Achievable Rates for the Fading Half-Duplex Single Relay Selection Network Using Buffer-Aided Relaying
aa r X i v : . [ c s . I T ] A p r Achievable Rates for the Fading Half-Duplex SingleRelay Selection Network Using Buffer-AidedRelaying
Nikola Zlatanov,
Student Member, IEEE,
Vahid Jamali,
Student Member, IEEE, and Robert Schober,
Fellow, IEEE
Abstract —In the half-duplex single relay selection network,comprised of a source, M half-duplex relays, and a destination,only one relay is active at any given time, i.e., only one relayreceives or transmits, and the other relays are inactive, i.e., theydo not receive nor transmit. The capacity of this network, whenall links are affected by independent slow time-continuous fadingand additive white Gaussian noise (AWGN), is still unknown, andonly achievable average rates have been reported in the literatureso far. In this paper, we present new achievable average rates forthis network which are larger than the best known average rates.These new average rates are achieved with a buffer-aided relayingprotocol. Since the developed buffer-aided protocol introducesunbounded delay, we also devise a buffer-aided protocol whichlimits the delay at the expense of a decrease in rate. Moreover,we discuss the practical implementation of the proposed buffer-aided relaying protocols and show that they do not requiremore resources for channel state information acquisition thanthe existing relay selection protocols. Index Terms —Buffer-aided relaying, half-duplex, relay selec-tion, achievable rate.
I. I
NTRODUCTION C OOPERATIVE communication has recently gainedmuch attention due to its ability to increase the through-put and/or reliability of wireless networks. The basic ideabehind cooperative communication is that each node canact as a relay and help the other nodes of the network toforward their information to their respective destination nodes.Because of the high complexity inherent to the investigation ofgeneral cooperative networks, and to get insight into the basicchallenges and benefits of cooperative communication, re-searchers have mainly considered relatively simple cooperativenetworks. Although simple, these basic cooperative networksreveal the gains that can be accomplished by cooperationamong network nodes. Moreover, because of their simplicity,these basic cooperative networks can be easily integrated intothe current communication infrastructure. One basic networkwhich has shown great potential in terms of utility andperformance is the half-duplex (HD) single relay selection
Manuscript received August 26, 2014; revised January 18, 2015; acceptedApril 1, 2015. This paper has been presented in part at IEEE Globecom 2014,Austin, TX, December 2014.N. Zlatanov is with the Department of Electrical and Computer Engineering,University of British Columbia, Vancouver, BC, V6T 1Z4, Canada, E-mail:[email protected]. Jamali and R. Schober are with the Institute for Digital Communi-cation, Friedrich-Alexander University, Erlangen 91054, Germany (e-mail:[email protected]; [email protected]). network proposed in [1]. In this network, only one relay isactive at any given time, i.e., one relay receives or transmits,and the other relays are inactive, i.e., they do not receive nortransmit. Because of the large achievable performance gains,this network has recently attracted considerable interest, see[1]-[11] and references therein. Although well investigated,the capacity of this network is still unknown when all linksare affected by independent slow time-continuous fading andadditive white Gaussian noise (AWGN). So far, only achiev-able average rates have been reported in the literature, see[10], [11]. In fact, to the best of the authors’ knowledge,the achievable average rates in [10] and [11] are the largestaverage rates reported in the literature for this network. Theserates are based on the relay selection protocol in [1], where, ineach time slot, the relay with the strongest minimum source-to-relay and relay-to-destination channel is selected to forwardthe information from the source to the destination. In thispaper, we will show that these rates can be surpassed. Inparticular, we develop a buffer-aided relaying protocol whichachieves average rates which are significantly larger than therates reported in [10] and [11]. Since the proposed buffer-aided protocol introduces unbounded delay, we also devise asecond buffer-aided protocol which limits the average delay atthe expense of a decrease in rate. Moreover, we show that theproposed buffer-aided relaying protocols do not require moreresources for channel state information (CSI) acquisition thanthe existing relay selection protocols.Buffer-aided HD relaying with adaptive switching betweenreception and transmission was proposed in [12] for a sim-ple three-node relay network without source-destination link.Later, buffer-aided relaying was further analyzed in [13] and[14] for adaptive and fixed rate transmission, respectively.Buffer-aided relaying protocols were also proposed for two-way relaying in [15], [16], the multihop relay network in[17], two source and two destination pairs sharing a singlerelay in [18], secure communication for two-hop relaying andrelay selection in [19] and [20], respectively, and amplify-and-forward relaying in [21]. For the considered relay selectionnetwork, relaying with buffers was investigated in [8] and[9]. However, the protocols in [8] and [9] are limited to thecase when all nodes transmit with fixed rates and all source-to-relay and relay-to-destination links undergo independentand identically distributed (i.i.d.) fading. These protocols were The “average rate” is also referred to as “expected rate” in the literature. developed for improving the outage probability performanceof the network. In order to use the protocols in [8] and [9] asperformance benchmarks, we modify them such that all nodestransmit with rates equal to their underlying channel capacities.However, the modified protocols are still only applicable tothe case when all links are affected by i.i.d. fading and willcause data loss due to buffer overflow for independent non-identically distributed (i.n.d.) fading. We note however thatthis drawback is not caused by our modifications since thephenomenon of buffer overflow also occurs for the originalprotocols in [8] and [9] for fixed rate transmission when thelinks of the network are i.n.d.This paper is organized as follows. In Section II, weintroduce the system model. In Section III, we present theproposed buffer-aided protocol for transmission without delayconstraints. In Section IV, we discuss the implementation ofthe proposed protocol. In Section V, we propose a protocolfor delay-limited transmission. In Section VI, we providenumerical examples comparing the achievable rates of theproposed protocols and the benchmark protocols. Finally,Section VII concludes the paper.II. S
YSTEM M ODEL
In the following, we introduce the system model of theconsidered relay network. Furthermore, as benchmark scheme,we briefly review the conventional non-buffer-aided relayselection protocol in [1].
A. System Model
The HD relay selection network consists of a source S , M HD decode-and-forward relays R k , k = 1 , ..., M , and adestination D , as shown in Fig. 1. The source transmits itsinformation to the destination only through the relays, i.e.,because of high attenuation there is no direct link between thesource and the destination, and therefore, all the informationthat the destination receives is first processed by the relays.We assume that the transmission is performed in N timeslots, where N → ∞ . The relays in the network are HDnodes, i.e., they cannot transmit and receive at the same time.Furthermore, in each time slot, only one relay is active, i.e.,it receives or transmits, and the other relays are inactive, i.e.,they do not receive nor transmit. Each relay is equipped witha buffer of unlimited size in which it stores the informationthat it receives from the source and from which it extractsthe information that it transmits to the destination. We assumethat all nodes transmit their codewords with constant power P and that the noise at all receivers is independent AWGN withvariance σ n . We assume transmission with capacity achiev-ing codes. Hence, the transmitted codewords are Gaussiandistributed, comprised of n → ∞ symbols, and span onetime slot. Moreover, we assume that each source-to-relay andrelay-to-destination channel is affected by independent slowtime-continuous fading such that the fading remains constantduring a single time slot and changes from one time slotto the next. We assume that the fading is an ergodic andstationary random process. Let | h Sk ( i ) | and | h kD ( i ) | denotethe squared amplitudes of the complex channel gains of thePSfrag replacements S DR R R M γ S ( i ) γ S ( i ) γ SM ( i ) γ D ( i ) γ D ( i ) γ MD ( i ) Fig. 1. System model for buffer-aided relay selection. source-to- k -th-relay and k -th-relay-to-destination channels inthe i -th time slot, respectively, and let Ω Sk = E {| h Sk ( i ) | } and Ω kD = E {| h kD ( i ) | } denote their mean values, respec-tively, where E {·} denotes expectation. Then, the signal-to-noise ratios (SNRs) of the source-to- k -th-relay and k -th-relay-to-destination channels are given by γ Sk ( i ) = Pσ n | h Sk ( i ) | and γ kD ( i ) = Pσ n | h kD ( i ) | , (1)respectively. Furthermore, we denote the average SNRs of thesource-to- k -th-relay and k -th-relay-to-destination channels by ¯ γ Sk = E { γ Sk ( i ) } and ¯ γ kD = E { γ kD ( i ) } , respectively. Using(1), the capacities of the source-to- k -th-relay and k -th-relay-to-destination channels in the i -th time slot, denoted by C Sk ( i ) and C kD ( i ) , respectively, are given by C Sk ( i ) = log (cid:0) γ Sk ( i ) (cid:1) (2) C kD ( i ) = log (cid:0) γ kD ( i ) (cid:1) . (3) B. Conventional Relay Selection Protocol
For comparison purpose, we briefly review the conventionalnon-buffer-aided relay selection protocol [1] and its corre-sponding achievable average rate [10], [11].The conventional relay selection protocol selects the relay k with the maximum min { C Sk ( i ) , C kD ( i ) } for forwarding theinformation from the source to the destination in the i -th timeslot [1]. The channel coding scheme adopted for conventionalrelaying is as follows. In the first half of time slot i , the sourcesends a codeword with rate min { C Sk ( i ) , C kD ( i ) } to the k -th relay. The k -th relay can successfully decode the receivedcodeword since the rate of the codeword is smaller than orequal to C Sk ( i ) . Then, in the second half of time slot i , therelay re-encodes the decoded information and sends it to thedestination with rate min { C Sk ( i ) , C kD ( i ) } . The destinationcan successfully decode the received codeword since the rateof the codeword is smaller than or equal to C kD ( i ) . Hence, theoverall rate transmitted from source to destination during timeslot i is min { C Sk ( i ) , C kD ( i ) } . Thereby, during N → ∞ time slots, the average rate achieved with conventional relay-ing, denoted by ¯ R conv , is obtained as [10], [11] ¯ R conv = 12 E (cid:8) max k min { C Sk ( i ) , C kD ( i ) } (cid:9) . (4)In the following, we present the proposed buffer-aided pro-tocols for the considered relay selection network and thecorresponding achievable rates.III. B UFFER -A IDED R ELAYING P ROTOCOL WITHOUT D ELAY C ONSTRAINT
In this section, we develop a buffer-aided relaying protocolwithout delay constraints which maximizes the achievableaverage rate for the considered network. To this end, we firstintroduce the instantaneous transmission rates at the nodes ineach time slot, and then derive the corresponding achievableaverage rate. Next, we maximize the achievable average rateand derive analytical expressions for the maximum averagerate.
A. Instantaneous Transmission Rates
In the considered HD single relay selection network, ina given time slot, only one relay is selected to receive ortransmit, i.e., to be active. Without loss of generality, assumethat the k -th relay has been selected to be active in the i -th time slot . Then, if the active relay is selected to receive,the source maps nR Sk ( i ) bits of information to a Gaussiandistributed codeword comprised of n → ∞ symbols, whereeach symbol is generated independently according to a zero-mean complex circular-symmetric Gaussian distribution withvariance P , and transmits this codeword to the selected relay.The rate of this codeword R Sk ( i ) is set as R Sk ( i ) = C Sk ( i ) , (5)where C Sk ( i ) is the capacity of the source-to- k -th-relay chan-nel given in (2). As a result of (5), the active relay can suc-cessfully decode this codeword and stores the correspondinginformation in its buffer. Let Q k ( i ) denote the number ofbits/symbol in the buffer of the k -th relay at the end of timeslot i . Then, with this transmission, Q k ( i ) increases as Q k ( i ) = Q k ( i −
1) + C Sk ( i ) . (6)On the other hand, if the active relay is selected to transmit,it extracts nR kD ( i ) bits of information from its buffer, mapsit to a Gaussian distributed codeword comprised of n → ∞ symbols, where each symbol is generated independently ac-cording to a zero-mean complex circular-symmetric Gaussiandistribution with variance P , and transmits it to the destination.The rate of this codeword is R kD ( i ) , which is set as R kD ( i ) = min { Q k ( i − , C kD ( i ) } , (7)where C kD ( i ) is the capacity of the k -th-relay-to-destinationchannel given in (2). The minimum in the expression for rate R kD ( i ) is a consequence of the fact that the relay cannottransmit more information than what it has stored in its buffer, How exactly the active relay is selected is explained in Theorem 1. i.e., more than Q k ( i − . The destination can successfullydecode this codeword since R kD ( i ) ≤ C kD ( i ) holds, andstores the corresponding information. When the active relaytransmits, Q k ( i ) decreases as Q k ( i ) = Q k ( i − − R kD ( i ) . (8)In the following, we obtain the average rates of buffer-aidedsingle-relay selection. B. Average Transmission and Reception Rates
In order to derive the average rates of buffer-aided single-relay selection, we first have to model the reception andtransmission of the k -th relay. To this end, we introduce twobinary indicator variables r R k ( i ) ∈ { , } and r T k ( i ) ∈ { , } ,which indicate whether, in the i -th time slot, the k -th relayreceives or transmits, respectively. More precisely, r R k ( i ) and r T k ( i ) are defined as r R k ( i ) , (cid:26) if the k -th relay receives if the k -th relay does not receive , (9) r T k ( i ) , (cid:26) if the k -th relay transmits if the k -th relay does not transmit . (10)Since exactly one relay is active in each time slot, r R k ( i ) and r T k ( i ) must satisfy M X k =1 [ r R k ( i ) + r T k ( i )] = 1 , ∀ i. (11)Using r R k ( i ) and r T k ( i ) , the average rates received at andtransmitted by the k -th relay, denoted by ¯ R Sk and ¯ R kD ,respectively, can be expressed as ¯ R Sk = lim N →∞ N N X i =1 r R k ( i ) R Sk ( i )= lim N →∞ N N X i =1 r R k ( i ) C Sk ( i ) , (12) ¯ R kD = lim N →∞ N N X i =1 r T k ( i ) R kD ( i )= lim N →∞ N N X i =1 r T k ( i ) min { Q k ( i − , C kD ( i ) } . (13)Using ¯ R kD , ∀ k , the average rate received at the destination,denoted by ¯ R SD , can be expressed as ¯ R SD = M X k =1 ¯ R kD = lim N →∞ N N X i =1 M X k =1 r T k ( i ) min { Q k ( i − , C kD ( i ) } . (14)In the following, our goal is to maximize ¯ R SD . C. Maximization of the Average Rate
In (12) and (13), the only variables with a degree of freedomare r R k ( i ) and r T k ( i ) , ∀ i, k . Any choice of these variables willprovide an average rate. However, in order for an average rateto be achievable, i.e., for data loss not to occur, the buffersat all relays must remain stable . Moreover, among all theachievable average rates, there exists one rate which is thelargest. In order to obtain the largest achievable average rate,we have to find the optimal values of r R k ( i ) and r T k ( i ) , ∀ i, k ,which maximize the average rate in (14) when constraint (11)holds and when the buffers at all relays are stable. To this end,we introduce the following useful lemma. Lemma 1:
The achievable average rate is maximized when r R k ( i ) and r T k ( i ) , ∀ i , are chosen such that the followingcondition is satisfied for all k = 1 , ..., M lim N →∞ N N X i =1 r R k ( i ) C Sk ( i ) = lim N →∞ N N X i =1 r T k ( i ) C kD ( i ) . (15)Moreover, when (15) holds for the k -th relay, (13) simplifiesto ¯ R kD = lim N →∞ N N X i =1 r T k ( i ) C kD ( i ) , (16)and when (15) holds ∀ k relays, (14) simplifies to ¯ R SD = lim N →∞ N N X i =1 M X k =1 r T k ( i ) C kD ( i ) . (17) Proof:
Please refer to Appendix A.With Lemma 1, we have reduced the search space for themaximum achievable average rate to only those rates for which(15) holds ∀ k . Moreover, we have obtained an expression for ¯ R SD which is independent of Q k ( i ) , ∀ i, k . Now, in orderto find the maximum achievable average rate, we devise amaximization problem for the average rate, ¯ R SD , under theconstraints given in (15) and (11). This maximization problem,for N → ∞ , is given by Maximize : r R k ( i ) ,r T k ( i ) , ∀ i,k N P Ni =1 P Mk =1 r T k ( i ) C kD ( i )Subject to : C1 : N P Ni =1 r R k ( i ) C Sk ( i )= N P Ni =1 r T k ( i ) C kD ( i ) , ∀ k C2 : r R k ( i ) ∈ { , } , ∀ k, i C3 : r T k ( i ) ∈ { , } , ∀ k, i C4 : P Mk =1 [ r R k ( i ) + r T k ( i )] = 1 , ∀ i. (18)In (18), the restrictions in (15) and (11) are reflected inconstraints C1 and C4, respectively. Fortunately, (18) can besolved analytically. The solution reveals how the values of r R k ( i ) and r T k ( i ) are to be chosen optimally in each time slot i such that the maximum average rate of the buffer-aidedprotocol is achieved. Before providing the solution to (18), wefirst introduce some notations. Let µ k , k = 1 , ..., M , denoteconstants which are independent of the time slot i and the By a stable buffer we mean that there is no information loss in the bufferand the information that enters the buffer eventually leaves the buffer, i.e., noinformation is trapped inside the buffer. instantaneous CSI. The values of these constants depend onthe fading statistics and will be determined later, cf. Lemma 2.Then, for a given time slot i , we multiply each C Sk ( i ) with µ k and each C kD ( i ) with (1 − µ k ) , and collect these productsin set A ( i ) . Hence, A ( i ) is given by A ( i ) = (cid:8) µ C S ( i ) , µ C S ( i ) , ..., µ M C SM ( i ) , (1 − µ ) C D ( i ) , (1 − µ ) C D ( i ) , ..., (1 − µ M ) C MD ( i ) (cid:9) . (19)We are now ready to present the solution to (18) in thefollowing theorem, which represents the proposed protocol fortransmission without delay constraints. Theorem 1:
The optimal values of r T k ( i ) and r R k ( i ) , ∀ k, i which maximize the achievable average rate of the proposedprotocol are given by r T k ( i ) = 1 if (1 − µ k ) C kD ( i ) = max A ( i ) r R k ( i ) = 1 if µ k C Sk ( i ) = max A ( i ) r T k ( i ) = r R k ( i ) = 0 otherwise , (20)where the µ k , ∀ k , are chosen such that constraint C1 in (18)is satisfied ∀ k . The maximum achievable average rate of theproposed protocol is given by (17) when r R k ( i ) and r T k ( i ) areset as in (20), ∀ i, k . Proof:
Please see Appendix B.
Remark 1:
Theorem 1 reveals that the optimal values of r R k ( i ) and r T k ( i ) depend only on the instantaneous CSI of the i -th time slot, and are independent of the instantaneous CSIsof past and future time slots. D. Analytical Characterization of the Maximum AchievableRate
By inserting (20) into (14), we obtain the maximum achiev-able rate of the proposed protocol as an average over N → ∞ time slots, which may not be convenient from an analyticalpoint of view. Furthermore, Theorem 1 does not provide anexpression for obtaining constants µ k , ∀ k . In order to obtainuseful analytical expressions for the maximum achievableaverage rate and constants µ k , ∀ k , we exploit the assumedergodicity and stationarity of the fading, and write (15) (i.e.,constraint C1 in (18)) and (17) equivalently as E { log (1 + Γ S ( i )) } = E { log (1 + Γ D ( i )) } E { log (1 + Γ S ( i )) } = E { log (1 + Γ D ( i )) } ... E { log (1 + Γ SM ( i )) } = E { log (1 + Γ MD ( i )) } (21)and ¯ R SD = M X k =1 E { log (1 + Γ kD ( i )) } , (22)respectively, where Γ Sk ( i ) = r R k ( i ) γ Sk ( i ) and Γ kD ( i ) = r T k ( i ) γ kD ( i ) , with r R k ( i ) and r T k ( i ) as in (20). In the followingtwo lemmas, we provide simplified expressions for the max-imum average rate ¯ R SD and constants µ k , ∀ k . Thereby, wedrop index i since, due to the stationarity and ergodicity of the fading, the statistics of Γ Sk ( i ) and Γ kD ( i ) are independent of i . Lemma 2:
The optimal values of µ k , k = 1 , ..., M , denotedby µ ∗ k , which maximize ¯ R SD , can be obtained by solving thefollowing system of M equations R ∞ log (1 + x ) f Γ S ( x ) dx = R ∞ log (1 + x ) f Γ D ( x ) dx R ∞ log (1 + x ) f Γ S ( x ) dx = R ∞ log (1 + x ) f Γ D ( x ) dx ... R ∞ log (1 + x ) f Γ SM ( x ) dx = R ∞ log (1 + x ) f Γ MD ( x ) dx, (23)where, for x > , f Γ Sk ( x ) = f γ Sk ( x ) F γ kD (cid:16) (1 + x ) µk − µk − (cid:17) × M Y j =1 j = k F γ Sj (cid:16) (1 + x ) µkµj − (cid:17) F γ jD (cid:16) (1 + x ) µk − µj − (cid:17) , (24) f Γ kD ( x ) = f γ kD ( x ) F γ Sk (cid:18) (1 + x ) − µkµk − (cid:19) × M Y j =1 j = k F γ Sj (cid:18) (1 + x ) − µkµj − (cid:19) F γ jD (cid:18) (1 + x ) − µk − µj − (cid:19) . (25)Here, f γ α ( x ) and F γ α ( x ) denote the probability density func-tion (PDF) and cumulative distribution function (CDF) of γ α , α ∈ { Sk, kD } , respectively. Furthermore, if the fading onall source-to-relay and relay-to-destination links is i.i.d., thesolution to (23) is µ ∗ k = 1 / , ∀ k . Proof:
Please refer to Appendix C.
Remark 2:
For i.i.d. links, since µ ∗ k = 1 / , ∀ k , the pro-posed protocol, given by (20), always selects the link with thelargest instantaneous channel gain among all M availablelinks for transmission. Hence, for i.i.d. links this protocolbecomes identical to the protocol proposed in [8]. However,for i.n.d. links, the protocol in [8] will cause data loss due tobuffer overflow. In particular, applying the protocols in [8] and[9], the buffers at relays with Ω Sk > Ω kD suffer from overflowand receive more information than they can transmit. Hence, afraction of the source’s data is trapped inside the relay buffersand does not reach the destination, i.e., data loss occurs. On theother hand, our proposed protocol is applicable for all fadingstatistics. Lemma 3:
The maximum achievable average rate of theprotocol in Theorem 1 is given by ¯ R SD = M X k =1 Z ∞ log (1 + x ) f ∗ Γ kD ( x ) dx, (26)where f ∗ Γ kD ( x ) is obtained by inserting µ k = µ ∗ k found usingLemma 2 into f Γ kD ( x ) given by (25). For i.i.d. fading onall links, i.e., when f γ Sk ( x ) = f γ kD ( x ) = f γ ( x ) , ∀ k , and A system of nonlinear equations can be solved e.g. by algorithms basedon Newton’s method [22]. F γ Sk ( x ) = F γ kD ( x ) = F γ ( x ) , ∀ k , (26) simplifies to ¯ R SD = M Z ∞ log (1 + x ) f γ ( x ) ( F γ ( x )) M − dx. (27) Proof:
Let us insert the optimal µ ∗ k , ∀ k , found fromLemma 2, into f Γ kD ( x ) given in (25) and denote it by f ∗ Γ kD ( x ) . Eq. (26) is obtained by inserting f ∗ Γ kD ( x ) into (22),whereas (27) is obtained by inserting µ ∗ = 1 / into (26) andsimplifying the resulting expression.To get more insight, in the following we investigate the caseof i.i.d. Rayleigh fading. E. Special Case: I.i.d. Rayleigh Fading
In the following, we simplify the expression for the maxi-mum average rate in (27) for i.i.d. Rayleigh fading.The expression f γ max ( x ) = 2 M f γ ( x ) ( F γ ( x )) M − in(27) can be interpreted as the distribution of the largestrandom variable (RV) among M i.i.d. RVs with distributions f γ Sk ( x ) = f γ kD ( x ) = f γ ( x ) , ∀ k , see [23]. For i.i.d. Rayleighfading, i.e., when f γ Sk ( x ) = f γ kD ( x ) = e − x/ ¯ γ / ¯ γ , ∀ k , where ¯ γ is the average SNR of all source-to-relay and relay-to-destination links, f γ max ( x ) is given as [23] f γ max ( x ) = 2 M M − X k =0 ( − k (cid:18) M − k (cid:19) γ exp (cid:18) − x ¯ γ ( k + 1) (cid:19) . (28)Inserting (28) into (27) and integrating, we obtain the averagerate as ¯ R SD = M M − X k =0 (cid:18) M − k (cid:19) ( − k (1 + k ) ln(2) exp (cid:18) k ¯ γ (cid:19) × E (cid:18) k ¯ γ (cid:19) , (29)where E ( · ) is the first order exponential integral functiondefined as E ( x ) = R ∞ e − xt / ( t ) dt . On the other hand, forthe same case, i.e., for i.i.d. Rayleigh fading on all links, theachievable rate for conventional relay selection given in (4)can be written equivalently as [24, Eq. (26)] ¯ R conv = M M − X k =0 (cid:18) M − k (cid:19) ( − k (1 + k ) ln(2) exp (cid:18) k )¯ γ (cid:19) × E (cid:18) k )¯ γ (cid:19) . (30)In order to gain further insight, expressions (29) and (30)can be further simplified for low and high SNRs using thefollowing first order Taylor approximations exp( c/ ¯ γ ) E ( c/ ¯ γ ) = c ¯ γ , as ¯ γ → , (31) exp( c/ ¯ γ ) E ( c/ ¯ γ ) = − K EM − ln( c ) + ln(¯ γ ) , as ¯ γ → ∞ , (32)where K EM is the Euler-Mascheroni constant and its value is K EM ≈ . .
1) Low SNR:
Using (31), the rates in (29) and (30) can beapproximated as ¯ R SD → ¯ γ M X k =1 k , as ¯ γ → (33) ¯ R conv → ¯ γ M X k =1 k , as ¯ γ → . (34)Dividing (33) by (34), we obtain the following ratio ¯ R SD ¯ R conv = 2 P Mk =1 1 k P Mk =1 1 k . (35)For M = 1 and M → ∞ , the ratio in (35) is equal to and , respectively, which constitute the upper and lower boundsof (35) for ≤ M ≤ ∞ . Hence, for low SNRs, the averagerate of the proposed buffer-aided relay selection protocol is to times higher than the rate of conventional relay selection.
2) High SNR:
On the other hand, using (32), the rates in(29) and (30) can be approximated as ¯ R SD → ¯ γ − K EM − M M − X k =0 (cid:18) M − k (cid:19) ( − k log (1 + k )(1 + k ) , as ¯ γ → ∞ , (36) ¯ R conv → ¯ γ − K EM − M M − X k =0 (cid:18) M − k (cid:19) ( − k log (1 + k )(1 + k ) − , as ¯ γ → ∞ . (37)Subtracting (37) from (36), we obtain ¯ R SD − ¯ R conv = 12 + M M − X k =0 (cid:18) M − k (cid:19) ( − k log (1 + k )(1 + k ) − M M − X k =0 (cid:18) M − k (cid:19) ( − k log (1 + k )(1 + k ) . (38)For M = 1 and M → ∞ , the expression in (38) evaluates to and / , respectively, which constitute the upper and lowerbounds of (38) for ≤ M ≤ ∞ . Hence, for high SNRs,the average rate of the proposed buffer-aided relay selectionprotocol is between and / bits/symb larger than the rateof conventional relay selection.In the following, we discuss the implementation of theproposed buffer-aided HD relay selection protocol.IV. I MPLEMENTATION OF THE P ROPOSED B UFFER -A IDED P ROTOCOL
In this section, we discuss the implementation of the pro-tocol proposed in Theorem 1. The proposed protocol canbe implemented in a centralized or in a distributed manner.A centralized implementation assumes a central node whichselects the active relay in each time slot and decides whether it should receive or transmit. On the other hand, in the dis-tributed implementation, there is no central node and the relaysthemselves negotiate which relay should be active in each timeslot. In the following, we discuss both implementations.
A. Centralized Implementation
For the centralized implementation, we assume that thedestination is the central node. Hence, in each time slot, thedestination has to obtain the CSI of all links. To this end,at the beginning of each time slot, the source transmits pilotsymbols from which all relays acquire their respective source-to-relay CSIs. Then, each relay broadcasts orthogonal pilots,from which the source and destination learn all source-to-relayand relay-to-destination CSIs, respectively. Next, each relayfeedsback the CSI of its respective source-to-relay channelto the destination. With the acquired CSI, the destinationcomputes C Sk ( i ) and C kD ( i ) , ∀ k . In order to select the activerelay according to the protocol in Theorem 1, the destinationhas to construct set A ( i ) , given by (19). This requires thecomputation of the constants µ k , ∀ k . These constants canbe computed using Lemma 2, but this requires knowledge ofthe PDFs of the fading gains of all links before the start oftransmission. Such a priori knowledge may not be available inpractice. In this case, the destination has to estimate µ k , ∀ k , inreal-time using only the CSI knowledge until time slot i . Since µ k , ∀ k , are actually Lagrange multipliers obtained by solvingthe linear optimization problem in (51), an accurate estimateof µ k , ∀ k , can be obtained using the gradient descent method[25]. In particular, using C Sk ( i ) and C kD ( i ) , the destinationrecursively computes an estimate of µ k , denoted by µ ek ( i ) , as µ ek ( i ) = µ ek ( i −
1) + δ k ( i )( ¯ R ekD ( i − − ¯ R eSk ( i − , (39)where ¯ R eSk ( i − and ¯ R ekD ( i − are real-time estimates of ¯ R Sk and ¯ R kD , respectively, computed for i ≥ as ¯ R eSk ( i −
1) = i − i − R eSk ( i −
2) + r R k ( i − i − C Sk ( i − , (40) ¯ R ekD ( i −
1) = i − i − R ekD ( i −
2) + r T k ( i − i − C kD ( i − , (41)where R eSk (0) and R ekD (0) are set to zero ∀ k . In (39), δ k ( i ) isan adaptive step size which controls the speed of convergenceof µ ek ( i ) to µ k . In particular, the step size δ k ( i ) is someproperly chosen monotonically decaying function of i with δ k (1) < , see [25] for more details.Once the destination has C Sk ( i ) , C kD ( i ) , and µ ek ( i ) , ∀ k , itconstructs the set A ( i ) , and selects the active relay accordingto Theorem 1. The destination also has to keep track of thequeue length in the buffers at each relay in each time slot.To this end, using C Sk ( i ) , C kD ( i ) , r R k ( i ) , and r T k ( i ) , ∀ k , thedestination computes the queue length in the buffers at each This feedback can also be done using pilots. In particular, since thedestination already knows the channel between each relay and itself, eachrelay can broadcast pilots whose amplitude is equal to the channel gain ofthe channel from the source to the selected relay. relay using the following formula Q k ( i ) = Q k ( i −
1) + r R k ( i ) C Sk ( i ) − r T k ( i ) min { Q k ( i − , C kD ( i ) } . (42)Then, the destination broadcasts a control message to therelays which contains information regarding which relay isselected and whether it will receive or transmit. If the selectedrelay is scheduled to transmit, it extracts information bitsfrom its buffer, maps them to a codeword, and transmits thecodeword to the destination with rate R kD ( i ) = min { Q k ( i − , C kD ( i ) } . Otherwise, if the selected relay is scheduled toreceive, it sends a control message to the source which informsthe source which relay is selected. Then, the source transmitsthe information codeword intended for the selected relay withrate R Sk ( i ) = C Sk ( i ) .The destination may receive the information bits in an orderwhich is different from that in which they were transmitted bythe source. However, using the acquired CSI, the destinationcan keep track of the amount of information received andtransmitted by each relay in each time slot. This information issufficient for the destination to perform successful reorderingof the received information bits. B. Distributed Implementation
We now outline the distributed implementation of the pro-posed protocol using timers, similar to the scheme in [1].At the beginning of time slot i , source and destinationtransmit pilots in successive pilot time slots. This enables therelays to acquire the CSI of their respective source-to-relay andrelay-to-destination channels, respectively. Using the acquiredCSI, the k -th relay computes C Sk ( i ) and C kD ( i ) . Next, using C Sk ( i ) and C kD ( i ) , the k -th relay computes the estimate of µ k , µ ek ( i ) , using (39), (40), and (41). Using C Sk ( i ) , C kD ( i ) ,and µ ek ( i ) , the k -th relay turns on a timer proportional to / max { µ ek ( i ) C Sk ( i ) , (1 − µ ek ( i )) C kD ( i ) } . This procedure isperformed by all M relays. If max { µ ek ( i ) C Sk ( i ) , (1 − µ ek ( i )) C kD ( i ) } = µ ek ( i ) C Sk ( i ) and max { µ ek ( i ) C Sk ( i ) , (1 − µ ek ( i )) C kD ( i ) } = (1 − µ ek ( i )) C kD ( i ) , the k -th relay knows that if it is selected, then it will receiveand transmit, respectively. The relay whose timer expires first,broadcasts a packet containing pilot symbols and a controlmessage with information about which relay is selected andwhether the selected relay receives or transmits. From thepacket broadcasted by the selected relay, both source anddestination learn the channels from the selected relay to thesource and destination, respectively. They also learn whichrelay is selected and whether it is scheduled to receive ortransmit. If the selected relay is scheduled to transmit, thenit extracts bits from its buffer, maps them to a codeword andtransmits the codeword to the destination with rate R kD ( i ) =min { Q k ( i − , C kD ( i ) } . Otherwise, if the relay is scheduledto receive, then the source transmits to the selected relay acodeword with rate C Sk ( i ) . Again, the destination may receive the information bits inan order which is different from that in which they weretransmitted by the source. Therefore, in order for the desti-nation to reorder the received information bits, it should keeptrack of the amount of information received and transmittedby each relay in each time slot. If the selected relay transmits,by successful decoding the destination learns the amount ofinformation received. However, when the selected relay isscheduled to receive, the relay should feedback the amountof information that it received to the destination. Using thisinformation, the destination can perform successful reorderingof the received information bits. Remark 3:
We note that distributed relay selection proto-cols based on timers may suffer from long waiting times beforethe first timer expires. Moreover, collisions are possible whentwo or more relay nodes declare that they are the selectednode at approximately the same time. However, by choosingthe timers suitably, as proposed in [26], these negative effectscan be minimized.
C. Comparison of the Overhead of the Conventional and theProposed Protocols
The conventional relay selection protocol reviewed in Sec-tion II-B can also be implemented in a centralized or adistributed manner. In the following, we discuss the overheadsentailed by both implementations.For the centralized implementation, the destination controlsthe relay selection. To this end, the destination has to acquirethe CSI of all links in the network. Therefore, for centralizedimplementation, in each time slot, M + 2 pilot symboltransmissions are required for CSI acquisition, one controlpacket transmission by the destination is needed to informthe relays which relay is selected, and another control packettransmission is required for the selected relay to inform thesource which relay is selected. Moreover, the source has toacquire knowledge of min { C Sk ( i ) , C kD ( i ) } in order to selectthe rate of transmission. Hence, if min { C Sk ( i ) , C kD ( i ) } = C kD ( i ) , the selected relay has to feedback the CSI of theselected-relay-to-destination link to the source. As a result,in total M + 4 or M + 5 pilot symbol, feedback, andcontrol packet transmissions are needed in each time slot.On the other hand, for the centralized implementation of theproposed buffer-aided relaying protocol, also M +4 or M +5 pilot symbol, feedback, and control packet transmissions arerequired. Hence, both the conventional and the proposedbuffer-aided relaying protocols have identical overheads whenimplemented centrally.For conventional relay selection with distributed implemen-tation, each relay has to acquire the CSI of its source-to-relay and relay-to-destination links. To this end, two pilottransmissions, one from the source and the other from thedestination, are needed. Moreover, one packet with pilots anda control message from the selected relay are needed to informsource and destination which relay is selected, and to allowsource and destination to learn the CSI of the source-to-selected-relay and selected-relay-to-destination links, respec-tively. Furthermore, assuming relay k is the selected relay in time slot i , in order for the source to adapt its transmission rateto min { C Sk ( i ) , C kD ( i ) } and the destination to know whichcodebook to use for decoding in time slot i , both sourceand relay have to know min { C Sk ( i ) , C kD ( i ) } . Acquiring thisCSI knowladge requires feedback of the source-to-relay or therelay-to-destination channel from the relay to the destinationor the source, respectively. Hence, the distributed implemen-tation of conventional relay selection requires pilot symbol,feedback, and control packet transmissions. On the other hand,the distributed implementation of the proposed buffer-aidedrelaying protocol has the same overhead as conventional relayselection since it also requires pilot symbol, feedback, andcontrol packet transmissions.As can be seen from the above discussion, the proposedbuffer-aided protocol does not require more signaling over-head than the conventional relay selection protocol. We note,however, that the proposed protocol requires the computationof µ ek ( i ) and Q k ( i ) , ∀ k , which are not required for theconventional protocols. On the other hand, the computationalcomplexity of obtaining µ ek ( i ) and Q k ( i ) using (39)-(41) and(42), respectively, is not high since these equations requireonly one or two additions and one to three multiplications.V. B UFFER -A IDED R ELAYING P ROTOCOL WITH A D ELAY C ONSTRAINT
The protocol in Theorem 1, with the µ ∗ k , ∀ k , obtainedfrom Lemma 2, gives the maximum average achievable rate,but introduces unbounded delay. To bound the delay, in thefollowing, we propose a buffer-aided relaying protocol fordelay limited transmission. Before presenting the protocol, wefirst determine the average delay for the considered network. A. Average Delay
The average delay for the considered network, denoted by ¯ T , is specified in the following lemma. Lemma 4:
The average delay for the considered network isgiven by ¯ T = P Mk =1 ¯ Q k P Mk =1 ¯ R Sk , (43)where ¯ R Sk is the average rate received at the k -th relay andgiven by (12). Furthermore, ¯ Q k is the average queue size inthe buffer of the k -th relay, which is found as ¯ Q k = lim N →∞ N N X i =1 Q k ( i ) . (44) Proof:
Please refer to Appendix D.The queue size at time slot i can be obtained using (42).Due to the recursiveness of the expression in (42), it isdifficult, if not impossible, to obtain an analytical expressionfor the average queue size ¯ Q k for a general buffer-aided relayselection protocol. Hence, in contrast to the case without delayconstraint, for the delay limited case, it is very difficult toformulate an optimization problem for maximization of theaverage rate subject to some average delay constraint. As aresult, in the following, we develop a simple heuristic protocol for delay limited transmission. The proposed protocol is adistributed protocol in the sense that the relays themselvesnegotiate which relay should receive or transmit in each timeslot such that the average delay constraint is satisfied. We notethat the proposed protocol does not need any knowledge of thestatistics of the channels. B. Distributed Buffer-Aided Protocol
Before presenting the proposed heuristic protocol for delaylimited transmission, we first explain the intuition behind theprotocol.
1) Intuition Behind the Protocol:
Assume that we havea buffer-aided protocol which, when implemented in theconsidered network, enforces the following relation ¯ Q k ¯ R Sk = T , ∀ k, (45)i.e., the average queue length divided by the average arrivalrate in the buffer at the k -th relay is equal to T . If (45)holds ∀ k , then by inserting (45) into (43), we see that theaverage delay of the network will be ¯ T = T . Moreover,enforcing (45) at the k -th relay requires only local knowledge,i.e., only knowledge of the average queue length and theaverage arrival rate at the k -th relay is required. Hence, thisprotocol can be implemented in a distributed manner. Thereare many ways to enforce (45) at the k -th relay. Our preferredmethod for enforcing (45) is to have the k -th relay receiveand transmit when Q k ( i ) / ¯ R Sk < T and Q k ( i ) / ¯ R Sk > T occur, respectively. Moreover, we prefer a protocol in whichthe more Q k ( i ) / ¯ R Sk differs from T , the higher the chanceof selecting the k -th relay should be. In this way, Q k ( i ) / ¯ R Sk becomes a random process which exhibits fluctuation aroundits mean value T , and thereby achieves (45) in the long run.We are now ready to present the proposed protocol.
2) The Proposed Protocol for Delay-Limited Transmission:
Let T be the desired average delay constraint of the system.At the beginning of time slot i , source and destination transmitpilots in successive pilot time slots. This enables the relays toacquire the CSI of their respective source-to-relay and relay-to-destination channels. Using the acquired CSI, the k -th relaycomputes C Sk ( i ) and C kD ( i ) . Next, using C Sk ( i ) and theamount of normalized information in its buffer, Q k ( i − , the k -th relay computes a variable λ k ( i ) as follows λ k ( i ) = λ k ( i −
1) + ζ k ( i ) (cid:18) T − Q k ( i − R eSk ( i − (cid:19) , (46)where ¯ R eSk ( i − is a real-time estimate of ¯ R Sk , computedusing (40). In (46), ζ k ( i ) is the step size function, which issome properly chosen monotonically decaying function of i with ζ k (1) < . Now, using C Sk ( i ) , C kD ( i ) , Q k ( i − , and λ k ( i ) , the k -th relay turns on a timer proportional to { λ k ( i ) C Sk ( i ) , min { Q k ( i − , C kD ( i ) } /λ k ( i ) } . (47)This procedure is performed by all M relays. If max { λ k ( i ) C Sk ( i ) , min { Q k ( i − , C kD ( i ) } /λ k ( i ) } = λ k ( i ) C Sk ( i ) and max { λ k ( i ) C Sk ( i ) , min { Q k ( i − , C kD ( i ) } /λ k ( i ) } = min { Q k ( i − , C kD ( i ) } /λ k ( i ) , (48)the k -th relay knows that if it is selected, then it will receiveand transmit, respectively. The relay whose timer expiresfirst, broadcasts a control packet containing pilot symbols andinformation about which relay is selected and whether the se-lected relay receives or transmits. From the packet broadcastedby the selected relay, both source and destination learn thesource-to-selected-relay and the selected-relay-to-destinationchannels, respectively, and which relay is selected and whetherit is scheduled to receive or transmit. If the selected relay isscheduled to transmit, then it extracts information from itsbuffer and transmits a codeword to the destination with rate R kD ( i ) = min { Q k ( i − , C kD ( i ) } . However, if the relay isscheduled to receive, then the source transmits a codewordto the k -th relay with rate R Sk ( i ) = C Sk ( i ) . In this case,the relay has to feedback its source-to-relay channel to thedestination. This fedback CSI is needed by the destinationto keep track of the amount of information that each relayreceives and transmits in each time slot so that the destinationcan perform successful reordering of the received informationbits. Moreover, exploiting (42), this information is used by thedestination to compute the queue length in the buffer at eachrelay, Q k ( i ) . Remark 4:
Note that with (46) we achieve the aforemen-tioned goal of increasing the probability of selecting the k -threlay when Q k ( i ) / ¯ R Sk differs more from T . More precisely,if Q k ( i ) / ¯ R Sk < T , then λ k ( i ) increases and /λ k ( i ) de-creases, giving the k -th relay a higher chance to be selectedfor reception. On the other hand, if Q k ( i ) / ¯ R Sk > T , then λ k ( i ) decreases and /λ k ( i ) increases, giving the k -th relaya higher chance to be selected for transmission. Remark 5:
The required overhead of the proposed dis-tributed delay-limited protocol is identical to the overheadof the proposed distributed protocol without delay constraint.Furthermore, the delay-limited buffer-aided protocol can alsobe implemented in a centralized manner, similar to the schemein Section IV-A. The centralized implementation of the delay-limited protocol has an overhead identical to the overheadof the centralized protocol without delay constraint, see Sec-tion IV-A. A summary of the overheads of conventionalrelay selection protocols and the proposed buffer-aided (BA)relaying protocols with and without delay constraint is givenin Table I. VI. N
UMERICAL E XAMPLES
We assume that all source-to-relay and relay-to-destinationlinks are impaired by Rayleigh fading. Throughout this sec-tion, we use the abbreviation “BA” to denote “buffer-aided”.In Fig. 2, we plot the theoretical maximum average rateobtained from Theorem 1, and Lemmas 2 and 3, for M = 5 relays and i.n.d. fading, where [Ω S , Ω S , Ω S , Ω S , Ω S ] = [0 . , , . , , . −5 0 5 10 15 20 25012345 A v e r ag e r a t e ( i nb i t s / s y m b ) P/ σ n (in dB) Proposed BA protocol without delay – TheoryProposed BA protocol without delay – SimulationProposed BA protocol, ¯ T = 5 time slots – SimulationProposed BA protocol, ¯ T = 10 time slots – SimulationRate in [10] and [11] Fig. 2. Achievable average rates for M = 5 , [Ω S , Ω S , Ω S , Ω S , Ω S ] =[0 . , , . , , . , and [Ω D , Ω D , Ω D , Ω D , Ω D ] =[3 , . , . , . , . .
200 400 600 800 1000 1200 1400 1600 1800 200000.20.40.60.81
Time slot i µ µ ek ( i ) µ k Start of search µ ek (0) = 0 . µ µ Fig. 3. Estimated µ e ( i ) and µ e ( i ) as a function of the time slot i . and [Ω D , Ω D , Ω D , Ω D , Ω D ] = [3 , . , . , . , . . We have also included simulation results for the proposedbuffer-aided protocol, where the µ ek ( i ) , k = 1 , ..., , are foundusing the recursive method in (39) with δ k ( i ) = 0 . / √ i , ∀ k .As can be seen, the simulated average rate coincides perfectlywith the theoretical average rate. As a benchmark, in Fig. 2,we show the average rate given in [10] and [11]. Moreover, wehave also included the average rates achieved using the delaylimited BA protocol introduced in Section V-B for an averagedelay of ¯ T = 5 and ¯ T = 10 time slots. For the delay limitedprotocol, in order to evaluate (46) we have used λ k (1) = 0 . and the step size function ζ k ( i ) = 0 . / √ i/ log (1 + P/σ n ) , ∀ k . As can be seen from Fig. 2, both the delay-unlimited andthe delay-limited BA protocols achieve higher rates than therate achieved in [10] and [11]. We note that we cannot usethe protocols in [8] and [9] as benchmarks in Fig. 2 sincethese protocols are not applicable in i.n.d. fading as the bufferswould become unstable. In particular, for the protocols in[8] and [9], the buffers at relays with Ω Sk > Ω kD wouldsuffer from overflow and receive more information than theycan transmit. Hence, a fraction of the source’s data would betrapped inside the buffers and does not reach the destination,i.e., data loss would occur.For the parameters adopted in Fig. 2, we show in Fig. 3 thecorresponding constants µ and µ obtained using Lemma 2, TABLE IN
UMBER OF PILOT SYMBOL , FEEDBACK , AND CONTROL PACKET TRANSMISSIONS REQUIRED FOR THE CONVENTIONAL AND THE PROPOSEDBUFFER - AIDED (BA)
PROTOCOLS PER TIME SLOT
Conventional BA protocols with and without delay constraintCentralized M + 4 or M + 5 2 M + 4 or M + 5 Distributed
Time slot i A v e r ag e d e l a y un t il t i m e s l o t i P/ σ n = 25 dB P/ σ n = 20 dB T = 5 Fig. 4. Average delay until time slot i for T = 5 and different P/σ n . Number of relays A v e r ag e r a t e ( b i t s / s y m b ) Proposed BA protocol and the protocol in [8]Rate in [10] and [11]Protocol in [9]
Fig. 5. Achievable average rates for Ω Sk = Ω kD = 1 , ∀ k , as a functionof the number of relays M . and the corresponding estimated parameters µ e ( i ) and µ e ( i ) obtained using the recursive method in (39) as functionsof time for P/σ n = 0 dB. As can be seen from Fig. 3,the estimated parameters µ e ( i ) and µ e ( i ) converge relativelyquickly to µ and µ , respectively.Furthermore, for the parameters adopted in Fig. 2, wehave plotted the average delay of the proposed delay-limitedprotocol until time slot i in Fig. 4, for the case when T = 5 time slots, and P/σ n = 20 dB and P/σ n = 25 dB. Theaverage delay until time slot i is computed based on (43) wherethe queue size and the arrival rates are averaged over the timewindow from the first time slot to the i -th time slot. Hence,for finite i , the average delay until time slot i is the averageof a random process over a time window of limited duration.Because of the assumed ergodicity, for i → ∞ , the size ofthe averaging window becomes infinite and the time averageconverges to the mean of this random process. However, for i < ∞ , the time average is still a random process. This isthe reason for the random fluctuations in the average delayuntil time slot i in Fig. 4. Nevertheless, Fig. 4 shows that theaverage delay until time slot i converges relatively fast to T as i increases. Moreover, after the average delay has reached T , it exhibits relatively small fluctuations around T . −5 0 5 10 15 20 25 300.511.522.533.544.55 P/ σ n (dB) A v e r ag e r a t e ( b i t s / s y m b ) BA, unbounded delay, M = 1BA, ¯ T = 5 time slots, M = 1Rate in [10] and [11], M = 1BA, unbounded delay, M = 2BA, ¯ T = 7 time slots, M = 2Rate in [10] and [11], M = 2BA, unbounded delay, M = 4BA, ¯ T = 10 time slots, M = 4Rate in [10] and [11], M = 4 Fig. 6. Achievable average rates for Ω Sk = Ω kD = 1 , ∀ k , vs P/σ n fordifferent number of relays M , and different delay. In Fig. 5, we plot the theoretical achievable average ratesfor BA relaying for i.i.d. fading with Ω Sk = Ω kD = 1 , ∀ k ,and P/σ n = 10 dB, as a function of the number of relays M . As can be seen from this numerical example, the growthrate of the maximum average rate is inversely proportional to M , i.e., the growth rate of the average data rate decreases as M increases. In particular, the largest increase in data rate isobserved when M increases from one to two relays, whereasthe increase in the maximum average rate when M increasesfrom 29 to 30 relays is almost negligible. This behavior canbe most clearly seen from the expression for the average ratefor low SNR given in (33). According to (33), the averagerate increases proportionally to / / ... + 1 / (2 M ) .Therefore, when M is large, adding one more relay to thenetwork has a negligible effect on the average rate. As bench-marks, we also show the average rate given in [10] and [11],and the average rates achieved with the protocols in [8] and[9]. For i.i.d. links, as explained in Remark 1, the protocol in[8] is identical to the protocol presented in Theorem 1, therebyleading to the same rate.In Fig. 6, we plot the achievable average rate for BArelaying without and with a delay constraint, as a functionof P/σ n , for i.i.d. fading and different numbers of relays M .This numerical example shows that, as the number of relaysincreases, the permissible average delay has to be increasedin order for the rate of the delay constrained protocol toapproach the rate of the non-delay constrained protocol. Moreprecisely, for a single relay network, an average delay of fivetime slots is sufficient for the rate of the delay constrainedprotocol to approach the rate of the non-delay constrainedprotocol. However, for a network with two and four relays,the corresponding required delays are and time slots,respectively. For comparison, we have also plotted the averagerate given in [10] and [11], which requires a delay of one timeslot. Fig. 6 shows that the average rate of the buffer-aided relaying protocol with five time slots delay and only one relaysurpasses the average rate in [10] and [11] for four relays.VII. C ONCLUSION
We have devised buffer-aided relaying protocols for theslow fading HD relay selection network and derived thecorresponding achievable average rates. We have proposed abuffer-aided protocol which maximizes the achievable averagerate but introduces an unbounded delay, and a buffer-aidedprotocol which bounds the average delay at the expense of adecrease in rate. We have shown that the new achievable ratesare larger than the rates achieved with existing relay selectionprotocols. We have also provided centralized and distributedimplementations of the proposed buffer-aided protocols, whichdo not cause more signaling overhead than conventional relayselection protocols for adaptive rate transmission and do notneed any a priori knowledge of the statistics of the involvedchannels. A
PPENDIX
A. Proof of Lemma 1
We denote the left and right hand sides of (15) as A k and D k , respectively, i.e., A k = lim N →∞ N N X i =1 r R k ( i ) C Sk ( i ) , (49) D k = lim N →∞ N N X i =1 r T k ( i ) C kD ( i ) . (50)There are three possible cases for the relationship between A k and D k , i.e., A k > D k , A k < D k , and A k = D k . If A k > D k then the buffer of the k -th relay is receiving more informationthan it transmits. Therefore, the average queue length in thebuffer grows with time to infinity, and, as a result, ¯ R kD = D k , for a proof please refer to [27, Section 1.5]. Whereas, if A k < D k , due to the conservation of flow, the buffer cannotemit more information than it receives, and therefore ¯ R kD = A k . We now prove that for A k > D k and A k < D k , ¯ R kD can always be increased by changing the values of r R k ( i ) and r T k ( i ) . As a result, the only remaining possibility is that ¯ R SD is maximized for A k = D k . Furthermore, since the achievablerate is given by ¯ R SD = P Mk =1 ¯ R kD , if ¯ R kD increases, ¯ R SD will also increase.Assume first that A k > D k . Then, we can always increase D k , and thereby increase ¯ R kD , by switching any r R k ( i ) = 1 for which Q k ( i − > holds, from one to zero and, forthe same i , switch r T k ( i ) from zero to one. On the other hand,if A k < D k then we can always increase A k , and therebyincrease ¯ R kD , by switching any randomly chosen r T k ( i ) = 1 from one to zero and, for the same i , switch r R k ( i ) from zero toone. Now, since ¯ R SD can always be improved when A k > D k or A k < D k , it follows that ¯ R SD is maximized for A k = D k .Furthermore, when the ¯ R kD are maximized ∀ k , then ¯ R SD isalso maximized. Moreover, for A k = D k the buffer at the k -threlay is stable since the information that arrives at the bufferalso leaves the buffer without information loss. On the otherhand, the proof that (16) holds when (15) is satisfied is given in [13, Appendix B]. Finally, considering (14), if (16) holds ∀ k , then (17) holds as well. This concludes the proof. B. Proof of Theorem 1
To solve (18), we first relax the binary constraints r T k ( i ) ∈{ , } and r R k ( i ) ∈ { , } in (18) to ≤ r T k ( i ) ≤ and ≤ r R k ( i ) ≤ , ∀ i , respectively. Thereby, we transform theoriginal problem (18) into the following linear optimizationproblem Maximize : r R k ( i ) ,r T k ( i ) , ∀ i,k N P Ni =1 P Mk =1 r T k ( i ) C kD ( i )Subject to : C1 : N P Ni =1 r R k ( i ) C Sk ( i )= N P Ni =1 r T k ( i ) C kD ( i ) , ∀ k C2 : 0 ≤ r R k ( i ) ≤ , ∀ k, i C3 : 0 ≤ r T k ( i ) ≤ , ∀ k, i C4 : 0 ≤ P Mk =1 [ r R k ( i ) + r T k ( i )] ≤ , ∀ k, i. (51)In the following, we solve the relaxed problem (51) and thenshow that the optimal values of r T k ( i ) and r R k ( i ) , ∀ i, k are atthe boundaries, i.e., r R k ( i ) ∈ { , } and r T k ( i ) ∈ { , } , ∀ i, k .Therefore, the solution of the relaxed problem (51) is also thesolution to the original maximization problem in (18).Since (51) is a linear optimization problem, we can solve itby using the method of Lagrange multipliers. The Lagrangianfunction for maximization problem (51) is given by L = M X k =1 N N X i =1 r T k ( i ) C kD ( i ) − M X k =1 µ k N N X i =1 r T k ( i ) C kD ( i ) − N N X i =1 r R k ( i ) C Sk ( i ) ! − M X k =1 N N X i =1 α T k ( i ) (cid:0) r T k ( i ) − (cid:1) + M X k =1 N N X i =1 β T k ( i ) r T k ( i ) − M X k =1 N N X i =1 α R k ( i ) (cid:0) r R k ( i ) − (cid:1) + M X k =1 N N X i =1 β R k ( i ) r R k ( i ) − N N X i =1 φ ( i ) M X k =1 [ r R k ( i ) + r T k ( i )] − ! + 1 N N X i =1 λ ( i ) M X k =1 [ r R k ( i ) + r T k ( i )] ! , (52)where µ k /N , α xk ( i ) /N , β xk ( i ) /N , for x ∈ { R , T } , φ ( i ) /N ,and λ ( i ) /N are Lagrange multipliers. These multipliers haveto satisfy the following conditions. Dual feasibility condition : The Lagrange multipliers forthe inequality constraints have to be non-negative, i.e., α R k ( i ) ≥ , α T k ( i ) ≥ , β R k ( i ) ≥ , β T k ( i ) ≥ ,φ ( i ) ≥ , λ ( i ) ≥ , ∀ i, k. (53)have to hold. Complementary slackness condition : If an inequality isinactive, i.e., the optimal solution is in the interior of the corresponding set, the corresponding Lagrange multipliers arezero. Therefore, we obtain α R k ( i ) (cid:0) r R k − (cid:1) = 0 , α T k ( i ) (cid:0) r T k − (cid:1) = 0 , ∀ i, k (54) β R k ( i ) r R k = 0 , β T k ( i ) r T k = 0 , ∀ i, k (55) φ ( i ) M X k =1 [ r R k ( i ) + r T k ( i )] − ! = 0 ,λ ( i ) M X k =1 [ r R k ( i ) + r T k ( i )] ! = 0 , ∀ i, k. (56)We now differentiate the Lagrangian function with respectto r R n ( i ) and r T m ( i ) , for n ∈ { , ..., M } and m ∈ { , ..., M } ,and equate the results to zero, respectively. This leads to thefollowing two equations µ n C Sn ( i ) = α R n ( i ) − β R n ( i ) + φ ( i ) − λ ( i ) (57) (1 − µ m ) C mD ( i ) = α T m ( i ) − β T m ( i ) + φ ( i ) − λ ( i ) . (58)We first show that for the optimal solution of r R n ( i ) and r R m ( i ) , < r R n ( i ) < and/or < r T m ( i ) < cannothold for any n, m ∈ { , ..., M } , and only r R n ( i ) ∈ { , } and r T m ( i ) ∈ { , } can hold ∀ n, m = 1 , ..., M . We provethis by contradiction. Assume that < r R n ( i ) < and < P Mk =1 [ r R k ( i ) + r T k ( i )] < . Then, according to (54), α R n ( i ) = β R n ( i ) = φ ( i ) = λ ( i ) = 0 must hold. Inserting thisinto (57a), we obtain µ n C Sn ( i ) = 0 . (59)Since C Sn ( i ) is an RV, (59) can hold only for µ n = 0 .However, if we assume µ n = 0 , and insert µ n = 0 in (57b)by setting m = n , we obtain C nD ( i ) = α T n ( i ) − β T n ( i ) . (60)Since C nD ( i ) is a non-negative RV, and since either α T n ( i ) or β T n ( i ) can be larger than zero but not both, in order for (60) tohold, β T n ( i ) must be zero and α T n ( i ) = C nD ( i ) . On the otherhand, if β T n ( i ) = 0 , it would mean that r T n ( i ) = 1 . However, if r T n ( i ) = 1 and < r R n ( i ) < hold jointly, this would violateour starting assumption that < P Mk =1 [ r R k ( i ) + r T k ( i )] < holds. Hence, < r R n ( i ) < and < P Mk =1 [ r R k ( i ) + r T k ( i )] < cannot hold.Now, let us assume that < r R n ( i ) < and P Mk =1 [ r R k ( i ) + r T k ( i )] = 1 . Since r R n ( i ) < , then at least one other variable r R k ( i ) or r T m ( i ) has to be larger than zero but smaller thanone, where k ∈ { , ..., M } , k = n , and m ∈ { , ..., M } . Letus assume that this variable is r R k ( i ) , where k = n . Hence, < r R k ( i ) < , for k = n . Then, according to (54), α R n ( i ) = β R n ( i ) = α R k ( i ) = β R k ( i ) = λ ( i ) = 0 , and φ ( i ) ≥ must hold.Inserting these values in (57a), we obtain µ n C Sn ( i ) = φ ( i ) = µ k C Sk ( i ) . (61)However, since C Sn ( i ) and C Sk ( i ) are independent RVs, (61)cannot hold for any arbitrarily chosen i . On the other hand, ifwe assume that instead of r R k ( i ) , the variable which is largerthan one is r T k ( i ) , we would have obtained that µ n C Sn ( i ) = φ ( i ) = (1 − µ k ) C kD ( i ) (62) must hold. Since (62) also cannot hold for any arbitrarilychosen i , we obtain that < r R n ( i ) < and P Mk =1 [ r R k ( i ) + r T k ( i )] = 1 cannot hold. Therefore, the only other possibilityis that r R n ( i ) ∈ { , } must hold.Following the same approach as above, we can also provethat r T m ( i ) ∈ { , } must hold. Moreover, due to constraint C4in (51), it is clear that if r R n ( i ) = 1 , for any n ∈ { , ..., M } ,then r R k ( i ) = 0 for all k = 1 , ..., M , k = n , and r T m ( i ) = 0 for all m = 1 , ..., M must hold. Similarly, if r T m ( i ) = 1 , forany m ∈ { , ..., M } , then r T k ( i ) = 0 for all k = 1 , ..., M , k = m , and r R n ( i ) = 0 for all n = 1 , ..., M must hold. In thefollowing, we investigate the conditions under which r R n ( i ) =1 and all other r R k ( i ) = 0 for k = 1 , ..., M , k = n , and allother r T m ( i ) = 0 for m = 1 , ..., M .Assume r R n ( i ) = 1 . Then, r R k ( i ) = 0 for k = 1 , ..., M , k = n , and r T m ( i ) = 0 for m = 1 , ..., M must hold. As aresult, according to (54), α R n ( i ) ≥ , β R k ( i ) ≥ , β T m ( i ) ≥ , φ ( i ) ≥ , and β R n ( i ) = α R k ( i ) = α R m ( i ) = λ ( i ) = 0 must hold,for k = 1 , ..., M , k = n , and m = 1 , ..., M . Inserting thesevariables in (57), we obtain the following µ n C Sn ( i ) = α R n ( i ) + φ ( i ) , (63) µ k C Sk ( i ) = − β R k ( i ) + φ ( i ) , ∀ k = n (64) (1 − µ m ) C mD ( i ) = − β T m ( i ) + φ ( i ) , ∀ m. (65)Subtracting (64) from (63) and subtracting (65) from (63), weobtain µ n C Sn ( i ) − µ k C Sk ( i ) = α R n ( i ) + β R k ( i ) , ∀ k = n (66) µ n C Sn ( i ) − (1 − µ m ) C mD ( i ) = α R n ( i ) + β R m ( i ) , ∀ m. (67)Since α R n ( i ) + β R k ( i ) ≥ and α R n ( i ) + β R m ( i ) ≥ hold, itfollows that r R n ( i ) = 1 when the following holds µ n C Sn ( i ) > µ k C Sk ( i ) , ∀ k = n AND µ n C Sn ( i ) > (1 − µ m ) C mD ( i ) , ∀ m. (68)Eq. (68) can be written in compact form as r R k ( i ) = 1 if µ k C Sk ( i ) = max A ( i ) , (69)where set A ( i ) is defined in (19). Following the same approachas above, we can prove that r T k ( i ) = 1 if (1 − µ k ) C kD ( i ) = max A ( i ) . (70)Combining (69) and (70), we obtain (20). This completes theproof of Theorem 1. C. Proof of Lemma 2
The optimal µ k , ∀ k , are found from the system of M equations given in (21). Using the definition of the expectedvalue, (21) can be written equivalently as (23), where the RVs Γ Sk and Γ kD are given by Γ Sk = (cid:26) γ Sk if µ k C Sk = max A if µ k C Sk = max A , Γ kD = (cid:26) γ kD if (1 − µ k ) C kD = max A if (1 − µ k ) C kD = max A . (71) Hence, to find the optimal µ k , ∀ k , we only have to find thePDFs of Γ Sk and Γ kD , f Γ Sk ( x ) and f Γ kD ( x ) , and insert theminto (23). In the following, we first derive the PDF of Γ Sk .Using (71), we can obtain the PDF of Γ Sk , f Γ Sk ( x ) , for x > , as f Γ Sk ( x ) = f γ Sk ( x )Pr (cid:8) µ k C Sk = max A (cid:9) , x > , (72)where Pr {·} denotes probability. Note that the distribution of f Γ Sk ( x ) for x = 0 , is not needed for the computation of theexpectations in (21) and (22). The only unknown in (72) isthe probability Pr (cid:8) µ k C Sk = max A (cid:9) . In the following, wederive this probability. To this end, we set γ Sk = x , and obtain Pr (cid:8) µ k C Sk = max A (cid:9) = Pr (cid:8) µ k log (1 + x ) = max A (cid:9) = M Y j =1 ,j = k Pr { µ j log (1 + γ Sj ) < µ k log (1 + x ) }× M Y j =1 Pr { (1 − µ j ) log (1 + γ jD ) < µ k log (1 + x ) } = M Y j =1 ,j = k Pr n γ Sj < (1 + x ) µkµj − o × M Y j =1 Pr n γ jD < (1 + x ) µk − µj − o = M Y j =1 ,j = k F γ Sj (cid:16) (1 + x ) µkµj − (cid:17) × M Y j =1 F γ jD (cid:16) (1 + x ) µk − µj − (cid:17) , (73)where F γ α ( x ) is the CDF of γ α , for α ∈ { Sk, kD } . Inserting(73) into (72), we obtain (24). Following a similar procedureas above, we obtain the distribution of Γ kD given in (25).Now, assume that all source-to-relay and relay-to-destination links are i.i.d. Then, f γ Sk ( x ) = f γ kD ( x ) = f γ ( x ) holds ∀ k . Moreover, F γ Sk ( x ) = F γ kD ( x ) = F γ ( x ) also holds ∀ k . As a result, (24) and (25) can be written for x > as f Γ Sk ( x ) = f γ ( x ) F γ (cid:16) (1 + x ) µk − µk − (cid:17) × M Y j =1 j = k F γ (cid:16) (1 + x ) µkµj − (cid:17) F γ (cid:16) (1 + x ) µk − µj − (cid:17) , (74) f Γ kD ( x ) = f γ ( x ) F γ (cid:18) (1 + x ) − µkµk − (cid:19) × M Y j =1 j = k F γ (cid:18) (1 + x ) − µkµj − (cid:19) F γ (cid:18) (1 + x ) − µk − µj − (cid:19) . (75)We observe that f Γ Sk ( x ) and f Γ kD ( x ) in (74) and (75), re-spectively, are both functions of µ k and show this explicitly byredefining them as f Γ Sk ( x, µ k ) and f Γ kD ( x, µ k ) , respectively.Moreover, from (74) and (75) we observe that f Γ kD ( x, µ k ) = f Γ Sk ( x, − µ k ) (76) ... S W I T CH BU FF E R ... S W I T CH Fig. 7. Equivalent single buffer model. holds. If we now insert (76) into (23), we obtain Z ∞ log (1 + x ) f Γ Sk ( x, µ k ) dx = Z ∞ log (1 + x ) f Γ Sk ( x, − µ k ) dx, ∀ k = 1 , ..., M. (77)Now, observe that (77) holds if and only if µ k = 1 − µ k , whichleads to µ k = 1 / . This concludes the proof. D. Proof of Lemma 4
The average delay for a system with M parallel queues iswell known, and given by [28, Eq. 11.69]. After changing thenotations in [28, Eq. 11.69] to our notations, we directly obtain(43). In the following, we give an alternative, more intuitiveproof of (43).The input-output dynamics at the M buffers in the con-sidered network during N time slots can be representedequivalently by a single buffer model, shown in Fig. 7. Thedifferent colors in this model correspond to the informationbits which are received/transmitted by the different relays. Forexample, the blue, green, and red colors correspond to the bitsthat are send/received via relay 1, 2, and 3, respectively. In thismodel, the buffer is filled in the same order as the order of thepackets that arrive at the buffers at the different relays. Whichpacket arrives at the equivalent buffer depends on the positionof the input switch in each time slot, which on the otherhand, depends on the values of r R k ( i ) , ∀ i, k . The extractionof the bits from the equivalent buffer also depends on theposition of the output switch in each time slot, which on theother hand, depends on the values of r T k ( i ) , ∀ i, k . Moreover,when the output switch is set to a line with a specific color,only bits with that color are extracted from the equivalentbuffer. Hence, the extraction order is different from the orderof filling the equivalent buffer. Nevertheless, since the averagedelay computed by Little’s formula [29], is independent of theorder of extracting from the buffer, see [30, pp. 89-91], for thesystem model in Fig. 7, the average delay ¯ T can be computedas [29] ¯ T = ¯ Q eq ¯ A eq , (78)where ¯ Q eq is the average queue size of the equivalent bufferand ¯ A eq is the average arrival rate of the equivalent buffer.Now, using the fact that ¯ Q eq = P Mk =1 ¯ Q k and ¯ A eq = P Mk =1 ¯ R Sk , we obtain (43). This concludes the proof.R EFERENCES[1] A. Bletsas, A. Khisti, D. Reed, and A. Lippman, “A Simple CooperativeDiversity Method Based on Network Path Selection,”
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Nikola Zlatanov (S’06) was born in Macedonia. Hereceived his Dipl.Ing. and M.S. degrees in electricalengineering from SS. Cyril and Methodius Univer-sity, Skopje, Macedonia, in 2007 and 2010, respec-tively. Currently, he is working toward his Ph.D.degree at the University of British Columbia (UBC),Vancouver, Canada. His current research interestsinclude wireless communications and informationtheory.Mr. Zlatanov received several awards for his workincluding UBC’s Four-Year Doctoral Fellowship in2010, UBC’s Killam Doctoral Scholarship and Macedonia’s Young Scientist ofthe Year Award in 2011, Vanier Canada Graduate Scholarship in 2012, DAADResearch Grant in 2013, and best paper award from the German InformationTechnology Society (ITG) in 2014.
Vahid Jamali (S’12) was born in Fasa, Iran, in 1988.He received his B.S. and M.S. degrees in electricalengineering from K. N. Toosi University of Tech-nology (KNTU), in 2010 and 2012, respectively.Currently, he is working toward his Ph.D. degree atthe Friedrich-Alexander University (FAU), Erlangen,Germany. His research interests include multiuserinformation theory, wireless communications, cog-nitive radio network, LDPC codes, and optimizationtheory.