Underlay Spectrum Sharing for NOMA Relaying Networks: Outage Analysis

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Underlay Spectrum Sharing for NOMA Relaying Networks:
Outage Analysis
Sultangali Arzykulov, Galymzhan Nauryzbayev, Theodoros A. Tsiftsis†, Behrouz Maham, Mohammad S. Hashmi
and Khaled M. Rabie◦
School of Engineering and Digital Sciences, Nazarbayev University, Nur-Sultan, Kazakhstan
†School of Electrical & Information Engineering and the Institute of Physical Internet, Jinan University, China
◦School of Engineering, Manchester Metropolitan University, Manchester, UK,
Email: {sultangali.arzykulov, galymzhan.nauryzbayev, behrouz.maham, mohammad.hashmi}@nu.edu.kz;
theo tsiftsis@jnu.edu.cn; k.rabie@mmu.ac.uk
Abstract—Non-orthogonal multiple access (NOMA) is recog-
nized as a promising multiple access technique for upcoming fifth
generation networks and is known for being able to accommodate
high system throughput, massive connectivity and low latency.
This paper investigates an underlay cognitive radio NOMA net-
work by adopting an amplify-and-forward (AF) relaying method.
The end-to-end outage probability (OP) is studied to evaluate
the performance of secondary NOMA users. Furthermore, the
OP performance for NOMA is compared with conventional
orthogonal multiple access to show the supremacy of the former.
Moreover, the proposed AF system is compared to the detect-
and-forward one. Finally, presented simulation results validate
the derived analytical expressions.
Index Terms—Amplify-and-forward (AF), cognitive radio (CR),
non-orthogonal multiple access (NOMA), outage probability (OP).
I. INTRODUCTION
Non-orthogonal multiple access (NOMA) has been recently
identified as encouraging multiple access (MA) technique for
the next fifth generation (5G) wireless networks [1]. Applying
NOMA, multiple users are permitted to utilize the whole
available resource, e.g., frequency, time, code, etc., and, as
a result, the SE of the network can be improved [2]. The
power-domain NOMA is the most popular NOMA technique,
where multiple users’ signals are superimposed at the transmit
node with specific power allocation (PA) factors depending
on the users’ channel quality. For example, to provide users
fairness, the transmitter allocates higher power to the users
with weak channel conditions, while less power is devoted to
the users with strong channel conditions. At the receive side,
users with weak channel quality apply successive interference
cancellation (SIC) to mitigate the severe effect of interference.
In [3], the authors studied the performance of randomly
allocated downlink (DL) NOMA users and proved that NOMA
is superior to OMA in means of ergodic sum rates. In addition,
the authors showed how the outage probability (OP) of the
NOMA network can be further improved by an appropriate
choice of PA factors. The authors in [2] investigated the OP
of the NOMA network with randomly distributed users with
partial channel state information (CSI), where the analytical
results showed the outperformance of the NOMA network.
Cooperative diversity [4] in wireless networks can be
achieved by using dedicated relays which create multiple paths
between source and destination nodes to combat fading. It
is encouraging to jointly study NOMA networks with relay
transmission due to the advantages of both NOMA concept
and relay method. Hence, the integration of relay networks
with NOMA was studied in [1], [5] where dedicated relay
nodes or users can operate as relays to enhance the SE and
the transmission reliability of NOMA networks. Moreover, an
approximated OP for a cooperative NOMA with the amplify-
and-forward (AF) relaying was provided in [6], where it was
shown that the proposed network achieves better coding gain
compared to the conventional cooperative OMA. Furthermore,
exact and asymptotic expressions for the ergodic sum-rate and
OP over Nakagami-mfading distributions were derived in [7],
where the authors considered multi-user NOMA networks with
a variable gain relay which main aim is to maintain the reliable
communication between the transmit node and NOMA users.
Cognitive radio (CR) network is a spectrum scarcity re-
solving technology [8], which consists of two types of users,
namely, primary users (PUs) and secondary users (SUs), which
can broadcast over the primary spectrum bands. The joint
investigation of the CR and NOMA techniques can accom-
modate further enhancement in effective spectrum utilization.
Considering this fact, the underlay CR-NOMA network was
investigated in [9], where the secondary transmitter sends a
superimposed signal to secondary NOMA destination users.
From the results, it was concluded that NOMA users can
obtain better performance if the PA factors and target rates are
accurately chosen. The authors in [10] derived a closed-form
expression for the OP in a DF relaying CR-NOMA and it was
concluded that the performance of both PUs and SUs were
improved by implementing the NOMA technique. Moreover,
underlay CR-NOMA with the DF relaying was studied in [11].
All the above mentioned papers studied the performance of
DF CR-NOMA network and no or limited research has been
carried out on the AF underlay CR-NOMA.
To the best of our knowledge, cooperative underlay CR-
NOMA networks with AF relaying have not been studied yet.
Therefore, in this paper, we investigate outage performance
and throughput of the dual-hop DL CR-NOMA model with
an AF relaying. Furthermore, to avoid harmful interference
in the primary network, interference temperature constraint
(ITC) is adopted at the primary destination (P D). The main
contributions of this paper are summarized as follows:
•We consider the system model with the ITC imposed by
the primary receiver node. This assumption is practical

and provides extra useful insights into the system perfor-
mance. Besides, considering ITC leads to the appearance
of an additional random variable (RV), which, in turn,
significantly complicates the performance analysis of the
system model. In spite of this, we obtain an end-to-end
exact OP expression for the proposed system model.
•The impact of the primary interference on the OP of
the NOMA secondary destination users is investigated.
Moreover, we obtain the PA factors based on the NOMA
users’ outage fairness (OF) and their channel conditions.
•A good agreement of the theoretical results with Monte
Carlo simulations verifies the correctness of our analysis.
II. SY ST EM MO DE L
We study a dual-hop CR-NOMA network with underlay
paradigm scenario (see Fig. 1), where a secondary source
(S) transmits information to two secondary destination users
(U1, U2) through a half-duplex AF-based secondary relay (R).
The channel gains between nodes, i.e., hSP ,hSR ,h1and h2,
are assumed to experience Rayleigh with path-loss exponent
ηand channel gains are constant withing every transmission
block T. The distances between nodes are accordingly given
as dSP ,dSR,d1and d2. Additionally, the interference from the
primary transmitters (PTs) to the secondary users is denoted
by IP1. We assume that Scan interfere with P D , whereas
Rcan not cause interference to the PUs as it is located far
away from P D (see Fig. 1). Hence, Scan transmit only if
its interference to P D is tolerable. Thus, the transmit power
at Sis constrained as PS≤min PIdη
SP
|hSP |2,¯
PS, where PI
and ¯
PSare the ITC at P D and maximum transmit power
available at S, accordingly. Taking into consideration source
power restriction, Stransmits to Ra superimposed signal
√PSP2
i=1 √αixi, where xi, with E(|xi|2) = 1, is the
intended data to Ui,i∈ {1,2}, while αiis the PA factor of
the respective user’s signal, with P2
i=1 αi= 1. Furthermore,
during time-slot (TS) 1, the received signal at Rcan be
expressed as follows
yR=sPS
dη
SR
hSR
2
X
j=1
√αjxj+IP+nR,(1)
where n(·)∼ CN (0, σ2
(·))denotes the additive white Gaussian
noise (AWGN) at each receiver. Hereafter, for mathematical
tractability, it is assumed that σ2
P=σ2
R=σ2
1=σ2
2=σ2.
We assume that h1< h2and α1> α2which means that S
allocates lower power portion to U2.
In TS 2, based on the AF protocol, Rconveys the signal
yRto both secondary destinations. Thus, the received signal
at Uican be written as
yi=sPSPR
dη
SR dη
i
hSR hi
2
X
j=1
√αjxjG
1Channel conditions of P T s are assumed to be unavailable at the SUs.
Hence, regarding to the central limit theorem [12], [13, §3.9.2], interference
caused by the P T s to the SUs may be considered as AWGN noise with
CN (0, τ σ2), where τis the scaling coefficient of IP.
hSR
hSP
h1
h2
Interference from the to the SUsPTs
Interference from the SUs to the P D
Secondary link
IP
IP
IP
SR
U2
U1
PD
PT1PT2PTN
Fig. 1. The proposed system model.
+sPR
dη
i
hinRG+sPR
dη
i
hiIPG+IP+ni
| {z }
effective noise
,(2)
where Gis the amplification factor at R, which can be written
as G=r1
PS
dη
SR
|hSR |2+σ2. This yields that U1, due to higher
α1, can treat x2as a noise and detect x1by the following
signal-to-interference-plus-noise ratio (SINR)
γ1=
PS
dη
SR |hSR |2PR
dη
1|h1|2G2α1
PS
dη
SR |hSR |2PR
dη
1|h1|2G2α2+PR
dη
1|h1|2G2σ2¯τ+σ2¯τ
=nsY n1Xα1
nsY n1Xα2+n1X¯τ+nsY¯τ+ ¯τ,(3)
where ρS=PS
σ2and ρR=PR
σ2are the transmit signal-to-noise
ratio (SNR) at Sand R, respectively; ns=ρS
dη
SR
,n1=ρR
dη
1,
¯τ=τ+ 1.Y=|hSR |2and X=|h1|2are the exponential
distributions. Next, according to the principle of NOMA [1],
U2detects x1, while treating its own signal as a background
noise, by the following SINR
γ2,1=
PS
dη
SR |hSR |2PR
dη
2|h2|2G2α1
PS
dη
SR |hSR |2PR
dη
2|h2|2G2α2+PR
dη
2|h2|2G2σ2¯τ+σ2¯τ
=nsY n2Qα1
nsY n2Qα2+n2Q¯τ+nsY¯τ+ ¯τ,(4)
where n2=ρR
dη
2and Q=|h2|2.After U2successfully removes
x1applying the SIC, it detects its own message x2from the
remaining signal with the SINR
γ2=
PS
dη
SR |hSR |2PR
dη
2|h2|2G2α2
PR
dη
2|h2|2G2σ2¯τ+σ2¯τ=nsY n2Qα2
n2Q¯τ+nsY¯τ+ ¯τ.(5)
Thus, the achievable rates of Uican be written as
Ri=1
2log2(1 + γi,),(6)
where 1
2indicates the half-duplex relaying.
III. OUTAG E ANALYSIS
The signal of user is assumed to be in outage if the received
SNR γis lower than a predefined threshold θ= 22Rth −1,

Fig. 2. The OP of Users 1and 2vs. the
transmit SNR when α1= 0.8,α2= 0.2
and τ= 0.
Fig. 3. The OP of Users 1and 2vs. the
transmit SNR when d1= 3d,α1= 0.8,
α2= 0.2and τ= 0.
Fig. 4. The OP of Users 1and 2vs.
the predefined SNR threshold at 20 dB
when d1= 3d.
i.e., P=Pr [γ < θ]. Thus, the OP of U1can be written as
P1(θ1) = 1 −Pr [γ1> θ1] = 1 −(1 −Fγ1(θ1)) ,Fγ1(θ1).
(7)
Furthermore, considering the ITC, the OP in (7) is derived as
in (8), which is shown at the top of the next page, where ¯
θ1=
¯τθ1,φ=α1−α2θ1,nP=ρIdη
SP
dη
SR
,c=ρIdη
SP
ρS,ρI=PI
σ2and
Z=|hSP |2. Moreover, the step εin (8) relies on α1> α2θ1,
otherwise, P1(θ1)=1regardless of the system SNR.
Proposition 1: The exact end-to-end OP expression for U1
is derived as
P1(θ1) =1 −e−u1(n1+ns)
n12√rK12√r1−e−c
−n1e−cn1(¯u1+1)+np¯u1
n1
¯u1(np¯u1+ 1)
∞
X
k=0 −c¯u1(np¯u1+1)
n1k
k!
×G1 3
3 2 n1(¯u1+ 1)
¯u1(np¯u1+ 1) 
0, k −1, k
0, k .(9)
Proof: See Appendix A. 
Remark Proposition 1 shows that the outage of U1occurs,
i.e., P1(θ1)∼1when θ1≥α1
α2. The reason for the outage
is that when θ1≥α1
α2, the message x1is dominated by the
interference from the message of U2and hence cannot be
decoded successfully by U1. Therefore, Proposition 1 reveals
that the minimum transmit power that the source should assign
for x1is PSθ1
θ1+1 .
Moreover, the OP for U2can be found as
P2(θ2) =1 −Pr [γ2> θ2],Fγ2(θ2).(10)
Furthermore, using (5), the OP for U2can be written as
P2(θ2) = Pr nsY n2Qα2
n2Q¯τ+nsY¯τ+ ¯τ< θ2, ρS<ρIdη
SP
Z
+Pr nP
ZY n2Qα2
n2Q¯τ+nP
ZY¯τ+ ¯τ< θ2, ρS>ρIdη
SP
Z
=Pr Y < ¯
θ2
nsα2
,u2, Q < Y nsu2+u2
n2(Y−u2), Z < c
| {z }
C
+Pr Y < Z¯
θ2
nPα2
,Z¯u2, Q < Y nP¯u2+Z¯u2
n2(Y−Z¯u2), Z > c
| {z }
D
.
(11)
Now, by following the same approach as in Appendix A, the
end-to-end exact OP expression for U2is written as
P2(θ2) =1 −e−u2(n2+ns)
n22√r2K1(2√r2)1−e−c
−n2e−cn2(¯u2+1)+np¯u2
n2
¯u2(np¯u2+ 1)
∞
X
k=0 −c¯u2(np¯u2+1)
n2k
k!
×G1 3
3 2 n2(¯u2+ 1)
¯u2(np¯u2+ 1) 
0, k −1, k
0, k ,(12)
where r2=u2(nsu2+1)
n2.
IV. NUMERICAL RES ULT S AN D DISCUSSIONS
The results for the above investigation are presented in
this section, which are validated by Monte Carlo simulations.
We assume Rayleigh fading channels and adopt the following
system parameters: PI= 20 dB, α1= 0.8,α2= 0.2,θ= 3
dB2,η= 2.7,d1={2d, 3d, 4d}, dSP =dSR =d2=d.
To focus on the OP results, with no loss of generality, dis
assumed to be unity.
The OP results for U1and U2are shown in Fig. 2. The PA
factors are taken as α1= 0.8and α2= 0.2. It is observed
that the OP for U2performs better than that for U1for all d1
distances. It is due to the fact that U2removes the unwanted
message of U1by applying the SIC technique which improves
the probability of errors in the data detection. On the other
hand, the OP saturation for U2starts at lower SNRs compared
with that for U2. Moreover, it is noticed that the OP of U1
2In the numerical results, we assume an equal predefined SNR threshold
for U1and U2, i.e., θ1=θ2=θ, in order to evaluate U1and U2from the
fairness point of view.

P1(θ1) =Pr nsY n1Xα1
nsY n1Xα2+n1X¯τ+nsY¯τ+ ¯τ< θ1, ρS<ρIdη
SP
Z
+Pr nP
ZY n1Xα1
nP
ZY n1Xα2+n1X¯τ+nP
ZY¯τ+ ¯τ< θ1, ρS>ρIdη
SP
Z
ε
=Pr n1XnsY(α1−α2θ1)−¯
θ1< nsY¯
θ1+¯
θ1, Z < ρIdη
SP
ρS
+Pr n1Xnp
ZY(α1−α2θ1)−¯
θ1<np
ZY¯
θ1+¯
θ1, Z > ρIdη
SP
ρS
=Pr Y < ¯
θ1
nsφ,u1, X < Y nsu1+u1
n1(Y−u1), Z < c
| {z }
A
+Pr Y < Z¯
θ1
nPφ,Z¯u1, X < Y nP¯u1+Z¯u1
n1(Y−Z¯u1), Z > c
| {z }
B
(8)
Fig. 5. he OP of Users 1and 2vs. the
PA factors at 20 dB transmit SNR
when τ= 0.
Fig. 6. The OP of Users 1and 2vs. the
transmit SNR considering OF-based PA
factors when τ= 0.
Fig. 7. The OP vs. the predefined
SNR threshold considering OF-based
based PA factors when τ= 0.
degrades as d1increases. However, the OP of U1starts to
saturate at higher SNRs when the value of d1increases. This
phenomenon can be explained by the fact that higher α1
is required when U1is located farther from R. Moreover,
Fig. 2 also compares the OP results derived for NOMA with
simulated results of OMA in the cases without ITC, i.e.,
PI=∞, and when PI= 20 dB. The proposed AF-based CR-
NOMA needs two TSs to transmit NOMA messages, while
cooperative OMA uses four TSs to serve the same number of
users. Thus, for a fair comparison, we set the data requirements
for CR-OMA as bi-fold of that for CR-NOMA.
In addition to this, we assume that the source and the
relay transmit power levels are normalized to P. Therefore,
in the OMA system, each transmit node allocates power of
1
2Pfor each data transmission. The rest parameters of the
CR-OMA system remain the same as those of the CR-NOMA
system mentioned above. In Fig. 2, within no ITC regime,
both NOMA users outperform the OMA users in terms of the
OP for all SNR values. However, when the ITC is applied
(PI= 20 dB), OMA users’ OP curves start their saturation
at higher SNR levels compared with NOMA users’ OP ones.
It is due to the fact that, regarding transmit power constraint,
the ITC level at the primary network is increased due to the
transmitted power of 1
2Pby secondary users.
In Fig. 3, the OP is plotted versus the transmit SNR for
U1and U2when d1= 3d. We compare the OP of the
proposed AF mode with that of the DF mode presented in
[14]. Moreover,the approximated OP results for both AF and
DF modes are also plotted, where we used an approximation
of e−t= 1 −tand K1(ω)∼1/ω for high SNR regimes. It
can be observed from the plot that the DF relaying performs
better than the AF one. Furthermore, the OP of the high SNR
approximation of the AF mode is consistent with that of the
DF mode, which means that the maximum optimal OP of the
AF can not exceed the outage performance of the DF mode.
In addition, Monte Carlo simulations results agree with the
derived analytical results, which confirm the accuracy of the
analytical exact and approximated OP expressions.
Fig. 4 illustrates the results of the OP versus the predefined
SNR threshold in the cases when τ={0,0.3,0.5}. The
observation is that the OP degrades as the predefined SNR
threshold value increases. Moreover, the OP for U2performs
better compared to that for U1. Thus, due to the remoteness of
U1from R(compared to the R-U2distance), the OP of x1
degrades dramatically after 3.5dB. in addition, it was noticed
that the primary interference degrades the performance of the
SUs. For example, when no primary interference exist, U1is
in an outage at about 5dB, while, with the PN interference,

Table I. The OF-based PA factors.
d11.5d2d4d
α10.79 0.84 0.945
α20.21 0.16 0.055
outage for U1starts at about 4.6dB.
The numerical results on the OF-based PA factors for
NOMA users are shown in Fig. 5. It is noted that, when
θ1>α1
α2, User 1is always in an outage, i.e., P1= 1, for all d1
values. When θ1<α1
α2, an increase of α1helps improve the
outage performance of User 1. On the other hand, an increase
of α1degrades the OP of User 2. It is also noticed that when
α1approaches 1, User 2is in outage, while User 1obtains the
best outage performance. This is due to the fact that User 1is
allocated with the maximum transmit power. Table I illustrates
all observed values of the OF-based PA factors.
Fig. 6 illustrates the OP versus the transmit SNR with the
OF-based (α1) and (α2) regarding different d1. It is observed
from the figure that, for all d1values, User 2shows better
outage performance compared to User 1at lower SNRs (<22
dB). However, at higher SNR levels, User 1is superior to User
2in terms of the outage performance. Furthermore, when the
value of d1rises, the OP saturation of User 1commences
at higher SNR levels. This is due to the fact that User 1
requires a larger value of α1when it is located farther from
R. Finally, at about 22 dB, the OP difference of both users is
the same, which confirms that OF-based PA factors provide
the OP fairness for NOMA users.
Fig. 7 illustrates the results on the OP versus the predefined
SNR threshold considering the OF-based PA factors for vari-
ous d1at 20 dB transmit SNR. It is worth mentioning that the
OF-based (α1) and (α2) improve the outage performance of
User 1(for all d1values) and provide fairness among NOMA
users, especially, at the predefined SNR threshold of 3dB.
V. CONCLUSION
In this paper, we studied the OP of the underlay CR-NOMA
AF relaying network. Exact end-to-end OP expressions for
the secondary NOMA users were derived and compared to
those of the corresponding OMA system. The comparison
result indicated that the cooperative CR-NOMA is obviously
superior to the cooperative CR-OMA. Moreover, it was shown
that properly evaluated PA factors guarantee the OP fairness
among NOMA users. Furthermore, the derived OP results
for the AF mode were compared to those for the DF one.
All derived analytical results were validated by the Monte
Carlo simulations. In future work, multiple numbers of CR-
NOMA users with wireless power transfer capability will
be considered, where both secondary source and relay cause
interference to primary networks.
APPENDIX A
PROO F OF PROPOSITION 1
The term Ain (8) can be further extended as
A=Zc
0
fZ(z)dzZu1
0
fY(y)dy
| {z }
A1
+Zc
0
fZ(z)dz
| {z }
A2
Z∞
u1
FXynsu1+u1
n1(y−u1)fY(y)dy
| {z }
A3
,(13)
where f(·)(·)and F(·)(·)denote the probability density func-
tion (PDF) and cumulative distribution function (CDF), re-
spectively. The PDFs in the term A1are independent from
each other, thus, the term A1can be written as
A1=Zc
0
fZ(z)dzZu1
0
fY(y)dy=1−e−c1−e−u1,
(14)
while A2can be written as
A2=Zc
0
fZ(z)dz=1−e−c.(15)
Then, the term A3in (13) can be further extended as
A3=Z∞
u1
e−ydy
| {z }
A31
−Z∞
u1
e−ynsu1+u1
n1(y−u1)−ydy
| {z }
A32
.(16)
The term A31 in (16) can be further rewritten as
A31 =Z∞
u1
e−ydy=e−u1.(17)
Moreover, by substituting t1=y−u1,A32 in (16) can be
rewritten as follows
A32 =Z∞
0
e−t1nsu1+nsu2
1+u1
n1t1−t1−u1dt1
=e−u1(n1+ns)
n1Z∞
0
e−u1(nsu1+1)
n1
1
t1−t1dt. (18)
Then, using [15, Eq. (3.324.1)], A32 can be further derived as
A32 =e−u1(n1+ns)
n12su1(nsu1+ 1)
n1
K1
2su1(nsu1+ 1)
n1
,
(19)
where K1(·)is the modified Bessel function of the second
kind of order 1. Further, the term A, after substituting (14),
(15), (17) and (19) into (13) is reformulated as
A=1−e−c1−e−u1+1−e−c
×e−u1−e−u1(n1+ns)
n12√r1K1(2√r1)
=1−e−c1−e−u1(n1+ns)
n12√r1K1(2√r1),(20)
where r1=u1(nsu1+1)
n1.

Now, we further extend the term Bin (8) by
B=Z∞
c
fZ(z)Zz¯u1
0
fY(y)dydz
| {z }
B1
+Z∞
c
fZ(z)Z∞
z¯u1
fY(y)FXnpy¯u1+z¯u1
n1(y−z¯u1)dydz
| {z }
B2
.
(21)
The term B1in (21) can be derived as
B1=Z∞
c
e−zZz¯u1
0
e−ydydz=Z∞
c
e−z1−e−z¯u1dz
=Z∞
c
e−zdz−Z∞
c
e−z(¯u1+1) dz=e−c−e−c(¯u1+1)
¯u1+ 1 .
(22)
Further, the term B2in (21) can be rewritten as
B2=Z∞
c
e−z



Z∞
z¯u1
e−ydy−Z∞
z¯u1
e−npy¯u1+z¯u1
n1(y−z¯u1)−ydy
| {z }
B21





dz.
(23)
Then, by using substitution p1=y−z¯u1and [15, Eq.
(3.324.1)], the term B21 in (23) is derived by
B21 =Z∞
0
e−np¯u1p1+np¯u2
1z+z¯u1
n1p1−p1−z¯u1dp1
=e−¯u1(np+z n1)
n1Z∞
0
e−z¯u1(np¯u1+1)
n1p1−p1dp1
=e−¯u1(np+z n1)
n12√s1K1(2√s1),(24)
where s1=z¯u1(np¯u1+1)
n1. Now, we rewrite (23) using (24) as
B2=Z∞
c
e−¯u1z−zdz−Z∞
c
e−¯u1(np+z n1)
n1−z
×2√s1K1(2√s1)dz=e−c(¯u1+1)
¯u1+ 1
−Z∞
c
e−zn1( ¯u1+1)+np¯u1
n12√s1K1(2√s1)dz
| {z }
B22
.(25)
Then, substituting q=z−cand after some algebraic
manipulations, the term B22 in (25) can be rewritten as
B22 =e−cn1(¯u1+1)+np¯u1
n1
×Z∞
0
e−q(¯u1+1) 2pl1K12pl1dq.
| {z }
Λ
(26)
where l1=q¯u1(np¯u1+1)+c¯u1(np¯u1+1)
n1. Further, after applying
[16, Eqs. (12) and (14)] and [17, Eq. (2.24.1.3)] to Λin (26)
and by substituting (22) - (25) into (21), the term Bcan be
written as
B=e−c−n1e−cn1(¯u1+1)+np¯u1
n1
¯u1(np¯u1+ 1)
∞
X
k=0 −c¯u1(np¯u1+1)
n1k
k!
×G1 3
3 2 n1(¯u1+ 1)
¯u1(np¯u1+ 1) 
0, k −1, k
0, k (27)
Finally, the exact OP expression for U1is derived by sub-
stituting (20) and (27) into (8) and written as in (9), where
θ1<α1
α2, otherwise, P1(θ1)∼1, which completes the proof.

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