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A hybrid MAC protocol for optimal channel allocation in largescale wireless powered communication networks
EURASIP Journal on Wireless Communications and Networking volume 2018, Article number: 9 (2018)
Abstract
This paper proposes a largescale wireless powered communication network (WPCN), which consists of a hybrid access point (HAP) and numerous nodes. The HAP broadcasts energy to all nodes, and the nodes harvest the energy and then use the harvested energy to transmit information. For the largescale WPCN, we also propose a novel hybrid multiple access protocol, termed hybrid multiple accessbased dual harvestthentransmit. The proposed protocol utilizes both time division multiple access (TDMA) and carriersense multiple access (CSMA), in order to deal with various traffic patterns and transmission reliability of different applications in largescale networks. We consider a dual wireless energy transfer (WET) at the HAP. The main WET is performed in TDMA and the other WET is performed at space holes in CSMA, to increase the channel utilization and harvested energy. For the considerations, we study the sumthroughput maximization in the largescale WPCN based on hybrid multiple accessbased dual harvestthentransmit. Simulation results show that the proposed protocol outperforms the conventional protocol in largescale WPCN.
1 Introduction
Recently, there has been a great interest in a largescale wireless network, such as internet of things (IoT) and a lowpower widearea network (LPWAN) due to the easiness of maintenance and the economic advantages for deployment compared to wired networks [1–3]. In these networks, one of major research issues is how to supply stable operating power for numerous endnodes, to improve network lifetime. To solve this problem, energy harvesting has received a great deal of attention as a replacement of traditional energy sources (e.g., batteries) in wireless networks. Especially, ambient radio signals have been regarded as a promising energy source for the energy harvesting, since the radio signals enable both wireless energy transfer (WET) and wireless information transfer (WIT). For this reason, wireless powered communication networks (WPCN) have been largely investigated, in which a radio signal is used for the WET [4–6].
A typical WPCN consists of a hybridaccess point (HAP) and multiple nodes, where the HAP performs the WET in downlink (DL) and the nodes carry out the WIT in uplink (UL) by using the harvested energy [7–10]. There have been many efforts to improve the performance of WPCN in terms of throughput, energy efficiency, resource allocation, etc. [11–18].
However, most existing WPCN models cannot be directly employed to largescale wireless networks, since nodes, in a typical WPCN, are designed to transmit information in every block. In addition, the existing WPCN models assume that all nodes can fully transmit their information in a single block, where each node transmits in rapid succession, using its own time slot. However, in largescale networks, there would be different traffic patterns or specialized requirements such as transmission reliability. Furthermore, the number of transmitting nodes can be larger than the number of available slots in a single block. In that case, some nodes will fail to acquire their slots then the nodes cannot perform the WIT.
Another scale issue in a WPCN is related to state information (SI) messages between the HAP and nodes, which are used to share status information, such as channel information, harvesting efficiency and traffic generation [18, 19]. In a largescale WPCN, the number of SI messages would be big enough to degrade the network performance significantly because of traffic overhead caused by numerous SI messages.
In this paper, we propose a novel protocol for a largescale WPCN, named hybrid multiple accessbased dual harvestthentransmit (HDHT). The HDHT has a hybrid multiple access structure which employs both time division multiple access (TDMA) and carriersense multiple access (CSMA), in order to manage different traffic patterns and requirements of nodes. We assume that there are two primary traffic patterns in the largescale WPCN: periodic and nonperiodic traffic. Note that transmission reliability is guaranteed for the periodic traffic, but not for the nonperiodic traffic. The periodic traffic can be allocated in the TDMA period (TP) for the reliable information transmission, i.e., the time slots in the TP are assigned to each node for their periodic traffic. Especially, we consider that the HAP can also perform the DL WIT in the TP when the HAP has information to transmit (not supported in [11–15]). On the other hand, the nonperiodic traffic, which is unconstrained from the transmission reliability, is allocated in the CSMA period (CP).
We also propose a novel WET method, named the dualWET, which improves the channel utilization and increases the amount of harvested energy at nodes. The HAP performs the mainWET during the TP and the additional WET, termed subWET, during the CP. As a typical CSMA is a contentionbased channel access protocol, channels are not always occupied and there can exist wasted time, called space holes, in wireless channels depending on the system of interest. However, in the HDHT, nodes can harvest additional energy by performing the subWET in space holes. As a result, the channel utilization can be improved without any throughput reduction.
Furthermore, we describe the procedure of the SI transmission between the HAP and nodes, where numerous SI messages could degrade the network performance in largescale networks [20]. In the proposed model, some nodes, which generate periodic traffic and require the transmission reliability, only report their states to the HAP, in order to obtain time slots in the TP. The other nodes, that generate nonperiodic traffic which is unconstrained from the transmission reliability, participate in the CP without any SI transmission. By the rate of transmission reliability, the ratio between TDMA and CSMA durations (RTC) can be calculated, which is motivated by the guaranteed time slots (GTS) allocation scheme in IEEE 802.15.4 network [20].
We also consider the minimum slot time (MST) in the DL and UL WIT. The HAP can estimate the number of slots for the DL and UL WIT to allocate time slots to nodes in the TP, by using the MST. It is worth noting that the number of received SIs cannot exceed the maximum number of slots in the TP. If the number of received SIs is larger than the maximum number of slots in the TP, the HAP will discard excessive SIs and inform the excess to corresponding nodes.
The main contributions of this paper are summarized as follows:

For largescale WPCN, we propose a protocol termed hybrid multiple accessbased dual harvestthentransmit protocol, which guarantees the transmission reliability by TDMA and the other transmission is performed by CSMA. In addition, the HAP can also perform the DL WIT.

In the proposed protocol, all nodes perform the dualWET, where the mainWET happens in the TDMA period and the subWET happens in the CSMA period at space holes. For that reason, the channel utilization can be improved and the amount of harvested energy at nodes can also be increased.

We propose the procedure of the SI transmission for largescale WPCN, in order to reduce the network overhead caused by SI messages. Nodes, that generate periodic traffic and require the transmission reliability, transmit the SI to the HAP for the slot allocation in the TDMA period.

We consider the minimum slot time for the DL and UL WIT. Based on the minimum slot time, the HAP can estimate the maximum number of nodes for the UL WIT.

With the proposed protocol, we present a system model for largescale WPCN, where the sumthroughput of nodes is maximized by the convex optimization technique. By comparing other multiple accessbased protocols, we investigate whether the hybrid multiple access is suitable for largescale WPCN or not.
The rest of this paper is organized as follows. Section 2 introduces related works and Section 3 describes the largescale WPCN model and the proposed HDHT protocol. Section 4 presents the problem formulation for the HDHT. Section 5 provides simulation results and discussion. Finally, Section 6 concludes the paper and discusses future work.
2 Related works
To improve WPCN performance, much research has been conducted. In [11], a time division multiple access (TDMA)based system model is presented for WPCN and a harvestthentransmit protocol is described, where nodes transmit in rapid succession, one after the other. To improve the amount of harvested energy at nodes, in [12] and [13], fullduplex WPCN models are researched, where the HAP broadcasts wireless energy and receives information from nodes simultaneously. However, the nodes can only perform UL WITs sequentially based on the TDMA protocol.
To apply different multiple access protocols, in [14], the authors propose an orthogonal frequency division multiplexing (OFDM)based system for WPCN. Based on OFDM, the HAP performs the DL WET in optimized subchannels, but the nodes still carry out the UL WIT in sequence over time. In [15], the spacedivision multiple access (SDMA) protocol is used for the DL WET and UL WIT. However, the UL WIT in [14, 15] is performed as the TDMAbased harvestthentransmit protocol [11]. In [16] and [17], the wireless powered cognitive radio networks are proposed for the secondary nodes to transmit information sequentially to the HAP when the primary nodes are inactive. Nonetheless, it is assumed that the HAP and nodes are aware of network information.
For the network information reception in WPCN, in [18], the authors propose a frame structure that utilizes state information (SI) of all nodes. Based on the SI, the HAP calculates the duration of WET and WIT in a block time. But the all or a part of the nodes still transmit information in rapid succession, based on TDMA. [19] proposes a multipleinputmultipleoutput system for WPCN based on the timedivisionduplexing (TDD) protocol. In addition, throughput maximization and rate fairness schemes are also investigated.
In addition, there are many researches for green communication focusing on energy efficiency. In [21], devicetodevice communication is studied, where devices can harvest energy from environmental energy sources, power beacons, which radiate power to devices, and ambient radio signals. In [22], cloud radio access networks (CRANs) are studied for the high energy efficiency benefit. In [23], green transmission technologies are introduced, focusing on how to utilize the degrees of freedom in different resource domains, as well as how to balance the tradeoff between energy and spectrum efficiency. Also, in [24], the energyefficient power allocation and wireless backhaul bandwidth allocation are studied in heterogeneous small cell networks. In [25], user association and power allocation in mmWavebased UDNs are studied regarding to load balance constraints, energy harvesting by base stations, user quality of service requirements, energy efficiency, and crosstier interference limits.
Optimal resource allocation is another issue, briskly researched for green communications. In [26] and [27], resource allocation, power control, and sensing time optimization problem in a cognitive small cell network are investigated. In [28], a resource allocation scheme for orthogonal frequency division multiple accessbased cognitive femtocells is proposed. In [29], resource allocation for energy efficiency optimization in heterogeneous networks is studied.
3 System model
In this section, we describe a largescale WPCN and propose the HDHT protocol. As illustrated in Fig. 1, a largescale WPCN consists of a single HAP and node i, ∀i∈{1,2,⋯,N}, where N is a positive integer and big as we consider a largescale network. We assume that the HAP equips one antenna for the DL WET/WIT and receiving information from nodes, one at a time. Nodes are also assumed to have a single antenna for harvesting energy and receiving information from the HAP, and the UL WIT, which cannot be performed at the same time. It is worth noting that the HAP and nodes operate over the same frequency band. The DL channel power gain from the HAP to node i, ∀i∈{1,2,⋯,N}, and the UL channel power gain from node i, ∀i∈{1,2,⋯,N}, to the HAP are denoted by h_{ i } and g_{ i }, respectively. We assume that all channels follow quasistatic flatfading, where h_{ i } and g_{ i } remain constant during each block time, denoted by T, but possibly can vary in different blocks depending on the system of interest. Furthermore, we assume that the HAP knows perfectly both h_{ i } and g_{ i }, ∀i∈{1,2,⋯,N}.
The proposed network adopts the HDHT protocol as shown in Fig. 2a, which consists of SI transmission, TDMA, and CSMA period in a block. Therefore, we have
where T_{ b }, T_{ t }, and T_{ c } are the SI transmission period (SIP), TP, and CP, respectively. For convenience, we normalize a block time as T=1 in the sequel; consequently, both the term of energy and power can be used interchangeably.
3.1 State information transmission period in the HDHT protocol
The SIP consists of the energy beacon period, SI transmission period, and command transfer period as shown in Fig. 2b. During the energy beacon period, the HAP broadcasts energy to all nodes in the proposed network and the nodes harvest the broadcasted energy for their SI transmission. It is worth noting that the broadcasted energy in the energy beacon period is sufficient for SI transmission.
During the SI transmission period, only nodes that generate periodic traffic report their SI to the HAP in order to request transmission reliability. Thus, we reduce a significant network overhead caused by numerous SI messages from all nodes in the largescale network. It is worth noting that although the nearfar problem [11] exists during the energy beacon and SI transfer periods, the excess energy can be neglected since the overall energy involved is very low. The node that transmitted their SI can be allocated in the TP for the UL WIT, but the other nodes, which are unconstrained from the transmission reliability, participate in the CP for the UL WIT.
During command transfer period, based on the received SI, the HAP decides the RTC in a block time and calculates the optimal time of nodes in the TP, considering the MST. Then, the HAP transforms the decision information into a command and broadcasts the command to all the nodes in the largescale WPCN.
Then, the SIP is obtained as
where τ_{ EB }, τ_{ SI }, and τ_{ CT } are the time of the energy beacon period, SI transfer period, and command transfer period, respectively. τ_{ SI } depends on the number of SIs from nodes. Since τ_{ EB } and τ_{ CT } are a fixed time and very moment, so we assume that τ_{ EB }, and τ_{ CT } are zero for the convenience in the sequel, as [18, 19].
3.2 The ratio between TDMA and CSMA
In the proposed model, we consider periodic and nonperiodic traffic from nodes in the largescale WPCN. In addition, we consider that some node i, ∀i∈{1,2,⋯,N}, might not generate any information in a block time, i.e., a part of the nodes can perform the UL WIT. Thus, K denotes the number of nodes, which generate transmitting information in a block time, and is obtained as
where P_{ d } is the average probability of generating information at the node i, ∀i∈{1,2,⋯,N}.
For the periodic traffic of nodes, we consider that transmission reliability is required. Some nodes that generate the periodic traffic among node i, ∀i∈{1,2,⋯,K}, transmits SI messages to the HAP. Thus, \(\hat {K}\) denotes the number of nodes that require transmission reliability in the a block time and is obtained as
where P_{ s } is the average probability of generating SI at the node i, ∀i∈{1,2,⋯,K}. By P_{ s }, the HAP calculates the RTC since P_{ s } means the rate of transmission reliability. Thus, we assume that the RTC is proportional to P_{ s }. It is worth noting that P_{ s } and the rate of transmission reliability is depending on the system of interest.
3.3 TDMA period in the HDHT protocol
In the TP, the HAP performs the mainWET and the DL WIT. The node i, \(\forall i \in \{1,2, \cdots, \hat {K} \}\), performs the UL WIT by using the harvested energy from the dualWET: the mainWET in the TP and subWET in the CP. It is worth noting that the DL WIT is performed in the TP, when the HAP is necessary to transmit information. τ_{0} denotes the time assigned to the HAP in the TP and is expressed as
where \(0 \leq \tau _{0_{WIT}} \leq \tau _{0}\) and \(0 \leq \tau _{0_{WET}} \leq \tau _{0}\) denote the time of the DL WIT and the mainWET at the HAP, respectively. If there is no transmitting information at the HAP, \(\tau _{0_{WIT}}\) can be zero. τ_{ i }, \(\forall i \in \{ 1, 2, \cdots, \hat {K} \}\), denotes the allocated time to node i, \(\forall i \in \{ 1, 2, \cdots, \hat {K} \}\), for the UL WIT in each block time as shown in Fig. 2c. Thus, the TP can be expressed as
For the reality, we assume that τ_{ i } is constrained to be larger than the required minimum slot time, denoted τ_{ mst }, where τ_{ i } converges to zero when the harvested energy is sufficiently large. It is worth noting that τ_{ mst } can be calculated by the maximum data rate and the frame length, depending on the system of interest. The constraint of τ_{ i } is expressed as
In the TP, the node i, ∀i∈{1,2,⋯,N}, performs energy harvesting during \(\tau _{0_{WET}}\). Then, the harvested energy at nodes in the TP is expressed as
where P_{ A } denotes the transmit power at the HAP, which is sufficiently stable to broadcast energy by wireless and large enough to ignore the receiver noise. 0≤ξ_{ i }≤1 denotes the energy harvesting efficiency for node i, ∀i∈{1,2,⋯,N}.
Also, in the CP, the node i, ∀i∈{1,2,⋯,N}, harvests the additional energy from the subWET at space holes, where the HDHT performs dualWET. \(E_{i}^{CSMA}\), ∀i∈{1,2,⋯,N}, denotes the harvested energy from the subWET, which is studied in the following subsection. Consequently, the total harvested energy from the dualWET is denoted by E_{ i } and is expressed as
where we assume that the node i, ∀i∈{1,2,⋯,N}, in the largescale WPCN, replenishes their energy from the dualWET
For the throughput maximization, E_{ i } at each node must be consumed for its transmission during τ_{ i }. We denote x_{ i } as the complex baseband signal transmitted by node i, ∀i∈{1,2,⋯,K}. In addition, we assume Gaussian inputs, i.e., \(x_{i}\sim \mathcal {CN}(0,P_{i}),\) where P_{ i } denotes the average transmit power at node i. Then, P_{ i } during the UL WIT at node i can be expressed as
where 0≤η_{ i }≤1 denotes the portion of the total harvested energy used for the UL WIT at node i, ∀i∈{1,2,⋯,K}, in steady state. For the purpose of exposition, we assume η_{ i }=1, ∀i∈{1,2,⋯,K}, in the sequel, i.e., all the energy harvested at each node i is used for its UL WIT. For the DL WIT, \(\overline {P_{A}}\) denotes the transmit power at the HAP and is expressed as
where 0≤η_{ A }≤1 denotes the utilization rate of the transmission power at the HAP and P_{ max } denotes the maximum transmission power available at the HAP. It is worth noting that since \(\tau _{0_{WIT}}\) converges zero if P_{ max } is unlimited, we assume that \(P_{i} \leq \overline {P_{A}} \leq P_{max}\) for the reality, where P_{ i } is the largest at ξ_{ i }h_{ i }η_{ i }=1. For the purpose of exposition, we assume \(\phantom {\dot {i}\!}P_{A} \tau _{0_{WET}} = \eta _{A} P_{MAX}\) in the sequel.
3.4 CSMA period in the HDHT protocol
In the CP, the HAP performs the subWET and the node i, \(\forall i \in \{ 1,2, \cdots, K\hat {K} \}\), performs the UL WIT. τ_{ i }, \(\forall i \in \{ 1, 2, \cdots, \hat {K} \}\), denotes the UL WIT time of node i, \(\forall i \in \{ 1, 2, \cdots, K\hat {K} \}\) in the CP as shown in Fig. 2d. Thus, the CP can be expressed as
The channel utilization of the contentionbased protocol, as CSMA, is degraded by space holes. However, in the HDHT, we consider that space holes in the CP can be utilized by the subWET to harvest additional energy and increase the sumthroughput. The harvested energy from the subWET can be expressed as
where \(\tau _{0_{CSMA}}\) denotes the total harvesting time in the CP. It is worth noting that \(E_{i}^{CSMA}\) is used for the UL WIT in the following block as Eq. (9) and can be zero when there is no space holes in the CP.
The node i, \(\forall i \in \{ 1,2, \cdots, K\hat {K} \}\), in the CP, performs the UL WIT without transmission reliability. We assume that each node i in the CP can sense the UL WIT of the other nodes. Also, we assume that the UL WIT in the CP cannot be performed within the fixed sensing delay, denoted by δT_{ s }, i.e., if two or more nodes initiate the UL WIT within δT_{ s }, there will be a collision. It is worth noting that if there is a collision, all of the information transmitted is assumed to be lost.
To describe the CP, we define two sets for the noncollision and collision transmission in Markovian model as Fig. 3 [30]. and denote noncollision and collision states, respectively.
where \(x_{i}^{j} \in \{ 0,1 \} \), \(\forall i \in \{ 1,2, \cdots, K\hat {K} \}\), denotes the link status at node i. \(x_{i}^{j} = 1\) represents an active transmission at node i and 0 represents waiting or performing energy harvesting by the subWET at node i. The link status is expressed as
In addition, we assume that the waiting time of node i, \(\forall i \in \{ 1,2, \cdots, K\hat {K} \}\), is exponentially distributed and λ^{−1} denotes the mean of the exponential distribution. The probability density function of the waiting time t_{ i } is given by
where \(\lambda _{i} \in \{ 1,2, \cdots, K\hat {K} \}\) denotes the transmitted frames per unit time [31].
Due to the sensing delay, δT_{ s }, experienced by the node i, \(\forall i \in \{ 1,2, \cdots, K\hat {K} \}\), the probability, that node i performs the UL WIT within δT_{ s } while starting the UL WIT from another node, is expressed as
by the memoryless property of the exponential random variable, i.e., CDF [31]. Thus, the rate of transition, , to one of the noncollision states in the Markov chain is defined as
The rate of transition, , to one of the collision states is given by
4 Problem formulation for the HDHT protocol
In this section, we study the sumthroughput maximization in the largescale WPCN based on the HDHT. Specifically, we aim to maximize the sumthroughput of the DL and UL WIT in the TP, and the UL WIT in the CP, respectively. In the following, we formulate the optimization problem for the TP and the CP.
4.1 TDMA period in the HDHT protocol
In the TP, the HAP and node i, \(\forall i \in \{ 1,2, \cdots, \hat {K} \}\), can perform the DL and UL WIT, respectively. \(R_{sum}^{TDMA}\) denotes the sumthroughput of the HAP and nodes allocated in the TP and can be expressed as
where ρ_{0} denotes the presence of the DL WIT at the HAP. If there is information transmission at the HAP, ρ_{0} is 1, or if not ρ_{0} is 0. R_{0} and R_{ i } denote the throughput of the DL WIT and UL WIT, respectively. R_{0} is obtained by
where \(\gamma _{0} = \frac {h_{i}}{\Gamma _{A}\sigma _{A}^{2}}\). In addition, R_{ i } is obtained by
where \(\gamma _{i} = \frac {g_{i}h_{i}\eta _{i}\xi _{i}P_{A}}{\Gamma \sigma ^{2}}\) and P_{ i } from Eq. (10). Γ denotes the signaltonoise ratio gap from the additive white Gaussian noise channel capacity as a modulation and coding scheme (MCS) use. σ^{2} represents the noise power at the HAP. For convenience, we assume ξ_{ i }h_{ i }P_{ A }=1 from Eqs. (8) and (13) in the sequel of this paper without loss of generality.
Consequently, from Eqs. (22) and (23), \(R_{sum}^{TDMA}\) can be expressed as
where \(\gamma _{0}^{\prime } = \frac {h_{i}P_{A}}{\Gamma _{A}\sigma _{A}^{2}}\).
To maximize the sumthroughput in the TP, \(R_{sum}^{TDMA}\), the optimal time allocation is investigated in the largescale WPCN. The throughput maximization is then expressed as the following problem:
where Eqs. (26) and (27) correspond to the original constraints from Eqs. (7) and (6), respectively. To solve (P1), we first present the following two lemmas.
Lemma 3.1
The optimal time allocation of (P1) must satisfy the constraint, Eq. (27), with equality, i.e., \(\sum _{i=0}^{\hat {K}}\tau _{i}^{*}=T_{t}\).
Proof
Please refer to Appendix A. □
Lemma 3.2
The objective function of (P1), Eq. (25), is a concave function of the allocated time for the HAP and node i, \(\forall i \in \{ 1, 2, \cdots, \hat {K} \}\), i.e., \(\boldsymbol {\tau }=\left [\tau _{0}, \cdots,\tau _{\hat {K}}\right ]^{T}\).
Proof
Please refer to Appendix B. □
Proposition 3.1
The optimal time allocation of (P1) is
where \(z^{*} = (\mathcal {F} 1)/\mathcal {W} \{ (\mathcal {F}1)/(\exp (1+\rho _{0} \cdot \mathcal {H}) \}\), \(\mathcal {F} \triangleq \sum _{j=1}^{K} \gamma _{j}\), and \(\mathcal {W}(\cdot)\) is the wellknown Lambert Wfunction [32].
Proof
Please refer to Appendix C. □
4.2 CSMA period in the HDHT protocol
In the CP, the HAP performs the subWET and node i, \(\forall i \in \{ 1, 2, \cdots, K\hat {K} \}\), performs the UL WIT. In Section 2, we describe the system model of the CP as continuous Markov chain model [33]. Now, we optimize the sumthroughput of nodes in the CP by maximizing the probability of being in the noncollision transmission states. For maximizing the probability, the stationary distribution of the continuous Markov chain, denoted by p(x^{i}), is defined as
where r_{ i } satisfies the detailed balance equation [34]. Eqs. 19 and (20) express r_{ i }, which is obtained as
where \(R_{i} \triangleq \frac {\lambda _{i}}{\mu _{i}}\) denotes the throughput of node i, \(\forall i \in \{ 1, 2, \cdots, K\hat {K} \}\). \(\frac {1}{\mu _{i}}\) denotes the mean transmission length of the information. It is worth noting that for the purpose of exposition, we assume λ=1. Hence, we can interchange the throughput of nodes in the CP and R_{ i } from Eq. (23), \(\forall i \in \{ 1, 2, \cdots, K\hat {K} \}\).
To quantify the sumthroughput of nodes, a loglikelihood function [35] is defined as the summation over all the noncollision transmission states, which can be expressed by
where \(\mathcal {A} \triangleq \ln \sum _{j} exp(r_{j})\). Then, is defined as
Then, the throughput maximization is then obtained by the following problem:
where Eqs. (38) and (39) correspond to the original constraints from Eqs. (7) and (12), respectively. To solve (P2), we first present the following lemma.
Lemma 3.3
The objective function of (P2) is a concave function of the allocated time for the HAP and node i, \(\forall i \in \{ 1, 2, \cdots, K\hat {K} \}\).
Proof
Please refer to Appendix D. □
Figure 4 further shows the objective function of (P2) by increasing nodes in the CP, i.e., Eq. (37) is a concave function over the number of nodes in the CP.
Proposition 3.1
The optimal time allocation of (P2) is
where \(q^{*} \triangleq 1+\frac {\gamma _{i} \mathcal {C}}{\tau _{i}^{*}}\), and q is calculated by the wellknown Lambert Wfunction [32] as Proposition 3.1.
Proof
Please refer to Appendix F. □
5 Simulation results and discussion
In this section, we present some simulation results to evaluate the performance of the proposed HDHT protocols in the largescale WPCN. The harvestthentransmit (HTT) protocol in [11] and the adaptive harvestthencooperate protocol (AHC) in [18] are used for the comparison. In the evaluation, the energy harvesting efficiency is assumed to be equal to one for all nodes, i.e., ξ_{ i }=1, ∀i∈{1,2,⋯,N}. The noise power at the receiver, σ^{2}, is assumed to be one, and the energy transmit power of the HAP for the DL WET, P_{ A }, is assumed to be 10 dB. We also assume i.i.d. Rayleigh fading for all channels in the network and the channel power gains of these channels are exponentially distributed, where the mean of the channel power gains is one, i.e., g_{ i }=1 and h_{ i }=1, ∀i∈{1,2,⋯,N}, respectively. Although we propose the HDHT protocol for largescale WPCN, in which there can be thousands of nodes like IoT or LPWAN networks, we assume that there are a hundred nodes for convenience. Also, we assume sensing delay to check channel state is 0.001 ms, and the network overhead caused by a single SI message is 0.5% in a block time.
Figure 5 shows the effect of the number of nodes to the ratio of the SI transmission period to a block time in the HTT, the AHC, and the proposed HDHT according to the average probability of generating SI message, P_{ s }, respectively. In the proposed HDHT protocol, the ratio of SI period to block time increases slightly as the number of nodes increases. We can see that the maximum ratio of the SI message transmission period to the block time is different according to the P_{ s }, since the P_{ s } affects the ratio of the TDMA period to the block time, which decides the maximum receptible number of SI messages. Due to the limitation of the maximum receptible number of SI messages, the increase of the ratio of the SI transmission period to the block time stops when the ratio reaches the maximum value. On the other hand, both the HTT and AHC protocols assume that all nodes in the WPCN transmit SI messages to the HAP, since we can see that the ratio of the SI message transmission period to the block time increases continuously according to the increase of the number of nodes.
Figure 6a shows the total harvested energy for UL WIT according to the ratio of TDMA period to block time. In the HTT and AHC protocols, a block time is entirely used for TDMAbased WET and WIT. Therefore, the total harvested energy does not change. However, in the proposed HDHT protocol, nodes harvest energy twice when the HAP performs a mainWET in the TP and a subWET in the CP within a block time. Especially, we assume that the HAP performs subWET when the wireless channel is in idle state in the CP. In general, the channel utilization rate of CSMA protocol for the wireless ad hoc network is very low compared to TDMA protocol [36]. Therefore, in Fig. 6a, we can see that the total harvested energy decreases according to the increase of the ratio of TDMA period to a block time. In addition, Fig. 6b shows that the harvested energy during the CP decreases according to the increase of the ratio of TDMA period to a block time because the subWET period decreases.
Figure 7 shows the sumthroughput according to the number of nodes in the HTT, AHC, and the proposed HDHT. As we can see in the Fig. 7, the throughput of both the HTT and AHC protocols decreases when the number of nodes increases, because of the network overhead caused by SI transmission. However, the sumthroughput of the HDHT protocol increases according to the number of nodes, because the HDHT protocol permit only the nodes, which require reliable data transmission, to send SI message to the HAP. Therefore, the network overhead caused by SI message transmission is relatively low compared to the HTT and AHC protocols. Also, the result shows that the sumthroughput increases when the P_{ s } increases, because high P_{ s } indicates that channel utilization rate increases.
In addition, the evaluation results show that the proposed HDHT protocol outperforms the conventional protocol, HTT and AHC, in terms of channel allocation. In the HDHT protocol, only the nodes which want to obtain a time slot of TP generate SI message. Therefore, in Fig. 5, we can see that the network overhead caused by SI is decreased in the HDHT compared to the conventional protocols, HTT and AHC. As a result, channel resource for WIT/WET is increased. Figure 6, also, shows that the proposed HDHT protocol outperforms the conventional protocols, HTT and AHC in terms of channel allocation. The HDHT protocol performs subWET at space holes in the CT, so there is no wasted time. As a result, in Fig. 6a, we can see that the nodes in the proposed HDHT protocol harvest more energy compared to nodes in the HTT and AHC, in most cases according to ratio of TDMA period to block time.
6 Conclusions
This paper proposes a novel protocol for a largescale WPCN, namely hybrid multiple accessbased dual harvestthentransmit (HDHT). The proposed HDHT protocol has a hybrid multiple access structure which employs both TDMA and CSMA, in order to manage different traffic patterns and requirements depending on applications. Thus, the HDHT protocol provides transmission reliability for transmitting information of nodes by TDMA. Furthermore, we propose a novel wireless energy transfer method, named dualWET, which improves the channel utilization and increases the amount of harvested energy at nodes. Specifically, from dualWET, the nodes can perform energy harvesting at space holes in CSMA. In addition, we describe the novel procedure of SI message transmission between the HAP and nodes to reduce network overhead caused by numerous SI messages. For the consideration, simulation results reveal that the HDHT outperforms the existing WPCN protocols in a largescale network, in terms of sumthroughput, SI transmission overhead, and an amount of harvested energy at nodes.
There are some research issues remained to improve the proposed system model for future work. Fullduplex WPCN models can be consider to the TP in order to improve harvested energy from mainWET, where the selfinterference is fully prevented. Additionally, we will expend the QoS in terms of receiving information as considering the energy efficiency.
7 \thelikesection Appendix
7.1 \thelikesubsection Appendix A: Proof of Lemma 3.1
This can be proved by contradiction. Suppose \(\boldsymbol {\tau }^{\prime }=\left [\tau _{0}^{\prime }, \cdots,\tau _{\hat {K}}^{\prime }\right ]^{T}\) is an optimal solution of (P1), and it satisfies that \(\sum _{i=0}^{\hat {K}}\tau _{i}^{\prime }<T_{t}\). It follows that \(\tau _{0}^{\prime }<T_{t}\sum _{i=1}^{\hat {K}}\tau _{i}^{\prime }\). The objective function given in Eq. (25) is a monotonic increasing function with respect to τ_{0}. Thus, the value of Eq. (25) under the vector \(\left [\tau _{0}^{\prime }, \cdots,\tau _{\hat {K}}^{\prime }\right ]^{T}\) is larger than that under τ^{′}. This contradicts with our presumption. Thus, the optimal τ^{∗} must satisfy \(\sum _{i=0}^{\hat {K}}\tau _{i}^{*}=T_{t}\).
This completes the proof of Lemma 3.1.
7.2 \thelikesubsection Appendix B: Proof of Lemma 3.2
According to [37], a function is concave if its Hessian is negative semidefinite. Thus, to show \(R_{sum}^{TMDA}(\boldsymbol {\tau })\) is a concave function of τ, we denote the Hessian of \(R_{sum}^{TDMA}(\boldsymbol {\tau })\) by H_{ i } and demonstrate that H_{ i } is a negative semidefinite. For any given real vector ν=[ν_{0},⋯,ν_{ K }]^{T}, it follows that
where the inequality follows from the fact that τ_{ mst }≤τ_{ i }. Thus, H_{ i } is negative semidefinite. Therefore, \(R_{sum}^{TDMA}(\boldsymbol {\tau })\) is a concave function of \(\boldsymbol {\tau }=\left [\tau _{0}, \cdots,\tau _{\hat {K}}\right ]^{T}\).
This completes the proof of Lemma 3.2.
7.3 \thelikesubsection Appendix C: Proof of Proposition 3.1
where λ≥0 denotes the Lagrange multiplier with the constraint in Eq. (27). The dual function of (P1) is thus given by
where \(\mathcal {D}\) is the feasible set of τ specified by Eqs. (26) and (27). Thus, strong duality holds for this problem thanks to the Slater’s condition.
Since (P1) is a convex optimization problem for which the strong duality holds, the KarushKuhnTucker (KKT) conditions are both necessary and sufficient for the global optimality of (P1), which are given by
where \(\tau _{i}^{*}\) and λ^{∗} denote the optimal primal and dual solutions of (P1), respectively. It can be easily be verified that \(\sum _{i=0}^{\hat {K}}\tau _{i}^{*} = T_{t}\) must hold for (P1) and thus from Eq. (27) without loss of generality, we assume λ>0. It is worth noting that for convenience, we normalize T_{ t } as 1 in the sequel.
where \(\mathcal {B} \triangleq \frac {\tau _{0}  \tau _{0_{WIT}}}{\tau _{0}  \tau _{0_{WET}}}\), \(\mathcal {C} \triangleq \tau _{0}  \tau _{0_{WIT}} + \tau _{0_{CSMA}}\), and \(\mathcal {H} \triangleq \ln (1+\gamma _{0}^{\prime } \mathcal {B}) + (1\mathcal {B}) \Big {(} \frac {\gamma _{0}^{\prime }}{1+ \gamma _{0}^{\prime } \mathcal {B}} \Big {)}\).
where \(\mathcal {X}(\alpha)\triangleq \ln (1+\alpha)  \frac {\alpha }{1+\alpha },~\alpha \geq 0\).
Given 1≤i, \(j \leq \hat {K}\), from Eq. (48) we have
It can be easily shown that \(\mathcal {X}(\alpha)\) is a monotonically increasing function of α≥0 since \(\frac {d \mathcal {X}(\alpha)}{d\alpha } \geq 0\) for α≥0. Therefore, equality in Eq. (50) holds if and only if \(\gamma _{i} \frac {\mathcal {C}}{\tau _{i}} = \gamma _{j} \frac {\mathcal {C}}{\tau _{j}}\), 1≤i, \(j \leq \hat {K}\), i.e.,
From Lemma 3.1 and \(\tau _{j}^{*} = \frac {\gamma _{j}}{\gamma _{i}}\tau _{i}^{*}\), \(\tau _{i}^{*}\) can be expressed as
where \(\mathcal {F}=\sum _{j=1}^{\hat {K}} \gamma _{j}\). In addition, it follows from Eqs. (47), (51), and (52) that
We can modify Eq. (53) as
where \(z=1+ \frac {F \cdot C}{1\tau _{0}}\). Therefore, the optimal time allocation is given by
From Lemma 3.1, τ_{ i } is obtained as
where z is calculated by the wellknown Lambert Wfunction [32] as
\(\tau _{0_{WIT}}^{*}\) is obtained as
since we assume that γ_{ i }=1, and
This thus proves Proposition 3.1.
7.4 \thelikesubsection Appendix D: Proof of Lemma 3.3
We may note that the proof of Lemma 3.3 is essentially equal to Lemma 3.2.
7.5 \thelikesubsection Appendix F: Proof of Proposition 3.2
where λ≥0 denotes the Lagrange multiplier with the constraint in Eq. (39). The dual function of (P2) is thus given by
where \(\mathcal {D}\) is the feasible set of τ specified by Eqs. (38) and (39). Thus, strong duality holds for this problem thanks to the Slater’s condition.
Since (P2) is a convex optimization problem for which the strong duality holds, the KarushKuhnTucker (KKT) conditions are both necessary and sufficient for the global optimality of (P2), which are given by
where \(\tau _{i}^{*}\) and λ^{∗} denote the optimal primal and dual solutions of (P2), respectively. It is worth noting that for convenience, we normalize T_{ c } as 1 in the sequel. Then, from Eq. (64), it follows that
where . First, consider the case of λ^{∗}>0, which corresponds to \(\sum _{i=i}^{K\hat {K}\tau _{i}^{*}} = 1\) from Eq. (39). Given 1≤i, \(j \geq K\hat {K}\) in the (65), we have equal result at i=j, since the (65) is monotonic function, i.e., \(\tau _{j}^{*} = \frac {\gamma _{j}}{\gamma _{i}}\tau _{i}^{*}\) and \(\tau _{i}^{*} = \frac {\gamma _{i}}{\sum _{j=1}^{K\hat {K}}}\gamma _{j}\) same as (51) and (52).
Next, for the case of λ^{∗}=0, the left side of (65) is zero. We can modify (65) as
where \(q \triangleq 1+\frac {\gamma _{i} \mathcal {C}}{\tau _{i}}\). Therefore, optimal time of the node i is obtained as
where q^{∗} is calculated by the Lambert Wfunction [32] as Proposition 3.1.
This completes the proof of Proposition 3.2.
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Acknowledgements
The SC, KL, and BK would like to acknowledge the encouragements and suggestions provided by Prof. Inwhee Joe during this research.
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SC and KL propose and analyze the HDHT protocol based in largescale WPCN. BK modified the English expressions. IJ organizes the whole paper as well as the proofreading. All authors read and approved the final manuscript.
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Cho, S., Lee, K., Kang, B. et al. A hybrid MAC protocol for optimal channel allocation in largescale wireless powered communication networks. J Wireless Com Network 2018, 9 (2018). https://doi.org/10.1186/s1363801710122
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DOI: https://doi.org/10.1186/s1363801710122