IEEE TRANSACTIONS ON COMMUNICATIONS 1
Buffer-Aided Non-Orthogonal Multiple Access
Relaying Systems in Rayleigh Fading Channels
Qi Zhang, Member,IEEE, Zijun Liang, Quanzhong Li, and Jiayin Qin
Abstract—Non-orthogonal multiple access (NOMA) is a
promising technology in future communication systems. In this
paper, we propose a buffer-aided NOMA relaying system which
consists of a source, a relay, and two destinations. In the
relaying system, the relay helps the source transmit packets
to two destinations simultaneously using NOMA scheme. We
theoretically derive outage probabilities of source-to-relay link
and relay-to-destinations links considering two scenarios that the
relay does and does not know the channel state information (CSI)
from itself to two destinations. When the relay knows CSI, the
obtained outage probability of relay-to-destinations links involves
integration operation. Thus, we derive an upper bound and two
lower bounds. Simulation results demonstrate that two lower
bounds approach exact outage probability at low and high signal-
to-noise ratios, respectively. We also propose a relay decision
scheme for the buffer-aided NOMA relaying system. Based on
the obtained system outage probability, we theoretically derive
the diversity order. It is found that no matter whether the relay
knows CSI or not, the diversity order of 2 can be achieved when
the buffer size is larger than or equal to 3.
Index Terms—Buffer-aided relaying, diversity order, non-
orthogonal multiple access (NOMA), outage probability, average
packet delay.
I. INT ROD UC TI ON
In decode-and-forward (DF) relaying system, conventional
relay without employing a data buffer can temporarily store
only one data packet. When the relay-to-destination link is
in outage and one data packet is stored at the relay, the
relay keeps in silence (neither transmits nor receives). When
the source-to-relay link is in outage and no data packet is
stored at the relay, the relay still keeps in silence. In buffer-
aided relaying system, a data buffer is employed at the relay.
The buffer enables the relay to transmit when the source-
to-relay link is in outage and to receive when the relay-to-
destination link is in outage. Thus, the system throughput can
be significantly enhanced [1]–[7].
This work was supported in part by the National Natural Science Foundation
of China under Grant 61472458 and Grand 61672549, in part by the
Guangdong Natural Science Foundation under Grant 2014A030311032, Grant
2014A030313111, and Grant 2014A030310374, in part by the Guangzhou
Science and Technology Program under Grant 201607010098, and in part
by the Fundamental Research Funds for the Central Universities under Grant
15lgzd10 and Grant 15lgpy15.
Q. Zhang, Z. Liang, and J. Qin are with the School of Electronics and
Information Technology, Sun Yat-sen University, Guangzhou 510006, Guang-
dong, China (e-mail: zhqi26@mail.sysu.edu.cn, kazuyaluffy@vip.qq.com, is-
sqjy@mail.sysu.edu.cn). J. Qin is also with Xinhua College of Sun Yat-
sen University, Guangzhou 510520, Guangdong, China. Q. Li is with the
School of Data and Computer Science, Sun Yat-sen University, Guangzhou
510006, Guangdong, China, also with the Guangdong Key Laboratory of
Information Security, Guangzhou, Guangdong, China, and also with the
Collaborative Innovation Center of High Performance Computing, National
University of Defense Technology, Changsha, Hunan, China (e-mail: li-
quanzhong2009@gmail.com).
For buffer-aided relaying system with multiple relays, Luo
et. al proposed a buffer state based relay selection scheme
which selects a relay node based on both the channel condi-
tions and the buffer states of the relays [6]. The main idea
in the buffer state based relay selection scheme is to make
all buffers of relays have only one packet, i.e., if its buffer is
empty, the relay should receive packets as long as it is possible
and if its buffer stores two or more packets, the relay should
transmit packets as long as it is possible.
All the aforementioned works consider orthogonal multiple
access (OMA) scheme, i.e., different signals from a source
to multiple destinations are orthogonal with one another,
either in time domain, or in frequency domain, or in code
domain. Non-orthogonal multiple access (NOMA) scheme
allows communication resources, such as time, frequency,
and code, to be shared by all users and achieves higher
transmission rate than OMA scheme [8]–[17]. The principle
of NOMA is that the transmitter transmits the superposition of
multiple signals to multiple receivers. The receivers close to
the transmitter which have better channel conditions are able
to employ successive interference cancellation (SIC) to detect
and remove the signals to receivers far from the transmitter
which have poorer channel conditions. After SIC, the receivers
close to the transmitter detect their own signals. In [18], [19],
NOMA relaying systems without buffering were proposed and
investigated.
In this paper, we propose a buffer-aided NOMA relay-
ing system which consists of a source, a relay, and two
destinations. In the relaying system, the relay helps the
source transmit packets to two destinations simultaneously
using NOMA scheme. It is noted that buffer-aided relaying
systems considering NOMA were also investigated in [20]–
[22]. The differences between our work and those in [20]–
[22] are listed as follows. In [20]–[22], resource allocation
optimization problem is investigated whereas here the system
outage probability, diversity order, and average packet delay
are theoretically derived. Furthermore, in [20]–[22], infinite
buffer size is assumed whereas here finite buffer size and
buffer state based relay decision scheme are considered.
The main contributions of this paper are listed as follows.
We theoretically derive outage probabilities of source-to-relay
link and relay-to-destinations links considering two scenarios
that the relay does and does not know the channel state
information (CSI) from itself to two destinations. When the
relay knows CSI, the obtained outage probability of relay-
to-destinations links involves integration operation. Thus, we
derive an upper bound and two lower bounds. Simulation
results demonstrate that two lower bounds approach exact out-
IEEE TRANSACTIONS ON COMMUNICATIONS 2
S R
D
1
Buffer
D
2
h
0
h
1
h
2
Fig. 1. System model of a buffer-aided NOMA relaying system where the
source, the relay and two destinations are denoted as S, R, D1, and D2,
respectively.
age probability at low and high signal-to-noise ratios (SNRs),
respectively. We also propose a relay decision scheme for the
buffer-aided NOMA relaying system. Based on the obtained
system outage probability, we theoretically derive the diversity
order. It is found that no matter whether the relay knows CSI
or not, the diversity order of 2 can be achieved when the buffer
size is larger than or equal to 3.
The benefit of applying the proposed buffer-aided NOMA
system is that when the channel conditions from relay to
destinations are good enough, the conventional buffer-aided
OMA systems may switch to the proposed buffer-aided NO-
MA mode and the throughput from relay to two destinations
doubles. Furthermore, NOMA provides another method for the
source and the relay to serve two destinations simultaneously
other than time-division multiple access (TDMA), frequency-
division multiple access (FDMA) and code-division multiple
access (CDMA) schemes.
The rest of the paper is organized as follows. Section
II describes the system model. Section III and Section IV
investigate outage probabilities of source-to-relay link and
relay-to-destinations links considering that the relay does
and does not know the CSI from itself to two destinations,
respectively. In Section V, we propose the relay decision
scheme. In Section VI, Section VII and Section VIII, system
outage probability, diversity order, and average packet delay
are theoretically derived, respectively. Simulation results are
shown and discussed in Section IX. We conclude our paper in
Section X.
II. SY ST EM MO DE L
Consider a buffer-aided NOMA relaying system which
consists of a source, a relay, and two destinations, as shown
in Fig. 1. We assume that the direct links between the source
and two destinations are sufficiently weak to be ignored. This
occurs when the direct links are blocked due to long-distance
path loss or obstacles.
A. Buffer-Aided Relaying
The network operates in the time-division duplex (TDD)
mode where time duration is partitioned into slots with equal
length of T. In each time slot, the source or the relay is
selected to transmit packets. If the source is selected, it
assembles information symbols intended for two destinations
into a packet with size of 2r0BT bits and transmits it to
the relay where r0denotes the target transmission rate and
Bdenotes the bandwidth of the system. A packet is divided
into two parts with equal size where part 1 and part 2 are
intended for destination 1 and destination 2, respectively.
The relay is equipped with a buffer. The buffer consists
of L≥2storage units and each storage unit can store
2r0BT bits data. The relay decodes the received information
symbols in a packet and store them into a storage unit. A
storage unit is divided into two parts with equal size where
part 1 and part 2 are for the storage of information symbols
intended for destination 1 and destination 2, respectively. If the
relay is selected to transmit, it retrieves information symbols
from a storage unit and reassembles them into two packets
with the same size of r0BT bits. The information symbols
of two packets are superimposed using NOMA scheme and
transmitted to two destinations.
Assume that the source always has information symbols to
transmit. At the relay, the number of storage units which have
stored information symbols is denoted as lwith 0≤l≤L. For
different values of l, we have different numbers of available
links for transmission:
•l= 0: Only the source-to-relay link is available because
the relay has no information symbol to transmit.
•0< l < L: Both the source-to-relay link and the relay-
to-destinations links are available.
•l=L: Only the relay-to-destinations links are available
because the relay has no free storage units to store the
received information symbols.
B. Channel Model
The channels are assumed to be flat Rayleigh block fading
channels which remain constant during one time slot and
change randomly from one time slot to anther. The channel
response from source to relay is denoted as h0. The channel
responses from the source to destination 1 and destination 2
are denoted as h1and h2, respectively. The channel responses
h0,h1, and h2are circularly symmetric complex Gaussian
distributed random variables with zero mean and variances of
Ω0,Ω1, and Ω2, respectively.
C. Performance Metrics
Our objective in this paper is to investigate the following
performance metrics of the buffer-aided NOMA relaying sys-
tem which are important to evaluate quality-of-service (QoS)
of the communication systems. Furthermore, the obtained
expressions of following performance metrics provide mean-
ingful objective functions and constraints for system parameter
optimizations.
•Target Transmission rate: Target transmission rate from
the relay to each destination is predefined to be r0>0.
Thus, the target transmission rate from the source to the
relay should be 2r0.
•Outage Probability: Outage probability of the system
is defined as the probability that neither the source-to-
relay link nor the relay-to-destinations links is available
for transmission to achieve the target transmission rate.
When an outage event happens, the relay cannot receive
information symbols transmitted from the source. Fur-
thermore, the relay cannot transmit information symbols
IEEE TRANSACTIONS ON COMMUNICATIONS 3
to two destinations using NOMA scheme. Note that when
the proposed system is in outage, the system may switch
to the conventional buffer-aided OMA relaying mode,
which will be discussed in detail in Section III.
•Throughput: Throughput of the system is defined as the
successful data transmission rate per unit time duration
from the source to two destinations.
•Average Packet Delay: Average packet delay of the sys-
tem is defined as over a long period of time, the average
time duration from the time that a packet is sent from the
source to the time that the packets are received by two
destinations.
III. OUTAG E PROBABILITY OF LINKS WH EN T HE RE LAY
KNOW S h1AN D h2
In a time slot, if the source is selected to transmit a packet,
the instantaneous SNR of the source-to-relay link is
γ0=Ps|h0|2
σ2
r
=β|h0|2(1)
where Psdenotes the transmit power of the source, σ2
rdenotes
the power of additive Gaussian noise at the relay and β=
Ps/σ2
r. Since the target transmission rate from source to relay
is 2r0, the outage probability of source-to-relay link is
P0=Pr {log2(1 + γ0)<2r0}
=1 −exp −22r0−1
β|h0|2.(2)
In a time slot, if the relay is selected to transmit a packet,
it transmits the superimposed information symbols,
x=√αx1+√1−αx2(3)
where x1and x2denote information symbols intended for
destination 1 and destination 2, respectively, with E[|x2
1|] =
E[|x2
2|] = 1, and 0< α < 1denotes the power allocation
coefficient. Note that α̸= 0 and α̸= 1 because the target
transmission rate from relay to each destinations, r0, is larger
than 0. In this section, we assume that the relay knows the
channel responses, h1and h2. Thus, the relay is able to
adjust the power allocation coefficient, α, to minimize outage
probability of relay-to-destinations links.
The received signal at destination i,i∈ {1,2}, is
yi=hiαPrx1+hi(1 −α)Prx2+ni(4)
where Prdenotes the transmit power of the relay, nidenotes
the additive Gaussian noise at destination iwith zero mean
and variance σ2
d.
If |h1|≥|h2|, the instantaneous signal-to-interference-and-
noise ratio (SINR) of the relay-to-destination-2 link is
γ2=(1 −α)Pr|h2|2
αPr|h2|2+σ2
d
=(1 −α)ρ|h2|2
αρ|h2|2+ 1 (5)
where ρ=Pr/σ2
dand the information symbol x1is considered
as interference at destination 2. At destination 1, since |h1| ≥
|h2|, we have
(1 −α)Pr|h1|2
αPr|h1|2+σ2
d
=(1 −α)ρ
αρ + (1/|h1|2)≥γ2.(6)
From (6), if the destination 2 is able to detect the information
symbol x2, so is destination 1. Applying SIC, destination 1 is
able to remove the detected information symbols x2from its
received signal. Thus, the instantaneous SNR of the relay-to-
destination-1 link is
γ1=αPr|h1|2
σ2
d
=αρ|h1|2.(7)
Since the target transmission rate from relay to each destina-
tion is r0, we have
log2(1 + min{γ1, γ2})≥r0.(8)
Note that if (8) is satisfied, from (6), it is guaranteed that
destination 1 is able to detect the information symbol x2
and apply SIC. From (8), the following inequalities should
be satisfied,
(1 −α)ρ|h2|2
αρ|h2|2+ 1 ≥b−1,(9)
αρ|h1|2≥b−1,(10)
where b= 2r0. Thus, we have
ζ
|h1|2≤α≤b−11−ζ
|h2|2(11)
where
ζ=b−1
ρ.(12)
Outage will not occur when there exists αwhich satisfies (11),
which is equivalent to
ζ
|h1|2≤b−11−ζ
|h2|2,(13)
i.e.,
|h2|2≥ζ|h1|2
|h1|2−bζ .(14)
Since |h1| ≥ |h2|, we have
|h1|2≥ζ|h1|2
|h1|2−bζ ,(15)
which is equivalent to
|h1|2≥(b+ 1)ζ. (16)
Since |h1|2and |h2|2are exponential distributed random
variables, the probability that |h1|≥|h2|and outage will not
occur is
P1=∞
(b+1)ζx
ζx
x−bζ
1
Ω2
exp −y
Ω2dy 1
Ω1
exp −x
Ω1dx.
(17)
After some mathematical manipulations, P1is simplified as
P1=1
Ω1
exp −ζΩ1+bζΩ2
Ω1Ω2∞
ζ
exp −x
Ω1−bζ2
Ω2xdx
−Ω2
Ω1+ Ω2
exp −ζ(Ω1+ Ω2)(b+ 1)
Ω1Ω2.(18)
We have the following proposition.
Proposition 1: The upper and lower bounds on P1are given
IEEE TRANSACTIONS ON COMMUNICATIONS 4
as follows
Pupper1
1= exp −ζΩ1+bζΩ2
Ω1Ω24ζ2b
Ω1Ω2
K1
4ζ2b
Ω1Ω2
−Ω2
Ω1+ Ω2
exp −ζ(Ω1+ Ω2)(b+ 1)
Ω1Ω2,(19)
Pupper2
1= exp −ζΩ1+ (b+ 1)ζΩ2
Ω1Ω2
−Ω2
Ω1+ Ω2
exp −ζ(Ω1+ Ω2)(b+ 1)
Ω1Ω2,(20)
Plower
1=Ω2
Ω1+ Ω2
exp −ζ(Ω1+ Ω2)(b+ 1)
Ω1Ω2,(21)
where K1(·)denotes the modified Bessel function of the
second kind of order 1 [23].
Proof : See Appendix A.
Similarly, the probability that |h2| ≥ |h1|and outage will
not occur is
P2=1
Ω2
exp −ζΩ2+bζΩ1
Ω1Ω2∞
ζ
exp −x
Ω2−bζ2
Ω1xdx
−Ω1
Ω1+ Ω2
exp −ζ(Ω1+ Ω2)(b+ 1)
Ω1Ω2.(22)
Thus, the outage probability of relay-to-destinations links is
P3= 1 −P1−P2(23)
whose lower and upper bounds are
Pupper
3= 1 −exp −ζ(Ω1+ Ω2)(b+ 1)
Ω1Ω2,(24)
Plower1
3= 1 + exp −ζ(Ω1+ Ω2)(b+ 1)
Ω1Ω2
−exp −ζΩ1+bζΩ2
Ω1Ω2+ exp −ζΩ2+bζ Ω1
Ω1Ω2
·4ζ2b
Ω1Ω2
K1
4ζ2b
Ω1Ω2
,(25)
Plower2
3= 1 + exp −ζ(Ω1+ Ω2)(b+ 1)
Ω1Ω2
−exp −ζΩ1+ (b+ 1)ζΩ2
Ω1Ω2
−exp −ζΩ2+ (b+ 1)ζΩ1
Ω1Ω2.(26)
Remark 1: In this paper, we consider that there exist two
destinations in the buffer-aided NOMA relaying system. Al-
though the system model is limited, it provides useful insights
on the performance of the system with multiple destinations,
especially on the diversity order derivations.
Remark 2: When the relay knows h1and h2, only varying
αis considered. We may also vary the transmit power of
relay and transmission rate. In this paper, our focus is on
performance analysis given the transmit power of relay and
transmission rate. Given the outage or delay performance
constraints, the optimization of system parameters such as
the transmit power of relay and transmission rate may be an
interesting future work.
Remark 3: In this paper, we define that an outage event
happens when one or both of the destinations cannot detect its
corresponding information symbol. The scheme is not optimal.
This is because the situation exists that one of the destinations
is able to detect its corresponding information symbol and
the other is not. When an outage event defined here happens,
the system may switch to the conventional buffer-aided OMA
relaying mode and the system still works. However, if we
consider the switch between the proposed buffer-aided NO-
MA relaying mode and the conventional buffer-aided OMA
relaying mode, i.e., if we define that outage event happens
when both of the destinations cannot detect its corresponding
information symbol, the performance analysis is very compli-
cated because of the coupled buffer-aided NOMA and OMA
relaying modes. Since the performance of the conventional
buffer-aided OMA relaying system was extensively analyzed
in the literature, we may focus on the situation where the
system works in buffer-aided NOMA relaying mode and draw
some meaningful conclusions.
Furthermore, insight into the system such as diversity order
may not be obtained if the coupled buffer-aided NOMA and
OMA relaying system is considered. It is shown in [6] the
diversity order of 2 can be achieved when the buffer size
is larger than or equal to 3 for buffer-aided OMA relaying
system. Thus, if the theoretically derived diversity order of
coupled buffer-aided NOMA and OMA relaying system is
2 when the buffer size is larger than or equal to 3, the
diversity order of solely buffer-aided NOMA relay system is
still unknown.
Remark 4: In practical buffer-aided relaying systems, the
switch between the proposed NOMA relaying mode and the
conventional OMA relaying mode should be allowed. If the
channel responses from the relay to two destinations are
good enough and the transmit power of relay is sufficient
large, the system should be switched to the proposed NOMA
relaying mode and thus the throughput from relay to two
destinations can be doubled. Furthermore, NOMA provides
another method for the source and the relay to serve two
destinations simultaneously other than TDMA, FDMA and
CDMA schemes.
It is noted that because we define an outage event happens
when one or both of the destinations cannot detect its cor-
responding information symbol here. The direct comparison
of the proposed NOMA relaying system and the conventional
OMA relaying system is improper. This is because when an
outage event defined here happens, the system may switch to
the conventional OMA relaying mode.
IV. OUTAG E PROBAB IL IT Y OF LINKS WH EN T HE RE LAY
DOE SN ’TKNOW h1A ND h2
In this section, we assume that the relay doesn’t know
the channel responses, h1and h2. The relay only knows the
variances of h1and h2, i.e., Ω1and Ω2. The relay adjusts
the power allocation coefficient, α, to minimize the outage
probability of relay-to-destinations links.
Without loss of generality, we assume that Ω1≥Ω2. At
the destinations, when |h1|≥|h2|, destination 1 is able to
IEEE TRANSACTIONS ON COMMUNICATIONS 5
detect the information symbol intended for destination 2 and
then remove the received information symbol from its received
signal. When (9) and (10) are satisfied, the outage will not
occur. It is noted that in this section, the power allocation
coefficient, α, is a constant instead of a variable as in Section
III. From (9) and (10), we have
|h1|2≥ζ
α(27)
and
|h2|2≥ζ
1−bα.(28)
Since |h1|2and |h2|2are exponential distributed random
variables, the probability that |h1|≥|h2|and outage will not
occur is
˜
P1=∞
ζ
αx
ζ
1−bα
1
Ω2
exp −y
Ω2dy 1
Ω1
exp −x
Ω1dx
= exp −ζ
(1 −bα)Ω2−ζ
αΩ1
−Ω2
Ω1+ Ω2
exp −ζΩ1+ζΩ2
αΩ1Ω2.(29)
Similarly, the probability that |h2| ≥ |h1|and outage will not
occur is
˜
P2= exp −ζ
(1 −bα)Ω1−ζ
αΩ2
−Ω1
Ω1+ Ω2
exp −ζΩ1+ζΩ2
αΩ1Ω2.(30)
Thus, the outage probability of relay-to-destinations links is
˜
P3= 1 −˜
P1−˜
P2.(31)
The optimal power allocation coefficient, denoted as α∗, which
minimizes the outage probability of relay-to-destinations links,
˜
P3, can be obtained by one-dimensional search. From (9), we
know
(1 −bα)ρ|h2|2≥b−1.(32)
Since b > 1, we have α < b−1. Thus, the upper and lower
bounds on αis
0< α < 2−r0.(33)
V. RELAY DECISION SCHEME
In this section, assuming L≥2, we propose the relay
decision scheme as shown in Table I to reduce the average
packet delay and improve the reliability of the system, where
“SR” denotes the source-to-relay link, “RD” denotes the relay-
to-destinations links, “out” denotes that the outage occurs in
the corresponding link(s), “suc” denotes that the outage does
not occur, “S” denotes that the relay keeps in silence (neither
transmits nor receives), “R” denotes that the relay chooses to
receive a packet, “T” denotes that the relay chooses to transmit
two packets to two destinations simultaneously using NOMA
scheme, PSR denotes the outage probability of the source-to-
relay link, PRD denotes the outage probability of the relay-to-
destinations links when the relay does or does not know h1
and h2,¯
PSR = 1 −PSR and ¯
PRD = 1 −PRD .
TABLE I
REL AY DECISION SCHEME
Case SR RD lDecision Probability
Case 1 out out S PSRPRD
Case 2 out l= 0 SPSR
Case 3 out l=LSPRD
Case 4 suc out l < L R¯
PSRPRD
Case 5 out suc l > 0TPSR ¯
PRD
Case 6 suc suc l≥2T¯
PSR ¯
PRD
Case 7 suc suc l≤1R¯
PSR ¯
PRD
From Sections III and IV, PSR =P0. When the relay knows
h1and h2,PRD =P3whereas when the relay doesn’t know
h1and h2,PRD =˜
P3. It is noted that Case 6 gives the relay
higher priority to transmit when both the source-to-relay link
and relay-to-destinations links are available, which aims to
reduce the packet delay. Case 7 aims to reduce the probability
that the buffer of the relay becomes empty, which increases
the number of available links of system and thus increases the
reliability of the system.
Remark 5: The proposed relay decision scheme in this
section, in the sense of outage probability minimization, is not
optimal. The optimal relay decision scheme should optimize
the value of lwhich differentiates Case 6 and Case 7 in Table I.
The optimal value of lin the outage probability minimization
problem may be found by the logarithmic moment generating
function and Lagrangian approach based method proposed in
[24]. However, to deal with complex expressions of P3and
˜
P3in (23) and (31) is difficult. An alternative method to find
the optimal value of lwhich differentiates Case 6 and Case 7
is exhaustive search, which is employed in the simulations in
the paper.
It is noted that the optimal value of lwhich minimizes
outage probability may not be optimal in the sense of average
packet delay minimization. Larger lwhich differentiates Case
6 and Case 7 results in larger average packet delay. Since
it will be shown in Section VII that the value of lwhich
differentiates Case 6 and Case 7 in the proposed relay decision
scheme is the minimum value of lwhich achieves the full
diversity order of 2. Thus, the proposed relay decision scheme
in this section leads to tradeoff between outage probability
minimization and average packet delay minimization. Fur-
thermore, it will be shown in the simulations that outage
probability achieved by the proposed relay decision scheme
is close to that achieved by the optimal relay decision scheme
which minimizes outage probability.
VI. OU TAGE PRO BABILITY ANA LYSI S OF T HE
BUFFER-AI DE D NOMA REL AYIN G SYSTEM
In this section, we analyze outage probability of the buffer-
aided relaying system which employs the relay decision
scheme proposed in Section V. Our analysis is based on the
Markov chain.
IEEE TRANSACTIONS ON COMMUNICATIONS 6
A. State Transition Matrix of the Markov Chain
In the buffer-aided relaying system, the number of storage
units which have stored information symbols, l, is described
by the states of Markov chain. Specifically, since 0≤l≤L,
lhas L+ 1 different values which decides that in the Markov
chain, there exists L+ 1 states. The nth state is defined as
sn,{l= (n−1),1≤n≤(L+ 1)}.(34)
Let Adenote the (L+ 1) ×(L+ 1) state transition matrix
of the Markov chain, in which the entry in the mth row and
the nth column, denoted as Amn, is the transition probability
to move from state snat time tto state smat time t+ 1, i.e.,
Amn =Pr (Xt+1 =sm|Xt=sn).(35)
The transition probability Amn depends on the status of the
buffer and the available links that can successfully transmit
packets.
For the buffer-aided relaying system with L= 4, the
corresponding state transition matrix Ais expressed as
A=
PSR PSR ¯
PRD 0 0 0
¯
PSR PSRPRD ¯
PRD 0 0
0¯
PSR PSRPRD ¯
PRD 0
0 0 ¯
PSRPRD PSR PRD ¯
PRD
0 0 0 ¯
PSRPRD PRD
.(36)
B. Steady State Distribution of the Markov Chain
From Table I, the state transition diagram of the Markov
chain is shown in Fig. 2.
0 1 2 L
RDSR
PP
RDSR
PP
SR
P
RD
P
RDSR
PP
SR
PSR
P
RDSR
PP
RDSR
PP
RD
PRD
P
RD
P
...
Fig. 2. State transition diagram of the Markov chain.
We have the following proposition to derive the steady state
distribution.
Proposition 2: The Markov chain with the state transition
matrix Ais irreducible and aperiodic.
Proof : Since in practical, we have 0< PSR <1and 0<
PRD <1, all states of the Markov chain communicate. Thus,
the Markov chain is irreducible. Furthermore, PSRPRD >0
means that the state of the Markov chain remains unchanged
when the system is in outage, i.e., the probability of staying
at any state is greater than zero. Thus, the Markov chain is
aperiodic.
According to [4], the steady state distribution of the Markov
chain, denoted by a column vector π= [π1, π2,···, πL+1 ], is
be expressed as
π= (A−I+B)−1b(37)
where b= [1,···,1]T,Idenotes the identity matrix and B
denotes an (L+ 1) ×(L+ 1) matrix with all elements to be
one.
C. Outage Probability of the System
Outage probability of the buffer-aided relaying system is
defined as the probability that the relay remains silence, i.e,
the relay neither transmits nor receives. When the buffer-aided
relaying system is in outage, the number of storage units in
the buffer of relay remains the same, i.e., the state of Markov
chain remains the same. Thus, from the relay decision scheme
proposed in Section V, outage probability of the system is
given by
Pout =
L+1
n=1
πnAnn.(38)
Remark 6: If PRD in Ann is replaced with Pupper
3,Plower1
3,
and Plower2
3, the upper bound and two lower bounds on Pout
when the relay knows h1and h2are obtained.
D. Throughput
The throughput of the system, which is equal to the through-
put of the relay, is the transmission rate r0multiplied by
the probability of successful transmission of the relay. If the
transmission of the relay is successful, the state snat time t
moves to state sn−1at time t+ 1. Thus, the throughput is
given by
η=r0
L+1
n=2
πnA(n−1)n.(39)
VII. DIV ER SI TY OR DE R OF T HE BUFFER-AID ED NOMA
REL AYIN G SYS TE M
The diversity order is defined as in [6]
d=−lim
¯γ→∞
log Pout
log ¯γ(40)
where ¯γis the SNR for each link.
A. Diversity Order When the Relay Knows h1and h2
When L= 2 and ¯γ→ ∞, from [6], we know that π1= 0,
π2=1
2, and π3=1
2. From Table I, the relay can only transmit
packets with l= 2 (i.e., n= 3). The relay can either receive
or transmit packets with l= 1 (i.e., n= 2). Thus, from (38),
we have
lim
¯γ→∞ Pout =1
2lim
¯γ→∞ P0+1
2lim
¯γ→∞ P0P3(41)
where P0is defined in (2). Since
−lim
β→∞
log P0
log β= 1,(42)
the diversity order when L= 2 is 1.
When L≥3and ¯γ→ ∞, from [6], we know that π1= 0,
πL+1 = 0, and L
n=2 πn= 1. From Table I, the relay can
either receive or transmit packets with 1≤l≤L−1(i.e.,
2≤n≤L). Thus, from (38), we have
lim
¯γ→∞ Pout = lim
¯γ→∞ P0P3.(43)
The diversity order is
d=−lim
β→∞
log P0
log β−lim
ρ→∞
log P3
log ρ.(44)
IEEE TRANSACTIONS ON COMMUNICATIONS 7
When ¯γ→ ∞ and the relay knows h1and h2, we have the
following proposition.
Proposition 3:
−lim
ρ→∞
log Plower2
3
log ρ= 1.(45)
Proof : See Appendix B.
On the other hand, it can be proved that
−lim
ρ→∞
log Pupper
3
log ρ= 1.(46)
Combining (45) and (46), we know
−lim
ρ→∞
log P3
log ρ= 1.(47)
Thus, the diversity order when L≥3is 2.
B. Diversity Order When the Relay Doesn’t Know h1and h2
When L= 2, the diversity order is still 1. When L≥3,
the diversity order is obtained by (44) where P3is replaced
by ˜
P3. To continue, we have the following proposition.
Proposition 4:
−lim
ρ→∞
log ˜
P3
log ρ= 1.(48)
Proof : The proof is similar to that of Proposition 3 and thus
is omitted for brevity.
From Proposition 4, we know that the diversity order when
L≥3is 2.
VIII. AVERAGE PACKET DELAY ANA LYSIS OF THE
BUFF ER -AID ED NOMA RELAYING SYST EM
Denote PTand PRas the probabilities that the relay
transmits and receives packets, respectively. Over a long period
of time, from the property that the sum of the packets received
by the relay should be equal to the sum of the packets
transmitted by it, we obtain
PT=PR.(49)
Since the relay either transmits packets, or receives packets,
or keeps silent when the whole system is in outage, we have
PT+PR+Pout = 1.(50)
Thus,
PT=PR=1−Pout
2.(51)
Denote ηSand ηRas the throughputs of source and relay,
respectively. We obtain
ηS=ηR=PT=(1 −Pout)
2.(52)
A. Average Packet Delay at the Source
Denote QSand QRas the average queuing lengthes (in num-
ber of time slots) at source and relay, respectively. According
to Little’s law [25], the average queuing length at source,
QS, is determined by how fast the packets are delivered by
the source. Since in each time slot, at most one packet is
transmitted from the source, we have
QS= 1 −PT=1 + Pout
2.(53)
The average packet delay at the source is
DS=QS
ηS
=1 + Pout
1−Pout
.(54)
B. Average Packet Delay at the Relay
The average queuing length at the relay is give by
QR=
L+1
n=2
πn(n−1).(55)
The average packet delay at the relay is
DR=QR
ηR
.(56)
C. Total Average Packet Delay
The total average packet delay of the system is
D=DS+DR.(57)
We have the following proposition.
Proposition 5: When ¯γ→ ∞, the total average packet delay
of the system is D= 4.
Proof : When ¯γ→ ∞, from Table I, only Case 6 and Case
7 can be chosen by the relay. Without loss of generality, we
suppose that the system begins in the state s1(i.e., l= 0).
Based on Case 7, the relay will receive packets in the first
and the second time slots and the system moves to the state
s3(i.e., l= 2). Then, based on Case 6, the relay will transmit
two packets to two destinations, respectively, and the system
moves to the state s2(i.e., l= 1). After that, based on Case
7, the relay will receive a packet and the system moves to the
state s3(i.e., l= 2) again. This process repeats and the buffer
state is cycling at s2(i.e., l= 1) and s3(i.e., l= 2). This
indicates that π2=π3=1
2.
Thus, from (55), we obtain
QR=π2(2 −1) + π3(3 −1) = 3
2.(58)
From (56), we have
DR=3
1−Pout
= 3 (59)
where Pout →0when ¯γ→ ∞. From (54), the average packet
delay at the source is DS= 1. Thus, we have D= 4.
IX. SIMULATION RESU LTS
In this section, we present simulation results to validate our
analysis and design. We compare our proposed relay decision
scheme with two schemes, i.e., scheme with the priority to
transmit [26] and scheme with the priority to receive [1], as
shown in Table II and Table III, respectively. The scheme with
the priority to transmit means that the relay always chooses to
transmit packets as long as both transmitting and receiving are
available. The scheme with the priority to receive means that
IEEE TRANSACTIONS ON COMMUNICATIONS 8
the relay always chooses to receive packets as long as both
transmitting and receiving are available. In the simulations, we
assume that Ω0= 1,Ω1= 1.2,Ω2= 0.5, and r0= 2 bps/Hz.
Furthermore, we assume that β=ρ= ¯γ. The number of
storage units in the buffer, i.e., the buffer size, if not specified,
is L= 50.
TABLE II
REL AY DECISION SCHEM E WI TH TH E PRI ORI TY TO TRANSMIT
Case SR RD lDecision
Case 1 out out S
Case 2 out l= 0 S
Case 3 out l=LS
Case 4 suc l > 0T
Case 5 suc suc l= 0 R
Case 6 suc out l < L R
TABLE III
REL AY DECISION SCHEM E WI TH TH E PRI ORI TY TO RE CE IVE
Case SR RD lDecision
Case 1 out out S
Case 2 out l= 0 S
Case 3 out l=LS
Case 4 suc l < L R
Case 5 suc suc l=LT
Case 6 out suc l > 0T
In Fig. 3, we present the outage probabilities of our pro-
posed relay decision scheme when the relay knows h1and h2
where “Analytical”, “Upper Bound”, “Lower Bound 1”, and
“Lower Bound 2” in the legend denote the theoretical results
obtained by replacing PRD in Ann of (38) with P3,Pupper
3,
Plower1
3, and Plower2
3, respectively. From Fig. 3, it is observed
that the “Analytical” results match the simulation results. From
Fig. 3, it is also found that at low SNR, “Lower Bound 1” is
closer to “Analytical” and simulation results whereas at high
SNR, “Lower Bound 2” is closer. This proves that at low SNR,
Pupper1
1is closer to P1whereas at high SNR, Pupper2
1is closer.
In Fig. 4 and Fig. 5, we present outage probability compar-
ison of our proposed relay decision scheme and the schemes
with the priority to transmit and receive [1], [5] when the
relay does and does not know h1and h2, respectively. From
Fig. 4 and Fig. 5, it is observed that the “Analytical” results
match the simulation results. From Fig. 4 and Fig. 5, it is also
found that our proposed relay decision scheme outperforms
other two schemes. This is because when the relay gives a
high priority to transmit or receive, the buffer state is more
likely to be either empty or full compared with our proposed
scheme. This results in that the number of the available links
are less than our proposed scheme where the waiting queue at
the buffer has high probability to be one or two.
In Fig. 4 and Fig. 5, we also present outage probability of
the optimal relay decision scheme which minimizes outage
probability, denoted as “Optimal” in the legends. The optimal
relay decision scheme is found by exhaustive search. From Fig.
4 and Fig. 5, it is observed that outage probability achieved by
5 10 15 20 25 30
10−4
10−3
10−2
10−1
100
¯γ(dB)
Outage Probability
Simulation
Analytical
Upper Bound
Lower Bound 1
Lower Bound 2
10 11 12 13 14 15
100
Fig. 3. Outage probability versus SNR, ¯γ; performance of our proposed
relay decision scheme when the relay knows h1and h2.
5 10 15 20 25 30
10−4
10−3
10−2
10−1
100
¯γ(dB)
Outage Probability
Optimal, Simulation
Proposed Relay Decision, Simulation
Proposed Relay Decision, Analytical
Priority to Transmit, Simulation
Priority to Transmit, Analytical
Priority to Receive, Simulation
Priority to Receive, Analytical
Fig. 4. Outage probability versus SNR, ¯γ; performance comparison of our
proposed relay decision scheme and the schemes with the priority to transmit
and receive when the relay knows h1and h2.
the proposed relay decision scheme is close to that achieved
by the optimal relay decision scheme.
In Fig. 6, we present throughput comparison of our proposed
relay decision scheme and the schemes with the priority
to transmit and receive when the relay knows h1and h2,
respectively. From Fig. 6, it is observed that with the increase
of SNR, ¯γ, throughput of the system increases because outage
probability of each link decreases. At high SNR, ¯γ, throughput
of the system approaches r0= 2 bps/Hz.
In Fig. 7, we present average packet delay comparison of
our proposed relay decision scheme and the schemes with the
priority to transmit and receive when the relay knows h1and
h2, respectively. From Fig. 7, it is found that at high SNR, ¯γ,
the simulation result validates Proposition 5, i.e., when ¯γ→
∞, the total average packet delay of the system is D= 4. It
is noted that for the scheme with priority to transmit, at high
SNR, ¯γ, the buffer state is circling in states “0” and “1”. The
IEEE TRANSACTIONS ON COMMUNICATIONS 9
5 10 15 20 25 30
10−4
10−3
10−2
10−1
100
¯γ(dB)
Outage Probability
Optimal, Simulation
Proposed Relay Decision, Simulation
Proposed Relay Decision, Analytical
Priority to Transmit, Simulation
Priority to Transmit, Analytical
Priority to Receive, Simulation
Priority to Receive, Analytical
Fig. 5. Outage probability versus SNR, ¯γ; performance comparison of our
proposed relay decision scheme and the schemes with the priority to transmit
and receive when the relay does not know h1and h2.
5 10 15 20 25
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
¯γ(dB)
Throughput (bps/Hz)
Proposed Relay Decision, Simulation
Proposed Relay Decision, Analytical
Priority to Transmit, Simulation
Priority to Transmit, Analytical
Priority to Receive, Simulation
Priority to Receive, Analytical
Fig. 6. Throughput versus SNR, ¯γ; performance comparison of our proposed
relay decision scheme and the schemes with the priority to transmit and receive
when the relay knows h1and h2.
average packet delay at the relay is 1 time slot. Thus, the total
average delay of the scheme with priority to transmit is 2 time
slots.
In Fig. 8, we show the effect of the buffer size, L, on
the outage probability of our proposed relay decision scheme
when the relay does and does not know h1and h2(denoted
as “w/ CSI” and “w/o CSI”, respectively). From Fig. 8, it
is observed that when L≥6, the outage probability of our
proposed relay decision scheme converges.
In Fig. 9, we compare the throughputs obtained by our
proposed NOMA relaying system, conventional OMA relay-
ing system, and hybrid NOMA and OMA relaying system,
denoted as “NOMA”, “OMA”, and “Hybrid” in the legend,
respectively. We assume that the relay knows h1and h2.
For conventional OMA system, we mean that the source
solely sends packets to destination 1 or destination 2, denoted
10 15 20 25 30
100
101
102
103
¯γ(dB)
Average Packet Delay (in numer of time slots)
Proposed Relay Decision
Priority to Transmit
Priority to Receive
Fig. 7. Average packet delay versus SNR, ¯γ; performance comparison of our
proposed relay decision scheme and the schemes with the priority to transmit
and receive when the relay knows h1and h2.
2 4 6 8 10 12 14 16 18 20
10−4
10−3
10−2
10−1
100
L
Outage Probability
¯γ= 10 dB, w/ CSI
¯γ= 10 dB, w/o CSI
¯γ= 15 dB, w/ CSI
¯γ= 15 dB, w/o CSI
¯γ= 20 dB, w/ CSI
¯γ= 20 dB, w/o CSI
¯γ= 30 dB, w/ CSI
¯γ= 30 dB, w/o CSI
Fig. 8. Outage probability versus the buffer size, L; performance of our
proposed relay decision scheme when the relay does and does not know h1
and h2.
as “SD1” or “SD2” in the legend, respectively. For fair
comparison, the OMA relaying system employs a relay with
the buffer size of 100. Target transmission rates for source
to destination, r0, of 1 bps/Hz and 2 bps/Hz are considered
for the OMA relaying system. For hybrid NOMA and OMA
relaying system, we mean that the relaying system is able to
switch between NOMA and OMA relaying modes, i.e., if one
of the destinations in our proposed NOMA relaying system is
able to detect its corresponding information symbols and the
other is not, the relaying system switches to the OMA relaying
mode. From Fig. 9, it is observed that the hybrid NOMA and
OMA relaying system outperforms any OMA relaying system.
In Fig. 10, we present the throughput versus target trans-
mission rate, r0, obtained by our proposed NOMA relaying
system, conventional OMA relaying system, and hybrid NO-
MA and OMA relaying system. We assume that ¯γ= 20 dB
IEEE TRANSACTIONS ON COMMUNICATIONS 10
0 5 10 15 20 25
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
¯γ(dB)
Throughput (bps/Hz)
Hybrid
NOMA
OMA, SD1, r0=1
OMA, SD2, r0=1
OMA, SD1, r0=2
OMA, SD2, r0=2
Fig. 9. Throughput versus SNR, ¯γ; performance comparison of our proposed
NOMA relaying system, conventional OMA relaying system, and hybrid
NOMA and OMA relaying system when the relay knows h1and h2.
0 2 4 6 8 10
0
0.5
1
1.5
2
2.5
3
3.5
r0 (bps/Hz)
Throughput (bps/Hz)
Hybrid
NOMA
OMA, SD1
OMA, SD2
Fig. 10. Throughput versus target transmission rate, r0; performance
comparison of our proposed NOMA relaying system, conventional OMA
relaying system, and hybrid NOMA and OMA relaying system when the
relay knows h1and h2.
the relay knows h1and h2. From Fig. 10, it is observed that
the highest throughput obtained by hybrid NOMA and OMA
relaying system is about 3.1 bps/Hz whereas that obtained by
conventional OMA system is lower than 2.4 bps/Hz.
X. CONCLUSION
In this paper, we have proposed a buffer-aided NOMA
relaying system. Considering that the relay does and does not
know the CSI from itself to destinations, we have theoretically
derived the outage probabilities of source-to-relay link and
relay-to-destinations links. When the relay knows the CSI,
the obtained outage probability of relay-to-destinations links
involves integration operation. Thus, we have derived an upper
bound and two lower bounds. Simulation results demonstrate
that two lower bounds approach exact outage probability at
low and high SNRs, respectively. For the buffer-aided NOMA
relaying system, we have proposed a relay decision scheme.
Based on the derived system outage probability, we have
theoretically derived the diversity order. It is found that no
matter whether the relay does or does not know the CSI, the
diversity order of 2 can be achieved when the buffer size is
larger than or equal to 3.
APPENDIX A
PROO F OF PROPOSITION 1
From [23], we have
∞
ζ
exp −x
Ω1−bζ2
Ω2xdx ≤∞
0
exp −x
Ω1−bζ2
Ω2xdx
=K1
4ζ2b
Ω1Ω2
.(60)
Substituting (60) into (18), we obtain
P1≤Pupper1
1(61)
where Pupper1
1is defined in (19). On the other hand, because
when ζ≤x < ∞,
exp −bζ2
Ω2x≤1(62)
due to the fact that bζ2
Ω2x>0, we have
∞
ζ
exp −x
Ω1−bζ2
Ω2xdx ≤∞
ζ
exp −x
Ω1dx
= Ω1exp −ζ
Ω1.(63)
Substituting (63) into (18), we obtain
P1≤Pupper2
1(64)
where Pupper2
1is defined in (20).
In the following, we will derive the lower bound on P1.
Since |h1| ≥ |h2|, we know
ζ
|h1|2≤ζ
|h2|2.(65)
If ζ
|h2|2≤b−11−ζ
|h2|2,(66)
i.e.,
|h2|2≥(b+ 1)ζ(67)
is satisfied, (13) is always satisfied. Thus,
P1≥∞
(b+1)ζx
(b+1)ζ
1
Ω2
exp −y
Ω2dy 1
Ω1
exp −x
Ω1dx.
(68)
After some mathematical manipulations, we have
P1≥Plower
1(69)
where Plower
1is defined in (21).
IEEE TRANSACTIONS ON COMMUNICATIONS 11
APPENDIX B
PROO F OF PROPOSITION 3
From (26), we have
Plower2
3= 1 + v−1
ρ
1−v−1
ρ
2−v−1
ρ
3(70)
where
v1= exp (b2−1) ·(Ω1+ Ω2)
Ω1Ω2,(71)
v2= exp (b−1)Ω1+ (b2−1)Ω2
Ω1Ω2,(72)
v3= exp (b−1)Ω2+ (b2−1)Ω1
Ω1Ω2.(73)
In (70), if v1=v2v3,Plower2
3can be rewritten as
Plower2
3=1−v−1
ρ
2·1−v−1
ρ
3.(74)
Thus,
−lim
ρ→∞
log Plower2
3
log ρ
=−lim
ρ→∞
log 1−v−1
ρ
2
log ρ−lim
ρ→∞
log 1−v−1
ρ
3
log ρ
=2.(75)
However, if v1=v2v3, we obtain b= 1, i.e., r0= 0 which
means that the buffer-aided NOMA relaying system is not an
applicable system. Therefore, v1̸=v2v3.
When v1̸=v2v3, by letting δ= 1/ρ, we have
−lim
ρ→∞
log Plower2
3
log ρ= lim
δ→0
log 1 + v−δ
1−v−δ
2−v−δ
3
log δ.(76)
Employing L’Hˆ
opital’s rule, we obtain
−lim
ρ→∞
log Plower2
3
log ρ
= lim
δ→0
δ−v−δ
1log v1+v−δ
2log v2+v−δ
3log v3
1 + v−δ
1−v−δ
2−v−δ
3
.(77)
Employing L’Hˆ
opital’s rule again, we obtain
−lim
ρ→∞
log Plower2
3
log ρ
= lim
δ→01 + δ−v−δ
1log2v1+v−δ
2log2v2+v−δ
3log2v3
−v−δ
1log v1+v−δ
2log v2+v−δ
3log v3
=1.(78)
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IEEE TRANSACTIONS ON COMMUNICATIONS 12
Qi Zhang (S’04-M’11) received the B.Eng. (with
Hons.) and M.S. degrees from the University
of Electronic Science and Technology of China
(UESTC), Chengdu, China, and the Ph.D. degree
in electrical and computer engineering from the
National University of Singapore (NUS), Singapore,
in 1999, 2002, and 2007, respectively.
He is currently an Associate Professor with the
School of Electronics and Information Technology,
Sun Yat-sen University, Guangzhou, China. From
2007 to 2008, he was a Research Fellow with
the Communications Laboratory, Department of Electrical and Computer
Engineering, NUS. From 2008 to 2011, he was with the Center for Integrated
Electronics, Shenzhen Institutes of Advanced Technology, Chinese Academy
of Sciences. His research interests include non-orthogonal multiple access,
wireless communications powered by energy harvesting, cooperative commu-
nications, and ultra-wideband (UWB) communications.
Zijun Liang received B.Eng. degree in electron-
ic and information engineering from South China
Normal University, Guangzhou, China, in 2014. He
is currently working toward the M.S. degree at the
School of Electronics and Information Technolo-
gy, Sun Yat-sen University, Guangzhou, China. His
research interests include non-orthogonal multiple
access, buffer-aided relaying, and cooperative com-
munications.
Quanzhong Li received the B.S. and Ph.D. degrees
from Sun Yat-sen University (SYSU), Guangzhou,
China, both in information and communications en-
gineering, in 2009 and 2014, respectively.
He is currently a Lecturer with the School of Data
and Computer Science, SYSU. His research inter-
ests include non-orthogonal multiple access, wire-
less communications powered by energy harvesting,
cognitive radio, cooperative communications, and
multiple-input multiple-output (MIMO) communica-
tions.
Jiayin Qin received the M.S. degree in radio physics
from Huazhong Normal University, China, and the
Ph.D. degree in electronics from Sun Yat-sen U-
niversity (SYSU), Guangzhou, China, in 1992 and
1997, respectively.
Since 1999, he has been a Professor with the
School of Electronics and Information Technology,
SYSU. From 2002 to 2004, he was the Head of
the Department of Electronics and Communication
Engineering, SYSU. From 2003 to 2008, he was the
Vice Dean of the School of Information Science and
Technology, SYSU. His research interests include wireless communication and
submillimeter wave technology.
Dr. Qin was the recipient of the IEEE Communications Society Heinrich
Hertz Award for Best Communications Letter in 2014, the Second Young
Teacher Award of Higher Education Institutions, Ministry of Education
(MOE), China in 2001, the Seventh Science and Technology Award for
Chinese Youth in 2001, and the New Century Excellent Talent, MOE, China
in 1999.
