# Resource Allocation in Wireless Powered Relay Networks: A Bargaining Game
Approach

Zijie Zheng1, Lingyang Song1, Dusit Niyato2, and Zhu Han3,
Part of the material in this paper has been accepted by IEEE International
Conference on Communications, Malaysia, May 2016. [1] 1School of Electrical
Engineering and Computer Science, Peking University, Beijing, China
2School of Computer Engineering, Nanyang Technological University, Singapore
3Electrical and Computer Engineering Department, University of Houston,
Houston, TX, USA

###### Abstract

Simultaneously information and power transfer in mobile relay networks have
recently emerged, where the relay can harvest the radio frequency (RF) energy
and then use this energy for data forwarding and system operation. Most of the
previous works do not consider that the relay may have its own objectives,
such as using the harvested energy for its own transmission instead of
maximizing transmission of the network. Therefore, in this paper, we propose a
Nash bargaining approach to balance the information transmission efficiency of
source-destination pairs and the harvested energy of the relay in a wireless
powered relay network with multiple source-destination pairs and one relay. We
analyze and prove that the Nash bargaining problem has several desirable
properties such as the discreteness and quasi-concavity, when it is decomposed
into three sub-problems: the energy transmission power optimization, the power
control for data transmission and the time division between energy
transmission and data transmission. Based on the theoretical analysis, we
propose an alternating power control and time division algorithm to find a
suboptimal solution. Simulation results clearly show and demonstrate the
properties of the problem and the convergence of our algorithm.

###### Index Terms:

RF Charging, Wireless Relay Network, Resource Allocation, Game Theory

##  I Introduction

Harvesting energy from the natural resources such as solar energy, wind energy
and thermal energy has been proved to be a promising method to prolong the
lifetime for networks without fixed power supply, such as wireless sensor
networks [2]. However, uncertainty and uncontrollability of environment
usually result in that energy harvesting has less stable energy supply and
might sometimes insufficient causing the outage of the equipments in the
networks [3]. To overcome uncertainty and randomness of energy harvesting from
environment, the technique of radio frequency (RF) energy transfer is adopted
recently and applied widely in wireless powered networks, such as in wireless
sensor networks [4], in wireless body area networks [5] and in wireless
charging systems [6]. With RF energy transfer, wireless receivers can harvest
energy through converting received signals from wireless transmitters into
electricity and store it in batteries [7]. Since the information can also be
transmitted using radio frequency in wireless networks, the concept of
simultaneous wireless information and power transfer (SWIPT) has become a
promising approach for energy and information delivery in wireless networks
[7] [8].

More recently, besides information and power transfer from a wireless node to
another, SWIPT has also been extended to wireless relay networks [9, 10, 12,
11, 13, 14, 15, 16, 17]. It is known that relays can expand the coverage of
the wireless networks, where the relays help forward the sources’ information
to the destinations far away. However, in reality, some relays, e.g., wireless
sensors and mobile phones, may have limited battery reserves and need external
energy supply to maintain the circuits system activeness and help transmit the
data [9]. With SWIPT, relays can at first harvest energy from sources or other
wireless transmitters, and then forward information to destinations with the
harvested energy [10].

The difficulty in realizing SWIPT effectively in such wireless powered relay
networks is that practical circuits on the relays cannot harvest energy and
extract data from wireless signals at the same time [11][12]. Therefore, one
of the key challenges is balancing the tradeoff between the quality of
information transmission and wireless energy transfer. For example, more time
or frequency resource allocated to source-to-relay information transmission
results in less time to harvest energy by the relays, which might lead to
relays’ turnoff or energy outage. To maximize the throughput and avoid the
energy outage of the relays, the time switching-based relaying (TSR) protocol
and power splitting-based relaying (PSR) protocol were proposed in [10] and
[12], respectively, deciding the energy harvesting time ratio and power
splitting ratio in the one-source-one-relay network. To cope with the system
that multiple sources, authors in [13] designed different encode and decode
protocols for multiple source energy and data transmission and compared their
signal-to-error-ratios (SERs). Moreover, TSR and PSR protocols were extended
in [14] to deal with the antenna selection problem for energy harvesting and
information transmission, when the relay was equipped with a MIMO system. The
relay selection problem was studied in [15] when the networks include multiple
relays, where the authors maximized the capacity under an energy transfer
constraint. The concept of self-looping energy recycling was proposed very
recently with full-duplex relays. The capacity was derived with the TSR
protocol in [16] with full-duplex relays and the average rate was optimized in
[17] when in combination of full-duplex relays and the MIMO system.

As mentioned above, most of the existing works considered the information
transmission and energy transfer tradeoff to achieve a common objective such
as maximizing the total capacity of the system. This setting is suitable only
when all nodes belong to the same authority. However, similar to the
conventional wireless relay networks [18, 19, 20], in reality, nodes in the
wireless powered relay networks have their own selfish goals, which usually
conflict with each other. As for wireless powered relay networks, each source-
destination pair cares about its information transmission efficiency, i.e.
bits successfully transmitted per Joule energy cost. As for the relay, when
the energy can be accumulated in the battery [22, 21, 23], the relay cares
mostly about its energy benefit, i.e., the harvested energy left after helping
sources transmit information. The residual harvested energy can be used by the
relay, e.g., for its own data transmission or system operation. Thus, source-
destination pairs and the relay hold contradicted goals where sources try to
guarantee their transmitted energy to the relay are used to transmit sources
data as much as possible. However, the relay wants to uses less harvested
energy to help sources and to keep more for the relay’s own uses. The
separated and contradicted individual objectives of source-destination pairs
and the relay require us to investigate how they negotiate and bargain with
each other. Thus, it is natural to apply game theory to balance the objectives
among different nodes [24, 25, 26, 27].

To balance the information transmission efficiency for source-destination
pairs and the residual harvested energy for the relay, in this paper, we
propose a Nash bargaining approach to obtain the Nash bargaining solution
[31], through optimizing power control and time allocation. We prove the non-
convexity of the bargaining problem and simplify it through decomposing the
problem into three sub-problems: the energy transmission power optimization,
the power control for information transmission and the time division between
information transmission and energy transmission. Based on the theoretical
analysis of the each problem, we design an alternating power control and time
division algorithm to reach a suboptimal solution of the Nash bargaining
solution.

The main contributions of this paper are as follows:

  * •

We are the first to introduce a Nash bargaining approach in wireless powered
relay networks to balance the tradeoff between the energy efficiency of
information transmission of source-destination pairs and the residual
harvested energy of the relay.

  * •

The bargaining problem decomposition is not only physically meaningful but
also mathematically tractable, where the subproblem of energy harvesting power
optimization is proved to be discretizable. Moreover, the subproblems of time
division and information transmission are all proved to be quasi-concave.

  * •

Simulation results corroborate the properties or subproblems and the
convergence of our algorithm. In addition, the utility imbalance in the Nash
bargaining solution is also illustrated through simulation.

The rest of this paper is organized as follows. The system model and problem
formulation are presented in Section II. In Section III, we theoretically
analyze and decompose the problem. Based on the analysis, we present our
algorithm in Section IV. Section V includes the numerical simulation results.
Finally, we conclude this paper in Section VI.

##  II System Model

Figure 1: System model for wireless powered relay networks.

We consider a wireless powered communication system as shown in Fig. 1. For
simplicity, we suppose that the system includes $N$ source-destination pairs
denoted by
$\mathcal{(S,D)}=\\{(s_{1},d_{1}),(s_{2},d_{2}),\ldots,(s_{N},d_{N})\\}$ and
an RF-powered cooperative relay, denoted by $r$. More complicated problems in
practical networks with multiple relays, such as relay selection, interference
control and spectrum resource allocation are not considered in this paper,
which is left for the future work. Sources transmit data to their destinations
with the help of this relay (which does not have its own data to transmit). In
this paper, the relay is RF-powered [8]. The half-duplex time-switching
relaying protocol (TSR) [10] is adopted to support the relay’s energy
harvesting from sources and data forwarding. Furthermore, the direct link is
assumed to be ignored between each source-destination pair [32]. More details
on the channel model, TSR protocol, utility of each node and problem
formulation are given as follows.

###  II-A Channel Model

The channel between each pair of nodes is modeled as a quasi-static flat
fading channel [33], where the channel is constant over a block time $T$ and
independent from one block to the next. Without loss of generality, the
channel fading in each block follows a Rayleigh distribution. The use of such
channels is motivated by prior research in the wireless powered networks [11,
12, 34]. For simplicity, it is also assumed that the channel state information
is available at the relay and destinations.

Figure 2: TSR protocol.

###  II-B TSR Protocol

Fig. 2 describes the block structure in the TSR protocol for energy harvesting
and information forwarding at the relay. It is a direct extension from the
protocol in [10] and [12] in support of the case where the relay is able to
assist multiple sources inside one time block $T$. The relay first harvests
the energy from its assisting sources simultaneously. Then the relay receives
signals from sources and then decodes and forwards (DF) the data to its
assisting destinations sequentially with the harvested energy [12, 28, 29,
30]. As shown in Fig. 2, $\alpha T$ denotes the fraction of the block time
where the relay $r$ harvests energy, which is determined by the energy
transmission time of each of its assisting source $s_{i}$, denoted by
$\alpha_{i}T$. We have $\alpha
T=\max_{(s_{i},d_{i})\in(\mathcal{S},\mathcal{D})}\alpha_{i}T$, which
indicates that the relay does not begin transmitting data before accomplishing
energy harvesting from all its assisting sources. $\beta_{i}T$ is the time for
source $s_{i}$ to relay $r$ data transmission, and $\gamma_{i}T$ is the time
for information forwarding from $r$ to $d_{i}$. Thus, the following constraint
is naturally satisfied:

$\max_{\alpha_{i}}~{}(\alpha_{i}T)+\sum_{(s_{i},d_{i})\in(\mathcal{S},\mathcal{D})}(\beta_{i}+\gamma_{i})T=T.$
(1)

In this paper, we assume that the following constraint must also be satisfied
to guarantee the quality of service for data transmission of each source-
destination pair. Each source-destination pair has at least $\theta_{0}T$ unit
time for data transmission in this system:

$(\beta_{i}+\gamma_{i})T\geq\theta_{0}T.$ (2)

###  II-C Utility of Each Node

####  II-C1 Source-Destination Pair Utility

For each source-destination pair $(s_{i},d_{i})$, we adopt the _data
transmission efficiency_ [35] as its gain, which is defined as the total
average number of bits successfully transmitted from $s_{i}$ to $d_{i}$ per
Joule unit of energy consumed by source $s_{i}$, in each block time. Assuming
that the relay The data transmission efficiency $U_{i}^{S}$ for
($s_{i},d_{i}$) can be calculated as follows:

$\small\left\\{\begin{aligned} &U_{i}^{S}=\frac{C_{i}\cdot
T}{P^{s1}_{i}\alpha_{i}T+P^{s0}_{i}\beta_{i}T}=\frac{C_{i}}{P^{s1}_{i}\alpha_{i}+P^{s0}_{i}\beta_{i}},\\\
&C_{i}=\min\left\\{\beta_{i}\log\left(1+\frac{P^{s0}_{i}|h_{i}|^{2}}{\sigma^{2}}\right),\gamma_{i}\log\left(1+\frac{P^{r}|g_{i}|^{2}}{\sigma^{2}}\right)\right\\},\\\
\end{aligned}\right.$ (3)

where $C_{i}$ is the average capacity for source $s_{i}$ transmitting data to
$d_{i}$ via relay $r$ with the TSR protocol, $P^{s0}_{i}$ is the transmission
power of source $s_{i}$ for data transmission and $P^{s1}_{i}$ is the
transmission power of source $s_{i}$ for energy transmission, which can be
different from $P^{s0}_{i}$, $P^{r}$ is the transmission power of relay $r$,
$h_{i}$ is the channel gain from $s_{i}$ to $r$, and $g_{i}$ denotes the
channel gain from $r$ to $d_{i}$. Since either in the energy transmission, or
in the data transmission, the transmission power cannot exceed the physical
power upper bound of each equipment, we have the power constraints as follows:

$\left\\{\begin{aligned} &0\leq P^{s1}_{i}\leq P^{s}_{0},\\\ &0\leq
P^{s0}_{i}\leq P^{s}_{0},\\\ &0\leq P^{r}\leq P^{r}_{0},\\\
\end{aligned}\right.$ (4)

where $P^{s}_{0}$ is the maximum energy (or information) transmission power
for each source, and $P^{r}_{0}$ is the maximum transmission power for the
relay.

####  II-C2 Relay Utility

For relay node $r$, its utility is the residual energy after energy harvesting
and data forwarding, which can be used to execute other applications for
itself, e.g., sensing or transmitting data in sensor networks. The harvested
energy at relay $r$ per block time can be calculated as

$E=\eta\sum_{(s_{i},d_{i})\in(\mathcal{S},\mathcal{D})}\alpha_{i}TP^{s1}_{i}|h_{i}|^{2},$
(5)

where $\eta\in(0,1)$ is the energy conversion efficiency which depends on the
energy harvesting circuits [11]. The energy cost in relay $r$ per each block
time can be calculated as follows:

$\varphi=\sum_{(s_{i},d_{i})\in(\mathcal{S},\mathcal{D})}\left(\gamma_{i}TP^{r}\right)+E_{0},$
(6)

where the first term is the energy cost for data transmission and $E_{0}$ is a
constant, which denotes the energy cost for signal decoding and recoding in
the assistance of data forwarding. Hence, the utility of relay $r$ is denoted
by $U^{R}$, which is calculated as follows:

$U^{R}=E-\varphi.$ (7)

###  II-D Game Theoretic Problem Formulation

Both the source-destination pairs and the relay aim to achieve the highest
utility. However, the more available energy at the relay requires more time
for energy transmission (with larger $\\{\alpha_{i}\\}$) and less time for
data forwarding ( with smaller $\\{\beta_{i}\\}$ and $\\{\gamma_{i}\\}$). This
usually results in the lower utilities for the source-destination pairs. Apart
from the influence of time division, the transmission power from sources (such
as $P^{s1}_{i}$ and $P^{s0}_{i}$ in (3) and (5)) and the transmission power
from the relay (such as $P^{r}$ in (3) and (5)) jointly affect the utilities
of source-destination pairs and the relay. Since the utilities of the sources
and the utility of the relay are obviously contradicted, a Nash bargaining
game [31] is adopted in this paper and the Nash bargaining solution [31] is
considered as a reasonable solution to balance the utilities of sources and
the utility of the relay.

In the Nash-bargaining game, the source-destination pairs and the relay are
modeled as $N+1$ players, where for convenience the relay is denoted as the
$({N+1})^{th}$ player. The strategy of each player (each source-destination
pair and the relay) consists of all the variables that may influence the value
of the player’s utility. Even though the utility of each player does not
directly contain all variables as in (3) and (7), the constraint in (1) and
the utility function in (3) make all variables correlated. Therefore, each
player needs to consider all the variables in its strategy. That is to say,
each player’s strategy consists of the transmission power of sources
$\\{P^{s1}_{i}\\}$, $\\{P^{s0}_{i}\\}$, the transmission of the relay $P^{r}$,
and the time division variables $\\{\alpha_{i}\\}$, $\\{\beta_{i}\\}$, and
$\\{\gamma_{i}\\}$. The strategy of the $n$’s player is denoted by
$\Upsilon_{n}$, which is given below.

$\Upsilon_{n}=\left\\{\\{P^{s1}_{i}\\},\\{P^{s0}_{i}\\},P^{r},\\{\alpha_{i}\\},\\{\beta_{i}\\},\\{\gamma_{i}\\}\right\\}_{n}.$
(8)

The objective function for a Nash bargaining game is defined as a Nash
bargaining solution, which can satisfy all the players in the game. In a Nash
bargaining solution, all players reach a consensus of their strategies to
maximize the product of utilities of all players[31]. Mathematically, the Nash
bargaining solution in this paper is defined as follows.

$\displaystyle\max_{\\{P^{s0}_{i}\\},\\{P^{s1}_{i}\\},P^{r},\\{\alpha_{i}\\},\\{\beta_{i}\\},\\{\gamma_{i}\\}}\Phi=\left(\prod_{(s_{i},d_{i})}{U^{S}_{i}}\right)U^{R},$
(9)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}s.t.~{}U^{S}_{i}>0,~{}U^{R}>0,$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Upsilon_{1}=\Upsilon_{2}=\ldots=\Upsilon_{N}=\Upsilon_{N+1}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\left\\{\\{P^{s1}_{i}\\},\\{P^{s0}_{i}\\},P^{r},\\{\alpha_{i}\\},\\{\beta_{i}\\},\\{\gamma_{i}\\}\right\\},$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{Constraints~{}in}~{}(\ref{equ:timesplitting}),(\ref{equ:timeconstraint}),(\ref{equ:powerconstraint}),$

where $U^{S}_{i}$ and $U^{R}$ are given in (3) and (7), respectively.

We emphasize that the main concerning in this paper is providing some static
insights in a Nash bargaining solution in the wireless powered relay networks
instead of designing a dynamic distributed process. Therefore, we assume that
throughout this paper, the bargaining process is simulated in a centralized
way on a randomly selected source. After the selected source achieves the
values of
$\\{P^{s0}_{i}\\},\\{P^{s1}_{i}\\},P^{r},\\{\alpha_{i}\\},\\{\beta_{i}\\}$ and
$\\{\gamma_{i}\\}$ in a Nash bargaining solution, it sends the values of
$P^{s0}_{i}$ and $P^{s1}_{i}$ to each source $s_{i}$ through the relay to tell
$s_{i}$ how to set the transmission power. In addition, the relay also achieve
the the values of $\alpha_{i}$, $\beta_{i}$, and $\gamma_{i}$ from the
selected source and sends them to each source $s_{i}$ to tell $s_{i}$ how to
allocate the time between the energy transmission and data transmission.

##  III Problem Analysis

In this section, we first highlight that the Nash bargaining problem in this
paper is not guaranteed to be a concave (convex) problem, which is hard to
obtain a strictly global optimal solution. Therefore, to find a suboptimal
solution, we decompose the problem into three parts: 1) energy transmission
power optimization, i.e., how to set $P_{i}^{s1}$; 2) data transmission power
optimization, i.e., how to set $P_{i}^{s0}$ and $P^{r}$; and 3) time division
optimization, i.e., how to set $\alpha_{i}$, $\beta_{i}$ and $\gamma_{i}$. We
find this decomposition is not only traditionally meaningful,111When the
scenario is complicated, it is a traditional trend of thoughts to decompose
the time division and power control problem in dealing with resource
allocation problems in wireless communication. but also mathematically
tractable that each problem holds its specific properties which can help
simplify the process of the algorithm design to find a suboptimal result of
the problem in (9).

###  III-A Non-Concavity of The Original Problem

In this subsection, we prove that the original problem is not a concave
(convex) problem (Property 1), which is hard to find a strictly optimal
solution [37].

###### Lemma 1

The objective function in (9) is maximized, if and only if the following
equation is satisfied for each source-destination pair:

$\beta_{i}\log\left(1+\frac{P_{i}^{s0}|h_{i}|^{2}}{\sigma^{2}}\right)=\gamma_{i}\log\left(1+\frac{P^{r}|g_{i}|^{2}}{\sigma^{2}}\right).$
(10)

That is to say, at the Nash bargaining solution, the relay transmits all the
data which it receives from its assisting sources.

###### Proof:

When
${\beta_{i}\log\left(1+\frac{P_{i}^{s0}|h_{i}|^{2}}{\sigma^{2}}\right)<\gamma_{i}\log\left(1+\frac{P^{r}|g_{i}|^{2}}{\sigma^{2}}\right)}$,
the relay can lower down its transmission time $\gamma_{i}T$, inducing a
larger
${\left(\eta\sum_{x_{i}=1}(\alpha_{i}TP^{s1}_{i}|h_{i}|^{2})-(\sum_{x_{i}=1}\gamma_{i}TP^{r}+E_{0})\right)}$
in (7). On the other hand, if
${\beta_{i}\log\left(1+\frac{P_{i}^{s0}|h_{i}|^{2}}{\sigma^{2}}\right)>\gamma_{i}\log\left(1+\frac{P^{r}|g_{i}|^{2}}{\sigma^{2}}\right)}$,
the source can lower down its transmission time $\beta_{i}T$, inducing a
larger $1/(\alpha_{i}P^{s1}+\beta_{i}P^{s0})$, which increases the objective
function. Overall, the objective function is maximized, if and only if
$\beta_{i}\log\left(1+\frac{P_{i}^{s0}|h_{i}|^{2}}{\sigma^{2}}\right)=\gamma_{i}\log\left(1+\frac{P^{r}|g_{i}|^{2}}{\sigma^{2}}\right)$
is satisfied. ∎

###### Property 1

The problem (9) is not guaranteed to be a concave (convex) optimization
problem with variables
$\\{\alpha_{i}\\},\\{\beta_{i}\\},\\{\gamma_{i}\\},\\{P^{s0}_{i}\\},\\{P^{s0}_{i}\\}$
and $P^{r}$.

###### Proof:

Based on Lemma 1 and the power constraint $P^{r}\leq P^{r}_{0}$ as mentioned
in Section II.C, we have the constraint set as follows:

$\frac{\sigma^{2}}{|g_{i}|^{2}}\left(\left(1+\frac{P^{s0}_{i}|h_{i}|^{2}}{\sigma^{2}}\right)^{\frac{\beta_{i}}{\gamma_{i}}}-1\right)\leq
P^{r}_{0}.$ (11)

It is not a convex set for $P^{s0}_{i}$ and $\beta_{i}$. Hence, the problem in
(9) is not a strict concave (convex) optimization problem [37]. ∎

###  III-B Dedicators and Enjoyers for Energy Transmission

In this subsection, we prove that at the Nash bargaining solution, the sources
can be divided into two groups as _dedicators_ and _enjoyers_. The definitions
of the dedicator and the enjoyer are given as follows:

###### Definition 1

Source $s_{i}$ is defined as a _dedicator_ when source $s_{i}$ transmits
energy with full transmission power, i.e., $P_{i}^{s1}=P_{0}^{s}$.

###### Definition 2

Source $s_{i}$ is defined as an _enjoyer_ when source $s_{i}$ never transmits
energy to the relay $r$, i.e., $P_{i}^{s1}=0$.

Then, we have the theorem as follows:

###### Theorem 1

The objective function in (9) is maximized if and only if
$P_{i}^{s1}=P_{0}^{s}$ or $P_{i}^{s1}=0$ for each source $s_{i}$. That is to
say, each source in the system is either a _dedicator_ or an _enjoyer_.

###### Proof:

See Appendix A. ∎

Then, we present Corollary 1 and Corollary 2, respectively, to emphasize the
energy transmission time property for the dedicators and which sources become
more likely to be dedicators. These two corollaries can help simplify the
process to decide the energy transmission time and the dedicator selection,
which is shown in the algorithm in Section IV.

###### Corollary 1

For each dedicator $s_{i}$ (with $P_{i}^{s1}=P_{0}^{s}$), its energy
transmission time is $\alpha_{i}T=\max_{j=1}^{N}\\{\alpha_{j}T\\}$. That is to
say, all dedicators spend the same time duration for energy transmission.

###### Proof:

According to Theorem 1, for $P_{i}^{s1}=P_{0}^{s}$, function (19) is a
monotonically increasing function for $P_{i}^{s1}$. Under this condition, it
is a monotonically increasing function with respect to $\alpha_{i}$. Thus,
each source $s_{i}$ tends to maximize $\alpha_{i}$ in maximizing the objective
function (9). Hence, we have $\alpha_{i}=\max_{j=1}^{N}\\{\alpha_{j}\\}$. ∎

###### Corollary 2

For each pair of sources $s_{i}$ and $s_{j}$, $P_{i}^{s1}\geq P_{j}^{s1}$ is
satisfied if and only if $\beta_{i}P_{i}^{s0}\geq\beta_{j}P_{j}^{s0}$ and
$|h_{1}|^{2}\geq|h_{2}|^{2}$. That is to say, the source is more likely to do
energy transmission as a dedicator, when it has a better channel condition
($|h_{i}|^{2}$) to the relay and holds more time ($\beta_{i}T$) and larger
power ($P_{i}^{s0}$) for data transmission.

###### Proof:

See Appendix B. ∎

Corollary 2 can also help to design an incentive mechanism, to encourage
sources to be dedicators instead of receivers. According to Corollary 2, in a
certain time block, sources who want to transmit more data (with a larger
$\beta_{i}T$) are more likely to be selected as dedicators. Therefore,
assuming that sources in the network have the similar amount of data in a
relatively long time (a number of time blocks), some sources have more data in
the first several time blocks are selected as dedicators and the others are
responsible to transmit energy in the rest ones. In addition, the channel
statement in the wireless network usually changes with time. This indicates
that sources as enjoyers are eventually selected as dedicators in the later
time blocks, when their channel statements between the relay become better.
Even though Corollary 2 can avoid the circumstance that some sources are
always working as dedicators and others are always selected as enjoyers, more
incentive mechanisms are needed to proactively encourage sources to work as
dedicators, which is left for the future work.

###  III-C Quasi-Concavity of Data Transmission Power Control

In this subsection, we prove that when the time division and the group of
_dedicators_ are fixed, the data transmission power control is a quasi-concave
optimization problem.

###### Theorem 2

Given fixed $\\{P_{i}^{s1}\\}$ and fixed
$\\{\alpha_{i}\\},\\{\beta_{i}\\},\\{\gamma_{i}\\}$, the problem in (9) is a
quasi-concave optimization problem with variables $\\{P^{s0}_{i}\\}$ and
$P^{r}$.

###### Proof:

See Appendix C. ∎

###  III-D Quasi-Concavity of Time Division

In this subsection, we prove that assuming the data transmission power control
and the group of _dedicators_ to be fixed, the time division problem is a
quasi-concave optimization problem.

###### Theorem 3

Given fixed $\\{P_{i}^{s1}\\}$ and fixed $\\{P^{s0}_{i}\\}$ and $P^{r}$, the
problem in (9) is a quasi-concave optimization problem with variables
$\\{\alpha_{i}\\},\\{\beta_{i}\\},\\{\gamma_{i}\\}$.

###### Proof:

See Appendix D. ∎

We conclude this section as follows. The original problem in (9) is not a
concave (convex) problem which can not guarantee to obtain a globally optimal
solution (Property 1). However, when we decompose the problem into three
parts: 1) energy harvesting power optimization, 2) data transmission power
optimization, and 3) time division, we find that the problem becomes simpler.
The energy transmission power optimization is a discrete optimization problem.
In particular, the sources can be divided into _dedicators_ and _enjoyers_
(Theorem 1), which are relevant to the channel conditions from the sources to
the relay and the transmission energy cost of the sources (Corollary 2). The
data transmission power optimization problem and time division problem are
both quasi-concave optimization problems (Theorem 2 and Theorem 3). Therefore,
there exist many optimization algorithms to solve [36][37], e.g. the gradient
ascent algorithm. The properties discussed in this section indicate that the
problem decomposition is not only traditionally meaningful, but also
reasonable from a mathematical perspective, which can guide us to design an
efficient algorithm in the next section.

##  IV Alternating Power Control and Time Division Algorithm

Input: $k=0$, $(\\{P_{i}^{s1}\\})^{L}$,
$(\\{P_{i}^{s0}\\})^{k}$,$(P^{r})^{k}$,$(\\{\alpha_{i}\\},\\{\beta_{i}\\},\\{\gamma_{i}\\})^{k}$

, Output:
$\max\Phi(\\{P_{i}^{s1}\\},\\{P_{i}^{s0}\\},P^{r},\\{\alpha_{i}\\},\\{\beta_{i}\\},\\{\gamma_{i}\\})$

1) Enumerating possible conditions of dedicator selection, at each condition
with $\\{P_{i}^{s1}\\}^{L}$:

k=0;

while  _Arbitral stopping criterion is not satisfied_ do

(a) Transmission power optimization:
$(\\{P_{i}^{s0}\\},P^{r})^{k+1}=\arg\max_{(\\{P_{i}^{s0}\\},P^{r})}\Phi_{1}^{k}$;

(b) Time division optimization:
$(\\{\alpha_{i}\\},\\{\beta_{i}\\},\\{\gamma_{i}\\})^{k+1}=\arg\max_{\\{\alpha_{i}\\},\\{\beta_{i}\\},\\{\gamma_{i}\\}}\Phi_{2}^{k}$;

end while

$\Phi^{L}=\Phi^{L}\left(\\{P_{i}^{s1}\\}^{L},(\\{P_{i}^{s0}\\},P^{r},\\{\alpha_{i}\\},\\{\beta_{i}\\},\\{\gamma_{i}\\})^{k+1}\right)$;

2) $\Phi=\max_{L}\Phi^{L}$;

* $(\cdot)^{L}$ is the $L^{th}$ condition to select dedicators, $(\cdot)^{k}$ is the value of the $k^{th}$ iteration.

Algorithm 1 Alternating Power Control and Time Division Algorithm

In this section, we design an alternating algorithm to solve the bargaining
problem in (9). The analysis in Section III enlightens the decomposed
algorithm design: 1) The outer loop is optimizing the energy transmission
power with enumerating all the possible conditions to select dedicators
(according to Theorem 1). 2) Under each condition, we alternatively and
iteratively solve data transmission power optimization and time division
problems (according to Theorem 2 and Theorem 3). We emphasize that since the
original problem is not a concave (convex) optimization problem (according to
Property 1), this kind of algorithm can only guarantee to reach a suboptimal
solution. Specifically, the alternating algorithm is shown in Algorithm 1,
where $\Phi_{1}^{k}$ and $\Phi_{2}^{k}$ are given as follows:

$\left\\{\begin{aligned}
&\Phi_{1}^{k}=\Phi(\\{P_{i}^{s1}\\}^{L},\\{P_{i}^{s0}\\},P^{r},\\{\alpha_{i}\\}^{k},\\{\beta_{i}\\}^{k},\\{\gamma_{i}\\}^{k}),\\\
&\Phi_{2}^{k}=\Phi(\\{P_{i}^{s1}\\}^{L},(\\{P_{i}^{s0}\\},P^{r})^{k+1},\\{\alpha_{i}\\},\\{\beta_{i}\\},\\{\gamma_{i}\\}),\\\
\end{aligned}\right.$ (12)

where $(\cdot)^{L}$ denotes the value at the $L$’s condition of energy
transmission power optimization, and $(\cdot)^{k}$ is the value at the $k$’s
iteration of data transmission power optimization and time division.
$\Phi_{1}^{k}$ equals to the objective $\Phi$ in (9) when $\\{P_{i}^{s1}\\}$
and $\\{\alpha_{i}\\},\\{\beta_{i}\\},\\{\gamma_{i}\\}$ are fixed.
$\Phi_{2}^{k}$ equals to the objective $\Phi$ when
$\\{P_{i}^{s1}\\},\\{P_{i}^{s0}\\},P^{r}$ are fixed.

More details on energy transmission power optimization (dedicator selection)
and the details on the alternating the data transmission power optimization
and time division process are given as below. In addition, we also discuss the
convergence and the complexity of our algorithm in this section.

###  IV-A Energy Transmission Power Optimization

In this subsection, we explain the dedicator selection in Algorithm 1.
According to Theorem 1, the energy transmission power optimization is first
transformed from a continuous problem into a discrete problem. Thus, what we
concern about is to decide which sources are selected to transmit energy as
dedicators and which are not. According to Theorem 1 and Corollary 1, we
emphasize that the power and time for energy transmission for all dedicators
are identical. For convenience, we define an indicator vector
$\Omega=\\{\omega_{1},\omega_{2},\ldots,\omega_{N}\\}\in\mathbb{B}$, where
$\omega_{i}=1$ when source $s_{i}$ is a dedicator. Otherwise $\omega_{i}=0$
and source $s_{i}$ is an enjoyer. $\mathbb{B}$ is the set of the all the
conditions of dedicator selection, where $|\mathbb{B}|=2^{N}$. We provide
Remark 1 and Remark 2 as below to help us decide which sources are more likely
to be selected as dedicators.

###### Remark 1

Given any index $K$, when $\sum_{i=1}^{N}\omega_{i}=K$ is not feasible for all
elements in $\mathbb{B}$, the conditions with $\sum_{i=1}^{N}\omega_{i}<K$ are
not feasible as well.

Remark 1 indicates that the transmitted energy is not enough with $K$
dedicators, it is also not enough when the number of dedicators is below $K$.
Thus, we enumerate the conditions from $N$ dedicators to $N-1,N-2,\ldots$
dedicators and stop when the result is not feasible for all conditions with
$K$ dedicators.

###### Remark 2

When $\sum_{i=1}^{N}\omega_{i}=K$, $\omega_{1}\geq\omega_{2}$ is _possibly_
satisfied when $|h_{i}|^{2}\geq|h_{j}|^{2}$.

According to Corollary 2, we have that the energy transmission power
$P_{i}^{s1}\geq P_{j}^{s1}$ is satisfied if and only if
$\beta_{i}P_{i}^{s0}\geq\beta_{j}P_{j}^{s0}$ and $|h_{1}|^{2}\geq|h_{2}|^{2}$.
However, $\beta_{i}$ and $P_{i}^{s0}$ for each source $s_{i}$ cannot be
determined before the alternating data transmission power and time division
process is accomplished in Algorithm 1. Thus, we can only decide $\omega_{i}$
based on $|h_{i}|^{2}$, which results in that the operation in Remark 2 might
miss the optimal point. However, the operation can observably reduce the
enumeration time with $K$ dedicators, from $\binom{N}{K}$ to $1$ (to select
$K$ sources as dedicators with the highest $|h_{i}|^{2}$).

###  IV-B Data Transmission Power Optimization and Time Division

In this subsection, we propose the concave (convex) optimization methods to
deal with the data transmission power optimization and time division, i.e.,
how to find the optimal $\\{P_{i}^{s0}\\}$, $P^{r}$, $\\{\alpha_{i}\\}$,
$\\{\beta_{i}\\}$, and $\\{\gamma_{i}\\}$. We decompose the data transmission
power optimization and time division and alternatively deal with the two
problems, i.e., the inner iteration in Algorithm 1, according to Property 1,
Theorem 2 and Theorem 3. The specific iteration of data transmission power
optimization and time division, and the arbitral stopping criterion are given
as follows.

  * •

Data Transmission Power Optimization: When the energy transmission power
$\\{P_{i}^{s1}\\}$ and the time division
$~{}\\{\alpha_{i}\\},~{}\\{\beta_{i}\\},~{}\\{\gamma_{i}\\}$ are fixed as the
values obtained from the latest iteration, the data transmission optimization
problem at the $k^{th}$ iteration is to maximize $\Phi_{1}^{k}$ in Algorithm
1, which is given as follows:

$\displaystyle\max_{P^{r}}\Phi_{1}^{k}$ (13)
$\displaystyle~{}~{}~{}~{}~{}~{}=\prod_{i=1}^{N}\left(\frac{\log(1+L_{1}^{i}P^{r})}{L_{3}^{i}((1+L_{1}^{i}P^{r})^{L_{2}^{i}}-1)+L_{4}^{i}}\right)\cdot\left(K_{1}-K_{2}P^{r}\right),$
$\displaystyle~{}~{}s.t.~{}0\leq P^{r}\leq[P_{0}^{r}]^{+},$

where
$[P_{0}^{r}]^{+}=\min\left\\{P_{0}^{r},\\{\frac{\sigma^{2}}{|g_{i}|^{2}}(1+\frac{P_{0}^{s}|h_{i}|^{2}}{\sigma^{2}})^{\frac{\beta_{i}}{\gamma_{i}}}-\frac{\sigma^{2}}{|g_{i}|^{2}}\\}\right\\}$
and $L_{1}^{i}$ to $L_{4}^{i}$ and $K_{1}$ to $K_{2}$ are given in (24) in
Appendix C. How to transfer the original problem in (9) into the above form is
also given in Appendix C. The Lagrangian function
$\mathcal{L}_{1}^{k}(P^{r},\rho_{1},\rho_{2})$ is given as follows:

$\displaystyle\mathcal{L}_{1}^{k}(P^{r},\rho_{1},\rho_{2})$ (14)
$\displaystyle=-\prod_{i=1}^{N}\left(\frac{\log(1+L_{1}^{i}P^{r})}{L_{3}^{i}((1+L_{1}^{i}P^{r})^{L_{2}^{i}}-1)+L_{4}^{i}}\right)\left(K_{1}-K_{2}P^{r}\right)$
$\displaystyle+\rho_{1}(0-P^{r})+\rho_{2}(P^{r}-[P_{0}^{r}]^{*}),$

where $\rho_{1}>0$ and $\rho_{2}>0$ are the Lagrange multipliers. Then,
$\mathcal{L}_{1}^{k}(P^{r},\rho_{1},\rho_{2})$ is minimized through the
typical gradient ascent algorithm (Chapter 3 of [37]). The details of the
gradient ascent algorithm for data transmission power optimization is given in
Appendix E. We obtain the optimal data transmission power of the relay as
$(P^{r})^{\star}$ and calculate the optimal data transmission power of the
sources as $(\\{(P^{s0}_{i})\\})^{\star}$ based on (10). Finally, we update
$P^{r}$ and $\\{P^{s0}_{i}\\}$ as $(P^{r})^{k+1}=(P^{r})^{\star}$ and
$(\\{P^{s0}_{i}\\})^{k+1}=(\\{P^{s0}_{i}\\})^{\star}$.

  * •

Time Division: When the energy transmission power $\\{P_{i}^{s1}\\}$ are fixed
and the data transmission power $P^{r}$ and $\\{P^{s0}_{i}\\}$ have already
been updated at the $k^{th}$ iteration, the time division problem at the
$k^{th}$ iteration is to maximize $\Phi_{2}^{k}$ in Algorithm 1, which is
given as follows:

$\displaystyle\max_{\\{\alpha_{i}\\},\\{\gamma_{i}\\}}\Phi_{2}^{k}$ (15)
$\displaystyle~{}~{}~{}~{}~{}~{}=\left(\prod_{P_{i}^{s1}=P_{0}^{s}}\frac{D_{1}^{i}\gamma_{i}}{D_{2}^{i}\alpha+D_{3}^{i}\gamma_{i}}\right)\cdot(F_{1}\alpha-
F_{2}-\sum_{i=1}^{N}D_{4}^{i}\gamma_{i}),$
$\displaystyle~{}~{}s.t.~{}\alpha+\sum_{i=1}^{N}(\gamma_{i}+D_{5}^{i}\gamma_{i})=1,$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}(\gamma_{i}+D_{5}^{i}\gamma_{i})T\geq\theta_{0}T,$

where $D_{1}$ to $D_{5}$ and $F_{1}$ to $F_{2}$ are given in (32) in Appendix
D. How to transform the original problem in (9) into the above form is also
given in Appendix D. The Lagrangian function
$\mathcal{L}_{2}^{k}(\alpha,\\{\gamma_{i}\\},\varrho_{1},\\{\varrho_{2}^{i}\\})$
is given as follows:

$\displaystyle\mathcal{L}_{2}^{k}(\alpha,\\{\gamma_{i}\\},\varrho_{1},\\{\varrho_{2}^{i}\\})$
(16)
$\displaystyle=-\left(\prod_{P_{i}^{s1}=P_{0}^{s}}\frac{D_{1}^{i}\gamma_{i}}{D_{2}^{i}\alpha+D_{3}^{i}\gamma_{i}}\right)\cdot(F_{1}\alpha-
F_{2}-\sum_{i=1}^{N}D_{4}^{i}\gamma_{i})$
$\displaystyle+\varrho_{1}\left(\alpha+\sum_{i=1}^{N}(\gamma_{i}+D_{5}^{i}\gamma_{i})-1\right)$
$\displaystyle+\sum_{i=1}^{N}\varrho_{2}^{i}\left(\theta_{0}T-(\gamma_{i}+D_{5}^{i}\gamma_{i})T\right),$

where $\varrho_{1}$ and $\\{\varrho_{2}^{i}\\}$ are the Lagrange multipliers.
Similar to the data transmission power optimization,
$\mathcal{L}_{2}^{k}(\alpha,\\{\gamma_{i}\\},\varrho_{1},\\{\varrho_{2}^{i}\\})$
is minimized through the typical gradient ascent algorithm (Chapter 3 of [37]
and find the optimal point as $\alpha^{\star}$, $(\\{\gamma_{i}\\})^{\star}$
and $(\\{\beta_{i}\\})^{\star}$. More details of the dual ascent algorithm for
time division is given in Appendix F. Finally we update the time division as
$\alpha^{k+1}=\alpha^{\star}$,
$\\{\gamma_{i}\\}^{k+1}=\\{\gamma_{i}\\}^{\star}$ and
$\\{\beta_{i}\\}^{k+1}=\\{\beta_{i}\\}^{\star}$.

  * •

Stopping Criterion: The stopping criterion in the optimization methods is used
to determine when the iteration stops [37]. Specifically, in Algorithm 1 the
stopping criterion is the rule to evaluate when the alternating data
transmission power optimization and the time division process should stop,
which is given as follows,

$\max\left\\{\frac{|\Phi_{1}^{k+1}-\Phi_{1}^{k}|}{\Phi_{1}^{k}},~{}\frac{|\Phi_{2}^{k+1}-\Phi_{2}^{k}|}{\Phi_{2}^{k}}\right\\}<\varepsilon,$
(17)

where $\varepsilon$ is a minor parameter to determine the convergence.

###  IV-C Convergence and Complexity

In this part, we discuss the convergence and the complexity of our proposed
algorithm. Specifically, we at first prove that the alternating data
transmission power optimization and time division process (the inner loop in
Algorithm 1). Then, we discuss the complexity of our algorithm, i.e., to
discuss the enumeration time of energy transmission power optimization (the
outer loop in Algorithm 1) and the iteration time of alternating data
transmission power optimization and time division process (the inner loop in
Algorithm 1).

We at first prove the convergence of this alternating data transmission power
optimization and time division process.

###### Theorem 4

The alternating data transmission power optimization and time division process
converges.

###### Proof:

It is easy to test that the singularity of the objective function $\Phi$ in
(9) exists, only when for any source $s_{i}$, $P_{i}^{s0}=0$ and
$P_{i}^{s1}=0$ simultaneously happen. However, $P_{i}^{s0}=0$ indicates that
$s_{i}$ does not transmit the data ($\beta_{i}=0$), which contradicts the
constraints in (2). Therefore, $\Phi$ has an upper bound in the feasible zone.
The Lagrangian methods in the alternating data transmission power optimization
and time division process guarantee that $\Phi_{1}^{k}$ and $\Phi_{2}^{k}$ are
non-decreasing at each iteration [37]. When the objective function $\Phi$
keeps increasing and has an upperbound, the stopping criterion will finally
become satisfied with the iteration execution [37], which indicates that the
iteration converges. ∎

We then discuss the complexity (the total number of iterations) of our
algorithm. Based on Corollary 2, Remark 1 and Remark 2, the enumeration time
of the outer loop is prominently reduced from $2^{N}$ to $N$. Thus, the
computation complexity can be expressed as $N\times T_{in}$, where $T_{in}$ is
the average number of iterations for the inner loop. As for the inner loop of
our algorithm, either for the data transmission power optimization or for the
time division, the general gradient ascent method usually guarantees a linear
convergence speed [38]. We use $O(\cdot)$ to represent the magnitude of
iteration time. Then, for the data transmission power optimization, the
iteration time can be expressed as $O(\frac{1}{\varepsilon_{1}})$, where
$\varepsilon_{1}$ is the parameter to constrain the accuracy of the data
transmission power optimization as given in (41) in Appendix E. For the time
division, the iteration time can be expressed as
$O(\frac{1}{\varepsilon_{2}})$, where $\varepsilon_{2}$ represents the
accuracy of time division as given in (43) in Appendix F. Thus, we can
calculate the computation complexity, denoted by $T$, as follows.

$\small T=N\times T_{in}=N\times N_{al}\times
O(\frac{1}{\varepsilon_{1}}+\frac{1}{\varepsilon_{2}}),$ (18)

where $N_{al}$ is the time of alternations between the data transmission power
optimization and the time division. We emphasize that even though the
alternating data transmission power optimization and the time division process
can be proved to converge, the strict closed-form relation between the
$N_{al}$ and the arbitral stopping criterion given in (17) is complicated to
derive. This is because the non-convexity (non-concavity) of the original
problem results in the distance between the point after each step of data
transmission power optimization or time division and the final result is
hardly to calculate. Thus, we left the mathematical derivation for the future
work.

We conclude this section as follows. We design an alternating power control
and time division algorithm to solve the bargaining problem in (9). The outer
loop in the Algorithm 1 is the energy transmission power optimization.
Properties in Remark 1 and Remark 2 can help to reduce the time of
enumeration. The inner iteration in the Algorithm 1 is the data transmission
power optimization and time division. At each iteration, we sequentially
optimize the data transmission power and deal with the time division, through
the Lagrangian optimization methods. The stopping criterion guarantees the
convergence of the inner iteration.

##  V Numerical Results

In this section, we provide the numerical results to illustrate the properties
of the problem and our alternating algorithm. We at first show the convergence
of our algorithm in Fig. 3. Then, we discuss the relation between objective in
(9) and the number of dedicators in Fig. 4. The sum capacity of all source-
destination pairs and the residual harvested energy in the relay are
illustrated in Fig. 5 and Fig. 6. The distribution of the utilities among
dedicators and enjoyers is shown in Fig. 7, which shows the necessity to
design incentive mechanisms in the future work. The simulation parameters are
given in Table I.

TABLE I: Simulation Parameters Network | Region Size ($L\times W$) | Constant $20$m$\times$$20$m
---|---|---
Compositions | Number of Source-Destination Pairs ($N$) | Constant $7$
| Distribution of Sources | Uniform Distribution in Square Region
|  |  from $(0,0)$ to $(0.5L,0.5W)$
| Distribution of Destinations | Uniform Distribution in Square Region
|  |  from $(0.5L,0.5W)$ to $(L,W)$
| Location of The Relay | Constant $(0.5L,0.5W)$
Data | The Highest Transmission Power of Each Source ($P_{0}^{s}$) | Constant 10mW
And | The Highest Transmission Power of The Relay ($P^{r}_{0}$) | Constant 10mW
Energy | Noise Power ($n^{2}$) | Constant -95dBm
Transmission | Channel Gain ($\gamma$) | Average Passloss: Distance-2
| Circuit Energy Cost Per Time Slot of The Relay ($\frac{E_{0}}{T}$) | Variable from $0$ to $0.2$mW
| Energy Harvesting Efficiency of The Relay ($\eta$) | Constant 0.5
| Time Division Parameter ($\theta_{0}$) | Constant 0.05
Algorithm | Convergence Decision Parameter ($\epsilon$) | Constant $10^{-4}$
Parameter |  |
Figure 3: Convergence of the iteration for data transmission power
optimization and time division.

In Fig. 3, we illustrate the convergence of the alternating data transmission
power optimization and time division presented in Algorithm 1. To clearly show
the process of the algorithm, we assume that all the sources are dedicators.
At each iteration, the first part (from $t$ to $t.5$) is the updating of the
data transmission power ($P^{r}$ and $\\{P^{s0}_{i}\\}$), and the second part
(from $t.5$ to $t+1$) is the updating of the time division. We can find that
with the iteration of alternating algorithm, the objective $\Phi$ keeps
increasing and finally converges, even though the original problem is neither
strictly convex nor strictly concave. In addition, it can be observed that the
data transmission power optimization and the time division steps improve the
objective function in the similar order of magnitude. This indicates that it
is hardly to say which step has a stronger influence on the objective
optimization.

Figure 4: The relation between the objective and the number of dedicators.

In Fig. 4, we show the relation between the objective ($\Phi$) and the number
of dedicators, when the energy for relay’s decoding and recoding ($E_{0}$)
varies. With each fixed $E_{0}$, we show a good representative condition for
intuitive illustration according to Remark 1 and Remark 2. We can find that
with fixed $E_{0}$, the objective $\Phi$ is infeasible when the number of
dedicators is small. This is because when the number of dedicators is not
enough, the transmitted energy is not enough for the relay’s data
transmission. However, when the number of dedicators is sufficient, less
dedicators leads to a larger $\Phi$, e.g., with 3 dedicators when $E_{0}$
equals $0.12$mW$\cdot T$.

Figure 5: The relation between the sum capacity and the number of dedicators.

In Fig. 5, we show the relation between the sum capacity
($\sum_{i=1}^{N}C_{i}$) of the source-destination pairs and the number of
dedicators, when the energy for relay’s decoding and recoding ($E_{0}$)
varies. We can find that with more sources acting as dedicators, the sum
capacity keeps increasing. This is because with more dedicators, the relay
harvests more energy with a shorter harvesting duration ($\alpha T$).
Correspondingly, the relay has a longer time
($\sum_{i=1}^{N}(\beta_{i}+\gamma_{i})T$) to forward data. Thus, the sum
capacity will be larger. This also illustrates the TSR protocol in Fig. 2,
where it is better that all sources work as dedicators to take full use of the
time resources ($\alpha T$), when we tend to maximize the sum capacity from a
centralized perspective.

Figure 6: The relation between the residual harvested energy in the relay and
the number of dedicators.

In Fig. 6, we show the relation between the residual harvested energy
($U_{r}$) in the relay and the number of dedicators. The residual harvested
energy $U_{r}$ equals to zero when number of dedicators is not enough, which
results in the harvested energy in the relay is not enough for data
forwarding. With more sources as dedicators, the residual harvested energy is
not guaranteed to be increasing. This indicates that in the Nash bargaining
game, to balance the utilities among sources and the relay, with more
dedicators the sources require the relay to forward more data where may result
in less residual harvested energy in the relay.

Figure 7: Utility distribution among source-destination pairs.

Fig. 7 illustrates the utility distribution among source-destination pairs,
with sources as dedicators and sources as enjoyers. We observe that the
dedicators’ utilities ($10^{-1}$) are dramatically lower than those of
enjoyers ($10^{3}$). The possible reasons are given as follows. To guarantee
quality of service of each source-destination pair’s data transmission, it is
reasonable to guarantee that each pair has a minimum duration in data
transmission ($\theta_{0}T$) as shown in (2). However, the constraints in (2)
possibly result in the utility imbalance among dedicators and enjoyers, where
all sources tend to be enjoyers instead of dedicators. This indicates that the
Nash bargaining solution can guarantee fairness only to a certain extent in
one time block. This fairness problem will subside and diminish in the long
term, since Corollary 2 and Remark 2 result in that the all sources tend to be
dedicators alternatively with their transmission channels changes. For
example, when source $s_{i}$ has a relatively larger $|h_{i}|^{2}$, it will
act as a dedicator. On the contrary, source $s_{j}$ will be a dedicator when
having a relatively larger $|h_{j}|^{2}$ in the next time block. That is to
say, enjoyers will eventually become dedicators according to Corollary 2 and
Remark 2. However, more effective incentive mechanisms still need to be
designed to guarantee fair utility distribution for more general cases, such
as when some sources always act as dedicators if their channel conditions
($|h_{i}|^{2}$) are always better than others. These points are left for the
future work.

##  VI Conclusion

In this paper, we have considered a wireless powered relay network with one
relay and multiple source-destination pairs from a Nash bargaining game
theoretical perspective. We have found that the problem is not a strictly
concave (convex) optimization problem, which is hard to solve. However, the
problem has been proved to be simplified when it is decomposed into three
parts: 1) energy transmission power optimization, 2) data transmission power
optimization and 3) time division. We have proved that with the Nash
bargaining solution, the sources can be divided into two groups as dedicators,
who do energy harvesting with the maximum power and enjoyers who do not
transmit energy to the relay at all. In addition, we have proved the quasi-
concavity of both transmission power control problem and transmission power
control problem. Based on the analysis on the problem, we have designed an
algorithm to find a suboptimal Nash bargaining solution. The convergence of
the algorithm and the properties have been illustrated in the simulation
results. In the simulation results, we have also shown the imbalance of
utilities between dedicators and enjoyers at the Nash bargaining solution,
which indicates that more effective incentive mechanisms are needed.

##  Appendix A Proof of Theorem 1

When we separately consider transmission power of energy harvesting
$P_{i}^{s1}$ for each source $s_{i}$, the problem in (9) can be rewritten as a
fractional programming problem, which is given as follows:

$\displaystyle\max\frac{A_{4}+A_{5}(\alpha_{i}P_{i}^{s1})}{A_{3}(A_{1}\alpha_{i}P_{i}^{s1}+A_{2})},$
(19) $\displaystyle~{}~{}s.t.~{}0\leq P_{i}^{s1}\leq P_{0}^{s},$

where $A_{1}$ to $A_{5}$ are independent from $P_{i}^{s1}$, which is given as
follows:

$\small\left\\{\begin{aligned} &A_{1}=T,\\\ &A_{2}=P_{i}^{s0}\beta_{i},\\\
&A_{3}=\frac{\prod_{k\neq
i}P_{k}^{s1}(P_{k}^{s1}\alpha_{k}T+P_{i}^{s0}\beta_{k}T)}{\prod_{1\leq k\leq
N}C_{i}},\\\ &A_{4}=\eta\sum_{k\neq i}P_{k}^{s1}\alpha_{k}T-E_{0}.\\\
&A_{5}=T,\\\ \end{aligned}\right.$ (20)

The objective function above is a monotonic function of $P_{i}^{s1}$. When
$A_{4}<A_{2}$, the objective function is a monotonically increasing function
of $P_{i}^{s1}$. The objective function is maximized only when
$P_{i}^{s1}=P_{0}^{s1}$. On the other hand, when $A_{4}\geq A_{2}$, the
objective function is a monotonically decreasing function of $P_{i}^{s1}$. The
objective function is maximized only when $P_{i}^{s1}=0$. Hence, we have
$P_{i}^{s1}$ equal to either $P_{0}^{s}$ or $0$ to maximize the objective in
(9). That is to say, the sources can be classified into two groups: the group
of dedicators with $P_{i}^{s1}=P_{0}^{s1}$ and the group of enjoyers with
$P_{i}^{s1}=0$.

##  Appendix B Proof of Corollary 2

When we set $P_{i}^{s1}=P_{0}^{s}$ and $P_{j}^{s1}=0$, the objective function
(9) can be written as follows:

$\Phi_{i}=\frac{B_{2}\alpha_{i}|h_{i}|^{2}P_{0}^{s}+B_{1}}{\beta_{j}P_{j}^{s0}(\alpha_{i}P_{0}^{s}+\beta_{i}P_{i}^{s0})},$
(21)

where $B_{2}>0$ and $B_{2}\alpha_{i}|h_{i}|^{2}P_{0}^{s}+B_{1}>0$ according to
the constraints in (9). On the other hand, we can rewrite (9) as
$\Phi_{j}=\frac{B_{2}\alpha_{j}|h_{j}|^{2}P_{0}^{s}+B_{1}}{\beta_{i}P_{i}^{s0}(\alpha_{j}P_{0}^{s}+\beta_{j}P_{j}^{s0})}$
when setting $P_{i}^{s1}=0$ and $P_{j}^{s1}=P_{0}^{s1}$. In addition,
according to Corollary 1, we have $\alpha=\alpha_{i}=\alpha_{j}$. Then, when
$\beta_{i}P_{i}^{s0}\geq\beta_{j}P_{j}^{s0}$ and $|h_{i}|^{2}\geq|h_{j}|^{2}$,
we have the inequation as follows:

$\displaystyle\Phi_{i}-\Phi_{j}$ (22)
$\displaystyle=\frac{\beta_{i}P_{i}^{s0}\alpha
P_{0}^{s}(B_{2}|h_{i}|^{2}P_{0}^{s}+B_{1})-\beta_{j}P_{j}^{s0}\alpha
P_{0}^{s}(B_{2}|h_{j}|^{2}P_{0}^{s}+B_{1})}{\beta_{i}P_{i}^{s0}\beta_{j}P_{j}^{s0}(\alpha
P_{0}^{s}+\beta_{j}P_{j}^{s0})(\alpha P_{0}^{s}+\beta_{i}P_{i}^{s0})}$
$\displaystyle-\frac{\beta_{i}P_{i}^{s0}\beta_{j}P_{j}^{s0}\alpha
P_{0}^{s}(|h_{i}|^{2}-|h_{j}|^{2})}{\beta_{i}P_{i}^{s0}\beta_{j}P_{j}^{s0}(\alpha
P_{0}^{s}+\beta_{j}P_{j}^{s0})(\alpha P_{0}^{s}+\beta_{i}P_{i}^{s0})}$
$\displaystyle\geq\frac{\beta_{i}P_{i}^{s0}\alpha
P_{0}^{s}(B_{2}|h_{i}|^{2}P_{0}^{s}+B_{1})-\beta_{j}P_{j}^{s0}\alpha
P_{0}^{s}(B_{2}|h_{j}|^{2}P_{0}^{s}+B_{1})}{\beta_{i}P_{i}^{s0}\beta_{j}P_{j}^{s0}(\alpha
P_{0}^{s}+\beta_{j}P_{j}^{s0})(\alpha P_{0}^{s}+\beta_{i}P_{i}^{s0})}$
$\displaystyle\geq\frac{\beta_{j}P_{j}^{s0}\alpha
P_{0}^{s}(B_{2}|h_{i}|^{2}P_{0}^{s}+B_{1})-\beta_{j}P_{j}^{s0}\alpha
P_{0}^{s}(B_{2}|h_{j}|^{2}P_{0}^{s}+B_{1})}{\beta_{i}P_{i}^{s0}\beta_{j}P_{j}^{s0}(\alpha
P_{0}^{s}+\beta_{j}P_{j}^{s0})(\alpha P_{0}^{s}+\beta_{i}P_{i}^{s0})}$
$\displaystyle=\frac{\beta_{j}(P_{j}^{s0})^{2}\alpha
P_{0}^{s}B_{2}(|h_{i}|^{2}-|h_{j}|^{2})}{\beta_{i}P_{i}^{s0}\beta_{j}P_{j}^{s0}(\alpha
P_{0}^{s}+\beta_{j}P_{j}^{s0})(\alpha P_{0}^{s}+\beta_{i}P_{i}^{s0})}\geq 0.$

Therefore, we have the source be more likely to perform energy transmission as
a dedicator (as $s_{i}$ to $s_{j}$), when it has better channel condition
($|h_{i}|^{2}$) to the relay, holds more time ($\beta_{i}T$) and larger power
($P_{i}^{s0}$) for data transmission.

##  Appendix C Proof of Theorem 2

According to Lemma 1, we can rewrite the problem in (9) with variable $P^{r}$
as follows:

$\displaystyle\max_{P^{r}}\Phi=\prod_{i=1}^{N}\left(\frac{\log(1+L_{1}^{i}P^{r})}{L_{3}^{i}((1+L_{1}^{i}P^{r})^{L_{2}^{i}}-1)+L_{4}^{i}}\right)\left(K_{1}-K_{2}P^{r}\right),$
(23) $\displaystyle~{}~{}s.t.~{}0\leq
P^{r}\leq\min\left\\{P_{0}^{r},\\{\frac{\sigma^{2}}{|g_{i}|^{2}}(1+\frac{P_{0}^{s}|h_{i}|^{2}}{\sigma^{2}})^{\frac{\beta_{i}}{\gamma_{i}}}-\frac{\sigma^{2}}{|g_{i}|^{2}}\\}\right\\},$

where for any $1\leq i\leq N$, $L_{1}^{i}$ to $L_{4}^{i}$ and $K_{1}$ to
$K_{2}$ are given as follows:

$\small\left\\{\begin{aligned} &L_{1}^{i}=\frac{|g_{i}|^{2}}{\sigma^{2}}\geq
0,\\\ &L_{2}^{i}=\frac{\gamma_{i}}{\beta_{i}}\geq 0,\\\
&L_{3}^{i}=\beta_{i}T\frac{\sigma^{2}}{{h_{i}}^{2}}\geq 0,\\\
&L_{4}^{i}=\alpha_{i}TP_{0}^{s}\geq 0,\\\
&K_{1}=\eta\sum_{i=1}^{N}\alpha_{i}P_{0}^{s}T-E_{0}\geq 0,\\\
&K_{2}=\sum_{i=1}^{N}\gamma_{i}T\geq 0,\\\ \end{aligned}\right.$ (24)

Since $\Phi$ in (23) is a smooth function when $P^{r}\geq 0$, we prove the
quasi-concavity of (23) through Step 1 to Step 2 as follows.

###### Step 1

The problem in (23) can be decomposed as follows:

$\displaystyle\max_{i}\log(\Phi)$ (25)
$\displaystyle~{}~{}=\sum_{i=1}^{N}\log\left(\frac{\log(1+L_{1}^{i}P^{r}_{i})}{L_{3}^{i}((1+L_{1}^{i}P^{r}_{i})^{L_{2}^{i}}-1)+L_{4}^{i}}\right)+\log\left(K_{1}-K_{2}P^{r}_{1}\right),$
$\displaystyle~{}~{}s.t.~{}0\leq
P^{r}\leq\min\left\\{P_{0}^{r},\\{\frac{\sigma^{2}}{|g_{i}|^{2}}(1+\frac{P_{0}^{s}|h_{i}|^{2}}{\sigma^{2}})^{\frac{\beta_{i}}{\gamma_{i}}}-\frac{\sigma^{2}}{|g_{i}|^{2}}\\}\right\\},$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}P^{r}_{i}=P^{r}_{1}.$

###### Step 2

At this step, we prove the decomposed problem in (25) to be a quasi-concave
problem. The quasi-concavity of the norm
$\log\left(\frac{\log(1+L_{1}^{i}P^{r}_{i})}{L_{3}^{i}((1+L_{1}^{i}P^{r}_{i})^{L_{2}^{i}}-1)+L_{4}^{i}}\right)$
for variable $P^{r}_{i}$ is easy to test when $i\neq 1$. When $i=1$, we prove
the quasi-concavity of the objective $\Phi$ for variable $P^{r}_{1}$ through
proving Lemma 2 and Lemma 3 as follows:

###### Lemma 2

$\nabla_{P^{r}_{1}}\Phi\geq 0$ when $P^{r}_{1}=0$ and
$\nabla_{P^{r}_{1}}\Phi\leq 0$ when $P^{r}_{1}=\frac{K_{1}}{K_{2}}$, where
$\nabla_{P^{r}_{1}}\Phi$ is the first-order derivative of $\Phi$ for variable
$P^{r}_{1}$.

###### Proof:

$\nabla_{P^{r}_{1}}\Phi$ is calculated as follows:

$\displaystyle\nabla_{P^{r}_{1}}\Phi=\Phi\cdot$
$\displaystyle\Bigg{(}\frac{L_{1}^{1}}{(1+L_{1}^{1}P^{r})\log(1+L_{1}^{1}P^{r}_{1})}$
(26)
$\displaystyle-\frac{L_{3}^{1}L_{2}^{1}L_{1}^{1}(1+L_{1}^{1}P^{r}_{1})^{L_{2}^{1}}}{(L_{3}^{1}((1+L_{1}^{1}P^{r}_{1})^{L_{2}^{1}}-1)+L_{4}^{1})(1+L_{1}^{1}P^{r}_{1})}$
$\displaystyle-\Phi\cdot\frac{K_{2}}{K_{1}-K_{2}P^{r}_{1}}\Bigg{)}$

It is easy to test that $\nabla_{P^{r}_{1}}\Phi\geq 0$ when $P^{r}=0$ and
$\nabla_{P^{r}_{1}}\Phi\leq 0$ when $P^{r}=\frac{D_{1}}{D_{2}}$. ∎

###### Lemma 3

There does not exist any local minimum inflection point of $\Phi$ when
$0<P^{r}_{1}\leq\frac{K_{1}}{K_{2}}$. That is to say,
$\nabla^{2}_{P^{r}_{1}}\Phi\geq 0$ guarantees $\nabla_{P^{r}_{1}}\Phi\leq 0$,
where $\nabla^{2}_{P^{r}_{2}}\Phi\geq 0$ is the second-order derivative
function of $\Phi$.

###### Proof:

We rewrite $\nabla_{P^{r}_{1}}\Phi$ as follows:

$\small\left\\{\begin{aligned}
\nabla_{P^{r}_{1}}\Phi&=\Phi\cdot\left(\zeta_{1}-\frac{K_{2}}{K_{1}-K_{2}P^{r}_{1}}\right),\\\
\zeta_{1}&=\frac{L_{1}^{1}}{(1+L_{1}^{1}P^{r})\log(1+L_{1}^{1}P^{r}_{1})}\\\
&-\frac{L_{3}^{1}L_{2}^{1}L_{1}^{1}(1+L_{1}^{1}P^{r}_{1})^{L_{2}^{1}}}{(L_{3}^{1}((1+L_{1}^{1}P^{r}_{1})^{L_{2}^{1}}-1)+L_{4}^{1})(1+L_{1}^{1}P^{r}_{1})}.\\\
\end{aligned}\right.$ (27)

Thus, we calculate the $\nabla^{2}_{P^{r}_{1}}\Phi$ as follows:

$\small\begin{split}\nabla^{2}_{P_{1}^{r}}\Phi=&\Phi\cdot\left(\zeta_{1}-\frac{K_{2}}{K_{1}-K_{2}P^{r}_{1}}\right)^{2}\\\
&-\Phi\cdot\nabla_{P^{r}_{1}}\left(\zeta_{1}-\frac{K_{2}}{K_{1}-K_{2}P^{r}_{1}}\right)\\\
=&-\Phi\cdot\frac{L_{1}^{1}}{1+L_{1}^{1}P^{r}_{1}}\zeta_{1}\\\
&-2\Phi\cdot\frac{L_{3}^{1}L_{2}^{1}L_{1}^{1}(1+L_{1}^{1}P^{r}_{1})^{L_{2}^{1}}}{(L_{3}^{1}((1+L_{1}^{1}P^{r}_{1})^{L_{2}^{1}}-1)+L_{4}^{1})(1+L_{1}^{1}P^{r}_{1})}\zeta_{1}\\\
&-2\Phi\cdot\frac{K_{2}}{K_{1}-K_{2}P^{r}_{1}}\zeta_{1}\\\
&-\Phi\cdot\frac{L_{3}^{1}(L_{2}^{1})^{2}(L_{1}^{1})^{2}(1+L_{1}^{1}P^{r}_{1})^{L_{2}^{1}}}{(L_{3}^{1}((1+L_{1}^{1}P^{r}_{1})^{L_{2}^{1}}-1)+L_{4}^{1})(1+L_{1}^{1}P^{r}_{1})^{2}}\\\
\leq&-\Phi\cdot\frac{L_{1}^{1}}{1+L_{1}^{1}P^{r}_{1}}\zeta_{1}\\\
&-2\Phi\cdot\frac{L_{3}^{1}L_{2}^{1}L_{1}^{1}(1+L_{1}^{1}P^{r}_{1})^{L_{2}^{1}}}{(L_{3}^{1}((1+L_{1}^{1}P^{r}_{1})^{L_{2}^{1}}-1)+L_{4}^{1})(1+L_{1}^{1}P^{r}_{1})}\zeta_{1}\\\
&-2\Phi\cdot\frac{K_{2}}{K_{1}-K_{2}P^{r}_{1}}\zeta_{1}\\\
=&-\Phi\cdot\xi_{1}\cdot\zeta_{1},\\\ \end{split}$ (28)

where $\xi_{1}>0$ is given as follows:

$\displaystyle\xi_{1}$
$\displaystyle=\frac{L_{1}^{1}}{1+L_{1}^{1}P^{r}_{1}}\zeta_{1}+2\cdot\frac{L_{3}^{1}L_{2}^{1}L_{1}^{1}(1+L_{1}^{1}P^{r}_{1})^{L_{2}^{1}}}{(L_{3}^{1}((1+L_{1}^{1}P^{r}_{1})^{L_{2}^{1}}-1)+L_{4}^{1})(1+L_{1}^{1}P^{r}_{1})}$
(29) $\displaystyle+2\cdot\frac{K_{2}}{K_{1}-K_{2}P^{r}_{1}}.$

When $\nabla^{2}_{P^{r}_{2}}\Phi\geq 0$, we have
$-\Phi\cdot\xi_{1}\cdot\zeta_{1}\geq 0$. Since $\Phi>0$ and $\xi_{1}>0$, we
have $\zeta_{1}\leq 0$. Therefore, we have the inequation as follows:

$\displaystyle\nabla_{P^{r}_{1}}\Phi$
$\displaystyle=\Phi\cdot(\zeta_{1}-\frac{K_{2}}{K_{1}-K_{2}P^{r}_{1}})\leq\Phi\cdot\zeta_{1}\leq
0.$ (30)

Hence, we conclude that $\nabla^{2}_{P^{r}_{2}}\Phi\geq 0$ guarantees
$\nabla_{P^{r}_{1}}\Phi\leq 0$. ∎

According to the decomposition in Step 1, Lemma 2 and Lemma 3 in Step 2 and
the linear constraint in (23), we conclude that the problem in (9) is a quasi-
concave problem with variables $\\{P_{i}^{s0}\\}$ and $P^{r}$.

##  Appendix D Proof of Theorem 3

According to Lemma 1 and Corollary 1, we can rewrite the problem in (9) with
variables $\alpha$ and $\\{\gamma_{i}\\}$ as follows:

$\displaystyle\max_{\\{\alpha_{i}\\},\\{\gamma_{i}\\}}\Phi=\left(\prod_{P_{i}^{s1}=P_{0}^{s}}\frac{D_{1}^{i}\gamma_{i}}{D_{2}^{i}\alpha+D_{3}^{i}\gamma_{i}}\right)\cdot(F_{1}\alpha-
F_{2}-\sum_{i=1}^{N}D_{4}^{i}\gamma_{i}),$ (31)
$\displaystyle~{}~{}s.t.~{}\alpha+\sum_{i=1}^{N}(\gamma_{i}+D_{5}^{i}\gamma_{i})=1,$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}(\gamma_{i}+D_{5}^{i}\gamma_{i})T\geq\theta_{0}T,$

where $\\{D_{1}^{i}\\}$ to $\\{D_{5}^{i}\\}$ and $F_{1}$ to $F_{2}$ are
positive, which are given as follows:

$\left\\{\begin{aligned}
&D_{1}^{i}=T\log\left(1+P^{r}\frac{|g_{i}|^{2}}{\sigma^{2}}\right),\\\
&D_{2}^{i}=P_{i}^{s1}T,\\\
&D_{3}^{i}=P_{i}^{s0}T\frac{\log(1+P^{r}|g_{i}|^{2}/\sigma^{2})}{\log(1+P_{i}^{s0}|h_{i}|^{2}/\sigma^{2})},\\\
&D_{4}^{i}=P^{r}T,\\\
&D_{5}^{i}=\frac{\log(1+P^{r}|g_{i}|^{2}/\sigma^{2})}{\log(1+P_{i}^{s0}|h_{i}|^{2}/\sigma^{2})},\\\
&F_{1}=\eta\sum_{P_{i}^{s1}=P_{0}^{s}}P_{0}^{s}T|h_{i}|^{2},\\\
&F_{2}=E_{0},\\\ \end{aligned}\right.$ (32)

where $D_{1}^{i}$ to $D_{5}^{i}$ and $F_{1}$ to $F_{2}$ are independent from
$\alpha$ and $\\{\gamma_{i}\\}$ and $\beta_{i}=D_{5}^{i}\gamma_{i}$. Since
$\gamma_{i}$ is separable in (31), it is easy to test that $\Phi$ is a concave
function for each $\gamma_{i}$. Then, the constraint in (31) are linear
constraints. Thus, the problem in (31) is a quasi-concave problem when $\Phi$
is a quasi-concave function of $\alpha$, which we prove it in Lemma 4 as
follows:

###### Lemma 4

$\nabla_{\alpha}^{2}\Phi\geq 0$ guarantees $\nabla^{1}_{\alpha}\Phi<0$, where
$\nabla_{\alpha}^{i}$ is the $i^{th}$-order derivative with variable $\alpha$.
That is to say, $\Phi$ has no local minimum inflection point of $\alpha$,
which indicates that the problem (31) is a quasi-concave problem of $\alpha$.

###### Proof:

We can calculate $\nabla_{\alpha}^{2}\Phi$ and $\nabla^{1}_{\alpha}\Phi$,
respectively, as follows:

$\small\left\\{\begin{aligned}
&\nabla^{1}_{\alpha}\Phi=\left(-\sum_{P_{i}^{s1}=P_{0}^{s}}\mu_{i}+\nu\right)\cdot\Phi,\\\
&\nabla^{2}_{\alpha}\Phi=\left(-\sum_{P_{i}^{s1}=P_{0}^{s}}\mu_{i}+\nu\right)^{2}\cdot\Phi+\left(\sum_{P_{i}^{s1}=P_{0}^{s}}(\mu_{i})^{2}-\nu^{2}\right)\cdot\Phi,\\\
&\mu_{i}=\frac{G_{1}^{i}}{G_{1}^{i}\alpha+G_{2}^{i}}>0,~{}~{}\nu=\frac{H_{1}}{H_{1}\alpha-
H_{2}}>0,\\\ \end{aligned}\right.$ (33)

where $\\{G_{1}^{i}\\}$, $\\{G_{2}^{i}\\}$, $H_{1}$ and $H_{2}$ are
independent from $\alpha$, which are given as follows:

$\left\\{\begin{aligned} &G_{1}^{i}=D_{2}^{i}=D_{2}^{i}=P_{i}^{s1}T,\\\
&G_{2}^{i}=D_{3}\gamma_{i}=P_{i}^{s0}\gamma_{i}T\frac{\log(1+P_{i}^{s0}|h_{i}|^{2}/\sigma^{2})}{\log(1+P^{r}|g_{i}|^{2}/\sigma^{2})},\\\
&H_{1}=F_{1}=\eta\sum_{P_{i}^{s1}=P_{0}^{s}}P_{0}^{s}T|h_{i}|^{2},\\\
&H_{2}=F_{2}+\sum_{k=1}^{N}D_{4}^{k}\gamma_{k}=E_{0}+\sum_{k=1}^{N}D_{4}^{k}\gamma_{k}.\\\
\end{aligned}\right.$ (34)

Therefore, when $\nabla^{2}_{\alpha}\Phi\geq 0$, we have the inequation as
follows:

$\displaystyle 0\leq\nabla^{2}_{\alpha}\Phi$ (35)
$\displaystyle=\left(\left(\sum_{P_{i}^{s1}=P_{0}^{s}}\mu_{i}\right)^{2}+\sum_{P_{i}^{s1}=P_{0}^{s}}(\mu_{i})^{2}-2\left(\sum_{P_{i}^{s1}=P_{0}^{s}}\mu_{i}\right)\nu\right)\Phi$
$\displaystyle<\left(2\left(\sum_{P_{i}^{s1}=P_{0}^{s}}\mu_{i}\right)^{2}-2\left(\sum_{P_{i}^{s1}=P_{0}^{s}}\mu_{i}\right)\nu\right)\Phi$
$\displaystyle=-2\left(\sum_{P_{i}^{s1}=P_{0}^{s}}\mu_{i}\right)\cdot\nabla^{1}_{\alpha}\Phi\Rightarrow\nabla^{1}_{\alpha}\Phi<0.$

Thus, the problem in (31) is a quasi-concave problem of $\alpha$. Therefore,
the problem in (9) is a quasi-concave problem with variables $\alpha$,
$\\{\beta_{i}\\}$, and $\\{\gamma_{i}\\}$.

∎

##  Appendix E Gradient Ascent for Data Transmission Power Optimization

The Lagrangian function for data transmission power optimization problem is
given as follows:

$\displaystyle\mathcal{L}_{1}^{k}(P^{r},\rho_{1},\rho_{2})$ (36)
$\displaystyle=-\prod_{i=1}^{N}\left(\frac{\log(1+L_{1}^{i}P^{r})}{L_{3}^{i}((1+L_{1}^{i}P^{r})^{L_{2}^{i}}-1)+L_{4}^{i}}\right)\left(K_{1}-K_{2}P^{r}\right)$
$\displaystyle+\rho_{1}(0-P^{r})+\rho_{2}(P^{r}-[P_{0}^{r}]^{*}).$

Thus ,the dual function is calculated as follows [37][39]:

$\displaystyle
g_{1}^{k}(\rho_{1},\rho_{2})=\inf_{P^{r}}\mathcal{L}_{1}^{k}(P^{r},\rho_{1},\rho_{2})$
(37)
$\displaystyle=\prod_{i=1}^{N}\left(\frac{\log(1+L_{1}^{i}[P^{r}]^{\sharp})}{L_{3}^{i}((1+L_{1}^{i}[P^{r}]^{\sharp})^{L_{2}^{i}}-1)+L_{4}^{i}}\right)\left(K_{1}-K_{2}[P^{r}]^{\sharp}\right)$
$\displaystyle~{}~{}-\rho_{2}[P_{0}^{r}]^{*},$

where $[P^{r}]^{\sharp}$ equals to $-\rho_{2}$.222Since all variables and all
parts in the Lagrangian function are real, we eliminate all the conjugate
operations.

The dual problem is as follows:

$\max_{\rho_{1},\rho_{2}}g_{1}^{k}(\rho_{1},\rho_{2}),$ (38)

with variables $\rho_{1}$ and $\rho_{2}$. We can achieve a dual optimal point
$\\{\rho_{1}^{*},\rho_{2}^{*}\\}$ through solve the dual problem above. Since
the data transmission power optimization problem in (13) is a quasi-concave
optimization problem, instead of a strict concave optimization problem, we can
only ’possibly’ obtain a primal optimal point $(P^{r})^{*}$ from a dual
optimal point $\\{\rho_{1}^{*},\rho_{2}^{*}\\}$ as

$(P^{r})^{*}=\mbox{arg}\min_{P^{r}}L_{1}^{k}(P^{r},\rho_{1}^{*},\rho_{2}^{*}).\\\
$ (39)

Since the $g_{1}^{k}$ is differentiable when $\rho_{1}>0$ and $\rho_{2}>0$, we
can update the primal variable $P^{r}$ and the Lagrange multipliers $\rho_{1}$
and $\rho_{2}$ as follows [37]:

$\left\\{\begin{aligned}
&(P^{r})^{t+1}=\mbox{arg}\min_{P^{r}}L_{1}^{k}(P^{r},\rho_{1}^{t},\rho_{2}^{t}),\\\
&\rho_{1}^{t+1}=\rho_{1}^{t}+\varsigma_{1}^{t}\left(0-(P^{r})^{t+1}\right),\\\
&\rho_{2}^{t+1}=\rho_{2}^{t}+\varsigma_{2}^{t}\left((P^{r})^{t+1}-[P_{0}^{r}]^{*}\right),\\\
\end{aligned}\right.$ (40)

where $\varsigma_{1}^{t}>0$ and $\varsigma_{2}^{t}>0$ are the step variables
for the $t^{th}$ step.

The iteration ends (the gradient ascent algorithm converges) at $t_{1}^{th}$
iteration when the primal variable $P^{r}$ satisfies the linear stopping
criterion [37], which is given as follows:

$\small|(P^{r})^{t_{1}+1}-(P^{r})^{t_{1}}|<\varepsilon_{1},$ (41)

where $\varepsilon_{1}$ is a minor parameter to determine the convergence.

Finally, we obtain the optimal $(P^{r})^{*}$ in (39) as
$(P^{r})^{*}=(P^{r})^{t_{1}+1}$. We emphasize that since the data transmission
power optimization problem is not strictly concave, we can obtain a suboptimal
solution with dual ascent method.

##  Appendix F Gradient Ascent for Time Division

Similarly with the gradient ascent method in Appendix E, we can update the
time division variables $\alpha$, $\\{\gamma_{i}\\}$ and the corresponding
Lagrange multipliers $\varrho_{1}$ and $\\{\varrho_{2}^{i}\\}$ in (16) as
follows:

$\left\\{\begin{aligned}
&(\alpha,\\{\gamma_{i}\\})^{t+1}=\mbox{arg}\min_{\alpha,\\{\gamma_{i}\\}}L_{2}^{k}(\alpha,\\{\gamma_{i}\\},\varrho_{1}^{t},(\\{\varrho_{2}^{i}\\})^{t}),\\\
&\varrho_{1}^{t+1}=\varrho_{1}^{t}+\tau_{1}^{t}\left(\alpha^{t+1}+\sum_{i=1}^{N}(\gamma_{i}^{t+1}+D_{5}^{i}\gamma_{i}^{t+1})-1\right),\\\
&(\varrho_{2}^{i})^{t+1}=(\varrho_{2}^{i})^{t}+(\tau_{2}^{i})^{t}\cdot\left(\theta_{0}T-(\gamma_{i}^{t+1}+D_{5}^{i}\gamma_{i}^{t+1})T\right),\\\
\end{aligned}\right.$ (42)

where $\tau_{1}^{t}>0$ and $\\{(\tau_{2}^{i})^{t}>0\\}$ are the step variables
for the $t^{th}$ step. Similar with the gradient ascent method in Appendix E,
we can set a primal stopping criterion when the time division converges at
$t_{2}^{th}$ iteration, which is given as follows:

$|\alpha^{t_{2}+1}-\alpha^{t_{2}}|<\varepsilon_{2},$ (43)

where $\varepsilon_{2}$ is a minor parameter to determine the convergence.

To avoid misleading the time division process, we obtain the
$(\alpha,\\{\gamma_{i}\\})^{t+1}$ at each $t^{th}$ iteration, through
alternatively update of $\alpha$ and $\\{\gamma_{i}\\}$, which is given as
follows:

$\left\\{\begin{aligned}
&\alpha^{\vartheta+1}=\alpha^{\vartheta}-(\kappa_{1})^{\vartheta}\nabla_{\alpha}L_{2}^{k}(\alpha,(\\{\gamma_{i}\\})^{\vartheta},\varrho_{1}^{t},(\\{\varrho_{2}^{i}\\})^{t}),\\\
&\gamma_{i}^{\vartheta+1}=\gamma_{i}^{\vartheta}-(\kappa_{2}^{i})^{\vartheta}\nabla_{\gamma_{i}}L_{2}^{k}(\alpha^{\vartheta},(\\{\gamma_{j}\\})_{j\neq
i}^{\vartheta},\gamma_{i},\varrho_{1}^{t},(\\{\varrho_{2}^{i}\\})^{t}),\\\
\end{aligned}\right.$ (44)

where $(\kappa_{1})^{\vartheta}>0$ and $\\{(\kappa_{2}^{i})^{\vartheta}\\}>0$
are the step size. To guarantee the constraints in (15) are always satisfied,
the following equation must be satisfied at $\vartheta^{th}$ update:

$\small\left\\{\begin{aligned}
&(\kappa_{1})^{\vartheta}\cdot\nabla_{\alpha}L_{2}^{k}(\alpha,(\\{\gamma_{i}\\})^{\vartheta},\varrho_{1}^{t},(\\{\varrho_{2}^{i}\\})^{t})\\\
&+\sum_{i=1}^{N}(\kappa_{2}^{i})^{\vartheta}\cdot\nabla_{\gamma_{i}}L_{2}^{k}(\alpha^{\vartheta},(\\{\gamma_{j}\\})_{j\neq
i}^{\vartheta},\gamma_{i},\varrho_{1}^{t},(\\{\varrho_{2}^{i}\\})^{t})=0,\\\
&\gamma_{i}^{\vartheta}-(\kappa_{2}^{i})^{\vartheta}\nabla_{\gamma_{i}}L_{2}^{k}(\alpha^{\vartheta},(\\{\gamma_{j}\\})_{j\neq
i}^{\vartheta},\gamma_{i},\varrho_{1}^{t},(\\{\varrho_{2}^{i}\\})^{t})>\frac{\theta_{0}}{1+D_{5}^{i}}.\\\
\end{aligned}\right.$ (45)

When the update ends at $\vartheta_{1}^{th}$ update, we have

$\left\\{\begin{aligned} &\alpha^{t+1}=\alpha^{\vartheta_{1}+1},\\\
&\gamma_{i}^{t+1}=\gamma_{i}^{\vartheta_{1}+1},~{}\forall 1\leq i\leq N.\\\
\end{aligned}\right.$ (46)

When the gradient ascent method converges at $t_{2}^{th}$ iteration, we can
obtain the optimal values as $\alpha^{\star}$ and $(\\{\gamma_{i}\\})^{\star}$
for the time division problem in (15), which is as follows:

$\left\\{\begin{aligned} &\alpha^{\star}=\alpha^{t_{2}+1},\\\
&(\\{\gamma_{i}\\})^{\star}=(\\{\gamma_{i}\\})^{t_{2}+1}.\\\
\end{aligned}\right.$ (47)

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