# Resource Allocation in Unmanned Aerial Vehicle (UAV)-Assisted Wireless-Powered Internet of Things
## 3 System Model and Problem Formulation
At this point, we can use the signal to interference noise ratio (SINR) to denote the revenue earned from the information transmission, which can be expressed in the form of the power level for information transmission as follows:  
\[\gamma_{k,i}=\frac{g_{k,i}(t)p_{k,i}(t)}{\sum\limits_{j\
eq i}g_{k,j}^{k,i}(t) p_{k,j}(t)+\sigma^{2}(t)}=\frac{g_{k,i}(t)p_{k,j}(t)}{l_{k,i}(t)} \tag{1}\]  
where \(k\) is the category number and \(k=\{H,L\}\). \(g_{k,i}(t)\) is the channel gain from transmit node \(i\) in category \(k\) to the UAV. \(\sigma^{2}(t)\) is the background noise, and \(\sum\limits_{j\
eq i}g_{k,j}^{k,i}(t)p_{k,j}(t)\) is the inter-cell interference from the other transmit node \(j\) in the category \(k\) to the transmit node \(i\). Then, \(l_{k,i}(t)=\sum\limits_{j\
eq i}g_{k,j}^{k,i}(t)p_{k,j}(t)+\sigma^{2}(t)\) is the total interference for the wireless node \(i\) in category \(k\), which is the overall sensed interference by the wireless node \(i\) in category \(k\). It is assumed that the nodes in our proposed system know their distance between the nodes and the UAV, and the distance among the nodes. The effect of distance on  
Figure 2: Leader-follower relationships between the UAV and the wireless nodes. transmission is uniformly represented by the channel gain. Based on the SINR definition given in (1), the cost function of energy transfer for the wireless nodes during the observation time \([0,T]\) is given as [27]:  
\[\max\int_{0}^{T}\left\{\left[\gamma_{t,i}\left(p_{k,i}(t)\right)-\gamma_{th} \right]^{2}-\pi_{k}(t)p_{k,i}(t)\right\}e^{-rt}dt \tag{2}\]  
where \(\gamma_{th}\) denotes the SINR threshold, which also means the QoS constraint; \(\pi_{k}(t)\) is the unit price of energy transfer controlled by the UAV at time \(t\); \(r\) is the discount rate; and \(e^{-rt}\) is the discount factor. Solving the above equation yields the optimal solutions of \(p_{k,i}(t)\):  
\[p_{k,i}(t)=\frac{1}{2}\pi_{k}(t)\frac{I_{k,i}^{2}(t)}{g_{k,i}^{2}(t)}+\gamma_{ th}\frac{I_{k,i}(t)}{g_{k,i}(t)} \tag{3}\]  
The wireless nodes in different categories will enjoy different energy supply services from the UAV, because the energy consumptions are different. Energy transferred to one kind of wireless node cannot be harvested by the other kind of node. Based on the dynamic game, the UAV should make its choice on the power level for energy transfer, and the wireless nodes in different categories should control their power level for information transmission. Let \(x(t)\) denote the state function of the proposed dynamic game model, which is the energy level of the whole IoT system at time \(t\). The dynamic variation of the state function can be described by the following differential equation:  
\[\frac{dx(t)}{dt}=\sum_{k=[H,L]}\eta_{k}p_{k}(t)+\sum_{i=1}^{N_{H}}p_{H,i}(t)+ \sum_{i=1}^{N_{L}}p_{L,i}(t)+\delta x(t) \tag{4}\]  
where \(p_{k}(t)\) is the allocated power level for energy transfer for the nodes in category \(k\); \(\eta_{k}\) denotes the energy conversion efficiency; and \(\delta\) denotes the energy consumption rate of the whole IoT system, which should be a constant value. The initial value of the system state is denoted by \(x_{0}=x(0)\). In this paper, we have divided the set of wireless nodes into two categories, which are the HEC nodes and the LEC nodes. Each category will be allocated different energy. For the UAV, once the energy is transferred to one kind of wireless node, there will be less energy available for the other kind of wireless node. In this paper, this kind of energy transfer effect is some kind of category of activity, and is considered the energy transfer cost for the system. In our model, the \"harvest-then-transmit\" protocol [28] is used, and the wireless energy transfer and wireless information transmission can operate at the same frequency. Then, information transmission will cause interferences to the UAV.