Buffer-Aided Relay Selection With Reduced Packet Delay in Cooperative Networks

Applying data buffers at relay nodes significantly improves the outage performance in relay networks, but the performance gain is often at the price of long packet delays. In this paper, a novel relay selection scheme with significantly reduced packet delay is proposed. The outage probability and average packet delay of the proposed scheme under different channel scenarios are analyzed. Simulation results are also given to verify the analysis. The analytical and simulation results show that, compared with non-buffer-aided relay selection schemes, the proposed scheme has not only significant gain in outage performance but also similar average packet delay when the channel signal-to-noise ratio (SNR) is high enough, making it an attractive scheme in practice.

Typical buffer-aided relay selection schemes include the max-max [7] and max-link [8] schemes.In max-max relay selection, at one time slot t, the best link among all sourceto-relay channels is selected, and a data packet is sent to the selected relay and stored in the buffer.At the next time slot (t + 1), the best link among all relay-to-destination channels is selected, and the selected relay (which is often not the same relay selected at time t) forwards one data packet from its buffer to the destination.The max-max scheme has significant coding gain over the traditional max-min scheme.In the maxlink scheme [8], the best link is selected among all available source-to-relay and relay-to-destination links.Depending on whether a source-to-relay or a relay-to-destination link is selected, either the source transmits a packet to the selected relay or the selected relay forwards a stored packet to the destination.As a result, the max-link relay selection not only has coding gain over the max-min scheme, but also higher diversity order than both the max-min and max-max schemes, making it more attractive than its max-max counterpart.
The performance gain of either the buffer-aided max-max or max-link schemes is however at the price of much increased packet delay.In the non buffer-aided relay selection scheme (e.g. the max-min scheme), it always takes two time slots for every packet passing through the network, corresponding to the source-to-relay and relay-to-destination transmission respectively.In the buffer aided approach, in contrast, when a packet is transmitted to a relay node, it is stored in the buffer and will not be forwarded to the destination until the corresponding relay-to-destination link is selected.As a result, different packets in the buffer-aided relay network may endure different delays.To be specific, in either the max-max or maxlink scheme, the average packet delay increases linearly with relay number and buffer size.On the other hand, in order to achieve high performance gain, relay number and buffer size in the max-max or max-min scheme are often set as high as possible.This makes the existing buffer-aided relay selection schemes unsuitable in most applications, particularly in 5G mobile systems which requires ultra-low latency.
While packet delay reduction has been investigated in adaptive link selection with infinite buffer size (e.g.[11]), little has been done for buffer-aided relay selection with finite buffer size.In this paper, we propose a novel buffer-aided relay selection scheme with significantly reduced packet delay.This is achieved by giving higher priority to select the relayto-destination than the source-to-relay links, so that the data queues at relay buffers are as short as possible.The main contributions of this paper are listed as follows: • Proposing a novel relay selection scheme.The proposed scheme provides a simple yet effective way to reduce the packet delay in the buffer-aided relay selection.• Deriving the closed-form expression for outage probability.The analysis is based on general asymmetric channel assumption that the source-to-relay and relay-todestination links may have different average gains.• Obtaining the closed-form expression for the average packet delay.Using Little's law, the average packet delay of the proposed scheme is analytically obtained.• Analyzing the asymptotic performance that the channel SNR goes to infinity.The asymptotic performances including diversity order, coding gain and average packet delay for infinite channel SNR are analyzed.
The remainder of the paper is organised as follows: Section II proposes the new relay selection scheme; Section III analyzes the outage probability; Section IV analyzes the average packet delay; Section V analyzes the asymptotic performance; Section VI shows simulation results; and Section VII concludes the paper.

DELAY
The system model of buffer-aided relay selection is similar to that shown in Fig. 1, except that every relay is equipped with a data buffer (1) respectively.We assume without losing generality that all transmission powers and noise variances are normalized to unity.We also assume that channel gains in either the sourceto-relay or relay-to-destination links are independent and identically distributed (i.i.d.), but in general γsr ̸ = γrd .
In the existing buffer-aided max-max and max-min relay selection schemes, the average packet delay increases linearly with relay number and buffer size.The large delay is due to the packets queuing at the buffers.This can be seen, for example, in the max-link scheme with relay number of N and buffer size of L > 2. Specifically, we assume that all buffers are empty initially and a packet s 1 is sent to relay R 1 at time t = 1.Then at the next time t = 2, except for R 1 which contains s 1 , all other buffers are still empty.Thus there are (N + 1) available links for selection in total: N from source-to-relay (S → R k for all k) links and one from relay-to-destination (R 1 → D) link.Because the max-link scheme always selects the strongest link among all available links, the probability that R 1 → D is selected and s 1 is forwarded to the destination is 1/(N +1)).In other words, it is more likely (with probability of N/(N + 1)) that s 1 remains in R 1 at t = 2, leading to one extra time slot in packet delay.It is clear that this extra delay may be avoided by forwarding s 1 to the destination immediately at t = 2, once the corresponding R 1 → D link is not in outage even though it is not the strongest link.This leads to a new principle of buffer-aided relay selection: that is to transmit the packets already in the buffers as fast as possible.This translates into giving higher priority to select the relay-to-destination links: only when no relay-to-destination link can be selected, are the source-to-relay links considered.As a result, the packet queuing lengths at the relay buffers are minimized, and so is the average packet delay.
To be specific, at time slot t, the link selection rule is as follows: 1) Choose the link with the highest channel SNR among all available relay-to-destination links (|h r k d (t)| 2 ).If the chosen link is not in outage, the corresponding relay forwards a packet from its buffer to the destination.2) Otherwise, if the selected link in step 1) is in outage or there are no available relay-to-destination links at time t, choose the link with the highest channel SNR among all available source-to-relay links (|h sr k (t)| 2 ).If the selected link is not in outage, the source transmits one packet to the corresponding relay and the packet is stored in the buffer.Otherwise outage occurs.The above proposed scheme is easy to implement as it requires the same knowledge as that in the existing buffer-aided max-max or max-min scheme.In the following 2 sections, the outage and delay performance of the proposed scheme will be analyzed respectively.

III. OUTAGE PROBABILITY
The numbers of data packets in all of the relay buffers form a"state".With N relays and buffer size of L, there are (L+1) N states in total.The l-th state vector is defined as where Every state corresponds to one pair of ), corresponding to the numbers of available source-to-relay and relay-to-destination links, respectively.A source-to-relay link is considered available when the buffer of the corresponding relay node is not full, and a relay-to-destination link is available when the corresponding relay buffer is not empty.At state s l , the total number of available source-torelay and relay-to-destination links are denoted as ≤ N .Specifically, if none of the buffers is full or empty, all links are available such that Considering all possible states, the outage probability of the proposed buffer-aided scheme can be obtained as where π l is the stationary probability for state s l , and p s l out is the outage probability at state s l .In the following two subsections, we derive p s l out and π l respectively.
A. p s l out : outage probability at state s l For independent Rayleigh fading channels, the instantaneous SNR for every channel, γ w (w ∈ {sr k , r k d}), is independently exponentially distributed.In the proposed scheme, outage occurs if all available source-to-relay links and relayto-destination links are in outage.Thus the outage probability at state s l is given by where where p S→R out and p R→D out are probabilities that all available source-to-relay links and relay-to-destination links are in outage respectively, r t is the target data rate and ∆ = 2 rt − 1.

B. π l : stationary probability of the state s l
We denote A as the (L + 1) N × (L + 1) N state transition matrix, where the entry A n,l = P (X t+1 = s n |X t = s l ) is the transition probability that the state moves from s l at time t to s n at time (t + 1).
We assume that at time slot t the state is at s l .The probability to select one relay-to-destination link is when not all of the available relay-to-destination links are in outage, or ) . ( On the other hand, because a source-to-relay link is selected only when all relay-to-destination links are in outage and not all source-to-relay links are in outage, the probability to select one source-to-relay link at state s l is given by ) .
(7) With these observations, the (n, l)-th entry of the state transition matrix A is expressed as where p s l out , p R→D s l and p S→R s l are given by ( 4), ( 6) and ( 7) respectively, U R→D s l and U S→R s l are the sets containing all states to which s l can move when a relay-to-destination link or a source-to-relay link is selected respectively.
Because the transition matrix A in ( 8) is column stochastic, irreducible and aperiodic1 , the stationary state probability vector is obtained as (see [15]) where T , I is the identity matrix and B n,l is an n × l all one matrix.Finally, substituting ( 8) and ( 9) into (3) gives the outage probability as where diag(A) is a vector consisting of all diagonal elements of A.

IV. AVERAGE PACKET DELAY
The delay of a packet in the system is the duration between the time when the packet leaves the source node and the time when it arrives the destination.Because it takes one time slot to transmit a packet from the source to a relay node, the average packet delay in the system is given by where Dr is the average delay at the relay nodes.Because the average delay through every relay node is the same, only the average delay through relay R k is analyzed below.Based on Little's Law [16], the average packet delay at relay R k is given by where Lk and ηl are the average queuing length and average throughput at R k respectively.
The average queuing length at R k is obtained by averaging the queueing lengths at buffer Q k over all states, or where Ψ l (Q k ) gives the number of packets (or the buffer length) of buffer Q k at state s l , and π l is given by ( 9).On the other hand, because the probabilities to select any of the relays are the same, the average throughput at relay R k is given by ηk where η is the average throughput of the overall system network.For delay-limited transmission, the average throughput η is obtained as (see [17], [18]) where R is the average data rate of the system (without considering the outage probability).In the proposed scheme, every packet requires two time slots (not necessarily consecutively) to reach the destination, we have R = 1/2 and thus Substituting ( 13) and ( 16) into (12), and further into (11), gives V. ASYMPTOTIC PERFORMANCE This section analyzes the asymptotic performance of the proposed scheme when the average channel SNR goes to infinity.The average channel SNRs for source-to-relay and relay-to-destination link can be respectively expressed as γsr = αγ and γrd = βγ, (18) where α and β are positive real constants, and γ is the normalized average channel SNR.Below we first derive the asymptotic outage probability for γ → ∞, from which the diversity order, coding gain and average packet delay are obtained.

A. Asymptotic outage probability
When γ → ∞, it is clear from (6) that This implies that, any packets in the relay buffers will be forwarded to the destination, and only after all buffers are empty, is a new packet transmitted to one of the relays.Thus when γ → ∞, the buffers can only be in two possible states: S (0) and S (1) , corresponding to the cases that all buffers are empty and only one of the buffers has on packet, respectively.It is then from (3) that where P (S (0) ) and P (S (1) ) are the probabilities that buffers are in states S (0) and S (1) respectively, and p S (0) out and p S (1)   out are the corresponding outage probabilities.Suppose at time t all buffers are empty so that the state is in S (0) .Then one packet will be transmitted to a relay at time (t + 1), and the state moves to S (1) .From (19), the packet in the buffer must be forwarded to the destination at (t + 2) and the state returns to S (0) .This process continues until all packets are transmitted.Thus we have When the buffers are in state S (0) , there are N available source-to-relay links and no available relay-to-destination links, or we have When the buffers are in state S (1) , there is one available relayto-destination link.And the number of available source-torelay links is denoted as K ∞ , where K ∞ = N − 1 or N , for buffer size L = 1 or larger respectively.Then we have Substituting ( 21), ( 22) and ( 23) into (20) gives

B. Diversity order
The diversity order can be defined as If the buffer size L = 1, substituting (24) into (25), and further noting that e x ≈ 1 + x for very small x, we have the diversity order for L = 1 as If the buffer size L ≥ 2, from (24), the asymptotic outage probability is given by the diversity order for L ≥ 2 can be obtained

C. Coding gain
The coding gain is defined as the SNR difference (in dB) between the traditional max-min and proposed schemes to achieve the same outage probability, or where d = N which is the diversity order, and (γ) are the outage probabilities at γ for the max-min and proposed schemes respectively.For fair comparison, the buffer size is set as L = 1 so that the diversity order for the max-min and proposed schemes are the same as d = N .
From (24), we have For the tradition max-min scheme, we have Substituting ( 32) and ( 33) into (31) gives Finally, substituting (34) into (30) gives the coding gain of the proposed scheme as It is interesting to observe that, for symmetric channel configuration with α = β, the coding gain is 3dB.

D. Average packet delay
We have shown that, when γ → ∞, the buffer states can only be in either S (0) or S (1) , or a buffer can only be empty or contains one packet.When all buffers are empty, a new packet is transmitted to a relay with probability of 1/N .Further from (21) that P (S (1) ) = 1/2, the probability that Q k contains one packet is given by Thus, when γ → ∞, the average buffer length at relay R k is given by From ( 16), and noticing that lim γ→∞ P out = 0, the average throughput at relay Q k is given by Finally, substituting (37) and ( 38) into (12), and further into (11), gives the average packet delay for γ → ∞ as It is clearly shown in (39) that, when SNR is high enough, the average packet delay of the proposed scheme is the same as that for the non-buffer-aided schemes.

E. Comparison between different schemes in symmetric channel configuration
For the symmetric channel configuration, Table I compares the diversity order, coding gain and average delay for the nonbuffer-aided max-min, traditional buffer-aided max-max and max-link, and the proposed schemes.
Table I shows that all buffer-aided schemes have 3dB coding gain over the max-min scheme.While the proposed link has slightly higher diversity order than the max-max scheme, but lower diversity order than the max-link scheme.In either the max-max or max-link scheme, the average packet delay increases linearly with relay number N and buffer size L. In the proposed scheme, when γ → ∞, the average delay is fixed at 2 which is the same as that for the non buffer-aided max-min scheme.
For asymmetric channels, the comparison between schemes is not as same as that shown in Table I and will be discussed in the following section.

VI. SIMULATIONS AND DISCUSSIONS
This section verifies the proposed scheme with numerical simulations, where the results for previous described max-link and non-buffer-aided max-min schemes are also shown for comparison.In the simulation below, the transmission rates in all schemes are set as r t = 2 bps/Hz, and simulation results are obtained with 1, 000, 000 Monte Carlo runs.Particularly in the proposed scheme, the simulation results always well match the theoretical analysis.

A. Symmetric channel configuration: γsr = γrd
In the first simulation, we consider symmetric channel scenario that the source-to-relay and relay-to-destination links have same average channel SNR-s.
Fig. 2 (a) and (b) compare the outage probabilities and average packet delays for the non-buffered max-min, traditional max-link and proposed schemes respectively, where the relay number is fixed at N = 3, and we let α = β = 1.5 and γ = 10 dB in (18) so that γsr = γrd = 15 dB.Fig. 2 (a) shows that, when the buffer size L = 1, the proposed and max-link have   the same outage probabilities, where both have significantly better outage performance than the traditional non-buffer-aided max-min scheme because of the 3dB coding gain.When the buffer size increases to L = 5, the proposed scheme has slightly better outage performance than that for L = 1.This well matches the asymptotic analysis that, when L ≥ 2, the diversity order is larger than N but smaller than (N + 1) for the proposed scheme.On the other hand, for the max-link scheme, the outage performance improves more significantly with larger buffer size.This is because that diversity order of the max-link scheme goes up with the buffer size, until it reaches 2N when L → ∞.Fig. 2 (b) shows that, even for L = 1, the average delay of the max-link scheme is at least twice as much that for the proposed scheme.When the buffer size increases to L = 5, the average packet delay of the proposed scheme still maintains at 2 in high SNR range, which is the same as that for L = 1.On the other hand, when L = 5, the average packet delay of the max-link scheme increases to 18 at high SNR-s, which is 9 times larger than that of the proposed scheme.
To further compare the delay performance of the max-link and proposed schemes in symmetric channels, Fig. 3 (a) and (b) show the average packet delay vs the buffer size and relay number respectively, where the average channel SNR-s in both schemes are set as 10 dB.In Fig. 3 (a), the relay number is fixed at N = 2, and the buffer size varies from 1 to 20.In Fig. 3 (b), the buffer size is fixed at L = 10, but the relay number varies from 1 to 10.It is clearly shown in both Fig. 3 (a) and (b) that, the average packet delay for the proposed scheme remains at a constant value of 2. On the other hand, the packet delay in the max-link scheme goes up linearly with either N or L.
In order to reveal the diversity order and coding gain of the proposed scheme, Fig. 4 compares the outage probabilities of the proposed and non-buffer-aided max-min scheme at very high SNR-s, where the relay number is set as N = 4 and all results are from theoretical analysis.First the coding gain is clearly 3 dB by comparing the max-min and proposed scheme with L = 1.For example, to achieve the outage probability of 10 −34 , the SNR-s for the max-min and proposed scheme with L = 1 are about 85 and 88dB respectively.The diversity order of the proposed scheme is also clearly shown to be (N, N +1) for L ≥ 2. For example, as is illustrated in the figure, for the proposes scheme with L = 8, the SNRs to achieve the outage probabilities of −315 and −340 dB are about 78 and 84 dB, respectively.Then according to the diversity order definition in (25), the diversity order is obtained as (340−315)/(84−78) = 4.17, which is clearly between N = 4 and N + 1 = 5.
It is very interesting to observe in Fig. 5 (a) that, for both L = 1 and L = 5, the outage performance of the proposed scheme is significantly better than the max-link scheme!This is because that, when the source-to-relay links are stronger than relay-to-destination links, the max-link scheme is more likely to select the source-to-relay links so that the buffers are more likely full.This effectively decreases the number of the available source-to-relay links, leading to fewer diversity order.On the other hand, in the proposed scheme, while the channel condition gives higher priority to the source-to-relay selection, the selection rule gives higher priority to the relayto-destination link selection.This leads to a more 'balanced' buffers at the relays, or fewer full or empty buffers, which again increases the diversity order.Fig. 5 (b) shows that, the average delay of the max-link even worse than that in symmetric channels.This is because the buffers are more likely to be full, or higher queuing length at buffers.On the contrary, the average delay for the proposed scheme is still as low as about 2 at high SNR range.
Therefore, when γsr > γrd , the proposed scheme has better performance in both outage probability and average delay than the max-link scheme.
C. Asymmetric channel configuration: γsr < γrd Fig. 6 assumes that the source-to-relay link is weaker than the relay-to-destination link in average, where we let α = 1, β = 2 and and γ = 10 dB in (18) so that γsr = 10 dB and γrd = 20 dB, and relay number is set as N = 3.
It is interesting to observe in Fig. 6 that, the max-link and proposed schemes have similarly performance both in outage and average delay.This is because that, stronger relay-todestination links 'naturally' give higher priority to select the relay-to-destination links.But even under this channel assumption, the average packet delay is still better constrained in the proposed scheme than in the max-link scheme, particularly in low SNR ranges.

VII. CONCLUSION
This paper proposed a novel buffer-aided relay selection scheme with significantly reduced packet delays.We have shown the outage and average delay performance under different channel configurations.To be specific, for symmetric S → R and R → D channels, the max-link scheme has better outage performance than the proposed.But when S → R links are stronger, the proposed scheme performs better in outage than the max-link.On the other hand, when R → D links are stronger, the max-link and proposed scheme have similar outage performance.Therefore, if the relay nodes are evenly spread within an area as in many practical systems, it is reasonable to expect that the outage performance of the proposed and max-link schemes are similar.This will be left for future study.We also highlight that, in all cases, the proposed scheme has significantly better outage performance than the non-buffer-aided schemes, making it an attractive scheme in practical applications.
finite size L. We assume relays apply decode-and-forward (DF) protocol.The channel coefficients for S → R k and R k → D links at time slot t are denoted as h sr k (t) and h r k d (t) respectively.All channels are Rayleigh fading, and the average channel gains for S → R k and R k → D links are given by γsr = E[|h sr k (t)| 2 ], γrd = E[|h r k d (t)| 2 ], for all k,

Fig. 1 .
Fig. 1.The system model of the relay selection system.
Delay vs buffer size, where N = 2 Delay vs relay number, where L = 10
10 log P