds to a local optimum of the problem in (12) but global optimality cannot be guaranteed [23]. Theorem 1 (Proposed Mode Selection Protocol): The optimal mode selection which maximizes the average rate of the considered BaD network with block fading is given by 1, if k = arg max{ k (t)} q k (t) = k=1,2,3,4 (13) 0, otherwise, where k (t) is referred to as the selection metric and is defined as 1 (t) μ 1 C (1) (t) + μ 2C (1) (t) (14a) 2 (t) (1 μ 1 )C (2) 1d (t) + (1 μ 2)C (2) 2d (t) (14b) 3 (t) μ 1 C (3) (t) + (1 μ 2)C (3) 2d (t) (14c) 4 (t) (1 μ 1 )C (4) 1d (t) + μ 2C (4) (t). (14d) Moreover, the power sharing variable α(t) is given by 1, if μ 1 =μ 2 γ (t) γ (t) 0, if μ 1 =μ 2 γ (t)<γ (t) [ ] μ2 γ (t) μ 1 γ (t) 1, if μ 1 =μ 2 γ (t) γ (t) (μ 1 μ 2 )γ (t)γ (t) [ 0 ] μ2 γ (t) μ 1 γ (t) 1 1, otherwise, (μ 1 μ 2 )γ (t)γ (t) 0 (15) whereas, the time sharing variable β(t), t, is given by β(t) = β 0, { ( )} if μ 1 >μ 2 R E q 4 (t)c(γ 1d (t)) C γ1d (t) 1+γ 2d (t) = { ( )}, if μ 1 = μ 2 E C(γ 1d (t)) C γ1d (t) 1+γ 2d (t) 1, otherwise (16) In (14)-(16), μ 1,μ 2,andβ are constants, and their optimal values depend on the statistics of the SNRs of the links. In particular, the optimal values of μ i (0, 1), i = 1, 2, can be obtained by a two-dimensional search or an iterative algorithm, cf. Section IV.C, such that constraints C1 and C2 in (12) hold. Proof: Please refer to Appendix B. Based on the instantaneous channel conditions and the channel statistics, the proposed protocol in Theorem 1 computes the optimal values of the selection metric parameters in order to find the best transmission mode for a given time slot. In particular, to obtain the numerical value of each selection metric k (t), t, the values of α(t), β, μ 1,andμ 2 are needed. The optimal value of α(t) has to be computed for each time slot. In contrast, since β, μ 1, and μ 2 are constants, they can be computed offline before the start of transmission. The optimal values of μ 1 and μ 2 can be found through e.g. a two-dimensional search such that constraints C1 and C2 in (12) hold while β is calculated from (16). We note that in order to calculate the average rates, R si and R id, in each search step, we can use numerical solvers such as Mathematica to compute the expectations in (7) by numerical integration of the arguments of the expectation multiplied by the probability density functions of the fading coefficients.

6 7362 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 11, NOVEMBER 2016 Alternatively, the expectations in (7) can be computed using Monte Carlo simulation, i.e., by averaging the arguments of the expectation over a large number of random realizations of the fading coefficients. We employed the latter approach in Section V in order to find constants μ 1 and μ 2. Remark 2: The queue stability for the protocol provided in Theorem 1 is taken into account by conditions C1 and C2 in the maximization problem in (12). In particular, according to queuing theory [24], the average arrival rate has to be less than or equal to the average departure rate to ensure that the queue is rate stable. Therefore, conditions C1 and C2 in (12) ensure that the queues at the relays are at the edge of stability, i.e., the queue is non-absorbing but is at the boundary of a non-absorbing and an absorbing queue, see [25, Th. 1] for further discussion. We emphasize that although the queues at the relays are rate stable for the protocol in Theorem 1, the average delay may grow with time. However, overflow does not occur as the buffer sizes are assumed to be infinite. IV. MODE SELECTION POLICIES UNDER PRACTICAL CONSTRAINTS In this section, we consider practical issues arising in the implementation of the proposed protocol, such as limiting the delay, interference cancellation without DPC, and adaptive computation of μ 1 and μ 2. A. Delay-Limited BaD Protocol According to Lemma 1, for rate maximization, the queues should be at the edge of stability, i.e., R si = C id, i = 1, 2. However, this condition does not guarantee a limited average delay at the buffers. We also note that the optimal protocol which maximizes the throughput for a given average delay requirement, i.e., which optimizes the throughput-delay tradeoff, is a difficult and untractable problem. One suboptimal approach to arrive at a protocol with a limited average delay is to limit the arrival rates at the queues by imposing condition R si + ɛ = C id, instead of R si = C id, i = 1, 2, in (12), where ɛ is a small non-negative number. We note that replacing constraints C1 and C2 in (11) with these new constraints does not change the optimal protocol in Theorem 1 except that now μ 1 and μ 2 have to guarantee R si = C id ɛ instead of R si = C id, i = 1, 2. Unfortunately, although this new protocol can guarantee a finite average delay by appropriately choosing the value of ɛ, the resulting loss in the average achievable rate compared to the protocol in Theorem 1 can be severe, particularly for small average delay requirements, cf. Fig. 2. Hence, in the following, we propose an alternative modified version of the optimal protocol in Theorem 1, which has the ability to limit the delay at the expense of a small reduction if the average rate. In this case, in order to limit the average delay, we limit the maximum buffer size to Q max i.the proposed delay-limited BaD (DL-BaD) protocol is identical to the optimal BaD protocol in Theorem 1 but with different selection metrics, i.e., instead of k (t), wenowuse ˆ k (t). The new selection metrics ˆ k (t) are defined as ˆ 1 (t) μ 1 R (1) (t) + μ 2 R (1) (t) (17a) Fig. 2. Performances comparison of the considered protocols for i.i.d channels where γ = γ = γ 1d = γ 2d = γ. ˆ 2 (t) (1 μ 1 )R (2) 1d (t) + (1 μ 2)R (2) ˆ 3 (t) μ 1 R (3) (t) + (1 μ 2)R (3) 2d (t) 2d (t) ˆ 4 (t) (1 μ 1 )R (4) 1d (t) + μ 2 R (4) (t) (17b) (17c) (17d) where C (k) id (t) are substituted by R(k) id (t) given in (3a), (3b), (5b), and (6b). Moreover, in this case, the transmission rates from S to R1 and R2 are also limited by the amount of space available in the respective buffers to store information, i.e., R (k) si (t) = min{c (k) si (t), Q max i Q i (t 1)}, i = 1, 2. (18) Note that although the buffer sizes are finite here, with the above protocol, buffer overflow cannot occur since the transmission rates are not only limited by the channel conditions, but also by the amount of space available in the relays buffers, cf. (18). Finally, the average delay can be controlled by appropriately setting the values of Q max i. Thereby, according to Little s law, the average delay is given by [26, Eq. (11.69)] D = E{Q 1(t)}+E{Q 2 (t)} E{A 1 (t)}+e{a 2 (t)}, (19) where A 1 (t) q 1 (t)r (1) (t) + q 3(t)R (3) (t) and A 2(t) q 1 (t)r (1) (t) + q 4(t)R (4) (t) are the rates received at R1 and R2 in time slot t, respectively. For sufficiently large values of Q max i the data rate of the proposed delay limited protocol approaches the upper bound in Theorem 1 with no delay constraint. Thereby, we obtain a finite upper bound on the average delay given by D = E{Q 1(t)}+E{Q 2 (t)} E{A 1 (t)}+e{a 2 (t)} Qmax 1 + Q max 2 R upp, (20) where R upp = R 1d + R 2d is the maximum rate achieved by the delay-unconstrained BaD protocol given by Theorem 1. B. Successive Interference Cancellation BaD Protocol For the transmission strategy assumed in Theorem 1, the inter-relay interference for Modes 3 and 4 was completely removed by using DPC, which is known to be

7 SIMONI et al.: BUFFER-AIDED DIAMOND RELAY NETWORK 7363 computationally complex. Therefore, in the following, we propose a modified version of the protocol in Theorem 1, which is based on an alternative coding scheme for interference reduction and avoids the complexity of DPC. Since the BC and MA schemes are not affected by inter-relay interference (they do not use DPC), Modes 1 and 2 remain unchanged, whereas, for Modes 3 and 4, we define new selection metrics 3 (t) and 4 (t), respectively. Let us consider Mode 3, where R1 receives data from S and simultaneously R2 transmits to D causing interference to R1. The SNR of the interfering signal at R1 is given by γ I 2 (t) = P 2 h rr (t) 2 /σr1 2,whereh rr(t) is the channel gain between the two relays. The interference at R1 can be treated in two ways, depending on the channel conditions. If the interference signal is strong, R1 decodes first the interference and then subtracts it from the received signal and recovers the source s codeword through successive cancellation. We define the condition of strong interference as ( ) γi 2 (t) C C(γ 2d (t)). (21) γ (t) + 1 In this case, since the maximum interference data rate that R1 can decode when ( the) source s codeword is treated as noise, given by C γi 2 (t) γ (t)+1, is greater than the data rate that R2 transmits to D, given by C(γ 2d (t)), R1 can decode the codewords of S and R2 as long as their data rates do not exceed C(γ (t)) and C(γ 2d (t)), respectively. On the other hand, if the strong interference condition in (21) does not hold, R1 can treat the interference as noise and decode the source s codeword. In this case, the( maximum) data rate of S that R1 can decode is given by C γ (t) γ I 2 (t)+1. We note that the drawback of this method is that the maximum data rate of S that R1 can decode decreases with increasing interference power. Because of this limitation, we enhance the proposed BaD protocol by allowing for adjustment of the interference level through power allocation. To this end, we introduce power control coefficients. For Mode 3, the power control variables which scale the transmit powers of S and R2 are defined as 0 ρ 1 (t) 1and0 ρ 2 (t) 1, respectively. In the following, we define the new transmission rates for Modes 3 as ( ) ρ2 (t)γ I 2 (t) C(ρ 1 (t)γ (t)), if C R (3) (t) C ( ) ρ1 (t)γ (t), otherwise ρ 2 (t)γ I 2 (t) + 1 ρ 1 (t)γ (t) + 1 C(ρ 2 (t)γ 2d (t)) (22a) R (3) 2d (t) min{c(ρ 2(t)γ 2d (t)), Q 2 (t 1)} (22b) Since Mode 4 is symmetric to Mode 3, similarly, the transmission rates for Mode 4 are defined as ( ) δ2 (t)γ I 1 (t) C(δ 1 (t)γ (t)), if C R (4) (t) C ( ) δ1 (t)γ (t), otherwise δ 2 (t)γ I 1 (t) + 1 δ 1 (t)γ (t) + 1 C(δ 2 (t)γ 1d (t)) (23a) R (4) 1d (t) min{c(δ 2(t)γ 1d (t)), Q 1 (t 1)}, (23b) where 0 δ 1 (t) 1and0 δ 2 (t) 1 are the corresponding control variables and γ I 1 (t) = P 1 h rr (t) 2 /σr2 2. We note that by decreasing the value of ρ 1 (t) from 1 to 0 in (22a), it is possible to( change the ) condition of strong interference in (22a) from C ρ2 (t)γ I 2 (t) ρ 1 (t)γ (t)+1 C(ρ 2 (t)γ 2d (t)) to C (γ I 2 (t)) C(γ 2d (t)), and still use successive cancellation. On the other hand, by changing this condition, the maximum data rate that R1 can reliably decode decreases. Similarly, when the interference is treated as noise, it is possible to reduce the interference by reducing the power (and data rate) of R2 by decreasing ρ 2 (t) and consequently allow S to transmit with a higher data rate. In the following, for the successive interference cancellation BaD (SIC-BaD) protocol, we provide the optimal values of the power control variables that maximize the data rate. Theorem 2 (Optimal Power Control Variables): The optimal power control coefficients that maximize the data rate for Mode 3 are given by ρ(t) = (ρ 1 (t), ρ 2 (t)) (24) (w 1 (t), 1), if γ I 2 (t) γ 2d (t) 3 (t) ρ(t)=(w1 (t),1) = 3 (t) ρ(t)=(1,w2 (t)) (1,w 2 (t)), otherwise where w 1 (t) = with a γi 2 2 (t)γ 2d(t)(1 μ 2 ) [ ] 1 γi 2 (t) γ 2d (t) γ (t)γ 2d (t) and w 2(t) = 0 b γ 2d (t) [ (1 μ 2 )(γ (t) + 2)γ I 2 (t) μ 1 γ I 2 (t)γ (t)γ 2d (t) ] [ b+ b 2 4ac 2a (25) ] 1 0 (26a) (26b) c (1 μ 2 )γ 2d (t)(γ (t) + 1) μ 1 γ (t)γ I 2 (t). (26c) The coefficients δ 1 (t) and δ 2 (t) for Mode 4 are obtained exactly in the same manner as ρ 1 (t) and ρ 2 (t) after replacing μ 1,γ (t), γ 2d (t),andγ I 2 (t) with μ 2,γ (t), γ 1d (t) and γ I 1 (t) respectively. Proof: Please refer to Appendix C. Moreover, the optimal values of q k (t), 1 (t), 2 (t), α(t), β(t), μ 1, and μ 2 are obtained in the same manner as in Theorem 1. However, 3 (t) and 4 (t) are obtained from (14c) and (14d), where the data rates for R (3) (t), R(3) 2d (t), R(4) (t), andr(4) 1d (t) are obtained from (22a), (22b), (23a), and (23b), respectively. Considering the condition for strong interference in (21), the optimal power control coefficients in Theorem 2 can be interpreted as follows. If (21) holds, then we obtain w 1 (t) = 1, the first condition in (24) holds, and the optimal power coefficients are (ρ 1 (t), ρ 2 (t)) = (1, 1). In other words, if (21) holds, both S and R1 transmit with maximum power and rate while R1 is still able to completely cancel the interference induced by R2. Now, if (21) does not hold, the optimal power control policy is one of the following options. i) R2 still transmits with its maximum power, i.e., ρ 2 (t) = 1, but S decreases its transmit power to ρ 1 (t) = w 1 (t) in order to

8 7364 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 11, NOVEMBER 2016 allow R1 to be able to decode R2 s codeword. Note that γ I 2 (t) γ 2d (t) has to hold for this option, otherwise, even with ρ 1 (t) = 0, R1 is still unable to decode the codeword of R2. ii) R1 transmits with its maximum power, i.e., ρ 1 (t) = 1, and R1 treats R2 s codeword as noise. Hereby, R2 decreases its transmit power to ρ 2 (t) = w 2 (t) in order to allow S to transmit with a higher rate to R1. Among these two options, the final optimal coefficient ρ(t) in (24) is the one which leads to a higher value of 3 (t). C. Adaptive Estimation of μ 1 and μ 2 Theorems 1 and 2 provide the optimal mode selection protocols, which maximize the average data rates of the considered network and for the respective modes. In order to select the optimal transmission modes, the values of μ 1 and μ 2 have to be computed, where perfect knowledge of the channel statistics is required. In practice, the CSI is estimated from a large number of channel measurements. The number of measurements depends on the desired level of accuracy and on how fast the channels change. If the number of measurements is not large enough with respect to the channel variations over time, this may result in an inaccurate estimation of the statistics. To overcome these issues, we propose an adaptive algorithm which directly computes estimates of μ 1 and μ 2 based on the well known subgradient method [27]. Let μ i (t) denote the estimated value of μ i in time slot t, and let us define the historical average data rates R (t a ), R (t a ), R 1d (t a ),and R 2d (t a ) from time slot 1 to t a as R (t a ) 1 t a R (t a ) 1 t a R 1d (t a ) 1 t a t a j=1 t a j=1 t a j=1 t a [q 1 ( j)r (1) ( j) + q 3( j)r (3) ( j)] [q 1 ( j)r (1) ( j) + q 4( j)r (4) ( j)] [q 2 ( j)r (2) 1d ( j) + q 4( j)r (4) 1d ( j)] (27a) (27b) (27c) R 2d (t a ) 1 [q 2 ( j)r (2) 2d t ( j) + q 3( j)r (3) 2d ( j)]. (27d) a j=1 Now, using (27), the elements of vector μ(t) =[ μ 1 (t), μ 2 (t)] for time slot t are obtained using the subgradient method [27] μ i (t) = μ i (t 1) + ζ i (t)[ R id (t 1) R si (t 1)], (28) where the step size ζ i (t) >0 is a properly chosen monotonically decreasing function. Eq. (28) tries to reduce the difference between R id (t) and R si (t) in each iteration. In particular, as t, we obtain R id (t) R si (t) 0, i.e., the equalities in the constraints C1 and C2 in (12) are satisfied, and μ 1 (t) μ 1 and μ 2 (t) μ 2. In practice, the updating process of μ(t) can be stopped in time slot t when μ(t) μ(t 1) <ɛ is satisfied for a sufficiently small ɛ. Furthermore, the gradient method is guaranteed to converge to the optimal Lagrange multipliers corresponding to a local optimum provided that the step sizes are chosen sufficiently small [23]. V. NUMERICAL RESULTS In this section, we numerically evaluate the achievable average rates of the proposed protocols and compare them with benchmark schemes. Throughout this section, we assume Rayleigh fading. Thereby, the instantaneous SNRs γ si (t) and γ id (t), i = 1, 2, follow exponential distributions with means γ si and γ id, respectively. In order to obtain simulation results for the proposed BaD protocol, we generated N = 10 5 channel realizations for each average SNR value, found the optimal q k (t), α(t), β(t), k, t, from Theorem 1, and plugged these values into (8). Moreover, μ 1 and μ 2 were estimated using the proposed adaptive estimation algorithm given in Section IV-C. The optimal parameter values were obtained in the same manner also for the SIC-BaD and DL-BaD protocols. A. Benchmark Schemes We compare the performance of the proposed BaD protocols with that of the multi-hopping decode-and-forward (MDF) protocol in [6] and the buffer-aided (BA) protocol for the relay selection network in [12]. The MDF protocol also employs the four transmission modes shown in Fig. 1. In particular, each time slot is divided into four sub-slots and each subslot is allocated to one of the four transmission modes. Then, based on the instantaneous CSI, the lengths of the sub-slots are optimized in order to obtain the maximum rate during each time slot. As the channel gains change from one time slot to the next, the sub-slot intervals have to be optimized in every time slot. Hence, considering that the BaD protocols in Theorems 1 and 2 introduce unbounded delay, for a fair comparison, we modify the MDF protocol to exploit buffering as well. Our goal is to keep the extended MDF protocol as similar as possible to the original MDF protocol. Therefore, we add buffers at the relays without changing the scheduling schemes of the MDF protocol. In particular, we assume that the source and the relays can transmit codewords which span (infinitely) many time slots, respectively. In order to obtain an upper bound on the performance of MDF with buffers, we assume that the buffer sizes at the relays are infinite. Furthermore, to evaluate the additional gains of the considered coding schemes used in the BaD protocol, we also compare our results with the BA protocol introduced in [12] for bufferaided relays. 1 Unlike the BaD and MDF protocols, the BA protocol in [12] selects in each time slot one of the two available relays to transmit or to receive, i.e., interference is avoided. Moreover, we compare the BaD protocol with the cut-set upper bound for the full-duplex diamond relay channel where the two relays are assumed to be connected via an errorfree link with infinite capacity, i.e., the two relay nodes can be seen as one virtual node with two distributed antennas. 2 Mathematically, this upper bound is given by R up fd = min(e{c(γ (t) + γ (t))}, 1 The rate achieved by the protocol in [12] is an upper bound for the achievable rates in [10] and [11]. Hence, for clarity of presentation, the results for the schemes [10] and [11] are not included in the figures. 2 We consider the upper bound for full-duplex relaying instead of the tighter upper bound for half-duplex relaying because the cut-set upper bound for the original half-duplex diamond relay channel is difficult to obtain.

9 SIMONI et al.: BUFFER-AIDED DIAMOND RELAY NETWORK 7365 Fig. 3. Performances comparison of the considered protocols with infinite delay for i.n.i.d. channels for three scenarios namely ( γ, γ 1d, γ 2d ) = (15 db, 15 db, 15 db), (15 db, 15 db, γ ), and ( γ, γ 10 db, γ 10 db). E{C(γ 1d (t) + γ 2d (t))}). (29) B. Performance Evaluation In Fig. 2, we show the average rate vs. the SNR γ (in db) for independent and identically distributed (i.i.d.) fading where γ si = γ id = γ. The results show that the proposed BaD protocol with unbounded delay (in Theorem 1) achieves gains of about 4 and 2 db compared to the MDF protocol with one time slot delay as in [6] and the modified MDF protocol with unbounded delay, respectively. The gain of the protocol proposed in Theorem 1 with respect to the MDF protocol with unbounded delay can be attributed to the adaptive mode selection and the additional gain obtained with respect to the MDF protocol with one time slot delay is due to the exploitation of the buffering capability. Furthermore, the BaD protocol approaches the full-duplex upper bound with a gap of less than 1 bit/symb for the scenario considered in Fig. 2. Although unbounded delay is impractical from an implementation point of view, the protocol in Theorem 1 provides an upper bound for the average rate of the considered BaD network. In Fig. 2, we also show the average rate achieved with the proposed DL-BaD protocol. Thereby, for an average delay of 4 time slots, the DL-BaD protocol proposed in Subsection IV-A which limits the queue sizes outperforms the DL-BaD protocol which limits the arrival rates at the queues and thereby starves the queues. Moreover, it achieves a gain of 2.5 db compared to the MDF protocol with one time slot delay and outperforms the MDF protocol with infinite delay. Furthermore, the results in Fig. 2 reveal that for an average delay of only 4 time slots, the proposed DL-BaD protocol suffers from an SNR loss of about 1.5 db compared to the upper bound achieved with the BaD protocol with infinite delay. In Fig. 2, the BA protocol achieves significantly lower data rates compared to the other protocols since it employs only one point-to-point transmission mode in each time slot, which results in a lower multiplexing gain compared to the other Fig. 4. Percentage of the selection of each mode (first row) and optimal values of μ 1 and μ 2 (second row) vs. γ [db] for i.n.i.d. channels for two scenarios namely ( γ, γ 1d, γ 2d ) = (15 db, 15 db, 15 db) and (15 db, 15 db, γ ). considered protocols. However, in the following, we show that for particular scenarios it can be convenient to switch to the BA protocol in order to obtain a similar performance with lower complexity. In Fig. 3, we show the average rate vs. SNR γ (in db) for independent but non-identically distributed (i.n.i.d) fading for three scenarios: Scenario 1 with SNRs ( γ, γ 1d, γ 2d ) = (15 db, 15 db, 15 db), Scenario 2 with SNRs (15 db, 15 db, γ ), and Scenario 3 with SNRs ( γ, γ 10 db, γ 10 db). For Scenario 1, the S-R1 link becomes unreliable for low γ values which makes the contribution of R1 negligible. Therefore, the BA and BaD protocols become identical as the only remaining selection options are the S-R2 and the R2-D point-to-point links. In this case, the BA protocol achieves the same multiplexing gain as the BaD protocol, and consequently, it outperforms the MDF protocol by the same margin as the BaD protocol. For Scenario 2, both the S-R1 and R2-D links become unavailable for small γ. Therefore, the average rate tends to zero for all protocols as γ 0. If the source node represents a base station, the relay nodes might have lower power budgets than the source node. This is reflected in Scenario 3. For this case, since the transmit powers of all nodes depend on γ,we obtain that as γ 0and γ, all data rates tend to zero and, respectively. In contrast, for Scenarios 1 and 2, when γ, all data rates will saturate at a certain level due to the bottleneck created by the links that have a constant average SNR. Nevertheless, for all considered scenarios, when the SNRs are sufficiently high, the MDF and BaD protocols achieve higher multiplexing gains than the BA protocol by exploiting all four transmission modes when both relays are available, and therefore, provide better performances. To further study the properties of the optimal protocol, in Fig. 4, we show the percentage of the selection of each transmission mode as well as the corresponding optimal values of the Lagrange multipliers vs. SNR γ (in db) for the Scenarios 1 and 2 considered in Fig. 3. In the following, we highlight some insights that Fig. 4 provides. At γ = 10 db, the bottleneck for transmission in Scenario 1 is the S-R1

10 7366 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 11, NOVEMBER 2016 Fig. 5. Data rates of the SIC-BaD protocol vs. γ for different levels of interference in i.i.d. channels. Fig. 6. Data rates of the SIC-BaD protocol vs. the interference for different values of γ. link. Therefore, Modes 2 and 4 are not selected, i.e., their selection probabilities are close to zero, whereas transmission Modes 1 and 3 are selected for the same percentages of time because the S-R2 and R2-D links have the same statistics. Furthermore, in order to enforce this optimal behavior, the protocol in Theorem 1 chooses μ 1 = 1 and μ which leads to the following selection metrics 1 (t) = C (1) (t) + 0.5C(1) (t) > 4(t) = 0.5C (4) (t) and 3(t) = C (3) (t)+0.5c(3) 2d (t) > 2(t) = 0.5C (2) 2d (t). On the other hand, for both scenarios, when SNR γ = 15 db, the channel is symmetric with respect to the first and second hops of each relay. In this case, we observe from Fig. 4 that Modes 1 and 2 (MAC and BC modes) are not selected whereas Modes 3 and 4 (successive relaying modes) are selected equiprobably. The respective optimal Lagrange multipliers are μ 1 = 0.5 and μ 2 = 0.5. In Fig. 5, we show how the average data rate of the SIC-BaD protocol changes for fixed values of the average interference SNR as a function of γ, where the average interference SNR is denoted by γ I γ I 1 = γ I 2. In this case, the interference is cancelled without DPC, and i.i.d channels and unbounded buffer sizes are assumed. As expected, when the interference link is stronger than the other links, the SIC-BaD protocol can completely remove the interference through successive cancellation and approach the upper bound given by BaD protocol which employs DPC. For example, for the curve for γ I = 40 db, for values of γ <20 db the SIC-BaD protocol approaches the curve of the BaD protocol. On the other hand, when γ increases compared to γ I, the data rates become lower with respect to the BaD protocol since the condition of strong interference does no longer hold. Indeed, in order to employ successive cancellation the data rate has to be lowered. In Fig. 6, the data rates for four different values of γ for i.i.d channels, where γ I varies from 10 db to 40 db are shown. The black dashed lines represent the data rates for the same SNR when DPC is employed. We note that since the BaD protocol completely removes the interference in transmission Modes 3 and 4 if DPC is used, the corresponding achievable rates do not depend on the value of the interference SNR γ I. From this figure, we note that when the interference level is either very high or very low compared to the SNR of the channels, the achievable rate of the low-complexity SIC-BaD protocol approaches that of the BaD protocol. On the other hand, when the interference level is the same as the SNR of the channels, the low-complexity SIC-BaD protocol suffers from the maximum rate loss compared to the BaD protocol. The reason for this behavior is that for very low values of γ I, the receiving relay in Modes 3 and 4 treats the interference as noise without severe degradation of the transmission rate of the source. On the other hand, for very high values of γ I,the receiving relay in Modes 3 and 4 can perfectly decode and remove the interference which causes no degradation in the transmission rate compared to the BaD protocol. However, if the interference SNR is at the same level as the SNR of the channels, the optimal power control policy in Theorem 2 reveals that either the source or the transmitting relay has to reduce their transmit power (and transmission rate) which may severely decrease the achievable rate of the SIC-BaD protocol. VI. CONCLUSIONS AND FUTURE WORK In this paper, we investigated the BaD relay network with block fading when there is no direct link between source and destination. Based on the half-duplex constraint, four transmission modes were considered. In order to exploit the diversity gain provided by the buffers, an optimal adaptive mode selection protocol was derived for the BaD network. Since the optimal BaD protocol introduces unbounded delay, we also proposed a delay-limited BaD protocol. We showed that the delay-limited BaD protocol with an average delay of only four time slots, achieves data rates which are close to the upper bound achieved by the optimal BaD protocol with unbounded delay. We also showed that the BaD protocol outperforms the considered benchmark schemes for the diamond network for both limited and unlimited delay. Moreover, we proposed a low-complexity SIC-BaD protocol which does not employ dirty paper coding for interference cancellation. Based

11 SIMONI et al.: BUFFER-AIDED DIAMOND RELAY NETWORK 7367 on numerical results, we showed that for the cases where the interference SNR is significantly higher or lower than the SNR of the other links, the SIC-BaD protocol can approach with lower complexity the same data rates as the BaD protocol. For the proposed protocol, we assumed that two decodeand-forward buffer-aided relays are available for transmission. An interesting extension of this work is to derive the optimal buffer-aided protocol, for the case, when the relay nodes can employ more efficient processing schemes, such as quantizemap-and-forward [28] and noisy-network-coding [29], or for the case, when an arbitrary number of relay nodes is available for transmission. To this end, the corresponding optimization problems have to be formulated and novel transmission modes need to be defined in order to optimally exploit the newly introduced degrees of freedom in the network. APPENDIX A PROOF OF LEMMA 1 In this appendix, we prove that the average rate is maximized when condition (9) holds. For infinite size buffers, the average transmit rates of the relays can be written independently of the dynamics of the queues as [30], [31] C id if R si > C id R id = R si if R si < C id (30) R si = C id if R si = C id. For example, if R si > C id holds, the average arrival rate that fills the buffer is larger than the average departure rate that flows out of the buffer, which means that the maximum average rate that the relay can transmit is C id, from which we obtain that R id = C id in (30). On the other hand, if R si < C id, then due to the law of the conservation of flow, the average arrival rate is equal to the average departure rate and we obtain R id = R si. Therefore, from these two conditions we obtain R id = min{ C id, R si }, thereby, if R si = C id holds, we obtain R id R si = C id. In order to find ( R 1d, R 2d ) based on (30), nine cases are possible according to the combinations of R C 1d and R C 2d. In the following, we investigate the optimality for two cases in detail. The other remaining cases can be analyzed in a similar manner. To this end, we define I k as the set comprising the time slots t for which q k (t) = 1, and k is one of the possible modes. Hence, I k contains the indices of all time slots for which mode k is active. Case 1: Assume that R > C 1d and R > C 2d hold for the optimal solution. Then, we obtain from (30) that R id = C id. In the considered case, if I 1 =, we can move some indices from I 1 to I 2 so that R and R decrease while R 1d and R 2d increase. Since rate R d = R 1d + R 2d can be further improved, Case 1 is suboptimal if I 1 =. IfI 1 = then we have I 3 = and I 4 =, otherwise, R or R would be 0 and this contradicts the condition of Case 1 that R > C 1d > 0 and R > C 2d > 0. In this case, we can move some indices from I 3 to I 2 and from I 4 to I 2, thereby reducing R and R and increasing C 2d and C 1d, and thereby increasing R 1d and R 2d, respectively. Thereby, since rate R d increases, Case 1 is sub-optimal. Case 2: Assume that R < C 1d and R < C 2d hold for the optimal solution. If I 2 =, we can move some indices from I 2 to I 1,insuchawaythat C 1d and C 2d decrease and R and R increase. Thereby, the conditions R < C 1d and R < C 2d when I 2 = are sub-optimal since the data rates R and R can be further improved, and thereby R 1d and R 2d increased, respectively. Now, consider the case when Mode 2 is not selected and I 2 =. Since the conditions for Case 2 state that R < C 1d and R < C 2d,then C 1d > 0and C 1d > 0 have to hold. Therefore, I 3 = and I 4 = also have to hold, otherwise C 1d = 0or C 2d = 0. For the conditions of Case 2, see (30), we have that R id = R si, thereby C 1d and C 2d can be decreased such that R = C 1d and R = C 2d.Tothis end, C 1d and C 2d can be decreased by moving some indices from I 3 and I 4 into I 1. Furthermore, it is possible to keep R and R unchanged by setting α(t) = 1andα(t) = 0forthe broadcast modes C (1) (t) and C(1) (t), respectively. This means that it is possible to obtain the same results by replacing the condition in Case 2 with R = C 1d and R = C 2d without changing the rate R d. Similar to the arguments for Case 1 and Case 2, for all the remaining cases except for the case when R = C 1d and R = C 2d, we can change some indices from one set I k to another, and either increase rate R d or reach the case R = C 1d and R = C 2d without reducing R d. Therefore, without loss of generality, rate R d can be maximized, if conditions C 1d = R and C 2d = R hold, cf. (9). Moreover, if (9) holds, according to (30), we obtain that R 1d = C 1d and R 2d = C 2d hold, cf. (11). This completes the proof. APPENDIX B PROOF OF THEOREM 1 In this appendix, we show the basic steps for solving the optimization problem in (12). First, we relax the binary constraint q k (t) ={0, 1} to 0 q k (t) 1 and find the solution of the relaxed problem using Lagrange dual formulation and the Karush-Kuhn-Tucker (KKT) conditions [23]. Then, we show that the solution of the relaxed problem is achieved at the boundary of the constraints, i.e., q k (t) ={0, 1}. Hence, the binary relaxation does not affect the optimal solution. Nevertheless, since the relaxed problem is still non-convex, the obtained solution corresponds to a local optimum point of the problem in (12) and not necessarily to a global optimum point. The relaxed optimization problem in a standard minimization form is given by minimize ( C 1d + C 2d ) q k (t),α(t),β(t) subject to C1: C 1d = C, C2: C 2d = C 4 C3: q k (t) 1 = 0, k=1 C4: q k (t) 0, C5: q k (t) 1 0 C6: α(t) 0, C7: α(t) 1 0 C8: β(t) 0, C9: β(t) 1 0. (31) The Lagrangian function for the above optimization problem

12 7368 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 11, NOVEMBER 2016 is given by L(q k (t), α(t), β(t), μ l,ν(t), η k (t), λ k (t), m l (t), n j (t)) = ( C 1d + C 2d ) + μ 1 ( C 1d C ) + μ 2 ( C 2d C ) ( 4 ) 4 + ν(t) q k (t) 1 η k (t)q k (t) k=1 k=1 λ k (t)(q k (t) 1) m 2 (t) (α(t) 1) k=1 m 1 (t)α(t) n 1 (t)β(t) n 2 (t) (β(t) 1), (32) where η k (t) and λ k (t) are the Lagrange multipliers for the relaxed conditions q k (t) 0andq k (t) 1, respectively, and μ 1, μ 2, ν(t), m 1 (t), m 2 (t), n 1 (t), andn 2 (t) are the Lagrange multipliers for constraints C1, C2, C3, C7, C8 and C9 in (31), respectively. A. Optimal q k (t) In order to determine the optimal selection policy, first we check the stationary conditions by calculating the derivatives of the Lagrangian function (32) with respect to q k (t) and setting them to zero. This leads to q k (t) = 1 N k(t) +[ν(t) η k (t) + λ k (t)] =0, (33) where k (t) is given in (14). In the following, we consider two possible cases depending on whether q k (t) assumes binary or non-binary values. Case 1 [Binary q k (t)]: If q k (t) = 1, then from constraint C3 in (12), q k (t) = 0, k = k has to hold. Referring to the complementary slackness conditions [23], [30], when an inequality constraint is inactive, the corresponding Lagrange multiplier has to be equal to zero. Considering the case where q k (t) = 1, the constraint q k (t) = 0 becomes inactive and the corresponding Lagrange multiplier η k (t) has to be zero. Therefore, from (33), we obtain that k (t) = N[ν(t)+λ k (t)] has to hold. Similarly for q k (t) = 0, where k = k, the constraint corresponding to Lagrange multiplier λ k (t) becomes inactive, therefore λ k (t) = 0 must hold. Similarly, we obtain from (33) k (t) = N[ν(t) η k (t)]. By subtracting k (t) from k (t), wehave k (t) k (t) = N[λ k (t) + η k (t)] 0 k = k, (34) where λ k (t) 0andη k (t) 0 must hold because of the dual feasibility condition [23]. Hence, the necessary condition for q k (t) = 1is k (t) k (t), k = k. Similarly, the necessary condition for q k (t) = 0isthat k = k for which k (t) k (t) holds. Case 2 [Non-Binary q k (t)]: If 0 < q k (t) <1, we obtain from constraint C3 in (12), that there exists at least another k = k for which 0 < q k (t) <1 holds. Similarly to Case 1, when 0 < q k (t) <1 holds for k {k, k }, due to conditions C4 and C5 in (31), the Lagrange multipliers λ k (t) and η k (t) become inactive, thus, λ k (t) = η k (t) = 0, k {k, k } must hold. Therefore, taking into account the Lagrange multipliers, we obtain from (33) that k (t) = k (t) = Nν(t). Sincewe assume that the fading has a continuous probability density function, the probability that k (t) = k (t) holds is zero. Therefore, selecting either mode k or mode k does not change the average rates. Hence, if k (t) = k (t) holds for a particular channel realization, we can select either mode k or mode k without compromising the maximum rate. From Case 1 and Case 2, we obtain the optimal selection policy in (13). B. Optimal α(t) To obtain α (t), we check again the stationary conditions by calculating the derivative of the Lagrangian function with respect to α(t) and setting it equal to zero. Assuming q 1 (t) = 1, the derivative of the Lagrangian function in (32) with respect to α(t) is given by { α(t) = A(α(t)) m 1 (t)+m 2 (t)=0, if γ (t) γ (t) A(1 α(t)) m 1 (t)+m 2 (t)=0, otherwise (35) μ where A(x) = 1 γ (t) (1+xγ (t))n log(2) + μ 2 γ (t) (1+xγ (t))n log(2).now,we check the Lagrange multipliers m 1 (t) and m 2 (t) with respect to constraints C6 and C7 in (31) and analyze (35) for α(t) = 0, α(t) = 1, and 0 <α(t) <1. In particular, if α(t) = 0 holds, the Lagrange multipliers have to be m 2 (t) = 0andm 1 (t) 0 which leads to condition A(α(t) = 0) 0. Similarly, if α(t) = 1 holds, the Lagrange multipliers have to be m 1 (t) = 0 and m 2 (t) 0 which leads to condition A(α(t) = 1) 0. Finally, if 0 <α(t) <1 holds, the Lagrange multipliers have to be m 1 (t) = m 2 (t) = 0 which leads to condition A(0 < α(t) <1) = 0. Assuming μ 1 = μ 2, conditions A(α(t) = 0) 0, and A(α(t) = 1) 0, A(0 <α(t) <1) = 0 simplify to γ (t) γ (t), γ (t) γ (t), and γ (t) = γ (t), respectively. We note that since we assume that the fading has a continuous probability density function, the probability that γ (t) = γ (t) occurs is zero. Therefore, without reducing the achievable rate, we set α(t) = 1 if γ (t) = γ (t) occurs. To conclude, if μ 1 = μ 2 holds, we obtain for the optimal power allocation variables α(t) = 0andα(t) = 1if γ (t) <γ (t) and γ (t) γ (t) hold, respectively. On the contrary, if μ 1 = μ 2 holds, conditions A(α(t) = 0) 0, { A(α(t) = 1) 0, and A(0 <α(t) <1) = 0 simplify to {x 0 γ (t) γ (t)} {x 1 γ (t) γ (t)} } {, {x 1 γ (t) γ (t)} {x 0 γ (t) γ (t)} },and 0 x 1, respectively, where x is given by A(x ) = 0 x = μ 2γ (t) μ 1 γ (t) γ (t)γ (t)(μ 1 μ 2 ). (36) In a compact form, if μ 1 = μ 2 holds, we can express the optimal α(t) as α(t) =[x ] 1 0 and α(t) = 1 [x ] 1 0 if γ (t) γ (t) and γ (t) <γ (t) hold, respectively, as given in (15).

13 SIMONI et al.: BUFFER-AIDED DIAMOND RELAY NETWORK 7369 C. Optimal β(t) Now, we differentiate the Lagrangian function with respect to β(t) and obtain [ ( ) ] β(t) = (1 μ γ1d (t) 1) C C (γ 1d (t)) 1 + γ 2d (t) [ ( )] γ2d (t) +(1 μ 2 ) C (γ 2d (t)) C 1 + γ 1d (t) n 1 (t) + n 2 (t) [ = (μ 1 μ 2 ) C (γ 1d (t)) + C (γ 2d (t)) ] C (γ 1d (t) + γ 2d (t)) n 1 (t) + n 2 (t) = 0. (37) We note that C (γ 1d (t))+c (γ 2d (t)) C (γ 1d (t) + γ 2d (t)) 0 holds for t. Moreover, we check the possible cases for the necessary conditions β(t) = 1, β(t) = 0, and 0 <β(t) <1. Case 1: When β(t) = 1, the Lagrange multipliers have to be n 1 (t) = 0andn 2 (t) 0. Therefore, we obtain from (37) that a necessary condition for β(t) = 1 to be the optimal value, is μ 1 <μ 2. Case 2: Similarly, for β(t) = 0, we obtain the necessary condition μ 1 >μ 2. Case 3: The last case is when 0 <β(t) <1, where consequently both constraints relative to the Lagrange multipliers are inactive, and we obtain as a necessary condition μ 1 = μ 2. In the following, we obtain the optimal β(t) when μ 1 = μ 2 holds. To this end, we rewrite the average data rate received at the destination in (12) as C 1d + C 2d = C (4) 1d + C (2) 1d + C (3) 2d + C (2) 2d = C (4) 1d + C (3) 2d + C sum. (38) where C (k) id = E{q k (t)c (k) id (t)}, and C sum = E{C(γ 1d (t) + γ 2d (t))}. From (4), we obtain that C (2) 1d + C (2) 2d = E{C(γ 1d(t) + γ 2d (t))} = C sum, (39) which means that the data rate at the destination does not depend on β(t). Therefore, as long as constraints C1 and C2 hold, any policy for β(t) is optimal with respect to rate maximization. In particular, μ = μ 1 = μ 2 is found such that R + R = C 1d + C 2d holds. Thereby, in order to ensure that constraints C1 and C2 hold, the role of β(t) is to ensure that one of constraints R = C 1d and R = C 2d holds because then the other one will also hold as we assumed μ is chosen such that R + R = C 1d + C 2d holds. It turns out that here even a fixed value of β(t) = β, t can satisfy constraints C1 and C2. For instance, the optimal value of β which satisfies constraint C1 (31) is given by { ( )} R E q 4 (t)c(γ 1d (t)) C γ1d (t) 1+γ 2d (t) β = { )}. (40) E C(γ 1d (t)) C ( γ1d (t) 1+γ 2d (t) D. Optimal μ 1 and μ 2 In the following, we first find the interval to which the optimal μ 1 and μ 2 belong. In particular, for given channel statistics (equivalently, given μ 1 and μ 2 ), the selection metrics k (t) should favour the selection of the links with the highest channel capacities. In fact, each selection metric is simply a weighted sum of the capacities of the involved links. Therefore, the weights have to be positive, i.e., μ i > 0and1 μ i > 0 have to hold which leads to μ i (0, 1), i = 1, 2. For μ i < 0 or μ i > 1, increasing the capacity of a link would reduce the probability of selecting the corresponding transmission modes which cannot be optimal. 3 Next, we show that the optimal values of μ 1 and μ 2 are unique. In particular, for a given local optimum, the optimal Lagrange multipliers are unique if and only if the gradients of the involved constraints, denoted by g 1 (q k (t), α(t), β(t)) and g 2 (q k (t), α(t), β(t)) where g 1 (q k (t), α(t), β(t)) = R C 1d and g 2 (q k (t), α(t), β(t)) = R C 2d, at the local optimum are linearly independent [33, Ch. 2, p. 8]. Here, vectors g 1 (q k (t), α(t), β(t)) and g 2 (q k (t), α(t), β(t)) have length 6N where N and the values of their elements depend on the random fading realizations. Hence, the probability that g 1 (q k (t), α (t), β (t)) is linearly dependent on g 2 (q k (t), α (t), β (t)) is zero which proves that the optimal μ 1 and μ 2 are unique. Therefore, using a two-dimensional search or an iterative algorithm, we can find the optimal μ 1 and μ 2 such that constraints C1 and C2 in (12) hold. This concludes the proof. APPENDIX C OPTIMAL ρ 1 (t) AND ρ 2 (t) In this section, we derive the optimal values of the power control variables ρ 1 (t) and ρ 2 (t) for Theorem 2. Based on the data rates given in (22), we note that the transmission rates and decoding strategies depend on the channel conditions. In particular, the decoding strategy for R1 changes based on the condition in (22a). We consider a minimization problem similar to the one in (31), with the difference that data rates (t)) and (R(4) (t), R(4) 1d (t)) in the constraints C1 and C2 are now given by (22) and (23), respectively. In order to avoid repetition, we do not restate the resulting Lagrangian function again. In the following, we analyze the derivative of the new Lagrangian in order to find the optimal values for ρ 1 (t) and ρ 2 (t). In the following, we derive the optimal power coefficients depending on whether R1 decodes the codeword of R2 or treats it as noise. Case 1 (R1 Decodes R2 s Codeword): In this case, the following condition has to hold (R (3) (t), R(3) 2d C ( ρ2 (t)γ I 2 (t) ρ 1 (t)γ (t) + 1 ) C(ρ 2 (t)γ 2d (t)). (41) When (41) holds, the derivative of the Lagrangian with respect to ρ 2 (t) is given by ρ 2 (t) = (1 μ 2) γ 2d (t) N ln(2) ρ 2 (t)γ 2d (t) + 1. (42) Since ρ 2 (t) is always negative, the Lagrangian decreases when ρ 2 (t) increases. Moreover, condition (41) does not change 3 A more rigorous proof can be developed by examining all possibilities for μ 1 and μ 2 being outside the interval (0, 1) and showing that these choices of μ 1 and μ 2 contradict the optimality with respect to rate maximization, see [32, Appendix, p. 1335] for a similar proof for bidirectional buffer-aided relaying.

14 7370 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 11, NOVEMBER 2016 if ρ 2 (t) varies, which means that we can increase the data rate of R (3) 2d (t) to its maximum by setting ρ 2(t) = 1 without changing the data rate of R (3) (t). Similarly, for the case when conditions (41) and ρ 2 (t) = 1 hold, we obtain the derivative of the Lagrangian with respect to ρ 1 (t) as ρ 1 (t) = μ 1 γ (t) N ln(2) ρ 1 (t)γ (t) + 1, (43) Again, the derivative ρ 1 (t) is negative, thereby the Lagrangian decreases when ρ 1 (t) increases. Differently from the previous case, condition (41) depends on ρ 1 (t), and the maximum value of ρ 1 (t) for which the condition still holds is given by [ ] γi 2 (t) γ 2d (t) 1 ρ 1 (t) =. (44) γ (t)γ 2d (t) 0 Thereby, when condition (41) holds, the optimal value of ρ 1 (t) is given by (44) and ρ 2 (t) = 1. Note that γ I 2 (t) γ 2d (t) has to hold for this case, otherwise, even with ρ 1 (t) = 0, condition (41) cannot hold. Case 2 (R1 Treats R2 s Codeword as Noise): In this case, the( transmission ) rate of S to R1 is given by R (3) (t) = C ρ1 (t)γ (t) ρ 2 (t)γ I 2 (t)+1. Thereby, by calculating the derivative with respect to ρ 1 (t), we obtain ρ 1 (t) = μ 1 γ (t) N ln(2) ρ 1 (t)γ (t) + ρ 2 (t)γ 2d (t) + 1. (45) Since ρ 1 (t) is always negative, we obtain that the optimum value is ρ 1 (t) = 1. Differentiating the Lagrangian with respect to ρ 2 (t), we obtain ρ 2 (t) = (1 μ 2) 1 γ 2d (t) ln(2) ρ 2 (t)γ 2d (t) + 1 μ 1 γ (t)γ I 2 (t) + ln(2)(ρ 2 (t)γ I 2 (t) + 1)(ρ 2 (t)γ I 2 (t) + γ (t) + 1). (46) In this case, we find the minimum of the Lagrangian by setting the derivative to zero and solving the equation with respect to ρ 2 (t). Furthermore, we check also the conditions at the boundaries for ρ 2 (t) = 0andρ 2 (t) = 1. After setting the derivative in (46) to zero and simplifying it with respect to ρ 2 (t), we obtain the following equation ρ2 2 (t)γ I 2 2 (t)γ 2d(t)(1 μ 2 ) (47) +ρ 2 (t)γ 2d (t) [ (1 μ 2 )γ I 2 (t)(γ (t) + 2) μ 1 γ I 2 γ (t)γ 2d (t) ] +(1 μ 2 )γ 2d (t)(γ (t) + 1) μ 1 γ (t)γ I 2 (t) = 0. (48) The above equation can be rewritten as a quadratic form as ax 2 +bx +c = 0, where x = ρ 2 (t), a γ 2 I 2 (t)γ 2d(t)(1 μ 2 ), b γ 2d (t)[(1 μ 2 )(γ (t)+2)γ I 2 (t) μ 1 γ I 2 (t)γ (t)γ 2d (t)], and c (1 μ 2 )γ 2d (t)(γ (t) + 1) μ 1 γ (t)γ I 2 (t), from which we obtain two possible solutions as x 1 = b b 2 4ac 2a and x 2 = b + b 2 4ac. (49) 2a Fig. 7. Derivative of the Lagrangian function with respect to ρ 2 (t). TABLE I ALL POSSIBLE SOLUTIONS FOR ρ 2 (t) Since the derivative of the Lagrangian is a quadratic function, where the second order coefficient a is always positive, then x 1 and x 2 are the local maximum and minimum of the Lagrangian function, respectively. In order to determine the value of ρ 2 (t) that minimizes the Lagrangian, based on the quadratic function of its derivative, we illustrate the solutions graphically. In Fig. 7(a), the Lagrangian function is depicted for the case when x 1 < x 2 < 0 holds. Since ρ 2 (t) can assume only values between 0 and 1, the function reaches its minimum only if ρ 2 (t) = 0. In Fig. 7(b) the case when x 1 < 0 < x 2 < 1 is depicted. In this case, the optimum value is obtained when ρ 2 (t) = x 2 holds. Similarly, we derive the solutions for ρ 2 (t) for the remaining cases as shown in Table I. Considering that 0 ρ 2 (t) 1 has to hold, we obtain ρ 2 (t) =[x 2 ] 1 0. We note that in all the cases where the solution is ρ 2 (t) = 0, only S transmits, which is equivalent to Mode 1 with α(t) = 1. In this case, the maximum data rate of Mode 3 will be less than or equal to the data rate of Mode 1 for the optimal α (t). Therefore, when ρ 2 (t) = 0 holds, it is clear that selecting transmission Mode 3 is sub-optimal since 1 (t) α (t) 3(t) ρ2 always holds. (t)=0 To summarize, we obtained the optimal power control coefficients for Case 1 and Case 2. Among these two cases, the final optimal coefficients are those which lead to a higher value of 3 (t). This leads to Theorem 2 and completes the proof. REFERENCES [1] E. C. van der Meulen, Three-terminal communication channels, Adv. Appl. Probab., vol. 3, no. 1, pp , [2] G. Kramer, M. Gastpar, and P. Gupta, Cooperative strategies and capacity theorems for relay networks, IEEE Trans. Inf. Theory, vol. 51, no. 9, pp , Sep [3] A. El Gamal and Y.-H. Kim, Network Information Theory. Cambridge, U.K.: Cambridge Univ. Press, [4] B. Rankov and A. Wittneben, Spectral efficient protocols for halfduplex fading relay channels, IEEE J. Sel. Areas Commun., vol. 25, no. 2, pp , Feb

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Rep., Renato Simoni received the B.S and M.S. degrees in engineering of telecommunications and the Ph.D. degree in telecommunication systems from the University of Florence, Italy, in 2009, 2011, and 2016, respectively, where the main fields of interest were co-operative networking, network coding, and buffer-aided relays. He is currently a Post-Doctoral Researcher with the Fraunhofer-Institut fuer Hochfrequenzphysik und Radartechnik FHR, Bonn, Germany. His current interests include automotive cognitive radars and space time adaptive algorithms. In 2012, he was a Researcher working in collaboration with the University of Florence and Telecom Italia on the formulation of new key performance indicators for the users perceived quality on the cellular data traffic. In 2011, he was a Researcher for Selex Galileo studying the performances of space-time adaptive processing algorithms for airborne radar systems. Vahid Jamali (S 12) was born in Fasa, Iran, in He received the B.S. and M.S. degrees (Hons.) in electrical engineering from the K. N. Toosi University of Technology, in 2010 and 2012, respectively. He is currently pursuing the Ph.D. degree with the University of Erlangen Nuremberg, Erlangen, Germany. His research interests include wireless communications, multiuser information theory, free space optical communications, molecular communications, cognitive radio network, LDPC codes, and optimization theory. He served as a member of the Technical Program Committees for the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS Special Issue on Recent Advances in Heterogeneous Cellular Networks in 2015 and the International Conference on Computing, Networking, and Communication in 2015 and He received the Best Paper Award from the IEEE International Conference on Communications in 2016, the Exemplary Reviewer of the IEEE COMMUNICATIONS LETTERS in 2014, and student travel grants for the SP Coding and Information School, Sao Paulo, Brazil, and Training School on Optical Wireless Communications, Istanbul, Turkey, in Nikola Zlatanov was born in Macedonia. He received the Dipl.-Ing. and master s degrees in electrical engineering from the Saints Cyril and Methodius University of Skopje, Skopje, Macedonia, in 2007 and 2010, respectively, and the Ph.D. degree from the University of British Columbia (UBC), Vancouver, Canada, in He is currently a Lecturer (Assistant Professor) with the Department of Electrical and Computer Systems Engineering, Monash University, Melbourne, Australia. His current research interests include wireless communications and information theory. He received several scholarships/awards for his work, including the UBC s Four Year Doctoral Fellowship in 2010, the UBC s Killam Doctoral Scholarship, Macedonia s Young Scientist of the Year in 2011, the Vanier Canada Graduate Scholarship in 2012, the Best Journal Paper Award from the German Information Technology Society in 2014, and the Best Conference Paper Award from the International Conference on Computing, Networking, and Communication in He is an Editor of the IEEE COMMUNICATIONS LETTERS.

16 7372 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 15, NO. 11, NOVEMBER 2016 Robert Schober (S 98 M 01 SM 08 F 10) was born in Neuendettelsau, Germany, in He received the Diploma (Univ.) and Ph.D. degrees in electrical engineering from the University of Erlangen Nuremberg, in 1997 and 2000, respectively. From 2001 to 2002 he was a Post- Doctoral Fellow with the University of Toronto, Canada, sponsored by the German Academic Exchange Service. Since 2002, he has been with the University of British Columbia (UBC), Vancouver, Canada, where he is currently a Full Professor. Since 2012, he is an Alexander von Humboldt Professor and the Chair of Digital Communication with the University of Erlangen Nuremberg, Erlangen, Germany. His research interests fall into the broad areas of communication theory, wireless communications, and statistical signal processing. Dr. Schober is a fellow of the Canadian Academy of Engineering and Engineering Institute of Canada. From 2012 to 2015 he served as an Editor-in-Chief of the IEEE TRANSACTIONS ON COMMUNICATIONS. He is currently the Chair of the Steering Committee of the IEEE TRANSACTIONS ON MOLECULAR, BIOLOGICAL AND MULTISCALE COMMUNICATION and a Member-at-Large on the Board of Governors of the IEEE Communication Society. He received several awards for his work, including the 2002 Heinz Maier Leibnitz Award of the German Science Foundation, the 2004 Innovations Award of the Vodafone Foundation for Research in Mobile Communications, the 2006 UBC Killam Research Prize, the 2007 Wilhelm Friedrich Bessel Research Award of the Alexander von Humboldt Foundation, the 2008 Charles McDowell Award for Excellence in Research from UBC, the 2011 Alexander von Humboldt Professorship, and the 2012 NSERC E.W.R. Steacie Fellowship. In addition, he received best paper awards from the German Information Technology Society, the European Association for Signal, Speech, and Image Processing (EURASIP), the IEEE ICC 2016, the IEEE WCNC 2012, the IEEE Globecom 2011, the IEEE ICUWB 2006, the International Zurich Seminar on Broadband Communications, and European Wireless Romano Fantacci (F 05) received the M.S. degree in electrical engineering and the Ph.D. degree in computer networks from the University of Florence, Florence, Italy. He is currently a Professor of Computer Networks with the University of Florence, where he is the Head of the Wireless Networks Research Laboratory. He is the Founding Director of Information Communication Technology Inc., with Finmeccanica S.p.A. and the Wireless Communications Research Centre with Telecom Italia. His current research interests encompass several fields of wireless engineering and computer communication networking, including in particular, performance evaluation and optimization of wireless networks, emerging generations of wireless standards, and cognitive wireless communications and networks. Dr. Fantacci served as an Area Editor of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, and an Associate Editor of the IEEE TRANS- ACTIONS ON COMMUNICATIONS, the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, and Telecommunications Systems. He has guest edited special issues for IEEE journals and magazines and served as the symposium chair of several IEEE conferences, including VTC, WCNC, PIRMC, ICC, and Globecom. He serves on the Board of Governors of the IEEE Sister Society AEIT and on the Advisory Board of the Security and Communication Networks Journal, and as an Associate Editor of the International Journal of Communication Systems and the IEEE Networks, and a Regional Editor of IET Communications. He received several awards for his research, including the IEE Benefactor Premium, the 2002 IEEE Distinguished Contributions to Satellite Communications Award, the 2015 IEEE WTC Recognition Award, the IEEE Sister Society AEIT Young Research Award, and the IARIA Best Paper Award. Laura Pierucci (SM 14) received the Degree in electronics engineering from the University of Florence, Italy, in She joined the Department of Information Engineering, University of Florence as an Assistant Professor in Her main research interests include digital signal processing, neural networks, radar signal processing, wireless communication systems in particular for the topics of 4G/5G systems, multiple-input and multiple-output antenna systems, co-operative communications, communication systems for emergency applications, and green ICT. She co-invented a patent on the use of RFID for health application. She has been involved in several national and European research projects on satellite communications, tele-medicine systems, wireless systems, and radar signal processing. She has been the Scientific Co-ordinator of the EU COST Action 252 and she has served as an Expert for the European Committee in the area of satellite communications. She is currently serving as an Associate Technical Editor of the Telecommunication Systems and the IEEE Communications Magazine.