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Energy Efficient Resource Allocation in
Machine-to-Machine Communications with Multiple
Access and Energy Harvesting for IoT
Zhaohui Yang, Wei Xu, Senior Member, IEEE, Yijin Pan, Cunhua Pan, and Ming Chen
Abstract—This paper studies energy efficient resource alloca-
tion for a machine-to-machine (M2M) enabled cellular network
with non-linear energy harvesting, especially focusing on two dif-
ferent multiple access strategies, namely non-orthogonal multiple
access (NOMA) and time division multiple access (TDMA). Our
goal is to minimize the total energy consumption of the network
via joint power control and time allocation while taking into
account circuit power consumption. For both NOMA and TDMA
strategies, we show that it is optimal for each machine type
communication device (MTCD) to transmit with the minimum
throughput, and the energy consumption of each MTCD is a
convex function with respect to the allocated transmission time.
Based on the derived optimal conditions for the transmission
power of MTCDs, we transform the original optimization prob-
lem for NOMA to an equivalent problem which can be solved
suboptimally via an iterative power control and time allocation
algorithm. Through an appropriate variable transformation, we
also transform the original optimization problem for TDMA to
an equivalent tractable problem, which can be iteratively solved.
Numerical results verify the theoretical findings and demonstrate
that NOMA consumes less total energy than TDMA at low circuit
power regime of MTCDs, while at high circuit power regime of
MTCDs TDMA achieves better network energy efficiency than
NOMA.
Index Terms—Internet of Things (IoT), machine-to-machine
(M2M), non-orthogonal multiple access (NOMA), energy har-
vesting, resource allocation.
I. INT ROD UC TI ON
Machine-to-machine (M2M) communications have been
considered as one of the promising technologies to realize
the Internet of Things (IoT) in the future 5th generation
network. M2M communications can be applied to many IoT
applications, which mainly involve new business models and
opportunities, such as smart grids, environmental monitoring
and intelligent transport systems [2]. Different from conven-
tional human type communications, M2M communications
have many unique features [3]. The unique features include
This work was supported in part by the National Nature Science Foundation
of China under grants 61471114, 61372106 & 61221002, in part by the
Six Talent Peaks project in Jiangsu Province under grant GDZB-005, in
part by the UK Engineering and Physical Sciences Research Council under
Grant EP/N029666/1, and in part by the Scientific Research Foundation of
Graduate School of Southeast University under Grant YBJJ1650. This paper
was presented at the IEEE Infocom Workshops 2017 in Atlanta, GA, USA
[1]. (Corresponding authors: Wei Xu; Cunhua Pan.)
Z. Yang, W. Xu, Y. Pan, and M. Chen are with the National Mobile
Communications Research Laboratory, Southeast University, Nanjing 210096,
China, (Email: {yangzhaohui, wxu, panyijin, chenming}@seu.edu.cn).
C. Pan is with the School of Electronic Engineering and Computer
Science, Queen Mary, University of London, London E1 4NS, U.K., (Email:
c.pan@qmul.ac.uk).
massive transmissions from a large number of machine type
communication devices (MTCDs), small bursty natured traffic
(periodically generated), extra low power consumption of
MTCDs, high requirements of energy efficiency and security.
A key challenge for M2M communications is access control,
which manages the engagement of massive MTCDs to the
core network. To tackle this challenge, various solutions have
been proposed, e.g., by using wired access (cable, DSL)
[4], wireless short distance techniques (WLAN, Bluetooth),
and wide area cellular network infrastructure (Long Term
Evolution-Advanced (LTE-A), WiMAX) [5]. Among all these
solutions, an effective approach is to deploy machine type
communication gateways (MTCGs) to act as relays of MTCDs
[3]. With the help of MTCGs, all MTCDs can be successfully
connected to the base station (BS) at the additional cost of
energy consumption [6]–[9]. To enable multiple MTCDs to
transmit data to the same MTCG, time division multiple access
(TDMA) was adopted in [10]. However, since there are a
vast number of MTCDs to be served, TDMA leads to large
transmission delay and synchronization overhead. By splitting
users in the power domain, non-orthogonal multiple access
(NOMA) can simultaneously serve multiple users at the same
frequency or time resource [11]. Consequently, NOMA based
access scheme yields a significant gain in spectral efficiency
over the conventional orthogonal TDMA [11]–[16]. This fa-
vorable characteristic makes NOMA an attractive solution for
supporting massive MTCDs in M2M networks. Considering
NOMA, [17] investigated an M2M enabled cellular network,
where multiple MTCDs simultaneously transmit data to the
same MTCG and multiple MTCGs simultaneously transmit
the gathered data to the BS.
Besides, another key challenge is the energy consumption of
MTCDs [18]–[20]. According to [3], the total system through-
put of an M2M network is mainly limited by the energy budget
of the MTCDs. To improve the system performance, energy
harvesting (EH) has been applied to wireless communication
networks [21]–[24]. In particular, direct and non-direct energy
transfer based schemes for EH were investigated in [23], while
in [24], the optimization of green-energy-powered cognitive
radio networks was surveyed. Recently, the downlink resource
allocation for EH in small cells was studied in [25]–[27]. By
using EH, MTCDs are able to harvest wireless energy from
radio frequency (RF) signals [28]–[31], and the system energy
can be significantly improved. Consequently, implementing
EH is promising in M2M communications especially with
MTCDs configured with low power consumption. In previous
2
wireless powered communication networks using relays [32]–
[34], it was assumed that an energy constrained relay node
harvests energy from RF signals and the relay uses that
harvested energy to forward source information to destination.
Due to the extra low power budget of MTCDs in M2M
communications, it is reasonable to let the MTCD transmit
data to an MTCG, and then the MTCG relays the information
while the MTCD simultaneously harvests energy from the
MTCG, which is different from existing works, e.g., [32]–[34].
Enabling the source node to harvest energy from the relay
node, a power-allocation scheme for a decode-and-forward
relaying-enhanced wireless system was proposed in [35] with
one source node, one relay node and one destination node.
The above-mentioned energy consumption models, consid-
ered in [1], [8]–[10], [17]–[19], [28]–[31], are only concerned
with the RF transmission power and ignore the circuit power
consumption of MTCDs and MTCGs. However, as stated in
[36], the circuit power consumption is non-negligible com-
pared to RF transmission power. Without considering the
circuit power consumption, energy saving can always benefit
from longer transmission time [37], [38]. While considering
the circuit power consumption which definitely exists in prac-
tice, the results vary significantly in that it can not be always
energy efficient for long transmission time due to the fact that
the total energy consumption becomes infinity as transmission
time goes infinity. Hence, it is of importance to investigate the
optimal transmission time when taking into account the circuit
power consumption in applications.
Given access control and energy consumption challenges in
M2M communications, both TDMA and NOMA based M2M
networks with EH are proposed in this work. The MTCDs first
transmit data to the corresponding MTCGs, and then MTCGs
transmit wireless information to the BS and wireless energy to
the MTCDs. To prolong the lifetime of the considered network,
the harvested energy for each MTCD is set to be no less
than the consumed energy in information transmission (IT)
stage. The main contributions of this paper are summarized as
follows:
•We formulate the total energy minimization problem for
the M2M enabled cellular network with non-linear energy
harvesting (EH) model via joint power control and time
allocation. In the non-linear EH model, we consider the
receiver sensitivity, on which energy conversion starts
beyond a threshold. Besides, we explicitly take into ac-
count the circuit energy consumption of both MTCDs and
MTCGs. All theses factors are critical in practical appli-
cations which inevitably affect the system performance.
Specifically, the non-linear EH model leads to a non-
smooth objective function and non-smooth constraints,
and the circuit energy consumption affects the optimal
transmission time of the system.
•For the NOMA strategy, we observe that: 1) it is optimal
for each MTCD to transmit with minimal throughput;
2) it is further revealed that the energy consumption of
each MTCD is a convex function with respect to the
allocated transmission time. Given these observations,
it indicates that a globally optimal transmission time
always exists that the optimal transmission time equals
the maximally allowed transmission time if it does not
exceed a quantified threshold derived in closed form.
•To solve the original total energy minimization problem
for the NOMA strategy, we devise a low-complexity
iterative power control and time allocation algorithm.
Specifically, to deal with the non-smooth EH function, we
introduce new sets during which MTCDs can effectively
harvest energy. Given new sets, the EH function of
MTCDs can be presented as a continuous one. Moreover,
to deal with nonconvex objective function, nonconvex
minimal throughput constraints, and nonconvex energy
causality constraints, we transform these nonconvex ones
into convex ones by manipulations with the optimal
conditions. The convergence of the iterative algorithm is
strictly proved.
•For the TDMA strategy, we verify that the two observa-
tions for NOMA are also valid. Although the original total
energy minimization problem for the TDMA strategy
is nonconvex, the problem can be transformed into an
equivalent tractable one, which can be iteratively solved
to its suboptimality. For the total energy minimization,
numerical results identify that NOMA is superior over
TDMA at small circuit power regime of MTCDs, while
TDMA outperforms NOMA at large circuit power regime
of MTCDs.
This paper is organized as follows. In Section II, we in-
troduce the system and power consumption model. Section III
and Section IV provide the energy efficient resource allocation
for NOMA and TDMA, respectively. Numerical results are
displayed in Section V and conclusions are drawn in Section
VI.
II. SY ST EM A ND POW ER CONSUMPTION MO DE L
A. System Model
Consider an uplink M2M enabled cellular network with
NMTCGs and MMTCDs, as shown in Fig. 1. Denote
the sets of MTCGs and MTCDs by N={1,·· · , N }
and M={1,·· · , M }, respectively. Each MTCG serves
as a relay for some MTCDs. Assume that the decode-and-
forward protocol [39] is adopted at each MTCG. Denote
Ji={Ji−1+ 1,·· · , Ji}as the specific set of MTCDs served
by MTCG i∈ N, where J0= 0,JN=M,Ji=i
l=1 |Jl|,
and |·|is the cardinality of a set.1To reduce the receiver
complexity at the MTCG, the maximal number of MTCDs
associated to one MTCG is set as four. Obviously, we have
i∈N Ji=M.
B. NOMA Strategy
In time constraint T, each MTCD has some payloads to
transmit to the BS. By using superposition coding at the
transmitter and successive interference cancellation (SIC) at
1In this paper, we assume that MTCDs are already associated to MTCGs by
using the cluster formation methods for M2M communications, e.g., in [40]–
[43]. Joint optimization of cluster formation and resource allocation in M2M
communications with NOMA/TDMA and EH can certainly further improve
the performance, but we leave it in future work in order to focus on the power
control and time allocation in the current submission.
3
ĂĂ
MTCG 1 MTCG N
MTCD 2
MTCD 1
BS
MTCD M-1
MTCD M
Information transmission (IT)
Energy harvesting (EH)
M
Fig. 1. The considered uplink M2M enabled cellular network.
ĂĂ
t1
MTCDs
tN+1
T
MTCGsMTCDs
tN+K
MTCGs
ĂĂ
MTCD uplink IT
MTCG uplink IT & MTCD EH
t1
tN
Fig. 2. Time sharing scheme for NOMA strategy during one uplink
transmission period.
the receiver, multiple MTCDs (or MTCGs) can simultaneously
transmit signals to the corresponding receiver using NOMA.
To reduce the receiver complexity and error propagation due
to SIC, it is reasonable for the same resource to be multiplexed
by a small number (usually two to four) of devices [44]. Con-
sidering the receiver complexity at the BS, the sets of MTCGs
are further classified into multiple small clusters. For MTCGs,
the set Nis classified into Kclusters. Let K={1,2,·· · , K}
be the set of clusters. Denote Ik={Ik−1+ 1,·· · , Ik}as
the specific set of MTCGs in cluster k∈ K, where I0= 0,
IK=N,Ik=k
l=1 |Il|, and |Ik| ≤ 4.2
Note that our major motivation of using NOMA is to
enhance the ability of serving more terminals simultaneously
[11]. However, the number of terminals occupying the same
resource cannot be arbitrarily large in order to make NOMA
effective in practice. Therefore, if there would be an even
higher number of IoT terminals, we believe that a number
of ways including NOMA should be further incorporated to
better address the access problem for higher number of IoT
terminals [47].
As depicted in Fig. 2, time Tconsists of N+Kuplink
transmission phases for MTCDs and MTCGs. NOMA is
adopted for MTCDs to transmit data to MTCGs in the first
Nphases. Both NOMA and EH are in operation in the
last K phases where MTCGs transmit data to the BS and
2A scheme for cluster formation for uplink NOMA is in [45]. According
to [45] and [46], one common scheme with two devices in each cluster is
the strong-weak scheme, i.e., the device with the strongest channel condition
is paired with the device with the weakest, and the device with the second
strongest is paired with one with the second weakest, and so on.
simultaneously MTCDs harvest energy from all MTCGs. In
the i-th (i≤N) phase with allocated time ti, all MTCDs
in Jisimultaneously transmit data to MTCG iaccording to
the NOMA principle and MTCG idetects the signal. In the
(N+k)-th (k≤K) phase with allocated time tN+k, all
MTCGs in Iksimultaneously transmit the decoded data from
the served MTCDs to the BS by using the NOMA strategy. As
a result, we have the following transmission time constraint
N+K
i=1
ti≤T. (1)
In the i-th (i≤N) phase, all MTCDs in Jisimultaneously
transmit data to MTCG ifollowing the NOMA principle. The
received signal of MTCG iis
yi=
Ji
j=Ji−1+1
hij √pjsj+ni,(2)
where hij is the channel between MTCD jand MTCG
i,pjdenotes the transmission power of MTCD j,sjis
the transmitted message of MTCD j, and nirepresents the
additive zero-mean Gaussian noise with variance σ2. Without
loss of generality, the channels are sorted as |hi(Ji−1+1)|2≥
·· · ≥ |hiJi|2. By applying SIC to decode the signals [14]–
[16], the achievable throughput of MTCD j∈ Jiis
rij =Btilog21 + |hij |2pj
Ji
l=j+1 |hil|2pl+σ2,(3)
where Bis the available bandwidth for transmission. Note
that we consider the case where MTCDs associated to dif-
ferent MTCGs are allocated with orthogonal time resource.
Therefore, the interference from other MTCDs associated to
different MTCDs is ignored.
In the (N+k)-th (k≤K) phase with allocated time tN+k,
after having successfully decoded the messages in the last N
phases, MTCGs in Iksimultaneously transmit the gathered
data to the BS based on the NOMA principle. Denote the
channel between MTCG iand the BS by hi. Without loss
of generality, the channels are sorted as |hIk−1+1|2≥ ·· · ≥
|hIk|2,∀k∈ K. Hence, the achievable throughput of MTCG
i∈ Ikcan be expressed as [14]–[16]
ri=BtN+ilog21 + |hi|2qi
Ik
n=i+1 |hn|2qn+σ2,(4)
where qiis the transmission power of MTCG i.
According to [3], MTCDs are always equipped with finite
batteries, which limit the lifetime of the M2M enabled cellular
network. To further prolong the lifetime, EH technology is
adopted for MTCDs to harvest energy remotely from RF
signals radiated by MTCGs [22]. Specifically, each MTCD
harvests energy when MTCGs transmit data to the BS. Since
the noise power is much smaller than the received power of
MTCGs in practice [48]–[50], the energy harvested from the
channel noise is negligible. Assume that uplink channel and
downlink channel follow the channel reciprocity [51]. The
4
0
Input RF power
Harvested power
Linear EH model
Non-linear EH model
0
P
Fig. 3. Comparison between the linear and non-linear EH model.
total energy harvested by MTCD jserved by MTCG ican
be evaluated as
EH
ij =
K
k=1
tN+ku
Ik
n=Ik−1+1 |hnj |2qn
,∀i∈ N, j ∈ Ji,
(5)
where Ik
n=Ik−1+1 |hnj |2qnis the received RF power of
MTCD jduring time tN+k, and function u(·)captures the EH
model which maps input RF power into harvested power. Two
commonly used EH models are shown in Fig. 3, i.e., linear
and non-linear EH models. According to [52] and [53], linear
EH model may lead to resource allocation mismatch. In order
to capture the effects of practical EH circuits on the end-to-
end power conversion, we adopt the more practical non-linear
EH model proposed in [52]:
u(x) = M(1+eab)
eab+e−a(x−2b)−M
eab ,if x≥P0
0,elsewise ,(6)
where a,b,Mand P0are positive parameters which capture
the joint effects of different non-linear phenomena caused by
hardware constraints. Note that P0is the receiver sensitivity
threshold of each MTCD, in which energy conversion starts.
Hence, it is possible that some MTCDs cannot effectively
harvest energy in some slots, since the received power is below
the receiver sensitivity threshold P0.
The total energy consumption of the M2M enabled commu-
nication network consists of two parts: the energy consumed
by MTCDs and MTCGs. For each part, the energy consump-
tion of a transmitter consists of both RF transmission power
and circuit power due to hardware processing [54]. According
to [55], the energy consumption when MTCDs or MTCGs are
in idle model, i.e., do not transmit RF signals, is negligible.
During the i-th (i≤N) phase, MTCD j∈ Jiserved by
MTCG ijust transmits data to MTCG iwith allocated time
tiand transmission power pj. Thus, the energy Eij consumed
by MTCD j∈ Jican be modeled as
Eij =tipj
η+PC,∀i∈ N, j ∈ Ji,(7)
where η∈(0,1] and PCdenote the power amplifier (PA)
efficiency and the circuit power consumption of each MTCD,
respectively. According to the energy causality constraint in
EH networks, Eij has to satisfy Eij ≤EH
ij . Summing the
energy consumed by all MTCDs in Ji, we can obtain the
energy Eiconsumed during the i-th phase as
Ei=
Ji
j=Ji−1+1
Eij ,∀i∈ N.(8)
During the (N+k)-th phase, the system energy consump-
tion, denoted by EN+k, is modeled as
EN+k=
Ik
i=Ik−1+1
tN+kqi
ξ+QC
−
N
i=1
Ji
j=Ji−1+1
tN+ku
Ik
n=Ik−1+1 |hnj |2qn
,(9)
where ξ∈(0,1] and QCare the PA efficiency and the circuit
power consumption of each MTCG, respectively. According
to the law of energy conservation [56], we must have
Ik
i=Ik−1+1
tN+kqi−
N
i=1
Ji
j=Ji−1+1
tN+ku
Ik
n=Ik−1+1|hnj |2qn
>0,
(10)
which is the energy loss due to wireless propagation.
Based on (5)-(9), the total energy consumption, ETot, of the
whole system during time Tcan be expressed as
ETot =
N+K
i=1
Ei
=
N
i=1
Ji
j=Ji−1+1
tipj
η+PC
+
K
k=1
Ik
i=Ik−1+1
tN+kqi
ξ+QC
−
K
k=1
N
i=1
Ji
j=Ji−1+1
tN+ku
Ik
n=Ik−1+1 |hnj |2qn
.(11)
C. TDMA Strategy
With the TDMA strategy, time Tconsists of M+Nuplink
transmission phases for MTCDs and MTCGs, as illustrated in
Fig. 4. All MTCDs transmit data to the corresponding MTCGs
in the first Mphases with TDMA, and all MTCGs transmit
the collected data to the BS in the last Nphases with TDMA.
Then, we obtain the following transmission time constraint
M+N
i=1
ti≤T. (12)
In the j-th (j≤M) phase, MTCD j∈ Jitransmits data to
its serving MTCG iwith achievable throughput
rij =Btjlog21 + |hij |2pj
σ2,∀i∈ N, j ∈ Ji.(13)
5
ĂĂ
t1
MTCD 1
tM+1
T
MTCG 1
tM
MTCD M
tM+N
MTCG N
ĂĂ
MTCD uplink IT
MTCG uplink IT & MTCD EH
Fig. 4. Time sharing scheme for TDMA strategy during one uplink
transmission period.
In the (M+i)-th phase, after having decoded all the messages
of its served MTCDs, MTCG itransmits the collected data to
the BS with achievable throughput
ri=BtM+ilog21 + |hi|2qi
σ2,∀i∈ N.(14)
Similar to (5), the total energy harvested of MTCD jserved
by MTCG iis
EH
ij =
N
n=1
tM+nu(|hnj |2qn),∀i∈ N, j ∈ Ji.(15)
According to (6), it is possible that some MTCDs cannot
effectively harvest energy in some slots, due to the fact that
the received power is below the receiver sensitivity threshold
P0.
As in (7) and (9), the energy consumption of a transmitter
includes both RF transmission power and circuit power [54].
With allocated transmission time tj, the energy Eij consumed
by MTCD j∈ Jican be modeled as
Eij =tjpj
η+PC,∀i∈ N, j ∈ Ji.(16)
With allocated transmission time tM+i, the system energy
consumption, denoted by EM+i, is modeled as
EM+i=tM+iqi
ξ+QC−
N
n=1
Jn
j=Jn−1+1
tM+iu(|hij |2qi),
(17)
where N
n=1 Jn
j=Jn−1+1 tM+iu(qi|hij |2)is the energy har-
vested by all MTCDs during the transmission time tM+ifor
MTCG ito transmit data to the BS.
According to (15)-(17), the total energy consumption, ETot,
of the whole system during time Tcan be expressed as
ETot =
N
i=1
Ji
j=Ji−1+1
Eij +
M+N
i=M+1
Ei
=
N
i=1
Ji
j=Ji−1+1
tjpj
η+PC+
N
i=1
tM+iqi
ξ+QC
−
N
i=1
Ji
j=Ji−1+1
N
n=1
tM+nu(|hnj |2qn).(18)
III. ENE RG Y EFFIC IE NT RESOURCE ALLOCATI ON F OR
NOMA
In this section, we study the resource allocation for an
uplink M2M enabled cellular network with NOMA and EH.
Specifically, we aim at minimizing the total energy consump-
tion via jointly optimizing power control and time allocation
for NOMA. The system energy minimization problem is
formulated as
min
p
p
p,q
q
q,t
t
tETot (19a)
s.t. rij ≥Dj,∀i∈ N, j ∈ Ji(19b)
ri≥
Ji
j=Ji−1+1
Dj,∀i∈ N (19c)
Eij ≤EH
ij ,∀i∈ N, j ∈ Ji(19d)
N+K
i=1
ti≤T(19e)
0≤pj≤Pj,0≤qi≤Qi,∀i∈ N, j ∈ Ji(19f)
t
t
t≥0
0
0,(19g)
where p
p
p= [p1,·· · , pM]T,q
q
q= [q1,·· · , qN]T,t
t
t= [t1,·· · ,
tN+K]T,Djis the payload that MTCD jhas to upload within
time constraint T,Pjis the maximal transmission power of
MTCD j, and Qiis the maximal transmission power of MTCG
i. It is assumed that all payloads are positive, i.e., Dj>0, for
all j. The objective function (19a) defined in (11) is the total
energy consumption of both MTCDs and MTCGs. Constraints
(19b) and (19c) reflect that the minimal required payloads for
MTCDs can be uploaded to the BS. The consumed energy of
each MTCD should not exceed its harvested energy in time T,
as stated in (19d). Constraints (19e) reflect that the payloads
for all MTCDs are transmitted in time T.
Note that problem (19) is nonconvex due to nonconvex
objective function (19a) and constraints (19b)-(19d). In gen-
eral, there is no standard algorithm for solving nonconvex
optimization problems. In the following, we first find the
optimal conditions for problem (19) by exploiting the special
structure of the uplink NOMA rate, and then provide an
iterative power control and time allocation algorithm.
A. Optimal Conditions
By analyzing problem (19), we have the following lemma.
Lemma 1: The optimal solution (p
p
p∗,q
q
q∗,t
t
t∗) to problem (19)
satisfies
r∗
ij =Dj,∀i∈ N, j ∈ Ji.(20)
This observation states that the minimal throughput leads
to more energy saving, which is similar to [17] and is also
widely known in the information theory community.
Lemma 1 states that the optimal transmit throughput for
each MTCD is required minimum. Note that the optimal
throughput for each MTCG is not always as its minimum
requirement, i.e., constraints (19c) are active at the optimum,
since MTCGs should transmit more power to maintain that the
harvested energy of each MTCD is no less than the consumed
energy.
6
Based on Lemma 1, we further have the following lemma
about the optimal transmission power of MTCDs.
Lemma 2: If (p
p
p∗,q
q
q∗,t
t
t∗) is the optimal solution to problem
(19), we have
p∗
j=
Ji
l=j+1
σ2
|hij |2e
al
t∗
i−1e
aj
t∗
i−1e
bjl
t∗
i
+σ2
|hij |2e
aj
t∗
i−1,∀i∈ N, j ∈ Ji,(21)
where
al=(ln 2)Dl
B, bjl =l−1
s=j+1(ln 2)Ds
B,∀i∈ N, j, l ∈ Ji.
(22)
Besides, the optimal transmission power p∗
jof MTCD j∈ Ji
is always non-negative and decreases with the transmission
time t∗
i.
Proof: Please refer to Appendix A.
From Lemma 2, large transmission time results in low
transmission power. This is reasonable as the minimal payload
is limited and large transmission time requires low achievable
rate measured in bits/s. It is also revealed from Lemma 2
that the optimal transmission power of MTCD j∈ Jiserved
by MTCG idepends only on the variable of the allocated
transmission time ti. As a result, the energy Eij consumed by
MTCD jin Jiis a function of the allocated transmission time
ti. Based on (7) and (21), we have
Eij =
Ji
l=j+1
σ2ti
η|hij |2e
al
ti−1e
aj
ti−1e
bjl
ti
+σ2ti
η|hij |2e
aj
ti−1+tiPC.(23)
Theorem 1: The energy Eij defined in (23) is convex with
respect to (w.r.t.) the transmission time ti. When PC= 0, the
energy Eij monotonically decreases with ti. When PC>0,
the energy Eij first decreases with tiwhen 0≤ti≤T∗
ij and
then increases with tiwhen ti> T ∗
ij , where T∗
ij is the unique
zero point of the first-order derivative ∂Eij
∂ti, i.e.,
∂Eij
∂titi=T∗
ij
= 0.(24)
Proof: Please refer to Appendix B.
Fig. 5 exemplifies the energy Eij given in (23) versus ti.
When PC= 0, i.e., the circuit energy consumption of MTCDs
is not considered, we come to the same conclusion as in [37]
and [38] that the consumed energy decreases as the transmis-
sion time increases according to Theorem 1. Without consid-
ering the circuit power consumption, R= log2(1 + SNR)and
the energy efficiency increases with the decrease of power.
Consequently, when PC= 0 and the minimal throughput
demand Djis given, the consumed energy Eij is a decreasing
function w.r.t. ti. This fundamentally follows the Shannon’s
law.
When PC>0, i.e., the circuit energy consumption of
MTCDs is taken into account, however, we find from Theorem
1 that the consumed energy first decreases and then increases
ij
E
C0P>
C0P=
*
ij
T
i
t
0
Fig. 5. The energy Eij versus transmission time ti.
with the transmission time, which is different from the pre-
vious conclusion in [37] and [38]. This is because that the
total energy contains two parts balancing each other, i.e., the
RF transmission energy part which monotonically decreases
with the transmission time and the circuit energy part which
linearly increases with the transmission time. In the following
of this section, we assume that the circuit power consumption
of MTCDs and MTCGs is in general positive, i.e., PC>0
and QC>0.
Theorem 2: If T≤max∀i∈N min∀j∈Ji{T∗
ij }, the optimal
time allocation t
t
t∗to problem (19) satisfies
N+K
i=1
t∗
i=T. (25)
If T≥TUpp, where TUpp is defined in (C.4), the optimal time
allocation t
t
t∗to problem (19) satisfies
N+K
i=1
t∗
i< T. (26)
Proof: Please refer to Appendix C.
From Theorem 2, it is observed that transmitting with the
maximal transmission time Tis optimal when Tis not large.
This is because that the reduced energy of RF transmis-
sion dominates the additional energy of circuit by increasing
transmission time. When the available time Tbecomes large
enough, Theorem 2 states that it is not optimal to transmit with
the maximal transmission time T. This is due to the fact that
the increased energy of circuit power dominates the power
consumption while the energy reduction of RF transmission
becomes relatively marginal.
B. Joint Power Control and Time Allocation Algorithm
Problem (19) has two difficulties: one comes from the
non-smooth EH function defined in (6), and the other one
is the non-convexity of both objective function (19a) and
constraints (19b)-(19d). To deal with the first difficulty, we
introduce notation Sij as the set of phases during which
7
MTCD j∈ Jican effectively harvest energy, i.e., Sij =
{k|Ik
n=Ik−1+1 |hnj |2qn> P0,∀k∈ K}. With Sij in hand,
the harvested power of MTCD j∈ Jican be presented by the
smooth function ¯u(x)defined in (28). To deal with the second
difficulty, we substitute (3)-(7), (11) and (21) into (19), and the
original problem (19) with fixed sets Sij ’s can be equivalently
transformed into the following problem:
min
q
q
q,t
t
t
N
i=1
Ji
j=Ji−1+1
Ji
l=j+1
σ2ti
η|hij |2e
al
ti−1e
aj
ti−1e
bjl
ti
+
N
i=1
Ji
j=Ji−1+1
σ2ti
η|hij |2e
aj
ti−1
+
N
i=1
Ji
j=Ji−1+1
tiPC
+
K
k=1
Ik
i=Ik−1+1
tN+kqi
ξ+QC
−
N
i=1
Ji
j=Ji−1+1
k∈Sij
tN+k¯u
Ik
n=Ik−1+1 |hnj |2qn
(27a)
s.t. |hi|2qi≥2
∑j∈JiDj
BtN+1 −1
Ik
l=i+1 |hl|2ql+σ2
,
∀k∈ K, i ∈ Ik(27b)
Ji
l=j+1
σ2
η|hij |2tie
al
ti−1e
aj
ti−1e
bjl
ti
+σ2
η|hij |2tie
aj
ti−1+tiPC
≤
k∈Sij
tN+k¯u
Ik
n=Ik−1+1|hnj |2qn
,∀i∈ N, j ∈ Ji
(27c)
Ik
n=Ik−1+1|hnj |2qn≥P0,∀i∈ N, j ∈ Ji, k ∈ Sij (27d)
N+K
i=1
ti≤T(27e)
Ji
l=j+1
σ2
|hij |2e
al
ti−1e
aj
ti−1e
bjl
ti
+σ2
|hij |2e
aj
ti−1≤Pj,∀i∈ N, j ∈ Ji(27f)
0≤qi≤Qi,∀i∈ N (27g)
t
t
t≥0
0
0,(27h)
where
¯u(x) = M(1 + eab)
eab +e−a(x−2b)−M
eab ,∀x≥0.(28)
Problem (27) is still nonconvex w.r.t. (q
q
q, t
t
t) due to nonconvex
objective function (27a) and constraints (27b)-(27c). Before
solving problem (27), we have the following theorem.
Theorem 3: Given transmission time τ
τ
τ= [tN+1,· ·· ,
tN+K]T, problem (27) is a convex problem w.r.t. (q
q
q, ¯
t
t
t), where
¯
t
t
t= [t1,·· · , tN]T. Given (q
q
q, ¯
t
t
t), problem (27) is equivalent to
a linear problem w.r.t. τ
τ
τ.
Proof: Please refer to Appendix D.
According to Theorem 3, problem (27) with given transmis-
sion time τ
τ
τcan be effectively solved by using the standard
convex optimization method, such as interior point method
[57]. Besides, problem (27) with given (q
q
q, ¯
t
t
t) is a linear prob-
lem, which can be optimally solved via the simplex method.
Based on Theorem 3, we propose an iterative power control
and time allocation for NOMA (IPCTA-NOMA) algorithm
with low complexity to obtain a suboptimal solution of prob-
lem (19). The idea is to iteratively update sets Sij’s according
to the power and time variables obtained in the previous step.
Algorithm 1: Iterative Power Control and Time Allocation for
NOMA (IPCTA-NOMA) Algorithm
1: Set S(0)
ij ={k|j∈ ∪n∈IkJn, k ∈ K},∀i∈ I,j∈ Jij ,
initialize a feasible solution (q
q
q(0), t
t
t(0)) to problem (27)
with S(0)
ij ’s, the tolerance θ, the iteration number v= 0,
and the maximal iteration number Vmax.
2: repeat
3: Set τ
τ
τ∗= [t(v)
N+1,· ·· , t(v)
N+K]T.
4: repeat
5: Obtain the optimal (q
q
q∗,¯
t
t
t∗) of convex problem (27)
with fixed τ
τ
τ∗and sets S(v)
ij .
6: Obtain the optimal τ
τ
τ∗of linear problem (27) with
fixed (q
q
q∗,¯
t
t
t∗) and sets S(v)
ij .
7: until the objective value (27a) with fixed sets S(v)
ij
converges.
8: Set v=v+ 1.
9: Denote q
q
q(v)=q
q
q∗,t
t
t(v)= [¯
t
t
t∗T,τ
τ
τ∗T]T.
10: Calculate the objective value (27a) with fixed sets S(v)
ij
as U(v)
Obj =ETot(q
q
q(v),t
t
t(v)).
11: Update S(v)
ij ={k|Ik
n=Ik−1+1 |hnj |2q(v)
n> P0,∀k∈
K},∀i∈ I, j ∈ Ji.
12: until v≥1and |U(v)
Obj −U(v−1)
Obj |/U(v−1)
Obj < θ or v > Vmax .
C. Convergence and Complexity Analysis
Theorem 4: Assuming Vmax → ∞, the sequence (q
q
q, t
t
t)
generated by the IPCTA-NOMA algorithm converges.
Proof: Please refer to Appendix E.
According to the IPCTA-NOMA algorithm, the major com-
plexity lies in solving the convex problem (27) with fixed τ
τ
τ.
Considering that the dimension of the variables in problem
(27) with fixed τ
τ
τis 2N, the complexity of solving problem
(27) with fixed τ
τ
τby using the standard interior point method is
O(N3)[57, Pages 487, 569]. As a result, the total complexity
of the proposed IPCTA-NOMA algorithm is O(LNOLIT N3),
where LNO denotes the number of outer iterations of the
IPCTA-NOMA algorithm, and LIT denotes the number of
inner iterations of the IPCTA-NOMA algorithm for iteratively
solving nonconvex problem (27) with fixed sets Sij ’s.
8
IV. ENERGY EFFIC IE NT RE SO UR CE AL LO CATI ON F OR
TDMA
In this section, we study the energy minimization for the
M2M enabled cellular network with TDMA. According to
(13)-(16) and (18), the energy minimization problem can be
formulated as
min
p
p
p,q
q
q,ˆ
t
t
t
N
i=1
Ji
j=Ji−1+1
tjpj
η+PC+
N
i=1
tM+iqi
ξ+QC
−
N
i=1
Ji
j=Ji−1+1
N
n=1
tM+nu(|hnj |2qn)(29a)
s.t. Btjlog21 + |hij |2pj
σ2≥Dj,∀i∈ N, j ∈ Ji
(29b)
BtM+ilog21 + |hi|2qi
σ2≥
Ji
j=Ji−1+1
Dj,∀i∈ N
(29c)
tjpj
η+PC≤
N
n=1
tM+nu(|hnj |2qn),∀i∈ N, j ∈ Ji
(29d)
N+1
i=1
ti≤T(29e)
0≤pj≤Pj,0≤qi≤Qi,∀i∈ N, j ∈ Ji(29f)
ˆ
t
t
t≥0
0
0,(29g)
where ˆ
t
t
t= [t1,·· · , tM+N]T.
Obviously, problem (29) is nonconvex due to nonconvex
objective function (29a) and constraints (29b)-(29d). In the
following, we first provide the optimal conditions for problem
(29), and then we propose a low-complexity algorithm to solve
problem (29).
A. Optimal Conditions
Similar to Lemma 1, it is also optimal for each MTCD to
transmit with the minimal throughput requirement. According-
ly, the following lemma is directly obtained.
Lemma 3: The optimal solution (p
p
p∗,q
q
q∗,ˆ
t
t
t∗) to problem (29)
satisfies
Bt∗
jlog21 + |hij|2p∗
j
σ2=Dj,∀i∈ N, j ∈ Ji.(30)
According to (30), the optimal transmission power of
MTCD jcan be presented as
pj=1
|hij |22
Dj
Btj−1,∀i∈ N, j ∈ Ji.(31)
Substituting (31) into (16) yields
Eij =tj
|hij |2η2
Dj
Btj−1+tjPC,∀i∈ N, j ∈ Ji.(32)
By analyzing (32), we can obtain the following theorem.
Theorem 5: The energy Eij defined in (32) is convex w.r.t.
tj. When PC= 0, the energy Eij monotonically decreases
with the transmission time tj. When PC>0, the energy Eij
first decreases with tjwhen 0≤tj≤T∗
ij and then increases
with tjwhen tj> T ∗
ij , where T∗
ij is the unique zero point of
the first-order derivative ∂Eij
∂tj, i.e.,
∂Eij
∂tjtj=T∗
ij
= 0.(33)
Since Theorem 5 can be proved by checking the first-order
derivative ∂Eij
∂tjas in Appendix B, the proof of Theorem 4
is omitted. Similar to Theorem 2 for NOMA, we come to
the similar conclusion for TDMA that transmitting with the
maximal transmission time Tis optimal when Tis not large,
while for large Tit is not optimal to transmit with the maximal
transmission time T.
B. Iterative Power Control and Time Allocation Algorithm
Similar to problem (19), problem (29) has two difficulties:
one is the non-smooth EH function in (6), and the other is
the nonconvex objective function (29a) and constraints (29b)-
(29d). To deal with the first difficulty, we introduce notation
Sij as the set of MTCGs from which MTCD j∈ Jican
effectively harvest energy, i.e., Sij ={n||hnj|2qn> P0,∀n∈
I}. With Sij in hand, the harvested power of MTCD j∈ Ji
from MTCG n∈ Sij can be presented by the smooth function
¯u(x)defined in (28). To tackle the second difficulty, we show
that problem (29) with fixed sets Sij ’s can be transformed into
an equivalent convex problem.
Theorem 6: The original problem in (29) with fixed sets
Sij ’s can be equivalently transformed into the following con-
vex problem as
min
ˆ
p
p
p,ˆ
q
q
q,ˆ
t
t
t
N
i=1
Ji
j=Ji−1+1 ˆpj
η+tjPC+
N
i=1 ˆqi
ξ+tM+iQC
−
N
i=1
Ji
j=Ji−1+1
n∈Sij
tM+n¯u|hnj |2ˆqn
tM+n(34a)
s.t. Btjlog21 + |hij |2ˆpj
σ2tj≥Dj,∀i∈ N, j ∈ Ji
(34b)
BtM+ilog21 + |hi|2ˆqi
σ2tM+i≥
Ji
j=Ji−1+1
Dj,∀i∈ N
(34c)
ˆpj
η+tjPC≤
n∈Sij
tM+n¯u|hnj|2ˆqn
tM+n,∀i∈N, j ∈Ji
(34d)
|hnj |2ˆqn> P0tM+n,∀i∈ N, j ∈ Ji, n ∈ Sij (34e)
N+1
i=1
ti≤T(34f)
0≤ˆpj≤Pjtj,∀i∈ N, j ∈ Ji(34g)
0≤ˆqi≤QitM+i,∀i∈ N (34h)
ˆ
t
t
t≥0
0
0,(34i)
where ˆ
p
p
p= [ˆp1,·· · ,ˆpM]Tand ˆ
q
q
q= [ˆq1,·· · ,ˆqN]T.
9
Proof: Please refer to Appendix F.
Based on Theorem 6, we propose an iterative power control
and time allocation for TDMA (IPCTA-TDMA) algorithm
with low complexity to obtain a suboptimal solution of prob-
lem (29). The idea is to iteratively update sets Sij’s according
to the power and time variables obtained in the previous step.
Algorithm 2: Iterative Power Control and Time Allocation for
TDMA (IPCTA-TDMA) Algorithm
1: Set S(0)
ij ={i},∀i∈ I,j∈ Jij , the tolerance θ, the
iteration number v= 0, and the maximal iteration number
Vmax.
2: repeat
3: Obtain the optimal (ˆ
p
p
p(v),ˆ
q
q
q(v),ˆ
t
t
t(v)) of convex problem
(34) with fixed sets S(v)
ij .
4: Calculate the objective value (34a) with fixed sets S(v)
ij
as U(v)
Obj =ETot(ˆ
p
p
p(v),ˆ
q
q
q(v),ˆ
t
t
t(v)).
5: Set v=v+ 1.
6: Update S(v)
ij =n
|hnj |2ˆq(v−1)
n
t(v−1)
M+n
> P0,∀n∈ I,∀i∈
I, j ∈ Ji.
7: until v≥2and |U(v)
Obj −U(v−1)
Obj |/U(v−1)
Obj < θ or v > Vmax .
8: Output p
p
p∗=ˆ
p
p
p(v),ˆ
t
t
t∗=ˆ
t
t
t(v), q∗
i= ˆq(v)
i/t(v)
M+i,∀i∈ I.
C. Convergence and Complexity Analysis
Theorem 7: Assuming Vmax → ∞, the sequence (ˆ
p
p
p, ˆ
q
q
q, ˆ
t
t
t)
generated by the IPCTA algorithm converges.
Theorem 7 can be proved by using the same method as in
Appendix E. The proof of Theorem 7 is thus omitted.
According to the IPCTA-TDMA algorithm, the major com-
plexity lies in solving the convex problem (34). Consid-
ering that the dimension of the variables in problem (34)
is 2(M+N), the complexity of solving problem (34) by
using the standard interior point method is O((M+N)3)
[57, Pages 487, 569]. As a result, the total complexity of
the proposed IPCTA-TDMA algorithm is O(LTD(M+N)3),
where LTD denotes the number of iterations of the IPCTA-
TDMA algorithm.
V. NUMERICAL RE SU LTS
In this section, we evaluate the proposed schemes through
simulations. There are 40 MTCDs uniformly distributed with
a BS in the center. We adopt the “data-centric” clustering
technique in [41] for cluster formation of MTCDs, the number
of MTCGs is set as 12, and the maximal number of MTCDs
associated to one MTCG is 4. For NOMA, all MTCGs are
classified into 6 clusters based on the strong-weak scheme
[45], [46].
The path loss model is 128.1+37.6 log10 d(dis in km) and
the standard deviation of shadow fading is 4dB [58]. The noise
power σ2=−104 dBm, and the bandwidth of the system is
B= 18 KHz. For the non-linear EH model in (6), we set
M= 24 mW, a= 1500 and b= 0.0014 according to [52],
which are obtained by curve fitting from the measurement
0.05 0.1 0.15 0.2
0
0.5
1
1.5
2
2.5 x 10−3
Transmission time t1 (s)
Energy E11 (J)
NOMA, PC=0
NOMA, PC=5 mW
NOMA, PC=10 mW
Fig. 6. Energy E11 consumed by MTCD 1 versus the transmission time t1
for NOMA strategy.
1 2 3 4 5 6 7 8 9 10
0
2
4
6
8
10
12
14
Number of iterations
Total energy (J)
NOMA
TDMA
Fig. 7. Convergence behaviors of IPCTA-NOMA and IPATC-TDMA.
data in [59]. The receiver sensitivity threshold P0is set as
0.1mW. The PA efficiencies of each MTCD and MTCG are
set to η=ξ= 0.9, and the circuit power of each MTCG
is QC= 500 mW as in [55]. We assume equal throughput
demand for all MTCDs, i.e., D1=·· · =DM=D, and
equal maximal transmission power for each MTCD or MTCG,
i.e., P1=·· · =PM=P, and Q1=·· · =QN=Q.
Unless otherwise specified, parameters are set as P= 5 mW,
PC= 0.5mW, Q= 1 W, D= 10 Kbits, and T= 5 s.
Fig. 6 depicts, for instance, E11 in (23) consumed by MTCD
1 served by MTCG 1 versus the transmission time t1for
NOMA. It is observed that E11 monotonically decreases with
t1when PC= 0. For the case with PC= 5 mW or PC= 10
mW, E11 first decreases and then increases with t1. Both
observations validate our theoretical findings in Theorem 1.
The convergence behaviors of IPCTA-NOMA and IPCTA-
TDMA are illustrated in Fig. 7. From this figure, the total
energy of both algorithms monotonically decreases, which
confirms the convergence analysis in Section III-C and IV-C.
It can be seen that both IPCTA-NOMA and IPCTA-TDMA
converge rapidly.
In Fig. 8, we illustrate the total energy consumption versus
the circuit power of each MTCD. According to Fig. 8, the total
energy of NOMA outperforms TDMA when the circuit power
of each MTCD is low, i.e., PC≤4mW in the test case. At
low circuit power regime, the total energy consumption of the
network mainly lies in the RF transmission power of MTCDs
10
01234567
1.5
2
2.5
3
3.5
4
4.5
5
Circuit power of each MTCD (mW)
Total energy (J)
NOMA
TDMA
Fig. 8. Total energy versus the circuit power of each MTCD.
and the energy consumed by MTCGs to charge the MTCDs
through EH. For the NOMA strategy, MTCDs served by the
same MTCG can simultaneously upload data and the MTCG
decodes the messages according to NOMA detections, which
requires lower RF transmission power of MTCDs than the
TDMA strategy. Thus, the total energy of NOMA is less than
TDMA for low circuit power of each MTCD. From Fig. 8, we
can find that the total energy of TDMA outperforms NOMA
when the circuit power of each MTCD becomes high, i.e.,
PC≥5mW in our tests. At high circuit power regime, the
total energy consumption of the network mainly lies in the
circuit power of MTCDs and the energy consumed by MTCGs
to charge the MTCDs through EH. For the NOMA strategy,
the transmission time of each MTCD with NOMA is always
longer than that with TDMA, which leads to higher circuit
power consumption of MTCDs than the TDMA strategy. As
a result, TDMA enjoys better energy efficiency than NOMA
for high circuit power of each MTCD.
Fig. 9 illustrates the total energy versus the maximal trans-
mission power of each MTCG. It is observed that the total
energy decreases with the increase of maximal transmission
power of each MTCG for both NOMA and TDMA. This is
because that the increment of maximal transmission power
of each MTCG allows the MTCG to transmit with large
transmission power, which leads to short EH time of MTCDs
to harvest enough energy and low total energy consumption
of the network. Moreover, it is found that the total energy of
NOMA is more sensitive to the maximal transmission power
of each MTCG than that of TDMA for high circuit power case
as PC= 5 mW for each MTCD. The reason is that MTCD
with low channel gain receives intra-cluster interference due to
NOMA and the energy consumption is hence especially large
for low maximal transmission power of each MTCG and high
circuit power of each MTCD.
The total energy versus the maximal transmission power
of each MTCD is shown in Fig. 10. It is observed that the
total energy decreases with growing maximal transmission
power of each MTCD for both NOMA and TDMA. This is
due to the fact that a larger maximal transmission power of
each MTCD ensures MTCDs can transmit with more power,
and the required payload can be uploaded in a shorter time,
which results in low circuit power consumption and low
0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
Maximal transmission power of each MTCG (W)
Total energy (J)
NOMA, PC=0.5 mW
TDMA, PC=0.5 mW
NOMA, PC=5 mW
TDMA, PC=5 mW
Fig. 9. Total energy versus the maximal transmission power of each MTCG.
1 2 3 4 5
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
Maximal transmission power of each MTCD (mW)
Total energy (J)
NOMA, PC=0.5 mW
TDMA, PC=0.5 mW
NOMA, PC=5 mW
TDMA, PC=5 mW
Fig. 10. Total energy versus the maximal transmission power of each MTCD.
energy consumption. It can be found that the total energy of
TDMA converges faster than that of NOMA as the maximal
transmission power of each MTCD increases. This is because
that the MTCDs served by the same MTCG simultaneously
transmit data for NOMA, and the required transmission power
of each MTCD for NOMA is always larger than that of MTCD
for TDMA.
Finally, in Fig. 11, we illustrate the total energy versus
the required payload of each MTCD. The figure shows that
the total energy increases with the required payload of each
MTCD. This is due to the fact that large payload of each
MTCD requires large energy consumption of MTCDs and
MTCGs, which leads to high energy consumption of the
network.
VI. CO NC LU SI ON S
This paper compares the total energy consumption between
NOMA and TDMA strategies in uplink M2M communications
with EH. We formulate the total energy minimization problem
subject to minimal throughput constraints, maximal transmis-
sion power constraints and energy causality constraints, with
the circuit power consumption taken into account. By applying
the conditions that it is optimal to transmit with the minimal
throughput for each MTCD, we transform the original problem
for NOMA strategy into an equivalent problem, which is
suboptimally solved through an iterative algorithm. By using
a proper variable transformation, we transform the nonconvex
problem for TDMA into an equivalent problem, which can
11
10 12 14 16 18 20
2
3
4
5
6
7
8
9
Required payload of each MTCD (kbits)
Total energy (J)
NOMA, PC=0.5 mW
TDMA, PC=0.5 mW
NOMA, PC=5 mW
TDMA, PC=5 mW
Fig. 11. Total energy versus the required payload of each MTCD.
be effectively solved. Through simulations, either NOMA
strategy or TDMA strategy may be preferred depending on
different circuit power regimes of MTCDs. At low circuit
power regime of MTCDs, NOMA consumes less energy,
while TDMA is preferred at high circuit power regime of
MTCDs since the energy consumption for NOMA increases
significantly as the circuit power of MTCDs increases.
APP EN DI X A
PROO F OF LE MM A 2
By inserting rij =Djinto (3) from Lemma 1, we have
2
Dj
Bti
Ji
l=j+1 |hil|2pl+σ22
Dj
Bti−1=
Ji
l=j|hil|2pl,(A.1)
for j=Ji−1+ 1,·· · , Ji. To solve those Ji−Ji−1equations,
we first define
uj=
Ji
l=j|hil|2pl,∀j∈ Ji.(A.2)
Applying (A.2) into (A.1) yields
uj= 2
Dj
Btiuj+1 +σ22
Dj
Bti−1,∀j∈ Ji.(A.3)
Denote u
u
ui= [uJi−1+1,··· , uJi]T,
v
v
vi=σ22
DJi−1+1
Bti−1,···, σ22
DJi
Bti−1T
,(A.4)
and
W
W
Wi=
0 2
DJi−1+1
Bti
0 2
DJi−1+2
Bti
......
0 2
DJi−1
Bti
0
.(A.5)
Equations in (A.3) can be rewritten as
(E
E
E−W
W
Wi)u
u
ui=v
v
vi,(A.6)
where E
E
Eis an identity matrix of size (Ji−Ji−1)×(Ji−Ji−1).
From (A.6), we have
u
u
ui= (E
E
E−W
W
Wi)−1v
v
vi.(A.7)
Before obtaining the inverse matrix (E
E
E−W
W
Wi)−1, we present
the following lemma.
Lemma 4: When l∈[1, Ji−Ji−1−1], the l-th power of
matrix W
W
Wican be expressed as
W
W
Wl
i=
0
0
0T
l2
∑Ji−1+l
s=Ji−1+1Ds
Bti
0 2
∑Ji−1+l+1
s=Ji−1+2Ds
Bti
......
0 2
∑Ji−1
s=Ji−lDs
Bti
0
0
0l
,
(A.8)
where 0
0
0ldenotes a l×1vector of zeros. When l=Ji−Ji−1,
W
W
Wl
i= 0
0
0(Ji−Ji−1)×(Ji−Ji−1), where 0
0
0(Ji−Ji−1)×(Ji−Ji−1)is a
(Ji−Ji−1)×(Ji−Ji−1)matrix of zeros.
Proof: Lemma 4 can be proved by the principle of
mathematical induction.
Basis: It can be verified that Lemma 4 is valid for l= 1.
Induction Hypothesis: For l∈[1, Ji−Ji−1−2], assume
that the l-th power of matrix W
W
Wican be expressed as (A.8).
Induction Step: According to (A.8), we can obtain
W
W
Wl+1
i=W
W
Wl
iW
W
Wi
=
0
0
0T
l2
∑Ji−1+l
s=Ji−1+1Ds
Bti
0 2
∑Ji−1+l+1
s=Ji−1+2Ds
Bti
......
0 2
∑Ji−1
s=Ji−lDs
Bti
0
0
0l
·
0 2
DJi−1+1
Bti
0 2
DJi−1+2
Bti
......
0 2
DJi−1
Bti
0
=
0
0
0T
l+1 2
∑Ji−1+l+1
s=Ji−1+1Ds
Bti
0 2
∑Ji−1+l+2
s=Ji−1+2Ds
Bti
......
0 2
∑Ji−1
s=Ji−l−1Ds
Bti
0
0
0l+1
,
which verifies that the (l+ 1)-th power of matrix W
W
Wican
also be expressed as (A.8). According to (A.8) and (A.5), it
is verified that
W
W
WJi−Ji−1
i=W
W
WJi−Ji−1−1
iW
W
Wi= 0
0
0(Ji−Ji−1)×(Ji−Ji−1).
(A.9)
Therefore, Lemma 4 is proved.
12
Now, it is ready to obtain the inverse matrix (E
E
E−W
W
Wi)−1.
Since
(E
E
E−W
W
Wi)
E
E
E+
Ji−Ji−1−1
l=1
W
W
Wl
i
=E
E
E−W
W
WJi−Ji−1
i=E
E
E,
(A.10)
we have
(E
E
E−W
W
Wi)−1=E
E
E+
Ji−Ji−1−1
l=1
W
W
Wl
i.(A.11)
Substituting (A.8) and (A.11) into (A.7) yields
uj=
Ji
l=j
σ22
Dl
Bti−12
∑l−1
s=jDs
Bti,∀j∈ Ji,(A.12)
where we define j−1
s=j2Ds
Bti= 20. From (A.12), we can obtain
uj=σ22
∑Ji
s=jDs
Bti−1. Since Ji
l=Ji+1 pj= 0, we have
uJi+1 = 0.(A.13)
From (A.2) and (A.13), we can obtain the transmission power
of MTCD jas
pij =uj−uj+1
|hij |2,∀j∈ Ji.(A.14)
Applying (A.12) and (A.13) to (A.14), we have
pj=
Ji
l=j
σ2
|hij |22
Dl
Bti−12
∑l−1
s=jDs
Bti
−
Ji
l=j+1
σ2
|hij |22
Dl
Bti−12
∑l−1
s=j+1 Ds
Bti
=
Ji
l=j+1
σ2
|hij |22
Dl
Bti−1
2
Dj
Bti−1
2
∑l−1
s=j+1 Ds
Bti
+σ2
|hij |22
Dj
Bti−12
∑j−1
s=jDs
Bti,
=
Ji
l=j+1
σ2
|hij |2e
al
ti−1e
aj
ti−1e
bjl
ti
+σ2
|hij |2e
aj
ti−1,∀j∈ Ji.(A.15)
where aland bjl are defined in (22). Since ex−1with x≥0
is non-negative and decreases with ti,pjis also non-negative
and decreases with tifrom (A.15).
APPENDIX B
PROO F OF TH EO RE M 1
To show that the energy Eij is convex w.r.t. ti, we first
define function
fijl (x) = (ealx−1)(eajx−1)ebjlx,∀x≥0.(B.1)
Then, the second-order derivative follows
f′′
ijl (x) = (a2
l+ 2albjl )(eajx−1)e(al+bjl)x
+2alaje(al+aj+bjl )x
+(a2
j+ 2ajbjl )(aalx−1)e(aj+bjl)x
+b2
jl (ealx−1)(eajx−1)ebjlx≥0,(B.2)
which indicates that fijl (x)is convex w.r.t. x. According
to [57, Page 89], the perspective of u(x)is the function
v(x, t)defined by v(x, t) = tu(x/t),dom v={(x, t)|x/t ∈
dom u, t > 0}. If u(x)is a convex function, then so is its
perspective function v(x, t)[57, Page 89]. Since fijl(x)is
convex, the perspective function
¯
fijl (x, ti) = tifijl x
ti(B.3)
is convex w.r.t. (x, ti). Thus, function ¯
fijl (1, ti)is also convex
w.r.t. ti.
Defining function
gij (x) = eajx−1,(B.4)
which is convex w.r.t. x. By using the property of perspective
function [57, Page 89], we also have that function
¯gij (x, ti) = tigij x
ti(B.5)
is convex w.r.t. (x, ti) and function ¯gij (1, ti)is accordingly
convex w.r.t. ti.
Substituting (B.3) and (B.5) into (23), we can obtain
Eij =
Ji
l=j+1
σ2
η|hij |2¯
fijl (1, ti) + σ2
η|hij |2¯gij (1, ti) + tiPC.
(B.6)
Due to the fact that both ¯
fijl (1, ti)and ¯gij (1, ti)are convex,
Eij is consequently convex w.r.t. tifrom (B.6).
According to (23), the first-order derivative of Eiw.r.t. ti
is expressed as
∂Eij
∂ti
=
Ji
l=j+1
σ2
η|hij |2e
al
ti−1e
aj
ti−1e
bjl
ti
−
Ji
l=j+1
σ2al
η|hij |2tie
aj
ti−1e
al+bjl
ti
−
Ji
l=j+1
σ2aj
η|hij |2tie
al
ti−1e
aj+bjl
ti
−
Ji
l=j+1
σ2bjl
η|hij |2tie
al
ti−1e
aj
ti−1e
bjl
ti
+σ2
η|hij |2e
aj
ti−1−σ2aj
η|hij |2ti
e
aj
ti+PC.(B.7)
Since Eij is convex w.r.t. ti, function ∂Eij
∂tiincreases with ti.
Because Djis positive for all j, we have al>0,bijl >0and
cijl >0from (22). To calculate the limit of ∂Eij
∂tiat ti= 0+,
13
we calculate the following limit,
lim
ti→0+
σ2
η|hij |2e
al
ti−1e
aj
ti−1e
bjl
ti
−σ2al
η|hij |2tie
aj
ti−1e
al+bjl
ti
= lim
x→+∞
σ2
η|hij |2(ealx−1) (eajx−1) ebjl x
−σ2alx
η|hij |2(eajx−1) e(al+bjl )x
= lim
x→+∞
σ2(1 −alx)
η|hij |2e(al+aj+bjl )x
=−∞.(B.8)
Thus, when tiapproaches zero in the positive direction, we
have
lim
ti→0+
∂Eij
∂ti
=−∞.(B.9)
When tiapproaches positive infinity, the limit of first-order
derivative ∂Eij
∂tican be calculated as
lim
ti→+∞
∂Eij
∂ti
=PC.(B.10)
If PC= 0, then ∂Eij
∂ti≤0for all ti≥0. In this case,
the energy Eij always decreases with ti. If PC>0, we
can observe that limti→+∞∂Eij
∂ti>0from (B.10). Since
limti→0+ ∂Eij
∂ti<0and ∂ Eij
∂tiincreases with ti, there exists
one unique solution T∗
ij satisfying (24), which can be solved
by using the bisection method. In particular, Eij decreases
with 0≤ti≤T∗
ij and increases with tij > T ∗
ij .
APPENDIX C
PROO F OF TH EO RE M 2
We first consider the case that T≤max∀i∈N
min∀j∈Ji{T∗
ij }. Without loss of generality, we denote T≤
T∗
nm = max∀i∈N min∀j∈Ji{T∗
ij }. Assume that the optimal
solution (p
p
p∗,q
q
q∗,t
t
t∗) to problem (19) satisfies N+K
i=1 t∗
i< T.
Due to that t∗
i≥0for all i, we can obtain that t∗
n≤T∗
nm ≤
T∗
nl,∀l∈ Jn. With all other power p∗
j,q∗
iand time t∗
kfixed,
j∈ M \ Jn,i∈ N,k∈ N \ {n}, we increase the time t∗
n
to t′
n=t∗
n+ϵby an arbitrary amount (0, T −N
i=1,i̸=nt∗
i].
Using (21), the corresponding power p∗
lstrictly deceases to p′
l,
l∈ Jn. According to Theorem 1, the energy Enl decreases
with the transmission time 0≤tn≤T∗
nl,∀l∈ Jn. As
a result, with new power-time pair (p′
Jn−1+1,··· , p′
Jn, t′
n),
the objective function (19a) decreases with all the constraints
satisfied. By contradiction, we must have N+K
i=1 t∗
i=Tfor
the optimal solution. This completes the proof of the first half
part of Theorem 2.
The last half part of Theorem 2 indicates that transmitting
with maximal transmission time is not optimal when Tis
larger than a threshold TUpp. This can be proved by using
the contradiction method. Specifically, assuming that total
transmission time of the optimal solution is the maximal
transmission time T, we can find a special solution with total
transmission time less than T, which strictly outperforms the
optimal solution. To obtain the threshold TUpp, we consider
a special solution that satisfies all the constraints of problem
(19) except the maximal transmission time constraint (19e).
Since Eij is convex w.r.t. tiaccording to Theorem 1, the
energy Ei=Ji
j=Ji−1+1 Eij consumed by all MTCDs in Ji
served by MTCG iis also convex w.r.t. ti. Based on the proof
of Theorem 1, we directly obtain the following lemma.
Lemma 5: The energy Eiconsumed by all MTCDs in Ji
first decreases with the transmission time tiwhen 0≤ti≤T∗
i
and then increases with the transmission time tiwhen ti> T ∗
i,
where T∗
iis the unique zero point of first-order derivative ∂Ei
∂ti,
i.e.,
∂Ei
∂titi=T∗
i
=
Ji
j=Ji−1+1
∂Eij
∂titi=T∗
i
= 0.(C.1)
Since Lemma 5 can be proved by checking the first-order
derivative ∂Ei
∂tias in Appendix B, the proof of Lemma 5 is
omitted here.
Set ˜
ti=T∗
i, and ˜pjcan be obtained from (21) with ˜
ti,
i∈ N,j∈ Ji. For the transmission power of the MTCGs,
we set ˜qi=Qi,i∈ N. Denote ˜
p
p
p= [˜p1,·· · ,˜pM]T,
˜
q
q
q= [˜q1,···,˜qN]T. With power (˜
p
p
p, ˜
q
q
q)and time (˜
ti,·· · ,˜
tN)
fixed for now, the energy minimization problem (19) without
constraint (19e) becomes
min
τ
τ
τ
K
k=1
Ik
i=Ik−1+1
tN+kQi
ξ+QC
−
K
k=1
N
i=1
Ji
j=Ji−1+1
tN+1u
Jk
n=Jk−1+1|hnj |2Qn
(C.2a)
s.t. BtN+ilog21+ |hi|2Qi
Ik
n=i+1|hn|2Qn+σ2≥
j∈Ji
Dj,
∀k∈ K, i ∈ Ik(C.2b)
˜
ti˜pj
η+PC
≤
K
k=1
tN+ku
Ik
n=Ik−1+1|hnj |2Qn
,
∀i∈ N, j ∈ Ji(C.2c)
τ
τ
τ≥0
0
0,(C.2d)
where τ
τ
τ= [tN+1,··· , tN+K]T. Problem (C.2) can be ob-
tained by substituting (11) into (19a), (4) into (19b), and (5)
and (7) into (19c). Problem (C.2) is a linear problem, which
can be optimally solved via the simplex method. Denote the
optimal solution of problem by τ
τ
τ∗= [T∗
N+1,··· , T ∗
N+K]T.
Denote ˜
tN+k=T∗
N+k,∀k∈ K, and ˜
t
t
t= [˜
t1,·· · ,˜
tN+K]T.
As a result, we obtain a special solution (˜
p
p
p, ˜
q
q
q, ˜
t
t
t) that satisfies
all the constraints of problem (19) except the maximal trans-
mission time constraint (19e). With solution (˜
p
p
p, ˜
q
q
q, ˜
t
t
t), the total
energy consumption ˜
ETot obtained from (11) can be expressed
14
as
˜
ETot =
N+K
i=1
˜
Ei
=
N
i=1
Ji
j=Ji−1+1
˜
ti˜pj
η+PC
+
K
k=1
Ik
i=Ik−1+1
˜
tN+k˜qi
ξ+QC
−
K
k=1
N
i=1
Ji
j=Ji−1+1
˜
tN+ku
Ik
n=Ik−1+1|hnj |2˜qn
,(C.3)
where ˜
Eiis the energy consumed by all MTCDs in Ji,i∈
N, and ˜
EN+kis the system energy consumption during the
(N+k)-th phase.
Denote an upper bound of maximal transmission time by
TUpp = max N+K
i=1
T∗
i, TAmp,(C.4)
where
TAmp =1 + N
i=1 βiK
k=1 ˜
EN+k
αQC,(C.5)
with αdefined in (C.8) and βidefined in (C.11). If T≥TUpp,
we show that optimal solution (p
p
p∗,q
q
q∗,t
t
t∗) to problem (19) must
satisfy constraint (26), i.e., (19e) is inactive, by contradiction.
Assume that
N+K
i=1
t∗
i=T. (C.6)
With (p
p
p∗,q
q
q∗,t
t
t∗), we denote E∗
ias the energy consumed by
all MTCDs in Ji,i∈ N,E∗
N+kas the system energy
consumption during the (N+k)-th phase, and E∗
Tot as the
total energy of the whole system. Thus, we have
E∗
Tot =
N+K
i=1
E∗
i
(a)
≥
N+1
i=1
˜
Ei+
K
k=1
E∗
N+k
(b)
=
N
i=1
˜
Ei+
K
k=1
t∗
N+k
Ik
n=Ik−1+1
q∗
n
ξ−
N
i=1
Ji
j=Ji−1+1
u
Ik
n=Ik−1+1|hnj |2q∗
n
+QC
K
k=1
t∗
N+k(Ik−Ik−1)
(c)
>
N
i=1
˜
Ei+αQC
K
k=1
t∗
N+k
(d)
≥
N+K
i=1
˜
Ei=˜
ETot,(C.7)
where inequality (a) follows from the fact that Eiachieves the
minimum when ti=T∗
iaccording to Lemma 5, equality (b)
holds from (5) and (9), and inequality (c) follows from (5),
(10), ξ∈(0,1] and
α,min
k∈K(Ik−Ik−1).(C.8)
To explain procedure (d), we substitute (5) and (7) into energy
causality constraints (19d) to obtain
t∗
ip∗
j
η+PC≤
K
k=1
t∗
N+ku
Ik
n=Ik−1+1 |hnj |2qn
,(C.9)
for all i∈ N, j ∈ Ji.
Considering that p∗
j≥0in the left hand side of (C.9) and
u(x)is a increasing function as well as q∗
i≤Qiin the right
hand side of (C.9), we have
t∗
i≤min
j∈Ji
K
k=1 t∗
N+kuIk
n=Ik−1+1 |hnj |2Qn
PC
,
≤βi
K
k=1
t∗
N+k,∀i∈ N,(C.10)
where
βi= min
j∈Ji
max
k∈K
uIk
n=Ik−1+1 |hnj |2Qn
PC
.(C.11)
Combining (C.6) and (C.8) yields
K
k=1
t∗
N+k≥T
1 + N
i=1 βi
.(C.12)
Hence, inequality (d) follows from (C.4) and (C.12).
According to (C.2) and (C.4), solution (˜
p
p
p, ˜
q
q
q, ˜
t
t
t) is a feasible
solution to problem (19). From (C.7), the objective value (19a)
can be decreased with solution (˜
p
p
p, ˜
q
q
q, ˜
t
t
t), which contradicts that
(p
p
p∗,q
q
q∗,t
t
t∗) is the optimal solution to problem (19). Hence, the
optimal solution to problem (19) must satisfy constraint (26).
APPENDIX D
PROO F OF TH EO RE M 3
We first show that the feasible set of problem (27) with
given τ
τ
τis convex. Obviously, constraints (27b), (27d), (27e)
and (27g) and (27h) are all linear w.r.t. (q
q
q, ¯
t
t
t). According to
(B.1) and (B.3), constraints (27c) and (27f) can be, respec-
tively, reformulated as
Ji
l=j+1
¯
fijl (1, ti) + tiPC≤
k∈Sij
tN+k¯u
Ik
n=Ik−1+1 |hnj |2qn
(D.1)
for all i∈ N, j ∈ Ji, and
Ji
l=j+1
fijl 1
ti≤Pj,∀i∈ N, j ∈ Ji.(D.2)
Based on (28), we have
¯u′′(x) = −Ma2(1 + eab)e−a(x−b)
eab(1 + e−a(x−b))3≤0,∀x≥0,(D.3)
15
which shows that ¯u(x)is concave, and −¯u(x)is convex.
Since ¯
fijl (1, ti)is convex w.r.t. tiaccording to Appendix B,
constraints (D.1) are convex, i.e., constraints (27c) are convex.
Due to that fijl (x)is convex and non-decreasing, and 1
xis
convex, fijl 1
xis also convex according to the composition
property of convex functions [57, Page 84]. Thus, constraints
(D.2) are convex, i.e., constraints (27f) are convex.
We then show that the objective function (27a) is convex.
Substituting (B.3) and (B.5) into (27a) yields
N
i=1
Ji
j=Ji−1+1
Ji
l=j+1
σ2
η|hij |2¯
fijl (1, ti)
+
N
i=1
Ji
j=Ji−1+1
σ2
η|hij |2¯gij (1, ti)
+
N
i=1
Ji
j=Ji−1+1
tiPC+
K
k=1
Ik
i=Ik−1+1
tN+kqi
ξ+QC
−
N
i=1
Ji
j=Ji−1+1
k∈Sij
tN+k¯u
Ik
n=Ik−1+1 |hnj |2qn
,(D.4)
which is convex w.r.t. (q
q
q, ¯
t
t
t) because ¯
fijl (1, ti),¯gij (1, ti)and
−¯u(x)are convex according to Appendix B and (D.3). As a
result, problem (27) is convex w.r.t. (q
q
q, ¯
t
t
t).
With given (q
q
q, ¯
t
t
t), constraints (27b) can be equivalently
transformed into
BtN+ilog21 + |hi|2qi
Ik
n=i+1 |hn|2qn+σ2≥
Ji
j=Ji−1+1
Dj,
(D.5)
which is linear w.r.t. tN+i. By replacing constraints (27b) with
(D.5), problem (27) with given (q
q
q, ¯
t
t
t) is a linear problem.
APP EN DI X E
PROO F OF TH EO RE M 4
The proof is established by showing that the total energy
value (27a) is non-increasing when sequence (q
q
q, t
t
t) is updated.
According to the IPCTA-NOMA algorithm, we have
U(v−1)
Obj =ETot(q
q
q(v−1), t
t
t(v−1))≥ETot(q
q
q(v),t
t
t(v)) = U(v)
Obj ,
(E.1)
where the inequality follows from that (q
q
q(v),t
t
t(v)) is a subop-
timally optimal solution of problem (27) with fixed sets S(v)
ij ,
while (q
q
q(v−1), t
t
t(v−1)) is the initial feasible solution of problem
(34) with fixed sets S(v)
ij . Furthermore, the total energy value
(27a) is always non-negative. Since the total energy value (27a)
is non-increasing in each iteration according to (E.1) and the
total energy value (27a) is finitely lower-bounded, the IPCTA-
NOMA algorithm must converge.
APPENDIX F
PROO F OF TH EO RE M 6
We first show that problem (29) can be equivalently trans-
formed into problem (34). We introduce a set of new non-
negative variables:
ˆpj=tjpj,ˆqi=tM+iqi,∀i∈ N, j ∈ Ji.(F.1)
Substituting (F.1) into problem (29), we show that problem
(29) is equivalent to problem (34).
Then, we show that problem (34) is a convex problem. Since
constraints (34e)-(34i) are all linear, we only need to check
that objective function (34a) and constraints (34b)-(34d) are
convex. Due to the fact that −¯u(|hnj |2ˆqn)is convex w.r.t. ˆqn
from (D.3), −tM+n¯u|hnj |2ˆqn
tM+nis convex w.r.t. (tM+n,ˆqn)
according to the property of perspective function [57, Page 89].
Thus, objective function (34a) is convex. Analogously, con-
straints (34b)-(34d) can also be shown convex. As a result,
problem (34) is a convex problem.
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