# Multi-Objective Optimized GPSR Intelligent Routing Protocol for UAV Clusters
## 3 Mathematical Modeling of Multi-Objective Routing Optimization Problems
### MDP Modeling of the Routing Forwarding Process  
(1) State space: let the state space of the node \(i\) where the packet is currently located be \(s_{t}\), \(s_{t}=\{s_{1,t},s_{2,t},...,s_{n,t}\}\), where \(s_{j,t}\) denotes the state information of the neighboring node \(j\) of \(i\) at the moment \(t\), and \(n\) denotes the number of all neighbor nodes of this node, and this study defines \(s_{j,t}\) as:  
\[s_{j,t}=\Big{\{}SNR_{i,j},E_{j}^{grade},T_{j}^{unit},d_{i,dst},d_{j,dst},P_{ hop_{2}},\cos_{j,d}\Big{\}},j\in NBR(i) \tag{5}\]  
where \(NBR(i)\) denotes the set of neighbors of the node; \(SNR_{i,j}\) denotes the signal-to-noise ratio (\(SNR\)) between the current node \(i\) and the neighbor node \(j\), reflecting the link quality; \(E_{j}^{grade}\) denotes the residual energy level of the candidate neighbor node; \(T_{j}^{unit}\) denotes the expected total waiting delay of the packet in the neighbor node \(j\); \(d_{i,dst}\) indicates the distance between node i and destination; \(d_{j,dst}\) denotes the distance between the neighbor node j and destination node; \(P_{hop_{2}}\) denotes the probability of a routing hole occurring in neighboring node j; \(\cos_{j,d}\) indicates \(j^{\prime}s\) relative movement trend with respect to the destination node. * Signal-to-noise ratio (\(SNR\)) \(SNR\) is an important indicator of the quality of a communication link. For wireless communication environments, a higher SNR indicates better signal quality. It is assumed that the energy of the received signal \(S(k)\) outside the packet \(k\) acceptance interval is 0. The received power of the signal is derived from the path loss propagation model as well as the signal transmit power. The calculation is based on the Friis free-space propagation model (Friis), which takes into account the attenuation of the signal with increasing propagation distance [32]. The calculation process is shown in (6):  
\[P_{l}(d)=P_{l}(d_{0})+n10\log_{10}(\frac{d}{d_{0}}) \tag{6}\]  
where \(P_{l}(d)\) is s the received power at distance d, is the reference distance, n is the path loss exponent (usually within 2 to 6), \(d_{0}\) is the reference distance, \(P_{l}(d_{0})\) is the received power at the reference distance of \(d_{0}\) (\(d_{0}\) usually set to 1 m), is the reference distance, and d is the actual distance between the receiver and the transmitter. Based on the Friis propagation model to define the calculation method of the reference distance received power \(P_{l}(d_{0})\):  
\[P_{l}(d_{0})=\frac{P_{l}G_{t}G_{r}\lambda^{2}}{16\pi^{2}d_{0}^{2}L} \tag{7}\]  
where \(P_{l}\) denotes the signal transmit power, \(G_{t}\) denotes the gain of the transmit antenna (default value 1 dBm), \(G_{r}\) denotes the gain of the receive antenna (default value 1 dBm), \(\lambda\) denotes the radio carrier wavelength, \(d_{0}\) denotes the reference distance, and \(L\) denotes the system loss (default value 1). The receiving end utilizes the formula (8) to calculate \(SNR(k)\), the signal-to-noise ratio of the packet \(k\):  
\[SNR(k)=10\lg(\frac{S_{k}}{N_{i}(k)+N_{f}})=10\lg(\frac{P_{l}(k)}{N_{i}(k)+N_{f }}) \tag{8}\]  
where \(N_{f}\) represents the noise floor, which is a characteristic constant of the receiving circuit. \(N_{i}(k)\) represents the interference noise, which is the sum of the energies of all other signals received on the same channel. \(N_{i}(k)\) can be expressed as:  
\[N_{i}(k)=\sum\
olimits_{m\
eq k}S(m) \tag{9}\]  
The base noise of the receiver needs \(N_{f}\) to take into account the thermal noise and the non-ideality of the receiver, which is calculated as follows:  
\[N_{f}=NF\cdot Nt \tag{10}\]where \(NF\) is the noise figure, \(Nt\) is the thermal noise power. \(Nt=kTB\), \(k=1.3803\times 10^{-23}\) is the Boltzmann constant in J/K, \(T=290\) K is the temperature, and \(B\) is the channel width in Hz. * Residual energy percentage  
Energy utilization is an important performance indicator for the operation of unmanned clustered networks, and optimizing the energy consumption problem is critical.