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Buffer-aided relaying improves both throughput and end-to-end delay Hajipour, Javad; Ruby, Rukhsana; Mohamed, Amr; M. Leung, Victor C Dec 15, 2015

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Hajipour et al. EURASIP Journal onWireless Communications andNetworking  (2015) 2015:261 DOI 10.1186/s13638-015-0482-3RESEARCH Open AccessBuffer-aided relaying improves boththroughput and end-to-end delayJavad Hajipour1*, Rukhsana Ruby1, Amr Mohamed2 and Victor C. M. Leung1AbstractBuffer-aided relaying has recently attracted a lot of attention due to the improvement in the system throughput.However, a side effect usually deemed is that buffering at relay nodes results in the increase of packet delays. In thispaper, we study the effect of buffering at relays on the end-to-end delay of users’ data, from the time they arrive at thesource until delivery to the destination. We use simple discussions to provide an insight on the overall waiting time ofthe packets in the system, taking into account the queue dynamics both in the source and relay. We analyze the end-to-end delay in the relay networks with Bernoulli data arrivals and channel conditions and prove that the data packetsexperience lower average end-to-end delay in the buffer-aided relaying system compared with the conventional one.Moreover, using intuitive generalizations, we conclude that the use of buffers at relays improves not only throughputbut ironically the average end-to-end packet delay. Through extensive simulations, we validate our analytical resultsfor the system when the data arrival and channel condition processes follow Bernoulli distribution. Furthermore, viathe simulations under the settings of practical systems, we confirm our intuition for the general scenarios.Keywords: Wireless relay networks, Buffering capability, Throughput, Delay1 IntroductionWireless relays have received significant attention in thepast decade because of their capability to enhance thecapacity and coverage of wireless networks. The authorsin [1] have investigated the bounds on the ergodic andoutage capacities for wireless relay channels in Rayleighfading environment. Similarly, Azarian et al. [2] have stud-ied amplify-and-forward (AF) and decode-and-forward(DF) relay channels and proposed variants of theseprotocols for reaching the bounds on achievable diversity-multiplexing trade-offs. Employing wireless relays in con-temporary cellular networks is also considered as apromising solution for meeting the growing demandsof users in these systems, due to the cost-effective andfast deployment possibility of relay stations [3]. There-fore, there has been extensive research in this area toidentify the challenges and address them accordingly[4–7]. In particular, Ng et al. [4] studied resource alloca-tion in an orthogonal frequency division multiple access(OFDMA)-based system with AF relays and proposed*Correspondence: hajipour@ece.ubc.ca1ECE Department, The University of British Columbia, Vancouver, CanadaFull list of author information is available at the end of the articleoptimal subchannel and power allocation for maximiz-ing the system goodput. In [5], the authors investigatedjoint relay selection and resource allocation taking linkasymmetry and imperfect channel state information (CSI)into account. Zhang et al. [6, 7] studied resource alloca-tion to provide quality of service (QoS) for the users withminimum rate or maximum packet delay requirements.Usually in the literature in this area, it is assumed thatrelaying procedure is performed in two consecutive sub-slots of a transmission interval; i.e., in the first subslot, thebase station (BS) transmits to the relay and in the secondone, the relay forwards the received data to the mobileterminal. We refer to this method as “conventional relay-ing” in which the end-to-end transmission rate in eachtransmission interval is limited by the poorest link qual-ity. Recently, it has been shown that using buffer in therelay node can improve the system throughput [8–11].This is achieved due to the fact that the buffering capa-bility allows the relay to store packets when its channelcondition is bad and transmit when it is good. Motivatedby this, several other works have studied buffer-aidedrelaying scheme in different areas [12–16]. While Krikidiset al. [12] have studied adaptive relay link selection in asingle-source multi-relay system, the authors of [13] have© 2015 Hajipour et al. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, andreproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to theCreative Commons license, and indicate if changes were made.Hajipour et al. EURASIP Journal onWireless Communications and Networking  (2015) 2015:261 Page 2 of 17investigated that for a multi-source multi-relay scenario.On the other hand, Ahmed et al. [14] have discussed theadvantages of buffer-aided relaying for the operation ofnodes with energy harvesting capability. Moreover, Liuet al. and Darabi et al. [15, 16] have confirmed the advan-tage of using buffer-aided relays in two-way relaying andcognitive radio networks, respectively.Any improvement in a system usually comes at a cost.In the case of buffer-aided relaying, the cost is usu-ally deemed to be the increase in packet delays dueto queueing in the relay. Consequently, the works in[8–11] have tried to investigate and discuss the trade-off between throughput and delay. This is however basedon the assumption of infinitely backlogged buffers in thesource (i.e., BS) and considering the queueing delay onlyat the relay buffer without taking into account the queuedynamics at the BS.In this paper, we aim at filling the abovementionedgap by taking into account the queue dynamics both inthe source node and the relay node. Whereas the exist-ing literature [8–11] mostly considers packet delay as thetime delay between packet transmission (departure) atthe source node and reception (arrival) at the destinationnode, in this paper, we study end-to-end packet delay, i.e.,the delay that data packets experience since their arrivalat the source node until reception at the destination node.The difference between the end-to-end delay consideredin this paper and the delay investigated in the aforemen-tioned works is that the end-to-end delay includes boththe queueing delay that data packets experience in thesource node and the time interval between their transmis-sion at the source and reception at the destination. Notingthat the delay perceived by the end user is affected bythe queueing at both the BS and the relay, in this paper,we investigate the effect of buffer-aided relaying on theend-to-end packet delay. For this, we first provide sim-ple reasoning and discuss the cause of queue formation ina simple queueing system. Based on that, we provide aninsight on the delay performance in the buffer-aided andconventional relaying systems. Then, we study the delayperformance when data arrival and channel conditionprocesses of the system follow Bernoulli distribution andderive closed form expressions for the average end-to-endpacket delay. Using these, we prove that the buffer-aidedrelaying system incurs lower average end-to-end packetdelay compared with the conventional one. Finally, we dis-cuss general scenarios and based on intuitive discussions,we conclude that buffering at relays improves the sys-tem throughput as well as the average end-to-end packetdelay. Using extensive simulations, we verify our analysisand demonstrate the validity of the presented perspec-tive. To the best of our knowledge, this is the first workthat discusses the effect of buffering at relays on the over-all waiting time in a relay-based network and providesthe above conclusion and insight. We note that the dis-cussions in this paper assume infinite buffer capacities inthe BS and relay. However, the insights provided can beused in future works to study the scenarios with limitedbuffer capacities, where the buffer overflow events canalso affect the end-to-end packet delays due to the needfor repeated transmissions. In such scenarios, for reducingbuffer overflow incidents, a buffer-aware source rate con-trol mechanism can be exploited to adjust the traffic loadin the network [17]. Also, buffer-aware resource alloca-tionmethods similar to [18] can be employed to efficientlyserve the system queues. Then, considering the discus-sions presented in this paper as well as the probability ofrepeated transmissions, the end-to-end packet delay canbe investigated for conventional and buffer-aided relayingsystems with finite buffer capacities in the BS and relay.The rest of this paper is organized as follows. Section 2provides a background on the queueing delay based on asimple queueing system. In Section 3, through the math-ematical analysis and generalized intuitions, we study theend-to-end delay performance of conventional and buffer-aided relaying systems. We validate our analytical studyand provide the results for general scenarios through theextensive simulations in Section 4, and finally Section 5concludes the paper.2 BackgroundIn this section, we study a simple queueing system and dis-cuss the cause of packet delays to provide a basis for thenext section, which studies the end-to-end packet delay inrelaying networks.Let us consider a single buffer, as shown in Fig. 1, whichis fed by a deterministic data arrival process and servedby a single server. We assume that time is divided intoslots with equal lengths, indexed by t ∈ {1, 2, . . .}. Thetotal number of data packets that arrive at the buffer isFig. 1 Simple queueing systemHajipour et al. EURASIP Journal onWireless Communications and Networking  (2015) 2015:261 Page 3 of 17N. Starting from t = 1, one packet arrives per time slot.Therefore, the last packet arrives at t = N . For simplicity,we assume that the arrivals occur at the beginning of timeslots. The server might be active or inactive in each timeslot. When it is active, it can serve only one packet pertime slot, where the service implies delivering the packetsuccessfully to the destination. When it is inactive, nopacket is served.We note that if the server is active in each time slot t ∈{1, . . . ,N}, each packet will be served immediately afterits arrival. In this case, there is no queue formed in thebuffer and consequently, each packet experiences an over-all delay of one time slot, which is due to the time spent inthe server. Accordingly, the packets will arrive in the des-tination at the beginning of time slots t ∈ {2, . . . ,N + 1}.However, if the server is inactive in the first time slot, thefirst packet has to wait in the buffer until time slot 2, toget served. Then, in time slot 2, when the second packetarrives, the server is busy with serving the first packet.Therefore, the second packet also experiences one slotdelay in the queue and one slot delay in the server. In asimilar manner, all the following packets incur the samequeueing and service delays. In other words, the delayedoperation of the server causes the nonzero queueing delayfor the first packet, which is transferred to the subsequentpackets as well.Based on the above discussion, if the server is inactive intime slot x ∈ {1, . . . ,N}, it adds one slot to the queueingdelay (and the overall waiting time) of every packet arrivedin slot x or afterward. In general, the packet which arrivedin time slot t will experience a queueing delay of nt andwillbe delivered at time slot t + nt + 1, where nt indicates thenumber of slots before and including t in which the serverwas inactive. It is clear that the cause of queue formationin such systems is the interruption in the operation of theserver, which is translated to queueing delays of the datapackets.3 Effect of buffer-aided relaying on theend-to-end packet delayIn this section, first we assume that data arrive in adeterministic manner and the availability of the chan-nels follows Bernoulli distribution and provide an insighton the end-to-end delay performance for conventionaland buffer-aided relaying systems. Then, we analyti-cally derive the average end-to-end packet delay forthese systems, in the case that both the data arrivalprocess and the availability of the channels followBernoulli distribution. Finally, we discuss general casesand present the intuitions about the end-to-end delayperformance.3.1 Relaying systems with deterministic data arrivals andBernoulli channel conditionsLet us consider a relay network, with one source node,i.e., the BS, one relay node and one destination (or user)node, where the relay works based on the DF technique.It is assumed that there is no direct link between the BSand the user, and the transmissions are done only throughthe relay. There is only one channel in the system, whichcan be used for transmissions either from the BS to therelay or from the relay to the user. We use c1 and c2 toindicate the BS channel condition (for the link betweenthe BS and relay) and relay channel condition (for the linkbetween the relay and user), respectively. These variablescan be either “Good” or “Bad”, meaning respectively that itis possible to transmit one or zero packet successfully onthe corresponding channel. It is assumed that the channelconditions remain constant during each time slot but varyindependently from one time slot to another. The prob-ability of being “Good” is s1 and s2 for the BS and relaychannel conditions, respectively. We assume that eachtime slot is further divided into two subslots, where theBS and relay can transmit a packet in the first and secondsubslots, respectively. The reason for considering subslotsis stated later in Remark 1.Figure 2a shows the queueing model for a conventionalrelaying system, where the relay does not have buffer andtherefore, if it receives a packet in a subslot, it has to trans-mit it immediately in the next subslot. The server 1 andserver 2 indicate the wireless channel from the BS to relayand from the relay to user, respectively. On the other hand,Fig. 2b indicates a relaying network, where the relay has abuffer which allows it to store the data packets and trans-mit whenever its channel is good. In both of the figures,the rectangle enclosed around the servers is to abstract theoverall serving behavior of the system from the time thatthe BS starts to transmit a data packet until it is deliveredto the user. Note that the works in [8–11] in fact study thedelay by considering only the time a packet spends insidethis rectangle and do not take into account the waitingtime in the BS queue, as they assume infinitely backloggedFig. 2 Queueing model of (a) conventional relaying system and (b) buffer-aided relaying systemHajipour et al. EURASIP Journal onWireless Communications and Networking  (2015) 2015:261 Page 4 of 17buffer for the BS. However, in practice, the packets arrivefinitely at the BS and experience a queueing delay beforethe transmission from the BS to the relay. Therefore, theoverall waiting time of a packet in the relay network,which we also refer to as end-to-end delay, includes bothits waiting time in the BS queue and the time it spendsinside the aforementioned rectangle (i.e., the time periodbetween the transmission at the BS and successful recep-tion at the destination). Note that the data arrivals at theBS might be due the packet generation in the BS itself, inwhich case the end-to-end packet delay might be referredto as “generation-to-delivery packet delay”; or it might bedue to exogenous packet arrivals from an external network(e.g., the Internet), in which case the end-to-end packetdelay might be referred to as “enter-to-exit packet delay”to specify the portion of the delay that a packet experi-ences since it enters the relay network until it exits it inthe destination. In this paper, for simplicity, we will usethe term “end-to-end” instead of “generation-to-delivery”or “enter-to-exit,” to specify the delay from the time thata packet arrives at the BS buffer until it is delivered to thedestination.In the following, we consider the data arrivals at the BSbuffer as the deterministic process, with N packets, men-tioned in the previous section. Taking the overall servicebehavior of the systems into account and based on the pre-vious section, we discuss the overall waiting time of datapackets in both the conventional and buffer-aided relayingsystems.Table 1 shows the different states for the joint condi-tions of the BS and relay channels, in which G and Bindicate “Good” and “Bad” conditions, respectively. Weassume that a central scheduler at the relay has the CSI ineach time slot, using the pilot signals transmitted by theBS and destination through error-free control channels atthe beginning of that time slot. Based on the CSI and thebuffering capability of relay, the scheduler decides aboutthe packet transmissions over the links and notifies theBS and the destination accordingly. These are explained indetail in the following.In the case of conventional relaying (no buffer in therelay), only when c1c2 = GG, the scheduler notifies the BSto transmit a packet in the first subslot. Then, the relay for-wards the received packet to the destination in the secondsubslot. In the other three cases, i.e., when one or both ofc1 and c2 are “Bad,” the packets remain in the BS bufferTable 1 Joint channel condition probabilitiesc1c2 ProbabilityGG s1s2GB s1(1 − s2)BG (1 − s1)s2BB (1 − s1)(1 − s2)and are not transmitted. Therefore, conventional relay-ing serves the packets with the probability of s = s1s2 ineach time slot. Consequently, based on the discussions inthe previous section, the overall server in the system isinactive with the probability ofunb = P(GB) + P(BG) + P(BB) = 1 − s = 1 − s1s2 (1)where unb indicates the interruption probability for theoverall server in the system without buffering at therelay. Considering this and the discussions in the previoussection, in each time slot, the probability of “increase ofone slot” in the overall waiting time of the packets presentin that time slot or arrived after that is unb = 1 − s1s2.Here, the increase in the overall waiting time is due to theincrease in the BS queueing delay of those packets.Remark 1. Now, we explain the reason for consideringsubslots in each time slot. First note that the conventionalrelaying protocol stated above takes into account the CSIto decide about the packet transmission. This is reason-able as the CSI is assumed to be available at the scheduler,irrespective of exploiting buffer or not in the relay, andusing it can prevent information loss in the case that suc-cessful delivery of packet is not possible (i.e., either one orboth of the channel conditions are “Bad”). Second, sincethe channel conditions might vary from one time slot toanother, the scheduler does not know what the CSI will bein the next time slot. Therefore, the CSI obtained in thebeginning of a time slot can only be used to decide aboutthe packet transmissions from the BS and relay duringthat time slot. Consequently, it is needed to have a sub-slot for the BS transmission and another one for the relaytransmission, in conventional relaying. Note that alter-natively, one might refer to subslots as slots, in whichcase the channel conditions remain constant over twotime slots and the BS transmissions happen in the odd-numbered slots and the relay transmissions happen ineven-numbered slots.Now consider the system where the relay has a buffer,but as before, the BS can only transmit in the first sub-slot and the relay can transmit in the second subslot. Wenote that if the channel conditions are as BB in time slot x,similar to the system with conventional relaying, no trans-mission will be scheduled and therefore, there will be anincrease of one slot in the overall waiting time of the pack-ets present in time slot x or arriving afterward. However,for the channel conditions as GB and BG, the case is dif-ferent. In order to clearly investigate these states, first weconsider the following example:• In time slot t = 1, the channel conditions are as GB.Therefore, in the first subslot, the BS transmission isHajipour et al. EURASIP Journal onWireless Communications and Networking  (2015) 2015:261 Page 5 of 17scheduled and packet 1 will be transmitted from theBS to relay; but due to the “Bad” channel condition ofrelay, it will not be transmitted to the user in thesecond subslot and will be stored in the relay buffer.• In time slot t = 2, the channel conditions are as BG.Therefore, in the first subslot, there will not be anytransmission from the BS to relay and the overallwaiting time of the packets 2, . . . ,N will be increasedby one slot. However, due to good condition of therelay channel, packet 1 will be transmitted from thebuffer of the relay to the user in the second subslot.In the above example, it is observed that packet 1 isserved by the relay in time slot t = 2 and therefore, it isdelivered to the user in time slot t = 3. This has becomepossible due to the queueing of that packet in the relaybuffer. Note that with conventional relaying, however, inthe above example, packet 1 would remain in the BS queuein both time slots t = 1 and t = 2, and the overallwaiting time would increase by two slots for all the pack-ets. Based on the above discussion and considering thenonzero probability of having channel conditions as GBand BG in two consecutive time slots, it can be concludedthat ub < unb, where ub is the interruption probabilityof the overall server in the buffer-aided relaying system.In other words, the buffering capability in relay reducesthe interruption probability of the overall server and con-sequently, it reduces the overall waiting time for the datapackets. This is achieved due to the fact that the queuesize in the BS is reduced, and the data packets transferredto the relay buffer enable the efficient use of the relaychannel.3.2 Relaying systems with Bernoulli data arrivals andchannel conditionsNow, we consider the relaying networks where both dataarrivals and channel conditions follow Bernoulli distribu-tion. We assume that in each time slot, the probability ofone packet arrival at the BS buffer is a, and, as before,the probability of “Good” channel condition for the BSand relay is equal to s1 and s2, respectively. It is assumedthat a < s1s2 and therefore, the system queues are sta-ble in the case of conventional and buffer-aided relaying[19, Chapter 2]. In the following, when we use subscripts band nb for the variables, we refer to them in the case withbuffering and without buffering at the relay, respectively.3.2.1 Buffer-aided relaying systemBased on [20, Section 7.5], Fig. 3 shows the MarkovChain model for the queue dynamics at the BS bufferfor the buffer-aided relaying network, where each staterepresents the number of packets in the queue. Let pn,n ∈ {0, 1, · · · } denote the probability that in steady state,there are n packets in the BS queue. Note that due toequilibrium in the steady state, we have:p0 = [1 − a(1 − s1)] p0 + s1(1 − a)p1,pn = a(1 − s1)pn−1 + [1 − {a(1 − s1) + s1(1 − a)}] pn+ s1(1 − a)pn+1, n = 1, 2, · · ·Based on the above equations, the probability of eachstate can be written aspn = ρnp0, n = 0, 1, 2, · · · , (2)whereρ = a(1 − s1)s1(1 − a) . (3)Considering the fact that∑∞n=0 pn = 1, we have:p0 = 1 − ρ. (4)Therefore, the expected number of packets in the BSqueue can be expressed byE(QBb) = ∞∑n=0npn = ρ1 − ρ =a(1 − s1)s1 − a . (5)Fig. 3Markov chain for the number of packets in the BS buffer in buffer-aided relaying systemHajipour et al. EURASIP Journal onWireless Communications and Networking  (2015) 2015:261 Page 6 of 17Note that when a new packet arrives at the BS buffer,its expected delay until completion of its service by theBS can be split into two parts. The first part is theexpected time that it has to wait until the packets alreadyin the queue are served, i.e., E(QBb)E(TBb), where E(TBb)is the expected delay imposed due the service of eachpacket when it is in the head of queue. The secondpart is the expected time since the packet itself getsto the head of the queue until its service is completed,which is denoted as E(TB∗b). Therefore, the expected wait-ing time of a packet in the BS, E(DBb), can be writtenas E(DBb) = E (QBb )E (TBb ) + E (TB∗b ). This is in factthe well known mean value approach which holds forqueueing systems with memoryless data arrival processes[21, Section 4.3].The interpretation of E(TBb)is as follows. The delaycaused due the service of a packet in the head of the queueis 1 slot with the probability of s1 (this is in the case thatthe BS channel is good at the time that the packet gets tothe head of queue). It is (1+1) slots with the probability of(1− s1)s1, (2+ 1) slots with the probability of (1− s1)2s1,(k + 1) slots with the probability of (1− s1)ks1, and so on.Therefore, the expected delay caused due to the service ofa packet in the head of queue is given byE(TBb) = s1 + (1 + 1)(1 − s1)s1 + (2 + 1)(1 − s1)2s1 + · · ·=∞∑k=0(1 − s1)ks1.(k + 1)= s1∞∑k=0(1 − s1)kk + s1∞∑k=0(1 − s1)k= s1(1 − s1)[dds1(−∞∑k=0(1 − s1)k)]+ s1 11 − (1 − s1)= − s1(1 − s1) dds11s1+ s1 1s1= 1 − s1s1 + 1= 1s1(6)On the other hand, we can compute E(TB∗b ) as follows.Considering that the packet has reached the head of thequeue, its delay until the departure from the BS is equal to0.5 with the probability of s1, (1+0.5)with the probabilityof (1 − s1)s1, (2 + 0.5) with the probability of (1 − s1)2s1,(k + 0.5) with the probability of (1 − s1)ks1, and soon. Hence, the expected waiting time of the packet afterreaching the head of the queue isE(TB∗b) = 0.5s1 + (1 + 0.5)(1 − s1)s1+ (2 + 0.5)(1 − s1)2s1 + · · ·=∞∑k=0(1 − s1)ks1.(k + 0.5)= s1∞∑k=0(1 − s1)kk + 0.5s1∞∑k=0(1 − s1)k= 1 − s1s1 + 0.5. (7)Based on the above discussions, the expected total delayof a packet in the BS is equal toE(DBb) = E (QBb )E (TBb )+ E (TB∗b )= a(1 − s1)s1 − a1s1+ 1 − s1s1 + 0.5= 1s1[a(1 − s1)s1 − a + 1]− 0.5= 1 − as1 − a − 0.5. (8)We note that in each time slot, either one or zero packetdeparts the BS. Therefore, the packet departures from theBS can be modeled as a Bernoulli process. Due to the sta-bility of the queues, the data departure rate from the BSis equal to the data arrival rate in its buffer. Consequently,the probability that one packet departs the BS, or, equiv-alently, the probability that one packet arrives at the relaybuffer, is equal to a. As a result, the average delay that apacket experiences in the relay can be computed in thesimilar manner as the average delay in the BS, which isexpressed byE(DRb) = 1 − as2 − a − 0.5. (9)Based on (8) and (9), the average waiting time of a packetin the buffer-aided relaying system is given byE(Db) = E(DBb)+ E (DRb) = 1 − as1 − a +1 − as2 − a − 1. (10)3.2.2 Conventional relaying systemNote that in the conventional relaying system, the BScan serve the packets in its buffer only when both itsown channel and the relay channel are in good condition.Hence, the service probability for serving the BS buffer iss1s2. Considering that, the average number of packets inthe BS buffer can be obtained by replacing s1 in (5) withs1s2, i.e., E(QBnb) = a(1−s1s2)s1s2−a . Similarly, the average delaycaused for a packet due to the service of each packet infront of it can be computed based on (6) and by usings1s2 instead of s1, i.e., E(TBnb) = 1s1s2 . Also, the averagedelay that a packet experiences when it gets to the headHajipour et al. EURASIP Journal onWireless Communications and Networking  (2015) 2015:261 Page 7 of 17of the queue can be obtained, based on (7), as E(TB∗nb) =1−s1s2s1s2 + 0.5. Therefore, we have:E(DBnb) = E (QBnb)E (TBnb)+E (TB∗nb ) = 1 − as1s2 − a −0.5.(11)On the other hand, when a packet arrives at the relay,it is immediately served without waiting in any buffer.Therefore, it only spends 0.5 of a slot in the relay, which isdue to the service time in the relay. Consequently, we have:E(DRnb) = 0.5. (12)Based on (11) and (12), the average waiting time of apacket in the conventional relaying system is given byE (Dnb) = E(DBnb)+ E (DRnb) = 1 − as1s2 − a . (13)In order to compare the delay performance of the con-ventional and buffer-aided relaying systems, Theorem 1states and proves the main result of this subsection.Theorem 1. Consider a relaying network where the dataarrival process at the BS and the channel availability pro-cesses follow Bernoulli distribution and the packet arrivalprobability satisfies stability condition a < s1s2. Then, theaverage end-to-end packet delay in the buffer-aided relay-ing system is less than or equal to that in the conventionalone. In other words, we have:E(Db) ≤ E(Dnb), (14)where the equality holds only in the case that the channelsare always in “Good” condition, i.e., s1 = s2 = 1.Proof. Please refer to the Appendix.3.3 General relaying systemNowwe consider a general scenario, where the data arrivaland channel condition processes follow general distribu-tions, and are stationary and ergodic. We assume thatthe data arrivals and transmission rates have finite meanand variance. We use rbr(t), rrd(t), and rbd(t) to showthe achievable transmission rate in time slot t betweenthe BS and relay, the relay and destination, and the BSand destination, respectively. Without buffering, the BSneeds to transmit to the relay in the first subslot, andthen the relay has to forward it immediately in the nextsubslot. We know that in this case, the end-to-end achiev-able rate between the BS and the user is rbd(t) = 12min{rbr(t), rrd(t)}. Therefore, the scheduler in the relaynotifies the BS in the beginning of each time slot to trans-mit with a rate that can be supported by both of the linksto lead to a successful reception at the destination. Due tothis, the transmission rate in each slot is limited by the linkwith the worst channel condition in that time slot.However, when the relay has a buffer, there is no neces-sity for the immediate forwarding of the data and theabovementioned limitation is relaxed; therefore, the BShas the opportunity for transmitting continuously to therelay when the channel condition from the BS to relayis good. Then, the relay can store them in the buffer totransmit when the channel from the relay to user is good.Because of this, the buffering makes it possible to improvethe system throughput as shown in [8–11]. Improvementin the throughput is equivalent to the improvement inthe average end-to-end service rate of the data arrived atthe BS buffer. In other words, the increase in the systemthroughput means that more data is transferred from theBS to the user, or equivalently, the same data is trans-ferred from the BS to the user in a less amount of time.Therefore, on average, packets experience lower end-to-end delay, i.e., the delay since their arrival at the BS untildelivery to the destination.Based on the above discussion, we make the conclu-sion as follows. Although buffer-aided relaying results inqueueing delay in the relay, it also facilitates data trans-fer from the BS to the user and leads to a large reductionin the queueing delay at the BS. Therefore, the over-all effect is the improvement of the average end-to-endpacket delay. In summary, we state this as follows.Proposition. Using buffer in the relay improves the sys-tem throughput, and therefore, it reduces the averageend-to-end packet delay.Remark 2. We note that the given proposition is aboutthe average end-to-end packet delay. There might besome packets that experience larger end-to-end delaysin buffer-aided relaying compared with the conventionalrelaying. However, reduction in the average end-to-endpacket delay indicates thatmost of the packets experienceless delay in the case of buffer-aided relaying comparedwith conventional relaying. This is confirmed in the nextsection, in Figs. 11 and 16. Moreover, we note that theabove discussions do not explain anything about the max-imum and minimum possible end-to-end packet delays inbuffer-aided relays. In general, considering the queueingdynamics in both the BS and relay, the maximum pos-sible end-to-end packet delay in both conventional andbuffer-aided relaying is infinite, which is due to the infi-nite buffer size of the BS and relay. However, simulationresults presented in the next section indicate that usuallythe maximum end-to-end packet delay is less in buffer-aided relying compared with the conventional relaying.On the other hand, the minimum possible end-to-endpacket delay in both of the relaying systems is one timeslot, which happens when there is no queue neither in theHajipour et al. EURASIP Journal onWireless Communications and Networking  (2015) 2015:261 Page 8 of 17BS buffer nor in the relay buffer; in such a case, when apacket arrives at the BS, it can be immediately transmittedto the relay and then, the relay can immediately transmitit to the destination. This will take totally two subslots orequivalently one time slot.In the previous subsection, even in the buffer-aidedrelaying system, we assumed that the BS and relay trans-missions are a priori scheduled to be done in the first andsecond subslots. In general, when buffering is exploitedin the relay, each subslot can be used dynamically for theBS transmission or relay transmission, if there are data intheir buffers. In this regard, a dynamic scheduling policyis required to stabilize the system queues. Specifically, ineach subslot, this policy should decide on allocating thechannel to the BS or relay such that the system queuesremain bounded. For this, the well-known Max-Weight(MW) algorithm can be used, which has the largest sta-bility region [19, 22, 23]. MW aims at maximizing theweighted rates of the links, where the weight of a link isconsidered equal to the difference of the queue sizes at thetwo ends of the link. Note that due to the interdependenceof the queue sizes and the scheduling decision in MW, itis highly intractable to derive the expressions for averagequeue sizes and delays under the MW policy. However,if the data arrival rate is inside the stability region (so itcan be supported by the network capacity), it is guaran-teed that scheduling the links by MW policy will result inbounded average queue sizes and delays [19, 22, 23]. MWis an attractive scheduling policy for stabilizing the queuesin buffer-aided relay networks as it works by utilizing justthe instantaneous queue and channel state information(QCSI) and does not require information about the prob-ability distribution of packet arrival processes and channelstates. Considering the abovementioned, we summarizethe costs of buffer-aided relaying in the following remark.Remark 3. Note that the costs for the improvementsbrought by buffer-aided relaying are the requirement for amemory to buffer data at the relay and the necessity for ascheduling algorithm to keep the queues stable.Remark 4. It is worth noting that the proposition statedabove can also be considered for the case of relay networkswith more than two hops or more than one relays. Forsuccessful data transmission with conventional relayingin a multihop network, it is needed to have the chan-nel states of all the hops from the source to destinationfavorable during the transmission interval. However, withbuffer-aided relaying, it is possible to use the channelsmore opportunistically. This is studied in [24], where itis shown that the outage (or unsuccessful packet recep-tion) is reduced in buffer-aided multihop networks, whichis equivalent to improvement in throughput. Similarly, itis shown in [25] that in a network with multiple relays,the system throughput is improved in buffer-aided relay-ing compared with the case without buffers at the relays.Therefore, based on those results and considering thequeueing delays both in the source and the relay, we con-clude that the packets that are successfully received at thedestination experience lower average end-to-end delay inthe aforementioned relaying systems when buffering usedin relays compared with the case without buffers in relays.The analysis for deriving the exact expressions of averageend-to-end packet delay in these scenarios needs moreinvestigation, as the effect of the relay selection policyshould also be taken into account, and is an interestingresearch topic for future works.4 Numerical resultsTo verify the presented discussions, we have conductedextensive Matlab simulations over 10,000 time slots andmore. We have investigated the cases that the data arrivaland channel condition processes follow Bernoulli distri-bution, as well as general cases with the settings of apractical system. We present the simulation results in thefollowing.4.1 Bernoulli data arrivals and channel conditionsIn order to validate the analysis provided in Subsec-tion 3.2, in Figs. 4, 5, and 6, we present the average packetdelay obtained from both the analytical expressions andthe simulation. In each of these figures, we have fixedthe values of s1 and s2 and have evaluated the effect ofincrease in a on the average end-to-end packet delay.In order to maintain the stability of the system queueswith both conventional and buffer-aided relaying, for eachfigure, we have considered a < s1s2. Figure 4 displaysthe case with high probability for the good channel con-ditions at the BS and relay, i.e., s1 = s2 = 0.9. It is clearthat the analytical results are quite close to the simulationones. Moreover, the results confirm that the buffer-aidedrelaying has lower packet delays compared with the con-ventional relaying. As expected, both of the systems incurlarger delay as the packet arrival probability increases.However, the delay in the conventional relaying systemincreases faster comparing with that in the buffer-aidedrelaying system.Furthermore, Figs. 5 and 6 show the results for the casesthat either one or both of the channels have relativelylower probability of being in good condition. It is observedthat in these cases, the conventional relaying results in sig-nificantly higher delays even at the lower data arrival rates.In particular, the performance difference of these relayingsystems is larger in Fig. 5 compared with Fig. 4 and thelargest in Fig. 6. This is because when the probability ofgood channel conditions is low, in the case of conventionalrelaying, the BS has to wait for a long time before havingboth the channels favorable for transmission. However,Hajipour et al. EURASIP Journal onWireless Communications and Networking  (2015) 2015:261 Page 9 of 17Fig. 4 Average end-to-end packet delay in the case of Bernoulli channel distribution with s1 = s2 = 0.9in the case of buffer-aided relaying, the BS can transmitto the relay even when the relay channel is bad. Then,the relay can buffer the received data and transmit in itssubslots whenever its channel is good.We have also conducted simulations to investigatethe average total queue sizes when the packet arrivalprobabilities get close to the stability region boundaries,in the case of s1 = 0.5, s2 = 0.5. We note that basedon [19, Chapter 2], the stability region boundary in con-ventional relaying is equal to s1s2 = 0.25 whereas it isequal to min[ s1, s2]= 0.5 in buffer-aided relaying. Also,note that in conventional relaying, the total queue size isFig. 5 Average end-to-end packet delay in the case of Bernoulli channel distribution with s1 = 0.5, s2 = 0.9Hajipour et al. EURASIP Journal onWireless Communications and Networking  (2015) 2015:261 Page 10 of 17Fig. 6 Average end-to-end packet delay in the case of Bernoulli channel distribution with s1 = s2 = 0.5the BS queue size whereas in buffer-aided relaying, thetotal queue size is the sum of the BS queue size and relayqueue size. Therefore, based on (5) and the discussionsin Subsection 3.2, the mathematical expression of aver-age total queue size is a(1−s1s2)s1s2−a in conventional relayingand a(1−s1)s1−a + a(1−s2)s2−a in buffer-aided relaying. The graphsfor these equations as well as the results of simulationsare shown in Figs. 7 and 8. It is observed that the analyti-cal results are close to the simulation ones. Furthermore,Fig. 7 shows that the average total queue size in conven-tional relaying system increases rapidly when the packetarrival probability gets close to 0.25 (the stability regionboundary for conventional relaying). This is due to the factthat the probability of having both the channel conditionsFig. 7 Average total queue size in conventional relaying; the case of Bernoulli channel distribution with s1 = s2 = 0.5 and the packet arrivalprobability close to the stability region boundaryHajipour et al. EURASIP Journal onWireless Communications and Networking  (2015) 2015:261 Page 11 of 17Fig. 8 Average total queue size in buffer-aided relaying; the case of Bernoulli channel distribution with s1 = s2 = 0.5 and the packet arrivalprobability close to the stability region boundary“Good” (to be able to serve the packets in the BS queue inconventional relaying) is 0.25.When the data arrival prob-ability gets close to this value, more packets have to waitin the queue until they get to the head of buffer and get thechance to be transmitted. On the other hand, it is observedin Fig. 8 that the similar effect happens in buffer-aidedrelaying in considerably larger packet arrival probability,i.e., 0.5 (the stability region boundary for buffer-aidedrelaying), and before that, the average total queue size issmall. This means that in the arrival probabilities largerthan 0.25 in conventional relaying, when a packet arrivesat the BS, it is expected to encounter an infinite queue size(and end-to-end delay) in its path to the destination. How-ever, even though in buffer-aided relaying, there are twobuffers in the path of packets from the BS to the destina-tion, it is expected that the packets will encounter a finitetotal queue size (and end-to-end delay) before reachingthe destination, as long as their arrival probability is lessthan 0.5.4.2 General scenarioNote that themathematical analysis presented in Subsection3.2 and the numerical results shown in Subsection 4.1are for Bernoulli data arrivals and channel conditions andprovide an insight on the effect of using a buffer in relay onthe end-to-end packet delay. In order to verify the discus-sions presented in Subsection 3.3 for general data arrivaland channel condition processes, we consider a scenariowith more realistic settings. For this scenario, the simula-tion parameters are shown in Table 2. It is assumed thatthe channel fading is flat over the system bandwidth andconstant during each time slot; however, it can vary fromone slot to another. For the link between the relay anduser, Rayleigh channel model is used, and for the link fromthe BS to relay, Rician channel model is used with κ fac-tor equal to 6 dB [26]. In the case of conventional relaying,the transmissions at the BS and relay are done in consecu-tive subslots. For buffer-aided relaying, we have used MWTable 2 Simulation parametersParameter name SettingCell radius 1000 mMin UE-BS distance 50 mBS antenna height 15 mRelay antenna height 10 mUser antenna height 1.5 mRelay distance from BS 1/2 cell radiusPathloss model From [27]Channel bandwidth 180 KHzTime slot duration 1 msNoise power spectral density −174 dBm/HzTraffic model PoissonPacket size 1 KbitsHajipour et al. EURASIP Journal onWireless Communications and Networking  (2015) 2015:261 Page 12 of 17Fig. 9 Average BS queue size over time at the arrival rate of 50 packets/secondpolicy [19, 22, 23] to decide in an adaptive way, about thetransmission in each subslot, either from the BS or fromthe relay buffer. The simulations were conducted for 100independent realizations of channel condition and dataarrival processes, each over 10,000 time slots.Figures 9 and 10 show the BS and relay average queuesizes over time, respectively, at the arrival rate of 50packets/second. The average queue size is obtained bytaking the average of queue sizes over 100 simulations. Itis observed that with buffer-aided relaying, although dataare queued in the relay, the average BS queue size in eachtime slot is reduced significantly. This results in loweraverage end-to-end packet delays in buffer-aided relay-ing compared with the conventional relaying, as shown inFig. 11. In particular, in this scenario, the average end-to-end packet delays are 11 ms and 30 ms in buffer-aidedFig. 10 Average relay queue size over time at the arrival rate of 50 packets/secondHajipour et al. EURASIP Journal onWireless Communications and Networking  (2015) 2015:261 Page 13 of 17Fig. 11 CDF of end-to-end packet delays at the arrival rate of 50 packets/secondand conventional relaying, respectively. Note that Fig. 11indicates that in general, the average end-to-end packetdelay is less in buffer-aided relaying. In other words, eventhough some packets might experience larger overall wait-ing time compared with the conventional relaying, mostof the packets experience lower delay since their arrivalat the BS until delivery to the destination. Moreover, it isobserved that the maximum end-to-end packet delay isless in the case of buffer-aided relaying.Next, we investigate the effect of increase in the packetarrival rate on the throughput and delay performance. Itis observed in Fig. 12 that the conventional relaying isFig. 12 Effect of packet arrival rate at the BS on the average throughput in each time slotHajipour et al. EURASIP Journal onWireless Communications and Networking  (2015) 2015:261 Page 14 of 17Fig. 13 Effect of packet arrival rate at the BS on the average end-to-end packet delayable to support data arrival rates up to 60 packets/second,in which range it results in the average throughput equalto data arrival rate at the BS. However after that, dueto low capacity, it starts to get saturated. This leads toqueue instability and large end-to-end delays for packets,as shown in Fig. 13. In contrast, the buffer-aided relayingis able to provide the average throughput equal to the dataarrival rate, in all the packet arrival rates, and thereforeleads to very low end-to-end packet delays.In order to have a complete picture, we also present thesystem performance in the arrival rate of 100 packets/second, in Figs. 14 and 15. Figure 14 shows that in conven-tional relaying, the average BS queue size grows unbounded;this is due to the low capacity of relaying channel whichFig. 14 Average BS queue size over time at the arrival rate of 100 packets/secondHajipour et al. EURASIP Journal onWireless Communications and Networking  (2015) 2015:261 Page 15 of 17Fig. 15 Average relay queue size over time at the arrival rate of 100 packets/secondis unable to serve all the arrived data. This leads to largeend-to-end packet delays as depicted in Fig. 16. On theother hand, as shown in Fig. 15, buffer-aided relayingleads to queueing in the relay buffer, which helps to uti-lize the channel variations efficiently. It allows to transferthe data from the BS buffer to relay buffer and from relaybuffer to user, when the corresponding channels havegood conditions, and therefore leads to low end-to-endpacket delays. In particular, in this scenario, the aver-age end-to-end packet delays are 20 ms and 1355 ms,respectively, in buffer-aided and conventional relaying.The above results confirm that using buffer in relayimproves the throughput as well as the average end-to-endpacket delay in the system.Fig. 16 CDF of end-to-end packet delays at the arrival rate of 100 packets/secondHajipour et al. EURASIP Journal onWireless Communications and Networking  (2015) 2015:261 Page 16 of 175 ConclusionsIn this paper, we have studied the effect of buffering at therelay on the end-to-end delay performance. Throughthe discussions about queueing delay, we have explainedthe cause of delay in a simple queueing system. Basedon that, we have provided an insight on the overall delayin the conventional and buffer-aided relaying networks.Moreover, for the case of Bernoulli data arrivals andchannel conditions, we have proved analytically that theaverage packet delay is lower in buffer-aided relaying sys-tem compared with the conventional one. Finally, basedon intuitive reasoning for general scenarios, we have con-cluded that employing buffer in the relay improves boththe system’s throughput and average end-to-end packetdelay. Using numerical results, we have verified our anal-ysis and discussions, and shown that using buffer in therelay leads to higher system throughput and lower averageend-to-end packet delay.AppendixIn order to prove that the buffer-aided relaying systemincurs equal or lower delay compared with the conven-tional one, it is required to prove E(Dnb) − E(Db) ≥ 0. Toshow this, note thatE(Dnb) − E(Db) = 1 − as1s2 − a −1 − as1 − a −1 − as2 − a + 1 (15)By adding and subtracting the term 1−as1−a1−as2−a and rear-ranging the equations, we haveE(Dnb) − E(Db) = 1 − as1 − a1 − as2 − a −1 − as1 − a −1 − as2 − a+ 1 + 1 − as1s2 − a −1 − as1 − a1 − as2 − a=( 1 − as1 − a − 1)( 1 − as2 − a − 1)+ 1 − as1s2 − a −1 − as1 − a1 − as2 − a . (16)Note that the packet arrival probability is nonzero andthe stability condition holds, i.e., 0 < a < s1s2. Since si ≤1, i = 1, 2, we have 0 < a < si, i = 1, 2, and 1−asi−a ≥ 1,i = 1, 2. Therefore, the first term in the right hand sideof (16) is non-negative. Hence, it suffices to show1 − as1s2 − a ≥1 − as1 − a1 − as2 − a . (17)By canceling 1 − a and cross-multiplying in (17), weobtain(s1 − a) (s2 − a) ≥ (1 − a) (s1s2 − a) . (18)After multiplying both sides out and canceling the com-mon terms of (18), we havea (1 − s1) (1 − s2) ≥ 0, (19)which is always true since s1 ≤ 1 and s2 ≤ 1.Competing interestsThe authors declare that they have no competing interests.AcknowledgementsThis work was supported by the Canadian Natural Sciences and EngineeringResearch Council through grants RGPIN-2014-06119 and RGPAS-462031-2014,and the National Natural Science Foundation of China through Grant No.61271182. The work of Amr Mohamed was supported by NPRP 5-782-2-322from the Qatar National Research Fund (a member of Qatar Foundation). Thestatements made herein are solely the responsibility of the authors.Author details1ECE Department, The University of British Columbia, Vancouver, Canada. 2CSEDepartment, Qatar University, Doha, Qatar.Received: 29 May 2015 Accepted: 12 November 2015References1. A Host-Madsen, J Zhang, Capacity bounds and power allocation forwireless relay channels. IEEE Trans. Inf. 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