Optimization of a Relay-Assisted Link with Buffer State ...

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Optimization of a Relay-Assisted Link with Buffer State Information at the Source Ahmed El Shafie, Student Member, IEEE, Mohammad Galal Khafagy, Student Member, IEEE, and Ahmed Sultan, Member, IEEE Abstract—This letter proposes a random-access scheme employed by a source in a three-node network composed of a source, a relay and a destination. The source knows the exact number of packets at the finite buffer of the relay. Based on this knowledge, the source probabilistically accesses the channel. We consider a fixed-rate transmission and study the cases with and without channel outage state information at the transmitting nodes. We maximize the throughput in both cases and provide numerical results to illustrate and verify the ideas presented in this work. Index Terms—Buffer-aided relay, BSI, CSI, linear-fractional program, Markov chain, queuing theory.

I. I NTRODUCTION Buffered relays can achieve higher throughput gains, compared to relays without buffers, due to their ability to either forward or buffer the received data packets depending on the channel and buffer states. Buffer-aided relaying has been addressed in recent literature, for instance in [1]–[3]. In this letter, we consider a buffered relay with a finite queue that maintains at most N packets. We assume the source knows the exact state of the buffer through the transmitted feedback messages (acknowledgment/negative acknowledgment (ACK/NACK) packets) from the relay and the destination. The source uses this buffer state information (BSI) in making its transmission decisions. Exploiting BSI has been used for instance in [2] and [4] to optimize transmission by excluding relays with full buffers from receiving and relays with empty buffers from transmitting. Note that the BSI has also been used for example in [5], albeit in noncooperative contexts. Unlike most of the existing work, e.g., [3], [6]–[8], we assume that the source has a direct link. First, we consider the case when the channel state information (CSI) is unknown at the source and relay. Afterwards, we tackle the case where the source and relay have knowledge of the channel connectivity states, i.e., whether or not the links are in outage. In both cases, we use throughput maximization as the design objective. The first case with BSI only is solved analytically, whereas the optimization problem of the second case is converted into a linear-fractional program, that is efficiently solved numerically. II. SYSTEM MODEL We consider a relay-assisted link with a source, a halfduplex decode-and-forward relay and a destination. The source and the relay are equipped with a single antenna and each transmits with power P Watts. There exists a direct link between the source and the single-antenna destination. The relay has a buffer that can store up to N data packets. The A. El Shafie is with Electrical Engineering Dept., University of Texas at Dallas, USA (e-mail: [email protected]). M. G. Khafagy and A. Sultan are with King Abdullah University of Science and Technology (KAUST), Thuwal, Makkah Province, Saudi Arabia (email: {mohammad.khafagy, ahmed.salem}@kaust.edu.sa).

size of a packet is B bits. The time is slotted and one time slot has a length of T seconds. We consider a backlogged source that always has a new packet to send at each time slot. Let t ∈ {1, 2, . . . } denote the time slot index. We assume block fading links, where the channel remains constant during a slot, but changes from slot to another according to a Rayleigh fading model. The squared magnitude of the complex fading coefficient between node j and node k, denoted by gjk (t), is exponentially distributed with mean σjk . Each link is perturbed by additive white Gaussian noise (AWGN) with zero mean and variance N . Channel outage occurs when the transmission rate exceeds the link capacity. For convenience, we denote the source as ‘s’, the relay as ‘r’ and the destination as ‘d’. Therefore,
j ∈ {s, r} and k ∈ {r, d}. Let Cjk (t) = P and Pjk = Pr{Cjk (t) ≤ rjk } denote, log2 1 + gjk (t) N respectively, the capacity and the outage probability of link j −k in time slot t. Assuming unit transmission bandwidth, the transmission rate employed by node j while communicating with node k is given by rjk = R = TB bits/sec/Hz. That is, we assume fixed-rate transmission over all links. Assuming R −1 P Rayleigh fading, Pjk = 1 − exp(− 2 γjk ) where γjk = σjk N is the average receive signal-to-noise-ratio. We note that Pjk increases with increasing R or decreasing γjk . We assume that the outage probabilities are known at the source. III. P ROPOSED S CHEME W ITHOUT T RANSMIT CSI We study here the proposed system assuming BSI only. When the relaying queue contains n packets, the channel is accessed by the source with probability 0 ≤ βn ≤ 1, while it is accessed by the relay when the source is inactive, i.e., with probability βn .1 If the buffer is empty, the source accesses the channel with probability 1, i.e., β0 = 1. Note that the source’s decision on accessing the channel can be sent to the relay at the beginning of the slot and this represents a slight overhead.2 Based on the ACK/NACK messages emitted by the destination near the end of the time slot, the relay decides whether or not to keep a correctly received packet in its buffer. Specifically, the relay keeps the packets that it decodes correctly, but are undecodable at the destination. If the relay can decode the source packet, it sends back an ACK; otherwise, it sends back a NACK. In case of failure in decoding the source’s packet at both the relay and the destination, a retransmission of the packet by the source is required in the following time slot. We can model the queueing dynamics of the relaying buffer as a birth-death Markov chain (as shown in Fig. 1). The birth-death 1 Throughout the letter, X = 1 − X . Note that X X = 1 − X X and 1 2 1 2 X 1X2 6= X1 X2 . 2 Alternatively, this can be exchanged via a separate low-rate control channel that is used to exchange all system-crucial information. Carrier sensing at the relay is also a possibility to ascertain the source’s state of activity.

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b1

b2

0

1

b3

bN −1

a2

aN −2

2

a0

a1

bN N-1

N aN −1

Fig. 1. Markov chain model for the relaying queue. State n denotes that the relaying queue contains n packets.

model is enforced by the fact that the relay neither receives nor transmits more than one packet in any given slot. In the Markov chain, the states represent the number of packets at the relaying queue. Let x(t) denote the number of packets in the queue at the beginning of time slot t. The transition probabilities of the states are given as follows. an = Pr{x(t) = n+1|x(t−1) = n} = βn Psr Psd , 0 ≤ n ≤ N −1 1≤n≤N

bn = Pr{x(t) = n−1|x(t−1) = n} = βn Prd ,

(1)

where Psr , Psd , and Prd are the probabilities that the s − r, s − d, and r − d links are in outage, respectively. The local balance equations of the Markov chain are given by (2)

n an = n+1 bn+1 , 0 ≤ n ≤ N − 1

where n is thePsteady state probability that the buffer has n N packets. Since n=0 n = 1, we have n = 0

n−1 Y `=0

a`



b`+1

, with 0 = 1 +

N n−1 X Y a` −1 b`+1 n=1

(3)

`=0

We define the source throughput, µs , as the probability of successful packet delivery to the relay/destination N −1 X

µs = Psr Psd

(4)

n βn + Psd N βN

n=0

The expression of µs can be explained as follows. A data packet is delivered successfully if either the s − d link or the s − r link is not in outage. If the buffer is full, it would not accept any extra packets from the source regardless of the s−r link outage state. In this case, the transmitted packet must be delivered only through the s − d link. Since the relay has a finite size buffer, we show in the Appendix that (4) is also equal to the average number of packets received successfully at the destination per time slot. Psd Let K = Psr . Using (1) and (3), the term n βn can be Prd written as n βn = 0

n−1 Y `=0

where Γn =

Qn

a` b`+1

βi i=1 βi .

βn = 0 K n

n Y βi = 0 K n Γn β i i=1

Note that Γ0 = 1, Γ1 =

β1 , β1

Γ3 = Γ2 ββ3 and so on. Since Γn = Γn−1 ββn , then 3

and

Γ2 =

(5)

Γ1 ββ2 , 2

(6) (7)

We can then write the throughput maximization problem as: max

{Γn}N n=0 ≥0

µs =

Psr Psd

PN −1

1+

Pn=0 N

K n Γn +Psd K N ΓN

n n=1 K (Γn +Γn−1 )

∂µs Psr Psd θ(η − γ) = ∂ΓN 1+K (θ + γΓN )2

, s.t. Γ0 = 1 (8)

−1 Note that {βn }N n=1 are obtained from the optimal values of

(9)

Note that given K, η and γ, it is straightforward to show that η < γ when Prd < Psd . In this case, the derivative with respect to ΓN is always negative and the maximum of µs is attained when ΓN = 0. This means that when Prd < Psd , the sr Psd regardless of θ. This has the optimal βN = 0, and µs = P1+K following intuitive explanation. When ΓN = 0, µs becomes independent of Γn and hence βn for all n ∈ {1, 2, . . . , N −1}. If N −1 {βn }n=1 are set to zero, then whenever the relay gets a packet, it will just attempt to transmit it to the destination while the source remains silent. Once it succeeds, the source transmits a new packet with probability one. If βn , n ∈ {1, 2, . . . , N −1}, are set to one, the source transmits with probability one in each time slot, any packet that fails to reach the destination but reaches the relay will be stored in the relay. Once the buffer is full, the relay transmits (and the source remains idle) with probability one. This renders the buffer size unimportant. If η > γ, or equivalently if Prd > Psd , then the derivative is always positive and the maximum of µs is attained when ΓN = ∞ (or equivalently βN = 1). It can be shown that, in this case, the corresponding optimal θ is 3 The optimal sr Psd η µs is thus given by P1+K γ = Psd , thereby indicating that the relay is not helpful to the source in this case. This has the following intuitive explanation. If the outage probability of the direct path is lower than that of the two-hop path, the source justifiably neglects the relay cooperation with the additional rate penalty it incurs due to two-hop transmission, where a packet needs at least two time slots to be delivered to the destination through the relay. Hence, it relies solely on the direct link which accordingly provides higher throughput. Therefore, the average throughput in this case is equal to the probability of the direct link being not in outage. To summarize, the optimal throughput with BSI and without transmit CSI is given by (

n

n Γn = Kn = K n (Γn−1 + Γn ) 0 βn Γn βn = , for n = 1, 2, ..., N Γn + Γn−1

−1 {Γn }N n=1 using Eqn. (7). Note also that queueing delay is not considered in this work. In what follows, we provide an analytical solution to this problem. The denominator of the objective function of (8) can PN −1 KN be rewritten as (1+K)(1+ n=1 K n Γn+ 1+K ΓN ). Therefore, θ+ηΓN sr Psd the objective function becomes µs = P1+K θ+γΓN , where PN −1 n KN θ = 1 + n=1 K Γn ≥ 1, η = P Psd K N , and γ = 1+K . sr Psd The first derivative of µs with respect to ΓN is given by

µs =

Psr Psd Prd , Prd+Psr Psd

Psd ,

if Prd ≤ Psd if Prd > Psd

(10)

The optimal throughput is attained when Prd ≤ Psd via setting N −1 β0 = 1 and βN = 0 for any values of {βn }n=1 , while it is attained at Prd > Psd when the relay is deactivated. IV. P ROPOSED S CHEME W ITH PARTIAL T RANSMIT CSI We assume here that the transmitters know the channel outage states in addition to the buffer state. This can be 3 This can be shown using ∂µs . If γ ≥ η, ∂µs is always negative ∂θ ∂θ regardless of the other parameters, e.g., {Γn }N the value of θ n=1 . Hence, P −1 n that maximizes µs is its lowest feasible value. Since θ = 1+ N n=1 K Γn , P −1 n and N K Γ ≥ 0, then the lowest feasible value of θ is equal to 1. n n=1

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attained, for instance, via one-bit feedback messages from the receivers. Due to the existence of three fading links, each with a binary state space, a channel can fall into one out of eight possible states. If the s − d link is not in outage, which covers four out of the eight possible channel states, then direct transmission is successful regardless of the state of the other two links. If the s − d and s − r links are in outage, then the relay transmits if the r − d link is not in outage and the buffer is nonempty. If the s − d and r − d links are in outage, then the source transmits if the s − r link is not in outage and the buffer is not full. If the s − d link is in outage while the s − r and r − d links are not, we use ξn to denote the probability of source transmission when the relay has n packets. Hence, the probability of relay transmission is ξn = 1 − ξn if the buffer has n packets. It is clear that ξ0 = 1 and ξN = 0.

According to the explanation above, the transition probabilities of the birth-death process for state increments and decrements are given, respectively, by
an = Psd Psr Prd + Prd ξn , 0 ≤ n ≤ N − 1
bn = Psd Prd Psr + Psr ξn , 1 ≤ n ≤ N

(11)

We aim at solving the same optimization problem as in the previous section with the source throughput given as µs = Psd + Psd Psr Prd (1 − N ) + Psd Psr Prd

N −1 X

ξn n (12)

n=0

with {n }N n=0 having the same form as in (3). The throughput can be written as µs = Psd + Psd Psr Prd + Psd Psr φs , where Prd + Prd

PN −1 n=1

φs =

1+

Q QN −1 a` ξn n−1 `=0 b`+1 −Prd `=0 PN Qn−1 a` n=1

a` b`+1

(13)

`=0 b`+1

Maximizing µs is equivalent to maximizing φs . We define a new set of variables, {Ψn }N n=1 , as Ψn =

n−1 Y `=0

a` for n = 1, 2, . . . , N b`+1

With the definition in (14), we can express (13) as PN −1 Prd + Prd n=1 ξn Ψn − Prd ΨN φs = PN 1 + n=1 Ψn

(14)

(15)

−1 Now, by expressing {ξn Ψn }N n=1 as a linear combination of N −1 {Ψn }n=1 , the objective function of the optimization problem becomes a linear-fractional function in {Ψn }N n=1 that can be efficiently solved. From (14), and ∀n ∈ {2, 3, · · · , N }, we get

Psr Prd +Prd ξn−1 Ψn an−1 Psr Prd +Prd ξn−1

= = = Ψn−1 bn Prd 1−Psr ξn Prd Psr +Psr ξn

Therefore, we get the following recurrence relation between Ψn ξn and Ψn−1 ξn−1 as

with a =

Ψn ξn = aΨn − bΨn−1 − Ψn−1 ξn−1

(16)

Ψ1 ξ1 = aΨ1 − α

(17)

1 Psr

and b =

Prd . Prd

Since Ψ1 =

a0 b1

from (14), then

where α =

1 . Prd

Starting from (17), we can obtain Ψn ξn as: Pn−1 Ψn ξn = aΨn + `=1 (−1)n−` (a + b)Ψ` + (−1)n α (18) Pl Noting that n=1 (−1)n = −1 when l is odd and 0 when l is even, we get N −1 X

n

Ψn ξn =

n=1

aΨN −1 − bΨN −2 + aΨN −3 − · · · − bΨ1 , N odd aΨN −1 − bΨN −2 + aΨN −3 − · · · + aΨ1 − α, N even

(19)

Substituting (19) in (13), we obtain the following linearfractional program: c† Ψ+d 1† Ψ+1

max Ψ

s.t. 0 ≤

†

cn Ψ+dn Ψn

(20)

≤ 1, ∀n ∈ {1, 2, · · · , N − 1}

cN † Ψ + dN = 0

where the superscript † denotes the vector transposition and

Ψ = [ΨN , ΨN −1 , · · · , Ψ2 , Ψ1 ]† , c = Prd [−b, a, −b, a, · · · ]†  Prd N odd d= Prd (1 − α) N even (21)

cn = [0, · · · , 0, a, −(a + b), (a + b), · · · ]† , dn = (−1)n α | {z } N −n zeros

Note that Ψ, c, and cn are N -dimensional column vectors, while d and dn are scalars. The linear-fractional program in (20) can be solved efficiently by converting it to a standard N −1 linear program [9]. Afterwards, we obtain {ξn }n=1 as ξn

=

cn † Ψ + d n , ∀n ∈ {1, 2, · · · , N − 1} Ψn

(22)

V. N UMERICAL R ESULTS AND C ONCLUSIONS In this section, we numerically evaluate the access schemes proposed in this letter. In Fig. 2a, we examine the case when no direct link exists (i.e., Psd = 1) and investigate the impact of knowing the BSI as compared to the CSI-only system.4 In this case, the throughput is ultimately limited to 0.5 packets/slot, or equivalently 0.5 bits/sec/Hz when R = 1 bits/sec/Hz, due to the half-duplex operation of the relay. The figure shows that knowing the BSI at the source with the availability of CSI improves the performance relative to the CSI-only and BSI-only cases. As shown in the figure, at low buffer size, specifically for 1 ≤ N ≤ 5, the BSI-only system outperforms the CSI-only system. The behavior is reversed for N > 5. This behavior can be interpreted as follows. Since in the CSI-only system the source is oblivious to the relay’s buffer state, a time slot is wasted when either 1) the source transmits and the buffer is full, or 2) the slot is dedicated for the relay at an empty buffer. At low buffer sizes, and for the same set of channel parameters, the probability of finding either a full or an empty buffer is higher than for those higher buffer sizes. Thus, knowing the BSI for the case of short buffer size is important to avoid wasted transmission instances. For the BSI/CSI case at N = 5, the optimal value of only ξ0 = 1, while it is in the order of 10−6 for all higher states, as compared to an access probability of ρ = 0.3193 4 The case of CSI-only is obtained from the system with BSI and CSI via setting βn = ρ ∈ [0, 1] for all n and finding ρ that maximizes the throughput.

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BSI/CSI CSI only BSI only No Relay

1.5 Throughput [bits/sec/Hz]

Throughput [bits/sec/Hz]

0.5 0.45 0.4 0.35

BSI/CSI CSI only BSI only

0.3

1

0.5

0.25 0.2

5

10

15

20

25

30

0

0

1

2

3

Buffer Size [packets]

Throughput [bits/sec/Hz]

(a) R = 1 bits/sec/Hz, γsr = 20 dB, γrd = 5 dB, and γsd = −10 dB.

0.8

0.75

BSI/CSI CSI only BSI only No Relay

0.7

5

10

15

20

25

30

Buffer Size [packets]

(b) R = 1 bits/sec/Hz, γsr = 10 dB, γrd = 3 dB and γsd = 4 dB. Fig. 2. Throughput vs. buffer size.

in the CSI-only case for all n. Since the second hop has much lower average gain than that of the first hop while the direct link experiences deep fading, this result agrees well with intuition where the BSI/CSI protocol allows the relay to drain its queue whenever it is possible so that it avoids building up and blocking future packets. This is further confirmed by knowing that the resulting full-buffer state probability, N , is equal to 0.008 in the CSI/BSI case as compared to 0.167 when CSI only is available. The figure is generated using R = 1 bits/sec/Hz, γsd = −10 dB, γsr = 20 dB, and γrd = 5 dB. In contrast to Fig. 2a, which shows the case of Psd > Prd , in Fig. 2b, we plot the throughput versus the buffer size for the case Psd ≤ Prd . The parameters used to generate the figure are: R = 1 bits/sec/Hz, γsr = 10 dB, γrd = 3 dB and γsd = 4 dB. As shown in Fig. 2b, the proposed scheme with BSI and with or without the availability of CSI outperforms the case of no relay. The CSI-only system outperforms the BSI-only system for the used parameters. Knowing the BSI and CSI provides further gains. Like Fig. 2a, the figure also shows the fact that the BSI with no CSI does not change with the buffer size of the relay. Consistent with the analytic result provided in Section III, the figure shows the fact that for Psd ≤ Prd , the relay provides no gains to the source when the BSI is known at the source and the CSI is unknown to transmitters. For the BSI/CSI case at N = 5, the optimal value of ξn is ξ0∗ = ξ1∗ = 1 and ξn∗ = 0 for n > 1, while the optimal access probability for the CSI-only case is ρ = 0.2272. In Fig. 3, we show the source throughput versus R. We can notice that the CSI-only and BSI-only scenarios exchange the performance superiority over different transmission rate ranges. Specifically, the CSI-only scenario always yields higher throughput both at low and high transmission rates. This is due to the fact that, at low rates, the buffer occupancy is very low due to low link outage probabilities. Also, at high rates, the outage probability becomes higher, which also alleviates

4

5

6

7

R [bits/sec/Hz]

Fig. 3. Throughput vs. rate for N = 4 packets, γsr = 15 dB, γrd = 15 dB and γsd = 2 dB.

the load on the available buffer, and thus reduces the buffer size effect on the performance. Contrarily, the BSI becomes of immense importance at the intermediate rate range due to the higher buffer occupancy, and hence the BSI-only scenario performs better as explained earlier in Fig. 2a. The figure is generated using the following parameters: N = 4 packets, γsr = 15 dB, γrd = 15 dB and γsd = 2 dB. A PPENDIX The average number of packets received successfully at the destination is given by µav = Psd

N X

n βn + Prd

n=0

N X

n βn

(23)

n=1

PN In (23), Psd n=0 n βn denotes the average number of packets directly PN delivered from the source to the destination, while Prd n=1 n βn denotes those delivered to the destination through the relay.P Summing bothPsides of (2) from n = 0 N −1 N to N − 1, we get n=0 n an = n=1 n bn . Substituting by an = βn Psr Psd and bn = βn Prd , we get the following Psd Psr

N −1 X

n βn = Prd

n=0

N X n=0

n βn + Prd

n βn

(24)

n=1

Using (24) in (4), we reach µs = Psd

N X

N X

n βn = µav

(25)

n=1

Hence, the source and the end-to-end throughput are equal. R EFERENCES [1] B. Xia, Y. Fan, J. Thompson, and H. V. Poor, “Buffering in a threenode relay network,” IEEE Trans. Wireless Commun., vol. 7, no. 11, pp. 4492–4496, Nov. 2008. [2] I. Krikidis, T. Charalambous, and J. S. Thompson, “Buffer-aided relay selection for cooperative diversity systems without delay constraints,” IEEE Trans. Wireless Commun., vol. 11, no. 5, pp. 1957–1967, May 2012. [3] N. Zlatanov and R. Schober, “Buffer-aided relaying with adaptive link selection–fixed and mixed rate transmission,” IEEE Trans. Inf. Theory, vol. 59, no. 5, pp. 2816–2840, May 2013. [4] G. Chen, Z. Tian, Y. Gong, Z. Chen, and J. Chambers, “Max-ratio relay selection in secure buffer-aided cooperative wireless networks,” IEEE Trans. Inf. Forensics Security, vol. 9, no. 4, pp. 719–729, Apr. 2014. [5] V. Majjigi, D. O’Neill, and J. Cioffi, “Buffer state information: Two-level water-filling for fixed rate applications,” in Proc. IEEE Glob. Telecomm. Conf., Nov. 2009. [6] N. Zlatanov, R. Schober, and P. Popovski, “Buffer-aided relaying with adaptive link selection,” IEEE J. Sel. Areas Commun., vol. 31, no. 8, pp. 1530–1542, Aug. 2013. [7] ——, “Throughput and diversity gain of buffer-aided relaying,” in Proc. IEEE Glob. Telecomm. Conf., 2011, pp. 1–6. [8] T. Islam, D. S. Michalopoulos, R. Schober, and V. Bhargava, “Delay constrained buffer-aided relaying with outdated CSI,” in Proc. IEEE WCNC, Apr. 2014. [9] S. Boyd and L. Vandenberghe, Convex optimization. Cambridge University Press, 2004.

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