Impact of Opportunistic Scheduling on Cooperative Dual-Hop Relay Networks

Abstract

This letter advocates the performance of a multiuser relay network (MRN) equipped with a single amplify-and-forward (AaF) relay over Rayleigh fading environments. We derive new expressions for the cumulative distribution function (CDF) of the highest instantaneous end-to-end signal-to-noise ratio (SNR) taking into consideration the two cases of fixed gain relays and variable gain relays. Relying on these statistical results, we derive new expressions for the outage probability and symbol error rate (SER), both of which are obtained in exact closed form. Furthermore, we derive simple asymptotic outage probability and SER. Our asymptotic results confirm that opportunistic scheduling has no impact on the diversity order. We further prove that the array gain is what determines the SNR advantage of opportunistic scheduling over the single user scenario.

Full text

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 3, MARCH 2011 689
Impact of Opportunistic Scheduling on
Cooperative Dual-Hop Relay Networks
Nan Yang, Student Member, IEEE, Maged Elkashlan, Member, IEEE, and Jinhong Yuan, Member, IEEE
Abstract—This letter advocates the performance of a multiuser
relay network (MRN) equipped with a single amplify-and-
forward (AaF) relay over Rayleigh fading environments. We
derive new expressions for the cumulative distribution function
(CDF) of the highest instantaneous end-to-end signal-to-noise
ratio (SNR) taking into consideration the two cases of fixed
gain relays and variable gain relays. Relying on these statistical
results, we derive new expressions for the outage probability
and symbol error rate (SER), both of which are obtained in
exact closed form. Furthermore, we derive simple asymptotic
outage probability and SER. Our asymptotic results confirm that
opportunistic scheduling has no impact on the diversity order.
We further prove that the array gain is what determines the
SNR advantage of opportunistic scheduling over the single user
scenario.
Index Terms—Cooperative transmission, fading channels, op-
portunistic scheduling.
I. INTRODUCTION
DEPLOYMENT of wireless relays in cooperative trans-
mission has recently appeared as an efficient alterna-
tive to extend coverage and combat multipath impairment
in wireless networks [1–3]. Among the proposed cooperative
strategies [3,4], amplify-and-forward (AaF) attracts consid-
erable attention due to its ease of implementation and low
power consumption. In AaF, the relay simply amplifies the
received signal from the source and retransmits a scaled copy
of the signal to the destination. As a further categorization,
AaF relays can be classified into two subcategories based
on the channel state information (CSI) available at the relay:
namely variable gain relays and fixed gain relays. Driven by
the potential application of such wireless relays, some seminal
works have examined the end-to-end performance of point-to-
point dual-hop links (i.e., single source and single destination
with single relay usage) [5, 6].
In point-to-multipoint multiuser applications, for example
the case of a cellular system, the base station can select the
mobile user with the strongest channel in a time/frequency
Paper approved by W. Yu, the Editor for Cooperative Communications and
Relaying of the IEEE Communications Society. Manuscript received June 2,
2010; revised October 8, 2010.
N. Yang is with the School of Information and Electronics, Beijing Institute
of Technology, Beijing 100081, China, and with the School of Electrical En-
gineering and Telecommunications, University of New South Wales, Sydney,
NSW 2052, Australia (e-mail: nan.yang@student.unsw.edu.au).
M. Elkashlan is with the Wireless Technologies Laboratory, CSIRO ICT
Centre, Marsfield, NSW 2122, Australia, and with the School of Electrical
Engineering and Telecommunications, University of New South Wales, Syd-
ney, NSW 2052, Australia (e-mail: maged.elkashlan@csiro.au).
J. Yuan is with the School of Electrical Engineering and Telecommu-
nications, University of New South Wales, Sydney, NSW 2052, Australia,
and with CSIRO ICT Centre, Marsfield, NSW 2122, Australia (e-mail:
j.yuan@unsw.edu.au).
Digital Object Identifier 10.1109/TCOMM.2011.122110.100133
Fig. 1. Illustrative system model for point-to-multipoint dual-hop links.
bin to schedule data transmission. This strategy, which has
come to be known as opportunistic scheduling [7–9], can
provide a potentially large performance improvement if ef-
ficiently utilized. More recently, the concept of cooperative
dual-hop transmission has been applied to multiuser wireless
downlinks [10, 11]. One example of this is point-to-multipoint
dual-hop links where a source (or equivalently base station)
communicates with many remote and/or geographically scat-
tered destinations (or equivalently mobile users) via a single
or multiple relays, as shown in Fig. 1. A few works have been
conducted on this architecture [12–15], the results from which
have focused only on the capacity and/or throughput perfor-
mance from the information-theoretic perspective. Despite the
demonstrated promised gains of wireless relay networks, the
impact of opportunistic scheduling on these networks has not
been thoroughly investigated and is not fully understood.
Motivated by this, we focus our attention on the bene-
fits conferred by opportunistic scheduling in relay-assisted
networks. In this letter, we refer to the system architecture
depicted in Fig. 1 as a multiuser relay network (MRN).
Assuming Rayleigh fading channels, we derive new closed-
form expressions for the outage probability and symbol error
rate (SER) for fixed and variable gain relays. In doing so, exact
expressions are derived for the cumulative distribution func-
tion (CDF) of the highest end-to-end SNR link associated with
the strongest destination. We further derive simple closed-form
expressions for the diversity order and array gain. We prove
that opportunistic scheduling does not affect the diversity
order, rather it increases the array gain and hence reduces
the SER. Finally, we demonstrate that increasing the number
of destinations shifts the optimal relay location towards the
source. In particular, we find that the shift in the optimal relay
location is considerable for variable gain relaying, however,
less noticeable for fixed gain relaying.
II. PRELIMINARIES AND SYSTEM MODEL
Consider a wireless relay-assisted communication system
shown in Fig. 1. A source communicates with 𝐾destinations
0090-6778/11$25.00 c
2011 IEEE

690 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 3, MARCH 2011
with the aid of a single AaF relay. The source has no direct
link with the destinations and the transmission is performed
only via the relay in a time-division multiple access (TDMA)
fashion with two signaling intervals. The source transmits its
data signal to the relay in a signaling interval, and in the
following signaling interval, the relay retransmits the amplified
signal to only one destination which has the most favorable
end-to-end channel quality.
Let the modulated signal transmitted by the source denoted
as 𝑠(𝑡). The received signal at the 𝑘th destination is given by
𝑦𝑘(𝑡)=𝐸RD𝑑−𝜂
RD𝑔𝑘𝐺𝐸SR 𝑑−𝜂
SRℎ𝑠(𝑡)+𝑛𝑟+𝑛𝑑𝑘,
(1)
where 𝐸SR and 𝐸RD denote the average symbol energies at
the source and the relay, respectively, and ℎand 𝑔𝑘represent
the channel complex fading coefficients between the source
and relay, and between the relay and the 𝑘th destination,
respectively. The symbols 𝑛𝑟and 𝑛𝑑𝑘represent the additive
white Gaussian noise (AWGN) components with one-sided
power spectral density 𝑁0at the relay and the 𝑘th destination,
respectively. 𝐺is defined as the scaling gain applied at the
relay. In this letter, the path loss is incorporated in the signal
propagation, where 𝑑SR is the distance between the source
and the relay, 𝑑RD is the distance between the relay and the
destinations, and 𝜂is the path loss exponent. Here we assume
that all destinations are equidistant from the relay, and hence
𝑑RD is constant for all destinations. The instantaneous end-to-
end SNR of the 𝑘th destination, 𝛾eq,𝑘, can be written as [5]
𝛾eq,𝑘 =𝛾1𝛾2,𝑘
𝛾2,𝑘 +1
𝐺2𝑁0
,(2)
where 𝛾1=∣ℎ∣2𝑑−𝜂
SR𝐸SR/𝑁0and 𝛾2,𝑘 =∣𝑔𝑘∣2𝑑−𝜂
RD𝐸RD /𝑁0
are the per-hop instantaneous SNRs associated with the chan-
nels ℎand 𝑔𝑘. Correspondingly, the per-hop average SNR is
given by 𝛾1=E[𝛾1]and 𝛾2,𝑘 =E[𝛾2,𝑘], respectively, where
E[⋅]is the expectation.
It is obvious from (2) that the choice of the relay gain
determines the instantaneous end-to-end SNR. Note that the
channel estimation error is inversely proportional to the input
SNR [16]. As such, when the source-relay link has a low
SNR, a fixed gain constraint of 𝐺2=1/(𝐶𝑁0)is applied at
the relay, where 𝐶=𝛾1+1 is a positive constant [6]. This
fixed gain relaying alleviates the requirement of full CSI while
offering a comparable performance to variable gain relaying.
The instantaneous end-to-end SNR of the 𝑘th destination
employing a fixed gain relay can be rewritten as
𝛾eq,𝑘,Fix =𝛾1𝛾2,𝑘
𝛾2,𝑘 +𝐶.(3)
Another choice of the relay gain is variable gain relaying,
which is applied for the case when the source-relay link fading
coefficients are precisely estimated at the relay. When the
relay does not account for the statistical noise, the variable
gain constraint is given by 𝐺2=1/∣ℎ∣2𝑑−𝜂
SR𝐸SR[5, 17].
The instantaneous end-to-end SNR of the 𝑘th destination
employing a variable gain relay can be rewritten as
𝛾eq,𝑘,Var =𝛾1𝛾2,𝑘
𝛾1+𝛾2,𝑘
.(4)
We assume that all the destinations are located in a ho-
mogeneous environment. In such an environment, the signals
from the relay to the 𝐾destinations experience independent
identically distributed (i.i.d.) Rayleigh fading where all the
destinations have the same per-hop average SNR, i.e., 𝛾2,𝑘 =
𝛾2. We also retain the practical consideration that the dual-
hop transmission is subject to independent but not necessarily
identically distributed (i.n.d.) Rayleigh fading, i.e., 𝛾1∕=𝛾2.
As a result, the per-hop instantaneous SNR, 𝑍={𝛾1,𝛾
2,𝑘},
follows an exponential distribution, with probability density
function (PDF) given by
𝑓𝑍(𝛾)= 1
𝛾𝑖
𝑒−𝛾
𝛾𝑖,(5)
where 𝑖=1,2. The corresponding CDF of 𝑍can be written
as
𝐹𝑍(𝛾)=1−𝑒−𝛾
𝛾𝑖.(6)
III. EXACT PERFORMANCE ANALYSIS
Opportunistic scheduling in MRN is achieved by selecting
the destination with the highest end-to-end instantaneous SNR
out of 𝐾destinations, at any particular point in time. The
highest instantaneous end-to-end SNR of the selected user
(i.e., strongest user), denoted as 𝛾𝑠, is determined by
𝛾𝑠=max
1≤𝑘≤𝐾{𝛾eq,𝑘}.(7)
It is assumed that CSI knowledge of the relay-destination links
of the 𝐾users are available at the relay. At the relay, channel
estimation is conducted based on a pilot sequence sent by the
𝐾users. The relay identifies and selects the strongest user.
The relay then feeds back the index of the strongest user to
the source.
A common metric for assessing the error performance is
the SER. In this letter, we adopt a CDF-based approach and
express the SER expression directly in terms of the CDF of
𝛾𝑠as [18]
𝑃𝑠=𝑎
2𝑏
𝜋∞
0
𝑒−𝑏𝛾
√𝛾𝐹𝛾𝑠(𝛾)𝑑𝛾. (8)
Our results in this section, apply for all general modulation
schemes that have an SER expression of the form 𝑃𝑠=
E𝑎𝑄 √2𝑏𝛾,whereE[⋅]is the statistical average operator.
Such modulation schemes include binary PSK (BPSK): 𝑎=1
and 𝑏=1, and quadrature PSK (QPSK): 𝑎=1and 𝑏=0.5.
A. Fixed Gain MRN
The outage probability 𝑃out is an important quality of
service measure, defined as the probability that 𝛾𝑠drops below
a certain specified SNR threshold 𝛾th. Considering MRN
equipped with a fixed gain relay, we can write its outage
probability 𝑃out,Fix as
𝑃out,Fix =Pr[𝛾𝑠,Fix <𝛾
th]=𝐹𝛾𝑠,Fix (𝛾th),(9)
where 𝐹𝛾𝑠,Fix (𝛾th)is the CDF of the highest instantaneous
end-to-end SNR of the strongest user for fixed gain relaying,
evaluated at 𝛾=𝛾th. To calculate the outage probability of
fixed gain MRN, we first obtain the CDF of 𝛾𝑠,Fix.

YANG et al.: IMPACT OF OPPORTUNISTIC SCHEDULING ON COOPERATIVE DUAL-HOP RELAY NETWORKS 691
Theorem 1: Since the CDF of each hop is given by (6), the
CDF of 𝛾𝑠,Fix can be expressed as
𝐹𝛾𝑠,Fix (𝛾)=1+2
𝐶𝛾
𝛾1𝛾2
𝑒−𝛾
𝛾1
𝐾−1

𝑖=0 𝐾
𝑖(−1)𝐾−𝑖
×√𝐾−𝑖𝐾12(𝐾−𝑖)𝐶𝛾
𝛾1𝛾2,(10)
where 𝐾𝑣(𝑥)denotes the 𝑣th-order modified Bessel function
of the second kind.
Proof: See Appendix A.
Using (9) and (10), the outage probability of MRN with
fixed gain relaying, 𝑃out,Fix, is obtained. In the special case
of single user dual-hop links, 𝑃out,Fix can be found by setting
𝐾=1in (10). This yields the same expression as that in [6,
eq. (9)].
Substituting (10) into (8), and using [19, eq. (6.614.5)], the
SER of fixed gain MRN is obtained in closed form as
𝑃𝑠,Fix =𝑎
2+𝑎
2𝑏𝛾1
1+𝑏𝛾1
𝐾−1

𝑖=0 𝐾
𝑖(−1)𝐾−𝑖𝑒𝜉
×𝜉(𝐾1(𝜉)−𝐾0(𝜉)) ,(11)
where 𝜉=𝐶(𝐾−𝑖)/2𝛾2(1 + 𝑏𝛾1). For the single user
scenario, the closed-form expression for 𝑃𝑠,Fix is found by
setting 𝐾=1in (11). Note that this result for single user
dual-hop links can also be derived using [6, eq. (10)]. Hence,
our result in (11) stands for a generalization of the single user
scenario.
By normalizing the total distance between the source and
the destinations to unity with 𝑑SR +𝑑RD =1and representing
𝛾1and 𝛾2with 𝑑SR,wefind that 𝑃𝑠,Fix in (11) is a convex
function of 𝑑SR. As such, the optimal relay location aiming at
minimizing the SER can be found by setting the derivative of
𝑃𝑠,Fix with respect to 𝑑SR to zero. Although it is intractable
to find a closed-form solution for this optimization problem,
the optimal relay location can be obtained via a simple line
search.
B. Variable Gain MRN
In this subsection, we analyze the performance of variable
gain MRN. The outage probability in this case is given by
𝑃out,Var =𝐹𝛾𝑠,Va r (𝛾th),(12)
where 𝐹𝛾𝑠,Var (𝛾)is the CDF of the highest instantaneous end-
to-end SNR of the strongest user for variable gain relaying.
Theorem 2: The CDF of each hop is shown in (6), and
consequently the CDF of 𝛾𝑠,Va r can be expressed as
𝐹𝛾𝑠,Var (𝛾)=1+ 2𝛾
√𝛾1𝛾2
𝐾−1

𝑖=0 𝐾
𝑖(−1)𝐾−𝑖√𝐾−𝑖
×𝑒−𝛾(1
𝛾1+𝐾−𝑖
𝛾2)𝐾12𝛾𝐾−𝑖
𝛾1𝛾2.(13)
Proof: Following the algebraic steps specified in Ap-
pendix A, the final result in (13) is derived.
The outage probability can be obtained by substituting (13)
into (12). Moreover, 𝑃out,Var for single user scenario can be
further simplified by setting 𝐾=1in (12). This result is the
same as that in [5, eq. (27)].
Substituting (13) into (8) and applying [20, eq. (2.16.1.3)],
the closed-form SER expression of variable gain MRN is
derived after some manipulations as
𝑃𝑠,Var =𝑎
2+𝑎
4𝑏𝛾1𝛾2
2Γ1
2Γ5
2𝐾−1

𝑖=0 𝐾
𝑖(−1)𝐾−𝑖
×𝜏−1
22𝐹11
4,3
4;2;1−4(𝐾−𝑖)𝛾1𝛾2
𝜏2,(14)
where 𝜏=𝛾2(1 + 𝑏𝛾1)+𝛾1(𝐾−𝑖),Γ(𝑥)represents the
gamma function, and 2𝐹1(𝑎, 𝑏;𝑐;𝑧)denotes the Gauss hy-
pergeometric function. For the single user scenario, 𝑃𝑠,Var
is obtained by substituting 𝐾=1into (14). We note that
this result for single user dual-hop links can also be derived
using [5, eq. (13)]. This demonstrates the accuracy and the
generality of our result in (14). Similar to fixedgainMRN,
we can find the optimal relay location for variable gain MRN.
IV. ASYMPTOTIC PERFORMANCE ANALYSIS
In this section, we derive asymptotic expressions for the
outage probability and SER. The new and relatively simple
expressions obtained are important to examine the effect of
opportunistic scheduling on the diversity order and array gain.
It was shown in [21] that the asymptotic SER can be derived
using the asymptotic outage probability. As such, we start our
analysis by characterizing the asymptotic outage probability,
followed by the asymptotic SER.
A. Fixed Gain MRN
To derive the asymptotic outage probability of fixed gain
relaying, we first note that 𝐶=1+𝛾1→𝛾1as 𝛾1, 𝛾2→∞.
As such, at high SNRs, by setting 𝛾2=𝜅𝛾1and 𝛾1=𝛾th /𝜆
in (9), 𝑃out,Fix can be rewritten as
𝑃out,Fix =𝑃(𝜆)
=1+𝑒−𝜆
𝐾−1

𝑖=0 𝐾
𝑖(−1)𝐾−𝑖𝜔𝐾1(𝜔),(15)
where 𝜔=2
((𝐾−𝑖)𝜆/𝜅)and 𝜆, 𝜅 ∈ℝ+. By comparing
(9) and (15) we find that the behavior of 𝑃out,Fix for large 𝛾1
and 𝛾2is equivalent to the behavior of 𝑃(𝜆)around 𝜆=0.
To proceed with our analysis we express the exponential and
Bessel functions in (15) in terms of the Taylor series expansion
around 𝜆=0. After further algebraic calculations, we obtain
𝑃(𝜆)=𝐴(𝜆)+𝐵(𝜆),(16)
where
𝐴(𝜆)=1+
𝐾−1

𝑖=0 𝐾
𝑖(−1)𝐾−𝑖
∞

𝑝=0
(−𝜆)𝑝
𝑝!,(17)
and
𝐵(𝜆)=
𝐾−1

𝑖=0 𝐾
𝑖(−1)𝐾−𝑖
∞

𝑝=0
∞

𝑞=0
(−1)𝑝𝐾−𝑖
𝜅𝑞+1
𝑝!𝑞!(𝑞+1)!
×𝜆𝑝+𝑞+1 ln (𝐾−𝑖)𝜆
𝜅−𝜓(𝑞+1)−𝜓(𝑞+2)
,(18)

692 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 3, MARCH 2011
where 𝜓(𝑥)is the psi function defined in [19, eq. (8.36.1)].
We next find the first nonzero derivative order of 𝐴(𝜆)and
𝐵(𝜆)and discard the higher order terms. After performing
some algebraic manipulations, we have
𝐴(𝜆)=𝜆+𝑜(𝜆),(19)
and
𝐵(𝜆)=𝜆
𝜅Ξ[𝐾]1+𝑜(𝜆),(20)
where Ξ[𝐾]1=𝐾−1
𝑖=0 𝐾
𝑖(−1)𝐾−𝑖(𝐾−𝑖)ln((𝐾−𝑖)𝜆/𝜅).
By substituting 𝜆=𝛾th /𝛾1into (19) and (20), the asymp-
totic outage probability is derived as
𝑃∞
out,Fix =𝛾th
𝛾11+ 1
𝜅Ξ[𝐾]2+𝑜𝛾−1
1,(21)
where Ξ[𝐾]2=∑𝐾−1
𝑖=0 (𝐾
𝑖)(−1)𝐾−𝑖(𝐾−𝑖)ln((𝐾−𝑖)𝛾th /𝛾2)
and 𝜅=𝛾2/𝛾1.
Using (21), we perform some basic algebraic manipulations
to obtain the asymptotic SER expression given by
𝑃∞
𝑠,Fix =(𝐺𝑎,Fix𝛾1)−𝐺𝑑,Fix +𝑜𝛾−𝐺𝑑,Fix
1,(22)
where the diversity order is 𝐺𝑑,Fix =1and the array gain is
𝐺𝑎,Fix =𝑎
4𝑏+𝑎
4𝑏𝜅Ξ[𝐾]3−1,(23)
where Ξ[𝐾]3=𝐾−1
𝑖=0 𝐾
𝑖(−1)𝐾−𝑖(𝐾−𝑖)ln((𝐾−𝑖)/𝛾2).
It is evident from (22) that for fixed gain relaying, the diversity
order is not influenced by opportunistic scheduling. However,
noting that Ξ[𝐾]3−Ξ[𝐾+1]
3>0for any 𝐾,weverify
that Ξ[𝐾]3is a monotonically decreasing function of 𝐾.As
such, the effect of opportunistic scheduling is to improve
the array gain, thereby reducing the SER. In Section V, we
demonstrate that an SNR improvement of at least 10 dB is
achieved with opportunistic scheduling.
B. Variable Gain MRN
To derive the asymptotic outage probability of variable gain
relaying, we first substitute 𝛾2=𝜅𝛾1and 𝛾1=𝛾th/𝜆 into
(12) to yield
𝑃out,Var =𝑄(𝜆)
=1 +
𝐾−1

𝑖=0 𝐾
𝑖(−1)𝐾−𝑖𝑒−𝜆−𝜒
4𝜆𝜒𝐾1(𝜒),(24)
where 𝜒=2𝜆(𝐾−𝑖/𝜅). Using the Taylor series expansion
of the exponential and Bessel functions, and following the
same algebraic steps as described in Section IV-A, we derive
the asymptotic outage probability as
𝑃∞
out,Var =𝛾th
𝛾11+ 1
𝜅+𝑜𝛾−1
1,𝐾=1,
𝛾th
𝛾1+𝑜𝛾−1
1,𝐾≥2,(25)
where 𝜅=𝛾2/𝛾1. By inserting (25) into (8), we derive the
asymptotic SER expression as
𝑃∞
𝑠,Var =(𝐺𝑎,Va r 𝛾1)−𝐺𝑑,Var +𝑜𝛾−𝐺𝑑,Va r
1,(26)
where the diversity order is 𝐺𝑑,Var =1and the array gain is
𝐺𝑎,Var =4𝑏𝜅
𝑎(1+𝜅),𝐾=1,
4𝑏
𝑎,𝐾≥2.(27)
The result in (26) confirms that for variable gain relaying,
opportunistic scheduling has no impact on the diversity order.
However, by inspecting (27), we see that the effect of oppor-
tunistic scheduling is to increase the array gain. This array
gain is as high as 18 dB, as we will show in Section V.
C. Fixed versus Variable Gain Relaying
We now compare the asymptotic SER of fixed gain relaying
with that of variable gain relaying. We first remark that both
fixed gain relaying and variable gain relaying achieve the same
diversity order. The main fundamental difference between the
two relaying protocols lies in the array gain, which is indicated
by (23) and (27). To characterize this difference, we present
the ratio of their array gains as
𝐺𝑎,Fix
𝐺𝑎,Var
=𝛾1+𝛾2
𝛾2+𝛾1ln𝛾2,𝐾=1
𝛾2
𝛾2+𝛾1Ξ[𝐾]3,𝐾≥2.(28)
By observing (28), we find that when 𝐾=1, the performance
gap between fixed and variable gain relaying is determined by
the values of 𝛾1and 𝛾2.When𝐾≥2, the ratio tends towards
one for large 𝐾. This reveals that the performance of fixed
gain relaying approaches that of variable gain relaying in the
large 𝐾limit.
V. N UMERICAL RESULTS
Numerical examples and simulations are carried out to
demonstrate the accuracy of our proposed analysis and the
effectiveness of opportunistic scheduling in Rayleigh fading
conditions. Throughout this section, we normalize the total
distance between the source and the destinations to unity such
that 𝑑SR +𝑑RD =1. The variance of the fading coefficients is
also normalized to unity with E∣ℎ∣2=1and E∣𝑔𝑘∣2=1.
Moreover, equal average energies are assumed at the source
and the relay, i.e., 𝐸SR =𝐸RD. Therefore, the first hop and the
second hop average SNR attenuates by 𝑑−𝜂
SR and (1 −𝑑SR)−𝜂,
respectively. In this section our results concentrate on 𝜂=4.
In addition, BPSK modulation is considered in all the figures.
Fig. 2 presents the outage probability for both fixed and
variable gain relaying as a function of 𝛾th. The curves
are plotted for 𝐾=1 and 𝐾=5.Wefind that the points
generated via Monte Carlo simulations match precisely with
the analytic curves, highlighting the accuracy of our analysis.
As expected, selecting the highest end-to-end SNR link in a
multiuser scenario (i.e., 𝐾=5) yields superior performance
compared to that of a single user scenario (i.e., 𝐾=1).
Fig. 3 presents the exact SER against 𝑑SR to investigate
the impact of the relay placement on the error performance.
For the single user scenario with 𝐾=1, we observe that the
optimal relay location for variable gain relaying is halfway
between the source and the destinations at 𝑑SR =0.5.
However, the optimal relay location for fixed gain relaying
is at 𝑑SR =0.6. This can be explained by the fact that the
instantaneous end-to-end SNR of fixed gain relaying given by

YANG et al.: IMPACT OF OPPORTUNISTIC SCHEDULING ON COOPERATIVE DUAL-HOP RELAY NETWORKS 693
0 10 20 30 40 50 60
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Variable gain (from (12))
Fixed gain (from (9))
Simulation points
γ
K = 1
K = 5
P
out
th
Fig. 2. Outage probability of MRN for fixed and variable gain relays:
𝐸SR/𝑁0=0dB,𝑑SR =0.4,and𝑑RD =0.6.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
10−4
10−3
10−2
10−1
100
10−5
Variable gain (from (14))
Fixed gain (from (11))
Simulation points
dSR
P
s
K = 1, 2, 3
Fig. 3. Exact SER of MRN for fixed and variable gain relays: 𝐸SR /𝑁0=
10 dB.
(3) is asymmetric caused by the constant 𝐶in the denominator.
For the multiuser scenario, we observe that the optimal relay
location is shifted towards the source. For example when
𝐾=3, the optimal relay location is shifted towards the source
at 𝑑SR =0.22 for variable gain relaying, and 𝑑SR =0.49
for fixed gain relaying. It is evident that with the increasing
number of users, the shift in the optimal relay location is more
salient in variable gain relaying than fixed gain relaying.
Figs. 4 and 5 present the SER of fixed gain and variable
gain relaying, respectively, versus 𝐸SR/𝑁0for various 𝐾.
The exact SER and the asymptotic SER are compared. As
expected, the asymptotic expressions well approximate the
exact expressions in the high SNR regime. As predicted from
the asymptotic expressions in (22) for fixed gain relaying
and (26) for variable gain relaying, we see that the diversity
order is unaffected by opportunistic scheduling. However, we
confirm that opportunistic scheduling has an obvious SNR
advantage over the single user scenario for both fixed and
variable gain relaying, which is explicitly indicated by the
−20 −10 0 10 20 30 40 50 60
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
E
SR
/
N0
P
s
K = 1, 2, 3, 4
Exact performance (from (11))
Asymptotic performance (from (22))
10 dB
Fig. 4. Exact and asymptotic SER of MRN for fixed gain relays: 𝑑SR =0.3
and 𝑑RD =0.7.
−20 −10 0 10 20 30 40 50 60
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10−1
100
E
SR
/
N0
P
s
Exact performance (from (14))
Asymptotic performance (from (26))
K = 1, 2, 3, 4
18 dB
Fig. 5. Exact and asymptotic SER of MRN for variable gain relays: 𝑑SR =
0.3and 𝑑RD =0.7.
array gain expressions in (23) and (27). At the SER of 10−6,
Fig. 4 shows that for fixed gain relaying, 𝐾=2outperforms
the single user scenario by 10 dB. Further increasing 𝐾brings
marginal benefits to the array gain. On the other hand, Fig. 5
shows that for variable gain relaying, 𝐾=2is superior by
18 dB to the single user scenario. However, further increasing
𝐾leads to array gain saturation.
VI. CONCLUSION
The performance gains offered by opportunistic scheduling
in a multiuser relay network have been examined for both
fixed and variable gain relays. Capitalizing on our new exact
closed-form expressions for the CDF of the strongest end-to-
end SNR link in a multiuser scenario, exact expressions for the
SER have been derived in closed form by following a unified
CDF-based approach. It has been shown that our analysis can
be viewed as a generalization of the single user dual-hop link.
Furthermore, by utilizing the behavior of the CDF, asymptotic
SER expressions have been derived in the high SNR regime

694 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 3, MARCH 2011
to explicitly reveal the impact of opportunistic scheduling on
the SER. We have proved that opportunistic scheduling has no
effect on the diversity order but leads to a noticeable increase
in the array gain.
APPENDIX A
PROOF OF THEOREM 1
We now calculate the CDF of 𝛾𝑠, Fix. Assuming that the 𝐾
relay-destination links undergo i.i.d. Rayleigh fading, the CDF
of 𝛾𝑠,Fix is given by
𝐹𝛾𝑠,Fix (𝛾)=∞
0
Pr [𝛾eq,𝑘,Fix <𝛾∣𝛾1]𝐾𝑓𝛾1(𝛾1)𝑑𝛾1
=∞
0
Pr 𝛾1𝛾2,𝑘
𝛾2,𝑘 +𝐶<𝛾∣𝛾1𝐾
𝑓𝛾1(𝛾1)𝑑𝛾1
=𝛾
0
Pr 𝛾2,𝑘 >𝐶𝛾
𝛾1−𝛾∣𝛾1𝐾
𝑓𝛾1(𝛾1)𝑑𝛾1
+∞
𝛾
Pr 𝛾2,𝑘 <𝐶𝛾
𝛾1−𝛾∣𝛾1𝐾
𝑓𝛾1(𝛾1)𝑑𝛾1
=𝐼1+𝐼2,(29)
where
𝐼1=𝛾
0
1
𝛾1
𝑒−𝛾1
𝛾1𝑑𝛾1=1−𝑒−𝛾
𝛾1,(30)
and
𝐼2=∞
𝛾1−𝑒−𝐶𝛾
𝛾2(𝛾1−𝛾)𝐾1
𝛾1
𝑒−𝛾1
𝛾1𝑑𝛾1.(31)
Using the binomial expansion in [19, eq. (1.111)], 𝐼2can
be evaluated as
𝐼2=1
𝛾1
𝐾

𝑖=0 𝐾
𝑖(−1)𝐾−𝑖∞
𝛾
𝑒−(𝐾−𝑖)𝐶𝛾
𝛾2(𝛾1−𝛾)𝑒−𝛾1
𝛾1𝑑𝛾1
=𝑒−𝛾
𝛾1+1
𝛾1
𝐾−1

𝑖=0 𝐾
𝑖(−1)𝐾−𝑖∞
𝛾
𝑒−(𝐾−𝑖)𝐶𝛾
𝛾2(𝛾1−𝛾)𝑒−𝛾1
𝛾1𝑑𝛾1
=𝑒−𝛾
𝛾1+1
𝛾1
𝐾−1

𝑖=0 𝐾
𝑖(−1)𝐾−𝑖𝐼3.(32)
By setting 𝜆=𝛾1−𝛾, we reexpress the integral 𝐼3as
𝐼3=𝑒−𝛾
𝛾1∞
0
𝑒−(𝐾−𝑖)𝐶𝛾
𝛾2𝜆𝑒−𝜆
𝛾1𝑑𝜆. (33)
Applying [19, eq. (3.324.1)] to (33), the integral 𝐼3can be
solved as
𝐼3=2𝑒−𝛾
𝛾1(𝐾−𝑖)𝐶𝛾1𝛾
𝛾2
𝐾12(𝐾−𝑖)𝐶𝛾
𝛾1𝛾2,(34)
and thus 𝐼2is obtained by inserting (34) into (32).
Finally, substituting (30) together with (32) into (29) leads
to 𝐹𝛾𝑠,Fix (𝛾)presented in (10) which is the desired result.
ACKNOWLEDGMENT
This paper is in part supported by Australian Research
Council (ARC) discovery Project (DP0987944).
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