# Computation Efficiency Maximization in Wireless-Powered Mobile Edge
Computing Networks

Fuhui Zhou, _Member, IEEE_ , and Rose Qingyang Hu, _Fellow, IEEE_ This paper
was presented in part at the IEEE Global Communications Conference (GLOBECOM),
Abu Dhabi, UAE, December 2018 [1]. Manuscript received March 3, 2019; revised
Aug. 6, 2019 and Dec. 16, 2019; accepted January 21, 2019. Date of publication
February ******; date of current version ******. This work was supported by
the Natural Science Foundation of China under Grants 61701214, in part by The
Open Foundation of The State Key Laboratory of Integrated Services Networks
under Grant ISN19-08, in part by the Intel Corporation, and in part by the US
National Science Foundation under Grant EARS1547312, in part by the Excellent
Youth Foundation of Jiangxi Province under Grant 2018ACB21012 and in part by
Young Elite Scientist Sponsorship Program by CAST. The associate editor
coordinating the review of this paper and approving it for publication was
Prof. Jemin Lee. (_Corresponding authors: Fuhui Zhou_) Fuhui Zhou is with the
College of Electronic and Information Engineering, Nanjing University of
Aeronautics and Astronautics, Nanjing, 210000, P. R. China, and also with the
State Key Laboratory of Integrated Services Networks, Xidian University, Xian
710071, China (e-mail: zhoufuhui@ieee.org). Rose Qingyang Hu is with the
Department of Electrical and Computer Engineering, Utah State University, USA.
(e-mail: rose.hu@usu.edu).

###### Abstract

Energy-efficient computation is an inevitable trend for mobile edge computing
(MEC) networks. Resource allocation strategies for maximizing the computation
efficiency are critically important. In this paper, computation efficiency
maximization problems are formulated in wireless-powered MEC networks under
both partial and binary computation offloading modes. A practical non-linear
energy harvesting model is considered. Both time division multiple access
(TDMA) and non-orthogonal multiple access (NOMA) are considered and evaluated
for offloading. The energy harvesting time, the local computing frequency, and
the offloading time and power are jointly optimized to maximize the
computation efficiency under the max-min fairness criterion. Two iterative
algorithms and two alternative optimization algorithms are respectively
proposed to address the non-convex problems formulated in this paper.
Simulation results show that the proposed resource allocation schemes
outperform the benchmark schemes in terms of user fairness. Moreover, a
tradeoff is elucidated between the achievable computation efficiency and the
total number of computed bits. Furthermore, simulation results demonstrate
that the partial computation offloading mode outperforms the binary
computation offloading mode and NOMA outperforms TDMA in terms of computation
efficiency.

###### Index Terms:

Mobile-edge computing, wireless power transfer, computation efficiency,
resource allocation, binary computation offloading, partial computation
offloading.

##  I Introduction

###  I-A Background and Related Works

The emerging intelligent applications (e.g., automatic navigation, face
recognition, unmanned driving, etc.) have imposed great challenges for mobile
devices since most of those applications have computation-intensive and
latency-sensitive tasks to be executed [1], [2]. However, most mobile devices
have low computing capability and finite battery capacity. Mobile edge
computing (MEC) can significantly augment the computing capability of mobile
devices by offloading tasks from mobile devices to the nearby MEC servers in a
low latency manner [2]. There exist two operation modes in MEC networks,
namely, partial computation offloading and binary computation offloading. In
the partial offloading mode, computation tasks can be divided into two parts,
one part is executed locally at mobile devices and the other part is offloaded
to the MEC server for computing. For the binary computation offloading mode,
computation tasks cannot be partitioned. The entire task is either locally
executed or completely offloaded to the MEC server for computing [2], [3]. In
this paper, both modes are considered.

To further improve the energy efficiency, wireless powered techniques that
exploit radio frequency (RF) signals as the energy sources for powering the
energy-limited mobile devices are considered promising and viable approaches
in MEC networks since they can provide stable and controllable amount of
energy and prolong the battery life of mobile devices [4], [5]. Recently, an
increasing attention has been paid to the wireless powered MEC networks [5].
It was shown that user quality of experience (QoE) can be improved by
integrating wireless powered techniques into MEC networks since the duration
of having MEC services is extended [6]. However, due to the ever-increasing
greenhouse gas emission concerns and the rapid growth of the operational cost,
future MEC networks will more and more focus on maximizing the computation
efficiency (CE) [7], which is defined as the ratio of the total computed bits
to the consumed energy. According to [7], information and communication
technologies account for about 2$\%$ of the greenhouse gas and 2$\%$ to 10$\%$
of global energy consumption. In order to achieve a sustainable and green
operation of MEC networks, it is crucial to design resource allocation
strategies for maximizing CE of MEC networks. To the authors’ best knowledge,
there have been only a few studies in this area. The related works are
summarized as follows.

The related works can be classified into three categories. The first category
has focused on designing energy-efficient resource allocation schemes in the
conventional MEC networks with orthogonal multiple access (OMA) in [8]-[14] or
with NOMA in [15]-[18]. In the second category resource allocation strategies
have been designed in wireless powered MEC networks [5], [6], [19]-[25]. The
third category has designed energy-efficient resource allocation schemes in
wireless-powered networks relying on either OMA or NOMA [26]-[31].

####  I-A1 Energy-Efficient Resource Allocation in The Conventional MEC
Networks

In the OMA-based MEC networks, efforts have been dedicated to jointly
optimizing the communication and computation resources for achieving energy-
efficient computing. Specifically, in [8], the consumed energy and execution
latency were minimized by jointly optimizing the local computing frequency and
offloading power of users. The authors in [9] and [10] extended the energy
consumption minimization problems into multi-user MEC networks with TDMA and
orthogonal frequency-division multiple access (OFDMA), respectively. In [9],
the computation performance of each user was guaranteed by optimizing the
offloading computation bits ratio and offloading time. In [10], it was shown
that the users with strong channel state information (CSI) prefer to offload
their computation task to the MEC server while users with weak CSI chooses to
perform local computing. In order to further reduce the energy consumption,
the authors in [11] studied the coexistence of MEC and cloud computing servers
and proposed optimal scheduling policies. Different form the works in
[8]-[11], multi-antenna techniques were exploited to improve the offloading
efficiency [12], [13]. The energy was minimized by jointly optimizing the
beamforming vector and the computation frequency. Recently, the authors in
[14] studied secure offloading in multi-user MEC networks. The works in
[8]-[14] focused on optimizing a single objective, which cannot achieve a good
tradeoff among different performance metrics. In [7], the authors studied CE
maximization problem in an MEC system with TDMA, which achieves a good
tradeoff between the achievable computation bits and the energy consumption.

Recently, in order to increase connectivity and reduce access latency,
resource allocation problems were studied in MEC networks with NOMA [15]-[18].
In [15], the impact of NOMA on offloading was analyzed in MEC networks. It was
proved that the application of NOMA can efficiently reduce the energy
consumption and offloading delay compared to OMA. The authors in [16]-[18]
designed resource allocation strategies for minimizing the consumption energy
of various MEC networks with NOMA. Specifically, the user clustering,
communication and computing resources were jointly optimized in multi-cell MEC
networks in [16] while the communication and computing resource and the
trajectory of the unmanned aerial vehicle (UAV) were jointly optimized in
[17]. In [18], the authors studied a multi-antenna MEC network with NOMA. It
was shown that the exploitation of multi-antenna techniques can significantly
improve the offloading efficiency.

####  I-A2 Resource Allocation in Wireless Powered MEC Networks

Recently, the authors in [5], [6], [19]-[25] studied resource allocation
problems in various wireless powered MEC networks. Specifically, in [19], the
central processing unit (CPU) frequency of the user and the mode selection
were jointly optimized for minimizing the consumed energy under both causal
and non-causal CSI conditions. The reinforcement learning and Lyapunov
optimization theory were exploited to design resource allocation strategies
for minimizing the system cost of wireless powered MEC networks in [20] and
[21], respectively. Although the computation performance can be improved by
integrating wireless powered techniques into MEC networks [19]-[21], the
performance improvement is limited by the harvested energy. In order to
improve the energy conservation efficiency, the authors in [5] exploited
multi-antenna techniques and jointly optimized the energy transmit beam-
former, the CPU frequencies and the offloading bits for minimizing the total
energy consumption. In [22], the authors extended the energy consumption
minimization problem into a cooperation-assisted wireless powered MEC network.
Different from the works in [5], [19]-[21], the authors in [23] and [24]
proposed optimal resource allocation strategies for maximizing the CE of
multi-user and full-duplex wireless powered MEC networks. In contrast to the
works in [5], [19]-[24], resource allocation strategies were designed for
maximizing the computation bits of multi-user and UAV-enabled wireless powered
MEC systems in [6] and [25] under the binary computation offloading mode.

####  I-A3 Energy-Efficient Resource Allocation in Wireless Powered Networks

In conventional wireless powered networks where the computing process was not
considered, the energy efficiency maximization problems have been well studied
in [26]-[31]. Specifically, the authors in [26] and [27] designed resource
allocation strategies for maximizing the system and user-centric energy
efficiency, respectively. In order to improve the achievable energy
efficiency, multiple-input multiple-out (MIMO) and massive MIMO techniques
have been applied in wireless powered networks in [28] and [29], respectively.
Different from the works in [26]-[29] where TDMA was applied for serving
multiple users, the authors in [30] and [31] have designed energy-efficient
resource allocation schemes in the wireless powered networks with NOMA. It was
interesting to find that NOMA does not necessarily guarantee to achieve a
better energy efficiency compared to TDMA.

###  I-B Motivations and Contributions

Note that the resource allocation strategies proposed in the conventional MEC
networks [8]-[18] cannot be readily applied to wireless powered MEC systems,
where the resource allocation problems should consider the EH causal
constraints and the relationship among the EH, offloading and computing
process. Moreover, the resource allocation strategies proposed for wireless
powered MEC networks in [5], [6], [19]-[25] are based on an ideally linear EH
model. These schemes may not achieve good performance in reality as the
practical EH circuits can result in a non-linear end-to-end wireless power
conversion [32]-[34]. Furthermore, resource allocation strategies designed for
maximizing the computation bits under the binary computation offloading mode
[6] and [25] and designed for maximizing CE under partial offloading mode
cannot guarantee to maximize the CE under the binary computation offloading
mode. Additionally, although energy-efficiency resource allocation strategies
have been designed in the conventional wireless powered networks [26]-[31],
only the efficiency of the computation task offloading process, i.e.,
communications process, was considered. They are inappropriate in wireless-
powered MEC networks for maximizing the CE since energy consumed in both the
offloading and local computation processes should be considered.

To the authors’ best knowledge, this is the first work that comprehensively
studies resource allocation problems for maximizing CE under both partial and
binary computation offloading modes. Please note that in [1], we only studied
the CE problem under the partial offloading mode with TDMA. Moreover, we did
not consider the effect of the power amplifier coefficient on CE [1]. The main
contributions of our work in this paper are summarized as follows.

  1. 1.

It is the first time that the CE maximization framework is formulated in
wireless powered MEC networks under both partial and binary computation
offloading modes. Both TDMA and NOMA are considered for offloading
transmission. With TDMA, the closed-form expressions for the optimal CPU
frequency and the optimal offloading power of users are derived under the
partial offloading mode and the close-form expression for the optimal
operational mode selection is given under the binary computation offloading
mode. An iterative algorithm and an alternative optimization algorithm are
proposed to solve the CE maximization under the partial offloading and binary
computation offloading mode, respectively.

  2. 2.

With NOMA, an iterative algorithm and an alternative optimization algorithm
based on the successive convex approximation (SCA) method are proposed for the
partial offloading mode and the binary computation offloading mode,
respectively. The closed-form expression for the operational mode selection on
whether users choose to locally compute or to offload tasks is derived. It is
shown that the selection of the operational mode depends on the trade-off
between the achievable computation bits and the energy consumption cost of
users.

  3. 3.

Simulation results show that our proposed resource allocation strategies can
improve fairness among users in terms of CE compared to the benchmark scheme.
Moreover, it is shown that the partial offloading mode and NOMA can achieve CE
gains compared with the binary computation offloading mode and TDMA,
respectively. Furthermore, the tradeoff between the achievable CE and the
computation bits is firstly elucidated.

The remainder of this paper is organized as follows. Section II presents the
system model. $\text{C}_{\text{eff}}$ maximization problems are investigated
for wireless powered MEC networks with TDMA in Section III. Section IV
presents the CE maximization problems in wireless powered MEC networks with
NOMA. Simulation results are presented in Section V. The paper is concluded in
Section VI.

##  II System Model

Figure 1: The system model.

A wireless powered MEC network is considered in Fig. 1, where the wireless
power station provides the wireless power transfer (WPT) services for $K$
users. Similar to the works in [6], [19] and [20], in order to clarify the
issues pertaining to CE and permit reaching meaningful insights into the CE
maximization problem, it is assumed that all devices are equipped with a
single antenna. In this paper, both partial and binary computation offloading
modes are considered. Similar to [5], [6], [21]-[25], local computation and
downlink WPT can be simultaneously executed while the downlink WPT and the
uplink computation offloading cannot be simultaneously performed. Thus, a
harvest-then-offload protocol is applied for downlink WPT and uplink
computation offloading. Moreover, both TDMA and NOMA protocols are exploited
for achieving multi-user offloading during the offloading process. All the
nodes and devices are equipped with a single antenna. Similar to [19]-[25],
all the channels have block fading thus the channel power gains are static
within each time frame but may change across time frames. In order to obtain
the upper bound of CE and provide theoretical support for the practical system
design, It is assumed that the perfect CSI can be obtained [5], [6],
[21]-[25].

Figure 2: The frame structure of the wireless powered MEC system.

The frame structure shown in Fig. 2 consists of four stages. The frame
duration is denoted by $T$, which is selected based on the correlated time of
the channel in order to guarantee that the channel power gains are constant
within one frame duration [5]. In the first stage, the wireless power station
transfers energy to $K$ users. In the second stage, users offload their
computation tasks to the MEC by using TDMA or NOMA protocol. In the third
stage, the MEC executes the computation tasks from users. In the fourth stage,
the MEC downloads the computation results to users. Similar to [19]-[25], the
computation time and the downloading time of the MEC are neglected as the MEC
has a strong computation capability compared with users and the number of the
bits related to the computation results is relatively small.

###### Remark 1

The major application scenarios for wireless powered MEC networks include two
cases. One is in the wireless sensor or wearable networks where the mobile
sensors and wearable computing devices are with milliwatt power consumption
while they need to perform computation tasks, such as environmental parameter
or physical condition monitoring [6]. The other one is in the areas, such as
wildernesses and complex terrains, where the government needs to keep
monitoring the environment so that the corresponding strategies can be taken
to protect the environment. In those areas, neither cable charging or battery
replacement can be conveniently established nor the cost for establishing
cable charging systems is affordable. The wireless powered MEC network becomes
a desirable alternative [19].

###  II-A Non-Linear Energy Harvesting Model

Let $\tau_{0}$ denote the duration of the WPT stage. In this paper, different
from the works in [5], [6], [19]-[25], a practical non-linear EH model is
applied while a sensitivity property is considered. Specifically, the
harvested energy is zero when the input RF power is smaller than the
sensitivity threshold. Based on the work in [32], the harvested energy of the
$k$th user denoted by ${\Phi_{k}}\left({{\tau_{0}},{P_{s}}}\right)$ can be
given as

$\displaystyle{\Phi_{k}}\left({{\tau_{0}},{P_{s}}}\right)=$
$\displaystyle{\tau_{0}}{\left[{\frac{{P_{k}^{\max}}}{{\exp\left({-\mu{P_{0}}+\psi}\right)}}\left({\frac{{1+\exp\left({-\mu{P_{0}}+\psi}\right)}}{{1+\exp\left({-\mu{h_{k}}{P_{s}}+\psi}\right)}}-1}\right)}\right]^{\rm{+}}},$
(1)

where $P_{s}$ is the transmit power of the power station; $P_{k}^{\max}$ is
the maximum harvested power of the $k$th user, $k\in{\cal K}$ and ${\cal
K}=1,2,\cdots,K$; $P_{0}$ is the sensitivity threshold; $\mu$ and $\psi$ are
the parameters for controlling the steepness of the function; $h_{k}$ is the
instantaneous channel power gain from the power station to the $k$th user;
${\left[a\right]^{+}}=\max\left({a,0}\right)$ and $\max\left({a,0}\right)$
denotes the bigger value of $a$ and $0$.

Figure 3: A comparison of the harvested power based on the linear EH model,
non-linear EH model and the measurement data [32].

In order to better illustrate the non-linear EH model, Fig. 3 compares the
harvested power obtained with the non-linear EH model [32] to those achieved
with the linear EH model and the experimental results given in [32]. The
harvested power of the linear EH model is $\varrho{h_{k}}{P_{s}}$, where
$\varrho$ is the energy conversion efficiency and selected as $0.5$. The
parameters in the non-linear EH model are set as $P_{k}^{\max}=0.004927$ W,
$P_{0}=0.000064$ W, $\mu=274$ and $\psi=0.29$.

###  II-B Partial Offloading

In this mode, the computation tasks of each user can be divided into two
parts, one for local computing and one for offloading.

####  II-B1 Local Computation

Similar to [5], [6], [19]-[25], each user can perform local computation in the
entire frame as each user can have the circuit architecture to separate the
computing unit and the offloading unit. $C$ cycles are required for computing
one bit of raw data at the user side CPU. Let $f_{k}$ denote the CPU frequency
of the $k$th user. Thus, the number of locally computed bits at the $k$th user
and the consumed energy are $Tf_{k}/C$ and $T{\gamma_{c}}f_{k}^{3}$,
respectively. $\gamma_{c}$ is the effective capacitance coefficient of the
processor’s chip, and $\gamma_{c}$ is dependent on the chip architecture.

####  II-B2 Offloading with TDMA

Let $\tau_{k}$ and $P_{k}$ denote the offloading time and the transmit power
for offloading of the $k$th user, respectively. Similar to the work in [6],
the offloaded task of the $k$th user consists of raw data and communication
overhead, such as the encryption and packet header. Let $v_{k}>1$ indicate the
communication overhead. According to [6], the number of bits that the $k$th
user offloads to the MEC server using TDMA is given as
$\frac{{B{\tau_{k}}}}{{{v_{k}}}}{\log_{2}}\left({1+\frac{{{g_{k}}{P_{k}}}}{{\sigma_{0}^{2}}}}\right),k\in{\cal
K},$ where $B$ is the communication bandwidth, $\sigma_{0}^{2}$ denotes the
noise power, and $g_{k}$ is the instantaneous channel power gain from the
$k$th user to the MEC server. Thus, the CE of the $k$th user is defined as

$\displaystyle\eta_{k}\left({\tau_{0},{\tau_{k}},{P_{k}},{f_{k}}}\right)=\frac{{R_{k}\left({{\tau_{k}},{P_{k}},{f_{k}}}\right)}}{{E_{k}\left({\tau_{0},{\tau_{k}},{P_{k}},{f_{k}}}\right)}},$
(2a) $\displaystyle
R_{k}\left({{\tau_{k}},{P_{k}},{f_{k}}}\right)={\frac{{T{f_{k}}}}{C}+\frac{{B{\tau_{k}}}}{{{v_{k}}}}{{\log}_{2}}\left({1+\frac{{{g_{k}}{P_{k}}}}{{\sigma_{0}^{2}}}}\right)},$
(2b) $\displaystyle
E_{k}\left({\tau_{0},{\tau_{k}},{P_{k}},{f_{k}}}\right)={{{\tau_{0}}{P_{r,k}}+\zeta{\tau_{k}}\left({P_{k}}+{P_{c,k}}\right)+T{\gamma_{c}}f_{k}^{3}}},$
(2c)

where $\eta_{k}\left({\tau_{0},{\tau_{k}},{P_{k}},{f_{k}}}\right)$ is the CE
of the $k$th user; $R_{k}\left({{\tau_{k}},{P_{k}},{f_{k}}}\right)$ and
$E_{k}\left({\tau_{0},{\tau_{k}},{P_{k}},{f_{k}}}\right)$ are the total number
of computed bits at the MEC server for the $k$th user and the consumption
energy of the $k$th user, respectively; $P_{r,k}$ and $P_{c,k}$ denote the
received power for the received signal processing during the WPT stage and the
constant circuit power consumption of the $k$th user during the computation
offloading process, respectively [27], and $\zeta$ denotes the amplifier
coefficient.

####  II-B3 Offloading with NOMA

Different from TDMA, NOMA enables all the users to simultaneously offload
their offloading tasks on the same frequency band so that offloading
throughput can be improved. Let $\tau_{1}$ denote the duration of the
offloading process. Without loss of generality, the channel gains for the NOMA
users have a descending order ${g_{1}}<\cdots<{g_{k}}<\cdots<{g_{K}}$. Thus,
using the simple decoding order based on the descending order of the channel
power channel [15]-[17], the CE of the $k$th user can be expressed as

$\displaystyle\eta_{k}\left({\tau_{0},{\tau_{1}},{P_{k}},{f_{k}}}\right)=\frac{{R_{k}\left({{\tau_{1}},{P_{k}},{f_{k}}}\right)}}{{E_{k}\left({\tau_{0},{\tau_{1}},{P_{k}},{f_{k}}}\right)}},$
(3a)
$\displaystyle{R_{k}}\left({{\tau_{1}},{f_{k}},{P_{k}}}\right)=\left\\{\begin{array}[]{l}\begin{array}[]{*{20}{c}}{\frac{{T{f_{k}}}}{C}+\frac{{B{\tau_{1}}}}{{{v_{k}}}}{{\log}_{2}}\left({1+\frac{{{g_{k}}{P_{k}}}}{{\sum\limits_{i=k+1}^{K}{{g_{i}}{P_{i}}}+\sigma_{0}^{2}}}}\right),}&\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{1\leq k\leq K-1},\end{array}\\\
\begin{array}[]{*{20}{c}}{\frac{{T{f_{k}}}}{C}+\frac{{B{\tau_{1}}}}{{{v_{k}}}}{{\log}_{2}}\left({1+\frac{{{g_{k}}{P_{k}}}}{{\sigma_{0}^{2}}}}\right),}&{k=K},\end{array}\end{array}\right.$
(3g) $\displaystyle
E_{k}\left({\tau_{0},{\tau_{1}},{P_{k}},{f_{k}}}\right)={{{\tau_{0}}{P_{r,k}}+\zeta{\tau_{1}}\left({P_{k}}+{P_{c,k}}\right)+T{\gamma_{c}}f_{k}^{3}}},$
(3h)

where $\eta_{k}\left({\tau_{0},{\tau_{1}},{P_{k}},{f_{k}}}\right)$ denotes the
CE of the $k$th user; ${R_{k}}\left({{\tau_{1}},{f_{k}},{P_{k}}}\right)$ and
$E_{k}\left({\tau_{0},{\tau_{1}},{P_{k}},{f_{k}}}\right)$ denote the total
number of computed bits at MEC and the total energy consumption of the $k$th
user respectively.

###  II-C Binary Offloading

Under the binary offloading mode, the computation task can be either
completely computed at the local device or completely offloaded to the MEC
server for computing. Let ${\cal M}_{1}$ and ${\cal M}_{0}$ denote the set of
users that choose to perform task offloading and the set of users that choose
to perform local computation, respectively. Thus, ${\cal K}={\cal
M}_{0}\cup{\cal M}_{1}$ and ${\cal M}_{0}\cap{\cal M}_{1}=\Theta$, where
$\Theta$ denotes the null set.

####  II-C1 Local Computation

In this case, all the harvested energy is used for local computing. Thus, the
CE of the $m$th user denoted by ${\eta_{m}}\left({{\tau_{0}},{f_{m}}}\right)$
can be given as

$\displaystyle{\eta_{m}}\left({{\tau_{0}},{f_{m}}}\right){\rm{=}}\frac{{\frac{{T{f_{k}}}}{C}}}{{{\tau_{0}}{P_{r,k}}+T{\gamma_{c}}f_{k}^{3}}}{\rm{=}}\frac{{T{f_{k}}}}{{C\left({{\tau_{0}}{P_{r,k}}+T{\gamma_{c}}f_{k}^{3}}\right)}},$
(4a) $\displaystyle{R_{m}}\left({{f_{m}}}\right){\rm{=}}\frac{{T{f_{m}}}}{C},\
m\in{\cal M}_{0},$ (4b)
$\displaystyle{E_{m}}\left({{\tau_{0}},{f_{m}}}\right){\rm{=}}{\tau_{0}}{P_{r,m}}+T{\gamma_{c}}f_{m}^{3},m\in{\cal
M}_{0},$ (4c)

where ${R_{m}}\left({{f_{m}}}\right)$ and
${E_{m}}\left({{\tau_{0}},{f_{m}}}\right)$ are the total number of locally
computed bits and energy consumption for computation of the $m$th user
respectively.

####  II-C2 Offloading with TDMA

With TDMA for offloading, the computation efficiency of the $n$th user denoted
by ${\eta_{n}}\left({{\tau_{0}},{\tau_{n}},{P_{n}}}\right)$ can be given as

$\displaystyle{\eta_{n}}\left({{\tau_{0}},{\tau_{n}},{P_{n}}}\right)=\frac{{{R_{n}}\left({{\tau_{n}},{P_{n}}}\right)}}{{{E_{n}}\left({{\tau_{0}},{\tau_{n}},{P_{n}}}\right)}},\
n\in{\cal M}_{1},$ (5a)
$\displaystyle{R_{n}}\left({{\tau_{n}},{P_{n}}}\right)=\frac{{B{\tau_{n}}}}{{{v_{n}}}}{\log_{2}}\left({1+\frac{{{g_{n}}{P_{n}}}}{{\sigma_{0}^{2}}}}\right),\
n\in{\cal M}_{1},$ (5b)
$\displaystyle{E_{n}}\left({{\tau_{0}},{\tau_{n}},{P_{n}}}\right)={\tau_{0}}{P_{r,n}}+\zeta{\tau_{n}}\left({{P_{n}}+{P_{c,n}}}\right),\
n\in{\cal M}_{1},$ (5c)

where ${R_{n}}\left({{\tau_{n}},{P_{n}}}\right)$ and
${E_{n}}\left({{\tau_{0}},{\tau_{n}},{P_{n}}}\right)$ are the number of
computed bits and energy consumption for offloading of the $n$th user
respectively.

####  II-C3 Offloading with NOMA

With NOMA, the CE of the $n$th user in ${\cal M}_{1}$ denoted by
${\eta_{n}}\left({{\tau_{0}},{\tau_{n}},{P_{n}}}\right)$ can be given as

$\displaystyle{\eta_{n}}\left({{\tau_{0}},{\tau_{n}},{P_{n}}}\right)=\frac{{{R_{n}}\left({{\tau_{n}},{P_{n}}}\right)}}{{{E_{n}}\left({{\tau_{0}},{\tau_{n}},{P_{n}}}\right)}},n\in{\cal
M}_{1},$ (6a)
$\displaystyle{R_{n}}\left({{\tau_{1}},{P_{n}}}\right)=\frac{{B{\tau_{1}}}}{{{v_{n}}}}{\log_{2}}\left({1+\frac{{{g_{n}}{P_{n}}}}{{\sum\limits_{i>n,i\in{M_{1}}}{{g_{i}}{P_{i}}}{\rm{+}}\sigma_{0}^{2}}}}\right),$
(6b)
$\displaystyle{E_{n}}\left({{\tau_{0}},{\tau_{n}},{P_{n}}}\right)={\tau_{0}}{P_{r,n}}+\zeta{\tau_{n}}\left({{P_{n}}+{P_{c,n}}}\right),n\in{\cal
M}_{1}.$ (6c)

##  III CE Maximization In Wireless Powered MEC Networks: TDMA based

###  III-A Partial Offloading Mode

####  III-A1 Problem Formulation

When the TDMA protocol and the partial computation offloading mode are
applied, the CE maximization problem is formulated under the max-min fairness
criterion as

$\displaystyle\mathbf{P}_{{1}}:{\mathop{\max}\limits_{{\tau_{0}},{\tau_{k}},{P_{k}},{P_{s}},{f_{k}}}}\
{\mathop{\min}\limits_{k\in{\cal K}}\
\eta\left({\tau_{0},{\tau_{k}},{P_{k}},{f_{k}}}\right)}$ (7a)
$\displaystyle\text{s.t.}\ \
C1:{\frac{{T{f_{k}}}}{C}+\frac{{B{\tau_{k}}}}{{{v_{k}}}}{{\log}_{2}}\left({1+\frac{{{g_{k}}{P_{k}}}}{{\sigma_{0}^{2}}}}\right)}\geq{R_{k,\min}},k\in{\cal
K},$ (7b) $\displaystyle
C2:{\tau_{0}}{P_{r,k}}+\zeta{\tau_{k}}\left({P_{k}}+{P_{c,k}}\right)+T{\gamma_{c}}f_{k}^{3}\leq{\Phi_{k}}\left({{\tau_{0}},{P_{s}}}\right),$
(7c) $\displaystyle C3:\sum\limits_{n=0}^{K}{{\tau_{n}}}\leq T,$ (7d)
$\displaystyle C4:0\leq{\tau_{0}}\leq T,0\leq{\tau_{k}}\leq T,\ k\in{\cal K},$
(7e) $\displaystyle C5:0\leq{P_{s}}\leq{P_{th}},{P_{k}}\geq 0,\ k\in{\cal K},$
(7f) $\displaystyle C6:{f_{k}}\geq 0,\ k\in{\cal K}.$ (7g)

$R_{k,\min}$ is the minimum number of computed bits required by the $k$th user
and $P_{th}$ is the maximum transmission power of the wireless power station.
The constraint $C1$ is the minimum computed bits constraint. The constraint
$C2$ is the EH causal constraint that the total consumption energy cannot be
larger than the harvested energy. The constraints $C3$ and $C4$ are the
constraints on the EH time and the computation offloading time.
$\mathbf{P}_{{1}}$ is a non-convex fractional optimization problem. It is
challenging to solve $\mathbf{P}_{{1}}$ due to the existence of coupling
relationship among different optimization variables, such as the coupling
between $t_{k}$ and $P_{k}$, as well as due to the non-convex constraints $C1$
and $C2$.

####  III-A2 Solution and Iterative Algorithm

In order to solve $\mathbf{P}_{{1}}$, Theorem 1 is presented as follows.

###### Theorem 1

In wireless powered MEC networks with TDMA under the partial computation
offloading mode, the maximum CE under the max-min fairness criterion is
achieved when $P_{s}=P_{th}$.

_Proof:_ Let
$\left\\{\tau_{0}^{*},{\tau_{k}^{*}},{P_{k}^{*}},{f_{k}^{*}},{P_{s}^{*}}\right\\}$
denote the optimal solution of $\mathbf{P}_{{1}}$. $\eta^{*}$ denotes the
maximum CE achieved under the max-min fairness criterion. It is not difficult
to prove that $\eta^{*}\geq 0$ and ${P_{s}^{*}}\geq P_{0}$. It is assumed that
${P_{s}^{*}}<P_{th}$. Let
$\left\\{\tau_{0}^{{\ddagger}},{\tau_{k}^{{\ddagger}}},{P_{k}^{{\ddagger}}},{f_{k}^{{\ddagger}}},{P_{s}^{{\ddagger}}}\right\\}$
denote another solution of $\mathbf{P}_{{1}}$ satisfying
${P_{s}^{{\ddagger}}}=P_{th}$, ${\tau_{k}^{{\ddagger}}}={\tau_{k}^{*}}$,
${P_{k}^{{\ddagger}}}={P_{k}^{*}}$, ${f_{k}^{{\ddagger}}}={f_{k}^{*}}$ and
${\Phi_{k}}\left({{\tau_{0}^{{\ddagger}}},{P_{s}^{{\ddagger}}}}\right)={\Phi_{k}}\left({{\tau_{0}^{*}},{P_{s}^{*}}}\right)$.
$\eta^{{\ddagger}}$ denotes the corresponding maximum CE. When ${P_{s}^{*}}$
is not large enough to achieve the maximum output power $P_{k}^{\max}$, one
has ${\tau_{0}^{*}}>\tau_{0}^{{\ddagger}}$ since
${P_{s}^{*}}<P_{s}^{{\ddagger}}$. It is quite evident that
$\left\\{\tau_{0}^{{\ddagger}},{\tau_{k}^{{\ddagger}}},{P_{k}^{{\ddagger}}},{f_{k}^{{\ddagger}}},{P_{s}^{{\ddagger}}}\right\\}$
satisfies all the constraints of $\mathbf{P}_{{1}}$. Since
$E_{k}\left({\tau_{0}^{{\ddagger}},{\tau_{k}^{{\ddagger}}},{P_{k}^{{\ddagger}}},{f_{k}^{{\ddagger}}}}\right)<E_{k}\left({\tau_{0}^{*},{\tau_{k}}^{*},{P_{k}}^{*},{f_{k}}}^{*}\right)$
and
$R_{k}\left({{\tau_{k}^{{\ddagger}}},{P_{k}^{{\ddagger}}},{f_{k}^{{\ddagger}}}}\right)=R_{k}\left({{\tau_{k}^{*}},{P_{k}^{*}},{f_{k}^{*}}}\right)$,
one has $\eta^{{\ddagger}}>\eta^{*}$. This contradicts the assumption that
$\left\\{\tau_{0}^{*},{\tau_{k}^{*}},{P_{k}^{*}},{f_{k}^{*}},{P_{s}^{*}}\right\\}$
is the optimal solution. Thus, ${P_{s}^{*}}=P_{th}$. When ${P_{s}^{*}}$ is
large to achieve the maximum output power $P_{k}^{\max}$, since
${\Phi_{k}}\left({{\tau_{0}^{{\ddagger}}},{P_{s}^{{\ddagger}}}}\right)={\Phi_{k}}\left({{\tau_{0}^{*}},{P_{s}^{*}}}\right)$,
one has ${\tau_{0}^{*}}=\tau_{0}^{{\ddagger}}$ and thus
$\eta^{{\ddagger}}=\eta^{*}$. Thus,
$\left\\{\tau_{0}^{{\ddagger}},{\tau_{k}^{{\ddagger}}},{P_{k}^{{\ddagger}}},{f_{k}^{{\ddagger}}},{P_{s}^{{\ddagger}}}\right\\}$
is also the optimal solution. Theorem 1 is proved.

###### Remark 2

It can be seen from Theorem 1 that the maximum CE achieved under the max-min
fairness criterion increases with the transmission power of the power station
in the wireless powered MEC networks with TDMA. If the transmission power
level of the power station is not large enough for achieving the maximum
harvested power of the user, the CE can be increased by increasing the
transmission power of the power station.

Motivated by the Dinkelbach’s method [26], Lemma 1 is given to transform
$\mathbf{P}_{{1}}$ to a tractable problem.

###### Lemma 1

The optimal solution of $\mathbf{P}_{{1}}$ can be obtained if and only if the
following equation holds.

$\displaystyle{\mathop{\max}\limits_{{\tau_{0}},{\tau_{k}},{P_{k}},{f_{k}}}}\
{\mathop{\min}\limits_{k\in{\cal K}}}\
R_{k}\left({{\tau_{k}},{P_{k}},{f_{k}}}\right)-\eta^{*}E_{k}\left({\tau_{0},{\tau_{k}},{P_{k}},{f_{k}}}\right)$
(8a) $\displaystyle={\mathop{\min}\limits_{k\in{\cal K}}}\
R_{k}\left({{\tau_{k}^{*}},{P_{k}^{*}},{f_{k}^{*}}}\right)-\eta^{*}E_{k}\left({\tau_{0}^{*},{\tau_{k}}^{*},{P_{k}}^{*},{f_{k}^{*}}}\right)=0,$
(8b)

where $\eta^{*}$ and $*$ denote the maximum CE and optimality, respectively.
The proof can be readily obtained from the generalized fractional programming
theory [36].

Based on Lemma 1, $\mathbf{P}_{{1}}$ can be solved by solving a parameter
problem, denoted by $\mathbf{P}_{{2}}$, given as

$\displaystyle\mathbf{P}_{{2}}:{\mathop{\max}\limits_{{\tau_{0}},{\tau_{k}},{P_{k}},{f_{k}}}}\
{\mathop{\min}\limits_{k\in{\cal K}}}\
R_{k}\left({{\tau_{k}},{P_{k}},{f_{k}}}\right)-\eta
E_{k}\left({\tau_{0},{\tau_{k}},{P_{k}},{f_{k}}}\right)$ (9a)
$\displaystyle\text{s.t.}\ \ C1-C6.$ (9b)

Here $\eta$ is a non-negative parameter. Although $\mathbf{P}_{{2}}$ is more
tractable, it is still non-convex and has coupling among optimization
variables. Auxiliary variables $y_{k}$ are further introduced, where
$y_{k}=\tau{{}_{k}}P_{k},k\in{\cal K}$. Using the auxiliary variables and
Theorem 1, $\mathbf{P}_{{2}}$ can be equivalently expressed as

$\displaystyle\mathbf{P}_{{3}}:{\mathop{\max}\limits_{{\tau_{0}},{\tau_{k}},{y_{k}},{f_{k}},\Upsilon}}\
\Upsilon$ (10a) $\displaystyle\text{s.t.}\
\frac{{T{f_{k}}}}{C}+\frac{{B{\tau_{k}}}}{{{v_{k}}}}{\log_{2}}\left({1+\frac{{{g_{k}}{y_{k}}}}{{{\tau_{k}}\sigma_{0}^{2}}}}\right)\geq{R_{k,\min}},k\in{\cal
K},$ (10b)
$\displaystyle{\tau_{0}}{P_{r,k}}+\zeta{y_{k}}+\zeta{\tau_{k}}{P_{c,k}}+T{\gamma_{c}}f_{k}^{3}\leq{\Phi_{k}}\left({{\tau_{0}},{P_{th}}}\right),$
(10c) $\displaystyle\ \ \ \ \ \
\frac{{T{f_{k}}}}{C}+\frac{{B{\tau_{k}}}}{{{v_{k}}}}{\log_{2}}\left({1+\frac{{{g_{k}}{y_{k}}}}{{{\tau_{k}}\sigma_{0}^{2}}}}\right)-$
$\displaystyle\eta\left[{{\tau_{0}}{P_{r,k}}+\zeta{y_{k}}+\zeta{\tau_{k}}{P_{c,k}}+T{\gamma_{c}}f_{k}^{3}}\right]\geq\Upsilon,k\in{\cal
K},$ (10d) $\displaystyle C3,C4,C6,y_{k}\geq 0,k\in{\cal K}.$ (10e)

###### Lemma 2

$\mathbf{P}_{{3}}$ is convex and can be efficiently solved by using the convex
optimization tool [37].

_Proof:_ Firstly, it is evident that the objective function and the
constraints $C3$, $C4$, $C6$ of $\mathbf{P}_{{3}}$ satisfy the conditions of a
convex problem since the objective function is linear and the constraints
$C3$, $C4$, $C6$ are linear inequality constraints. For the constraint given
by $\left(11\rm{b}\right)$, $\frac{{T{f_{k}}}}{C}$ is a linear function with
respect to $f_{k}$ and
$\frac{{B{\tau_{k}}}}{{{v_{k}}}}{\log_{2}}\left({1+\frac{{{g_{k}}{y_{k}}}}{{{\tau_{k}}\sigma_{0}^{2}}}}\right)$
is the perspective of
$\frac{{B}}{{{v_{k}}}}{\log_{2}}\left({1+\frac{{{g_{k}}{y_{k}}}}{{\sigma_{0}^{2}}}}\right)$,
which is a concave function of $y_{k}$. Since the perspective operation
preserves convexity [37],
$\frac{{B{\tau_{k}}}}{{{v_{k}}}}{\log_{2}}\left({1+\frac{{{g_{k}}{y_{k}}}}{{{\tau_{k}}\sigma_{0}^{2}}}}\right)$
is concave with respect to $\tau_{k}$ and $y_{k}$. Thus, it is easy to obtain
that the constraint given by $\left(11\rm{b}\right)$ is a convex constraint.
For the constraint
${\tau_{0}}{P_{r,k}}+\zeta{y_{k}}+\zeta{\tau_{k}}{P_{c,k}}+T{\gamma_{c}}f_{k}^{3}\leq{\Phi_{k}}\left({{\tau_{0}},{P_{th}}}\right)$,
the right side ${\Phi_{k}}\left({{\tau_{0}},{P_{th}}}\right)$ is a linear
function with respect to $\tau_{0}$ and the left side
${\tau_{0}}{P_{r,k}}+\zeta{y_{k}}+\zeta{\tau_{k}}{P_{c,k}}+T{\gamma_{c}}f_{k}^{3}$
is a linear function in regard to ${\tau_{0}}$, $y_{k}$ and $\tau_{k}$.
Moreover, since the local CPU frequency $f_{k}$ is nonnegative,
${\tau_{0}}{P_{r,k}}+\zeta{y_{k}}+\zeta{\tau_{k}}{P_{c,k}}+T{\gamma_{c}}f_{k}^{3}$
is a convex function with respect to $f_{k}$ when $f_{k}\geq 0$. Thus, the
constraint given by $\left(10\rm{c}\right)$ is also a convex constraint. Using
the same analysis method for the constraint given by $\left(10\rm{d}\right)$,
it is easy to prove that the constraint given by $\left(10\rm{d}\right)$ is
also a convex constraint. Thus, it is proved that $\mathbf{P}_{{3}}$ is
convex.

In this paper, in order to gain more meaningful insights, the optimal
solutions are obtained in closed forms by using the Lagrange duality method
[36]. Towards that end, let $f_{k}^{*}$ and $P_{k}^{*}$ denote the optimal
local computation frequency and the optimal offloading power of the $k$th
user, $k\in{\cal K}$, respectively. By solving $\mathbf{P}_{{3}}$, Theorem 2
can be stated as follows.

###### Theorem 2

In the wireless powered MEC systems with TDMA, the optimal local computation
frequency $f_{k}^{*}$ and the optimal offloading power $P_{k}^{*}$ of the
$k$th user for maximizing the CE under the max-min fairness criterion have the
following mathematical expressions:

$\displaystyle
f_{k}^{*}=\sqrt{\frac{{{\lambda_{k}}+{\theta_{k}}}}{{3C{\gamma_{c}}\left({{\rho_{k}}+{\theta_{k}}\eta}\right)}}};$
(11a) $\displaystyle
P_{k}^{*}=\left\\{\begin{array}[]{l}\begin{array}[]{*{20}{c}}{0,}&{\text{if}\
{\tau_{k}}=0}\end{array}\\\
\begin{array}[]{*{20}{c}}{{{\left[{\frac{{\left({{\lambda_{k}}+{\theta_{k}}}\right)B}}{{\zeta{v_{k}}\ln
2\left({{\rho_{k}}+{\theta_{k}}\eta}\right)}}-\frac{{\sigma_{0}^{2}}}{{{g_{k}}}}}\right]}^{+}},}&{\text{otherwise}};\end{array}\end{array}\right.$
(11f)

where $\lambda_{k}\geq 0$, $\rho_{k}\geq 0$ and $\theta_{k}\geq 0$ are the
dual variables corresponding to the constraints given by
$\left(7\rm{b}\right)$, $\left(7\rm{c}\right)$, and $\left(7\rm{d}\right)$,
respectively.

###### Proof:

See Appendix A. ∎

###### Remark 3

It can be seen from Theorem 2 that the $k$th user chooses to offload its
computation task only when the channel between the $k$th user and the MEC
server is good enough, namely, $g_{k}\geq{\left[{\sigma_{0}^{2}\zeta{v_{k}}\ln
2\left({{\rho_{k}}+{\theta_{k}}\eta}\right)}\right]}/{\left({{\lambda_{k}}+{\theta_{k}}}\right)B}$.
Moreover, the local computation frequency decreases with the increase of
$\eta$, which is related to the CE. It indicates that the users prefer to
offload their computation task to the MEC server in order to improve the CE.
Furthermore, it can be seen that the users prefer to offload their computation
task when the local CPU frequency is too high, namely,
$f_{k}^{*}\geq\sqrt{\frac{{\sigma_{0}^{2}\zeta{v_{k}}\ln
2}}{{{g_{k}}3C{\gamma_{c}}B}}}$. Additionally, when
$f_{k}^{*}<\sqrt{\frac{{\sigma_{0}^{2}\zeta{v_{k}}\ln
2}}{{{g_{k}}3C{\gamma_{c}}B}}}$, the optimal offloading power $P_{k}^{*}$ is
zero. It means that the users only perform local computation in order to
maximize their CE.

By solving $\mathbf{P}_{{3}}$, Theorem 3 is stated to clarify the
characteristic of the EH time $\tau_{0}$ and $\tau_{k}$.

###### Theorem 3

For the given $\lambda_{k}$, $\rho_{k}$, $\beta$ and $\theta_{k}$, in order to
maximize the Lagrangian of $\mathbf{P}_{{3}}$, the optimal EH time
$\tau_{0}^{*}$ and computation offloading time $\tau_{k}^{*}$ need to satisfy
the following equations.

$\displaystyle\tau_{0}^{*}=\left\\{\begin{array}[]{l}\begin{array}[]{*{20}{c}}{0,}&{\text{if}\
z\left({{\rho_{k}},\beta,{\theta_{k}}}\right)<0},\end{array}\\\
\begin{array}[]{*{20}{c}}{\in\left[{0,T}\right),}&{\text{if}\
z\left({{\rho_{k}},\beta,{\theta_{k}}}\right)=0},\end{array}\end{array}\right.$
(12e) $\displaystyle
z\left({{\rho_{k}},\beta,{\theta_{k}}}\right)=\sum\limits_{k=1}^{K}{{\rho_{k}}}\left({{P_{E,k}}-{P_{r,k}}}\right)-\beta-\sum\limits_{k=1}^{K}{{\theta_{k}}\eta}{P_{r,k}},$
(12f)
$\displaystyle{P_{E,k}}={\frac{{P_{k}^{\max}}}{{\exp\left({-\mu{P_{0}}+\psi}\right)}}\left({\frac{{1+\exp\left({-\mu{P_{0}}+\psi}\right)}}{{1+\exp\left({-\mu{h_{k}}{P_{th}}+\psi}\right)}}-1}\right)},$
(12g)
$\displaystyle\tau_{k}^{*}=\left\\{\begin{array}[]{l}\begin{array}[]{*{20}{c}}{{\rm
Z},}&{\text{if}\ \frac{{\sigma_{0}^{2}\zeta{v_{k}}{\theta_{k}}\eta\ln
2}}{{\left({{\lambda_{k}}+{\theta_{k}}}\right)B}}>\omega^{opt}},\end{array}\\\
\begin{array}[]{*{20}{c}}{\in\left[{0,{\rm Z}}\right],}&{\text{if}\
\frac{{\sigma_{0}^{2}\zeta{v_{k}}{\theta_{k}}\eta\ln
2}}{{\left({{\lambda_{k}}+{\theta_{k}}}\right)B}}=\omega^{opt}},\end{array}\\\
\begin{array}[]{*{20}{c}}{0,}&{\text{if}\
\frac{{\sigma_{0}^{2}\zeta{v_{k}}{\theta_{k}}\eta\ln
2}}{{\left({{\lambda_{k}}+{\theta_{k}}}\right)B}}<\omega^{opt}},\end{array}\end{array}\right.$
(13g) $\displaystyle{\rm
Z}=\frac{{{\Phi_{k}}\left({\tau_{0}^{opt},{P_{th}}}\right)-\tau_{0}^{opt}{P_{r,k}}-T{\gamma_{c}}{{\left({f_{k}^{opt}}\right)}^{3}}}}{{{P_{c,k}}+P_{k}^{opt}}},$
(13h)

where $\omega^{*}$ is the solution of the following equation.

$\displaystyle\frac{{B\left({{\lambda_{k}}+{\theta_{k}}}\right)}}{{{v_{k}}}}{\log_{2}}\left[{\frac{{\left({{\lambda_{k}}+{\theta_{k}}}\right)Bw}}{{\zeta{v_{k}}{\theta_{k}}\eta\sigma_{0}^{2}\ln
2}}}\right]-\frac{{\left({{\lambda_{k}}+{\theta_{k}}}\right)B}}{{{v_{k}}\ln
2}}$
$\displaystyle+\frac{{\zeta\left({{\rho_{k}}+{\theta_{k}}\eta}\right)\sigma_{0}^{2}}}{w}-\zeta\left({{\rho_{k}}+{\theta_{k}}\eta}\right){P_{c,k}}-\beta=0.$
(14)

###### Proof:

See Appendix B. ∎

By solving $\mathbf{P}_{{3}}$, Theorem 4 can be stated to obtain the maximum
CE denoted by $\Upsilon^{*}$.

###### Theorem 4

In the wireless powered MEC systems with TDMA under the partial computation
offloading mode, the maximum CE under the max-min fairness criterion is given
as

$\displaystyle\Upsilon^{*}=\left\\{\begin{array}[]{l}\begin{array}[]{*{20}{c}}{0,}&{\text{if}\
\sum\limits_{k=1}^{K}{{\theta_{k}}}>1,}\end{array}\\\
\begin{array}[]{*{20}{c}}{{\Lambda^{*},}}&{\text{if}\
\sum\limits_{k=1}^{K}{{\theta_{k}}}\leq 1;}\end{array}\end{array}\right.$
(15e) $\displaystyle\Lambda^{*}={\mathop{\min}\limits_{k\in{\cal K}}}\
R_{k}\left({{\tau_{k}^{*}},{P_{k}^{*}},{f_{k}^{*}}}\right)-\eta
E_{k}\left({\tau_{0}^{*},{\tau_{k}}^{*},{P_{k}}^{*},{f_{k}^{*}}}\right),$
(15f)

###### Proof:

Since $\mathbf{P}_{{3}}$ is convex and the Slater’s conditions are satisfied,
the Lagrangian of $\mathbf{P}_{{3}}$ should be upper-bound with respect to
$\Upsilon$. When $\sum\limits_{k=1}^{K}{{\theta_{k}}}>1$, the Lagrangian
decreases with $\Upsilon$. Thus, the maximum of Lagrangian is achieved with
$\Upsilon=0$ since $\Upsilon\geq 0$. When
$\sum\limits_{k=1}^{K}{{\theta_{k}}}\leq 1$, the maximum of Lagrangian is
obtained when the optimal solution is achieved. ∎

Finally, an iterative algorithm denoted by Algorithm 1 is given to obtain the
maximum CE. Specifically, when $\left|\Upsilon^{*,n}-\eta^{n}\right|=0$, the
optimal solution is obtained, where $n$ and $\Upsilon^{*,n}$ denote the
iterative number and the optimal solution achieved at the $n$th iteration,
respectively. Otherwise, an $\xi$-optimal solution is adopted with an error
tolerance $\xi$. In other words, the maximum CE is obtained when
$\left|\Upsilon^{*,n}-\eta^{n}\right|\leq\xi$, where $\left|\cdot\right|$
denotes the absolute operator. The details of Algorithm 1 are given in Table
I.

###### Remark 4

By using Theorem 1 and introducing auxiliary variables $y_{k}$, it is seen
that $\mathbf{P}_{{1}}$ is a generalized fractional programming problem [36].
Moreover, Algorithm 1 is proposed for solving $\mathbf{P}_{{1}}$ based on the
Dinkelbach’s method. Thus, according to [36], Algorithm 1 can converge when
updating $\eta^{n}$. The detail proof for the convergence can be seen in [36].

TABLE I: The iterative algorithm Algorithm 1: The iterative algorithm for
$\mathbf{P}_{\text{1}}$
---
1) Input settings:
the error tolerance $\xi>0$, $R_{k,\min}>0$ and $P_{th}$,
the maximum iteration number $N$.
2) Initialization:
EE $\eta^{n}=\eta_{0}$ and the iteration index $n=0$.
3) Optimization:
$\boldsymbol{\unrhd}$ for _n_ =1:N
solve $\text{P}_{\textbf{3}}$ by using CVX for the given $\eta^{n}$;
obtain the solution
$\left\\{{\tau_{0}^{*,n},{\tau_{k}}^{*,n},{P_{k}}^{*,n},{f_{k}^{*,n}},\Upsilon^{*,n}}\right\\}$;
if $\left|\Upsilon^{*,n}-\eta^{n}\right|\leq\xi$
the maximum CE $\Upsilon^{*,n}$ is obtained;
break;
else
update $n=n+1$ and $\eta^{n}=\Upsilon^{*,n}$;
end
$\boldsymbol{\unrhd}$ end

###  III-B Binary Offloading Mode

####  III-B1 Problem Formulation

Under the binary offloading mode, when the TDMA protocol is applied, the CE
maximization problem under the max-min fairness criterion can be formulated as
$\mathbf{P}_{{4}}$.

$\displaystyle{\mathop{\max}\limits_{{\tau_{0}},{\tau_{n}},{P_{n}},{P_{s}},{f_{m}}}}\
{\mathop{\min}\limits_{m\in{M_{0}},n\in{M_{1}}}}\left\\{{{\eta_{m}}\left({{\tau_{0}},{f_{m}}}\right),{\eta_{n}}\left({{\tau_{0}},{\tau_{n}},{P_{n}}}\right)}\right\\}$
(16a) $\displaystyle\text{s.t.}\ \
\frac{{T{f_{m}}}}{C}\geq{R_{m,\min}},m\in{\cal M}_{0},$ (16b)
$\displaystyle\frac{{B{\tau_{n}}}}{{{v_{n}}}}{\log_{2}}\left({1+\frac{{{g_{n}}{P_{n}}}}{{\sigma_{0}^{2}}}}\right)\geq{R_{n,\min}},n\in{\cal
M}_{1},$ (16c)
$\displaystyle{\tau_{0}}{P_{r,m}}+T{\gamma_{c}}f_{m}^{3}\leq{\Phi_{m}}\left({{\tau_{0}},{P_{s}}}\right),m\in{\cal
M}_{0}$ (16d)
$\displaystyle{\tau_{0}}{P_{r,n}}+\zeta{\tau_{n}}\left({{P_{n}}+{P_{c,n}}}\right)\leq{\Phi_{n}}\left({{\tau_{0}},{P_{s}}}\right),n\in{M_{1}}$
(16e) $\displaystyle\sum\limits_{n\in{{\cal M}_{1}}}{{\tau_{n}}}\leq
T,0\leq{\tau_{n}}\leq T,$ (16f) $\displaystyle{P_{n}}\geq 0,n\in{\cal
M}_{1},{f_{m}}\geq 0,m\in{\cal M}_{0},\ \text{and}\ 0\leq{P_{s}}\leq{P_{th}},$
(16g)

where the constraints given by $\left(16\rm{b}\right)$ and
$\left(16\rm{c}\right)$ are the requirements of the minimum computed bits. The
constraints given by $\left(16\rm{d}\right)$ and $\left(16\rm{e}\right)$ are
the EH causal constraints. $\mathbf{P}_{{4}}$ is challenging to solve.
Moreover, it is impractical to use the exhaustive search method for
determining the operational mode selection due to the extremely high
complexity, especially when the number of users is large.

####  III-B2 Alternative Optimization Algorithm

In order to solve $\mathbf{P}_{{4}}$, let ${{\alpha_{k}}}=0$ indicate that the
$k$th user performs local computation mode and ${{\alpha_{k}}}=1$ mean that
the $k$th user performs task offloading, where $k\in{{\cal K}}$. Moreover,
$\alpha_{k}$ is relaxed as a continuous sharing factor
$\alpha_{k}\in\left[0,1\right]$. Thus, $\mathbf{P}_{{4}}$ can be expressed as

$\displaystyle\mathbf{P}_{{5}}:{\mathop{\max}\limits_{{\tau_{0}},{\tau_{k}},{P_{k}},{P_{s}},{f_{k}},\alpha_{k}}}\
{\mathop{\min}\limits_{k\in{{\cal K}}}}\ \
{\eta_{k}}\left({{\tau_{0}},{\tau_{k}},{P_{k}},{f_{k}},{\alpha_{k}}}\right)=$
$\displaystyle\frac{{\left({1-{\alpha_{k}}}\right)\frac{{T{f_{k}}}}{C}+{\alpha_{k}}\frac{{B{\tau_{k}}}}{{{v_{k}}}}{{\log}_{2}}\left({1+\frac{{{g_{k}}{P_{k}}}}{{\sigma_{0}^{2}}}}\right)}}{{\left({1-{\alpha_{k}}}\right)\left({{\tau_{0}}{P_{r,k}}+T{\gamma_{c}}f_{k}^{3}}\right)+{\alpha_{k}}\left[{{\tau_{0}}{P_{r,k}}+{\tau_{k}}\zeta\left({{P_{k}}+{P_{c,k}}}\right)}\right]}}$
(17a) $\displaystyle\text{s.t.}\ \
\left({1-{\alpha_{k}}}\right)\frac{{T{f_{k}}}}{C}+{\alpha_{k}}\frac{{B{\tau_{k}}}}{{{v_{k}}}}{\log_{2}}\left({1+\frac{{{g_{k}}{P_{k}}}}{{\sigma_{0}^{2}}}}\right)\geq{R_{k,\min}},$
(17b)
$\displaystyle\left({1-{\alpha_{k}}}\right)\left({{\tau_{0}}{P_{r,k}}+T{\gamma_{c}}f_{k}^{3}}\right)+{\alpha_{k}}\left[{{\tau_{0}}{P_{r,k}}+{\tau_{k}}\zeta\left({{P_{k}}+{P_{c,k}}}\right)}\right]$
$\displaystyle\leq{\Phi_{k}}\left({{\tau_{0}},{P_{s}}}\right),\ k\in{\cal K},$
(17c)
$\displaystyle{\tau_{0}}{\rm{+}}\sum\limits_{k=1}^{K}{{\alpha_{k}}{\tau_{k}}}\leq
T,C4-C6,\ 0\leq{\alpha_{k}}\leq 1.$ (17d)

It can be seen from $\left(17\right)$ that $\mathbf{P}_{{5}}$ is similar to
$\mathbf{P}_{{1}}$. Thus, for a given $\alpha_{k}$, the method for solving
$\mathbf{P}_{{1}}$ can be applied to solve $\mathbf{P}_{{5}}$. Moreover, it is
easy to prove that $\mathbf{P}_{{5}}$ is a linear fractional optimization
problem when other optimization variables are given. Thus, an alternative
optimization algorithm denoted by Algorithm 2 is proposed. In order to tackle
the non-convexity of the constraint given by $\left(17\rm{c}\right)$, Lemma 2
is given.

###### Lemma 3

In wireless powered MEC systems with TDMA under the binary offloading mode,
the maximum CE under the max-min fairness criterion is achieved when
$P_{s}=P_{th}$.

Using Lemma $3$ and the same method for solving $\mathbf{P}_{{1}}$, for given
$\alpha_{k}$, $\text{P}_{{5}}$ can be solved by solving the following problem
$\mathbf{{P}_{6}}$.

$\displaystyle\mathbf{P}_{{6}}:{\mathop{\max}\limits_{{\tau_{0}},{\tau_{k}},{y_{k}},{f_{k}},\Upsilon}}\
\Upsilon$ (18a) $\displaystyle\text{s.t.}\ \
\left({1-{\alpha_{k}}}\right)\frac{{T{f_{k}}}}{C}+{\alpha_{k}}\frac{{B{\tau_{k}}}}{{{v_{k}}}}{\log_{2}}\left({1+\frac{{{g_{k}}{y_{k}}}}{{{\tau_{k}}\sigma_{0}^{2}}}}\right)\geq{R_{k,\min}},$
(18b)
$\displaystyle{\tau_{0}}{P_{r,k}}+{\alpha_{k}}\zeta{y_{k}}+{\alpha_{k}}\zeta{\tau_{k}}{P_{c,k}}+\left({1-{\alpha_{k}}}\right)T{\gamma_{c}}f_{k}^{3}$
$\displaystyle\leq{\Phi_{k}}\left({{\tau_{0}},{P_{th}}}\right),$ (18c)
$\displaystyle\left({1-{\alpha_{k}}}\right)\frac{{T{f_{k}}}}{C}+{\alpha_{k}}\frac{{B{\tau_{k}}}}{{{v_{k}}}}{\log_{2}}\left({1+\frac{{{g_{k}}{y_{k}}}}{{{\tau_{k}}\sigma_{0}^{2}}}}\right)$
$\displaystyle-\eta\left[{{\tau_{0}}{P_{r,k}}+{\alpha_{k}}\zeta{y_{k}}+{\alpha_{k}}\zeta{\tau_{k}}{P_{c,k}}+\left({1-{\alpha_{k}}}\right)T{\gamma_{c}}f_{k}^{3}}\right]\geq\Upsilon$
(18d) $\displaystyle C4,C6,\left(17\rm{d}\right),y_{k}\geq 0,k\in{\cal K}.$
(18e)

In the above $y_{k}={\tau_{k}}P_{k}$; $\Upsilon\geq 0$ and $\eta\geq 0$ are
auxiliary variables. Moreover, it can be seen that $\mathbf{P}_{{6}}$ is
convex in terms of ${\alpha_{k}}$ when other optimization variables are given.
Thus, using the Lagrange duality method, the operational mode selection
variables ${\alpha_{k}}$ can be obtained by using the following Theorem.

###### Theorem 5

In the wireless powered MEC systems with TDMA under the binary computation
offloading mode, the optimal operational mode selection index has the
following form

$\displaystyle{\alpha_{k}}^{*}=\left\\{\begin{array}[]{l}0,\text{if}\
{F_{1,k}}<{F_{2,k}},\\\ 1,\text{otherwise};\end{array}\right.$ (19c)
$\displaystyle{F_{1,k}}=\left({{\lambda_{{k}}}+{\chi_{k}}}\right)\frac{{B{\tau_{k}}}}{{{v_{k}}}}{\log_{2}}\left({1+\frac{{{g_{k}}{{{y}}_{k}}}}{{{\tau_{k}}\sigma_{0}^{2}}}}\right)$
$\displaystyle\ \ \ \ \ \ \
-\zeta\left({{\mu_{k}}+{\chi_{k}}\eta}\right)\left({{{{y}}_{k}}+{\tau_{k}}{P_{c,k}}}\right)-\upsilon{\tau_{k}},$
(19d)
$\displaystyle{F_{2,k}}=\left({{\lambda_{{k}}}+{\chi_{k}}}\right)\frac{{T{f_{k}}}}{C}-\left({{\mu_{k}}+{\chi_{k}}\eta}\right)T{\gamma_{c}}f_{k}^{3},$
(19e)

where $\lambda_{{k}}\geq 0$, $\mu_{k}\geq 0$, $\upsilon\geq 0$ and
$\chi_{k}\geq 0$ are the dual variables associated with the constraints given
by $\left(18\rm{b}\right)$, $\left(18\rm{c}\right)$, $\left(18\rm{d}\right)$
and $\left(17\rm{d}\right)$.

###### Proof:

Theorem 5 can be readily proved from the derivations of the Lagrangian with
respect to ${\alpha_{k}}$. The proof is omitted due to the space limit. ∎

It can be seen from Theorem 5 that the selection of the operation mode under
the binary computation offloading mode depends on the tradeoff between the
achievable computed bits and the cost. Specifically, when this tradeoff under
the local computing mode is better than that obtained under the complete
offloading mode, the user chooses to perform local computing, and vice versa.
Based on Algorithm 1 and Theorem 5, Algorithm 2 for solving
$\mathbf{P}_{\text{4}}$ is presented in Table 2.

###### Remark 5

The proof for the convergence of Algorithm 2 when updating $\alpha_{k}$ can be
obtained from two facts. One is that $\mathbf{P}_{{4}}$ is a generalized
fractional programming problem, which can be solved by using Algorithm 2 based
on the Dinkelbach’s method. Algorithm 2 can converge when iteratively updating
$\eta$ [36]. This indicates that the objective function of $\mathbf{P}_{{6}}$
is nondecreasing, which can be easily proved by using the property of the
generalized fractional programming problem [36]. The other is that
$\mathbf{P}_{{6}}$ is convex in terms of $\alpha_{k}$. Thus, in each
iteration, there is only an optimal solution of $\alpha_{k}$. Due to those two
facts, it is easy to prove that Algorithm 2 is converged when updating
$\alpha_{k}$.

TABLE II: The alternative algorithm Algorithm 3: The alternative algorithm for
$\mathbf{P}_{4}$
---
1) Input settings:
the error tolerance $\xi_{1}>0$, $\xi_{2}>0$, $R_{k,\min}>0$ and $P_{th}$,
the maximum iteration number $N$.
2) Initialization:
$\eta^{n}=\eta_{0}$ and the iteration index $n=0$.
3) Optimization:
$\boldsymbol{\unrhd}$ for _n_ =1:N
initialize the iteration index $j=1$ and ${\alpha_{k}^{j}}={\alpha_{k}^{1}}$;
Repeat:
solve $\text{P}_{\textbf{6}}$ by using CVX for the given $\eta^{n}$ and
${\alpha_{k}^{j}}$ ;
obtain the solution
$\left\\{{\tau_{0,j}^{*,n},{\tau_{k,j}^{*,n}},{P_{k,j}^{*,n}},{f_{k,j}^{*,n}},\Upsilon_{j}^{*,n}}\right\\}$;
use the subgradient method to update the dual variables;
update $j=j+1$ and ${\alpha_{k}^{j}}$;
if $\left|\Upsilon_{j}^{*,n}-\Upsilon_{j-1}^{*,n}\right|\leq\xi_{1}$
break;
end
end Repeat
if $\left|\Upsilon_{j}^{*,n}-\eta^{n}\right|\leq\xi_{2}$
the maximum CE $\Upsilon_{j}^{*,n}$ is obtained;
break;
else
update $n=n+1$ and $\eta^{n}=\Upsilon_{j}^{*,n}$;
end
$\boldsymbol{\unrhd}$ end

##  IV CE Maximization in Wireless Powered MEC Networks: NOMA Based

In this section, CE maximization problems are studied in the wireless powered
MEC networks with NOMA under both partial and binary computation offloading
modes. The CE achieved under the max-min fairness criterion is maximized by
jointly optimizing the CPU frequency, the EH time, the offloading power and
time of users. In order to tackle those non-convex optimization problems, an
iterative algorithm and an alternative optimization algorithm based on SCA are
proposed for solving the CE maximization problem under the partial and binary
computation offloading mode, respectively.

###  IV-A Partial Offloading Mode

####  IV-A1 Problem Formulation

When the NOMA protocol and the partial computation offloading mode are
considered, the CE maximization problem is formulated under the max-min
fairness criterion as

$\displaystyle\mathbf{P}_{{7}}:{\mathop{\max}\limits_{{\tau_{0}},{\tau_{1}},{P_{k}},{P_{s}},{f_{k}}}}\
{\mathop{\min}\limits_{k\in{\cal K}}\
\eta_{k}\left({\tau_{0},{\tau_{1}},{P_{k}},{f_{k}}}\right)}$ (20a)
$\displaystyle\text{s.t.}\ \
{R_{k}}\left({{\tau_{1}},{f_{k}},{P_{k}}}\right)\geq{R_{k,\min}},\ k\in{\cal
K},$ (20b) $\displaystyle\ \ \ \ \sum\limits_{i=0}^{1}{{\tau_{i}}}\leq T,$
(20c) $\displaystyle\ \ \ \ 0\leq{\tau_{0}}\leq T,0\leq{\tau_{1}}\leq T,$
(20d) $\displaystyle\ \ \ \ C2,C5\ \text{and}\ C6.$ (20e)

$\mathbf{P}_{{7}}$ is challenging to tackle due to the minimum computation bit
constraint given by $\left(21\rm{b}\right)$ and the non-convex constraint
$C2$. In order to tackle it, an iteration algorithm based on SCA is proposed.

####  IV-A2 Solution and The Iterative Algorithm

In order to tackle the constraint $C2$, Lemma 4 is given.

###### Lemma 4

In wireless powered MEC systems with NOMA under the partial computation
offloading mode, the maximum CE under the max-min fairness criterion is always
achieved when $P_{s}=P_{th}$.

It is easy to prove that ${P_{k}}$ and $\tau_{1}$ are larger than zero, where
$k\in{\cal K}$. Thus, in order to solve $\mathbf{P}_{{7}}$, auxiliary
variables $x_{k}$ and $d_{i}$ are introduced, where
${P_{k}}=\exp\left({{x_{k}}}\right)$ and
${\tau_{1}}=\exp\left({{d_{1}}}\right)$. Based on Lemma 1, using a similar
method as used for $\mathbf{P}_{{1}}$, $\mathbf{P}_{{7}}$ can be solved by
iteratively solving $\mathbf{P}_{{8}}$, given as

$\displaystyle\mathbf{P}_{{8}}:{\mathop{\max}\limits_{{\tau_{0}},{d_{1}},{x_{k}},{f_{k}},\Upsilon}}\
\Upsilon$ (21a) $\displaystyle\text{s.t.}\ \
\frac{{T{f_{k}}}}{C}+\frac{{B\exp\left({{{{d}}_{1}}}\right)}}{{{v_{k}}}}{\log_{2}}\left({1+\frac{{{g_{k}}\exp\left({{x_{k}}}\right)}}{{\sum\limits_{i=k+1}^{K}{{g_{i}}\exp\left({{x_{i}}}\right)}+\sigma_{0}^{2}}}}\right)$
$\displaystyle\geq{R_{k,\min}},\ 0\leq k\leq K-1,$ (21b)
$\displaystyle\frac{{T{f_{k}}}}{C}+\frac{{B\exp\left({{d_{1}}}\right)}}{{{v_{k}}}}{\log_{2}}\left({1+\frac{{{g_{k}}\exp\left({{x_{k}}}\right)}}{{\sigma_{0}^{2}}}}\right)\geq{R_{k,\min}},$
(21c)
$\displaystyle{\tau_{0}}{P_{r,k}}+\zeta\exp\left({{{{d}}_{1}}}\right)\left(\exp\left({{{{x}}_{k}}}\right)+{P_{c,k}}\right)+T{\gamma_{c}}f_{k}^{3}\leq{\Phi_{k}}\left({{\tau_{0}},{P_{th}}}\right)$
(21d)
$\displaystyle\frac{{T{f_{k}}}}{C}+\frac{{B\exp\left({{{{d}}_{1}}}\right)}}{{{v_{k}}}}{\log_{2}}\left({1+\frac{{{g_{k}}\exp\left({{x_{k}}}\right)}}{{\sum\limits_{i=k+1}^{K}{{g_{i}}\exp\left({{x_{i}}}\right)}+\sigma_{0}^{2}}}}\right)$
$\displaystyle-\eta\left[{{\tau_{0}}{P_{r,k}}+\zeta\exp\left({{{{d}}_{1}}}\right)\left(\exp\left({{{{x}}_{k}}}\right)+{P_{c,k}}\right)+T{\gamma_{c}}f_{k}^{3}}\right]\geq\Upsilon,$
(21e)
$\displaystyle\frac{{T{f_{k}}}}{C}+\frac{{B\exp\left({{d_{1}}}\right)}}{{{v_{k}}}}{\log_{2}}\left({1+\frac{{{g_{k}}\exp\left({{x_{k}}}\right)}}{{\sigma_{0}^{2}}}}\right)$
$\displaystyle-\eta\left[{{\tau_{0}}{P_{r,k}}+\zeta\exp\left({{{{d}}_{1}}}\right)\left(\exp\left({{{{x}}_{k}}}\right)+{P_{c,k}}\right)+T{\gamma_{c}}f_{k}^{3}}\right]\geq\Upsilon,$
(21f) $\displaystyle{\tau_{0}}+\exp\left({{{{d}}_{1}}}\right)\leq
T,0\leq{\tau_{0}}\leq T,{d_{1}}\leq\ln\left(T\right),\text{and}\ C6.$ (21g)

$\eta$ is a non-negative parameter and $\Upsilon\geq 0$ is an auxiliary
variable. In order to address those constraints, auxiliary variables
$\exp\left({{z_{k}}}\right)$ are introduced. By using SCA, $\mathbf{P}_{{8}}$
can be solved by iteratively solving $\mathbf{P}_{{9}}$.

$\displaystyle\mathbf{P}_{{9}}:{\mathop{\max}\limits_{{\tau_{0}},{d_{1}},{x_{k}},{z_{k}},{f_{k}},\Upsilon}}\
\Upsilon$ (22a) $\displaystyle\text{s.t.}\ \
\frac{{T{f_{k}}}}{C}+\exp\left({{{\overline{z}}_{k}^{j}}}\right)+\exp\left({{{\overline{z}}_{k}^{j}}}\right)\left({{z_{k}}-{{\overline{z}}_{k}^{j}}}\right)\geq{R_{k,\min}},$
(22b)
$\displaystyle\frac{{B\exp\left({{{{d}}_{1}}}\right)}}{{{v_{k}}}}{\log_{2}}\left({1+\frac{{{g_{k}}\exp\left({{x_{k}}}\right)}}{{\sum\limits_{i=k+1}^{K}{{g_{i}}\exp\left({{x_{i}}}\right)}+\sigma_{0}^{2}}}}\right)\geq\exp\left({{z_{k}}}\right),$
(22c)
$\displaystyle\frac{{B\exp\left({{d_{1}}}\right)}}{{{v_{k}}}}{\log_{2}}\left({1+\frac{{{g_{k}}\exp\left({{x_{k}}}\right)}}{{\sigma_{0}^{2}}}}\right)\geq\exp\left({{z_{k}}}\right),k=K,$
(22d)
$\displaystyle\frac{{T{f_{k}}}}{C}+\exp\left({{{\overline{z}}_{k}^{j}}}\right)+\exp\left({{{\overline{z}}_{k}^{j}}}\right)\left({{z_{k}}-{{\overline{z}}_{k}^{j}}}\right)$
(22e)
$\displaystyle-\eta\left[{{\tau_{0}}{P_{r,k}}+\zeta\exp\left({{{{d}}_{1}}}\right)\left(\exp\left({{{{x}}_{k}}}\right)+{P_{c,k}}\right)+T{\gamma_{c}}f_{k}^{3}}\right]\geq\Upsilon,$
(22f) $\displaystyle\left(22\rm{d}\right)\ \text{and}\ \left(22\rm{g}\right),$

where ${\overline{z}}_{k}^{j}$, $k\in{\cal K}$, are the given local points at
the $j$th iteration. It is not difficult to prove that $\mathbf{P}_{{9}}$ is
convex and can be readily solved by using the existing convex optimization
tool [37].

TABLE III: The iterative algorithm based on using SCA Algorithm 3: The
iterative algorithm for $\mathbf{P}_{\text{7}}$
---
1) Input settings:
the error tolerance $\xi_{1},\xi_{2}>0$, $R_{k,\min}>0$ and $P_{th}$,
the maximum iteration number $N$.
2) Initialization:
EE $\eta^{n}=\eta_{0}$ and the iteration index $n=0$.
3) Optimization:
$\boldsymbol{\unrhd}$ for _n_ =1:N
initialize the iterative number $j=1$ and ${\overline{z}}_{k}^{j}$;
Repeat:
solve $\text{P}_{\textbf{9}}$ by using CVX for the given $\eta^{n}$;
obtain the solution
$\left\\{{\tau_{0,j}^{*,n},{d_{1,j}}^{*,n},{x_{k,j}}^{*,n},{z_{k,j}}^{*,n},{f_{k,j}^{*,n}},\Upsilon_{j}^{*,n}}\right\\}$;
if $\left|{\Upsilon_{j}^{*,n}-\Upsilon_{j-1}^{*,n}}\right|\leq{\xi_{2}}$
break;
else
update $j=j+1$ and ${\overline{z}}_{k}^{j}={z_{k,j}}^{*,n}$;
end
end Repeat
if $\left|\Upsilon_{j}^{*,n}-\eta^{n}\right|\leq\xi$
the maximum CE $\Upsilon_{j}^{*,n}$ is obtained;
break;
else
update $n=n+1$ and $\eta^{n}=\Upsilon_{j}^{*,n}$;
end
$\boldsymbol{\unrhd}$ end

Finally, by iteratively solving $\mathbf{P}_{{9}}$, an iterative algorithm
based on SCA is proposed to solve $\mathbf{P}_{{7}}$, which is denoted by
Algorithm 3. The details for Algorithm 3 can be found in Table 3. In Algorithm
3, $\xi_{1}$ and $\xi_{2}$ are the error tolerances for the CE iteration and
the SCA iteration, respectively.

###  IV-B Binary Offloading Mode

When the binary computation offloading mode is applied, the CE maximization
problem is given as

$\displaystyle{\mathop{\max}\limits_{{\tau_{0}},{\tau_{1}},{P_{k}},{P_{s}},{f_{k}},\alpha_{k}}}\
{\mathop{\min}\limits_{k\in{{\cal K}}}}\ \
{\eta_{k}}\left({{\tau_{0}},{\tau_{1}},{P_{k}},{f_{k}},{\alpha_{k}}}\right)=$
$\displaystyle\frac{{{\left({1-{\alpha_{k}}}\right)\frac{{T{f_{k}}}}{C}+{\alpha_{k}}\frac{{B{\tau_{1}}}}{{{v_{k}}}}{{\log}_{2}}\left({1+\frac{{{g_{k}}{P_{k}}}}{{\sum\limits_{i=k+1}^{K}{{\alpha_{i}}{g_{i}}{P_{i}}}+\sigma_{0}^{2}}}}\right)}}}{{\left({1-{\alpha_{k}}}\right)\left({{\tau_{0}}{P_{r,k}}+T{\gamma_{c}}f_{k}^{3}}\right)+{\alpha_{k}}\left[{{\tau_{0}}{P_{r,k}}+{\tau_{1}}\zeta\left({{P_{k}}+{P_{c,k}}}\right)}\right]}}$
(23a) $\displaystyle\text{s.t.}\ \
\left({1-{\alpha_{k}}}\right)\frac{{T{f_{k}}}}{C}+{\alpha_{k}}\frac{{B{\tau_{1}}}}{{{v_{k}}}}{\log_{2}}\left({1+\frac{{{g_{k}}{P_{k}}}}{{\sum\limits_{i=k+1}^{K}{{\alpha_{i}}{g_{i}}{P_{i}}}+\sigma_{0}^{2}}}}\right)$
$\displaystyle\geq{R_{k,\min}},k\in{\cal K},$ (23b)
$\displaystyle\left({1-{\alpha_{k}}}\right)\left({{\tau_{0}}{P_{r,k}}+T{\gamma_{c}}f_{k}^{3}}\right)+{\alpha_{k}}\left[{{\tau_{0}}{P_{r,k}}+{\tau_{1}}\zeta\left({{P_{k}}+{P_{c,k}}}\right)}\right]$
$\displaystyle\leq{\Phi_{k}}\left({{\tau_{0}},{P_{s}}}\right),k\in{\cal K},$
(23c)
$\displaystyle{\tau_{0}}{\rm{+}}\sum\limits_{k=1}^{K}{{\alpha_{k}}{\tau_{k}}}\leq
T,\alpha_{k}\in\left\\{0,\right\\},\left(21\rm{c}\right),\left(21\rm{d}\right),C5,\
\text{and}\ C6,$ (23d)

where $\alpha_{k}$ are the operational mode selection variables for either
local computing or complete computation task offloading. $\mathbf{P}_{{10}}$
is a mixed integer non-convex fractional optimization problem. In order to
solve it, motivated by those algorithms for solving $\mathbf{P}_{{4}}$ and
$\mathbf{P}_{{7}}$, an alternative algorithm based on SCA can be proposed. Due
to the space limit, the details are not presented. The process iteratively
solves $\mathbf{P}_{{11}}$ for the given $\alpha_{k}$ and $\eta$ in the
following and updates $\alpha_{k}$ by using Theorem 6.

$\displaystyle\mathbf{P}_{{11}}:{\mathop{\max}\limits_{{\tau_{0}},{d_{1}},{x_{k}},{z_{k}},{f_{k}},\Upsilon}}\
\Upsilon$ (24a) $\displaystyle\text{s.t.}\ \
\left({1-{\alpha_{k}}}\right)\frac{{T{f_{k}}}}{C}+\exp\left({{{\overline{z}}_{k}^{j}}}\right)+\exp\left({{{\overline{z}}_{k}^{j}}}\right)\left({{z_{k}}-{{\overline{z}}_{k}^{j}}}\right)$
$\displaystyle\geq{R_{k,\min}},k\in{\cal K},$ (24b)
$\displaystyle{\alpha_{k}}\frac{{B\exp\left({{{{d}}_{1}}}\right)}}{{{v_{k}}}}{\log_{2}}\left({1+\frac{{{g_{k}}\exp\left({{x_{k}}}\right)}}{{\sum\limits_{i=k+1}^{K}{{g_{i}}\exp\left({{x_{i}}}\right)}+\sigma_{0}^{2}}}}\right)\geq\exp\left({{z_{k}}}\right),$
(24c)
$\displaystyle{\alpha_{k}}\frac{{B\exp\left({{d_{1}}}\right)}}{{{v_{k}}}}{\log_{2}}\left({1+\frac{{{g_{k}}\exp\left({{x_{k}}}\right)}}{{\sigma_{0}^{2}}}}\right)\geq\exp\left({{z_{k}}}\right),k=K,$
(24d)
$\displaystyle\left({1-{\alpha_{k}}}\right)\frac{{T{f_{k}}}}{C}+\exp\left({{{\overline{z}}_{k}^{j}}}\right)+\exp\left({{{\overline{z}}_{k}^{j}}}\right)\left({{z_{k}}-{{\overline{z}}_{k}^{j}}}\right)$
$\displaystyle-\eta\left[{\tau_{0}}{P_{r,k}}+{\alpha_{k}}\zeta\exp\left({{{{d}}_{1}}}\right)\left(\exp\left({{{{x}}_{k}}}\right)+{P_{c,k}}\right)\right.$
$\displaystyle\left.+\left({1-{\alpha_{k}}}\right)T{\gamma_{c}}f_{k}^{3}\right]\geq\Upsilon,k\in{\cal
K},$ (24e)
$\displaystyle{\tau_{0}}{P_{r,k}}+{\alpha_{k}}\zeta\exp\left({{{{d}}_{1}}}\right)\left(\exp\left({{{{x}}_{k}}}\right)+{P_{c,k}}\right)+\left({1-{\alpha_{k}}}\right)T{\gamma_{c}}f_{k}^{3}$
$\displaystyle\leq{\Phi_{k}}\left({{\tau_{0}},{P_{th}}}\right),$ (24f)
$\displaystyle\left(22\rm{g}\right)\ \text{and}\ \left(22\rm{h}\right),$ (24g)

where ${P_{k}}{\alpha_{k}}=\exp\left({{x_{k}}}\right)$ and
${\tau_{1}}=\exp\left({{d_{1}}}\right)$. $\Upsilon\geq 0$ and $z_{k}\geq 0$
are auxiliary variables. $\eta$ is a non-negative parameter and
${\overline{z}}_{k}^{j}$,$k\in{\cal K}$, are the given local points at the
$j$th iteration.

###### Theorem 6

In the wireless powered MEC systems with NOMA under the binary offloading
mode, the optimal operational mode selection index has the form given by eq.
(26) in the top of the next page,

$\displaystyle{\alpha_{k}}^{*}=\left\\{\begin{array}[]{l}0,\text{if}\
{F_{1,k}}<{F_{2,k}},\\\ 1,\text{otherwise};\end{array}\right.$ (25c)
$\displaystyle{F_{k,1}}=\left\\{{\begin{array}[]{*{20}{c}}\begin{array}[]{l}\begin{array}[]{*{20}{c}}{\exp\left({{d_{1}}}\right)\left[{\frac{{{\lambda_{k}}B}}{{{v_{k}}}}{{\log}_{2}}\left({1+\frac{{{g_{k}}\exp\left({{x_{k}}}\right)}}{{\sigma_{0}^{2}}}}\right)-\left({{\mu_{k}}+{\omega_{k}}\eta}\right)\left({\exp\left({{x_{k}}}\right)+{P_{c,k}}}\right)}\right],}&{k=K}\end{array}\\\
\exp\left({{d_{1}}}\right)\left[{\frac{{{\lambda_{k}}B}}{{{v_{k}}}}{{\log}_{2}}\left({1+\frac{{{g_{k}}\exp\left({{x_{k}}}\right)}}{{\sum\limits_{i=k+1}^{K}{{g_{i}}\exp\left({{x_{i}}}\right)}+\sigma_{0}^{2}}}}\right)-\zeta\left({{\mu_{k}}+{\omega_{k}}\eta}\right)\left({\exp\left({{x_{k}}}\right)+{P_{c,k}}}\right)}\right],\text{otherwise}\end{array}\end{array}}\right.$
(25h)
$\displaystyle{F_{k,2}}=\left({{\varpi_{{k}}}+{\omega_{k}}}\right)\frac{{T{f_{k}}}}{C}-\left({{\mu_{k}}+{\omega_{k}}\eta}\right)T{\gamma_{c}}f_{k}^{3},$
(25i)

where $\varpi_{{k}}\geq 0$, $\lambda_{{k}}\geq 0$, $\omega_{k}\geq 0$ and
$\mu_{k}\geq 0$ are the dual variables associated with the constraints given
by $\left(25\rm{b}\right)$ \- $\left(25\rm{g}\right)$, respectively.

It can been from Theorem 5 that in the wireless powered MEC networks with NOMA
under the binary offloading mode, the optimal operational mode selection also
depends on the tradeoff between the achievable computed bits and the energy
consumption cost.

Finally, the complexity analysis is presented. Note that there are no
references for analyzing the complexity of solving $\mathbf{P}_{\text{9}}$ and
$\mathbf{P}_{\text{11}}$ that involves the product of an exponential function
and logarithmic function, which involves the division of exponential
functions. We cannot provide the complexity analysis for Algorithm 3 and 4.
The complexity of Algorithm 1 comes from two parts. One is the for-loop
iteration required by using the Dinkelbach’s method. Let $L_{1}$ denote the
iteration numbers of the for-loop. The other is from the solution of
$\mathbf{P}_{\text{3}}$ by using CVX. In $\mathbf{P}_{\text{3}}$, there are
$3K+2$ variables, $3K+2$ linear matrix inequality (LMI) constraints of size 1,
$2K$ third-order inequality constraints and $K$ logarithm inequality
constraints given by $\left(11\rm{b}\right)$. According to the analysis in
[37]-[39], the complexity of Algorithm 1 is ${{\cal
O}}\left(n{L_{1}}\sqrt{9K+2+Kn\log\left(n\right)}\left[\left({5K+2}\right)+2{n^{2}}K\log\left(n\right)+\right.\right.$
$\left.\left.{n^{2}}\right]\right)$, where ${\cal O}\left({\cdot}\right)$ is
the big-${\cal O}$ notation and $n={{\cal O}}\left({3K+2}\right)$. The
complexity of Algorithm 2 are from four parts. Two parts are similar to those
of Algorithm 1. The solved problem is $\mathbf{P}_{\text{6}}$ instead of
$\mathbf{P}_{\text{3}}$. The third part is from the subgradient method and the
fourth part is from the alternative optimization. Let $L_{2}$ and $L_{3}$
denote the iteration numbers of the for-loop part and that of alternative
optimization, respectively. Let $\ell_{1}$ denote the tolerance error for the
subgradient method. Similar to the analysis for Algorithm 1, the complexity of
Algorithm 2 is ${{\cal
O}}\left(n{L_{2}}{L_{3}}\left[\sqrt{9K+2+Kn\log\left(n\right)}\left[\left({5K+2}\right)+2{n^{2}}K\log\right.\right.\right.\\\
\left.\left.\left.\left(n\right)+{n^{2}}\right]+1/\ell_{2}^{2}\right]\right)$.

##  V Simulation Results

TABLE IV: Simulation Parameters Parameters | Notation | Typical Values
---|---|---
Numbers of Users | $K$ | $5$
The maximum EH power | $P_{k}^{\max}$ |  $0.004927$ W
The sensitivity threshold | $P_{0}$ |  $0.000064$ W
The communication bandwidth | $B$ |  $2$ MHz
Circuit parameter | $\mu$ | $274$
Circuit parameter | $\psi$ | $0.29$
The noise power | $\sigma_{0}^{2}$ |  $10^{-9}$ W
The number of cycles for one bit | $C$ |  $10^{3}$ cycles/bit
The capacitance coefficient | $\gamma_{c}$ | $10^{-28}$
The tolerance error | $\varpi$ | $10^{-4}$
The minimum computation bits | $R_{k,\min}$ |  $10^{4}$ Bits
The amplifier coefficient | $\zeta$ | $3$
The received power | $P_{r,k}$ |  $5$ dbm
The constant circuit power | $P_{c,k}$ |  $5$ dbm

In this section, simulation results are presented to evaluate the proposed CE
maximization framework and compare its performance with the existing
computation bits (CB) maximization framework. The simulation parameters are
selected based on the works in [5], [6] and the parameters for the non-linear
EH model are selected based on [32]. Similar to [30]-[35], the reference
distance is set as 1 meter and the maximum services distance for users is 5
meters. The channel power gains are set the same as those in [30]. The details
for the parameters are given in Table IV.

(a)(b)

Figure 4: (a) CE under partial offloading versus the transmission power of the
wireless power station; (b) CE under binary offloading versus the transmission
power of the wireless power station.

Fig. 3 shows the CE achieved with the max-min fairness criterion versus the
transmission power of the wireless power station using the proposed CE
maximization framework and the CB maximization framework under both partial
and binary offloading modes. The CB maximization framework is to maximize the
number of computation bits under the max-min fairness criterion. It is seen
that the CB maximization framework cannot guarantee that the maximum CE can be
achieved. This indicates that the resource allocation schemes for maximizing
the number of CB are inappropriate to the wireless powered MEC network that
aim to achieve the maximum CE. Moreover, on one hand, in the CB maximization
framework, the CE firstly increases with the transmission power and then
decreases when the transmit power is large enough. On the other hand, the
number of CB always increases with the transmission power. Thus, it is found
that there is a tradeoff between the CE and the number of CB. It can also be
seen that the CE achieved with NOMA is larger than that obtained with TDMA,
irrespective of the selected offloading mode. This indicates that NOMA can
obtain a CE gain compared to TDMA. The reason is that the offloading
efficiency is higher when NOMA is applied compared to that of TDMA [15], [16].

Figure 5: The CE comparison achieved under different offloading modes and
multiple access schemes.

Fig. 4 comprehensively presents the CE comparison achieved under different
operation modes and different multiple access schemes. It can be seen that CE
achieved under all the cases is the same when the transmit power of the
wireless power station is small. The reason is that when the transmit power of
the wireless power station is small, all the users choose to perform local
computing even under the partial computation offloading mode due to the fact
that the harvested energy is very small. This is consistent with our
theoretical analysis presented in Section III. It can be also seen that the CE
achieved under the partial offloading mode is larger than that obtained under
the binary offloading mode, irrespective of the multiple access schemes. The
reason is that the partial offloading mode can flexibly allocate resources for
computation offloading and local computing while the resources under the
binary offloading mode can only be completely allocated either for local
computing or for computation offloading.

(a)(b)

Figure 6: (a) Fairness comparison under partial offloading mode with different
optimization objectives; (b) Fairness comparison under binary offloading mode
with different optimization objectives.

Fig. 5 compares the user fairness achieved with our proposed max-min fairness
criterion framework and the sum CE maximization framework under both the
partial and binary computation offloading modes. Note that the sum CE
maximization framework is to maximize the sum of CE of all users. The
transmission power of the wireless power station is $0.025$ W. It can be seen
that there is a tradeoff between the sum CE and the fairness among users. The
max-min fairness criterion framework can improve the fairness among users at
the cost of the sum CE. The reason is that our proposed resource allocation
schemes aim to maximize the minimum CE among all users while those schemes for
maximizing the sum CE allocate more resource to the user with a better
offloading efficiency.

Figure 7: CE value versus the number of iterations

Fig. 6 shows the CE versus the number of iterations of different algorithms.
It can be seen that less than 15 iterations are required for all algorithms to
converge to the maximum CE. This indicates that our proposed algorithm is
computationally efficient. Moreover, it can be seen that the number of
iterations of Algorithm 1 is less than that of other algorithms. It only needs
to update the CE while other algorithms need to update the operational mode
selection variable or perform SCA iteration.

##  VI Conclusions

In this paper a new performance metric called CE was defined and studied in a
wireless powered MEC network under both partial and binary offloading modes.
TDMA and NOMA were investigated for offloading transmission under a practical
non-linear EH model. The EH time, the CPU frequencies, the user offloading
times, and the user transmit powers were jointly optimized to maximize the CE
under the max-min fairness criterion. Two iterative algorithms and two
alternative optimization algorithms were proposed to tackle those challenging
non-convex optimization problems. It was shown that our proposed resource
allocation strategies are superior to other benchmark schemes in terms of CE.
It was also shown that the CE achieved with the partial computation offloading
mode outperforms that obtained with the binary computation offloading mode and
NOMA can always achieve a CE gain compared to TDMA. The study also elucidated
the performance tradeoff between CE and the CB.

##  Appendix A Proof of Theorem 2

Let $\lambda_{k}\geq 0$, $\rho_{k}\geq 0$, $\theta_{k}\geq 0$, and $\beta\geq
0$ denote the dual variables corresponding to the constraints given by
$\left(11\rm{b}\right)$-$\left(11\rm{d}\right)$ and the constraint C3,
respectively. Then, the Lagrangian of $\mathbf{P}_{{3}}$ can be given as

$\displaystyle{\cal{L}}\left(\Xi\right)=\sum\limits_{k=1}^{K}{{\lambda_{k}}}\left({\frac{{T{f_{k}}}}{C}+\frac{{B{\tau_{k}}}}{{{v_{k}}}}{{\log}_{2}}\left({1+\frac{{{g_{k}}{y_{k}}}}{{{\tau_{k}}\sigma_{0}^{2}}}}\right)-{R_{k,{\rm{min}}}}}\right)$
$\displaystyle+\sum\limits_{k=1}^{K}{{\rho_{k}}}\left({{\Phi_{k}}\left({{\tau_{0}},{P_{th}}}\right)-{\tau_{0}}{P_{r,k}}-\zeta{y_{k}}-\zeta{\tau_{k}}{P_{c,k}}-T{\gamma_{c}}f_{k}^{3}}\right)$
$\displaystyle+\beta\left({T-\sum\limits_{n=0}^{K}{{\tau_{n}}}}\right)+\sum\limits_{k=1}^{K}{{\theta_{k}}}\left(\frac{{T{f_{k}}}}{C}+\frac{{B{\tau_{k}}}}{{{v_{k}}}}{{\log}_{2}}\left({1+\frac{{{g_{k}}{y_{k}}}}{{{\tau_{k}}\sigma_{0}^{2}}}}\right)\right.$
$\displaystyle\left.-\eta\left[{{\tau_{0}}{P_{r,k}}+\zeta{y_{k}}+\zeta{\tau_{k}}{P_{c,k}}+T{\gamma_{c}}f_{k}^{3}}\right]-\Upsilon\right)+\Upsilon,$
(26)

where $\Xi$ denotes a collection of all the primal and dual variables related
to $\mathbf{P}_{{3}}$.

Based on the Lagrangian of $\mathbf{P}_{{3}}$, the derivations of the
Lagrangian with respect to $f_{k}$ and $y_{k}$, can be respectively given as

$\displaystyle\frac{{\partial{\cal{L}}\left(\Xi\right)}}{{{f_{k}}}}=\frac{{T{\lambda_{k}}}}{C}-3{\rho_{k}}T{\gamma_{c}}f_{k}^{2}+{\theta_{k}}\left({\frac{T}{C}-\eta
3T{\gamma_{c}}f_{k}^{2}}\right),$ (27a)
$\displaystyle\frac{{\partial{\cal{L}}\left(\Xi\right)}}{{{y_{k}}}}=\frac{{\left({{\theta_{k}}+{\lambda_{k}}}\right)B{\tau_{k}}{g_{k}}}}{{{v_{k}}\ln
2\left({{\tau_{k}}\sigma_{0}^{2}+{g_{k}}{y_{k}}}\right)}}-\zeta\left({{\theta_{k}}\eta+{\rho_{k}}}\right).$
(27b)

Let their derivations be zero. Thus, $\left(12\rm{a}\right)$ can be obtained,
and one has

$\displaystyle\
{y_{k}}={\left[{\frac{{\left({{\lambda_{k}}+{\theta_{k}}}\right)B{\tau_{k}}}}{{\zeta{v_{k}}\ln
2\left({{\rho_{k}}+{\theta_{k}}\eta}\right)}}-\frac{{{\tau_{k}}\sigma_{0}^{2}}}{{{g_{k}}}}}\right]^{+}}.$
(28)

When $\tau_{k}=0$, it is not difficult to prove that $P_{k}^{opt}=0$. Since
$y_{k}=\tau{{}_{k}}P_{k},k\in{\cal K}$, when $\tau_{k}\neq 0$,
$P_{k}^{opt}=y_{k}/\tau{{}_{k}}$. Thus, $\left(12\rm{b}\right)$ is obtained.
The proof for Theorem 2 is complete.

##  Appendix B Proof of Theorem 3

Let $z\left({{\rho_{k}},\beta,{\theta_{k}}}\right)$ and
$\Gamma\left({\lambda_{k},{\rho_{k}},\beta,{\theta_{k}},g_{k}}\right)$ denote
the derivation of the Lagrangian ${\cal{L}}\left(\Xi\right)$ with respect to
$\tau_{0}$ and $\tau_{k}$, respectively. They are respectively expressed as

$\displaystyle
z\left({{\rho_{k}},\beta,{\theta_{k}}}\right)=\sum\limits_{k=1}^{K}{{\rho_{k}}}\left({{P_{E,k}}-{P_{r,k}}}\right)-\beta-\sum\limits_{k=1}^{K}{{\theta_{k}}\eta}{P_{r,k}},$
(29a)
$\displaystyle\Gamma\left({\lambda_{k},{\rho_{k}},\beta,{\theta_{k}},g_{k}}\right)=$
$\displaystyle\frac{\left({{\lambda_{k}}+{\theta_{k}}}\right)B}{{{v_{k}}}}\left[{{{\log}_{2}}\left({1+\frac{{{g_{k}}{P_{k}}}}{{\sigma_{0}^{2}}}}\right)-\frac{{{g_{k}}{P_{k}}}}{{\ln
2\left({\sigma_{0}^{2}+{g_{k}}{P_{k}}}\right)}}}\right]$
$\displaystyle-\zeta\left({{\rho_{k}}+{\theta_{k}}\eta}\right){P_{c,k}}-\beta.$
(29b)

For the given $\lambda_{k}$, $\rho_{k}$, $\theta_{k}$ and $\beta$, it can be
seen from $\left(26\right)$ that ${\cal{L}}\left(\Xi\right)$ is a linear
function of $\tau_{0}$ and $\tau_{k}$. Since $\mathbf{P}_{{3}}$ is convex and
the Slater’s conditions are satisfied, ${\cal{L}}\left(\Xi\right)$ is upper-
bounded with respect to $\tau_{0}$ and $\tau_{k}$. Thus,
$z\left({{\rho_{k}},\beta,{\theta_{k}}}\right)\leq 0$. When
$z\left({{\rho_{k}},\beta,{\theta_{k}}}\right)<0$, the maximum of
${\cal{L}}\left(\Xi\right)$ is achieved when $\tau_{0}=0$. When
$z\left({{\rho_{k}},\beta,{\theta_{k}}}\right)=0$, $\tau_{0}$ can be any
arbitrary value within $\left[{0,T}\right)$ since ${\cal{L}}\left(\Xi\right)$
is constant with respect to $\tau_{0}$. Thus, $\left(13\rm{a}\right)$ is
obtained.

By using $\left(29\rm{b}\right)$ and substituting $\left(12\rm{b}\right)$ into
$\left(30\rm{b}\right)$, one has

$\displaystyle\Gamma\left({\lambda_{k},{\rho_{k}},\beta,{\theta_{k}},g_{k}}\right)=$
$\displaystyle\frac{\left({{\lambda_{k}}+{\theta_{k}}}\right)B}{{{v_{k}}}}\left[{{{\log}_{2}}\left({1+\frac{{{g_{k}}}}{{\sigma_{0}^{2}}}{{\left[{\frac{{\left({{\lambda_{k}}+{\theta_{k}}}\right)B}}{{\zeta{v_{k}}\ln
2\left({{\rho_{k}}+{\theta_{k}}\eta}\right)}}-\frac{{\sigma_{0}^{2}}}{{{g_{k}}}}}\right]}^{+}}}\right)}\right]$
$\displaystyle-\zeta\left({{\rho_{k}}+{\theta_{k}}\eta}\right)\left({{{\left[{\frac{{\left({{\lambda_{k}}+{\theta_{k}}}\right)B}}{{\zeta{v_{k}}\ln
2\left({{\rho_{k}}+{\theta_{k}}\eta}\right)}}-\frac{{\sigma_{0}^{2}}}{{{g_{k}}}}}\right]}^{+}}+{P_{c,k}}}\right)-\beta.$
(30)

It can be seen from $\left(30\right)$ that
$\Gamma\left({\lambda_{k},{\rho_{k}},\beta,{\theta_{k}},g_{k}}\right)$
increases with $g_{k}$ and decreases with $\rho_{k}$ when
$g_{k}\geq{\left[{\sigma_{0}^{2}\zeta{v_{k}}\ln
2\left({{\rho_{k}}+{\theta_{k}}\eta}\right)}\right]}/{\left({{\lambda_{k}}+{\theta_{k}}}\right)B}$.
Similar to the derivation for $\left(13\rm{a}\right)$, since $\tau_{k}\geq 0$,
one has $\tau_{k}=0$ when
$\Gamma\left({\lambda_{k},{\rho_{k}},\beta,{\theta_{k}},g_{k}}\right)<0$ and
$\tau_{k}\geq 0$ when
$\Gamma\left({\lambda_{k},{\rho_{k}},\beta,{\theta_{k}},g_{k}}\right)=0$.
Since $\rho_{k}\geq 0$,
$\Gamma\left({\lambda_{k},{\rho_{k}},\beta,{\theta_{k}},g_{k}}\right)$
achieves its maximum when $\rho_{k}=0$. Moreover,
$\Gamma\left({\lambda_{k},0,\beta,{\theta_{k}},g_{k}}\right)=-\zeta\left({{\rho_{k}}+{\theta_{k}}\eta}\right){P_{c,k}}-\beta<0$
when ${g_{k}}=\frac{{\sigma_{0}^{2}\zeta{v_{k}}{\theta_{k}}\eta\ln
2}}{{\left({{\lambda_{k}}+{\theta_{k}}}\right)B}}$, and
$\Gamma\left({\lambda_{k},0,\beta,{\theta_{k}},g_{k}}\right)$ tends to
$+\infty$ when $g_{k}$ goes to $+\infty$. This indicates that
$\Gamma\left({\lambda_{k},{0},\beta,{\theta_{k}},g_{k}}\right)$ always has
$w^{opt}$ that satisfies
$\Gamma\left({\lambda_{k},{0},\beta,{\theta_{k}},w^{opt}}\right)=0$. Thus,
$\left(15\right)$ is proved. Based on $\left(30\right)$ and $\left(15\right)$,
the following cases can be obtained. When $g_{k}<w^{opt}$, $t_{k}=0$ since
$\Gamma\left({\lambda_{k},{\rho_{k}},\beta,{\theta_{k}},g_{k}}\right)<0$; when
$g_{k}>w^{opt}$,
$\Gamma\left({\lambda_{k},{\rho_{k}},\beta,{\theta_{k}},g_{k}}\right)<0$ holds
when ${\rho_{k}}>0$. In this case, $t_{k}=0$ and
${\tau_{0}}{P_{r,k}}+\zeta{\tau_{k}}\left({P_{k}}+{P_{c,k}}\right)+T{\gamma_{c}}f_{k}^{3}<{\Phi_{k}}\left({{\tau_{0}},{P_{s}}}\right)$.
This contradicts the complementary slackness condition that
${{\rho_{k}}}\left({{\Phi_{k}}\left({{\tau_{0}},{P_{th}}}\right)-{\tau_{0}}{P_{r,k}}-\zeta{\tau_{k}}\left({P_{k}}+{P_{c,k}}\right)-T{\gamma_{c}}f_{k}^{3}}\right)=0$.
Thus, when $g_{k}>w^{opt}$ and $\rho_{k}>0$, one has
${{\Phi_{k}}\left({{\tau_{0}},{P_{th}}}\right)-{\tau_{0}}{P_{r,k}}-\zeta{\tau_{k}}\left({P_{k}}+{P_{c,k}}\right)-T{\gamma_{c}}f_{k}^{3}}=0$.
For $g_{k}=w^{opt}$, since
$\Gamma\left({\lambda_{k},{\rho_{k}},\beta,{\theta_{k}},g_{k}}\right)<\Gamma\left({\lambda_{k},{0},\beta,{\theta_{k}},g_{k}}\right)=0$
when $\rho_{k}>0$, $\tau_{k}=0$. In this case, the complementary slackness
condition that
${{\rho_{k}}}\left({{\Phi_{k}}\left({{\tau_{0}},{P_{th}}}\right)-{\tau_{0}}{P_{r,k}}-\zeta{\tau_{k}}\left({P_{k}}+{P_{c,k}}\right)-T{\gamma_{c}}f_{k}^{3}}\right)=0$
cannot be held. Thus, one has $\rho_{k}=0$ and
${{\rho_{k}}}\left({{\Phi_{k}}\left({{\tau_{0}},{P_{th}}}\right)-{\tau_{0}}{P_{r,k}}-\zeta{\tau_{k}}\left({P_{k}}+{P_{c,k}}\right)-T{\gamma_{c}}f_{k}^{3}}\right)\leq
0$. From the above analysis, $\left(14\right)$ is obtained. Thus, the proof
for Theorem 3 is complete.

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|  Fuhui Zhou has worked as a Senior Research Fellow at Utah State University.
He received the Ph. D. degree from Xidian University, Xian, China, in 2016. He
is currently a Full Professor at College of Electronic and Information
Engineering, Nanjing University of Aeronautics and Astronautics. His research
interests focus on cognitive radio, edge computing, machine learning, NOMA,
physical layer security, and resource allocation. He has published more than
90 papers, including IEEE Journal of Selected Areas in Communications, IEEE
Transactions on Wireless Communications, IEEE Wireless Communications, IEEE
Network, IEEE GLOBECOM, etc. He was awarded as Young Elite Scientist Award of
China. He has served as Technical Program Committee (TPC) member for many
International conferences, such as IEEE GLOBECOM, IEEE ICC, etc. He serves as
an Associate Editor of IEEE Transactions on Communications, IEEE Systems
Journal, IEEE Wireless Communications Letters, IEEE Access and Physical
Communications. He also serves as co-chair of IEEE Globecom 2019 and IEEE ICC
2019 workshop on Advanced Mobile Edge /Fog Computing for 5G Mobile Networks
and Beyond.
---|---
|  Rose Qingyang Hu (S’95-M’98-SM’06-F’20) is a Professor in the Electrical
and Computer Engineering Department and Associate Dean for research of College
of Engineering at Utah State University. She also directs Communications
Network Innovation Lab at Utah State University. Her current research
interests include next-generation wireless system design and optimization,
Internet of Things, Cyber Physical system, Mobile Edge Computing, V2X
communications, artificial intelligence in wireless networks, wireless system
modeling and performance analysis. Prof. Hu received the B.S. degree from the
University of Science and Technology of China, the M.S. degree from New York
University, and the Ph.D. degree from the University of Kansas. Besides a
decade academia experience, she has more than 10 years of R&D experience with
Nortel, Blackberry, and Intel as a Technical Manager, a Senior Wireless System
Architect, and a Senior Research Scientist, actively participating in
industrial 3G/4G technology development, standardization, system level
simulation, and performance evaluation. She has published extensively and
holds numerous patents in her research areas. Prof. Hu is currently serving on
the editorial boards of the IEEE Transactions on Wireless Communications, the
IEEE Transactions on Vehicular Technology, the IEEE Communications Magazine
and the IEEE Wireless Communications. She also served as the TPC Co-Chair for
the IEEE ICC 2018. She is an IEEE Communications Society Distinguished
Lecturer Class 2015-2018. She was a recipient of prestigious Best Paper Awards
from the IEEE GLOBECOM 2012, the IEEE ICC 2015, the IEEE VTC Spring 2016, and
the IEEE ICC 2016. Prof. Hu is fellow of IEEE and a member of Phi Kappa Phi
Honor Society.
---|---