Optimal time allocation for throughput maximization in backscatter assisted wireless powered communication networks

Integrating backscatter communication (BackCom) into wireless powered communication networks (WPCNs) makes it possible to transmit information and harvest energy simultaneously, which is regarded as a promising method to enhance the throughput. A very important issue is how to allocate the charging time and transmitting time to efﬁciently utilize the energy in hybrid access point (HAP) so as to improve the network performance. This paper adopts convex optimization to handle the time allocation for both WPCNs and BackCom users. A two-stage efﬁcient algorithm is proposed to solve the problem of sum-throughput maximization (STM) subject to the basic throughput requirements of Back-Com users, where the Lagrange duality method is applied at the ﬁrst-stage, while golden section and bisection method are jointly used at the second-stage. In order to solve the throughput unfairness among nodes in the STM problem, the common-throughput maximization (CTM; i.e. the worst node’s throughput) problem is further considered. This problem is decomposed into a master problem and a sub-problem, which are solved in a progressive manner. Simulation results show that the proposed methods obtain substantial improvement compared to the benchmark scheme.


INTRODUCTION
It is well known that the energy limitation of wireless nodes is one of great challenges in wireless networks [1][2][3][4][5]. Recently, energy harvesting (EH) is one of the key solutions to improve the energy efficiency of wireless communication system [6][7][8]. By using Radio-frequency (RF) energy harvesting technology [9][10][11], wireless powered communication networks (WPCNs) can overcome the deficiency of the traditional renewable energy sources (such as solar and wind) which are always uncontrollable and intermittent [12,13]. The harvest-then-transmit (HTT) protocol is proposed in [14] as one of the most common working mode of RF energy harvesting, where the users harvest energy from the hybrid access point (HAP) firstly and then use the harvested energy to transmit information to the HAP, which are called wireless energy transfer (WET) and wireless information transfer (WIT), respectively. Due to the benefits of WPCNs, researchers have done a lot of related studies. [7,8] considered network throughput maxi-related work before. WeIn our proposed syste studied the time allocation and energy allocation for both traditional and UAVenabled WPCN scenarios in [21][22][23]. However, the reception capability of the HAP is abandoned when it broadcasts energy to all nodes, in other words, in traditional WPCNs, the users don't fully use the time resource to transfer information [24].
In the meantime, ambient backscatter communication (Back-Com) has been brought up during the last decade, which allows low-power devices to communicate by utilizing ambient radio frequency (RF) signals without requiring active RF transmission [25]. The basic principle of BackCom is that ambient signals transmitted by RF sources can be reflected and modulated by backscatter transmitter to transmit data to the backscatter receiver several meters away [26,27]. Thus, this technique can potentially improve the energy efficiency of energy-constrained networks, and it has witnessed numerous Internet-of-Things (IoT) applications in practice such as remote switches, medical telemetry, and radio-frequency identification (RFID).
Considered as an outstanding approach for wireless communication network, BackCom has attracted a great deal of scholars to carry out relevant researches. The goodput maximization problem of BackCom system by jointly optimizing the time allocation, transmit power allocation and reflection ratio was studied in [28]. [29] addressed the physical layer security of BackCom system by injecting a noise-like signal into the carrier signal. Considering channel diversity, rate adaptation problem was studied in [30][31][32] to ensure high network throughput for backscatter communication. The authors of [33] focused on the signal detection with an energy detector, and the bit error rate of ambient modulated BackCom system. Based on the principle of BackCom, it can be integrated into WPCNs to reflect information when the HAP broadcasting energy to WPCN users, which can use WPCNs resource effectively by allowing RF sources to transmit data and charging simultaneously [24,[34][35][36].
The works in the literatures have shown that integrating BackCom into WET can efficiently and flexibly use the energy resources compared with the dedicated WET stage [34][35][36][37][38]. In [34], BackCom was firstly introduced in radio frequency powered cognitive radio networks to improve the overall transmission rate of the secondary system, where the secondary users could work in either BackCom mode or HTT mode. [38] studied hybrid device-to-device (D2D) communication paradigm by integrating BackCom with WPCN. [39] considered a system with one single-antenna reader and multiple hybrid backscatter-HTT transmitters, where the transmitters could choose freely between the HTT mode and the backscattering mode. The authors studied the total amount of time minimization problem by optimizing the transmitter group selection (i.e. the backscattering group and the HTT group) and the transmission time allocation of each HTT transmitters, and the results were then further extended to the case of massive MIMO reader. [36] studied a backscatter assisted wireless powered communication network with two users, where users can work both in the Back-Com and HTT modes, respectively. The optimal time allocation policy was derived to maximize the system throughput. The authors of [36] also studied another scenario where the multiple users could work in either HTT or BackCom mode in [37]. Followed by [24], the throughput maximization problem for Kb-BackCom users and Kt-WPCN users who could work either in HTT mode or BackCom mode was studied. It was proposed that only the BackCom user with the maximal transmission rate could be activated in the energy harvesting stage. That meant the other BackCom users had zero throughput, which was obviously very unfair. To solve this insufficiency, we aim to maximize the total throughput while reaching the basic throughput requirement of each BackCom user? Different from the available work [39], which studied the transmission time minimization problem by choosing the working mode of transmitters and optimizing time allocation of HTT transmitters, we study the throughput maximization problem by optimizing time allocation with the consideration of basic throughput requirements of backscatter transmitters. It is noticed that there may be unfairness among nodes because the WPCN users with good channel qualities can reach higher throughput compared to the WPCN users with bad channel qualities, and the throughput of each BackCom user is dramatically less than that of WPCN users as the transmission rate of BackCom user is relatively low, thus we further consider the common-throughout maximization (CTM; i.e. the worst node's throughput) problem. The main contributions of this paper are summarized as follows: • We consider sum-throughput maximization (STM) problem that meets the throughput requirements of BackCom users in backscatter assisted WPCNs (BAWPCN), where a two-stage algorithm is proposed. In the first-stage, we derive the closedform solution by solving its Lagrange duality problem without the explicit constraints. If its solution satisfies the explicit constraints, then this solution is the optimal solution. Otherwise, we move on to the second-stage, where we derive the numerical results by using bisection method to determine the optimal WIT time allocation under a given WET duration and use golden section search method to find the optimal WET duration. • Due to the throughput unfairness among nodes of the STM problem, the common-throughput maximization (CTM) problem is further considered. We solve this problem by decomposing it into a master problem and a sub-problem. The sub-problem is to determine whether a given commonthroughput is achievable, which is a feasibility problem. The master problem is to determine the maximum achievable common-throughput by bisection method. The sub-problem is solved by transforming it into a data collection time minimization problem. • Through extensive simulations, we demonstrate that, compared to the STM approach, a fairly larger commonthroughput can be achieved in the CTM problem, but at the cost of a reduction in the sum-throughput. And, our proposed methods are obviously superior to the benchmark scheme.
The rest of this paper is organized as follows. Section 2 describes the system model. Section 3 proposes a two-stage algorithm to obtain solutions to the optimal time allocation for FIGURE 1 Energy harvesting model. Firstly, WPCN users V j harvest energy from the HAP, BackCom users U i reflect information to the HAP. Then, WPCN users V j use the harvested energy to transmit information to the HAP the STM problem. Section 4 studies the CTM problem. Section 5 presents simulation results. Finally, section 6 concludes this paper.

SYSTEM MODEL
The scenario of the considered BAWPCN is shown as Figure 1, which consists of a single-antenna HAP and multiple singleantenna nodes, all the devices operate on the same frequency band. Our scenario mainly focuses on the short-distance communications, such as the healthcare body area networks. Among these nodes, we use U i to denote the BackCom user, where i = 1, 2, … , K . Similarly, we use V j to denote the WPCN user, where j = 1, 2, … , L. HAP works in half-duplex mode [12,24], it first broadcasts energy to users and then collects information from them. WPCN users have no embedded power supply available. We consider a quasi-static channel model based on a time block by assuming that the channel state is constant in a single time block and variable in different time blocks. Without loss of generality, we assume that the time block T is 1s. As mentioned previously [14], for the WPCN users, HTT protocol is employed. Furthermore, the channel power gain from HAP to V j and the channel power gain from V j to HAP are denoted by h j and g j , respectively. It is assumed that HAP knows the global channel state information (CSI). The CSI h j and g j can be obtained via efficient channel estimation and feedback methods [40].
The specific time structure diagram can be referred to Figure 2. We denote the time allocation of WPCN users as t = [t 0 , t 1 , … , t L ]. In the first time slot t 0 of the entire time block, HAP broadcasts energy to all users with a constant power P. During the remaining time t 1 , t 2 , … , t L , WPCN users use the harvested energy to transmit information to HAP via time divi-FIGURE 2 Block structure for BAWPCN. Firstly, during t 0 WPCN users V j harvest energy, BackCom users U i reflect information in i . Then, WPCN users V j transmit information to the HAP in t j sion multiple address (TDMA) [41]. The time allocated for WPCN user V j , denoted by t j , satisfies The energy harvested by WPCN user V j during t 0 is given by We assume that all WPCN users have the same energy harvesting efficiency, denoted by . We also assume that the energy harvested by the WPCN user V j is exhausted in the given time t j . The achievable throughput of V j is given by where W is the bandwidth, j = Ph j g j 2 , 2 is the noise power at the HAP.
During t 0 , each BackCom user U i reflects information to HAP via TDMA at the assigned time i . We denote the time allocation of BackCom users as = [ 1 , 2 , … , K ]. As shown in Figure 2, the time constraint should satisfy Once BackCom user U i is scheduled, it transmits information to HAP with a constant transmission rate B i [24,36,37], which depends on the setting of the RC (resistive capacitive) circuit elements [42]. In our proposed system model, there is no interference when the HAP receives backscatter signals and at the same time it transmits energy in the same frequency band. The reason is that when the backscatter node is activated by RF waves from the HAP, it then uses the carrier wave from the HAP to tune its antenna impedance to generate modulated signals. The HAP can decode the data from the modulated backscatter signal while emitting energy signals. In order to ensure the communication quality of BackCom users, it is assumed that the throughput of each BackCom user should meet the basic throughput requirements, which is written as where R i denotes the basic throughput requirement of Back-Com user U i .

Problem formulation
In this section, we study the STM problem of the BAWPCN and derive the optimal time allocation strategy. The total throughput of both BackCom users and WPCN users in the system is given by Based on the analysis above, we can formulate the STM problem as follows The optimal solution of P1 are expressed as Firstly, we present Lemma 1, which represents a useful conclusion.

Lemma 1. The optimal solution for problem P1 satisfies
i is allocated to any BackCom user which will lead to a larger throughput. It's also easy to understand that if is allocated to the HAP to broad energy for WPCN users, which will lead them harvest more energy and achieve a larger throughput. This completes proof. □ In order to achieve the throughput requirement of each Back-Com user, the minimum time assigned to each of them is expressed as And then we can get a further expression Based on Lemma 1, when t 0 is given, the optimal time allocation of can be easily obtained, which is presented via the Proposition 1.

Proposition 1. The optimal time policy for BackCom users satisfies
where k represents the serial number of the BackCom user with the maximum transmission rate. Proposition 1 reveals that after the basic time demand ′ i is allocated to each BackCom user, the remaining time

The first stage solution
According to Proposition 1, (6) can be re-expressed as Then, P1 can be rewritten as In order to obtain the optimal solution to P2, the Lemma 2 is introduced.
The optimal solutions for problem P2 satisfies Proof. We first prove that the problem P2 is a convex optimization problem. Firstly, we focus on the third term in the objective function of the problem P2. For any j = 1, 2, … , L, the second derivative of Furthermore, as the function t j W log 2 (1 + j t 0 t j ) is the perspective of the function W log 2 (1 + j t 0 ) and the perspective operation preserves concavity [43], t j W log 2 (1 + j t 0 t j ) is a jointly concave of t j and t 0 . According to the concavity preservation of concave function accumulation operation, is concave. Then, considering the first term is constant, the second term is linear, the objective function of the problem P2 is concave. Moreover, the constraint (13b) is affine, the constraints (13c) and (13d) are linear, the problem P2 is a convex optimization problem. We can get the solutions of the convex optimization problem P2 by solving the Lagrange dual problem. The Lagrange of P2 is expressed as where denotes the Lagrange multiplier associated with the constraint in (13b). The dual function of P2 is thus given by To derive g( ), we obtain the optimal t j and t 0 by taking the partial derivative of L sum (t , ) with respect to t i and t 0 , and set them to zero. Therefore, we get the following expressions where * denotes the optimal dual solution of P2. Let Based on Equation (18), (17) can be rewritten as It can be easily proved that t (x) is a monotonically function of Based where A = ∑ L j =1 j . In addition, it follows from (16), (20), (21) and (22), that where C is defined in (22). It can be obtained from (23) can be re-expressed as We use z * to represent the solution of (25), therefore the optimal time allocation of the problem P2 can be expressed as This thus proves Lemma 2.
The computation complexity of the first-stage algorithm is(log 2 (z max ∕ )), which is mainly caused by bisection method using in (24).
Then we examine the constraint (13d) of P2, if the (13d) is satisfied, the solution at this stage is the optimal solution of P2, otherwise, we turn to use another algorithm to obtain the optimal solution of P2 , which is presented as follows.

The second stage solution
In order to meet the throughput requirements of BackCom users, the total time threshold assigned to them can be expressed as For a given , we first find the optimal time allocation of WPCN users, then we reformulate the STM problem as where R j is given in (3), which denotes the throughput of WPCN user V j . Lemma 3 can enlighten us to search the optimal time allocation of WPCN users.

Lemma 3. The optimal time allocation of WPCN users satisfies the following property
Proof. We prove it by contradiction. Without loss of generality, assume that Because the throughput increases strictly monotonically with t j [21], let t ′ i = t * i + Δt and t ′ j = t * j − Δt , then the feasible solution (t 1 , t 2 , … t ′ i , … t ′ j , … t L ) has a larger throughput R, which contradicts to the assumption and it completes the proof. □ In order to solve the problem P3, we first present Lemma 4.

Lemma 4. P3 is a convex optimization problem with respect to t 0 .
Proof. The first derivative of the objective function of P3 with respect to t 0 is j t j W (t j + j t 0 ) ln 2 + B k , and the second derivative of the objective function of P3 with respect to is − 2 j ln 2t j W (t j + j t 0 ) 2 .
Obviously, − 2 j ln 2t j W (t j + j t 0 ) 2 ≤ 0, which proves the objective function of problem P3 is concave. Furthermore, the constraint (29b) for problem P3 is linear. Therefore, the problem P3 is a convex optimization problem with respect to t 0 . This completes proof. □ Based on Lemma 4, we solve this problem in a nested way. The golden section method is used to find the optimal t 0 at the outer layer, and the optimal value that satisfies ∑ L j =1 t j = 1 − t 0 is founded by bisection method at the inner layer under the given t 0 . The derivative of R j with respect to t j is given by Let z = 1 + j t 0 t j and R j t j = , (31) can be expressed as By taking the derivative of (32) , we get that z = 1 is the only stationary point of (32), so there is a unique solution z * when z > 1. The optimal time allocation can be given by To summarize, the algorithm for solving P3 is given in Algorithm 1.
Let us briefly analyze the complexity of the above optimal algorithm. The gold section search for the optimal t 0 has log 0.618 (t ′′ 0 − t ′ 0 ) iterations. Using the bisection method to find has log 2 ( max ∕ ) iterations. Above all, the computational complexity is (log 0.618 (t ′′ 0 − t ′ 0 ) * log 2 ( max ∕ ) * log 2 (z max ∕ )) , which is significantly higher than the computational complexity of the first-stage algorithm. This shows that in some appropriate cases, using the first-stage algorithm to solve the problem can improve efficiency. ALGORITHM 1 Determine the optimal time allocation to maximize sum-throughput We evaluate the two-stage algorithm for solving STM problem through the simulations. The simulation results show that when the power P of HAP is low, the transmission rates of BackCom users are high, and the throughput requirements of BackCom users are low, the first-stage execution algorithm is more likely to be selected. For example, we counted the results of 1000 randomly generated topologies with P = 20 dbm and P = 30 dbm. For the former case, 1000 topology solutions all selected the first-stage execution algorithm, and for the latter case, 745 topology solutions selected the first-stage execution algorithm. We also counted the results of 1000 randomly generated topologies with the transmission rates of BackCom users are 21k, 22k, 23k and that are 22k, 24k, 26k. For the former case, 744 topology solutions selected the first-stage execution algorithm, and for the latter case, 554 topology solutions selected the first-stage execution algorithm. Furthermore, we counted the results of 1000 randomly generated topology with the throughput requirements of BackCom users are 2, 1, 2 kbps and that are 4, 3, 4 kbps. For the former case, 1000 topology solutions all selected the first-stage execution algorithm, and for the latter case, 555 topology solutions selected the first-stage execution algorithm.

COMMON-THROUGHPUT MAXIMIZATION OF BAWPCN
It should be noted that the STM problem may lead to throughput unfairness among nodes. For example, when the transmission rates of the BackCom users are considerable low, they can only achieve the basic throughput requirements, which are far lower than the throughput of WPCN users. And, when the WPCN users are far away from the HAP, their throughput are lower than that of WPCN users close to the HAP. Therefore, we further consider the common-throughput maximization (CTM; i.e. the worst node's throughput) problem. We formulate the CTM problem as s.t. constraints (7c)-(7e), To make the problem easier to tackle, we introduce an auxiliary variableR to rewrite P4 into an equivalent form as Based on analysis, we find that if the constraint (35b) is satisfied, the algorithm proposed in [21] can be used to get the optimal time allocation of WPCN users. For simplicity, we will not elaborate on the algorithm proposed in [21]. After calling the algorithm proposed in [21] to calculate the best time allocation for WPCN users, we can easily derive the best time allocation for BackCom users. Obviously, when the CTM scheme achieves the best performance, the throughput of each BackCom user is equal. We present Lemma 5 in the following, which is used to calculate the optimal time allocation of BackCom users. ALGORITHM 2 Determine the optimal time allocation to maximize common-throughput Lemma 5. The optimal time allocation of BackCom users satisfies where R * represents the optimal throughput of each BackCom user. Based on (36), the optimal time allocation of BackCom users is given by * R * can be obtained by substituting (38) into (37), and then * can be obtained consequently. The algorithm for solving P4 is given in Algorithm 2.

SIMULATION RESULTS AND DISCUSSION
In this section, we conduct numerical simulations to compare the STM scheme with the CTM scheme, and we design a benchmark scheme to verify our proposed method. In the benchmark scheme, the transmission time for each BackCom user is allocated according to i = , and the energy harvesting time for WPCN users is allocated according to t 0 = ∑ K i=1 i . Then, calculate each WPCN user information transmission time using t j = 1−t 0 L . The parameters are set as follows. The bandwidth is set as W = 100khz. The channel power gains between the HAP and WPCN users are modeled as h j = g j = 10 −3 d −3 j , where d j is the distance between the HAP and WPCN user V j , and is randomly generated from 4m to 8m. For each WPCN user, the energy harvesting efficiency is set to be = 0.6. The noise power at the HAP is set as 2 = 1e − 12w. Each data point in the figures represents the average throughput of 1000 random topologies. Figure 3 shows the average sum-throughput and commonthroughput of two schemes compared to the benchmark scheme under different transmit power of HAP, with three BackCom users and three WPCN users. The transmission rates of the three BackCom users are 20k, 30k, 40k, and the basic throughput requirements of them are 3, 4, 6 kbps. From this figure, we can observe that the STM scheme and the CTM scheme are obviously superior to the benchmark scheme. This is because less time is allocated to the WPCN users to harvest energy in the benchmark scheme, which leads to the transmission power used by WPCN users to transmit information is decreased, so the total throughput is dramatically reduced  Figure 3. As observing from Figure 3, the average sum-throughput of STM scheme is greater than that of CTM scheme. However, the average common-throughput of CTM scheme is significantly better than that of STM scheme. This can be explained as follows.
In the STM scheme, the throughput of WPCN nodes is very unbalanced. For example, in the simulations we found that, in a certain topology, WPCN node close to the HAP has a throughput of up to 199 kbps, while the WPCN node far away from the HAP has a throughput of only about 4.8 kbps. This is because the nodes near to the HAP have better channel quality, resulting in harvesting more energy to the HAP and being allocated more time to transmit information to the HAP, so their throughput is obviously high. In other words, these nodes make a meaningful contribution to the STM problem. In the CTM scheme, the throughput of nodes is quite balanced. And the reason is that WPCN users with poor channel quality and BackCom users with low transmission rates are allocated more time to realize common-throughput maximization. Figure 4 further verifies the balance of the throughput of the CTM scheme compared to the STM scheme, and one interesting insight is provided, that is the throughput of both WPCN users and BackCom users is equal in the CTM scheme. For both STM scheme and CTM scheme, as the transmit power P of the HAP increases, both the average sum-throughput and common-throughput increase. It needs to be specially stated that in some cases, the growth rate is very slow, and even after reaching a certain point, the growth rate doesn't change, so some trend lines appear to be horizon. This is because with the increase of P, the energy captured by the WPCN users increases, and thus the transmission power used by WPCN users to transmit information increases, so the throughput increases. For the STM scheme, Figure 3b indicates that when P is very low, the throughput of WPCN user is less than that of BackCom user. This is because energy harvested by WPCN user is too small. And we can also see from Figure 3b that the average common-throughput of the STM scheme increases approximately horizontal. In the simu-FIGURE 5 Average sum-throughput and common-throughput versus the number of WPCN nodes, with the basic throughput requirements of BackCom users are 6.6, 5, 7 kbps lations, we found that when P reached a certain value, the average common-throughput remained constant and the value was the throughput of the BackCom user with the worst throughput requirement. This is because the transmission rate of Back-Com user is relatively low compared to WPCN user. In the simulations process, we also found that when P reached a certain level, the time allocated to the BackCom users is fixed and could only be maintained at the basic throughput requirements. This is because under this circumstances, the WPCN users can harvest more energy and achieve higher throughput, so they are critical to the goal of sum-throughput maximization and more transmission time should be allocated to them. Figure 5 shows the average sum-throughput and commonthroughput of two schemes under different number of WPCN users, with three BackCom users, P = 20 dbm. The Average sum-throughput and common-throughput vs. The transmission rates of BackCom users, with the basic throughput requirements of BackCom users are 5, 1, 4 kbps transmission rates of the three BackCom users are 21k, 42k, 40k, and the basic throughput requirements of them are 6.6, 5, 7 kbps. It is observed from Figure 5a that for both STM scheme and CTM scheme, as the number of the WPCN users increases, the average sum-throughput increases. Additionally, Figure 5b shows that as the number of WPCN users increases, the average common-throughput decreases. Because as the number of WPCN users increases, the proportion of nodes located at the edge of the topology also increases, so the worst channel power gain in this topology may be worse and leads to a lower common-throughput. For the STM problem, Figure 5b shows that the average common-throughput is the basic throughput requirement of the BackCom user with the lowest transmission rate at the early stage, but as the number of WPCN users increases, the average common-throughput begins to drop. This is because as the number of WPCN users increases, information transmission time allocated for each WPCN user decreases and the probability of poor channel state FIGURE 7 Average sum-throughput and common-throughput versus energy harvest efficiency, with the basic throughput requirements of BackCom users are 5, 1, 4 kbps increases, which results in the fact that a WPCN user becomes the worst node with the lowest throughput. Figure 6 shows the average sum-throughput and commonthroughput of two schemes under different transmission rates of BackCom users, with three BackCom users, three WPCN users, P = 20 dbm. The basic throughput requirements of BackCom users are 5, 1, 4 kbps. We set 5 group of transmission rates for BackCom users, which are (21k, 22k, 23k), (31k, 32k, 33k), (41k, 42k, 43k), (51k, 52k, 53k), and (61k, 62k, 63k), respectively. We can observe from Figure 6 that as the transmission rates of BackCom users increase, the average sum-throughput and the common-throughput increase. For the STM scheme, the average common-throughput remains constant, this is because the transmission rates of BackCom users are rather low, they are regarded as the worst throughput nodes in the system, thus the average common-throughput is not affected by the number of WPCN users. The average commonthroughput 1 kbps of the STM scheme in the Figure 6a is exactly the throughput of BackCom user with the lowest throughput requirement. When the benchmark transmission rate of Back-Com users increases, for example, we increased it to 600 kbps, the trend curve began to decline. Because at this time, the node with the worst throughput became the WPCN user. However, in actual scenario, the transmission rate of the backscatter user can hardly reach such a high criteria, so we did not show the relevant figure. Figure 6b shows that the average sum-throughput changes gently for STM problem. This is because WPCN users play an important role in network performance and greatly affect the sum-throughput. Figure 7 shows the average sum-throughput and commonthroughput of two schemes under different energy harvest efficiency, with three BackCom users, three WPCN users, and P = 20 dbm. The basic throughput requirements of BackCom users are 5, 1, 4 kbps, and the transmission rates of them are 22k, 24k, 26k. As observing form Figure 7a, the average sumthroughput increases with the increase of energy harvest efficiency. This is because when energy harvest efficiency becomes larger, more energy will be harvested by WPCN users and more transmit power can be used by WPCN users to transmit information, thus the average sum-throughput increases. We can also observe from Figure 7b, for the CTM scheme, the average common-throughput increases with the increase of energy harvest efficiency. However, for the STM scheme, the average common-throughput remains unchanged when the energy harvest efficiency becomes larger, which is the lowest throughput requirement among BackCom users. This is because the transmission rates of BackCom users are really low compared to WPCN users and they are the common-throughput nodes, therefore, the common-throughput is not affected by energy harvest efficiency of WPCN users.

CONCLUSIONS
This paper considered a WPCN system with the deployment of BackCom nodes. We analyzed the throughput of the BackCom assisted WPCN (BAWPCN) and formulated the time allocation as an optimization problem. We proposed a two-stage algorithm to derive the optimal time allocation for the STM problem. Furthermore, we studied the CTM problem to guarantee the throughput fairness among nodes. The simulation results shown that compared with STM scheme, CTM scheme can achieve much balanced throughput among nodes at the cost of reduced sum-throughput, and our proposed methods are superior to the benchmark scheme.