Mode Selection and Resource Allocation Algorithm in Energy-Harvesting D2D Heterogeneous Network

With the rapid increase of smart mobile devices and the improvement of people’s demand for wireless communication, the insufficient system capacity and shortage of spectrum resources is gradually emerging. The current wireless communication technology is facing enormous challenges. Device-to-Device (D2D) technology has become a key technology in the future 5G network, with its flexible working mode, low energy consumption, low latency and high capacity, etc. At the same time, D2D communication devices are usually powered by batteries and have a limited lifetime. The lack of spectrum resources and the single power supply for D2D User Equipment (DUE) have become an important issue for mobile communication. Energy harvesting (EH) can provide the power supply. The joint problem of mode selection and resource allocation is challenging issues in energy harvesting D2D heterogeneous networks (EH-DHNs). In this paper, mode selection and resource allocation problem with DUEs multiplexing cellular user equipments (CUEs) uplink spectrum resources for EH-DHNs is investigated. Our object is to maximize the system throughput, and the energy harvesting constraint and the quality of service of CUEs are considered. In order to tackle these issues. The mode selection and resource allocation system throughput maximization problem in EH-DHN is formulated. The formulated problem is solved by the convex optimization theory and greedy strategy. We proposed a Mode Selection and Resource Allocation algorithm (MSRA). In order to illustrate the advantage of the MSRA algorithm, we compared it with other algorithms. We also analyzed the convergence of the MSRA and the performance with different system configurations on system throughput. Comparison shows that the MSRA algorithm can improve system throughput effectively.

SECTION I. Introduction

With the increasing of mobile devices and mobile data traffic, cellular networks are facing challenging issues for high data rate. Device-to-Device (D2D) communications share cellular resources, which have been considered as a promising technology to alleviate the challenging issues [1]. In D2D communications, D2D User Equipments (DUEs) communicate with each other directly instead of the interaction of the Base Stations (BSs) [2]. The advantage is that communications can be possible in poorly covered areas, and the spectrum efficiency and energy efficiency can be improved [3], [4].

Due to the battery lifetime budget of DUEs, the system throughput is limited. In order to prolongate the lifetime of D2D heterogeneous networks, Energy Harvesting (EH) has been applied to D2D heterogeneous networks, which can further improve the spectrum efficiency of D2D heterogeneous networks by using the EH capability to recharge DUEs [5]. DUEs first harvest energy from various kinds of energy sources, and then use the harvested energy multiplex the spectrum resource for D2D communication in EH-based D2D heterogeneous networks (EH-DHNs).

The strategies of mode selection and resource allocation are considered as an efficient way to improve the spectrum efficiency in D2D communication. However, owing to the DUEs with EH capability, mode selection and resource allocation are facing new challenging issues in EH-DHNs.

There are optimized couplings between the amount of the harvested energy or the duration of EH and spectrum efficiency [6], which means that, even if more energy is harvested, higher spectrum efficient is not necessarily obtained. If the DUEs are exhausted or with very low energy, the DUEs could not multiplex the spectrum resources of the CUEs. For the purpose of maximizing the spectrum efficiency and guaranteeing that the DUEs can be supplied with sufficient energy for communication in EH-DHNs, the optimal transmission time duration and the EH time duration for D2D communication should be obtained. Therefore, the joint problem of mode selection and resource allocation with the goal of maximizing spectrum efficiency in EH-DHNs is investigated in this paper. The resource allocation problem involves EH time duration allocation of DUEs, transmission power allocation and spectrum resource allocation, and these three factors are inter-coupled with each other.

In recent years, various DUEs mode selection and resource allocation schemes and algorithms for D2D communication have been proposed for different purposes [7].

From the point of view of interference management, The joint problem of power control and channel allocation in D2D cellular network is studied. Two distributed power control algorithms for link establishment and a distributed adaptive power control scheme for link maintenance are proposed in [8]. In [9], the power control and channel allocation problem with the scenario of D2D links sharing resources with multiple cellular users is studied, the mathematical tool of stochastic geometry is used to solve the problem. In [10], the joint of physical-layer secure transmission and resource allocation in D2D communications is studied. The objective is to maximize system sum rate in a socially aware D2D network, and each D2D pair can access multiple CUEs’ resources. Moreover, the multiple D2D pairs can share single CUE resources in the scenario of multiple eavesdroppers. In [11], underlaying a non-orthogonal multiple access (NOMA) cellular network, the power control and channel allocation problem in D2D communications is studied. The goal is to maximize the sum rate of D2D pairs while guaranteeing the minimum rate requirements of NOMA-based cellular users. In [12], in the scenario of a cell with one D2D pair and multiple CUs, under the condition of partial complete channel state information, a joint mode selection, user scheduling, and rate adaptation policy problem is investigated. In [13], the joint problem of mode selection and resource allocation for D2D-enabled NOMA cellular networks is investigated. In order to maximize the system sum rate while meeting the successive interference cancellation (SIC) decoding constraint, a joint D2D mode selection and resource allocation scheme with interlay mode was proposed. In [14], a SINR-aware mode selection analytical framework based on the markov-chain theory is developed for single-user case. A joint MS, resource allocation (RA), and scheduling optimization problem is investigated for multi-user communication scenario.

The common point of above existing mode selection and resource allocation algorithms in D2D networks is that the EH function of DUE is not considered. The optimal coupling relationship among the spectrum efficiency, the EH time and the transmission time are not considered as well. The mode selection and resource allocation algorithms and schemes without considering the EH capability are not suit for EH-DHNs owing to the new challenging introduced by EH technology. Consequently, several research works have been done in this area so far.

For instance, in [15], the joint problem of the EH time allocation, spectrum resource allocation and transmission power allocation is investigated with the aim of maximizing the energy efficiency in EH-DHNs. An iterative algorithm based on Lagrangian multiplier is proposed to solve the formulated mixed-integer nonlinear constraints optimization problem. However, the mode selection is not considered. In [16], considering of DUEs RF harvesting energy for relay transmissions in machine-type communication, the spectral efficiency of energy harvesting-based D2D-assisted MTC communications using stochastic geometry is analyzed. In [17], with the aim of maximizing the sum rate of systems, the problem of D2D-CU matching, transmission time and power allocation in EH-DHNs is investigated. However, the mode selection is not considered as well. In [18], a joint problem of subcarrier assignment and power allocation in EH-DHNs with the goal of maximizing the overall sum rate is investigated. The sum rate is the sum of the data rate of DUEs and CUEs. In [19], the random spectrum access policy and prioritized spectrum access policy are investigate in the cognitive and energy harvesting D2D networks. The outage probability for D2D and cellular users is evaluated using stochastic geometry tool. In [20], the channel allocation and power allocation problem with the goal of maximization sum-rate of CUEs and DUEs in EH-powered D2D cellular network is investigated. In [21], the energy harvesting process of a spatial random EH-D2D network is analyzed. A Markov chain model for success probability of the proposed four schemes is developed. In [22], the EH time and power allocation problem with the objective of maximizing the throughput in unmanned aerial vehicle-assisted D2D networks is studied. In [23], with the goal of maximizing the data rate of the cognitive spectrum sharing-based D2D communication system, a joint time division and power splitting-based information and energy cooperation transmission scheme is studied. In [24], with the goal of maximizing the sum rate, the joint problem of resource allocation and power allocation in energy harvesting relay-aided D2D networks is investigated.

The spectrum efficiency and energy efficiency are very important goals for fog and edge computing [25]–​[29], cognitive radio [3], [30], internet of things [31]–​[37] and wireless powered communication networks [38], [39]. However, with the objective of maximizing the spectrum efficiency, the joint problem of mode selection and resource allocation in EH-DHNs has not been investigated yet. The contributions of our work can be summarized as follows.

1. We considered the spectrum efficiency mode selection and resource allocation problem in the uplink EH-DHNs.

2. We formulated the joint problem of mode selection, EH time duration allocation of DUEs, transmission power allocation and spectrum resource allocation with the objective of maximizing the system throughput.

3. We proposed a Mode Selection and Resource Allocation algorithm (MSRA) based on convex optimization theory and greedy strategy to solve the formulated problem.

The rest of this paper is organized as follows. Section II and Section III present the system model and problem formulation, respectively. In Section IV, the joint optimization algorithm including the equivalent transformation and the detail of the proposed algorithm is discussed. Section V shows the performance of the proposed algorithm. Finally, we conclude our work in Section VI.

SECTION II. System Model

We consider an uplink D2D heterogeneous network scenario, the coverage radius of the network is $r$ , there is a base station and $N_{d}$ pair of D2D devices, $D=\left \{{{1,2,\cdots,N_{d}} }\right \}$ in the network. Each D2D pair has a transmission device and a receiving device. Denote $C=\left \{{{1,2,\cdots,N_{c}} }\right \}$ as the uplink cellular users.

The system model is shown in Fig. 1. In order to simplify the description of the problem, we assume that each cellular user $i$ is assigned a cellular uplink channel $i$ in advance, and each cellular channel is orthogonal to each other, and the channels are represented by a set $CH=\left \{{{1,2,\cdots,N_{c}} }\right \}$ . We assume that the D2D devices are with energy harvesting feature. The DUEs can harvest the RF energy in the environment to work, and the energy arrival model obeys the Poisson distribution. In order to better illustrate the problem in this paper, the time slot mechanism is used in this paper. There are $t$ time slots, and the duration of each time slot is $\tau _{t}$ .

FIGURE 1.

System model.

Let $\rho _{i,j}^{t} \in \left \{{{0,1} }\right \}\left ({{i\in CH,j\in D,t\in M} }\right)$ indicates whether the $j$ -$th$ D2D link reuses the $i$ link spectrum resource block in the time slot $t$ . We assume that any D2D link can only reuse at most one spectrum resource block in the same time slot. The constraint is given by

$$\begin{equation*} \sum \limits _{i=1}^{N_{c}} {\rho _{i,j}^{t}} \le 1 \quad \forall t\in M, ~j\in D\tag{1}\end{equation*}$$

Meanwhile, we assume that any spectrum resource block can only be reused by one D2D link at most, formulated as

$$\begin{equation*} \sum \limits _{j=1}^{N_{d}} {\rho _{i,j}^{t}} \le 1 \quad \forall t\in M, ~i\in CH\tag{2}\end{equation*}$$

Let $V_{j}^{t}$ denote the average energy harvesting rate of the $j$ -$th$ D2D transmission device in the time slot $t$ , which can be expressed as

$$\begin{equation*} V_{j}^{t} \in Poisson\left ({\lambda }\right)\quad \forall t\in M, ~j\in D\tag{3}\end{equation*}$$

We define $\theta _{i,j}^{t}$ as the transmission time of the $j$ -$th$ D2D link reusing the $i$ resource block in the time slot $t$ , and denote $E_{j}^{t}$ as the total energy harvested by the D2D transmission device in the time slot $t$ , which is given by

$$\begin{equation*} E_{j}^{t} =\rho _{i,j}^{t} V_{j}^{t} \left ({{\tau _{t} -\theta _{i,j}^{t}} }\right)\quad \forall t\in M, ~j\in D, ~i\in CH\tag{4}\end{equation*}$$ where $\lambda$ represents the average energy arrival rate. We denote $x_{j}^{t}$ as the communication mode of the $j$ -$th$ D2D link in the time slot $t$ , which is expressed as $$\begin{equation*} x_{j}^{t} = \begin{cases} 1 & \textrm {D2D Mode} \\ 0 & \textrm {Cellular Mode} \\ \end{cases} \quad \forall t\in M, ~j\in D\tag{5}\end{equation*}$$

We denote $\left |{ {h_{i}} }\right |$ as the distance between the $i$ -$th$ CUE and the base station in the system, and denote $\left |{ {g_{j}} }\right |$ as the distance from the D2D transmission device to the base station, respectively. Let $\left |{ d }\right |$ denote the distance between the D2D pairs, and $\left |{ {h_{i,j}} }\right |$ denote the distance between the CUE and the D2D receiver. The D2D link reuses the cellular uplink channel resource for communication. The path loss of the channel gain and channel interference gain exponents are 3 and 4, respectively. The path loss is considered, without considering any fading in this paper, which is the same as the existing work in [15]. Let $\left |{ {h_{i}} }\right |^{-3}$ denote the channel gain between the $i$ -$th$ CUE device and the base station. Let $\left |{ {g_{j}} }\right |^{-3}$ denote the channel gain between the $j$ -$th$ D2D link and the base station, where the D2D device selects cellular mode communication. Denote $\left |{ {g_{j}} }\right |^{-4}$ as the interference gain from the $j$ -$th$ D2D transmission device to the $i$ -$th$ CUE link. Denote $\left |{ {h_{i}} }\right |^{-4}$ as the interference gain from the $i$ -$th$ CUE device to the $j$ -$th$ D2D link. We denote $\left |{ d }\right |^{-3}$ as the channel gain from the D2D transmission to the D2D receiver while the D2D mode communication is selected. We denote $\left |{ {h_{i,j}} }\right |^{-4}$ as interference gain from the CUE to the D2D receiver.

When the $j$ -$th$ D2D reuses $i$ spectrum block resource in time slot $t$ , the date rate of the $i$ -$th$ CUE can be expressed as

$$\begin{equation*} R_{i}^{t}\!=\! B\log _{2} \left ({\!{1+\frac {p_{i}^{t} \left |{ {h_{i}} }\right |^{-3}}{N_{0} B+p_{i,j}^{t} \left |{ {g_{j}} }\right |^{-4}}} \!}\right) \quad \forall t\!\in \! M, ~i\!\in \! CH\tag{6}\end{equation*}$$ where $B$ is the channel bandwidth, and $N_{0}$ is the noise power, $p_{i}^{t}$ and $p_{i,j}^{t}$ are the transmission power of the $i$ -$th$ CUE and the transmission power of the $j$ -$th$ DUE transmission device in the time slot $t$ .

When the D2D communication mode is used by the $j$ -$th$ D2D link, the data rate of the $j$ -$th$ D2D link can be expressed as

$$\begin{align*} R_{j,D}^{t}\!=\!\sum \limits _{i=1}^{N_{c}} {\rho _{i,j}^{t} B\log _{2} \left ({\!{1\!+\!\frac {p_{i,j}^{t} \left |{ d }\right |^{-3}}{N_{0} B\!+\!p_{i}^{t} \left |{ {h_{i,j}} }\right |^{-4}}}\! }\right)} \quad \forall t\!\in \! M, ~j\!\in \! D\!\! \\{}\tag{7}\end{align*}$$

However, when the cellular communication mode is used by the $j$ -$th$ D2D link, the data rate of the $j$ -$th$ D2D link can be given by

$$\begin{align*} R_{j,C}^{t} \!=\!\sum \limits _{i=1}^{N_{c}} {\rho _{i,j}^{t} B\log _{2} \left ({\!{1\!+\!\frac {p_{i,j}^{t} \left |{ {g_{j}} }\right |^{-3}}{N_{0} B+p_{i}^{t} \left |{ {h_{i}} }\right |^{-4}}}\! }\right)} \quad \forall t\!\in \! M, ~j\!\in \! D\!\!\!\! \\{}\tag{8}\end{align*}$$

SECTION III. Proble Formulation

We study the joint problem of spectrum resource block allocation, power control, mode selection and transmission time allocation in energy harvesting D2D heterogeneous networks. The goal is to maximize the throughput of all D2D links. We denote $R^{t}$ as the total throughput of all D2D links in the time slot $t$ , which is given by

$$\begin{equation*} R^{t}=\sum \limits _{j=1}^{N_{d}} {\left ({{x_{j}^{t} R_{j,C}^{t} +\left ({{1-x_{j}^{t}} }\right)R_{j,D}^{t}} }\right)}\quad \forall t\in M\tag{9}\end{equation*}$$ where $x_{j}^{t}$ , $R_{j,D}^{t}$ and $R_{j,C}^{t}$ are given in (4), (6), (7), respectively. We denote $\Re$ as the total throughput of the system in all time slots, which is expressed as $$\begin{equation*} \Re =\sum \limits _{t=1}^{M} {R^{t}}\tag{10}\end{equation*}$$

The joint optimization problem of maximizing the total throughput of all D2D links within the time $M$ is formulated as

$$\begin{align*}&P1\max \limits _{\rho _{i,j}^{t},x_{j}^{t},p_{i,j}^{t},\theta _{i,j}^{t},p_{i}^{t}} \Re \tag{11a}\\&\qquad \quad ~\mathrm {Subject~to} \\&\hphantom {\qquad \quad ~}R_{i}^{t} =B\log _{2} \left ({{1+\frac {p_{i}^{t} \left |{ {h_{i}} }\right |^{-3}}{N_{0} B+p_{i,j}^{t} \left |{ {g_{j}} }\right |^{-4}}} }\right)\ge \delta _{i}^{\min } \\&\hphantom {\qquad \quad ~}\forall t\in M,\quad i\in CH\tag{11b}\\&\hphantom {\qquad \quad ~}\sum \limits _{i=1}^{N_{c}} {\rho _{i,j}^{t}} \le 1\quad \forall t\in M, ~j\in D\tag{11c}\\&\hphantom {\qquad \quad ~}\sum \limits _{j=1}^{N_{d}} {\rho _{i,j}^{t}} \le 1\quad \forall t\in M, ~i\in CH\tag{11d}\\&\hphantom {\qquad \quad ~}\sum \limits _{\tau =1}^{t} {\sum \limits _{i=1}^{Nc} {\rho _{i,j}^{\tau } p_{i,j}^{\tau } \theta _{i,j}^{\tau }}} \le E_{j}^{0} +\sum \limits _{\tau =1}^{t-1} {E_{j}^{\tau }} \\&\hphantom {\qquad \quad ~}\forall t\in M,\quad j\in D, ~i\in CH\tag{11e}\\&\hphantom {\qquad \quad ~}\sum \limits _{i=1}^{N_{c}} {\rho _{i,j}^{t} \theta _{i,j}^{t}} \le \tau _{t}\quad \forall t\!\in \! M,~j\!\in \! D,~i\!\in \! CH\tag{11f}\\&\hphantom {\qquad \quad ~}x_{j}^{t} \in \left \{{{0,1} }\right \},\quad \rho _{i,j}^{t} \in \left \{{{0,1} }\right \}, ~p_{i}^{t} \ge 0, p_{i,j}^{t} \ge 0, \\&\hphantom {\qquad \quad ~}0\le \theta _{i,j}^{t} \le \tau _{t}\quad \forall t\in M, ~i\in CH, ~j\in D,~i\in C \\{}\tag{11g}\end{align*}$$

Equation (11b) indicates that the QoS constraints of each cellular user should be greater than $\delta _{i}^{\min }$ in any time slot $t$ . Equation (11c) and (11d) are the constraints of spectrum resource block. Equation (11c) represents each D2D link can multiplex one spectrum resource block of CUE at most. Equation (11d) means that each spectrum resource blocks can be multiplexed by one D2D link at most. Equation (11e) shows that the energy used by the $j$ -$th$ D2D transmission device in the time slot $t$ cannot exceed the remaining energy at the start of the time slot $t$ . We assume that the DUEs transmit data first in the each time slot, then harvest energy in the remaining time slot in this paper. $E_{j}^{0}$ is the initial energy for each DUE. $\sum \limits _{\tau =1}^{t-1} {E_{j}^{\tau }}$ is the total harvested energy from time slot 1 to time slot ${t}$ -1. Equation (11f) means that the time of transmitting the information cannot exceed the slot length $\tau _{t}$ , in case of the D2D transmission device multiplexes the spectrum resource block of the $i$ -$th$ CUE link.

SECTION IV. Joint Optimization Algorithm

A. Equivalent Transformation

Since $p_{i,j}^{t}$ , $\theta _{i,j}^{t}$ and $p_{i}^{t}$ are continuous variables, and $\rho _{i,j}^{t}$ and $x_{j}^{t}$ are binary variables, the problem (11) is a mixed integer nonlinear programming problem. In order to solve the formulated problem. The problem (11) is transformed into an equivalent form. We assume that the $j$ -$th$ D2D link is multiplexed the spectrum resource block of the $i$ -$th$ CUE in the time slot $t$ . From equation (11b), we can obtain

$$\begin{equation*} p_{i}^{t} \ge \left |{ {h_{i}} }\right |^{3} z_{i} \left ({{p_{i,j}^{t} \left |{ {g_{j}} }\right |^{-4}+N_{0} B} }\right)\tag{12}\end{equation*}$$ where $z_{i} =2^{\delta _{i}^{\min } B^{-1}}-1$ .

Since the objective function (11a) is monotonically decreasing in $p_{i}^{t}$ for a fixed $p_{i,j}^{t}$ . In order to obtain the maximum throughput, the optimal transmission power of the CUE $p_{i}^{t\ast }$ should be given as

$$\begin{equation*} p_{i}^{t\ast }=\left |{ {h_{i}} }\right |^{3} z_{i} \left ({{p_{i,j}^{t} \left |{ {g_{j}} }\right |^{-4}+N_{0} B} }\right)\tag{13}\end{equation*}$$

We can modify the objective function (11a) from equation (13) by substituting $p_{i}^{t\ast }$ , then we get an equivalent problem form of problem P1, which is given by

$$\begin{align*}&\Re ^{\ast }=\max \limits _{X} \sum \limits _{t=1}^{M} {R^{t}}\tag{14a}\\&\qquad \quad \mathrm {Subject~to} \\&\hphantom {\qquad \quad }\sum \limits _{i=1}^{N_{c}} {\rho _{i,j}^{t}} \le 1\quad \forall t\in M,~j\in D\tag{14b}\\&\hphantom {\qquad \quad }\sum \limits _{j=1}^{N_{d}} {\rho _{i,j}^{t}} \le 1\quad \forall t\in M,~i\in CH\tag{14c}\\&\hphantom {\qquad \quad }\sum \limits _{\tau =1}^{t} {\sum \limits _{i=1}^{Nc} {\rho _{i,j}^{\tau } p_{i,j}^{\tau } \theta _{i,j}^{\tau }}} \le E_{j}^{0} +\sum \limits _{\tau =1}^{t-1} {E_{j}^{t}} \\&\hphantom {\qquad \quad } \forall t\in M,\quad j\in D,~i\in CH\tag{14d}\\&\hphantom {\qquad \quad }\sum \limits _{i=1}^{N_{c}} {\rho _{i,j}^{t} \theta _{i,j}^{t}} \le \tau _{t}\quad \forall t \!\in \! M,~j \! \in \! D,~i\!\in \! CH\tag{14e}\\&\hphantom {\qquad \quad }x_{j}^{t} \in \left \{{{0,1} }\right \}, \rho _{i,j}^{t} \in \left \{{{0,1} }\right \}, p_{i}^{t} \ge 0, p_{i,j}^{t} \ge 0, \\&\hphantom {\qquad \quad }0\le \theta _{i,j}^{t} \le \tau _{t}\quad \forall t\in M, ~i\in CH, ~j\in D, ~i\in C \\{}\tag{14f}\end{align*}$$ where $$\begin{equation*} \begin{cases} \alpha _{i} =N_{0} B\left ({{1+\left |{ {h_{i}} }\right |^{-1} z_{i}} }\right), \\ u_{i,j} =\left |{ {h_{i}} }\right |^{-1} z_{i} \left |{ {g_{j}} }\right |^{-4}, \\ e_{i,j} =\left |{ {h_{i}} }\right |^{-1}\left |{ {g_{j}} }\right |^{-4} z_{i} +\left |{ {g_{j}} }\right |^{-3}, \\ \end{cases} \begin{cases} k_{i,j} = N_{0} B\\ \left ({{1\!+\!\left |{ {h_{i}} }\right |^{3} \!z_{i} \left |{ {h_{i,j} } }\right |^{-4}} }\right), \\ s_{i,j} = \left |{ {h_{i}} }\right |^{3} \!z_{i} \left |{ {h_{i,j}} }\right |^{-4} \\ \left |{ {g_{j}} }\right |^{-4}, \\ m_{i,j} = \left |{ d }\right |^{-3}\\ \!+\left |{ {h_{i}} }\right |^{3}\! z_{i}\! \left |{ {h_{i,j}} }\right |^{-4}\!\left |{ {g_{j}} }\right |^{-4}\!. \\ \end{cases}\end{equation*}$$

It can be seen that in question (14), the optimization variable is reduced to $X=\left \{{{p_{i,j}^{t},\rho _{i,j}^{t},x_{j}^{t},\theta _{i,j}^{t}} }\right \}$ . Since the variables $\rho _{i,j}^{t}$ and $x_{j}^{t}$ are binary, the problem is not easy to solve, so we will relax the variables $\rho _{i,j}^{t}$ and $x_{j}^{t}$ as continuous variables in the interval $\left [{ {0,1} }\right]$ .

Theorem 1:

$\Re \left ({X }\right)$ is convex jointly in variables $\left \{{{p_{i,j}^{t},x_{j}^{t}} }\right \}$ .

Proof:

Please refer to Appendix A.

In order to solve the problem, we introduce the variables $\Upsilon _{i,j}^{t} =\rho _{i,j}^{t} p_{i,j}^{t}$ , $\textrm {X}_{i,j}^{t} =\rho _{i,j}^{t} x_{j}^{t}$ , and replace the variables in the equivalence problem (14), and transform the original problem into (15a), which is expressed as

$$\begin{align*}&\max \limits _{X} \sum \limits _{t=1}^{M} \sum \limits _{j=1}^{N_{d}} \sum \limits _{i=1}^{N_{c}} w_{j}^{t} \rho _{i,j}^{t} B \\&\qquad \times \,\left [{ {\begin{array}{l} \dfrac {\textrm {X}_{i,j}^{t}}{\rho _{i,j}^{t}}\log _{2} \left ({{\dfrac {\alpha _{i} +e_{i,j} \frac {\Upsilon _{i,j}^{t}}{\rho _{i,j}^{t}}}{\alpha _{i} +u_{i,j} \frac {\Upsilon _{i,j}^{t}}{\rho _{i,j}^{t}}}} }\right) \\ +\left ({{1-\dfrac {\textrm {X}_{i,j}^{t}}{\rho _{i,j}^{t}}} }\right)\log _{2} \left ({{\dfrac {k_{i,j} +m_{i,j} \frac {\Upsilon _{i,j}^{t}}{\rho _{i,j}^{t}}}{k_{i,j} +s_{i,j} \frac {\Upsilon _{i,j}^{t}}{\rho _{i,j}^{t}}}} }\right) \\ \end{array}} }\right] \qquad \quad \tag{15a}\\&\mathrm {Subject~to} \\&\hphantom {\mathrm {Subject~to}}\sum \limits _{i=1}^{N_{c}} {\rho _{i,j}^{t}} \le 1\quad \forall t\in M, ~j\in D\tag{15b}\\&\hphantom {\mathrm {Subject~to}}\sum \limits _{j=1}^{N_{d}} {\rho _{i,j}^{t}} \le 1\quad \forall t\in M, ~i\in CH\tag{15c}\\&\hphantom {\mathrm {Subject~to}}\sum \limits _{\tau =1}^{t} {\sum \limits _{i=1}^{Nc} {\Upsilon _{i,j}^{t} \theta _{i,j}^{\tau }}} \le E_{j}^{0} +\sum \limits _{\tau =1}^{t-1} {E_{j}^{t}} \\&\hphantom {\mathrm {Subject~to}}\forall t\in M,\quad j\in D, ~i\in CH\tag{15d}\\&\hphantom {\mathrm {Subject~to}}\sum \limits _{i=1}^{N_{c}} {\rho _{i,j}^{t} \theta _{i,j}^{t}} \le \tau _{t}\quad \forall t\in M,~j\in D,~i\in CH \\{}\tag{15e}\end{align*}$$

Theorem 2:

The objective function (15a) is a convex function on the variable $\left \{{{\rho _{i,j}^{t},p_{i,j}^{t},x_{j}^{t}} }\right \}$

Proof:

Please refer to Appendix B.

Similar to Theorem 2, all constraints in the transformed problem (15) are jointly convex, so the original problem (14) is a jointly convex optimization problem.

B. Proposed Algorithm

Since the transformed problem (14) is a jointly convex problem, the convex optimization theory can be used to solve the problem. According to the objective function (14a) and the constraints (14b), (14c), (14d), (14e), we can obtain its Lagrangian equation, which is given by

$$\begin{align*} L\left ({{\textrm {X},\Lambda } }\right)=&-Max\sum \limits _{t=1}^{M} {\sum \limits _{j=1}^{N_{d}} {\sum \limits _{i=1}^{N_{c}} {w_{j}^{t}}}} B\rho _{i,j}^{t} \\&\times \,\left [{ {\begin{array}{l} x_{j}^{t} \log _{2} \left ({{\dfrac {\alpha _{i} +e_{i,j} \times p_{i,j}^{\tau }}{\alpha _{i} +u_{i,j} \times p_{i,j}^{\tau }}} }\right) \\ + \left ({{1-x_{j}^{t}} }\right)\log _{2} \left ({{\dfrac {k_{i,j} +m_{i,j} \times p_{i,j}^{\tau }}{k_{i,j} +s_{i,j} \times p_{i,j}^{\tau }}} }\right) \\ \end{array}} }\right] \\&+ \sum \limits _{t=1}^{M} {\sum \limits _{i=1}^{N_{c}} {\lambda _{1,i}^{t}} \left ({{\sum \limits _{j=1}^{N_{d}} {\rho _{i,j}^{t}} -1} }\right)} \\&+ \sum \limits _{t=1}^{M} {\sum \limits _{j=1}^{N_{d}} {\lambda _{2,j}^{t}} \left ({{\sum \limits _{i=1}^{N_{c}} {\rho _{i,j}^{t}} -1} }\right)} \\&+\sum \limits _{t=1}^{M} {\sum \limits _{j=1}^{N_{d}} {\lambda _{3,j}^{t}} \left ({\! {\sum \limits _{\tau =1}^{t} {\sum \limits _{i\in CH} {\rho _{i,j}^{\tau } p_{i,j}^{\tau } \theta _{i,j}^{\tau }}} \!-\!E_{j}^{0} \!-\!\!\sum \limits _{\tau =1}^{t-1} {E_{j}^{\tau }}}\! }\right)} \\&+\sum \limits _{t=1}^{M} {\sum \limits _{j=1}^{N_{d}} {\lambda _{4,j}^{t}}} \left ({{\sum \limits _{i\in CH} {\rho _{i,j}^{t} \theta _{i,j}^{t}} -\tau _{t}} }\right)\tag{16}\end{align*}$$ where $\lambda _{1,i}^{t} \ge 0$ , $\lambda _{2,j}^{t} \ge 0$ , $\lambda _{3,j}^{t} \ge 0$ and $\lambda _{4,j}^{t} \ge 0$ represent the Lagrangian multipliers corresponding to the constraints (14b), (14c), (14d) and (14e), respectively. Then we obtain the dual function $$\begin{equation*} D\left ({\Lambda }\right)=\min \limits _{\textrm {X}} L\left ({{\textrm {X},\Lambda } }\right)\tag{17}\end{equation*}$$ where $\Lambda =\left ({{\lambda _{1,i}^{t},\lambda _{2,j}^{t},\lambda _{3,j}^{t},\lambda _{4,j}^{t}} }\right)$ and $\textrm {X}=\left \{{{p_{i,j}^{t},\rho _{i,j}^{t},x_{j}^{t},\theta _{i,j}^{t}} }\right \}$ .

We assume that the $j$ -$th$ D2D link reuses the spectrum resource blocks of the $i$ -$th$ cellular link in the time slot $t$ . According to Karush-Kuhn-Tucker (KKT), we can obtain the optimal transmission power, which is expressed as

$$\begin{equation*} p_{i,j}^{t\ast }=\max \left [{ {0,\frac {\lambda _{4,j}^{t}}{\lambda _{3,j}^{t}}} }\right]\tag{18}\end{equation*}$$

According to the constraints (14b) and (14c), each D2D link can only multiplex one CUE spectrum resource block at most in the same time slot, and a CUE spectrum resource block can only be reused by a D2D link at most in the same time slot. At the same time, since the variables $\rho _{i,j}^{t}$ and $x_{j}^{t}$ are binary variables previously, we get the closed-form expression of the variable $\rho _{i,j}^{t\ast }$ according to the greedy strategy, which is given by

$$\begin{equation*} \rho _{i,j}^{t\ast }= \begin{cases} 1, & \left ({{i,j} }\right)=\mathop {\arg \max (\xi _{i,j}^{t})}\limits _{i\in SET\_{}CH,j\in SET\rho \_{}D} \\ 0, & otherwise \\ \end{cases}\quad \forall t\in M\tag{19}\end{equation*}$$

The set of all the available channels for D2D link multiplexing is $SET\_{}CH$ , and the set of D2D links without allocated channel is$SET_{\rho } \_{}D$ . The variable $\xi _{i,j}^{t}$ in equation (19) is expressed as (20).

$$\begin{equation*} \xi _{i,j}^{t}\!=\!\max \left ({{\log _{2} \frac {\alpha _{i} \!+\! e_{i,j} \times p_{i,j}^{t}}{\alpha _{i} \!+\! u_{i,j} \!\times \! p_{i,j}^{t}},\log _{2} \frac {k_{i,j} \!+\! m_{i,j} \!\times \! p_{i,j}^{t}}{k_{i,j} +s_{i,j} \!\times \! p_{i,j}^{t}}} }\right)\tag{20}\end{equation*}$$

Similarly, the closed-form expression of the variable $x_{j}^{t\ast }$ is given by

$$\begin{equation*} x_{j}^{t\ast }\!=\! \begin{cases} 1, & \rho _{i,j}^{t} \vartheta _{j}^{t} \!>\!0 \\ 0, & otherwise \\ \end{cases}\quad \forall t\!\in \! M, ~i\!\in \! CH,~j\!\in \! SET_{x} \_{}D\tag{21}\end{equation*}$$ where $SET_{x} \_{}D$ indicates all D2D link sets that do not have a selected communication mode, the variable $\vartheta _{j}^{t}$ is expressed as $$\begin{equation*} \vartheta _{j}^{t} =\log _{2} \frac {\alpha _{i} +e_{i,j} \times p_{i,j}^{t}}{\alpha _{i} +u_{i,j} \times p_{i,j}^{t}}-\log _{2} \frac {k_{i,j} +m_{i,j} \times p_{i,j}^{t}}{k_{i,j} +s_{i,j} \times p_{i,j}^{t}}\tag{22}\end{equation*}$$

After obtaining the D2D links transmission power $p_{i,j}^{t\ast }$ , the channel allocation variable $\rho _{i,j}^{t\ast }$ , and the D2D link communication mode selection variable $x_{j}^{t\ast }$ , the transmission time variable $\theta _{i,j}^{t\ast }$ can be obtained as follows:

$$\begin{equation*} \theta _{i,j}^{t} =\frac {w_{j}^{t} B\left [{ {x_{j}^{t} \left ({{g_{i,j}^{t}} }\right)^{\prime }+\left ({{1-x_{j}^{t}} }\right)\left ({{h_{i,j}^{t}} }\right)^{\prime }} }\right]}{\lambda _{3,j}^{t}}\tag{23}\end{equation*}$$ where: $$\begin{align*} \left ({{g_{i,j}^{t}} }\right)^{\prime }=&\frac {\partial }{\partial p_{i,j}^{t}}\left ({{\log _{2} \frac {\alpha _{i} +e_{i,j} \times p_{i,j}^{t}}{\alpha _{i} +u_{i,j} \times p_{i,j}^{t}}} }\right)\tag{24}\\ \left ({{h_{i,j}^{t}} }\right)^{\prime }=&\frac {\partial }{\partial p_{i,j}^{t}}\left ({{\log _{2} \frac {k_{i,j} +m_{i,j} \times p_{i,j}^{t}}{k_{i,j} +s_{i,j} \times p_{i,j}^{t}}} }\right)\tag{25}\end{align*}$$

The constraint (14f) will be satisfied during the problem solving steps. The remaining energy of the $j$ -$th$ D2D link in the time slot $t$ for the constraint (14d) is expressed as

$$\begin{align*}&\hspace {-0.5pc}F_{3,j}^{t} =E_{j}^{0} +\sum \limits _{\tau =1}^{t-1} {E_{j}^{\tau }} -\sum \limits _{\tau =1}^{t} {\sum \limits _{i\in CH} {\rho _{i,j}^{\tau } p_{i,j}^{\tau } \theta _{i,j}^{\tau }}} \ge 0 \\&\qquad \qquad \qquad \qquad \qquad ~ \quad {t\in M,\quad j\in D,~i\in CH}\tag{26}\end{align*}$$

The remaining energy $F_{3,j}^{t}$ should larger than zero.

The energy harvesting time of the $j$ -$th$ D2D link in the time slot $t$ for the constraint (14e) is expressed as

$$\begin{equation*} F_{4,j}^{t} \!=\!\tau _{t} \!-\!\sum \limits _{i\in CH} {\rho _{i,j}^{t} \theta _{i,j}^{t}} \ge 0 \quad t\in M,~j\in D,~i\in CH\tag{27}\end{equation*}$$

The energy harvesting time $F_{4,j}^{t}$ should larger than zero as well.

Due to the Lagrangian dual problem, the dual variable $\left ({{\lambda _{3,j}^{t},\lambda _{4,j}^{t}} }\right)$ is convex. While the constraints (14b), (14c) and (14f) will be satisfied in the solving process, we use equations (26) and (27) as boundaries to determine whether $\left ({{\lambda _{3,j}^{t},\lambda _{4,j}^{t}} }\right)$ is an interior point. In the iteration of the k-th finding interior point of each time slot, when $\rho _{i,j}^{t}$ and $x_{j}^{t}$ of each D2D link are determined, the original problem can be decomposed into $N_{d}$ sub-problems according to the D2D link due to the additional of the convex function. We denote $\beta _{j}$ as the step coefficient of finding the interior point of the j-th link, take a positive value less than 1, and $\varepsilon _{1}$ is the step coefficient that finding the interior point. For all D2D links, let $\Lambda _{k} =\left ({{\lambda _{3,j,k}^{t},\lambda _{4,j,k}^{t}} }\right)^{T}$ , in case of the j-th D2D link has a channel allocated, if one of $F_{3,j,k}^{t}$ and $F_{4,j,k}^{t}$ is negative, then the ($k+ 1$ )-th iteration is performed, then calculate $\Lambda _{m,k+1},m\in \left \{{{1,2,3,4} }\right \}$ , which is given by

$$\begin{equation*} \Lambda _{m,k+1} ={\begin{cases} \Lambda _{m=1,k+1} =\left [{ {{\begin{array}{c} {\lambda _{3,j,k+1}^{t}} \\ {\lambda _{4,j,k+1}^{t}} \\ \end{array}}} }\right]\\ =\left [{ {{\begin{array}{cc} {1+\beta _{j,k}} & 0 \\ 0 & {1+\beta _{j,k}} \\ \end{array}}} }\right]\left [{ {{\begin{array}{c} {\lambda _{3,j,k}^{t}} \\ {\lambda _{4,j,k}^{t}} \\ \end{array}}} }\right] \\ \Lambda _{m=2,k+1} =\left [{ {{\begin{array}{c} {\lambda _{3,j,k+1}^{t}} \\ {\lambda _{4,j,k+1}^{t}} \\ \end{array}}} }\right]\\ =\left [{ {{\begin{array}{cc} {1+\beta _{j,k}} & 0 \\ 0 & {\beta _{j,k})} \\ \end{array}}} }\right]\left [{ {{\begin{array}{c} {\lambda _{3,j,k}^{t}} \\ {\lambda _{4,j,k}^{t}} \\ \end{array}}} }\right] \\ \Lambda _{m=3,k+1} \\ =\left [{ {{\begin{array}{c} {\lambda _{3,j,k+1}^{t}} \\ {\lambda _{4,j,k+1}^{t}} \\ \end{array}}} }\right]\\ =\left [{ {{\begin{array}{cc} {\beta _{j,k}} & 0 \\ 0 & {1+\beta _{j,k})} \\ \end{array}}} }\right]\left [{ {{\begin{array}{c} {\lambda _{3,j,k}^{t}} \\ {\lambda _{4,j,k}^{t}} \\ \end{array}}} }\right] \\ \Lambda _{m=4,k+1} \\ =\left [{ {{\begin{array}{c} {\lambda _{3,j,k+1}^{t}} \\ {\lambda _{4,j,k+1}^{t}} \\ \end{array}}} }\right]\\ =\left [{ {{\begin{array}{cc} {\beta _{j,k}} & 0 \\ 0 & {\beta _{j,k})} \\ \end{array}}} }\right]\left [{ {{\begin{array}{c} {\lambda _{3,j,k}^{t}} \\ {\lambda _{4,j,k}^{t}} \\ \end{array}}} }\right] \\ \end{cases}}\tag{28}\end{equation*}$$ where $\Lambda _{m=1,k+1}$ means the increase of $\lambda _{3,j}^{t}$ and $\lambda _{4,j}^{t}$ , and $\Lambda _{m=2,k+1}$ means the increase of $\lambda _{3,j}^{t}$ and the decrease of $\lambda _{4,j}^{t}$ , and $\Lambda _{m=3,k+1}$ means the decrease of $\lambda _{3,j}^{t}$ and the increase of $\lambda _{4,j}^{t}$ , and $\Lambda _{m=4,k+1}$ means the decrease of $\lambda _{3,j}^{t}$ and $\lambda _{4,j}^{t}$ , respectively. According to $\Lambda _{m,k+1},m\in \left \{{{1,2,3,4} }\right \}$ , $\left ({{F_{3,j,k+1}^{t,m},F_{4,j,k+1}^{t,m}} }\right),m\in \left \{{{1,2,3,4} }\right \}$ is calculated according to equations (26) and (27) in turn. After the remaining energy $F_{3,j}^{t}$ and the energy harvesting time $F_{4,j}^{t}$ are calculated, the remaining energy constraint and The energy harvesting time constraint should be checked for each of $m\in \left \{{{1,2,3,4} }\right \}$ sequentially, which is expressed as $$\begin{align*}&(F_{3,j,k+1}^{t,m} >0\vert \vert (F_{3,j,k+1}^{t,m} >F_{3,j,k}^{t,m})) \\[8pt]&\& \& (F_{4,j,k+1}^{t,m} >0\vert \vert (F_{4,j,k+1}^{t,m} >F_{4,j,k}^{t,m})) \\[8pt]&\& \& (F_{4,j,k+1}^{t,m} \le \tau _{t})\tag{29}\end{align*}$$

We always have to ensure that equations (26) and (27) are satisfied for each link in any time slot. If $\left \{{{p_{i,j}^{t},\rho _{i,j}^{t},x_{j}^{t},\theta _{i,j}^{t}} }\right \}$ is not within the feasible domain due to $\left ({{\lambda _{3,j,k}^{t},\lambda _{4,j,k}^{t}} }\right)$ , then equations (26) or (27) is less than 0. If $\left \{{{p_{i,j}^{t},\rho _{i,j}^{t},x_{j}^{t},\theta _{i,j}^{t}} }\right \}$ is within the feasible domain due to $\left ({{\lambda _{3,j,k}^{t},\lambda _{4,j,k}^{t}} }\right)$ , then both equations (26) and (27) are greater than zero. The finding interior point algorithm is described in Algorithm 1.

Algorithm 1 Finding Interior Point Algorithm

Output:

$\lambda _{3,j,n}^{t}$ , $\lambda _{4,j,n}^{t}$

Initialize:

$k=1$ , $0< \beta _{j,k} < 1$ , $0< \varepsilon _{1} \ll 1$ , $j=1$ .

$\lambda _{3,j,k}^{t} =\lambda _{3,j,n}^{t};\lambda _{4,j,k}^{t} =\lambda _{4,j,n}^{t}$ ;

while $(F_{3,j,k}^{t} < 0\vert \vert F_{4,j,k}^{t} < 0$ )&&($\sum \limits _{i=1}^{N_{c}} {\rho _{i,j}^{t}} =1\boldsymbol{),}\forall j\in D$ do

if (($j\le N_{d}$ )&&($F_{3,j,k}^{t} < 0\vert \vert F_{4,j,k}^{t} < 0$ ) then

Calculate Eq. (28)

if (the $m^{\mathrm {th}}\Lambda _{m,k+1}$ makes Eq. (29) true) then

$\Lambda _{k+1} =\Lambda _{m,k+1}^{\ast }$ ; calculate $p_{i,j}^{t}$ , $\theta _{i,j}^{t}$ , $E_{j}^{t}$ ; $j=j+1$ ;

else

$\Lambda _{k+1} =\Lambda _{m,k}^{\ast };\beta _{j,k+1} =\beta _{j,k} \times \beta _{j,k}; j=j+1$ ;

end if

else if($\beta _{j,k+1} < \varepsilon _{1}$ ) then

$p_{i,j}^{t} =0$ ; calculate Eq. (19), (21), (23); $j=1$ ;

end if

end while

$\lambda _{3,j,n}^{t} =\lambda _{3,j,k}^{t};\lambda _{4,j,n}^{t} =\lambda _{4,j,k}^{t}$ ; j=1;

After the Interior point is obtained, we start to find the optimal value. Since (14a) is convex, the feasible falling direction detection method and the variable step size is used to solve the problem. Since the inner point has been found, the objective function is a concave function. In order to maximize the throughput, the direction in which the objective function increases is the feasible iteration direction. We set the convergence condition of the optimization iteration as:

$$\begin{equation*} \left |{ {\Re _{n+1}^{t} -\Re _{n}^{t}} }\right |< \varepsilon _{2}\tag{30}\end{equation*}$$ where $0< \varepsilon _{2} < 1$ is the convergence threshold of the objective function and $n$ is the number of optimization iterations. The convergence threshold $\varepsilon _{2}$ is $10^{-6}$ in the simulation. We denote $step_{j,n}, step_{j,n} >0$ as the iteration step size of each D2D link in the $n$ -th optimization iteration, and denote $\alpha$ as the step update coefficient. Let $\Lambda _{n} =\left ({{\lambda _{3,j,n}^{t},\lambda _{4,j,n}^{t}} }\right)^{T}$ , $\Lambda _{n+1,m},m\in \left \{{{1,2,3,4} }\right \}$ is calculated in next iteration, which is given by $$\begin{align*} \Lambda _{n+1,m} =\begin{cases} \Lambda _{n+1,m=1} =\left [{ {{\begin{array}{c} {\lambda _{3,j,n+1}^{t}} \\[5pt] {\lambda _{4,j,n+1}^{t}} \\[5pt] \end{array}}} }\right]=\left [{ {{\begin{array}{c} {\lambda _{3,j,n}^{t} +step_{j,n}} \\[5pt] {\lambda _{4,j,n}^{t} +step_{j,n}} \\[5pt] \end{array}}} }\right] \\[0.8pc] \Lambda _{n+1,m=2} =\left [{ {{\begin{array}{c} {\lambda _{3,j,n+1}^{t}} \\[5pt] {\lambda _{4,j,n+1}^{t}} \\[5pt] \end{array}}} }\right]=\left [{ {{\begin{array}{c} {\lambda _{3,j,n}^{t} -step_{j,n}} \\[5pt] {\lambda _{4,j,n}^{t} -step_{j,n}} \\[5pt] \end{array}}} }\right] \\[0.8pc] \Lambda _{n+1,m=3} =\left [{ {{\begin{array}{c} {\lambda _{3,j,n+1}^{t}} \\[5pt] {\lambda _{4,j,n+1}^{t}} \\[5pt] \end{array}}} }\right]=\left [{ {{\begin{array}{c} {\lambda _{3,j,n}^{t} +step_{j,n}} \\ {\lambda _{4,j,n}^{t} -step_{j,n}} \\ \end{array}}} }\right] \\[0.8pc] \Lambda _{n+1,m=4} =\left [{ {{\begin{array}{c} {\lambda _{3,j,n+1}^{t}} \\ {\lambda _{4,j,n+1}^{t}} \\ \end{array}}} }\right]=\left [{ {{\begin{array}{c} {\lambda _{3,j,n}^{t} -step_{j,n}} \\ {\lambda _{4,j,n}^{t} +step_{j,n}} \\ \end{array}}} }\right] \\ \end{cases}\!\!\!\!\!\!\!\!\!\! \\{}\tag{31}\end{align*}$$

After Eq.(31) is calculated, the corresponding $m$ target values $\Re _{n+1,m}^{t},m\in \left \{{{1,2,3,4} }\right \}$ is calculated according to $\Lambda _{n+1,m},m\in \left \{{{1,2,3,4} }\right \}$ . The Mode Selection and Resource Allocation (MSRA) algorithm based on convex optimization theory and greedy strategy is proposed in this section. Which is described in Algorithm 2.

Algorithm 2 Mode Selection and Resource Allocation algorithm

Input:

$h_{i}$ , $h_{j}$ , $h_{i,j}$ , $b$ , $r$ , $N_{0}$ , $V_{j}^{t}$ , $M$ , $\tau _{t}$ , $\delta _{i}^{\min }$ , $B$ ,

Output:

$p_{i,j}^{t}$ , $\rho _{i,j}^{t}$ , $x_{j}^{t}$ , $\theta _{i,j}^{t}$ , $p_{i}^{t}$

Initialize:

$\lambda _{3,j,n}^{t} =300$ , $\lambda _{4,j,n}^{t} =400$ , $0< step_{j,n}$ , $0< \varepsilon _{2} \ll 1$ , $0< \alpha < 1$ , $\Re _{n=0}^{t} =0$ , $n=1$ , $a=0$ , $j=1$ .

call Algorithm 1;

while $\left |{ {\Re _{n+1}^{t} -\Re _{n}^{t}} }\right |>\varepsilon _{2}$ do

if $j\le N_{d}$ then

Calculate Eq. (31), and $\Re _{n+1,m}^{t},m\in \left \{{{1,2,3,4} }\right \}$

if ($max\left ({{\Re _{n}^{t},\Re _{n+1,m}^{t}} }\right)=\Re _{n+1,m}^{t\ast })$ then

$\left ({{\lambda _{3,j,n+1}^{t},\lambda _{4,j,n+1}^{t}} }\right)^{T}=\Lambda _{n+1,m}^{\ast }; step_{j,n+1}=\left ({{1+\alpha } }\right)\times step_{j,n}$ ;

$a=a+1$ ;

else if($max\left ({{\Re _{n}^{t},\Re _{n+1,m}^{t}} }\right)=\Re _{n}^{t})$ then

$\left ({{\lambda _{3,j,n+1}^{t},\lambda _{4,j,n+1}^{t}} }\right)^{T}=\Lambda _{n,m}^{\ast }; step_{j,n+1}=\left ({{1-\alpha } }\right)\times step_{j,n}$ ;

calculate $p_{i,j}^{t}$ , $\theta _{i,j}^{t}$ , $E_{j}^{t}$ ;

end if

j=j+1;

else

calculate $p_{i,j}^{t}$ , $\rho _{i,j}^{t}$ , $x_{j}^{t}$ , $\theta _{i,j}^{t}$ , $E_{j}^{t}$ ;

if $a=0$ then

$step_{j,n+1} =\alpha \times step_{j,n},\forall j\in D$ ;

else

call Algorithm 1;

end if

end if

$j=1$ ;

end while

SECTION V. Simulation Results

A. Parameter Settings

As shown in Figure.1, the basic parameters of the generated scenario are shown in Table 1.

TABLE 1 Scenario Parameters

We will analyze the convergence of the proposed MSRA algorithm firstly. We will compare the system throughput with different number of D2D devices $N_{d}$ , different D2D device distances $d$ , different number of cellular channels $N_{c}$ , QoS, and energy arrival rates $\lambda$ as well, and also compare the proposed MSRA algorithm with MSRA-E algorithm. The transmission time in the MSRA-E algorithm is fixed $0.5\tau _{t}$ in this paper.

B. Convergence Analysis

In order to analyze the convergence of the proposed algorithm, we set up four groups of different parameters for comparison experiments, and compare and analyze the convergence energy of the MSRA algorithm. The same parameters for four group simulation are that D2D device pair distance $d=20m$ , $N_{c} =10$ , $N_{d} =10$ . The different simulation parameters of four groups are QoS, and energy arrival rate $\lambda$ . The first group is $QoS={12bps}/ {Hz}$ , $\lambda =3$ ; the second group is $QoS={20bps}/{Hz}$ , $\lambda =3$ ; the third group is $QoS={12bps}/ {Hz}$ , $\lambda =8$ ; the fourth group is $QoS={20bps}/{Hz}$ , $\lambda =8$ . The convergence curve of one time slot is selected in four groups randomly. The simulation results are shown in Figure 2.

FIGURE 2.

Algorithm convergence of D2D links sum rate in one slot (Mbps).

We can see from Figure 2. In the case of $QoS={12bps} / {Hz}$ , $\lambda =3$ , the algorithm convergence begins with 40 iterations; in the case of $QoS={20bps} / {Hz}$ , $\lambda =3$ , the algorithm convergence begins near 35 iterations; in the case of $QoS={12bps}/ {Hz}$ , $\lambda =8$ , the algorithm convergence begins near 35 iterations; in the case of $QoS={12bps}/ {Hz}$ , $\lambda =8$ , the algorithm convergence begins nearly 34 generations, respectively. It can be seen that the proposed algorithm can solve the joint problem of the transmission power, transmission time, mode selection and channel allocation in different network environments. It can quickly and effectively reach the convergence state to obtain the optimal throughput.

C. System Throughput With Different D2D Distance

In this section, the system throughput with different D2D distance are compared, and the proposed MSRA and the MSRA-E algorithm are compared as well. The simulation results are in Figure 3 and Figure 4.

FIGURE 3.

Comparison of throughput under different D2D distances, $\lambda =3$ .

FIGURE 4.

Comparison of throughput under different D2D distances, $\lambda =8$ .

Figure 3 and Figure 4 plot the system throughput with different D2D distances. The average energy arrival rate is $\lambda =3$ mJ/s and $\lambda = 8$ mJ/s, respectively. It can be seen that the system throughput decreases with D2D distances. This is because the channel gains decrease with the D2D distances, the system throughput decreases with the decreasing of the channel gains. The proposed MSRA can achieve higher system throughput than MSRA-E. The reason is that MSRA can obtain an optimal energy harvesting time, which is not $0.5\tau _{t}$ used by MSRA-E. MSRA with $N_{d} =10$ , $N_{c} =16$ and MSRA with $N_{d} =16$ , $N_{c} =10$ can obtain higher system throughput than MSRA with $N_{d} =10$ , $N_{c} =10$ . We can see that the higher system throughput can be achieved with more number of channels and the same number of D2D devices. This is because the more channel resource can be allocated. The higher system throughput can be obtained with more number of D2D devices and the same number of channels as well. However, MSRA with $N_{d} =16$ , $N_{c} =10$ can obtain higher system throughput than MSRA with $N_{d} =10$ , $N_{c} =16$ , which illustrates that the higher system throughput can be obtained through increasing the number of D2D links. As illustrated in Figure 3 and Figure 4, the higher system throughput can be obtained with higher energy arrival rate ($\lambda =8$ mJ/s). The reason is that the harvested energy increases with the energy arrival rate.

D. System Throughput With Different QoS

Figure 5 and Figure 6 compare the system throughput with different QoS thresholds. The average energy arrival rate is $\lambda = 3$ mJ/s and $\lambda =8$ mJ/s, respectively. The system throughput decreases with D2D distances. The reason is that the transmission power of CUE increases with the QoS constraints, and the interference from CUE to DUE increases with transmission power of CUE. The proposed MSRA can achieve higher system throughput than MSRA-E as well. The reason is that MSRA can obtain an optimal energy harvesting time. With the increasing of QoS thresholds, MSRA with $N_{d} =10$ , $N_{c} =16$ and MSRA with $N_{d} =16$ , $N_{c} =10$ can obtain higher system throughput than MSRA with $N_{d} =10$ , $N_{c} =10$ . We can see that the higher system throughput can be obtained with more number of channels and the same number of D2D devices as well. MSRA with $N_{d} =16$ , $N_{c} =10$ can obtain higher system throughput than MSRA with $N_{d} =10$ , $N_{c} =16$ as well with the same reason. Figure 5 and Figure 6 show that the higher system throughput can be achieved with higher energy arrival rate ($\lambda =8$ mJ/s).

FIGURE 5.

Comparison of throughput under different QoS, $\lambda =3$ .

FIGURE 6.

Comparisonof throughput under different QoS, $\lambda =8$ .

E. System Throughput With Different Number of Channels

Figure 7 and Figure 8 illustrate the system throughput with different number of channels. The D2D device distance is $d=20m$ and $d=40m$ , respectively. The proposed MSRA can obtain higher system throughput than MSRA-E with the same QoS thresholds. This is because MSRA can obtain an optimal energy harvesting time. It can be seen that system throughput of MSRA with Qos=12bps/Hz and MSRA-E with Qos=12bps/Hz increase with the number of channels. The reason is that the more channel resource can be allocated for DUE. MSRA with Qos=12bps/Hz can obtain higher system throughput than MSRA with Qos=20bps/Hz and MSRA with Qos=28bps/Hz, which shows that the system throughput increases with the decreasing of the QoS thresholds. The system throughput of MSRA with Qos=20bps/Hz increases with the number of channels from 8 to 10, and does not change at all after the number of channel equaling to 10. MSRA with Qos=28bps/Hz is with the same system throughput, which means that the system throughput does not change with higher QoS thresholds. Figure 7 and Figure 8 show that the higher system throughput can be obtained with shorter D2D device distance ($d=20m$ ).

FIGURE 7.

Comparison of throughput under different channel numbers, d=20m.

FIGURE 8.

Comparison of throughput under different channel numbers, d=40m.

F. System Throughput With Different Number of DUE

Figure 9 and Figure 10 show the system throughput with different number of D2D links. The average energy arrival rate is $\lambda = 3$ mJ/s and $\lambda = 8$ mJ/s, respectively. The proposed MSRA can obtain higher system throughput than MSRA-E with the same QoS thresholds. The reason is that MSRA can obtain an optimal energy harvesting time with different number of D2D links. As we can see from the figures, when QoS=12bps/Hz, the higher system throughput are obtained by MSRA and MSRA-E with the increasing of the number of D2D links. When QoS=20bps/Hz and the number of D2D links is less than 10, the system throughput increases with the number of D2D links. When it is larger than 10, the system throughput increases very slightly. When QoS=28bps/Hz, the system throughput increases very slightly all the time. Which means that the system throughput increases very little with higher QoS thresholds. MSRA with Qos=12bps/Hz can obtain higher system throughput than MSRA with Qos=20bps/Hz and MSRA with Qos=28bps/Hz. Which shows that the system throughput increases with the decreasing of the QoS thresholds under condition of different number of D2D links. Figure 9 and Figure 10 present that the higher system throughput can be obtained with higher energy arrival rate ($\lambda = 8$ mJ/s).

FIGURE 9.

Comparison of throughput with different number of D2D links, $\lambda =3$ .

FIGURE 10.

Comparison of throughput with different number of D2D links, $\lambda =8$ .

G. System Throughput With Different Energy Arrival Rate

Figure 11 and Figure 12 compare the system throughput with different energy arrival rates. The D2D device distance is $d=20m$ and $d =40m$ , respectively. The proposed MSRA can obtain higher system throughput than MSRA-E with the same QoS thresholds. This is because the optimal energy harvesting time can be obtained by MSRA with different energy arrival rates. From the four curves, MSRA with QoS=12bps/Hz and MSRA-E can obtain higher system throughput than MSRA with QoS=20bps/Hz and MSRA with QoS=28bps/Hz. The reason is that the transmission power of CUEs increase with the QoS thresholds, and the interference from CUEs to DUEs increase with the transmission power.

FIGURE 11.

Comparison of throughput at different energy arrival rates, d=20m.

FIGURE 12.

Comparison of throughput at different energy arrival rates, d=40m.

When QoS=12bps/Hz, the higher system throughput are obtained by MSRA and MSRA-E with the increasing of the energy arrival rates. When QoS=20bps/Hz, the higher system throughput obtained by MSRA increases very slightly all the time. When QoS=28bps/Hz, the system throughput is not changed at all. Figure 11 and Figure 12 show that the higher system throughput can be obtained with shorter D2D device distance ($d=20m$ ).

SECTION VI. Conclusion

In this paper, we investigated the joint problem of mode selection and resource allocation in EH-DHN. We formulated a maximizing system throughput problem, which is a mixed-integer optimization problem. The energy harvesting constraint and the quality of service of CUEs are considered. In order to handle the formulated problem. We proposed a Mode Selection and Resource Allocation algorithm (MSRA) based on Lagrangian dual decomposition. In order to illustrate the advantage of the MSRA algorithm, we compared it with other algorithms. We also analyzed the convergence of the MSRA and the performance with different system configurations on system throughput. Simulation results show that the MSRA algorithm can improve system throughput effectively.

Appendix

SECTION A. Proof of the Theorem 1

First, according to OF (14a), we define the function $R\left ({{p,\rho,x} }\right)$ as:

$$\begin{align*}R\left ({{p,\rho,x} }\right)=&\rho B\Biggl [{ x\log _{2} \left ({{\frac {\alpha +e\times p}{\alpha +u\times p}} }\right)} \\&\qquad \quad {{+\,\left ({{1-x} }\right)\log _{2} \left ({{\frac {k+m\times p}{k+s\times p}} }\right) }\Biggr ]}\tag{32}\end{align*}$$

Eq. (32) is a multivariate function about variables $\left ({{p,\rho,x} }\right)$ . We can prove (33) is a convex function about variables $\left ({{p,x} }\right)$ by proving inequality (34) is true.

$$\begin{align*}&\hspace {-0.5pc} R_{1} =\frac {1}{2}\left [{ {R(p_{1},x_{1})+R(p_{2},x_{2})} }\right] \\&\qquad \qquad \qquad \qquad {\le R\left ({{\frac {p_{1} +p_{2}}{2},\frac {x_{1} +x_{2}}{2}} }\right)=R_{2}}\tag{33}\end{align*}$$

First of all, we take two points $\left ({{p_{1},x_{1}} }\right)$ and $\left ({{p_{2},x_{2}} }\right)$ substitute Eq. (33). The left side of inequality is expressed as

$$\begin{align*}&\hspace {-0.5pc} R_{1} \!=\!\frac {\rho B}{2} \\&\times \,\left [{\!\! {\begin{array}{l} x_{1} \log _{2} \left ({{\dfrac {\alpha \!+\!e\!\times \! p_{1}}{\alpha \!+\!u\!\times \! p_{1}}} }\right)\!+\!\left ({{1\!-\!x_{1}} }\right)\log _{2} \left ({{\dfrac {k\!+\!m\!\times \! p_{1}}{k+s\!\times \! p_{1}}} }\right) \\ +x_{2} \log _{2} \left ({{\dfrac {\alpha \!+\!e\times p_{2}}{\alpha \!+\!u\!\times \! p_{2}}} }\right)\!+\!\left ({{1\!-\!x_{2}} }\right)\log _{2} \left ({{\dfrac {k\!+\!m\!\times \! p_{2}}{k\!+\!s\!\times \! p_{2}}} }\right) \\ \end{array}}\!\! }\right] \\{}\tag{34}\end{align*}$$

The right side of inequality is given by

$$\begin{align*} R_{2}\! =\!\frac {\rho B}{2}\left \{{\!\!{\begin{array}{c} \left ({{x_{1} +x_{2}} }\right)\log _{2} \left ({{\dfrac {2\alpha +e\times p_{1} +e\times p_{2}}{2\alpha +u\times p_{1} +u\times p_{2}}} }\right) \\ +\left ({{2-x_{1} -x_{2}} }\right)\log _{2} \left ({{\dfrac {2k+m\times p_{1} +m\times p_{2}}{2k+s\times p_{1} +s\times p_{2}}} }\right) \\ \end{array}}\!\! }\right \}\!\!\!\!\! \\{}\tag{35}\end{align*}$$

We take Eq. (34) minus Eq. (35), which can be expressed as

$$\begin{align*}&\hspace {-1pc}\frac {1}{2}\left [{ {\textrm {R}(p_{1},x_{1})+\textrm {R}(p_{2},x_{2})} }\right]-\textrm {R}\left ({{\frac {p_{1} \!+\!p_{2}}{2},\frac {x_{1} \!+\!x_{2}}{2}} }\right) \\=&\frac {\rho Bx_{1}}{2}\left ({{\log _{2} \left ({{\frac {\alpha \!+\! e\times p_{1}}{\alpha \!+\! u\!\times \! p_{1}}} }\right)\!-\!\log _{2} \left ({{\frac {2\alpha \!+\! e\!\times \! \left ({{p_{1} \!+\! p_{2}} }\right)}{2\alpha \!+\! u\!\times \! \left ({{p_{1} \!+\! p_{2}} }\right)}} }\right)} }\right) \\&+\,\frac {\rho Bx_{2}}{2}\left ({\! {\log _{2} \left ({\! {\frac {\alpha \!+\! e\times p_{2}}{\alpha \!+\! u\!\times \! p_{2}}} \!}\right)\!-\!\log _{2} \left ({\! {\frac {2\alpha \!+\! e\!\times \! \left ({{p_{1} \!+\! p_{2}} }\right)}{2\alpha \!+\!u\!\times \! \left ({{p_{1} \!+\! p_{2}} }\right)}} \!}\right)} \!}\right) \\&+\,\frac {\rho B\left ({{1-x_{1}} }\right)}{2}\Biggl ({\log _{2} \left ({{\frac {k+m \times p_{1}}{k+s \times p_{1}}} }\right)} \\&{-\, \log _{2} \left ({{\frac {2k+m \times \left ({{p_{1} + p_{2}} }\right)}{2k+s \times \left ({{p_{1} + p_{2}} }\right)}} }\right) }\Biggr ) \\&+\,\frac {\rho B\left ({{1-x_{2}} }\right)}{2}\Biggl ({\log _{2} \left ({{\frac {k+m \times p_{2}}{k+s \times p_{2}}} }\right)} \\&{-\,\log _{2} \left ({{\frac {2k+ m \times \left ({{p_{1} + p_{2}} }\right)}{2k+s \times \left ({{p_{1} + p_{2}} }\right)}} }\right) }\Biggr )\tag{36}\end{align*}$$

It can prove Eq. (33) is a concave function with variables $\left ({{p,x} }\right)$ through the Eq. (36). In [2], we can know that a concave function plus a concave function is still a concave function, therefore Eq. (14a) is a concave function about variables $\left \{{{p_{i,j}^{t},x_{j}^{t}} }\right \}$ .

SECTION B. Proof of the Theorem 2

It can be seen that each constraint in problem (15) is a convex function with variables $\left \{{{\rho ^{t}_{i,j},\Upsilon _{i,j}^{t},{\mathrm {X}}_{i,j}^{t}}}\right \}$ . And than, we know that $\Re \left ({{\Upsilon _{i,j}^{t},{\mathrm {X}}_{i,j}^{t}} }\right)$ is a concave function about variables $\left \{{{\Upsilon _{i,j}^{t},\textrm {X}_{i,j}^{t}} }\right \}$ , since $\Re \left ({{\rho _{i,j}^{t},\Upsilon _{i,j}^{t},{\mathrm {X}}_{i,j}^{t}} }\right)$ is the perspective function of $\Re \left ({{\Upsilon _{i,j}^{t},{\mathrm {X}}_{i,j}^{t}} }\right)$ [2]. Therefore, function $\Re$ is jointly concave in $\rho _{i,j}^{t}$ , $p_{i,j}^{t}$ and $x_{j}^{t}$ , it means Eq. (14a) preserves the concavity and convexity of the problem.