In this section, we describe a large-scale WPCN and propose the H-DHT protocol. As illustrated in Fig. 1, a large-scale WPCN consists of a single H-AP and node *i*, ∀*i*∈{1,2,⋯,*N*}, where *N* is a positive integer and big as we consider a large-scale network. We assume that the H-AP 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 H-AP, and the UL WIT, which cannot be performed at the same time. It is worth noting that the H-AP and nodes operate over the same frequency band. The DL channel power gain from the H-AP to node *i*, ∀*i*∈{1,2,⋯,*N*}, and the UL channel power gain from node *i*, ∀*i*∈{1,2,⋯,*N*}, to the H-AP are denoted by *h*_{
i
} and *g*_{
i
}, respectively. We assume that all channels follow quasi-static flat-fading, 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 H-AP knows perfectly both *h*_{
i
} and *g*_{
i
}, ∀*i*∈{1,2,⋯,*N*}.

The proposed network adopts the H-DHT protocol as shown in Fig. 2a, which consists of SI transmission, TDMA, and CSMA period in a block. Therefore, we have

$$ T = T_{b} + T_{t} + T_{c}, $$

(1)

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.

### State information transmission period in the H-DHT 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 H-AP 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 H-AP in order to request transmission reliability. Thus, we reduce a significant network overhead caused by numerous SI messages from all nodes in the large-scale network. It is worth noting that although the near-far 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 H-AP decides the RTC in a block time and calculates the optimal time of nodes in the TP, considering the MST. Then, the H-AP transforms the decision information into a command and broadcasts the command to all the nodes in the large-scale WPCN.

Then, the SIP is obtained as

$$ T_{b} = \tau_{EB} + \tau_{SI} + \tau_{CT}, $$

(2)

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].

### The ratio between TDMA and CSMA

In the proposed model, we consider periodic and non-periodic traffic from nodes in the large-scale 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

$$ K = N \cdot P_{d}, $$

(3)

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 H-AP. Thus, \(\hat {K}\) denotes the number of nodes that require transmission reliability in the a block time and is obtained as

$$ \hat{K} = K \cdot P_{s} $$

(4)

where *P*_{
s
} is the average probability of generating SI at the node *i*, ∀*i*∈{1,2,⋯,*K*}. By *P*_{
s
}, the H-AP 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.

### TDMA period in the H-DHT protocol

In the TP, the H-AP performs the *main*-WET 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 *dual*-WET: the *main*-WET in the TP and *sub*-WET in the CP. It is worth noting that the DL WIT is performed in the TP, when the H-AP is necessary to transmit information. *τ*_{0} denotes the time assigned to the H-AP in the TP and is expressed as

$$ \tau_{0} = \tau_{0_{WIT}} + \tau_{0_{WET}}, $$

(5)

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 *main*-WET at the H-AP, respectively. If there is no transmitting information at the H-AP, \(\tau _{0_{WIT}}\) 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

$$ \sum_{i=0}^{\hat{K}}\tau_{i} \leq T_{t}. $$

(6)

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

$$ \tau_{mst} \leq \tau_{i},~\forall i \in \{1,2, \cdots, N \}. $$

(7)

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

$$ E_{i}^{TDMA}=\xi_{i}h_{i}P_{A}{\tau}_{0_{WET}},~\forall i \in\{1,2, \cdots, N\}, $$

(8)

where *P*_{
A
} denotes the transmit power at the H-AP, 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 *sub*-WET at *space holes*, where the H-DHT performs *dual*-WET. \(E_{i}^{CSMA}\), ∀*i*∈{1,2,⋯,*N*}, denotes the harvested energy from the *sub*-WET, which is studied in the following subsection. Consequently, the total harvested energy from the *dual*-WET is denoted by *E*_{
i
} and is expressed as

$$ E_{i}=E_{i}^{TDMA} + E_{i}^{CSMA},~\forall i \in\{1,2, \cdots, N\}, $$

(9)

where we assume that the node *i*, ∀*i*∈{1,2,⋯,*N*}, in the large-scale WPCN, replenishes their energy from the *dual*-WET

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

$$ P_{i}=\frac{\eta_{i}E_{i}}{\tau_{i}}, ~ \forall i \in \{1,2, \cdots, K\}, $$

(10)

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 H-AP and is expressed as

$$\begin{array}{*{20}l} \overline{P_{A}}=&\frac{\eta_{A}P_{max}}{\tau_{0_{WIT}}} \end{array} $$

(11)

where 0≤*η*_{
A
}≤1 denotes the utilization rate of the transmission power at the H-AP and *P*_{
max
} denotes the maximum transmission power available at the H-AP. 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.

### CSMA period in the H-DHT protocol

In the CP, the H-AP performs the *sub*-WET 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

$$ \sum_{i=1}^{K-\hat{K}}\tau_{i} \leq T_{c}. $$

(12)

The channel utilization of the contention-based protocol, as CSMA, is degraded by *space holes*. However, in the H-DHT, we consider that *space holes* in the CP can be utilized by the *sub*-WET to harvest additional energy and increase the sum-throughput. The harvested energy from the *sub*-WET can be expressed as

$$ E_{i}^{CSMA}= \xi_{i}h_{i}P_{A}\tau_{0_{CSMA}},\forall i \in\{1,2, \cdots, N\}, $$

(13)

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 *δ**T*_{
s
}, i.e., if two or more nodes initiate the UL WIT within *δ**T*_{
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 non-collision and collision transmission in Markovian model as Fig. 3 [30]. and denote non-collision 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 *sub*-WET at node *i*. The link status is expressed as

$$ x_{i}^{j}= {\begin{cases} 1 ~& if~\text{node~\textit{i}~in~state~\textit{j}~transmits~data},\\ 0 ~& otherwise.\\ \end{cases}} $$

(16)

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

$$ f(t_{i};\lambda_{i})= {\begin{cases} \lambda_{i} exp(-\lambda_{i} t_{i}) & if~t_{i} \geq 0,\\ 0 & if~t_{i} < 0,\\ \end{cases}} $$

(17)

where \(\lambda _{i} \in \{ 1,2, \cdots, K-\hat {K} \}\) denotes the transmitted frames per unit time [31].

Due to the sensing delay, *δ**T*_{
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 *δ**T*_{
s
} while starting the UL WIT from another node, is expressed as

$$ p_{i} \triangleq 1 - exp(-\lambda_{i} \delta T_{s}), $$

(18)

by the memoryless property of the exponential random variable, i.e., CDF [31]. Thus, the rate of transition, , to one of the non-collision states in the Markov chain is defined as

The rate of transition, , to one of the collision states is given by