A hybrid MAC protocol for optimal channel allocation in large-scale wireless powered communication networks

3.1 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.

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

where P _d is the average probability of generating information at the node i, ∀i∈{1,2,⋯,N}.

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.

3.3 TDMA period in the H-DHT protocol

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}.

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

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.

3.4 CSMA period in the H-DHT protocol

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 non-collision and collision transmission in Markovian model as Fig. 3 [30]. and denote non-collision and collision states, respectively.

(14)

(15)

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

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

(19)

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

(20)

4.1 TDMA period in the H-DHT protocol

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 signal-to-noise ratio gap from the additive white Gaussian noise channel capacity as a modulation and coding scheme (MCS) use. σ² represents the noise power at the H-AP. 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.

where \(\gamma _{0}^{\prime } = \frac {h_{i}P_{A}}{\Gamma _{A}\sigma _{A}^{2}}\).

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.

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 well-known Lambert W-function [32].

4.2 CSMA period in the H-DHT protocol

where r _i satisfies the detailed balance equation [34]. Eqs. 19 and (20) express r _i , which is obtained as

(33)

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 sum-throughput of nodes, a log-likelihood function [35] is defined as the summation over all the non-collision transmission states, which can be expressed by

(34)

(35)

where \(\mathcal {A} \triangleq \ln \sum _{j} exp(r_{j})\). Then, is defined as

(36)

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.

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.

where \(q^{*} \triangleq 1+\frac {\gamma _{i} \mathcal {C}}{\tau _{i}^{*}}\), and q is calculated by the well-known Lambert W-function [32] as Proposition 3.1.

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 τ₀. 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

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 \(\mathcal {D}\) 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.

where \(\mathcal {X}(\alpha)\triangleq \ln (1+\alpha) - \frac {\alpha }{1+\alpha },~\alpha \geq 0\).

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 \(\mathcal {D}\) 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.

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).

where q^∗ is calculated by the Lambert W-function [32] as Proposition 3.1.

This completes the proof of Proposition 3.2.