## 3 Performance Analysis  
The performance of the proposed approach has been evaluated by means of numerical simulations conducted in the MATLAB framework. Specifically, the simulated nodes are based on IEEE 802.15.4 standard and are divided in sensors and actuators. A sensor is  
Figure 5: Planar distribution of RF power converted in DC (I\({}_{\text{DC}}\) in uA). capable to observe homogeneous physical parameters, while an actuator can gather data from other devices and perform specific tasks. The simulated IWSAN nodes have been modelled considering the data sheet of a commercial sensor node as indicated in Table 1. Specifically, we considered a initial battery level of 2500 mAh, corresponding to 2 AAA batteries, a keep alive consumption of 1 uA, a radio transmission consumption of 17 mA and finally a reception consumption of 20 mA. The transmission power mainly depends on the transmission duration, which is correlated to the packet size and to the adopted data rate. The former was set to 50 bytes, while the latter was set equal to the maximum rate allowed in the IEEE 802.15.4 standard, that is 250 kbps. Finally, the overall node consumption has been completed adding a fixed value of 15 mA for sensing operations. The CPU consumption has not been considered because it is negligible with respect to the transmission and reception power consumption. The WPT has been modeled considering the result shown in Figure 5 and a constant recharging period of 5 s. This value has been evaluated considering an information publishing rate equal to 2 packet/s and a maximum tolerable packet loss of 10 packets. Moreover, we adopted the MQTT-SN standard as the application protocol since it represents a solution suited for IIoT in the presence of resources constrained devices, but we further embodied also the WPT module in the design of the proposed WPT-MQTT-SN. For the sake of comparison, we have considered the CoAP protocol performance. The network has been partitioned in logical clusters, each managed by its own CH that specifically handles the traffic from or to the ES, where the PTA is executed. The overall system performance has been evaluated in terms of network lifetime, which indicated the network duration. In particular, we adopted a strict definition, that is the time elapsed since the first node in the network depletes its battery. Results have been obtained considering a initial battery level of 150 mAh. Furthermore, the proposed system has been tested over three different nodes deployment: (i) Random Mesh, (ii) Two-Tier, and (iii) Unbalanced Random Mesh and, for each of them we evaluated how the data transmission frequency impacts on the WPT procedure. Finally, the overall number of devices has been assumed to be equal to 40, in compliance with most common IIoT scenarios. First of all, we evaluated the performance in the presence of a Random Mesh nodes deployment, which is the most common IoT scenario. In this case, a mesh topology network is originated, in which the number of hops depends on the nodes spatial distribution \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Min** & **Nom** & **Max** & **Unit** \\ \hline Supply voltage during radio operation & 2.1 & 3.6 & V \\ \hline RF Frequency Range & 2400 & & 2483.5 & MHz \\ \hline Transmit bit & 250 & & 250 & Kbps \\ \hline Current consumption: Radio & & 17.4 & mA \\ Transmitting at 0 dBm & & & \\ \hline Current consumption: Radio receiving & 19.7 & mA \\ \hline Current consumption: Radio on, & & 365 & \(\mu\)A \\ Oscillator on & & & \\ \hline Current consumption: Idle mode, & & 20 & \(\mu\)A \\ Oscillator off & & & \\ \hline Current consumption: Sleep Mode & & 1 & \(\mu\)A \\ \hline Active current at Vcc = 3 V, 1 MHz & 500 & 600 & \(\mu\)A \\ \hline Current consumption during & & 15.4 & mA \\ sensing operation & & & \\ \hline Voltage regulator current draw & 13 & 20 & 29 & \(\mu\)S \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters adopted to model a IWSAN node in simulation campaign.