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Massive NOMA Enhanced IoT Networks with
Partial CSI
Tianwei Hou∗, Yuanwei Liu†, Xin Sun∗, Zhengyu Song∗, Yue Chen†, and Jianjun Hou∗
∗School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing, CN
†School of Electronic Engineering and Computer Science, Queen Mary University of London, London, UK
Abstract—This paper investigates a massive non-orthogonal
multiple access (NOMA) enhanced Internet of Things (IoT)
network. In order to provide massive connectivity, a novel clus-
ter strategy is proposed, where massive devices can be served
simultaneously. New channel statistics are derived. The exact
and the asymptotic expressions in terms of coverage probability
are derived. In order to obtain further engineering insights,
short-packet communication scenarios are investigated. From our
analysis, we show that the performance of NOMA enhanced IoT
networks is capable of outperforming orthogonal multiple access
(OMA) enhanced IoT networks.
I. INTRODUCTION
In recent years, a growing number of Internet of Things
(IoT) devices are being connected to the internet at an unprece-
dented rate [1]. Aiming to provide connectivity for massive
devices, two potential solutions have been proposed [2]. New
techniques from the existing wireless networks, i.e., non-
orthogonal multiple access (NOMA), received considerable
attention for providing access services to massive machine-
to-machine (M2M) communications or massive machine-type
communications (MTC) [3]. It is estimated that by the year
2020, more than 50 billion IoT devices will be connected
as components of the IoT networks [4]. However, given the
constraint of scarce bandwidth resources, it is still challenging
to serve massive IoT devices simultaneously in the uplink
scenarios by conventional orthogonal multiple access (OMA)
techniques.
In order to solve this problem, massive non-orthogonal
multiple access (NOMA) stands as a promising solution to
provide massive connectivity by efficiently using the avail-
able bandwidth resource [3]. To be more clear, in NOMA
enhance uplink scenarios, the base station (BS) receives the
signal from multiple devices simultaneously by power domain
multiplexing within the same frequency, time and code block.
The potentials and limitations of massive NOMA assisted IoT
networks were discussed in [5], which indicates that massive
NOMA assisted IoT network is capable of providing higher
throughput compared with conventional orthogonal multiple
access techniques, i.e., time-division multiple access (TDMA)
and frequency-division multiple access (FDMA). Wu et al. [6]
proposed a NOMA enhanced wireless powered IoT network.
The performance gap between NOMA and OMA enhanced
wireless powered IoT network has been compared. Further-
more, a massive NOMA assisted IoT network was proposed
for the case that IoT devices have strict latency requirements
and no retransmission opportunities are available [7]. Shirvan-
imoghaddam et al. [8] proposed a massive IoT scenario in
cellular networks, where the throughput and energy efficiency
in a NOMA scenario with random packets arrival model were
evaluated.
In order to provide massive connectivity to massive devices,
a novel clustering strategy based on stochastic geometry tools
is proposed, where massive devices can be simultaneously
served by utilizing NOMA technique, and the BS can simply
decode the signal of massive devices from the nearest device
to the farthest device. In practice, obtaining the channel state
information (CSI) at the transmitter or receiver is not a trivial
problem, which requires the classic pilot-based training pro-
cess. Thus, it is not possible to evaluate the accurate CSI
for massive devices due to the unacceptable computational
complexity. Massive NOMA enhanced networks design has to
tackle two additional challenges: i) Having massive NOMA
devices imposes additional intra-pair interference at the BS;
ii) The massive connected devices dramatically increase the
analyse complexity. In this article, aiming at tackling the
aforementioned issues, we propose a massive NOMA enhanced
IoT network, where only partial CSI, distance information, is
required to cluster multiple devices. It is also worth noting that
the proposed massive NOMA network is a good solution for
the delay sensitive IoT devices. The transmission can be started
after synchronize immediately.
In this paper, we consider a novel massive NOMA enhanced
network, where only partial CSI is required. Based on the
proposed network, the primary theoretical contributions can
be summarized as follows: 1) We develop a novel clustering
strategy for the massive NOMA enhanced IoT networks, where
only distance information is required to cluster massive devices.
We then develop a potential scenario to address the impact of
massive NOMA on the network performance, where stochastic
geometry approaches are invoked to model the locations of
devices. 2) We derive the new channel statistics for devices.
The closed-form expressions of clustered devices in terms of
coverage probability are derived. Additionally, we derive the
general expressions in terms of coverage probability for the
OMA enhanced IoT networks. 3) Simulation results confirm
our analysis, and illustrate that by setting coverage radius and
targeted rate properly, the proposed massive NOMA enhanced
network has superior performance over its OMA counterparts in
terms of coverage probability, which demonstrates the benefits
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NOMA Uplink
Vehicle Device
Device
Fig. 1: Illustration of a typical NOMA enhanced IoT network.
of the proposed strategy. Our analytical results also illustrate
that the proposed massive NOMA enhanced network is not in
need of a larger transmit power for increasing the coverage
probability due to the fact that the coverage probability ceiling
occurs in the high SNR regimes. For the case of finite packet
length, it is demonstrated that the impacts of packet length on
the achievable rate are getting stronger with increased number
of devices.
II. SYSTEM MODEL
Consider a NOMA enhanced uplink communication sce-
nario in which multiple terrestrial devices equipped with a
single transmitting antenna each are communicating with a BS
equipped with a single omni receiving antenna. Fig. 1 illustrates
the NOMA enhanced wireless communication model with a
single BS.
The terrestrial devices are located in the different power
zones according to homogeneous Poisson point process
(HPPP), which is denoted by Ψgand associated with the
density λg. It is assumed that Mterrestrial devices transmit
their signal to the BS via NOMA protocol, where Mdevices are
located in different power zones. Without loss of generality, the
disc R2with the radius Ris equally separated to Mdifferent
power zones by distance, e.g. the radius of the i-th power zone
is between (i−1)R
Mto iR
Mfor the case of 1<i≤M. In this
article, we define the device located in the i-th power zone as
user i.
Consider the use of a composite channel model with two
parts, large-scale fading and small-scale fading. Ldenotes the
large-scale fading between the BS and devices. It is assumed
that large-scale fading and small-scale fading are independently
and identically distributed (i.i.d.). In this article, large-scale
fading represents the path loss between the BS and devices,
which can be expressed as
Lg,i(d)=d−αg
g,i ,ifd
g,i >r
0
r−αg
0,otherwise ,(1)
where dg,i denotes the distance between the BS and terrestrial
device i, and αgdenotes the path loss exponent for terrestrial
devices. The parameter r0avoids a singularity when the dis-
tance is small. For simplicity, it is assumed that the minimum
radius of the disc and space are greater than r0.
Due to the fact that the strong scattering between the BS
and terrestrial devices, the small-scale fading of device iis
defined by Rayleigh fading, which is denoted by |hg,i|2, and
the probability density functions (PDFs) can be expressed as
f(x)=e−x.(2)
Since large-scale fading is the dominate component of attenu-
ations, the BS only needs partial CSI, the distance information
between devices and the BS, to group multiple devices in a
NOMA cluster. Thus, given the channel gain relationship of
multiple devices, we have d−αg
g,1|hg,1|2>d
−αg
g,2|hg,2|2>···>
d−αg
g,M |hg,M|2at the BS. In this article, it is assumed that the
transmit power of multiple devices are the same, and therefore
the BS can decode multiple devices from the nearest device to
the farthest device.
In uplink transmission, the BS receives the signal for multiple
terrestrial devices simultaneously. Thus, the received power for
the BS is given by
PB=
M
i=1
PgLg,i|hg,i|2+σ2,(3)
where σ2denotes the additive white Gaussian noise (AWGN)
power, Pgdenotes the transmit power of terrestrial devices.
Note that the proposed design cannot guarantee the optimal
performance for the NOMA enhanced network. More sophis-
ticated designs on transmit power levels can be developed for
further enhancing the attainable performance of the network
considered, but this is beyond the scope of this treatise. Besides,
it is assumed that the CSI of terrestrial devices are partly
known, where the information of small-scale fading is unknown
at the BS.
III. MASSIVE NOMA ENHANCED IOTNETWORKS
We first discuss the performance of the massive NOMA
enhanced IoT networks. New channel statistics and coverage
probabilities are illustrated in the following subsections.
1) New Channel Statistics: In this subsection, we derive new
channel statistics for the NOMA enhanced networks, which
will be used for evaluating the coverage probabilities in the
following subsections.
Lemma 1. Assuming that terrestrial devices are i.i.d. located
according to HPPPs in the disc R2of Fig. 1. In order to provide
access services to devices simultaneously by NOMA technique,
multiple users located in different power zones are grouped.
Therefore, the PDFs of terrestrial device ican be given by
fg,i(r)=2M2r
R2−M2r2
0,i=1,r
0<r< R
M
2M2r
(2i−1)R2,1<i≤M, (i−1)R
M<r< iR
M
,(4)
where M≥2.
Proof: We first focus on the nearest device, who is located
in the disc with the radius roto R
M. According to HPPP, the
PDF of the nearest device can be derived by
fg,1(r)= λgΨg2πr
λgΨgπR
M2−π(ro)2.(5)
Again, according to HPPPs, the PDF of terrestrial devices
ican be given by
fg,i (r)= λgΨg2πr
λgΨgπiR
M2−π(i−1)R
M2,(6)
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if i>1. After some algebraic manipulations, the proof of
Lemma 1 is complete.
2) Coverage Probability: In this subsection, we derive the
coverage probability for terrestrial devices. The coverage prob-
ability is defined as the probability that the BS can successfully
decode the multiplexed signal via SIC technique with a certain
pre-determined SINR threshold. As such, the coverage proba-
bility for the device iis given in the following Lemma.
Lemma 2. For the proposed NOMA enhanced network with
Mdevices, the overall transmission coverage probability for
device iwith M≥iis given by
Pg,i,cov(τi)=
i
b=1
Pg,b(τb),(7)
where Pg,b(τb)denotes the coverage probability for decoding
the signal of device b.
Remark 1. The results in (7) illustrate that the coverage
probability of device iis depending on the devices located
nearer than the device i.
Remark 2. The results in (7) indicate that if the decoding for
device bwith b<iis failed, the coverage probability of device
iis zero.
We then focus on analyzing the coverage probability for
decoding the signal of terrestrial device i, which can be
expressed as
Pg,i(τi)=fg,i (r)Pr{Blog2(1+SINRg,i)>R
i}dr,
(8)
where Bdenotes the bandwidth of terrestrial device i, and the
SINR threshold can be given by τi=2Ri
B−1,Rirepresents the
target rate of the device i. Thus, the SINR of terrestrial device
ican be expressed as
SINRg,i =Pgd−αg
g,i |hg,i|2
M
c=i+1
Pgd−αg
g,c |hg,c|2+σ2
.(9)
Based on (8) and (9), one can obtain
Pg,i(τi)=Pr|hg,i|2>τiσ2
Pg
dαg
g,i +τidαg
g,i
Pg
Ig
=e−ρiσ2Lg,i (ρi),
(10)
where ρi=τirαg
Pg,Ig=
M
c=i+1
Pgd−αg
g,c |hg,c|2,Lg,i (ρi)repre-
sents the Laplace transform of the power density distributions
of interference from the terrestrial devices.
We then turn our attention to obtaining the Laplace transform
of intra-pair interference in (10).
Lemma 3. Assuming that Mterrestrial devices are i.i.d.
located according to HPPPs in the disc R2. The Laplace
transform of terrestrial interference for terrestrial device ican
be given by
Lg,i (s)=
M
c=i+1
1
(2c−1) c22F11,−δg;1−δg;−sPgM
cR αg
−(c−1)22F11,−δg;1−δg;−sPgM
(c−1)Rαg,
(11)
where δg=2
αg, and 2F1(·,·;·;·)represents the Gauss hyper-
geometric function [9, eq. (3.194.2)].
Proof: Please refer to Appendix A.
Based on derived results in Lemma 3, we can obtain the
coverage probability in the following Theorem.
Theorem 1. Assuming that the devices are located in the differ-
ent power zones according to HPPPs, the coverage probability
of terrestrial device ican be expressed as follows:
Pg,i(τi)= 2M2
(2i−1)R2iR
M
(i−1)R
M
re−ρiσ2Lg,i (ρi)dr, (12)
for i≥2, and
Pg,1(τ1)= 2M2
(R2−M2r2
0)R
M
r0
re−ρ1σ2Lg,1(ρ1)dr, (13)
for i=1.
Proof: Based on the derived results in Lemma 3,wecan
first express the coverage probability of terrestrial device ias
follows:
Pg,i(τi)=
R2
i
fg,i(dg,i)e−ρi,gσ2Lg,i (ρi,g )d(dαg
g,i),(14)
where ρi,g =τidαg
g,i
Pg. For simplicity, R2
irepresents the ring for
terrestrial devices i. Upon changing to polar coordinates, we
can obtain the desired results in (12) and (13). Thus, the proof
is complete.
It is hard to obtain engineering insights from (12) and (13)
directly, and thus we derive the following corollary.
Corollary 1. Assuming that the devices are located in the
different power zones according to HPPPs, and r0<< R
M,
the coverage probability of device ican be approximated to
Pg,i(τi)≈M
(2i−1)Rωn
N
n=1
ξnlne−ρn,gσ2Lg,i (ρn,g ),(15)
for i>1, where ωn=π
N,ξn=1−ν2
n,νn=cos2n−1
2Nπ,
ln=R(νn+2i−1)
2M,ρn,g =τilαg
n
Pg,Ndenotes the Gaussian-
Chebyshev parameter.
The coverage probability of the nearest terrestrial device can
be expressed as
Pg,1(τ1)≈M
R+Mr0
ωn
N
n=1
ξnl1e−ρ1,gσ2Lg,i (ρ1,g ),(16)
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for i=1,l1=(R−Mr0)νn+2R
R−Mr0−1
2M,ρ1,g =τ1lαg
1
Pg. Here,
Lg,i(·)is given by (11).
Proof: By utilizing Gauss-Chebyshev Quadrature, we can
obtain the desired results in (15) and (16). Thus, the proof is
complete.
In order to obtain further engineering insights, we also derive
the following corollary in the case of low target rate scenario.
Corollary 2. Assuming that the devices are located in the
different power zones according to HPPPs, and r0<< R
M.
It is assumed that the target rate of terrestrial device iis lower
than the bandwidth, i.e., τi<1, the closed-form expression in
terms of coverage probability of device ican be approximated
to
Pg,i(τi)=φ1φ−n−δg
2
αgγn+δg+1,φ
2iR
Mαg
−γn+δg+1,φ
2(i−1)R
Mαg,
(17)
where
φ1=2M2
(2i−1)R2
M
m=i+1
1
(2m−1)
N
n=0
(1)n(−δg)n
(1 −δg)nn!
×−τiM
Rαgnm−nαg+2 −(m−1)−nαg+2,
φ2=τiσ2
Pg, and (x)nrepresents rising Pochhammer symbol
with (x)n=Γ(x+n)
Γ(x).
Proof: Please refer to Appendix C.
In order to glean further engineering insights, the coverage
probability of terrestrial device iin the OMA scenario, i.e.,
TDMA, is also derived in the following Corollary. In the
OMA scenario, multiple terrestrial devices obey the same
distance distributions and small-scale fading channels. The
OMA benchmark adopted in this treatise is that by dividing
the multiple users in equal time/frequency slots.
Corollary 3. In the OMA scenario, assuming that the devices
are located in the different power zones according to HPPPs,
the overall coverage probability of device ican be approxi-
mated as follows:
PO
g,i,cov(τO
i)= 2M2φ−δg
2,O
(2i−1)R2αgγδg+1,φ
2,OiR
Mαg
−γδg+1,φ
2,O(i−1)R
Mαg,
(18)
where τO
i=2MRi
B−1, and φ2,O =τO
iσ2
Pg.
Proof: We first derive the coverage probability expression
of terrestrial device iin the OMA case as follows
Pr B
Mlog2(1+SINRi,O)>R
i,(19)
where SINRi,O =Pgd
−αg
g,i |hg,i|2
σ2. Following the similar steps
in Appendix C, the result in (18) can be readily proved.
Remark 3. The results in (18) indicate that the coverage prob-
ability of multiple devices in the OMA scenario is independent
on other devices, whereas the coverage probability of devices
depend on the nearer devices in the proposed NOMA enhanced
networks.
We also want to provide some benchmark schemes in
TABLE I. We use RBs to represent the required number of
resource blocks for the case that the amount number of devices
is set to 1000. For the proposed NOMA networks, the required
number of RBs is 200 by setting M=5
, which indicates that
the proposed NOMA network is more efficient on the RBs.
TABLE I:
REQUIRED NUMBER OF RBs (1000 DEVICES)
Access Mode RBs
Conventional OMA 1000
SCMA [10] 667
Proposed NOMA 1000
M
A. NOMA Enhanced IoT Networks with Finite Packet Length
One of the negligible advantages of the proposed NOMA
enhanced IoT networks is low latency, and thus information
may also conveyed in short-packets with finite blocklength.
However, the finite blocklength results in a non-negligible
decreasing of achievable rate at the BS [11,12]. We then
analyse achievable rate for the case of finite packet length. In
this article, we denote Ri,f as the maximum achievable rate
of device iin the case of finite blocklength, which can be
expressed as:
Ri,f =log
2(1 + SINRi)−Vi
Nf
Q−1(Pi)
ln 2 ,(20)
where Nfdenotes the packet length, Q−1(x)represents the
inverse of Q(x)=∞
x
1
√2πexp −t2
2dt,Videnotes the
channel dispersion with Vi=1−(1 + SIN Ri)−2, and Pi
is the overall outage probability of device i[13].
Remark 4. Based on the expression in (20), one can know
that the packet length has impact on the achievable rate
dramatically, where higher packet length results in larger
achievable rate. We can also observe that for the case of
Nf≈∞, the achievable rate can be maximized as Ri,f =
log2(1 + SINRi).
IV. NUMERICAL STUDIES
In this section, numerical results are provided to facilitate
the performance evaluation of massive NOMA enhanced IoT
networks. Monte Carlo simulations are conducted for verifying
analytical results. It is assumed that the bandwidth is B= 125
kHz as one of the most common setting-ups for IoT networks.
The power of AWGN is set to σ2=−174 + 10log 10(B)dBm.
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-50 -40 -30 -20 -10 0 10 20 30 40 50
Power of device P g (dBm)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Coverage probability
Solid lines: The exact result
Dash lines: The asymptotic result
1st device, M=3
2nd device, M=3
3rd device, M=3
1st device, M=2
2nd device, M=2
Simulation
Fig. 2: Coverage probability of NOMA devices versus the
transmit power with different number of devices. The threshold
is τi=0.5.
R/B (BPCU)
Distance of the disc radius (m)
0
2000
0.2
0.4
0.6
Coverage probability
0.8
0
1
0.5
1500 1
1.5
2
2.5
1000 3
NOMA
OMA
Fig. 3: Coverage probability of both the massive NOMA and
OMA enhanced network versus disc radius and target rate in
the case of M=2. The transmit power of devices is Pg=0
dBm.
The minimum distance is r0=1m. The path loss exponents is
set to αg=4. The radius of the disc is set to R= 1000m.
1) Impact of the Number of Devices: Fig. 2 plots the
coverage probability with different number of accessed devices.
The coverage probability of two-device case is plotted as the
benchmark schemes. As we can see in the figure, coverage
probability ceilings occurs, which meet the expectation due
to the strong intra-pair interference. Therefore, the proposed
network is not in need of a larger transmit power for increasing
the coverage probability. We can also see that the coverage
probabilities for the first device and the second device in the
case of M=2are the same in the high SNR regime. This
is due to the fact that in the high SNR regime, the coverage
probability of far devices approaches one.
2) Performance Comparing with OMA: In Fig. 3, we
evaluate the coverage probability of both NOMA and OMA
enhanced networks with different disc radius and target rate
in two-device scenario. The coverage probability is derived
by Pg,1,cov ×Pg,2,cov. The two-device scenario of OMA
enhanced network in terms of coverage probability is derived
by PO
g,1,cov ×PO
g,2,cov. As can be seen from Fig. 3, the coverage
probability of NOMA enhanced network is higher than the
OMA enhanced networks, which implies that NOMA enhanced
networks is capable of providing better network performance
than OMA.
-40 -30 -20 -10 0 10 20 30
Transmit Power P g (dBm)
0
10
20
30
40
50
60
SINR (dB)
Near device, 2 devices case
Near device, 3 devices case
Near device, 4 devices case
Near device, 5 devices case
Fig. 4: SINR of the nearest NOMA device versus transmit
power Pgwith different number of serving devices.
-20 -15 -10 -5 0 5 10 15 20
Transmit power P (dBm)
0
5
10
15
20
25
30
Network throughput (BPCU)
Solid lines: The result of infinite-packet length
Dash lines: The result of finite-packet length with N
f=100
Dotted lines: The result of finite-packet length with N
f=300
Terrestrial networks with M=5
Terrestrial networks with M=7
Terrestrial networks with M=15
Fig. 5: Network throughput of massive NOMA enhanced
network versus the transmit power with Pg=P. The target
rate of the nearest device and far devices are set as R1=1.5
BPCU and R2,··· ,R
M=1BPCU, respectively.
3) Impact of the Number of Devices: Fig. 4 plots the SINR
threshold with different number of devices. On the one hand, in
the low transmit power regime, the SINR performance for the
nearest devices in the case of five-device case is better than the
two-device case. This is because that the distance of the nearest
devices in five-device case is much smaller than two-device
case. Observe that in the high transmit power regime, the SINR
of two-device case is higher than other cases, which indicates
that the interference of two-device case is the minimum case.
Additionally, there is a cross point of curves, which indicates
that there exists an optimal point for the proposed scenario. It
is also not in need of a larger transmit power for increasing the
coverage probability due to the fact that the coverage ceilings
occur in the high transmit power regime.
4) Achievable Throughput with Finite Packet Length: Fig. 5
plots the network throughput versus the transmit power with
different packet length. The achievable rate of each device is
derived by (20), where the system throughput is derived by
the summation of massive devices. The solid curve, dashed
curve and dotted curve are the achievable rate of the infinite
packet length Nf≈∞, finite packet length with Nf= 100 and
Nf= 300, respectively. It is observed that the larger packet
length is capable of providing higher achievable rate. Based
on blue curves and red curves, one can observe that the gap of
network throughput between infinite packet length scenario and
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finite packet length scenario increases, which indicates that the
network throughput is sensitive on the packet length. Observe
that the network throughput supported by massive NOMA
technique are nearly the same for the case of Nf= 100, which
indicates that the network throughput cannot be enhanced for
the short packet scenario. This observation shown that NOMA
technique may not be a good solution for short packet scenario.
V. C ONCLUSIONS
In this article, the application of a massive NOMA enhanced
IoT networks was proposed. Specifically, a novel clustering
strategy was proposed, where only partial CSI is required.
Stochastic geometry tools were invoked for modeling the
spatial randomness. Additionally, new closed-form expressions
in terms of coverage probability were derived for characterizing
the network performance. The performance of OMA enhanced
networks was also derived as the benchmark schemes. It was
analytically demonstrated that the NOMA enhanced networks
are capable of outperforming its OMA counterparts.
APPENDIX A: PROOF OF LEMMA 3
Recall that the intra-pair interference received at the BS for
decoding terrestrial device ican be expressed as
Ig,i=
M
c=i+1
Pgd−αg
g,c |hg,c|2.(A.1)
Therefore, the expectation for the intra-pair interference can be
calculated as follows:
Lg,i (s)=Eexp −s
M
c=i+1
Pgd−αg
g,c |hg,c|2
(a)
=
∞
0
M
c=i+1
exp −sPgd−αg
g,c exp(−x)dx
=EM
c=i+1
1
1+sPgd−αg
g,c .
(A.2)
where (a) can be gleaned by the fact that |hg,m|follows
Rayleigh distribution.
Recall that the distance PDFs of terrestrial interferences
follow (4), and thus the Laplace transform can be transformed
into
Lg,i (s)=
M
c=i+1
fg,c (x)cR
M
(c−1)R
M
x
1+sPgx−αgdx
(b)
=
M
c=i+1
2M2(sPg)δg
(2c−1)R2αgsPg((c−1)R
M)−αg
sPg(cR
M)−αg
t−δg−1
1+tdt,
(A.3)
where (b) is obtained by using t=sPgx−αg, and by apply-
ing [9, eq. (3.194.2)], we can obtain the Laplace transform in
an elegant form in (11). The proof is complete.
APPENDIX B: PROOF OF COROLLARY 2
We first expand Gauss hypergeometric function as follows
2F1(a, b;c;z)=
N
n=0
(a)n(b)n
(c)nn!zn.(C.1)
Therefore, the result in (11) can be further transformed into
Lg,i (ρi)=
M
c=i+1
1
(2c−1)
N
n=0
(1)n(−δg)n
(1 −δg)nn!
×−τiM
Rαgnc−nαg+2 −(c−1)−nαg+2rnαg.
(C.2)
By substituting (C.2) into (12), one can obtain
Pi(τi)=φ1iR
M
(i−1)R
M
rnαg+1e−φ2rαgdr. (C.3)
By using t=φ2rαg, and by applying [9, eq. (8.350.1)], the
coverage probability in (17) can be obtained. The proof is
complete.
ACKNOWLEDGMENT
This work was supported by Beijing Natural Science Foun-
dation under Grant 4194087.
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