1 Buffer–Aided Relaying with Outdated CSI Toufiqul Islam, Member, IEEE, Diomidis S. Michalopoulos, Senior Member, IEEE, Robert Schober, Fellow, IEEE, and Vijay Bhargava, Life Fellow, IEEE

Abstract—Adaptive link selection for buffer–aided relaying can Over the last few years, there has been an increasing interest provide significant performance gains compared to conventional in buffer–aided relaying protocols [5]- [15], in particular for relaying with fixed transmission schedule. For fixed rate transmis- delay tolerant applications. Two buffer–aided relaying schemes sion, significant gains in error rate and/or throughput performance have been observed compared to conventional relaying when were proposed in [5], where the relay is allowed to receive perfect channel state information (CSI) is available for link for a fixed number of time slots before it re–transmits to the selection. However, in practice, link selection may have to be destination. These schemes show a throughput improvement performed based on outdated CSI, because of infrequent feedback compared to receiving and re–transmitting in subsequent time of CSI. In this paper, we study the effect of outdated CSI on slots. A joint cross–layer scheduling and buffer–equipped relay the error rate performance of adaptive link selection for a three node decode–and–forward (DF) relay network with fixed–rate selection problem was considered in [6], where a considerable transmission. In particular, we propose two protocols for link throughput improvement was reported compared to relaying selection. The first protocol does not require knowledge of the without buffers. Throughput optimization for adaptive link reliability of the CSI estimates whereas the second protocol does. selection with buffer–aided relaying was studied in [7] and For both protocols, we provide a unified error-rate analysis in [8] for adaptive and fixed rate transmission, respectively. The terms of a decision threshold β, which can be adjusted to maintain buffer stability. In particular, we obtain closed–form expressions authors in [9] observed that if some packet delay is tolerated, for the error rates, and derive asymptotic approximations which the asymptotic throughput can be improved by exploiting relay reveal the diversity and coding gains. Since packet transmission buffering and relay mobility. In [10], buffer–aided HD relaying delay is unavoidable for opportunistic link selection, we analyze was shown to outperform ideal full duplex relaying in some the average delay considering both finite and infinite buffer size. special cases, when the relay employs multiple antennas. We We also calculate the throughput for the proposed protocols. The average delay and throughput are functions of the decision note that [5]– [9] focused on throughput optimization only threshold β which can be optimized to minimize the error rate and the error rate was not considered. In [11] and [12], relay while satisfying average delay and/or throughput constraints. selection was considered, i.e., the relays with the best S → R Numerical results manifest that even with outdated CSI, adaptive and the best R → D channels were selected for reception and link selection provides a significant coding gain advantage over transmission, respectively, but adaptive link selection was not conventional DF relaying. Furthermore, we show that a diversity gain of two can be achieved for perfect CSI and the optimum error employed. The authors in [13] extended the work in [11] and rate can be approached with small delay and/or high throughput. showed that joint relay and link selection can provide additional diversity gains. In [14], the authors reported that buffering at the Index Terms—Cooperative diversity, link selection, buffer–aided relaying, average delay, outdated CSI. relay is useful for achieving multihop diversity for independent and identicaly distributed (i.i.d.) multihop links. The authors I.INTRODUCTION in [15] proposed a packet selection based transmission scheme for buffer–aided amplify–and–forward relaying and showed that Conventional decode–and–forward (DF) relay networks [2], outage probability can be improved compared to traditional [3] employ half–duplex (HD) relays and a pre–fixed transmis- relaying with fixed schedule. A critical requirement for the link sion schedule. As the link quality is not taken into account selection protocols reported in [7], [13]–[15] is the availability for scheduling the transmissions, the end–to–end performance of perfect channel state information (CSI) for link selection. is limited by the bottleneck link. However, if the transmitting However, in practice, the CSI exploited for link selection may nodes can be selected opportunistically based on the link be outdated because CSI may not be fed back frequently as it quality, a better error rate and/or throughput performance is incurs overhead. It is not clear from the literature whether the expected compared to conventional relaying. This opportunistic protocols developed for perfect CSI would also yield optimum scheduling requires buffers at the relays such that the received performance when only outdated CSI is available. In particular, message can be stored until a good quality transmit channel is if the reliability of the outdated CSI is known, we expect that observed [4]. by exploiting this information for link selection, a better error rate performance can be achieved. The impact of outdated CSI Manuscript received June 26, 2014; revised October 26, 2014 and August 30, 2015; accepted October 18, 2015. The review of this paper was coordinated for relay selection has been considered in [16]–[19]. However, by Dr. Chia–Han Lee. This work has been presented in part at IEEE WCNC the effect of outdated CSI on link selection for buffer–aided 2014, Istanbul, Turkey [1]. relaying has not been considered in the literature, yet. Toufiqul Islam was with The University of British Columbia, Vancouver where this work was conducted. He is now at the Huawei Canada Research On the other hand, one important aspect of all opportunistic Center in Ottawa, Canada (email: toufi[email protected]). Diomidis S. Michalopou- link selection protocols is the increased end–to–end delay. los is with Nokia Networks, Munich, Germany (email: [email protected]). Robert Despite its importance, most of the previous works [11]– Schober is with The Friedrich–Alexander–University Erlangen–Nuremberg, Germany (email: [email protected]). Vijay Bhargava is with The University of [13] on link selection did not consider delay constrained British Columbia, Vancouver, Canada. (email: [email protected]) transmission. In [8], the authors proposed delay constrained 2 link selection protocols for throughput optimization for fixed II.SYSTEM MODEL and mixed rate transmission. In [20], the authors considered fixed rate transmission and analyzed the coded error rate and The considered system consists of a source terminal S, a the delay of a buffer–aided relaying protocol operating over DF relay R which is equipped with a buffer, and a destination frequency–selective channels. It was shown that full diversity terminal D. The direct link between S and D is not exploited and minimum error rate are observed only when a large delay due to heavy attenuation and/or simplicity of implementation. can be tolerated. However, both [8] and [20] assumed perfect In our model, S and R transmit at a fixed rate determined by the CSI and the effect of the unavoidable delay in the feedback adopted modulation scheme and S has always data to transmit. link and/or infrequent feedback of CSI was not taken into We assume uncoded transmission over flat–fading links and account. Furthermore, in [8] and [20], the average system each transmission frame consists of two phases: a handshaking delay was calculated considering the queueing delay only. phase and a data transmission phase, cf. Fig. 1. Furthermore, However, as the buffer at the relay can become empty or we assume that the link selection is performed by a control full, it is possible that successful transmission cannot always unit (CU) which collects the CSI of both links and feeds the take place and silent time slots may occur. This causes an result of the selection process back to the transmitting nodes. additional end–to–end delay which has not been investigated Note that depending on the scenario, the source, the relay, the for adaptive link selection yet. Moreover, it is not clear from destination or an external node can serve as the CU. During the available literature whether full diversity in terms of the the handshaking phase, the CSI required for link selection error rate performance can be realized if only a small delay is acquired by the CU and the decision regarding the link can be tolerated for buffer–aided link selection. However, selection is fed back to the nodes. Pilot symbols are inserted even though the objective is to analyze the error rate in this between chunks of data symbols in the data transmission phase. work, a constraint on throughput is also important to ensure a These pilots are used at the receiver, i.e., the relay or the minimum end–to–end data rate. Hence, studying adaptive link destination, to accurately estimate the channel for coherent selection for a delay and throughput constrained buffer–aided detection. To reduce the signalling overhead, the link selection relay network with outdated CSI and a corresponding error–rate is performed only at the beginning of each frame. Hence, analysis are of great practical value. the CSI used for link selection may become outdated during In this paper, we consider a three–node relay network where the data transmission phase, and the link selection decision the relay is equipped with a buffer. We propose two link selec- may not be optimal for the transmitted data symbols. As the tion protocols for buffer–aided DF relaying for the cases when data transmission phase can be longer than the handshaking the reliability of the outdated CSI is known and not known, phase, the correlation between the channel states used for link respectively. To evaluate the proposed protocols, we analyze selection and data detection may change over time. However, the error rate, delay, and throughput as relevant performance taking into account different correlation values during data transmission would make the delay analysis in this paper highly metrics. Similar to [20], we introduce a decision threshold β in the link selection criterion which can be tuned to ensure stable complicated. Hence, for convenience and to gain basic insight buffer operation. For both proposed link selection protocols, for system design, we assume a fixed correlation value between we provide a unified end–to–end error rate analysis and derive the channels used for link selection and the channels during data transmission1. Note that, the channel gains in different frames the corresponding diversity and coding gain expressions for 2 outdated and perfect CSI. Calculation of the error rate is not are assumed to be independent . affected by the silent time slots, which, however, influence A packet, consisting of data and pilot symbols, cf. Fig. 1, is the delay and throughput. We derive an expression for the transmitted from S to R at time ts and at a later time tr > ts, average delay of the system considering both the delay due the decoded packet is re–transmitted from R to D. Here, each to queueing at the buffer and the delay due to the silent time buffer element at the relay contains one packet. To facilitate slots caused by the proposed protocols. Moreover, we also the understanding of the analysis, in the sequel we refer to the show that the silent time slots constitute outage events and duration of each data transmission phase as ‘time slot’. In the are responsible for a loss in throughput. Interestingly, the three following, we present the CSI model, the assumptions regarding considered performance metrics are connected through and can the buffer operation, and the link selection schemes. be controlled via the decision threshold for link selection, β, and our analyses provide insight into this connection. We 1The assumption of a fixed correlation value is accurate for time division also provide guidelines for choosing β for achieving buffer multiple access (TDMA) based relaying systems, where a relay is shared by stability and for minimizing the error rate for a target delay several users for forwarding their information to the base station. The relay has separate buffers for all users to temporarily store the information. Without and throughput constraint. loss of generality, we can assume a block of one ‘pilot’ and one contiguous ‘data symbols’ in Fig. 1 comprises transmission of one user and one frame Notation: In this paper, E{·} and | · | denote statistical is composed of the handshaking phase and data transmission of all users in expectation and the magnitude of a scalar or the cardinality of a TDMA mode. The information of each user is transmitted according to the 2 1 R ∞ − u link selection decisions obtained at the beginning of transmission (i.e., during set, respectively. √ 2 is the Gaussian – Q(x) = 2π x e du Q the handshaking phase). In this set up, each user can be assumed to observe function and N (µ, σ2) denotes the Gaussian distribution with a fixed correlation value for the channel estimates used for link selection and 2 + data detection. However, correlation values used for different users may not be mean µ and variance σ . R denotes the set of positive real the same, of course. numbers, “r.h.s.” denotes the right hand side, “w.r.t.” stands for 2 . Similar assumptions in the context of relay selection for outdated CSI were with respect to, and “=” denotes asymptotic equivalence. made in [16]. 

       

   Ͳ   U  

    3     Link selection +  data symbols  data symbols  data symbols  pilots pilots feedback  of decision  pilots 

 handshaking phase  data transmission phase  

 one frame 

Fig. 1. Each frame consists of two phases: a handshaking phase and a data transmission phase.

A. CSI Model  and different transmission rates can be incorporated into the We denote the flat–fading complex Gaussian channel gain be- analysis. The received packets at the relay are appended to the  queue regardless of decoding errors3. We assume the queue tween nodes A and B at time tz as hAB(tz), AB ∈ {SR,RD}, operates in a First–In–First–Out (FIFO) mode4. z ∈ {s, r}. γAB(tz) refers to the instantaneous SNR of link  2 From queueing theory, we know that for stable operation of A → B at time tz, where γAB(tz) , PA|hAB(tz)| /N0, P and N denotes the transmit power of node A and the the buffer, the length of the queue must not grow over time, A 0  additive white Gaussian noise variance, respectively. For the i.e., we require the departure rate D to be equal to or higher rest of the paper, we drop the time index for convenience than the arrival rate A, i.e., D ≥ A. For maximization of  and it is implicitly assumed that transmission of a packet over the throughput, it was shown in [7] that the queue needs to the S → R link takes place earlier than the transmission of operate at the edge of stability, i.e., when A = D. However,  for error rate minimization, this condition may not be optimal, the same packet over the R → D link. γAB is modelled as as will be shown in Section IV. Furthermore, we can control an exponentially distributed random variable (RV) with mean 2 the arrival rate to limit the average delay. It was shown in [20] γ¯AB = PAσABγ¯, AB ∈ {SR,RD}, where γ¯ , 1/N0 and 2 2 that for fixed rate transmission, the average delay is infinite channel gain variance σAB , E{|hAB| }. when A = D holds. Hence, we need to operate the queue in The outdated channel gain used for selection, hˆAB, is mod- elled as [16]  the region D > A to achieve a tolerable delay. Now, taking q the above considerations into account, we propose two link ˆ − 2 (1) hAB = ρABhAB + 1 ρABwAB, selection protocols. Analytical expressions for A and D for where wAB is a circularly symmetric complex Gaussian RV finite and infinite buffer sizes are presented in Section IV, where having the same variance as hAB and ρAB is the correlation co- we study the end–to–end delay for the system. efficient between hˆAB and hAB, which, following Jakes’ auto– correlation model [21], is given by ρAB = J0(2πfd,ABTd). C. Link Selection Protocols Here, J0(·) denotes the zeroth order Bessel function of the For the development of the proposed link selection protocols, first kind, fd,AB denotes the Doppler shift on the A → B we assume the relay is equipped with a buffer of finite size. link, and Td is the time difference between the link selection Depending on whether the reliability of the CSI estimates ρAB, process and data transmission. ρSR and ρRD can have different AB ∈ {SR,RD}, is known or not, we formulate two link values since the relative speeds of the relay with respect to the selection protocols. Protocol 1 depends on the instantaneous source and the destination are not necessarily identical, which outdated SNR only, whereas Protocol 2 considers both the causes the Doppler shifts to be different. The estimated SNR instantaneous outdated SNR and the reliability of the estimates. 2 is given by γˆAB = PA|hˆAB| /N0. Although γˆAB and γAB 1) Protocol 1: For M–ary phase shift keying (PSK), M– are not necessarily identical, they follow the same distribution ary quadrature amplitude modulation, M–ary frequency shift with average value γ¯AB. The joint probability density function keying, and high values of SNR, the symbol error rate (SER) (PDF) of γAB and γˆAB is given by [16] of the A → B link can be expressed as [24] − x+y γ¯ 1−ρ2 √ √ AB ( AB )   e 2 xyρAB PB(e) = EγAB {CQ( ηγAB)}, (3) fγ ,γˆ (x, y) = I0 , AB AB γ¯2 (1 − ρ2 ) γ¯ (1 − ρ2 ) AB AB AB AB 3Allowing the relay to forward erroneous packets is delay efficient and is (2) not uncommon in the literature. For conventional DF relay networks, it was where I0(·) denotes the zeroth order modified Bessel function shown in [22], [23] that with intelligent combining at the destination and/or of the first kind. link adaptation, full diversity can still be achieved even if the relays forward erroneous packets. Note that it is also possible that the relay appends packets to the queue only if the decoding is successful. However, this assumption will B. Operation of the Buffer completely change the link selection procedure and delay analysis and is not considered here. As the buffering capability at the relay is critical, a sta- 4We note that the FIFO mode is appropriate if the packets do not have ble buffer operation has to be maintained throughout the individual delay requirements. If the packets do have individual delay require- link selection procedure. For simplicity, we assume that the ments, packets with more stringent delay requirements can be moved to the head of the queue to decrease their end–to–end delay. This does not affect the transmission rates of source and relay, denoted by S0 and error rate, delay, and throughput averaged over all packets, i.e., the analyses R0, respectively, are equal. However, this can be generalized presented in Sections III–V are still valid. 4

TABLE I LINK SELECTION PROTOCOLS.

Case Buffer status Link quality (Protocol 1) Link quality (Protocol 2) Selected link 1 Empty γˆSR < βγˆRD γˆSR < l1 for Case a None γˆRD ≥ l2 for Case b 2 Not empty γˆSR < βγˆRD γˆSR < l1 for Case a R → D γˆRD ≥ l2 for Case b 3 full γˆSR ≥ βγˆRD γˆSR ≥ l1 for Case a None γˆRD < l2 for Case b 4 Not full γˆSR ≥ βγˆRD γˆSR ≥ l1 for Case a S → R γˆRD < l2 for Case b where C and η are modulation dependent parameters. As P (e|γˆ ) is smaller than P (e|βγˆ ), i.e., √ R SR D RD Q( ηγAB) monotonically decreases with increasing γAB, ac- γˆ ρ2 βγˆ ρ2 − SR SR − RD RD 1 γ¯ (1−ρ2 ) 1 γ¯ (1−ρ2 ) tivating the link with the highest γAB minimizes the instan- e SR SR ≤ e RD RD , (7) taneous error rate. However, due to buffer stability issues, we µSR µRD cannot perform link selection based on instantaneous CSI only. otherwise the R → D link is selected. Note that in (7), we Instead, we select the S → R link when use β in such a way that the decision depends on a scaled version of γˆ , similar to Protocol 1. The appropriate choice γ ≥ βγ , (4) RD SR RD of β will be discussed in Section VI. Here, β = 1 means otherwise, the R → D link is selected. Parameter β ∈ R+ in that the link is selected based on the ‘actual’ conditional error (4) can be adjusted to balance the selection of both links to rates. However, as argued for Protocol 1, we choose the value ensure buffer stability. Furthermore, by tuning β, we can also of β such that buffer stability is maintained, and delay and/or guarantee certain average delay and throughput constraints as throughput constraints are satisfied. Even though (4) and (7) can will be discussed in Section VI. However, in practice, perfect be identical for some special cases, as will be shown below, in CSI may not be available, and we have to rely on outdated CSI most cases different values of β may be required for the two for link selection. Hence, for outdated CSI, the S → R link is protocols to observe a target end–to–end delay or throughput. selected when After some manipulations from (7), we obtain the criterion γˆ > βγˆ , (5) SR RD for selecting the S → R link as otherwise, the R → D link is selected. Furthermore, the buffer 2 2 γˆSRρSR βγˆRDρRD may be either empty or full at times. If an empty (full) buffer 2 + log(µSR) ≥ 2 + log(µRD). γ¯SR(1 − ρSR) γ¯RD(1 − ρRD) is encountered and the R → D (S → R) link is selected, no (8) transmission takes place and a silent time slot results. Here, We study (8) for the following two special cases: Case a) unlike the protocol in [20], we do not force the source or the µSR ≤ µRD and Case b) µSR > µRD. At high SNR, we relay to transmit during the silent time slots because in this 2 can further simplify the cases to: Case a) γ¯SR(1 − ρSR) ≤ case, transmission would take place over the weaker link and 2 2 2 γ¯RD(1 − ρRD) and Case b) γ¯SR(1 − ρSR) > γ¯RD(1 − ρRD). consequently, the error rate performance would degrade. We After some basic modifications, we summarize the condition summarize the proposed protocol in Table I. for selecting the S → R link as: Case a) 2) Protocol 2: Now, we develop a link selection policy that 2  2   takes into account the reliabilities of the CSI in both links, γ¯SR(1 − ρSR) βγˆRDρRD µRD γˆSR ≥ 2 2 + log , l1, i.e., ρSR and ρRD. In particular, we develop a link selection ρSR γ¯RD(1 − ρRD) µSR policy based on the error rates in both links conditioned on the (9) outdated CSI. The following Proposition provides an expression Case b) for the conditional error rate. γ¯ (1 − ρ2 )  γˆ ρ2  µ  γˆ < RD RD SR SR + log SR l , → RD 2 2 , 2 Proposition 1: The error rate of the A B link condi- βρRD γ¯SR(1 − ρSR) µRD tioned on γˆAB can be approximated as (10)

γˆ ρ2 − AB AB otherwise the R → D link is selected, cf. Table I where 1 γ¯ (1−ρ2 ) PB(e|γˆAB) ≈ e AB AB , (6) Protocol 2 is summarized. µAB 2 Remark 1: If the links have the same average SNR and the where µAB = . same reliabilities of the CSI estimates (symmetric links), i.e., r 2 γ¯AB η(1−ρAB ) 1 − 2 γ¯SR =γ ¯RD and ρSR = ρRD, we observe from (8) that the 2+¯γAB η(1−ρ ) AB selection rule simplifies to

Proof: Please refer to the Appendix A. γˆSR > βγˆRD, (11) Now, we are ready to introduce the link selection criterion. which is identical to Protocol 1 where only instantaneous Again using parameter β, we select the S → R link if outdated CSI was considered for selection. Note that (11) also 5

5 holds when {ρSR, ρRD} → 0 and the links have the same taken into account, and (12) is a tight approximation for high average SNR. SNR. Note that silent time slots do not contribute to the error Remark 2: When the reliability of the CSI estimates is high, rate, which is similar to the DF relaying scheme in [25]. In i.e., {ρSR, ρRD} → 1, the selection rule in (8) simplifies to (11) the following, we derive exact and asymptotic expressions for as well. In this case, the selection rule depends on instantaneous P (e) in (12) for both considered protocols. CSI only. A. Protocol 1 Remark 3: Note that β can be negative for Protocol 2 unlike for protocol 1. When the reliabilities of the CSI estimates are The end–to–end error rate for Protocol 1 is provided in the poor, {ρSR, ρRD} → 0, and γ¯SR < γ¯RD (cf. Case a, (9)) following proposition. holds, a negative β is required to balance the selection of the Proposition 2: For Protocol 1, the end–to–end error rate links. On the other hand, β ∈ + always holds for Case b in P (e) is given by (13) at the top of next page. For γ¯SR = aγ¯, R 2 2 Protocol 2. γ¯RD = bγ¯, γ¯ → ∞, where a , PSσSR and b , PRσRD, we Remark 4: For both protocols, we can observe from (5) and obtain the following asymptotic expression for P (e) (7)/(8) that an increase in β also increases the probability of ( 3C (a+bβ)(b+aβ) . 2 2 2 2 , if ρSR = ρRD = 1 selecting the R → D link over the S → R link. P (e) = 4η a b βγ¯ (14) ASR + ARD , if ρ < 1 or ρ < 1 Remark 5: Note that when β = 1, the rule in (7) of Protocol γ¯ γ¯ SR RD 2 only ensures that the link with the minimum error rate is where ASR selected at any time slot. β = 1 does not guarantee that the 2 2 C(a + bβ)(1 − ρSR) C(a + bβ)(1 − ρRD) end–to–end error rate will be minimized, unless the links are , ARD . , 2aη(bβ + a(1 − ρ2 )) , 2bη(a + bβ(1 − ρ2 )) symmetric. The dependence of the error rate on β will be SR RD (15) discussed in detail in Sections VI and VII. For independent and non–identically distributed (i.n.d.) fading with dissimilar Proof: Please refer to the Appendix B. correlation coefficients, the value of β required for Protocol 2 From the asymptotic expression for P (e) in (14), we can to achieve a target delay or throughput may be different from obtain the diversity and coding gains, which are provided in that for Protocol 1. In this work, we compare the performance the following corollary. of the two protocols for a fixed delay or throughput, see Section Corollary 1: The diversity gain Gd is given by VII. We note that Protocol 2 may not always yield a lower error  2, if ρSR = ρRD = 1 rate compared to Protocol 1 over the feasible range of β. Gd = (16) 1, if ρSR < 1 or ρRD < 1 Remark 6: From Table I, we observe that the proposed protocols do not allow for transmission during silent time slots, and the coding gain Gc is given by ( −1/2 which occur if the buffer is full (empty) and the S → R  3C (a+bβ)(b+aβ)  2 2 2 , if ρSR = ρRD = 1 (R → D) link is selected for transmission, such that trans- Gc = 4η a b β −1 mission over a poor link can be avoided. On the other hand, (ASR + ARD) , if ρSR < 1 or ρRD < 1 silent time slots degrade the throughput and the delay, as the (17) system is losing possible transmission opportunities. Hence, Proof: The expressions for Gd and Gc can be easily −G under certain conditions, it may be beneficial to modify the obtained by expressing P (e) in (14) in the form of (Gcγ¯) d . proposed protocols such that S (R) transmits if the buffer is empty (full) when the S → R (R → D) link SNR is larger Hence, we observe that for perfect CSI, P (e) decays at than some pre–defined threshold. In Section VII, we discuss the rate of γ¯−2, i.e., buffer–aided relaying with link selection this modification of the proposed protocols and simulate its achieves a diversity gain of two6 (which is the maximum performance. possible diversity gain as there are only two flat–fading links to choose from), whereas for outdated CSI, only a diversity gain of one is observed. This is because in the latter case the probability III.ERROR RATE ANALYSIS of selecting the worst link becomes non–negligible which leads In this section, we evaluate the end–to–end symbol error rate to a diversity gain loss. Note that conventional half–duplex DF (SER) of both proposed protocols. In our model, a symbol is relaying without link selection also achieves a diversity gain of received in error at D if a) the S → R link causes an error one [2]. However, it will be shown in Section VII that buffer– and the R → D link is error free, b) R receives the symbol aided relaying with adaptive link selection achieves a higher correctly but the R → D link causes an error, and c) both links coding gain compared to conventional DF relaying, even if the cause errors but the errors do not cancel each other. Thus, the CSI is outdated. end–to–end error rate can be formulated as B. Protocol 2 P˜(e) ≤ PR(e)(1 −PD(e)) + PD(e)(1 − PR(e)) +PR(e)PD(e) If the reliability of the CSI estimates is known, we can exploit this information in Protocol 2. As outlined in Section II-C2, we ≈ PR(e) + PD(e) , P (e), (12) 5 The term PR(e)PD(e) decays faster with increasing SNR than PR(e) and where PR(e) and PD(e) denote the SER of the S → R and PD(e), respectively. R → D links, respectively. Here, P (e) is an upper bound 6In terms of outage probability, a diversity gain of two was also observed because the case where the errors in both links cancel is not in [8] for fixed rate transmission. 6

1  q  q  2 − 2 γ¯SRη γ¯SRη 2 2ρSR γ¯SR 1 − − βγ¯RD + βγ¯RD 1 + + 2 2+¯γSRη 2+¯γSRη γ¯SRη γ¯ η(1−ρ )+βγ¯ η P (e) = C SR SR RD 2¯γSR 1  q  q  2 − 2 γ¯RD η γ¯RD η 2 2βρRD βγ¯RD 1 − − γ¯SR +γ ¯SR 1 + + 2 2+¯γRD η 2+¯γRD η γ¯RD η γ¯ η+βγ¯ η(1−ρ ) + C SR RD RD (13) 2βγ¯RD

   2 µ −1/ρSR 2 RD ρ2 (1−ρ2 )  1−ρRD µSR SR RD  2 2 2 2 2 −  C(1−ρ )(βρ (1−ρ )+ρ (1−ρ )) µRD µ (ρ2 +β(1−ρ2 )ρ2 )  SR RD SR SR RD SR SR SR RD  2 2 2 + C 2 , Case a  µSR(βρ +ρ (1−ρ )))  µ 1−1/ρ  RD SR RD RD SR ρ2 (1−ρ2 )  µ SR RD  1− SR  βρ2 +ρ2 (1−(1+β)ρ2 ) RD SR RD P (e) =   2 2 (18) µ (−1+ρRD )/βρRD 2 β SR ρ2 (1−ρ2 )  1−ρ µ RD SR  −1/β  SR − RD µSR 2 2 2 2  µ µ (ρ2 +(β−ρ2 )ρ2 ) C (1−ρRD )(βρRD +ρSR(1−(1+β)ρRD ))  SR SR SR SR RD µRD  C 2 2 + 2 2 2 , Case b   µ (−1+ρ )/βρ µRD (βρ +ρ (1−βρ ))  β SR RD RD ρ2 (1−ρ2 ) RD SR RD  µ RD SR  1− RD  βρ2 +ρ2 (1−(1+β)ρ2 ) RD SR RD

2 study two possible cases: Case a) γ¯SR(1 − ρSR) ≤ γ¯RD(1 − cases. We note that for both cases in Protocol 2, perfect CSI 2 2 2 ρRD) and Case b) γ¯SR(1−ρSR) > γ¯RD(1−ρRD), and derive yields the same error rate as Protocol 1, as expected, and a the error rate expressions for each case separately7 diversity gain of two. Note that for perfect CSI, the conditional Proposition 3: For Protocol 2, the end–to–end error rate error rate of the links is simply given by a Q–function, as P (e) is given by (18). For γ¯SR = aγ¯, γ¯RD = bγ¯, γ¯ → ∞, we shown in (3), and, the rule in (4) holds. obtain an asymptotic expression for P (e) in (18) as

 3C (a+bβ)(b+aβ) IV. DELAY ANALYSIS 2 2 2 2 , if ρ = ρ = 1  4η a b βγ¯ SR RD  A1,SR A1,RD In this section, we derive a closed–form expression for the  + , if ρSR < 1 , ρRD < 1, Case a  γ¯ γ¯ average delay of the considered buffer–aided relaying scheme, . A1,RD P (e) = γ¯ , if ρSR → 1 , ρRD < 1, Case a which is valid for both protocols. In Fig. 2, we show the state  A2,SR A2,RD  γ¯ + γ¯ , if ρSR < 1 , ρRD < 1, Case b transition diagram for the queue operation at the relay for a  A  2,SR buffer of size L. PRD and PSR denote the probabilities of γ¯ , if ρSR < 1 , ρRD → 1, Case b → → (19) selecting the R D and the S R links, respectively, and PRD = 1−PSR holds. For convenience, we assume S0 = R0 = where A1,SR 1 in this section. If the buffer is empty (full), the source (relay) 2 2 2 2 does not transmit, unlike the operation in [20]. The arrival and C(1 − ρSR)(βρRD + ρSR(1 − (1 + β)ρRD)) , 2 2 2 2 , (20) departure rates are given by A = (1 − PRD)(1 − Pfull) and 2bη(1 − ρRD)(βρRD + ρSR(1 − ρRD)) D = PRD(1 − Pempty), respectively, where Pfull and Pempty −1/ρ2  b(1−ρ2 )  SR ρ2 RD denote the probabilities of full and empty buffer, respectively. SR a(1−ρ2 ) 1 SR 2ηb − 2ηa(ρ2 +β(1−ρ2 )ρ2 ) The expressions for PSR, PRD, Pfull, and Pempty will be A SR SR RD (21) 1,RD , C 1−1/ρ2 , provided later.  b(1−ρ2 )  SR ρ2 (1−ρ2 ) RD SR RD a(1−ρ2 ) SR In [8] and [20], the average delay of link selection schemes 1 − 2 2 2 βρRD +ρSR(1−(1+β)ρRD ) for fixed–rate transmission was defined as the average time (−1+ρ2 )/βρ2  a(1−ρ2 )  RD RD βρ2 SR it takes for a symbol to be received at the destination, once RD b(1−ρ2 ) 1 RD the symbol was transmitted by the source. For the proposed 2aη − 2aη(ρ2 +(β−ρ2 )ρ2 ) A SR SR RD (22) 2,SR , C (−1+ρ2 )/βρ2 , protocols, we may observe silent time slots depending on the  a(1−ρ2 )  RD RD β(1−ρ2 )ρ2 SR SR RD b(1−ρ2 ) status of the buffer. The silent time slots, when no successful RD 1 − 2 2 2 βρRD +ρSR(1−(1+β)ρRD ) transmission occurs, also contribute to the end–to–end delay. These events account for the cases when the R → D link is and A 2,RD , chosen while the buffer is empty, or the S → R link is chosen  2 −1/β when the buffer is full, so no successful transmission takes a(1−ρSR) 2 2 2 2 (βρRD + ρSR(1 − (1 + β)ρRD)) b(1−ρRD ) place. In this work, we consider for the delay the time it takes C 2 2 2 . (23) 2ηb(βρRD + ρSR(1 − βρRD)) for a symbol that awaits transmission at the source until it is Proof: Please refer to the Appendix C. received at the destination. Hence, we take into account both The error rate expressions for both Cases a and b are in the queueing delay and the delay due to silent time slots to closed form and can be readily evaluated. Following Corollary compute the average system delay T . 1, we can easily obtain the diversity and coding gains for both A. Queueing Delay 7From (9), we observe that the log term is positive (negative) for Case 1 In the following proposition, we derive the queueing delay (Case 2). Hence, the integration regions for γˆSR and γˆRD are quite different for the two cases. TQ for the considered buffer operation. 7

P P P P SR SR SR SR P RD P G G G G G G SR 0 1 2 … L-2 L-1 L

P P P P RD RD RD RD Fig. 2. State transition diagram for queue operation.

Proposition 4: For the considered protocols, the average do not force the source (relay) to transmit if the buffer is queueing delay for a buffer of size L is given by empty (full), which in turn results in silent time slots, i.e., no transmission takes place. Hence, the number of silent time slots L(1 − P )L(2P − 1) + P ((1 − P )L − P L ) RD RD RD RD RD increases as and grow. There are two cases when TQ = L L . Pempty Pfull PRD(2PRD − 1)((1 − PRD) − PRD) (24) a silent time slot is observed: Case 1) the buffer is full and the → link is selected, and Case 2) the buffer is empty and for L → ∞, S R 1 and the R → D link is selected. Note that Case 1 contributes TQ = , (25) 2PRD − 1 to the queueing delay, however Case 2 does not. Hence, the delay incurred for Case 2 has to be computed separately. In where P is the probability of selecting the R → D link. RD the following proposition, we provide an expression for the Proof: Following the Appendix of [20], we can easily average delay due to observing silent time slots T for Case 2. derive the steady state probability of the buffer being in state S Proposition 5: For the considered protocols, the average Gi, denoted by PGi , i ∈ {0,...,L}, cf. Fig. 2. The average delay due to observing silent time slots for Case 2, i.e., the queueing delay TQ of the queue is given by buffer is empty and the R → D link is selected, is given by -1 E{ } -1 Q BICM OFDM L TQ = , (26) (2PRD − 1)PRD A TS = (29) (1 − P )(P L − (1 − P )L) where the average queue size, E{Q} = P iP , after some RD RD RD i Gi Buffer manipulations, is obtained as E{Q} and for L → ∞, 2PRD − 1 TS = . (30) 1 − P 1 − P P L+1 − (1 − P )L(L(2P − 1) + P ) RD = RD RD RD RD RD , L+1 L+1 Proof: For the considered queue operation, Pempty is given 2PRD − 1 PRD − (1 − PRD) (27) by L PRD(2PRD − 1) and A = (1 − P )(1 − P ), where P = P denotes the Pempty = PG0 = L+1 , (31) RD full full GL P − (1 − P )L+1 probability of observing a full buffer and is given by RD RD L and for L → ∞, we obtain (2PRD − 1)(1 − PRD) 1 Pfull = . (28) Pempty = 2 − . (32) L+1 L+1 PRD PRD − (1 − PRD) 1 + We use Pfull to calculate A, which we substitute, along with For PRD → 2 and L → ∞, we note that Pempty → 0. E{Q} from (27), into (26) to obtain the expression shown in This is expected as both the links are selected with equal (24). For L → ∞, Pfull → 0 holds and we obtain the average probability and the buffer almost always contains packets in − delay as shown in (25). steady state. On the other hand, if PRD → 1 , the buffer Remark 7: From (25), we observe that as PRD → 1 (i.e., the would be empty most of the time as the departure rate is departure rate is very high), TQ approaches the minimum value very high, which in turn results in a large number of unused of one, i.e., on average a packet experiences a queueing delay time slots. We consider a sequence of the buffer statuses 1 + { }, where we want to of one time slot only. On the other hand, when PRD → 2 empty, empty,..., empty, not empty (i.e., when PRD → PSR), a large queueing delay results. This calculate the average number of time slots the buffer remains is because departure rate and arrival rate are nearly equal which empty before R receives a packet. Here, the number of time causes a packet to be held in the buffer for a large number of slots the buffer remains empty before S transmits follows a time slots. In general, for infinite buffer size, we need PRD ≥ geometric distribution with mean value Pempty/(1 − Pempty) 8 PSR to ensure that the buffer is stable . [26].

Plugging in the values for PG0 = Pempty from (31), we B. Silent Time Slots obtain the expression for TS in (29). For L → ∞, TS in (30) Note that for finite buffer size, the probability that the buffer is easily obtained after plugging in Pempty from (32). 1 + is full or empty is not zero. According to our protocol, we Remark 8: For PRD → 2 , we note that TS → 0, as expected. On the other hand, for PRD > PSR, TS increases 8 For S0 6= R0, we require R0PRD > S0PSR to ensure that the buffer is as the probability of observing silent time slots increases. Note stable. Furthermore, the arrival and departure rates are given by A = S (1 − 0 that when L → ∞, silent time slots only occur when the buffer PRD)(1 − Pfull) and D = R0PRD(1 − Pempty), respectively, and the state P S R 1 probabilities Gi depend on 0 and 0 as well. is empty, as Pfull = 0 for any PRD > 2 . 8

Corollary 2: For a buffer of size L, the average system average queue size remains unchanged. As the arrival rate is 9 delay T in time slots is given by T = TQ + TS also unaffected by this, the average delay expressions presented in this section are still valid, but the delivery of individual L(1 − P )L(2P − 1) + P ((1 − P )L − P L ) RD RD RD RD RD packets will be faster or slower due to the reordering, of course. = L L PRD(2PRD − 1)((1 − PRD) − PRD) (2P − 1)P L + RD RD (33) V. THROUGHPUT ANALYSIS − L − − L (1 PRD)(PRD (1 PRD) ) In this section, we investigate the achievable throughput for and for L → ∞, T = TQ + TS the proposed link selection protocols. It is obvious that we achieve the maximum throughput when silent time slots are 1 2P − 1 2 − 5P + 4P 2 = + RD = RD RD . (34) avoided. This can be achieved when both links are selected with 2PRD − 1 1 − PRD (1 − PRD)(2PRD − 1) equal probability. We denote this throughput as τ0. However, Proof: By combining the expressions for TQ and TS from in practice, we may want to choose β such that the R → D Propositions 4 and 5, we can easily obtain T as shown in (33) link is selected more often than the S → R link in order to and (34). achieve a finite delay, which in turn results in some silent time To calculate the delay in (33) and (34), we obtain appro- slots. These silent time slots constitute outage events11 which priate values for PRD according to the chosen protocol in the lower the system throughout. The throughput in the presence following corollary. of silent time slots is denoted as τ and is given by Corollary 3: For Protocol 1, PRD is given by τ = τ0(1 − Fsilent), (38) βγ¯RD PRD = , (35) where Fsilent denotes the probability of observing a silent (¯γSR + βγ¯RD) time lot. For finite buffer size, we have Fsilent = and for Protocol 2, P is given by RD PemptyPRD + PfullPSR, whereas for infinite buffer size, 2   µ 1−1/ρSR . Recall from Section IV-B that RD ρ2 (1−ρ2 ) Fsilent = PemptyPRD A =  µSR SR RD  1 − 2 2 2 , Case a PSR(1−Pfull)R0 and D = PRD(1−Pempty)R0. The maximum βρRD +ρSR(1−(1+β)ρRD ) PRD = 2 2 (36) 12  µ (−1+ρ )/βρ throughput τ = R /2 can be achieved for infinite buffer size β SR RD RD (1−ρ2 )ρ2 0 0  µRD SR RD  2 2 2 , Case b if there are no silent time slots and both links are selected with βρRD +ρSR(1−(1+β)ρRD ) equal probability, i.e., PRD = PSR. Below, we calculate the Proof: For Protocol 1, we calculate PRD = 1−PSR in (35) throughput for the two proposed protocols for both finite and by using (74). For protocol 2, Case a, PRD = Pr(ˆγSR < l1) is infinite buffer sizes. obtained from (82). Following a similar approach as for Case For a buffer of finite size, we obtain from (28), (31), (38) a (cf. Appendix C), the expression for PRD = Pr(ˆγRD > l2) shown in (36) can be obtained for Case b. τ = τ0(1 − PemptyPRD − PfullPSR) L L Combining (33) and (34) with (35) and (36), we obtain 2(1 − PRD)PRD(PRD − (1 − PRD) ) = τ0 (39) closed–form expressions for the average delays for both finite L L+1 L (1 − PRD) PRD + P − (1 − PRD) and infinite buffer sizes for both protocols. For example, for RD Protocol 1, we obtain a simple expression for the average delay where we use PSR = 1 − PRD. Note that τ = A holds as well for infinite buffer size as and (39) can also be obtained from 2 2 2 2¯γ − βγ¯ γ¯ + β γ¯ τ = R0(1 − PRD)(1 − Pfull) T = − SR SR RD RD , (37) 2  L  γ¯SR − βγ¯SRγ¯RD (2PRD − 1)(1 − PRD) = 2τ0(1 − PRD) 1 − . (40) L+1 L+1 where we use (34) and PRD from (35). PRD − (1 − PRD) In Section VI, we discuss the choice of β and how delay For a buffer of infinite size, we obtain from (32), (38), (39), depends on P and β for both protocols and also focus on RD 1 optimization of the error rate for a target delay constraint. τ = τ0(1 − PemptyPRD) = 2τ0(1 − PRD), when PRD ≥ , 2 Remark 9: As PSR and PRD are not affected by outdated (41) 10 CSI in Protocol 1, the average delay in (37) holds for all or, τ = A = R0PSR = 2τ0(1 − PRD), as Pfull = 0 when values of ρSR and ρRD. However, the delay for Protocol 2 L → ∞. depends on ρAB, AB ∈ {SR,RD}, which stems from the fact Remark 11: The maximum achievable throughput for finite that the link selection probabilities PSR = 1 − PRD and PRD buffer size is less than that for infinite buffer size. We observe are functions of ρAB, cf. (36). that for PRD = 1/2, we have τ = τ0 (Pempty = 0 holds when Remark 10: We note that reordering the queue at the buffer PRD = 1/2) in (41), whereas PemptyPRD + PfullPSR in (39) to give priority to individual packets does not affect the average is never zero for any value of PRD. This also relates to the delay. In particular, if the queue is reordered, the total number fact that we strictly need PRD ≥ 1/2 for infinite buffer size to of packets remaining in the queue does not change and the ensure stability, whereas finite size buffers are never unstable.

9Note that we assume that the queueing at the buffer and silent time slots 11Here, an outage event corresponds to no transmission, i.e., an unused time are the dominant sources of packet delay for the considered protocol, and other slot. 12 forms of delay such as transmission delay are negligible. We note that for S0 6= R0, the maximum throughput is given by τ0 = 10 Recall that the actual and the outdated CSI follow the same distribution. S0R0/(S0 + R0). 9

   L  L βγ¯RD βγ¯RD βγ¯RD 2βγ¯RD 1 − − 1 − γ¯SR+βγ¯RD γ¯SR+βγ¯RD γ¯SR+βγ¯RD

τ = τ0 L , (42)  βγ¯RD  !  L+1  L βγ¯RD 1− βγ¯RD βγ¯RD γ¯SR+βγ¯RD (¯γSR + βγ¯RD) − 1 − + γ¯SR+βγ¯RD γ¯SR+βγ¯RD γ¯SR+βγ¯RD

From (41), we observe that throughput monotonically de- As a special case, we obtain the achievable throughput for creases as PRD increases from 1/2 to 1. From Remark 4, we perfect CSI and infinite buffer size by replacing ρAB = 1 in can intuitively observe that throughput decreases if β increases (44) in the region 1 . Below, we discuss the achievable 2¯γSR PRD > 2 τ = τ0 . (46) throughput for the two proposed protocols and also compute γ¯SR + βγ¯RD the decision threshold β to achieve maximum throughput τ0. As expected, the throughput for perfect CSI is identical to (43), For Protocol 2 and finite buffer size, we do not show the final i.e., the throughput of Protocol 1. This is because both actual closed–form throughput for brevity, but the expression can be and outdated CSI follow the same distribution. easily obtained using (39) and substituting PRD. We provide the expressions for infinite buffer sizes for both cases. VI.TRADE–OFF AND CHOICESFOR β The parameter β has to be properly chosen to maintain A. Protocol 1 buffer stability and to obtain a trade–off among three relevant In the following corollary, we obtain end–to–end average performance metrics: error rate, delay, and throughput. In this throughput for Protocol 1. section, we study the dependence of the error rate, delay, and Corollary 4: For Protocol 1, throughput τ for buffer size throughput on β in detail 13. First, we discuss the convexity L is given by (42) and for L → ∞ and/or monotonicity of the performance metrics w.r.t. β, which will be useful later when we optimize the error rate by tuning 2¯γSR τ = τ0 . (43) β under delay and throughput constraints. γ¯SR + βγ¯RD Proof: Using (39), (41), and P from (35), we obtain RD A. SER vs. β τ as shown in (42) and (43). We need β ≥ γ¯SR/γ¯RD in (43) to ensure buffer stability and β =γ ¯ /γ¯ yields maximum For both protocols, P (e) is in general not convex in β, SR RD 14 except for ρSR = ρRD = 1 and high SNR . The value of throughput τ0. For finite buffer size, the value of β, which β which yields minimum SER is not necessarily the same for minimizes PemptyPRD + PfullPSR, also maximizes τ in (42) and is obtained numerically. both protocols. Below, we discuss special cases when β = 1 results in the minimum SER. B. Protocol 2 Corollary 6: For perfect CSI, i.e., when ρAB = 1, β = 1 results in minimum SER at high SNR. For Protocol 2, the end–to–end throughput for infinite buffer Proof: For γ¯ = aγ¯, γ¯ = bγ¯, γ¯ → ∞, where a and size is provided in the following corollary. SR RD b are positive constants, the asymptotic expression for P (e) Corollary 5: For Protocol 2, throughput τ for L → ∞ and when ρ = 1 is provided in (14). Taking the first derivative P ≥ 1/2 is given by τ = AB RD of P (e) w.r.t. β yields   1−1/ρ2 ! µRD SR ρ2 (1−ρ2 ) 2
 µSR SR RD dP (e) 3C β − 1 2τ0 βρ2 +ρ2 (1−(1+β)ρ2 ) , Case a = . (47)  RD SR RD dβ 4abβ2γ¯2η2 2 2 (44)  µ (−1+ρ )/βρ ! β SR RD RD (1−ρ2 )ρ2 dP (e)  µRD SR RD Solving = 0, we obtain β = {1, −1}. As β = −1 is not 2τ0 1 − βρ2 +ρ2 (1−(1+β)ρ2 ) , Case b dβ  RD SR RD feasible, β = 1 holds. Proof: Combining (36) and (41), we obtain the expressions Corollary 7: For symmetric links, i.e., ρSR = ρRD = ρ of τ for both cases, as shown in (44). From (36), we solve and γ¯SR =γ ¯RD, β = 1 results in minimum SER for both protocols. PRD = 1/2 for β for Case a and obtain β = l3 which yields Proof: For γ¯SR =γ ¯RD =γ ¯, and ρSR = ρRD = ρ, i.e., τ = τ0, where l3 is given by µSR = µRD = µ, the expressions for P (e) in (18) are identical  1−1/ρ2   µ  SR for both cases. Taking the first derivative of of Case a in 2 RD − 1 ρ2 (1 − ρ2 ) P (e) µSR SR RD (18) w.r.t. β yields l3 , 2 2 . (45) ρRD(1 − ρSR) dP (e) (β2 − 1)ρ4(2 − ρ2)(1 − ρ2)2 = 2 2 2 2 . (48) Here, β ≥ l3 ensures a stable buffer operation. For Case b, PRD dβ µ(1 + β − ρ ) (−1 + β(−1 + ρ )) in (36) is non–linear in β. Hence, β is obtained numerically 13 for P ≥ 1/2. The choice of β is also affected by the transmission rates S0 and R0, if RD they are unequal. As mentioned in Section II, in this paper, we assume identical Remark 12: From (42)–(44), we observe that for Protocol transmission rates and focus on how the optimal β is affected by the channel 1, the throughput does not depend on the reliability of the CSI fading statistics. 2 2 14 ρ = ρ = 1 P = 3C ( a +b + 1 + estimates, ρAB, AB ∈ {SR,RD}. However, for Protocol 2, it From (14), for SR RD we obtain e 4η2γ¯2 ab βab β + does. ab ), where the sum inside the parenthesis is convex in β ∈ R . 10

dP (e) Solving dβ = 0, we obtain β = {1, −1}. As β = −1 is holds. not feasible, β = 1 holds. Using (13), similar results can be Next, we study the delay as a function of β. It is difficult, obtained for Protocol 1. if not impossible, to analytically prove the convexity or quasi– We can intuitively justify Corollary 7 as follows: The link convexity of T ( cf. (33)) in β and obtain a closed–form solution selection criterion for symmetric links is given by (11). β = 1 for β for any buffer size L. However for some special cases, means that the link will be selected based on the instantaneous we can prove the convexity of the delay for the two protocols, CSI only, i.e., the link is selected based on max(ˆγSR, γˆRD). as discussed in the following propositions. We also calculate Hence, given that the reliabilities of the CSI estimates of the the value of β which yields minimum delay. For Protocol 2, two links are identical, this rule implies that selecting the we only show results for Case a because in Case b, PRD is strongest link in terms of the outdated instantaneous SNR is highly non–linear in β, and it is difficult to prove convexity of optimal. the delay in β or solve for β analytically to achieve a target For other cases, we find the β yielding minimum SER for the delay. two protocols numerically. For a particular set of γ¯AB and ρAB, ∀ , the value of which provides minimum SER can be Proposition 7: For infinite buffer size, delay T is convex AB β 2d−c different for the two protocols. However, note that the value of in β for β > γ¯SR/γ¯RD in Protocol 1 and for β > g in β, which achieves minimum SER, may not always be a feasible Protocol 2 Case a, where {c, d, g} are positive constants and choice because it may incur buffer instability and entail a large defined as 1− 1 delay and/or low throughput.   ρ2 2 2 µRD SR 2 2 c , ρSR(1 − ρRD), d , ρSR(1 − ρRD), µSR B. Delay vs. β 2 2 g , ρRD(1 − ρSR). (53) In practice, there may be a limit on the tolerable system √ + delay. Unless mentioned otherwise, we assume infinite buffer Furthermore,√ β = (1 + 2)ξ, where ξ , γ¯SR/γ¯RD ∈ R and (2+ 2)d−c √ size in this section and provide some insight on the delay β = g results in a minimum delay of Tmin = 1+2 2 performance of the considered protocols. To ensure buffer time slots for Protocol 1 and Protocol 2 Case a, respectively. stability, we need to operate in the region PRD ≥ 1/2, which shrinks the search space for the optimal β minimizing the Proof: From (37), we obtain SER. However, from Section IV, we know that PRD = 1/2 β 2ξ introduces infinite delay, because both links are selected with T = + , (54) equal probability. Below, we examine the behaviour of the ξ −ξ + β average delay as a function of β in the region PRD > 1/2. where the r.h.s. of (54) is convex15 in β only if β > ξ. On the The following proposition holds for both protocols. other hand, PRD for Protocol 2 Case a is given by (36), which Proposition 6: For infinite buffer size, the delay T is can be simplified as convex in P√RD for PRD > 1/2. A minimum delay of 1 d time slots can be observed when √ . Tmin = 1 + 2 2 PRD = 2 PRD = 1 − , (55) Proof: From (34), we obtain c + gβ 2 where {c, d, g} are defined above. Now, delay T in (34) can dT 2PRD − 1 = 2 2 , (49) be expressed as dPRD (2PRD − 1) (1 − PRD) 2 c − d gβ 2d d T 2 8 T = + + , (56) 2 = 3 + 3 , (50) d 2d −2d + c + gβ dPRD (1 − PRD) (2PRD − 1) d2T 1 where, using a similar argument as for Protocol 1, the r.h.s. of where ≥ , when ≥ . Hence, is convex in 2d−c dP 2 0 PRD 2 T (56) is convex in β only if β > . Furthermore, calculating RD g √ PRD, when the R → D link is selected more often than the ∗ dT/dβ = 0 from√ (54) and (56), we obtain β = (1 + 2)ξ S → R link. Furthermore,√ solving for dT/dPRD =√ 0, we ∗ (2+ 2)d−c and β = g for Protocol 1 and Protocol 2 Case a, obtain PRD = 1/ 2 and this results in Tmin = 1 + 2 2. 2 2 respectively, which satisfy the condition d T/dβ√ |β=β∗ > 0. Furthermore, to obtain a target delay T , we solve for PRD ∗ in (34) Plugging in β into T , we obtain Tmin = 1 + 2 2 time slots. P = {P ,P } RD RD,1 RD,2 Note that both β > ξ and β > 2d−c in Proposition 7 ( √ √ ) g 5 + 3T − T 2 − 2T − 7 5 + 3T + T 2 − 2T − 7 correspond to P > 1/2. Recall from Section II that P = , . RD RD 4(T + 2) 4(T + 2) monotonically increases with β. Exploiting Proposition 6, we (51) can calculate the value of β corresponding to PRD,1 and √ PRD,2 to achieve a target delay T for both protocols by using Note that PRD,1 ≤ 1/ 2 ≤ PRD,2 holds and both values appropriate expressions of PRD as function of β (cf. Corollary of PRD are valid since both {PRD,1,PRD,2} ensure buffer 3 where PRD is shown as a function of β). We refer to them stability, i.e., they are larger than 1/2 because in (51) √ 5 + 3T − T 2 − 2T − 7 1 lim = (52) T →∞ 4(T + 2) 2 15Note that r/(s + x) is convex in x for r > 0 and x ∈ (−s, ∞). 11

as β1 and β2. For Protocol 1, we obtain β = For other values of L, we obtain β numerically to achieve a n ξ p ξ p o target delay. As PRD is a non–linear function of β for Protocol (T + 1+ −7− 2T + T 2), (T +1 − −7 − 2T + T 2) , 2 and Case b, cf. (36), it is difficult to obtain a closed–form 2 2 | {z } | {z } expression for β. We exploit the monotonicity of PRD in β, cf. β1 β2 Remark 4, and we calculate β and β numerically for a target (57) 1 2 delay T , by using (36) and (51), for Case b. p 2 where β1, β2 > ξ, as T > −7 − 2T + T . (58) C. Throughput vs. β Similarly, for Protocol 2 Case a, we obtain β = From (41) in Section V, we observe throughput τ decreases 1 ( √ monotonically with PRD, as expected. Note that PRD = 2 −c + d + d(T − −3 − 2T + T 2) − , yields the maximum throughput τ0, and as PRD → 1 , g τ approaches zero. Furthermore, considering Remark 4, we | {z } β1 conclude that τ monotonically decreases with β. √ ) The following Corollaries provide some insight, regrading −c + d + d(T + −3 − 2T + T 2) , (59) the achievable throughput if the end–to–end delay is minimized. g Corollary 8: For infinite buffer size and T = T = | {z } √ √ min β2 1 + 2 2, the achievable throughput is τ = (2 − 2)τ0. where √ Proof: We observed from Proposition 6 that P√RD = 1/ 2 2d − c p 2 β1, β2 > , as T > −3 − 2T + T . (60) yields the minimum delay. Plugging P = 1/ 2 into τ = g RD √ τ0(2−2PRD) (cf. (41)), we obtain τ = (2− 2)τ0 = 0.5858τ0. Remark 13: Both β1 and β2 in (58) and (60) ensure stable ∗ buffer operation and β1 ≤ β ≤ β2 holds. The intuitive reason Corollary 9: For L = 1 and T = Tmin = 3, the achievable for having two solutions for the same delay is because as throughput is τ = τ0/2. 1 + + 2d−c PRD → 2 , i.e., β → ξ for Protocol 1 or β → g for − Protocol 2 Case a (PRD → 1 , i.e., β  ξ (Protocol 1) Proof: We observed from Proposition 8 that PRD = 1/2 2d−c or β  g for Protocol 2 Case a, TQ increases and TS yields the minimum delay for L = 1. From (39), we obtain decreases (and vice–versa). This inverse behaviour of the two τ = 2τ0(1 − PRD)PRD for L = 1. Plugging in PRD = 1/2, delay components with respect to β also explains why an aver-√ we obtain τ = τ0/2. age system delay below a certain value, i.e., Tmin = 1 + 2 2 The values of β corresponding to Corollaries 8 and 9, are in our case, is not possible. derived in Propositions 7 and 8 for Protocol 1 and Protocol Next, in the following Proposition, we focus on L = 1 as a 2 Case a, respectively. For Protocol 2 Case b, we calculate β special of the finite buffer size. numerically, as indicated before. + Proposition 8: For L = 1, the delay is convex in β ∈ R Remark 15: From Corollary 9, we observe that for the d−c for Protocol 1 and in β > g for Protocol 2 Case a.A proposed protocols, the maximum achievable throughput for minimum delay of Tmin = 3 time slots is achieved when L = 1 is half the value that can be obtained if a buffer of 1 ∗ ∗ 2d−c PRD = 2 or β = ξ (Protocol 1) and β = g . infinite size is employed. In other words, as the buffer size increases, we expect an increase in the throughput. Proof: Substituting L = 1 in (33), we obtain 1 P β ξ D. Optimization of SER T = + RD = + + 1, (61) PRD 1 − PRD ξ β In practice, there may be constraints on the minimum achiev- able throughput and the maximum end–to–end delay. Here, we where P = β is used, cf. Corollary 3. Clearly, T is convex RD ξ+β formulate an optimization problem for β and our objective is in β ∈ R+. Furthermore, solving dT/dβ = 0, we obtain β = 2 2 2 to minimize the SER for a maximum allowable delay Tmax and {−ξ, ξ}. Discarding β = −ξ, we get d T/dβ |β=ξ = 2 > 0. ξ a minimum required throughput τmin, i.e., ∗ 1 Hence, β = ξ or PRD = 2 yields the minimum T . After plugging this value into (61), we obtain . Similarly, min P (e) (63) Tmin = 3 β for Protocol 2 Case a, we obtain  T ≤ Tmax c gβ d s.t. , T = + + , (62) τ ≥ τmin d d c − d + gβ We cannot adopt tools from convex optimization here, as d−c where T is convex in β if β > g . Following a similar ap- P (e) is not convex in β in general. However, we perform ∗ 2d−c proach as adopted for Protocol 1, we can show that β = g a simple one–dimensional search over β within the range yields the minimum delay of Tmin = 3. specified by the delay and throughput constraints, and minimize Remark 14: In views of Proposition 7 and 8, we conjecture P (e) over this range. Let {β1, β2} yield delay T = Tmax, that√ the minimum achievable delay increases from three to 1 + where β2 > β1 holds, cf. (58), (60). We assume β = β3 2 2, as the buffer size increases from one to ∞. We validate results in τ = τmin, cf. (42)–(44) and β = β4 corresponds this claim in Section VII. to PRD = 1/2, i.e., τ = τ0. Then, we search in the interval 12

ρ =ρ =1, ξ = 1, P1 SR RD 0 10 ρ =ρ =0.99, ξ = 1, P1 SR RD −1 ρ =ρ =0.50, ξ = 1, P1 10 SR RD

−1 ρ = 0.3, ρ =0.9, ξ = 2, P1 10 SR RD ρ = 0.3, ρ =0.9, ξ = 2, P2 G = 1 SR RD d Conv. DF, ξ = 1 −2 10 −2 10 BER −3 BER 10 Proposed Scheme, T = 4 Simulation Scheme 2, T = 4 Analysis (Asymptotic) Scheme 1, T = 4 −4 −3 10 Analysis (Approx.) 10 Scheme 2, T = 6 Scheme 1, T = 6 Proposed Scheme, T = 6 G = 2 −5 10 d

−4 10 0 5 10 15 20 25 30 0 2 4 6 8 10 12 14 16 18 20 22 γ¯, dB γ¯ (dB)

Fig. 3. BER vs. SNR (γ¯) performance without delay constraints. P1 and P2 Fig. 4. BER vs. SNR (γ¯) performance with delay constraints. refers to Protocol 1 and Protocol 2, respectively. A. BER vs. SNR max{β4, β1} ≤ β ≤ min{β2, β3}. Hence, we transform the problem as In Fig. 3, the BER vs. SNR (γ¯) is shown when link selection is performed without any delay constraints, i.e., when P = min P (e), (64) RD βl≤β≤βu 1/2 and maximum throughput τ0 is observed. We consider two values of ξ =γ ¯ /γ¯ ∈ {1, 2} in Fig. 3. Here, β is chosen where β max{β , β } and β min{β , β }. If β < β , SR RD l , 4 1 u , 2 3 3 1 such that P = 1/2, cf. Section VI. A wide range of values then the delay and throughput constraints cannot be simulta- RD for {ρ , ρ } are considered. We also show the analytical neously satisfied. This could happen when only a very small SR RD results along with simulation results to corroborate the claims delay can be afforded ({β , β } → β∗) while a very high 1 2 made in Section III. In Fig. 3, “approx.” and “asymptotic” refer throughput is required (β → β ). For infinite buffer size 3 4 to (13) and (14) for Protocol 1 and (18) and (19) for Protocol β = β holds as β > β is valid. However, for finite buffer l 1 1 4 2. For perfect CSI, we observe a diversity gain of two, as size, we may have β < β . Recall from Proposition 8 and 1 4 predicted in Section III. When the CSI is outdated, a loss in Corollary 9 that for L = 1, the value of β which yields diversity occurs, and all BER curves for ρ , ρ < 1 exhibit minimum delay also provides maximum throughput. In view SR RD a diversity gain of one at high SNR, similar to conventional of the discussions above and Section V, we can obtain β and l (conv.) DF relaying, where the relay receives and transmits β analytically for some cases (e.g., infinite buffer size and u in consecutive time slots. A higher coding gain compared Protocol 1 or Protocol 2 Case a), in other cases, we have to to conventional DF relaying is observed for perfect CSI and obtain them numerically. In Section VII, we show an example outdated CSI, especially when the actual and outdated CSI are for the considered optimization problem. strongly correlated. We compare Protocol 1 with Protocol 2 for the case when the ρSR and ρRD are dissimilar and differ by a VII.SIMULATION RESULTS large margin, e.g., ρSR = 0.3, ρRD = 0.9. Protocol 2 exploits In this section, we illustrate the performance of the link se- this dissimilarity by incorporating it into the link selection rule lection schemes for outdated and perfect CSI for the considered and achieves superior performance (close to 2dB coding gain) three node network. Each node transmits a BPSK modulated compared to Protocol 1 at high SNR. (C = 1, η = 2) symbol and the average delay is expressed In Fig. 4, the BER vs. SNR (γ¯) is shown when link selection in time slots, unless otherwise stated. Note that, τ0 = 1/2 is performed for delay constrained transmission. We consider holds for the adopted modulation scheme. We assume equal i.i.d. fading here, i.e., ξ = 1. β is adjusted to account for the power allocation PS = PR = P . First, we study the BER delay constraint, cf. Section VI-B. Here, we assume perfect CSI as a function of the transmit SNR γ¯ , P/N0 with/without (i.e., ρSR = ρRD = 1), and focus on the merits of the proposed delay constraints and assume γ¯RD =γ ¯ and infinite buffer protocol in delay constrained transmission compared to other sizes, unless otherwise mentioned. Next, for delay constrained existing buffer–aided relaying protocols for the same average transmission, we investigate the performance of a modified link delay. We show here results for T = {4, 6} and compare with selection protocol, which aims to improve throughput and delay two existing buffer–aided relaying schemes. In Scheme 1 [5], by allowing threshold–based transmission during silent time the relay receives for T time slots, before it transmits for T slots. Finally, we focus on the dependence of BER, delay, and time slots. In Scheme 2 [20], there is no silent time slots and throughput on β, and constrained BER optimization. the source (relay) is forced to transmit if the buffer at the relay 13

TABLE II MODIFIED LINK SELECTION PROTOCOL 1 WITHTHRESHOLD–BASEDTRANSMISSIONINSILENTTIMESLOTS.

Case Buffer status Link quality Selected link 1a Empty γˆSR < βγˆRD AND γˆSR > θγ¯SR S → R 1b Empty γˆSR < βγˆRD AND γˆSR ≤ θγ¯SR None 2 Not empty γˆSR < βγˆRD R → D 3a full γˆSR ≥ βγˆRD AND γˆRD > φγ¯RD R → D 3b full γˆSR ≥ βγˆRD AND γˆRD ≤ φγ¯RD None 4 Not full γˆSR ≥ βγˆRD S → R

TABLE III THROUGHPUT AND DELAY OBSERVED WITH DIFFERENT SNR THRESHOLDS.BPSK TRANSMISSION, γ¯SR = 0.2¯γ, γ¯RD =γ ¯, AND γ¯ = 15 DB ARE ASSUMED.THEMAXIMUMTHROUGHPUTIS τ0 = 0.5 AND ξ = 0.2 IS VALID.

β SNR threshold θ Throughput τ Delay T BER 0.1 0.483 2 0.01171 1 0.355 2.81 0.004 0.6 10 0.251 3.93 0.0032 ∞ 0.25 4.0 0.003 0.1 0.478 1.376 0.01403 1 0.303 2.545 0.0044 1.5 10 0.117 6.548 0.004 ∞ 0.11 7.8 0.004

compared to Schemes 1 and 2, where always either the source θ = 0.1 or the relay transmits and an effective average throughput of θ = 2 τ0 = 1/2 is achieved, as there are no silent time slots. For −1 10 T = 4, the throughput for the proposed scheme is 66% of τ θ = 3 0 for β = 2, whereas for T = 6, it increases to 82% of τ0 for θ = ∞ β = 1.4416.

B. Threshold–based Transmission Next, we consider a modification of Protocol 1, which allows −2 10 BER for a threshold–based transmission during the silent time slots, thus improving throughput and delay at the expense of a degradation of the error rate performance, cf. Table II. In the modified protocol, transmission during a silent time slot is allowed if the outdated instantaneous link SNR is larger than a scalar times the average SNR 17. A similar modification can be −3 10 applied for Protocol 2, but is not shown here. We study the case of β > ξ =γ ¯SR/γ¯RD (so that silent time slots are observed)

0 2 4 6 8 10 12 14 16 18 and infinite buffer size. θ and φ are the scaling parameters in γ¯ (dB) the threshold for the S → R and R → D links, respectively, cf. Table II. Hence, θ = φ = ∞ denotes the original scheme Fig. 5. BER vs. SNR (γ¯) for different threshold values, θ ∈ {0.1, 2, 3, ∞}. proposed in Section II. Note that φ is not needed here (cf. Table II), as a full buffer cannot occur In Fig. 5, we show the simulated BER for different SNR because of the assumed infinite buffer size. thresholds, θ = {0.1, 2, 3, ∞}, for i.i.d. channels, perfect CSI, and adopt β = 2 and ξ = 1. For θ = {2, 3}, the BER is slightly is empty (full). In Scheme 2, β is calculated using (35) and worse than for θ = ∞, i.e., no threshold–based transmission [20, Eq. (39)]) to achieve a target delay T . Similar to the during silent time slots. On the other hand, for θ = {2, 3, ∞}, proposed scheme, the relays in Schemes 1 and 2 forward the throughputs of τ = {0.42, 0.34, 0.33} are observed, i.e., lower detected bits to the destination, regardless of decision errors at 16The values of β are obtained from (57). Note that (57) has two solutions the relay. We observe that for both T = 4 and T = 6, the for β for achieving a target delay T . We choose the solution for β that yields proposed protocol achieves full diversity, whereas Schemes 1 larger throughput. 17We note that alternatively a threshold which is a power of the average and 2 achieve a diversity gain of one only, as the best link is c SNR could be adopted, i.e., γAB ≥ γ¯AB , where c is a positive constant. not always exploited because of forced transmissions. On the However, this modification does not affect the maximum achievable diversity other hand, the proposed scheme suffers from a throughput loss gain (which is two). 14

−3 x 10 6 20

18 ξ = 1 5 ξ Protocol 1 16 = 0.5 Protocol 2 ξ = 2 Case 2 4 Case 3 14

12

Protocol 1 BER 3

10 Finite buffer 8 2 Case 1

6 Average System Delay (time slots) L = {1,3,∞} 1 4 Protocol 2

0 1 2 3 4 5 6 7 8 9 10 0 5 10 15 20 25 β β

Fig. 6. Average delay vs. β for two protocols. Results are shown for different Fig. 7. BER vs. β for the proposed protocols. Case 1) 2¯γSR =γ ¯RD = 0.5¯γ, ρ = ρ = 1, Case 2) 3¯γ =γ ¯ = 0.75¯γ, ρ = 0.9, ρ = 0.7, choices of ξ =γ ¯SR/γ¯RD. For ξ = 0.5, delay results for Protocol 2 are SR RD SR RD SR RD Case 3) γ¯ = 2¯γ =γ ¯, ρ = 0.7, ρ = 0.9. γ¯ = 20 dB is assumed shown for ρSR = 0.9 and ρRD = 0.6. Delay results for finite buffer sizes SR RD SR RD L ∈ {1, 3} are shown for Protocol 1, when ξ = 2. Square markers denote and square markers denote simulated results. simulated results. vs. curves shift to the left (right) of that for when threshold values yield a higher throughput due to the reduced β ξ = 1 ( ). This is because increasing also increases number of silent time slots at the expense of an increased BER. ξ < 1 ξ > 1 β the probability of selecting the R → D link (cf. Remark 4) This behaviour can also be observed in Table III, where and when the R → D link is stronger, i.e., ξ < 1, we require we show the throughput and delay of threshold–based trans- smaller values of β to achieve a certain delay compared to the mission for i.n.d. fading. A lower value of θ implies that S cases when ξ ≥ 1, and vice–versa. transmits more often when silent time slots are observed due to empty buffers. As θ increases, the delay and throughput In Fig. 7, we show BER vs. β for both protocols for i.i.d. performances of the modified protocol approach those of the and i.n.d. links. We consider three cases, as specified in the original scheme. Hence, we conclude from Fig. 5 and Table III caption of Fig. 7. We observe that when β = 1, the lowest that with an appropriate choice of the threshold, improved delay BER is achieved for perfect CSI (Case 1), cf. Section VI-A. and throughput performances can be achieved for threshold– In other words, when β approaches one (i.e., link selection based transmission at the expense of a small loss in the BER is performed based on the instantaneous link quality only), compared to the case where threshold–based transmission is the BER performance improves. We also show the asymptotic not applied. approximation for the error rate for Case 1 (dotted line) which is convex as claimed in Section VI-A. For i.n.d. fading with dissimilar correlation coefficients, β =6 1 yields minimum C. Performance Trade–offs BER. For Case 2, (9) holds for Protocol 2 which yields better In Fig. 6, we show that low average delay can be obtained by performance compared to Protocol 1. For Case 3, (10) holds the proposed protocols for both finite and infinite buffer sizes. for Protocol 2. Note that for some values of β, Protocol 1 In particular, we show√ that for both i.i.d. and i.n.d. fading, a results in a lower BER compared to Protocol 2, even though delay as low as 1 + 2 2 ≈ 3.81 time slots can be observed the minimum of the BER for Protocol 2 is lower than that by choosing β properly for infinite buffer size, as predicted of Protocol 1. However, if we want to compare the BERs in Section IV. We observe a sharp rise in the average delay of the two protocols for a certain target delay or throughput, γ¯SR when β → for infinite buffer size, i.e., TQ becomes very different values of β are needed (cf. Remark 5). For example, γ¯RD large. This fact was also alluded to in [20] for the queueing in Case 3, β = 2 (β = 0.88) yields maximum throughput delay. Another interesting observation√ is that for finite buffer for Protocol 1 (Protocol 2) and β = 3.5 (β = 1.4) yields an sizes, average delays lower than 1 + 2 2 can be achieved, and average delay of T = 4.41 time slots for Protocol 1 (Protocol for L = 1, a minimum delay of three time slots is observed, 2). Note that Protocol 2 yields a lower BER for this target cf. Proposition 8. This result is supported by the fact that the delay and throughput. On the other hand, for large delay and/or average queueing delay is proportional to the average queue low throughput, Protocol 1 may yield similar or lower BER size, which, with appropriate selection of β, can be made compared to Protocol 2. For example, β = 10 (β = 3.3) smaller for a finite size buffer than for an infinite buffer size. corresponds to 33% of maximum throughput for Protocol 1 We also note from Fig. 6 that the delay is convex in β over the (Protocol 2), and both protocols perform similarly in terms of considered region, as shown in Section VI. Note that the delay BER. 15

1 L = 1 min−BER 0.9 L = 2 min−delay

−1 max−throughput L = 5 10 0.8 L = ∞ opt−BER

0.7 Protocol 2: Case 1

0.6 0 τ / BER τ 0.5 −2 10

0.4

0.3 Protocol 1 0.2

−3 0.1 Protocol 2: Case 2 10 0 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 β γ¯ (dB)

Fig. 8. Normalized throughput vs. β for the two considered protocols for Fig. 9. BER vs. SNR (γ¯) optimization for infinite buffer size, under different different buffer sizes L ∈ {1, 2, 5, ∞}. Protocol 1: ξ = 1, Protocol 2 Case objectives. For min–BER, min–delay, max–throughput, β is chosen to achieve a: ξ = 1, ρSR = 0.9, ρRD = 0.6, Protocol 2 Case b: ξ = 1, ρSR = 0.6, the minimum BER, minimum delay, and maximum throughput, respectively. ρRD = 0.9. γ¯ = 20 dB is assumed. For opt–BER, β is chosen to minimize BER for a target delay and throughput constraint.

In Fig. 8, we show the normalized throughput vs. β for the min–BER and lower than the min–delay and max–throughput two proposed protocols and different buffer sizes. We observe BERs. that the maximum normalized throughput for L = 1, is 0.5, and it increases to 1 for increasing buffer size. As β increases, the probability of selecting the R → D link PRD increases which VIII.CONCLUSIONS corresponds to a decrease in the throughput. In Fig. 9, we study BER optimization for Protocol 2 with delay and throughput constraints. We assume infinite buffer size In this paper, we studied adaptive link selection for both perfect and outdated CSI for a three node network where the and γ¯SR = 2¯γRD =γ ¯, ρSR = 0.6, ρRD = 0.9. For opti- relay was equipped with a buffer. In particular, we proposed mization, we assume Tmax = 6 time slots and τmin = 0.8τ0, cf. Section VI-D. For comparison, we have shown the BER two link selection protocols based on the availability of the for three other cases where β is chosen to: a) minimize BER reliability information of the CSI estimates. We provided a unified error rate analysis for outdated and perfect CSI and without any delay or throughput constraints (min–BER),√ b) minimize delay (min–delay), i.e., when T = 1 + 2 2 time showed that for perfect CSI, a diversity gain of two can be achieved, even if only a small delay can be tolerated. When the slots, and c) maximize throughput, i.e., when PRD = 1/2 and links are asymmetric, Protocol 2, which exploits the reliability τ = τ0. The chosen channel parameters correspond to Case b of Protocol 2, cf. Section (II-C). In Fig. 9, at γ¯ = 20 information of the CSI estimates of the links, can result in dB, β = 1.614 (β = 0.80) is numerically obtained for a lower error rate compared to Protocol 1, for a certain target √ delay or throughput. We also provided closed–form expressions PRD = 1/ 2 (PRD = 1/2) which corresponds to minimum √ for the average delay and throughput for the proposed protocols delay Tmin = 1 + 2 2 (maximum throughput τ0 = 1/2). On the other hand, for min–BER, we obtain β = 0.4 which for both finite and infinite buffer sizes. We showed that the minimizes P (e) in (18) for Case b for γ¯ = 20 dB. In Fig. decision threshold β can be chosen to satisfy different objec- 9, the error rate for min–BER is calculated analytically and tives, such as achieving minimum delay, maximum throughput, shown for comparison purpose only, as we cannot operate at and minimum SER under minimum throughput and maximum β < 0.8018 to avoid buffer instability. Hence, min–BER serves delay constraints. Our results demonstrate that by appropriately as a lower bound for other protocols. At γ¯ = 20 dB, the value choosing the value of β, it is possible to maintain buffer of β which gives the optimal BER (opt–BER) in Fig. 9 is 1.1, stability and operate close to the optimal SER at low delay with only a slight loss in throughput. for which T = 5 and τ = 0.8τ0 are obtained. Note that min– delay and max–throughput BERs correspond to τ = 0.5τ0 and Interesting topics for future work include the analysis of T = ∞, respectively. We observe that opt–BER is very close to the proposed modified protocol, which allows for threshold- based transmission during silent time slots, and the extension 18 β < 0.80 would cause PRD < 1/2, i.e., the S → R link would be of the proposed protocols to multi-relay networks where relays chosen more often than the R → D link. forward only correctly regenerated bits. 16

1  q  q  2 − 2 γ¯SRη γ¯SRη 2 2ρSR γ¯SR 1 − − βγ¯RD + βγ¯RD 1 + + 2 2+¯γSRη 2+¯γSRη γ¯SRη γ¯SRη(1−ρSR)+βγ¯RD η PR(e) = C . (75) 2¯γSR

√ APPENDIX A , E{CQ( ηγAB) ∩ γˆSR > βγˆRD} PROOFOF PROPOSITION 1 v2 y Z ∞ Z η Z ∞ Z β = C f (x, y)f (z) We calculate the error rate conditioned on the outdated CSI γSR,γˆSR γˆRD v=0 x=0 y=0 z=0 as × fV (v)dz dy dx dv, (72) Z ∞ Z ∞ 2 1 − w 2 −z/γ¯ PB(e|γˆAB) = C fγAB |γˆAB (x|y)√ e dwdx, RD √ where fγˆRD (z) = (1/γ¯RD)e , and fγSR,γˆSR (x, y) is x=0 w= xη 2π given in (2). After evaluating the integrals in (72), we obtain (65)  q  q γ¯SRη γ¯SRη γ¯SR 1 − − βγ¯RD where fγ |γˆ (x|y) is given by [18] 2+¯γSRη 2+¯γSRη AB AB J1 = C x+ρ2 y 2(¯γSR + βγ¯RD) − AB γ¯ 1−ρ2 √ AB ( AB )   Cβγ¯RD e 2 xyρAB + r . (73) fγ |γˆ (x|y) = I0 . 2ρ2 AB AB − 2 − 2 2(¯γ + βγ¯ ) 1 + 2 + SR γ¯AB (1 ρAB) γ¯AB (1 ρAB) SR RD γ¯ η γ¯ η(1−ρ2 )+βγ¯ η (66) SR SR SR RD From (65), we obtain Furthermore, for the denominator in (71), we obtain ∞ ∞ Z ∞ 2 r  Z Z 1 w 1 xη − 2 √ e dw = erfc . (67) PSR Pr(ˆγSR > βγˆRD) = fγˆSR (x)fγˆRD (y)dx dy √ , w= xη 2π 2 2 y=0 x=βy γ¯ Now, combining (65)–(67), evaluating the following integral = SR . (74) γ¯SR + βγ¯RD x+ρ2 y − AB γ¯ 1−ρ2 √ ∞ AB ( AB )   r  Now, combining (71), (73) and (74), we obtain PR(e) in (75). Z e 2 xyρAB 1 xη C I erfc dx Similarly, we can derive the expression for PD(e). Combining − 2 0 − 2 x=0 γ¯AB (1 ρAB) γ¯AB (1 ρAB) 2 2 and , we obtain in (13). The error rate for (68) PR(e) PD(e) P (e) perfect CSI can be obtained by setting in in closed–form is difficult, if not impossible. Hence, we ap- ρSR = ρRD = 1 (13). This concludes the proof. proximate the Bessel function in (66) as  √   √ 2 2 xyρAB ρAB xy I0 2 ≈ 1 + 2 , (69) γ¯AB (1 − ρAB) (1 − ρAB)¯γAB APPENDIX C which simplifies the calculation, and leads to the simple high PROOFOF PROPOSITION 3 SNR approximation for PB(e|γˆAB) as shown in (6). Here, we provide the proof of Proposition 3. In particular, APPENDIX B we provide the derivation for Case a. The expressions for Case PROOFOF PROPOSITION 2 b can be obtained following a similar approach and are omitted here. We obtain the unconditional error rate P (e) by averaging For a RV V ∈ N (0, 1), P (e), B ∈ {R,D}, can be R B P (e|γˆ ) (cf. (6)) as P (e) expressed as [24] R SR R n n 2 oo E{PR(e, γˆSR ≥ l1)} J2 C V = Eγˆ ,γˆ {P (e|γˆSR ≥ l1)} = = , PB(e) = Eγ Pr γAB < SR RD AB 2 η Pr(ˆγSR ≥ l1) PSR (76) Z ∞ v2 = C F ( )f (v)dv, (70) γAB V where J is given by 0 η 2 where FU (u) denotes the cumulative distribution function of J2 = E{PR(e, γˆSR ≥ l1)} v2 1 − ∞ ∞ yρ2 U and fV (v) = √ e 2 . Z Z − SR 2π 1 γ¯ (1−ρ2 ) = C e SR SR fγˆ (y)fγˆ (z)dydz, Conditioned on the event γˆSR > βγˆRD, the average error SR RD z=0 y=l1 µSR rate of the S → R link is given by (77) √ PR(e) = E{CQ( ηγAB)|γˆSR > βγˆRD} where l was defined in (9). Evaluating the integrals in (77), √ 1 E{CQ( ηγAB) ∩ γˆSR > βγˆRD} we obtain J2 as (71) = . 2 Pr(ˆγSR > βγˆRD)  −1/ρSR 2 µRD 2 2 ρSR µ (1 − ρSR)(1 − ρRD) As mentioned in Section II-A, perfect CSI is assumed available J = C SR . (78) 2 2 2 − 2 for decoding at the receiver; only the link selection procedure µSR(βρRD + ρSR(1 ρRD)) has to rely on imperfect CSI. Now, for the numerator in (71), Similar to (74) for Protocol 1, we obtain PSR, the probability we obtain J1 of selecting the S → R link, for Protocol 2 and Case a as 17

2  −1/ρSR ρ2 µRD (1 − ρ2 ) [14] C. Dong, L.-L. Yiang, and L. Hanzo, “Performance Analysis of Multihop– SR µSR RD Diversity–Aided Multihop Links,” IEEE Trans. Veh. Technol., vol. 61, pp. PSR Pr(ˆγSR ≥ l1) = . , βρ2 + ρ2 (1 − (1 + β)ρ2 ) 2504–2516, Jul. 2012. RD SR RD [15] G.-X. Li, C. Dong, D. Liu, G. Li, and Y. Zhang, “Outage Analysis (79) of Dual–Hop Transmission with Buffer Aided Amplify–and–Forward Similarly, the unconditional error rate of the R → D link is Relay,” in Proc. IEEE VTC–Fall, Vancouver, Sep. 2014. [16] J. L. Vicario, A. Bel, J. A. Lopez-Salcedo, and G. Seco, “Opportunistic obtained as Relay Selection with Outdated CSI: Outage Probability and Diversity Analysis,” IEEE Trans. Wireless Commun., vol. 8, pp. 2872–2876, Jun. E{PD(e, γˆSR < l1)} J3 PD(e) = = , (80) 2009. Pr(ˆγSR < l1) PRD [17] M. Torabi, D. Haccoun, and J. F. Frigon, “Impact of Outdated Relay Selection on the Capacity of AF Opportunistic Relaying Systems with where J3 Adaptive Transmission over Non–Identically Distributed Links,” IEEE ∞ l βzρ2 Trans. Wireless Commun., vol. 10, pp. 3626–3631, Nov. 2011. Z Z 1 − RD 1 γ¯ (1−ρ2 ) [18] D. S. Michalopoulos, H. A. Suraweera, G. K. Karagiannidis, and RD RD ,C e fγˆSR (y)fγˆRD (z)dydz R. Schober, “Amplify–and–Forward Relay Selection with Outdated Chan- z=0 y=0 µRD nel Estimates,” IEEE Trans. Commun., vol. 60, pp. 1278– 1290, Jun. 2  −1/ρSR 2012. µRD 2 2 ! 2 ρSR(1 − ρRD) 1 − ρRD µSR [19] B. Zhong, Z. Zhang, X. Zhang, Y. Li, and K. Long, “Impact of Partial = C − 2 2 2 . (81) Relay Selection on the Capacity of Communications Systems with µRD µSR(ρSR + β(1 − ρSR)ρRD) Outdated CSI and Adaptive Transmission Techniques,” EURASIP Journal on Wireless Communications and Networking, 2013, 2013:24, 13 pages. Next, the probability of selecting the R → D link PRD = [20] T. Islam, A. Ikhlef, R. Schober, and V. K. Bhargava, “Diversity and Delay Pr(ˆγSR < l1) for Protocol 2 and Case a) is obtained as Analysis of Buffer–Aided BICM–OFDM Relaying,” IEEE Trans. Wireless Commun., vol. 12, pp. 5506–5519, Nov. 2013. 2  1−1/ρSR [21] E. W. C. Jakes, Jr., Microwave Mobile Communications. New Jersey: µRD ρ2 (1 − ρ2 ) µSR SR RD Prentice–Hall, 1974. PRD = Pr(ˆγSR < l1) = 1 − 2 2 2 . [22] T. Wang, A. Cano, G. B. Giannakis, and J. N. Laneman, “High- βρRD + ρSR(1 − (1 + β)ρRD) Performance Cooperative Demodulation with Decode-and-Forward Re- (82) lays,” IEEE Trans. Commun., vol. 55, pp. 1427– 1438, Jul. 2007. Now, combining (76)–(82), the end–to–end unconditional SER [23] T. Wang, G. Giannakis, and R. Wang, “Smart Regenerative Relays for can be obtained as shown in (18). Link-Adaptive Cooperation,” IEEE Trans. Commun., vol. 56, pp. 1950– 1960, Nov. 2008. [24] Y. Zhao, R. Adve, and T. Lim, “Symbol Error Rate of Selection Amplify– and–Forward Relay Systems,” IEEE Commun. Lett., vol. 10, pp. 757–759, EFERENCES R Nov. 2006. [25] P. Kalansuriya and C. Tellambura, “Performance Analysis of Decode– [1] T. Islam, D. S. Michalopoulos, R. Schober, and V. Bhargava, “Delay and–Forward Relay Network under Adaptive M–QAM,” in Proc. IEEE Constrained Buffer–Aided Relaying with Outdated CSI,” in Proc. IEEE ICC, Dresden, Germany, Jun. 2009. WCNC, Istanbul, Turkey, Apr. 2014. [26] A. Papoulis, Probability, Random Variables and Stochastic Processes. [2] J. Laneman, D. Tse, and G. Wornell, “Cooperative Diversity in Wireless New York: McGraw–Hill, 1984. Networks: Efficient Protocols and Outage Behavior,” IEEE Trans. Inform. Theory, vol. 50, pp. 3062–3080, Sep. 2004. [3] A. Sendonaris, E. Erkip, and B. Aazhang, “User Cooperation Diversity – Parts I and II,” IEEE Trans. Commun., vol. 51, pp. 1927–1948, Nov. 2003. Toufiqul Islam (S’10, M’15) received his B.Sc. and [4] N. Zlatanov, A. Ikhlef, T. Islam, and R. Schober, “Buffer–Aided Coop- M.Sc. degrees in Electrical and Electronic Engineer- erative Communications: Opportunities and Challenges,” IEEE Commun. ing from Bangladesh University of Engineering and Magazine, vol. 52, pp. 146–153, Apr. 2014. Technology, Dhaka, Bangladesh, in 2006 and 2008, [5] B. Xia, Y. Fan, J. Thompson, and H. V. Poor, “Buffering in a Three Node respectively. He obtained the Ph.D. degree in Electri- Relay Network,” IEEE Trans. Wireless Commun., vol. 7, pp. 4492–4496, cal and Computer Engineering at The University of Nov. 2008. British Columbia (UBC), Vancouver, Canada in 2014. [6] L. Ding, M. Tao, F. Yang, and W. Zhang, “Joint Scheduling and Relay He is now a Research Engineer at the Huawei Canada Selection in One– and Two–Way relay Networks with Buffering,” in Proc. Research Center, Ottawa, Canada. IEEE ICC, Dresden, Germany, 2009. His research interests include flexible air–interface [7] N. Zlatanov, R. Schober, and P. Popovski, “Buffer–Aided Relaying with for 5G, wireless caching, buffer–aided relaying, and Adaptive Link Selection,” IEEE J. Select. Areas Commun., vol. 31, pp. applications of machine learning techniques in communications. He has won 1530–1542, Aug. 2013. the Donald N. Byers Prize, the highest graduate student honor at the UBC. He [8] N. Zlatanov and R. Schober, “Buffer–Aided Relaying with Adaptive Link was also the recipient of the prestigious Vanier Canada Graduate Scholarship, Selection – Fixed and Mixed Rate Transmission,” IEEE Trans. Inform. the Killam Doctoral Fellowship, and the UBC Four Year Fellowship in 2011, Theory, vol. 59, pp. 2816 – 2840, May 2013. 2010, and 2009, respectively. He was one of the six finalists for the 2015 [9] R. Wang, V. K. Lau, and H. Huang, “Opportunistic Buffered Decode– Governor General Gold Medal at the UBC. Sponsored by a competitive Wait–and–Forward (OBDWF) Protocol for Mobile Wireless Relay Net- German Academic Exchange Service (DAAD) research grant and the NSERC works,” IEEE Trans. Wireless Commun., vol. 10, pp. 1224–1231, Apr. Michael Smith Foreign Study Supplements program, he visited the Institute 2011. of Digital Communications at the Friedrich Alexander University (FAU), [10] N. Zlatanov and R. Schober, “Buffer–Aided Half–Duplex Relaying Can Erlangen, Germany during the Summer of 2013. His paper was awarded 2nd Outperform Ideal Full–Duplex Relaying,” IEEE Commun. Lett., vol. 17, prize in the IEEE Region 10 undergraduate student paper contest in 2006. He pp. 479–482, Mar. 2013. also won the IEEE Globecom 2012 travel grant award. [11] A. Ikhlef, D. S. Michalopoulos, and R. Schober, “Max–Max Relay Selection for Relays with Buffers,” IEEE Trans. Wireless Commun., vol. 11, pp. 1124–1135, Mar. 2012. [12] A. Ikhlef, J. Kim, and R. 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Diomidis S. Michalopoulos (S’05, M’10, SM’15) Vijay K. Bhargava (M’74, SM’82, F’92) Vijay was born in Thessaloniki, Greece, in 1983. He re- Bhargava was born in Beawar, India in 1948. He came ceived the Diploma in engineering (5 year studies) to Canada in 1966 and obtained BASc, MASc and and the Ph.D. degree from the Electrical and Com- PhD degrees from Queen’s University at Kingston in puter Engineering Department, Aristotle University of 1970, 1972 and 1974 respectively. He is a Profes- Thessaloniki, in 2005 and 2009, respectively. In 2009, sor in the Department of Electrical and Computer he joined the University of British Columbia, Canada, Engineering at the University of British Columbia as a Killam postdoctoral fellow, and in 2011, he in Vancouver, where he served as Department Head was awarded a Banting postdoctoral fellowship. From during 2003–2008. Previously he was with the Uni- Feb. 2014 to Apr. 2015 he was with the University versity of Victoria (1984–2003), Concordia University of Erlangen–Nuremberg, Germany, as researcher and (1976–1984), the University of Waterloo (1976) and teaching instructor. Since May 2015 he is with Nokia Networks, Germany, as the Indian Institute of Science (1974-1975). He has held visiting appointments Radio Systems Research Engineer. at Ecole Polytechnique de Montreal, NTT Research Lab, Tokyo Institute of His research interests span the broad area of digital wireless communica- Technology, the University of Indonesia, the Hong Kong University of Science tions, with emphasis on physical layer as well as radio access aspects. Dr. and Technology, and Tohoku University. He is an Honourary Professor at Michalopoulos received the Marconi Young Scholar award from the Marconi UESTC, Chengdu and a Gandhi Distinguished Professor at IIT Bombay. Society in 2010, the Killam postdoctoral fellow research prize for excellence Currently he is on sabbatical leave at the Friedrich-Alexander University in research in the University of British Columbia in 2011, and the Best Erlangen–Nuremberg. He is in the Institute for Scientific Information (ISI) Paper Award of the Wireless Communications Symposium (WCS) in the Highly Cited list. IEEE International Conference on Communications (ICC’07). He has served as Vijay served as the Founder and President of “Binary Communications Inc.” member of Technical Program Committees for major IEEE conferences such (1983–2000). He is a co-author (with D. Haccoun, R. Matyas and P. Nuspl) as Globecom, WCNC, and VTC. He is also an Associate Editor of the IEEE of “Digital Communications by Satellite” (New York: Wiley: 1981) which was Communications Letters. translated in Chinese and Japanese. He is a co-editor (with S. Wicker) of “Reed Solomon Codes and their Applications” (IEEE Press: 1994), a co-editor (with H.V. Poor, V. Tarokh and S. Yoon) of “Communications, Information and Network Security” (Kluwer: 2003) a co-editor (with E. Hossain) of “Cognitive Wireless Communication Networks” (Springer: 2007), a co-editor (with E. Robert Schober (M’01, SM’08, F’10) was born Hossain and D.I Kim) of “Cooperative Wireless Communications Networks”, in Neuendettelsau, Germany, in 1971. He received (Cambridge University Press: 2011) and a co–editor (with E. Hossain and G. the Diplom (Univ.) and the Ph.D. degrees in elec- Fettweis) of “Green Radio Communications Networks” (Cambridge University trical engineering from the University of Erlangen– Press 2012). Nuermberg in 1997 and 2000, respectively. From May Vijay is a Fellow of the IEEE, The Royal Society of Canada, The Canadian 2001 to April 2002 he was a Postdoctoral Fellow at Academy of Engineering and the Engineering Institute of Canada. He is a the University of Toronto, Canada, sponsored by the Foreign Fellow of the National Academy of Engineering (India) and has served German Academic Exchange Service (DAAD). Since as a Distinguished Visiting Fellow of the Royal Academy of Engineering May 2002 he has been with the University of British (U.K.). He is a recipient of the 2015 Killam prize in Engineering awarded Columbia (UBC), Vancouver, Canada, where he is by the Canada Council and a Humboldt Research Award from the Alexander now a Full Professor and Canada Research Chair (Tier von Humboldt Stiftung. II) in Wireless Communications. Since January 2012, he is an Alexander von Vijay has served as the editor–in–chief (2007–2009) of the IEEE Trans- Humboldt Professor and the Chair for Digital Communication at the Friedrich actions on Wireless Communications. He is a past President of the IEEE Alexander University (FAU), Erlangen, Germany. His research interests fall Information Theory Society and a Past President of the IEEE Communications into the broad areas of Communication Theory, Wireless Communications, Society. and Statistical Signal Processing. Dr. Schober received several awards for his work including the 2002 Heinz Maier–Leibnitz 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 2000. Dr. Schober is a Fellow of the IEEE, the Canadian Academy of Engineering, and the Engineering Institute of Canada. He is currently the Editor–in–Chief of the IEEE TRANSACTIONS ON COMMUNICATIONS.