# Resource Allocation in Wireless Powered IoT System: A Mean Field Stackelberg Game-Based Approach
## 3 System Model and Problem Formulation
### System State  
In this paper, as we concentrate on controlling the power for both the HAP and SNs, we use the energy as the system state for mean field game (MFG) construction. There are two energy state variables for the proposed IoT systems, the energy level of the HAP denoted by \(x_{0}(t)\), and the energy level of SN \(i\) denoted by \(\{x_{i}(t),1\leq i\leq N\}\).  
(1) Energy of the HAP: the energy of the HAP is mainly dominated by the power level for energy transfer. Assuming the energy is transferred by the HAP in a unique frequency, to avoid interference to information transmission and the power level for energy transfer is denoted by \(p_{0}(t)\) at time instant \(t\). For the HAP, the energy level can be described by the following differential equation:  
\[dx_{0}(t)=[\alpha_{0}x_{0}(t)+\beta_{0}p_{0}(t)]dt \tag{1}\]  
where \(x_{0}(t)\) is the energy level of the HAP, with an initial energy state \(x_{0}(0)\). \(\alpha_{0}\) is a random coefficient of energy degradation brought by the system consumption, an \(\alpha_{0}x_{0}(t)\) denotes the energy brought by the system consumption. Generally, \(\alpha_{0}x_{0}(t)\) can be represented as [21; 22]:  
\[\alpha_{0}x_{0}(t)=(P_{HC}+P_{RF}+P_{RP})\delta_{0}\]  
Figure 3: Stackelberg game.  
where \(P_{HC}\) is the power consumption of the hardware circuit, \(P_{RF}\) is the power consumption of the RF module, and \(P_{RP}\) is the power consumption of packets exchanged by the HAP with controller. \(\delta_{0}\) is the duration/slot for the energy transfer.  
\(\beta_{0}\) is a random efficiency coefficient of energy transfer, which depends on the energy transfer circuit. The energy transfer process should be a broadcast process. The initial state of the HAP is independent of the SNs with mean \(Ex_{0}(0)=\overline{x}_{0}\).  
(2) Energy of the SNs: the energy of each SN is dominated by the energy from the HAP and the power for information transmission. Assuming the power for information transmission is denoted by \(\{p_{i}(t),1\leq i\leq N\}\). For any specific SN, the evolution of the energy is described by:  
\[dx_{i}(t)=[\alpha_{i}x_{i}(t)+\beta_{i}p_{i}(t)+\rho_{i}h_{i}p_{0}(t)]dt \tag{2}\]  
where \(\{x_{i}(t),1\leq i\leq N\}\) is the energy level of the SN \(i\). Each SN has an initial energy state, which is denoted by \(\{x_{i}(0),1\leq i\leq N\}\), which are independent of each other with mean \(\{Ex_{i}(0)=\overline{x},1\leq i\leq N\}\). In the current analysis, \(N\) is taken to be large so that MFG analysis may be applied. \(\alpha_{i}\) is a random coefficient of energy degradation caused by the system consumption, which includes the power consumption of the hardware circuit and the RF module [23]. The power consumed in sensing and processing are also included in this coefficient [24]. \(\beta_{i}\) is a random efficiency coefficient of information transmission, which depends on the information transmission circuit. \(\rho_{i}\) is the conversion efficiency coefficient of energy transfer, and \(h_{i}\) is the channel power gain from the HAP to SN \(i\).