6240 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 11, NOVEMBER 2015

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1 6240 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 11, NOVEMBER 2015 Achievable Rate of the Half-Duplex Multi-Hop Buffer-Aided Relay Channel With Block Fading Vahid Jamali, Student Member, IEEE, Nikola Zlatanov, Student Member, IEEE, Hebatallah Shoukry, and Robert Schober, Fellow, IEEE Abstract The half-duplex (HD) multi-hop relay channel consists of a source, multiple HD relays connected in series, and a destination where links are present only between adjacent nodes. In this paper, we focus on decode-and-forward relays and assume that the links are impaired by block fading and additive white Gaussian noise. We design a new protocol which, unlike the conventional protocols for the multi-hop relay channel, does not adhere to a fixed and predefined pattern of using the transmit, receive, and silent states of the nodes. In particular, the proposed protocol selects the optimal states of the nodes and the corresponding optimal transmission rates based on the instantaneous channel state information (CSI) of the involved links in each fading block such that the achievable average rate from source to destination is maximized. To enable adaptive scheduling of the states of the nodes, the relay nodes have to be equipped with buffers for temporary storage of the information received from the preceding node. Additionally, we discuss and address two practical challenges arising in the implementation of the optimal protocol, namely the unconstrained end-to-end delay due to data buffering at the relays and the required CSI overhead. Numerical results confirm the superiority of the proposed buffer-aided protocols compared to existing multi-hop relaying protocols. Index Terms Buffer-aided relaying, multi-hop diversity, half-duplex constraint, block fading, average delay, distributed protocol. I. INTRODUCTION FUTURE generations of communication networks are expected to include some form of cooperative, relay-based communication [1]. Relaying networks offer several benefits over traditional non-cooperative networks such as better connectivity because of their increased coverage area, and an improved quality-of-service because of their higher throughput and/or reliability [1] [3]. The relay channel was first introduced and studied by Van der Meulen [4] for the case of a simple three-node relay channel comprised of a source, a relay, and a destination and was later investigated comprehensively by Manuscript received January 9, 2015; revised May 7, 2015; accepted June 25, Date of publication July 1, 2015; date of current version November 9, The associate editor coordinating the review of this paper and approving it for publication was Y. Zeng. V. Jamali, H. Shoukry, and R. Schober are with Chair for Digital Communication at the Friedrich-Alexander University (FAU), Erlangen 91058, Germany ( vahid.jamali@fau.de; hebashoukry13@gmail.com; robert.schober@ fau.de). N. Zlatanov is with the Department of Electrical and Computer Engineering, University of British Columbia, Vancouver, BC V6T 1Z4, Canada ( zlatanov@ece.ubc.ca). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TWC Cover and El Gamal [5]. As the distance between the source and the destination increases, more than one relay have to be deployed between source and destination in order to combat the large path loss. A suitable network architecture that can be employed for such a scenario is the multi-hop or cascade relay channel. The multi-hop relay channel is comprised of a source, multiple relays connected in series, and a destination. Multihop relaying has attracted considerable research interest due to potential practical applications such as connecting remote areas and sensor networks [2], [6] [12]. The relays in the multi-hop relay channel can operate in two modes, namely full-duplex (FD) and half-duplex (HD). FD relay nodes can transmit and receive at the same time and in the same frequency band. The capacity of the ideal FD multi-hop relay channel is known when only the links between adjacent nodes are present, and can be derived using the degraded relay channel model presented in [5]. However, due to the self-interference caused by concurrent transmission and reception, the design of ideal FD relays is very challenging in practice [13] and demands precise and expensive components [14], [15]. As a result, HD relaying is often preferred for practical cooperative networks. HD relay nodes are not allowed to transmit and receive at the same time and in the same frequency band simultaneously. For the HD multi-hop relay channel, several transmission protocols have been proposed in the literature [2], [6] [10]. A common strategy for HD communication is to define a time-division schedule a priori. For this case, achievable rates for the HD multi-hop relay channel have been obtained in [2], [6] [8]. One of the protocols considered in [2], [6] is the so called multi-hop relaying without spatial reuse protocol where only one node is transmitting at a given time. Thereby, the achievable rate decreases considerably with increasing number of hops. 1 This rate degradation is avoided if non-adjacent relays are allowed to transmit simultaneously whichisreferredtoas multi-hop relaying with spatial reuse [6] [8]. For example, in the protocol proposed in [7], each relay receives in one time slot and forwards the received information in the next time slot. More generally, the protocol in [8] activates the nodes according to a common reuse schedule, i.e., the protocol in [8] contains the protocol in [7] as a special case for a reuse period of two. We note that spatial reuse leads to a more efficient use of bandwidth at the cost of possibly inducing 1 This protocol is a suitable choice if the nodes are close to each other and the link between the non-adjacent nodes are not negligible. In this case, simultaneous transmission of non-adjacent nodes can induce strong interference and hence, orthogonal transmission may be preferable IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

2 JAMALI et al.: ACHIEVABLE RATE OF THE HALF-DUPLEX MULTI-HOP BUFFER-AIDED RELAY CHANNEL 6241 inter-hop interference. Thereby, for a given receiving node, two types of interference may be distinguished for the multi-hop relay channel: i) feedback interference which is caused by the transmit signals of the subsequent nodes, and ii) feedforward interference which is caused by the transmit signals of the preceding nodes [8], [9]. Exploiting the channel state information (CSI) of the interfering links, feedback interference can be completely canceled as the preceding nodes have full knowledge of the information transmitted by the subsequent nodes in the multi-hop relay channel. Moreover, because of the HD constraint, a node may induce feedforward interference only to receiving nodes that are at least three hops away [8]. Therefore, if the path loss is severe, the feedforward interference becomes negligible and spatial reuse is an attractive option. Motivated by these practical considerations, in this paper, we focus on the HD multi-hop relay channel with arbitrary number of relays where links are present only between adjacent nodes. In wireless networks, environmental effects such as path loss, shadowing, and multipath fading cause time varying fluctuations of the signals received at the nodes. Thereby, as shown in [16] and [17] for the unidirectional and bidirectional three-node relay channels, respectively, using relays which have a fixed and predefined pattern of transmission and reception, e.g., switching successively between reception and transmission as in [7], can considerably degrade the achievable rate. In particular, in [16] and [17], the authors showed that the achievable average rates can be significantly improved if the relay s switching between reception and transmission is conditioned on the instantaneous qualities of the receiving and transmitting links in the network. Such conditional switching was first proposed in [16] and can be employed if the relays are equipped with a buffer for temporary storage of information until the quality of the transmit link becomes strong enough for the relay to forward the stored information. The aim of this paper is to develop a new protocol for the HD multi-hop relay channel with block fading based on the concept of buffer-aided relaying and to derive the corresponding achievable rates. In particular, we allow full spatial reuse by considering all feasible (according to the HD constraint) and non-redundant combinations of the states of the nodes, namely, the transmit, receive, and silent states, as possible transmission modes. We design a buffer-aided relaying protocol for the considered channel which selects the optimal transmission mode and the corresponding optimal transmission rates based on the instantaneous CSI of the involved links in each fading block such that the achievable average rate from source to destination is maximized. Additionally, we discuss and address two practical challenges for the implementation of the optimal protocol, namely the unconstrained end-to-end delay due to data buffering at the relay nodes and the required CSI overhead. In particular, we develop sub-optimal but efficient delay-constrained and distributed buffer-aided protocols to overcome the aforementioned challenges. Our numerical results reveal that the proposed protocols outperform the existing protocols available in the literature by a considerable margin. We note that the buffer-aided protocols proposed in [16], [17] for the two-hop relay channel cannot be straightforwardly extended to the general multi-hop HD relay channel considered in this paper. In fact, there are several new challenges that appear in the multi-hop relay channel including a rapidly increasing number of transmission modes, the different network architecture, and practical issues regarding CSI overhead and delay. In addition to [16], [17], the benefits of buffer-aided relaying have been studied in the literature also for the unidirectional relay channel [18] [20], the bidirectional relay channel [19], [21] [23], the multi-hop relay channel [11], [12], [24], the interference channel [25], the diamond relay network [26], [27], cognitive radio network [28], and a network with two sources, two destinations, and a single relay [29], see [3] for a comprehensive overview of buffer-aided relaying protocols. Since the buffer-aided protocols in [18] [23], [25] [29] were proposed for different network architectures, they are not applicable to the considered general multi-hop HD relay channel. Moreover, the buffer-aided multi-hop relaying protocols in [11], [12] are not optimal for rate maximization and were designed for uncoded and/or fixed rate transmission. In Section V, we use modified versions of these protocols as benchmark schemes for the proposed protocols. In the conference version of this paper [24], only the three-hop network architecture was considered and extending the approach used in [24] to the general multihop case is not straightforward. Hence, in order to obtain a concise optimal protocol in this paper, we employ a new problem formulation. The remainder of this paper is organized as follows. In Section II, the system model and the possible transmission modes are introduced, and the nodes transmission rates and the CSI knowledge required for the proposed protocol are discussed. In Section III, the rate maximization problem is formulated and the resulting optimal protocol is provided. In Section IV, challenges arising from the implementation of the proposed protocol are discussed and addressed. Numerical results are given in Section V, and conclusions are drawn in Section VI. Notations: We use the following notations throughout this paper: E x { } denotes expectation with respect to random variable x,and stands for the cardinality of a set and the absolute value of a scalar. Bold letters are used to denote vectors. Furthermore, a =[a i ] denotes a vector with elements a i, i, and a 0 denotes a vector with non-negative elements, i.e., a i 0, i. a denotes the Euclidean norm of vector a. For notational convenience, we use the definitions C(x) log 2 (1+x) and [x] + max{0, x}. II. SYSTEM MODEL In this section, we introduce the channel model, the adopted coding scheme, and the dynamics of the queues. Furthermore, we specify the possible transmission modes of the HD multihop relay channel and discuss the CSI knowledge assumed for the development of the proposed protocols. A. Channel Model We consider a multihop HD relay channel consisting of a source, a destination, and M decode-and-forward HD relays, as shown in Fig. 1. For notational simplicity, we also refer to the source, relay m, and the destination as nodes 0, m, andm + 1,

3 6242 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 11, NOVEMBER 2015 ergodic, and stationary random processes with means m = E γm {γ m }, m = 0, 1,...,M. Furthermore, we assume that the instantaneous SNRs have continuous probability density functions denoted by f γm (γ m ), m = 0, 1,...,M. Fig. 1. Multi-hop buffer-aided relay channel consisting of a source, a destination, and M decode-and-forward HD relays. respectively. We assume that only the links between adjacent nodes are present, i.e., the links between nodes m and m + 1 for m = 0, 1,...,M. This is a practical assumption when non-adjacent nodes are located far from each other and, hence, because of high attenuation, the non-adjacent links are weak. Therefore, in order to transmit a message from the source to the destination, the message has to be forwarded from one node to the next until it finally reaches the destination. We assume that time is divided into blocks of equal length indexed by b = 1,...,B and that each node transmits codewords which span one block and are comprised of N symbols indexed by n = 1,...,N. We assume that all links are impaired by complex additive white Gaussian noise (AWGN) and block fading, i.e., the channel coefficients are constant during one block and change from one block to the next. Throughout this paper, all signals and systems are represented by their complex baseband equivalents. Assuming that node m is in the receive mode, the received signal at node m can be modelled as h m 1 [b]x Ym n m 1 n [b]+h m[b]xm+1 n [b]+zn m [b], [b] = for m = 1, 2,...,M 1 h m 1 [b]xm 1 n [b]+zn m [b], for m = M, M + 1 (1) where Xm n [b], m = 0, 1,...,M, Yn m [b], m = 1, 2,...,M + 1, and Zm n [b], m = 1, 2,...,M + 1, denote the channel input of node m, the channel output of node m, and the noise at node m for the n-th symbol interval in the b-th block, respectively. Furthermore, h m [b], m = 0, 1,...,M, denote the complex-valued channel coefficient between node m and node m + 1inthe b-th fading block. We assume that the noises at the nodes are mutually independent and independent from the transmitted codewords. In addition, we impose an average per-node power constraint across the symbols in one block as follows 1 N Xm n N [b] 2 P m, m = M + 1, b, (2) n=1 i.e., P m represents the maximum transmit power of the codewords of node m. Letγ m [b] = P m h m [b] 2, m = 0, 1,...,M, denote the SNR of the link between node m and node m + 1in σm+1 2 the b-th block, where σm+1 2 = E Zm+1 n { Zn m+1 [b] 2 } is the noise variance at receiving node m + 1. We introduce set G m which contains the possible values that SNR γ m can assume, i.e., γ = [γ 0,...,γ M ] G 0 G M. 2 Moreover, the instantaneous SNRs in vector γ are assumed to be mutually independent, 2 In this paper, we drop block index b in instantaneous SNR vector γ for notational simplicity. B. Coding Scheme and Dynamics of the Queues We assume that the source intends to send its information to the destination over B fading blocks where in each block, N transmit symbols are sent and B, N. Before transmission starts, the source represents its information in binary form (information bits) and stores it in its infinite-size buffer B 0. Each of the other nodes m = 1, 2,...,M + 1 is also equipped with an infinite-size buffer B m, in which the information received from the preceding node m 1 is stored. Moreover, Q m [b], m = 1, 2,...,M, denotes the amount of normalized information in bits/symbol available in buffer B m at the end of the b-th block. Let R m [γ ] 0, m = 0, 1,...,M, denote the transmission rate from node m to node m + 1intheb-thfading block. Using these notations, the encoding at the transmitting nodes, the decoding at the receiving nodes, and the dynamics of the queues of all nodes are described in the following: Encoding: The transmitting nodes employ Gaussian codebooks, i.e., symbol Xm n [b] is generated independently according to a zero-mean rotationally invariant complex Gaussian distribution with variance P m. Moreover, we assume that the employed codebooks are known to both transmitting and receiving nodes. Thereby, if node m = 0, 1,...,M is selected for transmission, it extracts NR m [γ ] bits of information from its buffer and encodes them into a codeword taken from a codebook with rate R m [γ ]. Note that the source is assumed to be fully backlogged, i.e., it has always enough information to transmit, while the relay nodes cannot transmit more information bits than they have stored in their buffers. In other words, the transmission rates of the relay nodes have to satisfy R m [γ ] Q m [b 1], m = 0, M + 1. (3) Thereby, after the completion of the transmission in the b-th block, the amount of information in buffer B m decreases to Q m [b] =Q m [b 1] R m [γ ]. (4) Note that relay node m has to temporarily store these NR m [γ ] bits of information extracted from buffer B m in another buffer. In fact, as will be explained in the next paragraph, when relay node m receives in some future blocks, it requires this information for interference cancellation if relay node m + 1is transmitting in the same block. Decoding: Receiving node m = 1, 2,...,M + 1 decodes the codeword transmitted by the preceding node m 1 based on its channel output in (1) in two steps. In the first step, node m performs interference cancellation by removing the contribution of the codeword transmitted by node m + 1. This is possible since node m has perfect knowledge of the information stored in the buffers of the subsequent nodes as it transmitted this information in previous blocks. If node m + 1 does not transmit or m = M + 1, node m omits the first step. In the second step, node m decodes the desired codeword transmitted by node

4 JAMALI et al.: ACHIEVABLE RATE OF THE HALF-DUPLEX MULTI-HOP BUFFER-AIDED RELAY CHANNEL 6243 m 1. For reliable decoding at node m, assuming N,the transmission rate of node m 1 has to satisfy R m 1 [γ ] log 2 (1 + γ m 1 ) C(γ m 1 ), m = 0. (5) Thereby, after node m has successfully decoded the codeword transmitted by node m 1, it stores the decoded information from this codeword in its buffer B m. Hence, the amount of information in its buffer at the end of the b-th block increases to Q m [b] =Q m [b 1]+R m 1 [γ ]. (6) If node m is in the silent state in the b-th block, it does not perform any encoding, i.e., Xm n [b] =Xn m 1 [b] =0, n, nor decoding and the state of its queue remains unchanged during the b-th block, i.e., Q m [b] =Q m [b 1]. C. Transmission Modes In general, the nodes in the network can assume one of three possible states in each block, namely the transmit, the receive, and the silent states. We denote the nodes state vector by K where element K[m] {t, r, s}, m, specifies whether node m transmits, receives, or is silent. In order to avoid information loss, state vector K has to satisfy the HD constraint, i.e., if node m transmits, it cannot receive, hence node m 1must not transmit and node m + 1 has to receive. Moreover, without loss of generality, we only consider non-redundant state vectors throughout this paper where a redundant state vector is defined as a state vector which can be obtained from another state vector by changing the states of some of transmitting nodes to the silent state, i.e., by setting the transmit rates of some of the transmitting nodes to zero. In other words, by removing the redundant state vectors, we ensure that if node m has the opportunity to transmit, i.e., node m 1 and node m + 1do not transmit, it actually transmits. Considering only the nonredundant state vectors is possible since the nodes transmit rates are optimization variables in our problem formulation. Thereby, a node is allowed to transmit with zero rate, which is equivalent to the silent state. Hence, without loss of generality, we can remove redundant state vectors to simplify the problem formulation and the resulting optimal protocol, cf. Theorem 1 and Lemma 2. Based on these considerations, the set of all possible non-redundant state vectors, denoted by K,isgivenby K = {K C1 : K[m 1] = t K[m + 1] =r, if K[m] =t; C2 : K[m] =t, if K[m 1] = t K[m + 1] = t} (7) where constraint C1 guarantees the HD constraint and constraint C2 ensures the non-redundancyof the state vectors. Now, let us assume that M is given and K =K, i.e., there are K nonredundant node state vectors. Then, we refer to a given state vector K k K, k = 1,...,K, as a transmission mode denoted by M k. Fig. 2 illustrates all possible transmission modes for M = 1, 2, 3, and 4. In the following lemma, we specify the number of possible non-redundant transmission modes, K, as a function of the number of relays M. Fig. 2. Possible transmission modes for the multi-hop relay channel for different numbers of relays M = 1, 2, 3, and 4. TABLE I NUMBER OF RELEVANT TRANSMISSION MODES FOR THE HD MULTI-HOP RELAY CHANNEL Lemma 1: For a multi-hop relay channel with HD nodes, the number of non-redundant transmission modes as a function of the number of relay nodes is given by 2, if M = 1, 2 K(M) = 3, if M = 3 f (M) + 2f (M 1) + f (M 2), otherwise (8) with f (M) given by f (M) = a 1 x M 1 + a 2x M 2 + a 3x M 3, (9) where x =[x 1, x 2, x 3 ] and a =[a 1, a 2, a 3 ] are the solutions of the following equations x 3 x 1 = 0 x [1.3247, i0.5622, i0.5622] (10) x 1 x 2 x 3 x 2 1 x 2 2 x 2 a 1 a 2 = 0 1 x 3 1 x 3 2 x 3 3 a 3 1 a [0.4114, i0.1381, i0.1381], (11) respectively. Proof: Please refer to Appendix A. For convenience, for M = 1, 2,...,10 relay nodes, we show the number of transmission modes in Table I. Note that although due to constraints C1 and C2 in (7), many combinations of the states of the nodes are not allowed, Lemma 1 reveals that the number of non-redundant transmission modes is still an exponential function of the number of relay nodes. Nevertheless, for practical numbers of relay nodes, the number of transmission modes is moderate, e.g., K(M) <2M holds for M < 10. In Appendix A, a method for systematically constructing the transmission modes is introduced. Using this method, the possible transmission modes for M = 1, 2,...,10 relay nodes are shown in Fig. 12.

5 6244 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 11, NOVEMBER 2015 D. CSI Requirements Throughout this paper, we assume that transmitting node m has perfect instantaneous CSI of the link between node m and node m + 1 in order to adapt its transmission rate. Receiving node m has to have perfect instantaneous CSI of the link between node m 1 and node m and possibly of the link between node m and node m + 1 for successful coherent decoding and interference cancellation, respectively. Moreover, for the proposed optimal protocol, cf. Theorem 1 and Lemma 2, we assume that one central node, e.g., the destination, has full knowledge of the instantaneous CSI of all links. Based on the CSI and the proposed protocol, the central node determines the optimal transmission mode and the corresponding transmission rates and informs the results to the other nodes. The assumption of full CSI knowledge at a central node is relaxed in Section IV, where a distributed protocol is proposed which requires only local instantaneous CSI knowledge. Moreover, we assume that the channel states change slow enough such that the signaling overhead caused by channel estimation and feedback is negligible compared to the amount of transmitted information. In Section IV, CSI acquisition and the resulting signaling overhead of the proposed protocol are discussed in more detail. III. ACHIEVABLE LONG-TERM RATE OF THE HD MULTI-HOP RELAY CHANNEL In this section, we first formally present the problem formulation for average rate maximization of the HD multi-hop relay channel. Subsequently, we determine the optimal protocol by solving the formulated optimization problem. A. Problem Formulation In this paper, we focus on the achievable average rate of the HD multi-hop relay channel averaged over all fading states. This achievable long-term rate is denoted by τ. In particular, our goal is to propose a protocol which selects the optimal transmission mode in each fading block based on the quality of the involved links such that the average rate is maximized. For transmission mode selection, we introduce binary variables q k [γ ] {0, 1}, γ, k, whereq k [γ ]=1ifthek-th transmission mode is selected in the b-th fading block and q k [γ ]=0 if it is not selected. Furthermore, since in each fading block only one of the transmission modes can be selected, only one of the mode selection variables is equal to one and the others are zero, i.e., K k=1 q k [γ ]=1, γ, holds. Thereby, the average transmission rate from node m to node m + 1 is denoted by R m and is given by 1 B R m = lim q k [γ ]R m [γ ], m = M + 1, B B b=1 k K k [m]=t (12) where R m [γ ] has to satisfy (3) and (5). Note that in the second summation in (12), we incorporate only those transmission modes for which node m transmits, i.e., k K k [m] =t. Furthermore, the average rate that the destination receives from the source is identical to the average rate it receives from relay M, i.e., τ = R M. Therefore, the maximum achievable rate of the proposed protocol for the HD multi-hop relay channel is obtained by solving the following maximization problem P1: maximize τ = lim 1 q Q,R R B B B b=1 k K k [M]=t q k [γ ]R M [γ ], (13) where q =[q k [γ ]], γ, k, includes all mode selection variables and Q ={q q k [γ ] {0, 1}, γ, k K k=1 q k [γ ]=1, γ } is the feasible set of q. Moreover, R =[R m [γ ]], γ, m, collects the transmission rates of all nodes and R ={R R m [γ ] satisfies (3) and (5)} is the feasible set of R. We note that although τ = R M, the throughput is affected by R m, m < M, through the limiting effects of the queues on the transmission rates in each block, cf. (3). In fact, by the law of conservation of flow, the buffers cannot supply more information than they receive, i.e., the following relation holds R M R M 1... R 1 R 0. (14) Finding the optimal mode selection policy from (13) is quite involved due to the recursive dynamics of the queues, cf. (3). Nevertheless, we can obtain an upper bound on the achievable average rate of the proposed protocol by neglecting the effect of the queues on the transmission rates in (3). In order to formally state this upper bound and for clarity of presentation, we define the following average capacity rates 1 C m = lim B B = E γ B b=1 k K k [m]=t k K k [m]=t q k [γ ]C(γ m ) q k [γ ]C(γ m ), m = M + 1. (15) In fact, for a given q, C m is the maximum average transmission rate that can be sent reliably from node m to node m + 1. Note that C m is only a function of the links fading distributions and does not depend on the dynamics of the queues. Upper Bound τ upp : The achievable average rate of the proposed protocol with adaptive mode selection for the Gaussian HD multi-hop relay channel with block fading is upper bounded by τ upp obtained from the following optimization problem P2: maximize q Q,τ upp 0 τ upp subject to τ upp C m, m = M + 1. (16) Solving the optimization problem for the upper bound in (16) is significantly simpler than solving the optimization problem in (13) as the recursive nature of the dynamics of the queues in (3) is avoided. In the following subsections, we solve (16) and show that this upper bound is indeed achievable. We provide a corresponding protocol by specifying the optimal policies for mode selection, q, and rate selection, R.

6 JAMALI et al.: ACHIEVABLE RATE OF THE HALF-DUPLEX MULTI-HOP BUFFER-AIDED RELAY CHANNEL 6245 B. Upper Bound In this subsection, we determine τ upp and the corresponding optimal mode selection policy as the solution of (16). Due to the binary constraints q k [γ ] {0, 1}, γ, k, the optimization problem in (16) is an integer programming problem which belongs to the category of non-deterministic polynomial-time hard (NP hard) problems. To make the problem tractable, we relax the binary constraints to 0 q k [γ ] 1 which in general implies that the solution of the relaxed problem might not be in the feasible set of the original problem as the relaxed problem has a larger feasible set than the original problem. However, for the optimization problem in (16), one of the solutions always lies at the boundaries of 0 q k [γ ] 1, thus, this solution of the relaxed problem solves the original problem as well. For future reference, let Q ={q q k [γ ] [0, 1], γ, k Kk=1 q k [γ ]=1, γ } denote the feasible set of q for the relaxed constraint. We solve the considered problem via the dual formulation of relaxed problem (16). As the relaxed problem is linear in the optimization variables, the duality gap is zero and the solution of the primal problem can be found from the solution of the dual problem [30]. Before formally stating the optimal mode selection policy, we introduce some variables that we require for specifying the solution of (16), i.e., the optimal protocol. In particular, the optimal protocol involves weighted sums of the instantaneous capacities C(γ m ) with constant weights μ m, m, which are referred to as selection weights. 3 The value of μ =[μ m ], m, depends on the channel statistics and for given channel statistics, the optimal value of μ is unique. Theorem 1: The solution of (16), i.e., the optimal mode selection policy which maximizes the upper bound τ upp for the considered Gaussian HD multi-hop relay channel with block fading, is given by 1, if k = arg max k [γ ] q k [γ ]= k (17) 0, otherwise where k [γ ] is referr ed to as selection metric and is given by k [γ ]= μ m C(γ m ). (18) m K k [m]=t Moreover, μ is constant and its optimal value can be obtained offline before transmission starts using iterative Algorithm 1 at the top of the next page with the following update equations [ μm [i] λ m [i] ( C m μ m [i + 1] = [i] C 0 [i])] +, if m = 0 [ 1 + M v=1 μ v [i + 1]], if m = 0 (19) where i is the iteration index and λ m [i], m, are appropriately chosen step size parameters. For a given μ[i] =[μ m [i]], m, in the i-th iteration, q k [γ ], γ, k, is obtained from (17). There- 3 Appendix B reveals that the selection weights μ m, m, are in fact Lagrange multipliers (dual variables) corresponding to the constraint τ upp C m in (16). fore, using the distribution of the link SNRs f γm (γ m ), m, the elements of C [i] =[ C m [i]], m, are computed analytically as C m [i] =E γ q k [γ ]C(γ m ) (20) = k K k [m]=t γ G m m k K k [m]=t q k [γ ]C(γ m ) f γm (γ m )dγ. m Employing the optimal μ obtained from Algorithm 1 and the optimal q k [γ ], γ, k, from (17), and substituting them into (20) to compute C m, upper bound τ upp is obtained as τ upp = min m Algorithm 1 Gradient algorithm for μ C m. (21) Initialize i=0, μ[0]=[μ m [0]], m, and a desired small ε>0 repeat 1. Compute C [i] from (20) for μ[i] 2. Update μ[i + 1] based on (19) 3. Set i = i + 1 until μ[i + 1] μ[i] <ε return μ =μ[i] and C = C [i] and compute τ upp from (21) Proof: Please refer to Appendix B. Remark 1: In (18), the capacity of each link C(γ m ) is weighted by constant μ m and the metric for each transmission mode, k [γ ], is simply a sum of the weighted capacities of all involved links, cf. (18). Moreover, since the fading states have continuous probability density functions, the probability that k [γ ]= k [γ ], k = k, holds is zero. Therefore, the selection policy in (17) indicates that, for any instantaneous SNR γ,the choice of the optimal transmission mode is unique. In other words, for a given fading state, it is sub-optimal to share the resources between the transmission modes and only one of the transmission modes should be used. Hence, adaptive mode selection is the key to improve the achievable average rate of the multi-hop relay channel with HD nodes and block fading. Remark 2: The gradient method in Algorithm 1 is guaranteed to converge to the optimal dual variable μ provided that the step sizes λ m [i], m, i, are chosen sufficiently small [30]. Common choices of the step sizes include: i) Constant step size, i.e., λ m [i] =λ m, m, i, ii) square summable but not absolutely summable step size, i.e., i=1 λ 2 m [i] < and i=1 λ m [i], m,andiii) nonsummable diminishing step size, i.e., lim λ m[i] =0 and i i=1 λ m [i], m. For a comprehensive study of the different choices of step sizes and the resulting convergence properties, we refer to [30]. C. Optimal Protocol In this subsection, we discuss the achievability of the upper bound τ upp and provide the corresponding protocol in the following lemma.

7 6246 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 11, NOVEMBER 2015 Lemma 2: The upper bound τ upp is achievable for the considered Gaussian HD multi-hop relay channel with block fading provided that the optimal values of q given in Theorem 1 are employed and the following rate selection policy is used R m [γ ]=min {ρ m C(γ m ), Q m [b 1]} (22) where ρ m is a constant and given by τ upp, if m = 0 C m ρ m = [ ] τ upp, 1, otherwise C m (23) and C m is obtained from (20) for the optimal μ from Algorithm 1. Proof: Please refer to Appendix C. Remark 3: The optimal rate selection policy in Lemma 2 indicates that, provided there are enough bits in the buffers, each node transmits with a constant fraction of the capacity rate in each fading block. Thereby, the central node can calculate constants ρ m, m, offline and inform the result to the nodes. Subsequently, according to Lemma 2, each transmitting node is able to determine its optimal transmission rate based on its local CSI, which is needed to calculate the respective transmitting link capacity, C(γ m ), and the status of the buffer, Q m [b 1]. Remark 4: The purpose of constants ρ m, m, istomakethe average transmission rates on all links equal to min m C m in (21). For the source, the constant is unique, i.e., ρ 0 = τ upp, while for C 0 the relay] nodes, we have a degree of freedom in choosing ρ m [ τ upp, 1. Based on the proof of Lemma 2 in Appendix C, we C m discuss the following cases to provide some intuition: Case 1: If we choose ρ m = τ upp, m, the rate selection C m policy proposed in Lemma 2 operates the queue of buffer B m such that the probability that buffer B m does not have enough information to supply, due to the constraint in (3), vanishes as B. Therefore, the effect of the queues at the relay nodes becomes negligible as B and the upper bound τ upp can be achieved. Case 2: If we choose ρ m > τ upp C m, m = 0, the probability that buffer B m does not have enough information to supply does not approach zero even if B. However, compared to Case 1, relay node m transmits with higher rates in some blocks since ρ m is larger compared to Case 1 but it transmits with lower transmission rates in other blocks because buffer B m does not have enough information to supply. Nevertheless, the average transmission rate of relay node m for both Case 1 and Case 2 is identical to τ upp, i.e., the upper bound in (21) is achievable by the optimal rate selection policy given in Lemma 2. Remark 5: Constants ρ m in Lemma 2 are required to stabilize the queues especially for the case of asymmetric channels. For instance, if the relay M-to-destination link is the bottleneck of the network, i.e., M m, m = M, the source cannot transmit with its capacity rate when it is selected to transmit. Otherwise, some information bits would be trapped in buffer B M and would never be forwarded to the destination. In order to avoid this information loss, the source transmits only with a fraction of the capacity rate such that relay M is able to forward this information to the destination in some later blocks. Note that, as discussed for Case 2 in the previous remark, the relay nodes can choose ρ m > τ upp, m = 0, in this case, the limiting C m effect of the queues, cf. (3), will stabilize the buffers. Remark 6: The optimal protocol proposed in this paper, i.e., the mode selection in Theorem 1 and the rate selection in Lemma 2, contains the following protocols available in the literature as special cases: One Relay Node (M = 1): The optimal protocol for this case was derived in [16]. Thereby, only two transmission modes are possible: i) the source-to-relay link, and ii) the relayto-destination link, see Fig. 2. By substituting the selection metrics for these two modes (links), i.e., 0 [γ ]=μ 0 C(γ 0 ) = (1 μ 1 )C(γ 0 ) and 1 [γ ]=μ 1 C(γ 1 ), into (17), we obtain the optimal link selection policy in [16, Theorem 3]. Moreover, in this case, τ upp = C 0 = C 1 holds which leads to ρ 0 = ρ 1 = 1 from Lemma 2. Hence, both source and relay transmit with the capacity rate when they are selected for transmission. Moreover, for the optimal policy, the relay node is never silent when M = 1. Two Relay Nodes (M = 2): The optimal protocol for this case was recently derived in a conference version of this paper [24]. Thereby, it was assumed that the nodes always transmit with the capacity rates. In order to stabilize the buffers at the relays, four transmission modes, M i, i = 1,...,4, were considered: M 1 : both the source and relay 2 transmit and relay 1 and the destination receive, M 2 : relay 1 transmits, relay 2 receives, and the source and the destination are silent, M 3 : relay 2 transmits, the destination receives, and the source and relay 1 are silent, and M 4 : the source transmits, relay 1 receives, and relay 2 and the destination are silent. We note that the extension of the problem formulation in [24] to the general multi-hop HD relay channel with arbitrary number of relay nodes leads to a very complicated optimal mode selection protocol. Hence, in this paper, we introduced the concept of non-redundant transmission modes in (7) where the redundant transmission modes were removed to simplify the problem formulation. Thereby, for M = 2, only transmission modes M 1 and M 2 in [24] are non-redundant according to (7), see Fig. 2. Nevertheless, both the protocol proposed in this paper and the protocol in [24] lead to the same maximum throughput τ = τ upp. For symmetric channels, the protocol proposed in [24] selects only transmission modes M 1 and M 2, and hence becomes identical to the protocol proposed in this paper since ρ m = 1, m, holds for symmetric channels according to Lemma 2. We note that the problem formulation and the approach used to obtain the optimal solution in this paper are quite different from those used in [16], [24]. For example, the upper bound, the achievability proof, and the concept of non-redundant transmission modes are not employed in [16], [24]. Therefore, although in special cases, the proposed protocol reduces to the protocols in [16], [24], it is a non-straightforward generalization of these protocols. IV. STRATEGIES TO OVERCOME PRACTICAL CHALLENGES In this section, we discuss some of the challenges arising in the implementation of the optimal protocol and propose possible strategies to overcome these challenges.

8 JAMALI et al.: ACHIEVABLE RATE OF THE HALF-DUPLEX MULTI-HOP BUFFER-AIDED RELAY CHANNEL 6247 A. Delay-Constrained Transmission The optimal protocol proposed in the previous section is delay-unconstrained and provides a performance upper bound for delay-constrained transmission. For the delayunconstrained protocol the average delay tends to infinity as B. In fact, the advantages of buffering and adaptive mode selection come at the expense of an increased end-to-end delay. However, our numerical results in Section V show that with the simple heuristic modifications proposed in this subsection, the protocol for delay-unconstrained transmission can be also employed for delay-constrained transmission at the expense of a small performance degradation due to the delay constraint. Let T[b] denote the waiting time (delay) that a bit transmitted from the source in the b-th block experiences until it reaches the destination. In other words, if a bit is transmitted in the b-th block by the source and is decoded in the b -th block by the destination, the delay for this bit is T[b] =b b. Hence, intuitively, for the HD multi-hop relay channel, the delay is lower bounded by M. Moreover, according to Little s Law [31], the average waiting time/delay is given by T = M m=1 Q Mm=1 m (a) Q m =, (24) R m 1 R 0 where Q m = B 1 Bb=1 Q m [b] is the average queue size at buffer m and R m is given in (12). Moreover, equality (a) in (24) holds for non-absorbing queues where R 0 = R 1 = = R M holds. In fact, if there existed an m for which R m > R m+1 held, the queue at buffer B m+1 would be absorbing. Moreover, R m < R m+1 cannot hold due to the law of conservation of flow, cf. (41). Therefore, for non-absorbing queues, we must have R 0 = R 1 = = R M. The basic idea behind the delay-limited protocol proposed in this subsection is to force the queues at the relay nodes to operate around certain threshold values Q thr m, m = 1, 2,...,M. Thereby, the proposed delay-constrained protocol does not permit the queues to grow infinitely large as B.Inthefollowing, we formally introduce the proposed delay-constrained protocol. Delay-Constrained Protocol: The following mode and rate selection policies lead to a delay-constrained transmission for the considered Gaussian HD multi-hop relay channel with block fading: The transmission mode is selected according to (17) in each block by using the following modified selection metric k [γ ]= m K k [m]=t μ m min {Q m [b 1]}, { [ } + Q thr m+1 Q m+1[b 1]], C(γm ), (25) and the corresponding rate selection policy is given by { { min ρ 0 C(γ m ), [ Q thr 1 R m [γ ]= Q 1[b 1] ] + }, if m = 0 min {C(γ m ), Q m [b 1]}, otherwise (26) Furthermore, μ and ρ 0 for the delay-constrained protocol are obtained from Algorithm 1 and (23), respectively, with the following modification: Instead of computing C m [i] analytically from (20), we compute it numerically by independently generating B realizations of SNR vector γ according to f γm (γ m ), m, and inserting them into C m [i] = 1 B B b=1 k K k [m]=t q k [γ ]C(γ m ), (27) where q k [γ ], k, γ, is obtained from (17) with the new metrics given in (25). The above protocol selects the transmission modes such that the queue at buffer B m, m = 2, 3,...,M, operates around threshold Q thr. Moreover, the proposed protocol does not permit m the queue at buffer B 1 to exceed threshold Q thr 1 by limiting the transmission rate of the source. This policy effectively limits the average delay as will be shown in Section V. Remark 7: In the following, we illustrate that the proposed delay-constrained protocol with a practically large buffer size and delay requirement can approach the performance of the optimal delay-unconstrained protocol with infinite buffer size. To show this, let Q max and T req denote the finite buffer size of the relay nodes and the average delay requirement, respectively. Thereby, for large values of T req, we obtain the following upper bound on the average delay T = Mm=1 Q m R 0 (a) Mm=1 Q m τ upp (b) MQmax, (28) τ upp where (a) follows from the assumption of a large delay requirement and (b) follows from Q m Q max. Hence, if we choose buffer size Q max = τ upp T req M, the average delay constraint will be satisfied, i.e., T T req. For example, Fig. 9 in Section V shows that for M = 2, the proposed delay-limited protocol can closely approach τ upp for an average delay of more than T = 20 blocks. To obtain insight regarding the range of Q max, assume M = 2, τ upp = 1.8 bits/symbol (corresponding to γ = 10 db in Fig. 9), T req = 100 to ensure (a) in (28), and that each codeword is comprised of 2000 symbols [32]. Thereby, we obtain Q max = = 180 kbits which considering current memory technology [33], is a quite practical value for a buffer size. Although, for a large delay requirement, Q max = τ upp T req guarantees that T T req holds, in order to achieve M a particular average delay, i.e., T = T req,wehavetofindan appropriate value for threshold Q thr m < Qmax numerically. B. CSI Overhead For the implementation of the optimal protocol introduced in Theorem 1 and Lemma 2, a central node has to collect the instantaneous CSI of all links in each fading block to compute q k [γ ] and R m [γ ] from (17) and (22), respectively. In fact, the protocol proposed in Theorem 1 and Lemma 2 constitutes a performance upper bound assuming that the perfect CSI of all links is available. However, as M increases, collecting the CSI of all links at a central node might not be possible for many practical scenarios. Hence, in the following, exploiting

9 6248 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 11, NOVEMBER 2015 Fig. 3. Schematic representation of the local CSI acquisition for reciprocal and non-reciprocal channels with/without using feedback in the multi-hop HD relay channel. Solid arrows indicate pilot transmissions whereas dot arrows indicate feedback. the intuition that the optimal protocol provides, we propose a sub-optimal distributed protocol which requires only local instantaneous CSI at the nodes and full statistical CSI of all links. Since the statistics of the channel change much more slowly than the instantaneous CSI, the proposed distributed protocol entails a much lower signaling overhead compared to the optimal protocol. For the statement of the distributed protocol, we define d m [b] {t, r, s} as a decision variable for node m where if d m [b] =t, d m [b] =r, andd m [b] =s hold, node m transmits, receives, and is silent, respectively, in the b-th block. Distributed Protocol: The proposed distributed protocol comprises the following two phases: Phase 1 (Local Preference): At the beginning of each block, all relay nodes obtain an estimate of their local link SNR, where the local link SNRs for relay node m are γ m 1 [b] and γ m [b]. Thereby, each relay node m locally determines its state preference, denoted by d m [b] {t, r, s}, based on the mode selection policy in (17) after substituting all C(γ m ) for the non-adjacent links by their expected values E γ {C(γ m )}. Phase 2 (Final Decision): All nodes m > 1 inform their preceding node about their state preferences. Consecutively, each node determines its final state based on the final decision of its preceding node, its own state preference, and the state preference of its subsequent node according to { t, if d 1 [b] =t d 2 [b] =r d 1 [b] = (29a) r, otherwise t, if d m 1 [b] = t {{ d m [b] =t d m+1 [b] =r} d m [b] m =1,M = { d m [b] =r d m+1 [b] =t}} r, if d m 1 [b] =t s, otherwise (29b) { t, if d M 1 [b] =r d M [b] = (29c) r, otherwise After obtaining d m [b] from (29), relay m sends its decision d m [b] to relay m + 1. Hence, at the end of Phase 2, the states of all the nodes are determined. Furthermore, μ and ρ for the distributed protocol are obtained from Algorithm 1 and (23), respectively, with a similar modification as applied for the delay-constrained protocol. In particular, instead of computing C m [i] analytically from (20), we compute it numerically from (27) after determining the corresponding q k [γ ], k, γ, based on d m [b], b, obtained from (29). Remark 8: The rationale behind the decision exchange strategy employed in Phase 2 of the proposed distributed protocol is as follows. The state preference of the relay nodes in Phase 1 based the local CSI knowledge may violate the HD constraint. To resolve this problem, the relay nodes exchange their decisions according to (29) as explained for relay 1, cf. (29a), in detail in the following. i) If [ d 1 [b], d 2 [b]] = [r, t], [t, r], then the HD constraint is not violated and relay 1 chooses its state preference as the final decision. ii) If [ d 1 [b], d 2 [b]] = [r, r], [t, t], the HD constraint is violated. Now, our strategy to resolve this problem is to choose the best link with the highest expected (weighted) link capacity. Thereby, if [ d 1 [b], d 2 [b]] = [r, r] holds, it is to be expected that the source-to-relay 1 link is better than the relay 1-to-relay 2 link (since d 1 [b] =r) andthe relay 1-to-relay 2 link is better than the relay 2-to-relay 3 link (since d 2 [b] =r). Therefore, relay 1 chooses to receive in order to exploit the good source-to-relay 1 link, i.e., d 1 [b] =r. If [ d 1 [b], d 2 [b]] = [t, t] holds, it is to be expected that the sourceto-relay 1 link is worse than the relay 1-to-relay 2 link (since d 1 [b] =t) and the relay 1-to-relay 2 link is worse than the relay 2-to-relay 3 link (since d 2 [b] =t). Therefore, relay 1 decides to receive in order to let relay 2 exploit the good relay 2-to-relay 3 link, i.e., d 1 [b] =r. This leads to (29a). The policies in (29b) and (29c) follow a similar rational. Remark 9: Fig. 3 schematically illustrates the local CSI acquisition in the multi-hop HD relay channel. In particular, the CSI acquisition has to be conducted in several stages. In order to avoid interference from the adjacent nodes, the directions of pilot transmission for the active links have to be changed in an alternating manner, see Fig. 3. Recall that we assume channel reciprocity in the system model in Section II. Thereby, if feedback is allowed, two stages of pilot transmissions are sufficient, i.e., Fig. 3(a). Otherwise, two other pilot transmissions are required to acquire the local CSI at all nodes, i.e., Fig. 3(b). However, if channel reciprocity does not hold, e.g., due to hardware impairments, the local CSI acquisition can be achieved by using four stages of pilot transmissions and feedback as shown in Fig. 3(c). Remark 10: Note that for the proposed protocols, CSI acquisition, mode selection, informing the nodes, and data transmission must occur within the coherence time of the channel, denoted by T c, i.e., the block length has to be identical to T c. For typical wireless networks with fixed/slow users, the coherence time may span 0.1 s to 1 s or more [34]. Let

10 JAMALI et al.: ACHIEVABLE RATE OF THE HALF-DUPLEX MULTI-HOP BUFFER-AIDED RELAY CHANNEL 6249 T o denote the total overhead time for CSI acquisition, mode selection, and informing the nodes. The value of T o depends on the number of relay nodes, the adopted channel estimation technique, and the length of pilot sequence. Here, we focus only on the overhead required for local CSI acquisition at all nodes, denoted by TCSI,l o, and global CSI acquisition to gather the instantaneous of all the links at a central node, denoted by TCSI,g o,i.e.,wehaveto = TCSI,l o + To CSI,g. For instance, assuming the four-stage local CSI acquisition in Fig. 3(b), the minimum overhead of pilot transmission is given by TCSI,l o = W 4 [35] where W is the bandwidth, i.e., the duration of the pilot sequence is assumed to be one symbol interval. Note that the local CSI overhead does not scale with M. To compute TCSI,g o, let s assume that the value of each link SNR is quantized to n b bits. Thereby, assuming the destination is the central node, SNR γ m has to be sent over M m hops to the destination. Hence, M 1 M(M+1) m=0 [M m] = 2 total transmissions are needed to convey all the estimated link SNRs to the destination where in each hop n b bits have to transmitted. To obtain some insight, let s assume that the average capacities of all links are equal to C bits/channel use. Then, the overhead for global CSI acquisition at the destination is roughly given by TCSI,g o = M(M+1) 2 n b C W 1. Furthermore, the effective rate of the proposed protocol after accounting for the overhead time is given by τ eff = Tc T o T c τ. Assuming practical values for the involved parameters, i.e., T c = 1s,W = 15 khz, M = 5, n b = 8 bits, and C = 3 bits/channel use (corresponding to γ 10 db) [34], [35], we obtain τ eff = τ. In this paper, as performance upper bound, we assume that the value of T o is negligible compared to that of T c, i.e., τ eff τ. Remark 11: We emphasize that the proposed delayconstrained and distributed protocols are heuristic and suboptimal. However, in Section V, our numerical results show that these sub-optimal protocols are indeed efficient when compared with the performance upper bound τ upp in Theorem 1. The derivation of the optimal delay-constrained/distributed protocols is beyond the scope of this paper and constitutes an interesting topic for future research. V. A NALYTICAL AND SIMULATION RESULTS In this section, we present performance results when all links in the network undergo independent Rayleigh fading, i.e., the { SNR gains } follow exponential distributions f γm (γ m ) = 1 m exp γ m m, m. Moreover, we assume that the nodes are placed on a straight line with distance r m, m = 0, 1,...,M between node m and node m + 1. Hence, the average link SNRs are given by m = P m 1 σm+1 2 rm κ,whereκ is the path loss exponent and depends on the environment. In this section, unless specified otherwise, we use κ = 3 for all the analytical and simulation results. For notational convenience, we also introduce vector r =[r m ], m, which collects all link distances, and assume that the transmit SNRs of all nodes are identical and given by γ = P m. In the following, we first introduce bench- σm+1 2 mark schemes which are used to evaluate the performance of the proposed protocols. Subsequently, we provide performance results for the proposed protocols and the benchmark schemes. A. Benchmark Schemes As benchmarks, we employ modified versions of the conventional protocols proposed in [6] and [10], where the nodes use a prefixed schedule of transmission and reception, as well as the protocols with adaptive link selection proposed in [11] and [12]. In particular, a non-buffer-aided protocol for the HD multi-hop relay channel is obtained by dividing each block into M + 1 sub-blocks in which nodes 0, 1,..., M transmit consecutively to nodes 1, 2,...,M + 1, respectively. If the sub-block lengths are optimized for throughput maximization in each fading block and B, the achievable rate of this protocol approaches [6], [10] τ non-buffer conv [ M = E γ m=0 ] 1 1 C(γ m ). (30) In order to apply the conventional protocol in [7] for the block fading model used in this paper, we have to modify the protocol such that the relay nodes are equipped with buffers. Thereby, we assume that in the first α fraction of each fading block, nodes 0, 2, 4,... transmit and in the remaining 1 α fractionof theblock, nodes1, 3, 5,...transmit. For the optimal value of α as B, the achievable rate of this scheme approaches [7] τconv buffer min m=0,...,m 1 [ ] 1 E γm {C(γ m )} E γm+1 {C(γ m+1 )} (31) The protocols in [11] and [12] were proposed for fixed rate transmission. Hence, in order to employ them as benchmarks, we modify these protocols such that each node transmits with a rate equal to the capacity of its underlying transmit channel. In the protocol in [11], only the link with the highest SNR is activated in each fading block. Hence, this protocol is mainly applicable to the case when all links are affected by independent and identically distributed (i.i.d.) fading and may cause data loss due to buffer overflow for independent and non-identically distributed (i.n.d.) fading. On the other hand, the protocol in [12] activates the link whose SNR cumulative distribution function (CDF) gives the highest ordinate value among all the links. Hence, the protocol in [12] is applicable to both i.i.d. and i.n.d. fading. The achieved average rates of the protocol in [11] (only for i.i.d. fading) and the protocol in [12] are given by τmax-link buffer = 1 { } M + 1 E γ max C(γ m) and (32) m=0,...,m τcdf-link buffer = 1 M + 1 E γ {C(γ m )}, where m = arg max F γm (γ m ), (33) m=0,...,m respectively, where F γm (γ m ) is the CDF of the SNR γ m. In the following, we denote the achievable throughputs of the proposed optimal, delay-limited, and distributed protocols by τopt-mode buffer, τ del-mode buffer,andτbuffer dis-mode, respectively. For all considered protocols, Table II summarizes the CSI requirement in terms of local instantaneous CSI (LI-CSI), global instantaneous

11 6250 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 11, NOVEMBER 2015 TABLE II CSI AND BUFFER REQUIREMENTS AS WELL AS DELAY PROPERTIES OF THE PROPOSED PROTOCOLS AND BENCHMARK SCHEMES (THE PROTOCOLS ARE SPECIFIED BY THEIR ACHIEVABLE THROUGHPUTS) Fig. 5. Achievable average rate in bits/symbol vs. transmit power γ in db for M = 3 relay nodes and symmetric channels with r =[1, 1, 1, 1]. Fig. 4. [ Dual variable μ versus iteration number i for different step sizes, μ[0] = 13, 1 3, 1 ] 3, M = 2, and r =[1.05, 1, 0.95]. CSI (GI-CSI), and global statistical CSI (GS-CSI) as well as the need for buffers and the resulting delay. B. Performance Evaluation For the benchmark schemes, the results shown in Figs. 5 and 6 are analytical results, cf. (30) (32). For the proposed optimal protocol, we present both analytical and simulation results in Figs. 5 and 6 and only simulation results in Figs. 7 and 8. In particular, the analytical results are obtained from (21). For the simulation results, we generate B = 10 6 independent fading realizations and choose the optimal transmission mode and the corresponding transmission rates for each fading block according to Theorem 1 and Lemma 2. The achievable throughput is in fact the average number of information bits/symbol which are received at the destination during B fading blocks. Finally, for the proposed delay-limited and distributed protocols, only simulation results are shown in Figs. 9 and 10, respectively. Before presenting the results for the achievable rate, in Fig. 4, we show the convergence behavior of Algorithm 1. We plot the value of dual variable μ versus iteration number [ i for ] different step sizes λ[i] =0.01, 0.1, 1 i i, i, μ[0] = 13, 1 3, 1 3, M = 2, r =[1.05, 1, 0.95], andκ = 3. We also solved the relaxed version of optimization problem P2 in (16) using the Fig. 6. Achievable average rate in bits/symbol vs. transmit power γ in db for M = 3 relay nodes and asymmetric channels with r =[2, 1, 1, 2]. CVX solver [36] and include the dual variable found by CVX in Fig. 4 as a benchmark. Fig. 4 shows that for the considered choices of the step sizes, the dual variables obtained by Algorithm 1 converge to those obtained with CVX. Moreover, as expected, the convergence behavior of Algorithm 1 depends on the choice of the step size, e.g., for the constant step size λ[i] =0.01, the convergence is slower and for the adaptive step sizes λ[i] = 0.1, 1 i i, we observe oscillation for i < 15. Note that if the channel gains are not stationary processes, i.e., the channel statistics vary due to e.g. a changing mobility of the nodes, μ has to be updated on a regular basis in order to track the possible changes of the channel statistics. For this case, a constant step size is preferable over adaptive step sizes. In Figs. 5 and 6, we show the achievable average rate versus the transmit SNR of the nodes, γ in db, for symmetric channels (r =[1, 1, 1, 1]) and asymmetric channels (r =[2, 1, 1, 2]), respectively. We observe a perfect match between analytical and simulation results for the proposed

12 JAMALI et al.: ACHIEVABLE RATE OF THE HALF-DUPLEX MULTI-HOP BUFFER-AIDED RELAY CHANNEL 6251 Fig. 7. Achievable average rate in bits/symbol vs. the number of relay nodes M for γ = 10 db and two scenarios for symmetric channels (r m = r, m): i) r = 1andii) M m=0 r m = 1. protocol. Moreover, Fig. 5 reveals that the multiplexing gain (i.e., the slope of the curves as γ ) of the protocols with spatial reuse, i.e., τopt-mode buffer and τ buffer conv, is higher than the multiplexing gain of the protocols without spatial reuse, i.e., τconv non-buffer, τmax-link buffer buffer,andτcdf-link. Moreover, for symmetric channels, we obtain τmax-link buffer = τ cdf-link buffer, since both protocols in [11] and [12] select the same link in each block. The protocols with a higher multiplexing gain outperform the protocols with a lower multiplexing gain in the high SNR regime. The proposed protocol outperforms all benchmark schemes for all SNRs. For asymmetric channels, the same behavior in terms of the multiplexing gain is observed in Fig. 6 as for symmetric channels in Fig. 5. However, the performance gain of the proposed protocol with optimal mode selection over the benchmark schemes is higher for asymmetric channels than for symmetric channels. Moreover, as it takes into account the channel statistics, the adaptive link selection policy in [12] significantly outperforms the policy in [11]. In Figs. 7 and 8, we show the achievable average rate versus the number of relay nodes for symmetric channels (r m = r, m) and asymmetric channels (r m = r if m = 2, 3, 5, 7, 8, 9, 11 and r m = 2r if m = 1, 4, 6, 10), respectively, and γ = 10 db. Moreover, we consider the following two scenarios: i) r = 1 and ii) M m=0 r m = 1. For scenario i), by increasing the number of relay nodes, we increase the coverage area. In this case, we observe from Figs. 7 and 8 that τopt-mode buffer and τ buffer conv are almost constant whereas τmax-link buffer, τ cdf-link buffer,andτnon-buffer conv decrease as M increases. For scenario ii), the end-to-end distance between the source and the destination is fixed for all numbers of relay nodes. In this case, we observe from Figs. 7 and 8 that τopt-mode buffer and τconv buffer considerably improve as the number of relay nodes increases due to the improved quality of the links. 4 In contrast, τmax-link buffer, τ cdf-link buffer,andτnon-buffer conv decrease as M increases since 4 We note that for scenario ii) in Figs. 7 and 8, we assume that although M increases, still only the adjacent links are present. However, in practice, this assumption may not be valid for large M and our system model may not be applicable any more. Fig. 8. Achievable average rate in bits/symbol vs. the number of relay nodes M for γ = 10 db and two scenarios for asymmetric channels (r m = r if m = 2, 3, 5, 7, 8, 9, 11 and r m = 2r if m = 1,4,6,10): i) r = 1andii) M m=0 r m = 1. Fig. 9. Achievable average rate in bits/symbol vs. transmit power γ in db for M = 2, r m = 1, m and different average delay requirements T req = 3, 5, 10, 20,. these protocols do not employ spatial reuse. Thereby, the effect of not employing spatial reuse dominates the effect of improving the link quality for τmax-link buffer, τ cdf-link buffer,andτnon-buffer conv. In Fig. 9, we show the achievable average rate of the proposed delay-constrained protocol versus transmit power γ in db for M = 2, r m = 1, m, and different delay requirements T req = 3, 5, 10, 20,. We assume Q thr m = Qthr, m,and choose Q thr for each SNR such that the targeted average delay requirement is satisfied. We first recall that the minimum possible delay for the HD multi-hop relay channel is T min = M which is two blocks for the example considered in Fig. 9. Let us define the SNR gap as the difference between the SNRs required for two protocols to achieve a certain average rate. For the parameters considered in Fig. 9 and τ =2.5 bits/symbol, the SNR gaps of the proposed delay-constrained protocol compared to the upper bound for the delay-unconstrained protocol with T req are limited to 4 db, 2 db, 1 db, and 0.5 db for

13 6252 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 11, NOVEMBER 2015 Fig. 10. Achievable average rate in bits/symbol vs. transmit power γ in db for M = 5, symmetric channels (r =[1, 1, 1, 1, 1, 1]), and asymmetric channels (r =[1, 2, 1, 1, 2, 1]). average delay requirements of 3, 5, 10, and 20 blocks, respectively. Therefore, the proposed sub-optimal delay-constrained protocol is indeed efficient for sufficiently large delay requirements, e.g., more than ten blocks in this example. In Fig. 10, the achievable average rates τ buffer opt-mode, τ buffer dis-mode, and τconv buffer versus transmit power γ in db are plotted for M = 5 and both symmetric channels (r =[1, 1, 1, 1, 1, 1]) and asymmetric channels (r =[1, 2, 1, 1, 2, 1]). We only show results for the best benchmark scheme, i.e., τconv buffer, for clarity of presentation. We observe that the centralized protocol has a SNR gain of up to 2 db over the distributed protocol at the expense collecting the instantaneous CSI of all links. For symmetric channels and τ = 3 bits/symbol, the SNR gaps of the proposed distributed protocol and our modification of the conventional protocol with buffering capability to the optimal protocol are 1.3 db and 1.9 db, respectively. Therefore, for symmetric channels, both the proposed distributed protocol and the conventional buffer-aided protocol are efficient and the gain of the proposed distributed protocol over the conventional bufferaided protocol is relatively small. In contrast, for asymmetric channels and τ = 3 bits/symbol, the SNR gaps of the proposed distributed protocol and the conventional buffer-aided protocol to the optimal protocol are 1.5 db and 6 db, respectively. Hence, we can conclude that the proposed distributed protocol is still efficient for asymmetric channels and outperforms the conventional buffer-aided protocol by a large margin. Recall that only the adjacent links are assumed to be present in the system model adopted in this paper. In order to verify the validity of this assumption, in Fig. 11, we plot the achievable rate of the proposed protocol for path loss exponents κ = [2, 3, 4] for the case when the interference links between all the nodes are present. 5 Assuming the availability of CSI, the feedback interference can be removed at all receiving nodes, cf. Section II-B. We observe from Fig. 11 that, in this case, 5 We note that for r m = 1, m, the path loss exponent κ does not affect the path loss between the adjacent nodes. However, the path loss between the nonadjacent nodes is affected by κ. Fig. 11. Achievable average rate in bits/symbol vs. transmit power γ in db for M = 3 relay nodes, symmetric channels with r =[1, 1, 1, 1],andκ =[2, 3, 4]. the presence of feedforward interference only does not severely deteriorate the performance of the proposed protocol even for small values of κ. The reason for this robustness is that the feedforward interference at a receiving node originates from a transmitting node which is at least three hops way. However, if collecting the required CSI is not possible, the feedback interference cannot be canceled. Hence, in Fig. 11, we also show the performance of the proposed protocol if both feedforward and feedback interference are present and not mitigated. In this case, we observe an SNR loss of approximately 2.5 db since the feedback interference originates in part from the neighboring preceding node. VI. CONCLUSION In this paper, we presented new achievable average rates for the HD multi-hop relay channel with AWGN and block fading for arbitrary numbers of relay nodes. In particular, we designed a buffer-aided relaying protocol which determines the optimal transmission modes and the corresponding optimal node transmission rates in each block based on the quality of the involved links. We also proposed a sub-optimal delayconstrained buffer-aided relaying protocol which can guarantee a bounded delay and a sub-optimal distributed protocol in which each node requires only the local instantaneous CSI of the links to determine the best mode. Our numerical results showed that the maximum average rates achieved with the developed buffer-aided protocols are strictly larger than the average rates achieved with existing protocols for the HD multihop relay network. APPENDIX A The proof is based on induction. From Fig. 2, it is obvious that the number of relevant transmission modes for M = 1, 2 is two. For M > 2, the states of the two relays closest to the destination can only be [t, r], [r, t], and[s, t], otherwise,at least one of the constraints C1 and C2 in (7) is violated. We refer to the aforementioned modes as type-one, type-two, and

14 JAMALI et al.: ACHIEVABLE RATE OF THE HALF-DUPLEX MULTI-HOP BUFFER-AIDED RELAY CHANNEL 6253 with constant coefficients of order three [37]. The characteristic equation for the recursive formula in (35) is given by (10). Using the roots of (10), we obtain f (M) = a 1 x M 1 + a 2x M 2 + a 3x M 3, where a 1, a 2,anda 3 have to satisfy the initial conditions f (1) = 0andf (2) = f (3) = 1 and are given in (11). This completes the proof. APPENDIX B Fig. 12. Illustration of the construction of the possible transmission modes in the HD multi-hop relay channel with M relays. type-three transmission modes. Let f tr (M), f rt (M), andf st (M) denote the number of type-one, type-two, and type-three transmission modes, respectively, for a HD multi-hop relay channel with M relays. Now, we explain how we can construct the modes for the HD multi-hop relay channel with M + 1 relays from that with M relays. Note that K(M) can be rewritten as K(M) = f tr (M) + f rt (M) + f st (M). (34) In the following, we first show how we can recursively obtain f tr (M + 1), f rt (M + 1), andf st (M + 1) as a function of f tr (M), f rt (M), andf st (M). Type-One Modes ( f tr (M + 1)): Type-one modes for a network with M + 1 relays are obtained by adding a relay in receive state between relay M and the destination for type-two and type-three modes in a network with M relays, see Fig. 12. Hence, we obtain f tr (M + 1) = f rt (M) + f st (M). Type-Two Modes ( f rt (M + 1)): Type-two modes for a network with M + 1 relays are obtained by adding a relay in transmit state between relay M and the destination for type-one modes in a network with M relays, see Fig. 12. Thus, we obtain f rt (M + 1) = f tr (M). Type-Three Modes ( f st (M + 1)): Type-three modes for a network with M + 1 relays are obtained by adding a relay in transmit state between relay M and the destination for typetwo modes in a network with M relays and then changing the state of relay M to the silent state, see Fig. 12. Thus, we obtain f st (M + 1) = f rt (M). Defining f (M) f rt (M) and plugging f tr (M), f rt (M), and f st (M) as a function of f (M) into K(M) given in (34) leads to (8). Moreover, f (M) can be recursively written as f (M) = f (M 2) + f (M 3), M 4, (35) where f (1) = 0, f (2) = f (3) = 1. Next, using (35), we obtain a closed-form expression for f (M), M 4 as a function of M. In particular, (35) is a linear homogeneous recurrence relation In this appendix, our aim is to find the optimal mode selection policy as a solution of the relaxed version of the optimization problem given in (16). Since the cost function and the constraints in (16) are affine in the optimization variables q Q and τ upp 0 and the feasible set is nonempty, Slater s condition is satisfied. Hence, the duality gap is zero [30]. Therefore, the solution of the primal problem in (16) can be found from the solution of the dual problem of (16) [30]. Denoting the Lagrange multipliers corresponding to τ upp C m by μ m,the Lagrangian function for the optimization problem in (16) is obtained as L(q,τ upp, μ) = τ upp + M μ m ( C m τ upp ), (36) m=0 where μ =[μ m ], m. The dual problem is then given by minimize μ 0 D(μ) where D(μ) = maximize q Q,τ upp 0 where D(μ) is the dual function as is given by L(q,τ upp, μ). (37) D(μ) = maximizel(q,τ upp, μ). (38) q Q,τ upp 0 To solve (16) using the dual problem in (37), we first obtain the primal variables q and τ upp for a given dual variable μ to find D(μ). The optimal μ is obtained by solving the dual problem in (37). A. Optimal Primal Variables The optimal q k [γ ], γ, k, andτ upp are either the boundary points of the feasible sets, i.e., q Q and τ upp 0, or the stationary points which can be obtained by setting the derivatives of the Lagrangian function in (36) with respect to τ upp and q k [γ ] to zero. In order to be able to calculate the derivative of L with respect to q k [γ ], we implicitly assume that the instantaneous SNRs can take discrete values from set γ G m where Pr{γ } is the probability of SNR γ.note m that the continuous distribution f γm (γ m ) can be modelled by G m, m. The derivatives of the Lagrangian function in (36) are given by L M τ upp = 1 m=0 L q k [γ ] = Pr{γ } μ m m K k [m]=t μ m C(γ m ) Pr{γ } k [γ ]. (39a) (39b)

15 6254 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 11, NOVEMBER 2015 L If the derivative τ upp is non-zero, the optimal value of τ upp is at the boundary of its feasible set, i.e., τ upp or τ upp 0, which cannot be the optimal solution. Therefore, the derivative L τ upp in (39a) has to be zero which leads to M m=0 μ m = 1. L For a given channel state, the derivative q k [γ ] is always positive. Moreover, for ergodic fading with continuous probability density function, the probability that k [γ ]= k [γ ], k = k, holds is zero. Therefore, since k q k[γ ]=1, γ, holds, the optimal mode is the one with the largest selection metric k [γ ]. This leads to the optimal mode selection policy in (17). B. Optimal Dual Variables The optimal value of μ is obtained by solving the dual problem in (37). In particular, substituting the optimal values of q in (17) for a given μ into (36), we obtain the dual function as D(μ) = M μ m C m (a) M = C 0 + ( μ m C m C 0), (40) m=0 m=1 where equality (a) is obtained by substituting μ 0 = 1 Mm=1 μ m since M m=0 μ m = 1 has to hold from (39a). In order to solve the dual problem in (37), we use the well-known sub-gradient method [30]. The sub-gradient method reduces to a gradient-based search if the functions in the optimization problem are differentiable. The main idea is to minimize D(μ) by updating all the components of μ at the same time along the sub-gradient search directions. The updates are performed as follows [ μ m [i + 1] = μ m [i] λ m [i] D(μ) ] + μ m = [ μ m [i] λ m [i] ( C m [i] C 0 [i])] +, (41) where [ ] + is due to the dual feasibility condition as μ m 0 has to hold. This leads to Algorithm 1 with the updates given in (19). The sub-gradient method is guaranteed to converge to optimal dual variable μ provided that the step sizes λ m [i] are chosen sufficiently small [30]. This completes the proof. APPENDIX C The transmission rates of the relays are limited by the amount of information available in their buffers. In this appendix, we show that using the optimized mode selection variables in Theorem 1, there exists a rate selection policy that achieves the upper bound as B. To this end, we first express the bounds on the average transmission rates of the relays independent of the dynamics of the queues as [38] C m, if R m 1 > C m R m R m 1 = C m, if R m 1 = C m (42) R m 1, if R m 1 < C m In particular, if R m 1 > C m holds, i.e., the average rate flowing into buffer B m is larger than the average capacity of the respective departure channel, then, as B,bufferB m always has enough information to supply because the amount of information in the queues increases over time and we obtain bound R m C m. On the other hand, if R m 1 < C m holds, i.e., the average information flowing into buffer B m is less than the average capacity of the respective departure channel, then by the law of conservation of flow, we obtain R m R m 1. Finally, if R m 1 = C m holds, we obtain R m R m 1 = C m. In the following, using the bounds in (42) and depending on the transmitting nodes and whether C m >τupp holds or C m = τ upp, we propose a rate selection policy which achieves the upper bound τ upp. Note that according to (21), C m <τupp cannot hold. Source: If C 0 >τupp holds, if we let the source transmit with the capacity rate, we obtain R 0 = C 0 >τupp.however, since τ upp is the upper bound for the achievable rate, some of the transmitted information bits of the source are trapped in the buffers and will never be forwarded to the respective destination. To avoid this information loss, we let the source transmit with a fraction of the capacity rate ρ 0 = τ upp. With this C 0 transmission rate, we obtain R 0 = τ upp. Similarly, if C 0 = τ upp holds, we obtain ρ 0 = 1, i.e., the source transmits with the capacity rate. Relay 1: If C 1 >τupp holds, we obtain the bound R 1 R 0 = τ upp from (42). Here, we let relay 1 transmit with a fraction of the capacity rate τ upp ρ C With this strategy, the bound R 1 R 0 = τ upp holds with equality. If C 1 = τ upp holds, we obtain R 1 C 1 = R 0 = τ upp from (42). Therefore, we let relay 1 transmit with the capacity rate to achieve R 1 = τ upp. Relay 2 m M: The analysis is similar to that provided for relay 1. Hence, if C m >τupp, relay m transmits with a fraction of the capacity rate τ upp ρ C m m 1andif C m = τ upp, relay m transmits with the capacity rate ρ m = 1. 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San Rafael, CA, USA: Morgan & Claypool, Vahid Jamali (S 12) was born in Fasa, Iran, in He received the B.S. and M.S. degrees (both with first class honors) in electrical engineering from K. N. Toosi University of Technology (KNTU), in 2010 and 2012, respectively. Currently, he is working toward the Ph.D. degree at the Friedrich-Alexander University (FAU), Erlangen, Germany. His research interests include multiuser information theory, wireless communications, free space optical communications, cognitive radio network, LDPC codes, and optimization theory. He was awarded as Exemplary Reviewer of the IEEE COMMUNICATIONS LETTERS in 2014 and received student travel grant for SP Coding and Information School, Brazil, in Nikola Zlatanov (S 06) was born in Macedonia. He received the Dipl.Ing. and M.S. degrees in electrical engineering from SS. Cyril and Methodius University, Skopje, Macedonia, in 2007 and 2010, respectively. Currently, he is working toward the Ph.D. degree at the University of British Columbia (UBC), Vancouver, Canada. His current research interests include wireless communications and information theory. Mr. Zlatanov received several awards for his work including UBC s Four-Year Doctoral Fellowship in 2010, UBC s Killam Doctoral Scholarship and Macedonia s Young Scientist of the Year Award in 2011, Vanier Canada Graduate Scholarship in 2012, DAAD Research Grant in 2013, and best paper award from the German Information Technology Society (ITG) in Hebatallah Shoukry was born in Alexandria, Egypt, in She received the B.Sc. degree (with Honors) in electrical engineering from Alexandria University, Egypt, in 2006 and two M.Sc. degrees in electrical engineering from University of Central Florida, Orlando, FL, USA, and Friedrich-Alexander University, Erlangen, Germany, in 2010 and 2013, respectively. She worked as a Research Assistant at Friedrich-Alexander University, Erlangen, Germany, from 2013 to 2014.

17 6256 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 14, NO. 11, NOVEMBER 2015 Robert Schober (S 98 M 01 SM 08 F 10) was born in Neuendettelsau, Germany, in He received the Diploma (Univ.) and the Ph.D. degrees in electrical engineering from the University of Erlangen-Nuermberg in 1997 and 2000, respectively. From May 2001 to April 2002, he was a Postdoctoral Fellow at the University of Toronto, Canada, sponsored by the German Academic Exchange Service (DAAD). Since May 2002, he has been with the University of British Columbia (UBC), Vancouver, Canada, where he is now a Full Professor. Since January 2012, he is an Alexander von Humboldt Professor and the Chair for Digital Communication at the Friedrich Alexander University (FAU), Erlangen, Germany. His research interests fall into the broad areas of Communication Theory, Wireless Communications, and Statistical Signal Processing. Dr. Schober received several awards for his work including the 2002 Heinz MaierLeibnitz Award of the German Science Foundation (DFG), 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, a 2011 Alexander von Humboldt Professorship, and a 2012 NSERC E.W.R. Steacie Fellowship. In addition, he received best paper awards from the German Information Technology Society (ITG), the European Association for Signal, Speech and Image Processing (EURASIP), IEEE WCNC 2012, IEEE Globecom 2011, IEEE ICUWB 2006, the International Zurich Seminar on Broadband Communications, and European Wireless He is a Fellow of the Canadian Academy of Engineering and a Fellow of the Engineering Institute of Canada. He is currently the Editor-in-Chief of the IEEE TRANSACTIONS ON COMMUNICATIONS.